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BEMPS –
Bozen Economics & Management
Paper Series
NO 102/ 2024
Nowcasting Inflation at Quantiles:
Causality from Commodities
Sara Boni, Massimiliano Caporin,
Francesco Ravazzolo
Nowcasting Inflation at Quantiles: Causality from
Commodities
Sara Boni†Massimiliano Caporin‡Francesco Ravazzolo†∗
†Free University of Bolzano-Bozen
‡University of Padova
∗BI Norwegian Business School
Abstract
This paper proposes a non-parametric test for Granger causality in quantiles to detect causal-
ity from a high-frequency driver to a low-frequency target. In an economic application, we
examine Granger causality between inflation, as a low-frequency macroeconomic variable, and
a selection of commodity futures, including gold, oil, and corn, as high-frequency financial vari-
ables. We find that logarithmic returns on given commodity futures are a prima facie cause of
inflation at the lower quantiles of the distribution and marginally around the median. In the
context of a nowcasting exercise, we find that incorporating commodity futures in the model
with a polynomial function enhances short-term forecasting accuracy, leveraging timely data for
more precise nowcasting of inflationary trends. 123
1Corresponding Author: Sara Boni. Email: sboni@unibz.it Address: Piazza Universit`a 1, 39100, Bolzano, BZ,
Italy. (Other authors emails: massimiliano.caporin@unipd.it; francesco.ravazzolo@unibz.it)
2We would like to express our sincere gratitude to the participants to the 3rd Dolomiti Macro Meetings, the
2023 Annual Conference of the International Association for Applied Econometrics (IAAE), the Junior Workshop in
Econometrics and Applied Economics at Luiss Guido Carli, the 3rd International PhD Conference in Econometrics
at the Erasmus University Rotterdam. Additionally, we extend our appreciation to the ifo Institut and the Warwick
Business School and its senior and junior researchers for giving us the chance to present our work at various research
seminars. Finally, our heartfelt thanks go Domenico Giannone, Michael W. McCracken, Giovanni Ricco, Giorgio
Primiceri, Tommaso Proietti, Ivan Petrella, and Fabrizio Ghezzi for their valuable comments and guidance.
3Ethics and Integrity Policies. Data Availability: The authors do not have permission to share all the data used
in the present study. The data can be recovered from the reported sources under subscription; Funding Statement:
Francesco Ravazzolo acknowledges financial support from the Italian Ministry MIUR under the PRIN project Econo-
1
Keywords: MIDAS Quantile, Granger Causality, Commodities, Inflation, Nowcasting.
JEL Classification: C12, C14, C58, E31, Q02.
metric and Macro-Financial Models of Climate Change: Transition, Policies and Extreme Events (grant 20225J7H4K);
Conflict of Interest: none
2
1 Introduction
A burgeoning body of research accentuates the pivotal role of financial market indicators in shaping
macroeconomic trajectories and in channeling shocks to the real economy. Leading contributions
include the contribution by Claessens et al. (2012) emphasizing the interconnections between finan-
cial and business cycles, particularly highlighting the direct correspondence between recessions and
financial upheavals. Additionally, the contribution by Jermann and Quadrini (2012) elucidates how
events within the financial sector manifest as constrictions in firms’ financing conditions, a key factor
contributing to the 2008-2009 recession. Furthermore, the contribution by Tamakoshi and Hamori
(2012) posits that energy commodity prices, due to their causal linkages, serve as significant infor-
mational indicators for the overall price levels. Moreover, the very recent contribution by Foroni
et al. (2023) underscores the necessity of incorporating macroeconomic variables when forecasting
electricity prices to enhance accuracy in short-term point and density estimations.
Consequently, the information encapsulated in financial data or disseminated through financial
markets holds substantial significance for comprehending macroeconomic shifts. The different nature
of financial and macroeconomic variables coupled with the mismatch in their sampling frequen-
cies certainly lie at the core of this relevance. Indeed, while most macroeconomic variables have a
backward-looking nature and are low frequency, financial indicators are often forward-looking and
are available at a much higher frequency. Accordingly, evaluating the potential impact of fluctuations
in financial variables on macroeconomic indicators could yield valuable insights for policymakers and
investors. This is mostly the case since the majority of macroeconomic variables are available only
after the end of the reference month or quarter, whereas financial data often offers a degree of fore-
sight prior to the release of macroeconomic indicators.
The framework proposed by Granger (1969) stands as a renowned approach to examining causal-
ity between economic variables, extensively studied and applied in research. The majority of research
results focused on evaluating Granger causality in the conditional mean with data sampled at the
same frequency. Jeong et al. (2012) were among the first authors proposing a non-parametric Granger
causality test in quantiles, allowing for non-linearities to be considered when evaluating causal rela-
tions between two time series. Their contribution has been particularly valuable since the conditional
3
mean might pose interpretative challenges, especially when the involved variables exhibit fat-tailed
and non-elliptical distributions — characteristics commonly observed in financial returns. Indeed,
existing studies suggest that while Granger causality might exhibit significance in tail quantiles, it
might not necessarily manifest in the mean (Lee and Yang,2014). While similar works have been
conducted, the robustness and flexibility of the results offered by Jeong et al. (2012) remain unpar-
alleled. 4
While Jeong et al. (2012)’s work marked an important step forward in evaluating Granger causal-
ity, the mismatch in sampling frequencies of different variables remained unaddressed. The paramount
significance of preserving the inherent high-frequency nature of financial data is crucial for leveraging
the comprehensive information embedded within. Indeed, as highlighted by Ferrara et al. (2022),
aggregation mechanisms employed to ensure the same frequency are likely to lead to biased estimates
unless the underlying data-generating process features a flat aggregation scheme from high to low
frequencies. Notably, to the best of our current knowledge, no test of Granger causality in quantiles
with mixed frequency data is available yet.
Building upon the methodology proposed by Jeong et al. (2012), we present an updated version
of their test, adapting it to accommodate both high- and low-frequency data without necessitating
aggregation procedures. To perform the Granger causality test in quantiles with mixed-frequency
data, we need to estimate the polynomial weights that allow the high-frequency variable to be accu-
rately collapsed into a low-frequency one; this step would allow placing the causality testing within
the framework of Jeong et al. (2012). To this aim, we estimate a Quantile-MIDAS model, which, in
a second step, also allows us to evaluate the magnitude of the relationship between a financial and
a macroeconomic variable. The seminal work by Ghysels et al. (2004), followed by Ghysels et al.
(2006) and Ghysels et al. (2007), first introduced the idea of regressions involving time series data
sampled at different frequencies, specifying conditional expectations as a distributed lag of regressors
recorded at higher sampling frequencies. While a limited number of authors among which Lei et al.
(2019), Xu et al. (2021), Ferrara et al. (2022) and Yang et al. (2023) resorted to Quantile-MIDAS
models in their empirical works, the literature on this model is still at a relatively early stage yet.
Thus, our first contribution is on the methodological side, as we introduce a practical approach to
4Other prominent contributions include Hong et al. (2009), Troster (2018) and Song and Taamouti (2021).
4
detect Granger causality at quantiles and, as a bi-product, to measure the intensity of the causality.
We apply our mixed-frequency Granger causality test in quantiles and our Quantile-MIDAS model
to investigate whether futures on commodities, including gold, crude oil, and corn do impact inflation,
that is if futures on commodities can be early warning indicators of inflation. As a matter of fact,
while commodity prices are often regarded as a relevant cause for inflation risk, how to practically
model a formal link between commodity prices and inflation remains an unresolved question (Garratt
and Petrella,2022). We believe that our contribution is relevant in filling this gap and addressing
this much-debated question. We find that logarithmic returns on commodity futures are a prima
facie cause for inflation in the lower quantiles of the distribution and marginally around the median.
Particularly, starting with the future contract on gold, we find that the effect is larger in the left tail
than in the center of the inflation distribution. This means that when inflation is very low, (i.e. in the
5th and 25th quantile of the distribution), if returns on gold futures increase by 1%, inflation increases
by 0.06 units and 0.11 units, respectively. The application to future contracts on oil yields similar
results. In the 5th and 25th quantile of the inflation distribution, if returns on oil futures increase
by 1%, inflation increases by 0.31 units and 0.24 units. While there is causality between futures
on agricultural commodities and inflation, the magnitude of the relationship is rather small. We
conclude that precious metals and energy commodities have the most prominent impact on inflation.
As a further step, we conduct a nowcasting exercise to assess how well commodity futures predict
future inflation compared to using solely the inflation series. We find that incorporating futures
on commodities significantly improves the predictive performance of inflation forecasts. Particularly
integrating the whole monthly history without resorting to an aggregation mechanism, that is using
a MIDAS specification as opposed to a specification that merely considers the monthly average, re-
turns more accurate forecasts. This integration enables a more precise understanding of short-term
fluctuations in inflation by leveraging the timely and informative nature of commodity futures data,
thereby contributing to more accurate nowcasting of inflationary trends.
The remainder of the paper is organized as follows. Section 2illustrates our methodology, where
Section 2.1 describes the estimation of our Quantile-MIDAS model, Section 2.2 presents the Granger
causality test in quantiles with mixed frequency data, and Section 2.3 shows how to conduct inference
between two time series sampled at different frequencies. Section 3considers the causal relations and
5
inference between inflation and futures on commodities, where Section 3.1 looks at future contracts
on gold, Section 3.2 at contracts on oil, and 3.3 at contracts on corn and wheat, as an economic
application. Section 4concludes.
2 Methodology
Our methodology consists of three main steps. Before delving into detailed explanations for each, we
offer a brief overview to provide context and enhance clarity.
Jeong et al. (2012) developed a non-parametric test of Granger causality in quantile for data
sampled at the same frequency. Our proposed methodology aims to extend their test to accom-
modate mixed-frequency data, enabling the detection of causality from a high-frequency driver to
a low-frequency target quantile. To this aim, we collapse the high-frequency variable to the same
frequency as the target variable. In order to mitigate potential biases inherent in simple aggregation
mechanisms, we utilize a procedure reminiscent of finite one-side polynomials typically found in a
MIDAS specification. Specifically, we use a functional form with two parameters as suggested by
Ghysels et al. (2005). Defining this polynomial function requires the parameters that link a high-
frequency variable to a low-frequency target and drive its behavior. We derive these parameters,
which we will denote as (θ1, θ2), through the estimation of a Quantile-MIDAS model, which enables
us to gain reasonable insights into their potential values. We store a grid ranging from the minimum
to the maximum of the estimated value of each parameter, and we utilize it for the subsequent anal-
ysis stages.
Upon retrieving the parameters allowing to collapse the high-frequency driver into the same fre-
quency as the low-frequency target, we are finally able to extend the test by Jeong et al. (2012),
accommodating the inclusion of mixed-frequency data. Once we have defined our test statistic, we
compute it for every possible combination of (θ1, θ2) in the grid that we stored and mark which pair
returns the maximum value of our test statistic.
Finally, building on the two previous steps, we are able to perform a causal inference. We esti-
mate the Quantile-MIDAS model again, fixing the parameters of the polynomial function at those
values that maximize our test statistic. In this way, we are able to infer the relationship between a
6
low-frequency macroeconomic variable and a high-frequency financial variable. 5
Figure 1stylizes our outlined methodology placing it into three main steps. The first two steps
concern the methodology to retrieve the test statistic, while the final step centers on conducting the
causal inference.
In the following section, we go through each of these explained steps in detail.
2.1 Preliminary Quantile-MIDAS Estimation
The first preliminary step of our methodology encompasses retrieving the parameters of the polyno-
mial function, which is needed to collapse the high-frequency variable to the same frequency as the
low-frequency target. To ascertain the potential ranges for these parameters, we estimate a Quantile-
MIDAS model. This ensures that the parameters are based on logical and sought-after values.
We start from the standard quantile regression first introduced by Koenker and Bassett (1978).
Building on Ghysels et al. (2004) and Ghysels et al. (2007), we add a MIDAS component, repre-
sented by ˜
B(c;θ(τ)) in Equation (1) below. This function serves to align high-frequency financial
data with macroeconomic data frequency and our focus in this preliminary step lies in determining
the parameters θ(τ) governing this function. Since we are considering time series, it is reasonable
to incorporate some lags of the dependent macroeconomic variable ytinto our model specification.
To select the best autoregressive structure for yt, we resort to the pseudo-R-Squared approach for
quantile regressions suggested by Koenker and Machado (1999). To this aim, we availed ourselves of
the quantreg function in MATLAB for quantile regressions with bootstrapping confidence intervals
and we modified it including the calculation of the Pseudo-R-Squared dropping the polynomial order
in the original function and in the code accordingly. 6
Our Quantile-MIDAS model has the following specification:
yt=β0(τ) +
N
X
n=1
[β1(τ)yt−n+β2(τ)|yt−n|] + β3(τ)
C−1
X
c=0
˜
B(c;θ(τ))Γc/mwt+ϵ(τ)t(1)
5The whole code has been written in MATLAB and will be made available
6Aslak Grinsted (2008). quantreg(x,y,tau,order,Nboot) (https://www.mathworks.com/matlabcentral/fileexchange/32115-
quantreg-x-y-tau-order-nboot), MATLAB Central File Exchange. Retrieved January 24, 2023.
7
where τindicates the quantile considered; ytis the low-frequency macroeconomic variable and, ac-
cordingly, yt−nits lagged value with nbeing the lags; ˜
B(c;θ(τ) is a weighting function normalized
to sum up to 1 that depends on a vector of parameters θ(τ), and a lag order c= 0, ..., C −1 treat-
ing the high-frequency nature of w;wtis the high-frequency financial explanatory variable; Γc/m is
the standard lag operator, with mbeing the number of high-frequency observation available in one
low-frequency period; and ϵ(τ)tis the residual of the model.
To parsimoniously estimate the model in Equation (1), the lagged coefficients of ˜
B(c;θ(τ)) needs
to be parametrized. Otherwise, the number of lags of wtwould be quite large, and major drawbacks
from the elevated number of parameters would arise (Breiman and Freedman,1983). One of the
more parsimonious fashions to parametrize the lags of ˜
B(c;θ(τ)) are finite one-sided polynomials.
In the distributed lag literature, the exponential Almon lag polynomial is one of the most common
(Almon,1965) since it has two important properties. Namely, it provides positive coefficients and
it sums up to unity, which are both desirable properties for economic and financial applications. 7
Equation (2) defines the exponential Almon lag polynomial. Following Ghysels et al. (2005), we use
a functional form with two parameters, that is θ= (θ1, θ2):
˜
B(k;θ(τ)1;θ(τ)2) = eθ(τ)1k+θ(τ)2k2
PK
k=1 eθ(τ)1k+θ(τ)2k2(2)
where kis an index tracking every high-frequency observation in one period, that is ranging from
1 to k, and (θ1, θ2) are the parameters to be estimated which differ for every quantile τthat one
might consider. 8Through the function in (2) we are able to significantly decrease the number of
parameters to be estimated and therefore the model becomes parsimonious. In fact,the numbers of
parameters to be estimated for the high-frequency variable of our model reduces to only two, that is
the polynomial weights θ1and θ2. The weighting function in Equation (2) yields a vector of weights
with dimensions mx 1, with mbeing the number of high-frequency observations corresponding to
one low-frequency observation. Multiplying the matrix of the high-frequency information times this
vector of weights allows us to insert both components in our model, collapsing the high-frequency
7Another common finite one-sided polynomial is the Beta Lag polynomial. This function is though less used in
the literature and therefore we stick to the Almon Lag.
8If low-frequency data are available monthly and high-frequency data daily, k=1:22, if we assume that there are
22 days in a month where information is available; if low-frequency data are available quarterly, k=1:60; etc.
8
variable to the same frequency of the low-frequency dependent variable.
At this point, having inserted in the model the MIDAS component, we can estimate the condi-
tional quantiles. We proceed as suggested by Koenker and Hallock (2001) and minimize a sum of
asymmetrical weighted absolute residuals, assigning different weights to negative and positive resid-
uals. That is, positive residuals are multiplied by the quantile τand negative residuals by τ−1.
To obtain conditional quantiles we solve the optimization problem in Equation 3replacing absolute
values by ρτ(·):
min
β∈RρXρτ(yi−ξ(xi, β)) (3)
where ρτ(·) is the tilted absolute value function yielding the τth sample quantile as its solution and
ξ(xi, β) is a parametric function.
2.2 Mixed-Frequency Granger Causality Test in Quantiles
In the first preliminary step of our methodology described in Section 2.1 we retrieved the possible
ranges that the parameters governing the weights could take. At this point, we can move to the second
step of our methodology which concerns performing the Granger causality test for mixed-frequency
data in quantiles. This test is non-parametric, that is it depends on the aggregation weights, which
we just estimated. Since the exact values are not known, we consider a values grid for the weights
and we determine which combination of weights yields the maximum value of the test statistic at
every quantile.
Following Jeong et al. (2012), a time series wtdoes not Granger cause another time series ytin
the τth quantile as to zt={yt−1, ..., yt−p, wt−1, ..., wt−q}if
Qτ(yt|zt) = Qτ(yt|xt) (4)
On the contrary, wtis a prima facie cause in the τth quantile if:
Qτ(yt|zt)=Qτ(yt|xt) (5)
9
where xt={yt−1, ..., yt−p}and Qτ(yt|·) is the τth conditional quantile of ytgiven ·. Accordingly, to
establish causality, we need to test the following set of hypotheses:
H0 : P{Fy|z(Qτ(xt)|zt) = τ}= 1 (6)
H1 : P{Fy|z(Qτ(xt)|zt) = τ}<1 (7)
The null hypothesis in Equation (6) is only true if 1{yt⩽Qτ(xt)}=τ+ϵt, where Fy|z(·) is the
conditional distribution function of ytgiven zt(xt) and 1(·) is the indicator function.
While different distant measures exist to test this hypothesis, we follow Jeong et al. (2012) and
consider J≡E[{Fy|z(Qτ(xt)|zt)−τ)}fz(zt)], where fz(zt) is the marginal density function of zt. Since
Fy|z(Qτ(xt)|zt)−τ) = E(ϵt|zt), we can rewrite:
J=E{ϵtE(ϵt|zt)fz(zt)}(8)
Jin Equation (8) above can be used to test the hypotheses in Equation (6) and (7) since it is strictly
positive and the equality in Equation (8) holds if and only if the H0 in Equation (6) is true.
At this point, we estimate the density-weighted conditional expectation E(ϵt|zt)fz(zt) with the
kernel method:
ˆ
E(ϵt|zt)ˆ
fz(zt) = 1
(T−1)hd
T
X
s=t
Ktsϵs(9)
where d=p+qis the dimension of z;Kts =K{(zt−zs)
h}is the kernel function, and his the bandwidth,
which we optimally choose for xtand derive for ztaccordingly. Replacing Equation (9) in Equation
(8), we obtain the test-statistic:
JT=1
T(T−1)hd
T
X
t=1
T
X
s=t
Ktsϵtϵs
=1
T(T−1)hd
T
X
t=1
T
X
s=t
Kts[1{yt⩽Qτ(xt)} − τ][1{ys⩽Qτ(xs)} − τ]
(10)
To have a feasible kernel based JT, we can estimate Qτ(xs), the τth conditional quantile of ysgiven
xs, using the nonparametric kernel method Jeong et al. (2012). Accordingly, the error ϵis computed
10
as ˆϵt=I{yt⩽ˆ
Qτ(xt)} − τand replaced in Equation (10). 9
Having defined the test statistic JT, we now need to deal with the high-frequency nature of the
financial variable wtand insert it into the test. We start from the estimation of the Quantile-MIDAS
model described in Section 2.1 and perform the estimation for every quantile τ∈[0.05 : 0.95]
with a step of 0.05. This allows us to gain extensive insights into potential polynomial weights
for the weighting function in Equation (2). This is particularly important since the test is rather
sensible to the polynomial weights since they determine the value of the collapsed high-frequency
variable. Deriving them from the Quantile-MIDAS estimation allows us to assign them based on
some presumable knowledge and information. Considering the minimum and maximum value of the
estimated polynomial weights, we define a range for (θ1, θ2) and we specify a value grid for every
combination of both weights. 10 We consider each and every one of these combinations to collapse the
high-frequency variable wt, ending up with different possible versions of the collapsed wt. We compute
the test statistic JTin Equation (10) for every version of the collapsed high-frequency variable and
we store a matrix with the maximum value of JTat every quantile τ. At this point, considering
the standard critical value of 1.96 we check if the high-frequency variable causes the low-frequency
variable at various quantiles.
2.3 Causal Inference
At this point, we can move to the third part of our methodology to finally infer about the relationship
between a low-frequency macroeconomic variable and a high-frequency financial variable.
The procedure described in Section 2.1 allows us to estimate a Quantile-MIDAS model at various
quantiles along with the respective polynomial weights. The Granger causality test in quantiles
adapted to mixed-frequency data described in Section 2.2 allows us to establish a formal link between
a macroeconomic low-frequency variable and a high-frequency financial variable, ascertaining if there
is a causal relation between the two variables. This is a crucial step since establishing Granger
causality justifies forecasting one variable based on another. The last step of our methodology aims
9Up to this point we replicated the code written by Jeong et al. (2012) in MATLAB. We successfully reproduced
their code and their empirical application to confirm that our version of their code was correct and in line with their
work. Minor modifications have been applied to smooth the code, but by and large, we remained faithful to their
version. What comes next is our original contribution.
10In defining a value range, we consider a reasonable step ending up with approximately 300 combinations of (θ1, θ2)
11
at inferring the magnitude of the relationship between a low-frequency macroeconomic variable and
a high-frequency financial variable. To do so, we estimate the Quantile-MIDAS model in Equation
(1) again, fixing (θ1, θ2) to those value maximizing the test-static JTat every quantile considered.
First we extract the combinations of polynomial weights as follow:
JT(wt(ˆ
θ)) = max(JT) (11)
where JTis the test statistic of Equation (10) computed at every τ∈[0.01 : 0.99]; ˆ
θis the estimated
combination of (θ1, θ2) that yields the maximum value of JT; and wtis the high-frequency variable.
Then, we re-estimate the model in Equation (1) with these polynomial weights, thus estimating only
the parameters related to the lagged values of the low-frequency dependent variable and the one
applied to the explanatory high-frequency variable without its weights:
yt=β0(τ) +
N
X
n=1
[β1(τ)yt−n+β2(τ)|yt−n|] + β3(τ)
C−1
X
c=0
˜
B(c;ˆ
θ(τ))Γc/mwt+ϵ(τ)t(12)
In this way, we are able to quantify the relationship between these two variables at different quantiles,
that is considering the whole distribution of the low-frequency dependent variable.
3 Empirical Analysis
Having defined how to model a formal link between a low-frequency macroeconomic variable and a
high-frequency financial variable, we now apply our framework to a much-debated question in the
literature. That is, do commodity prices contain helpful insights to predict inflation? Can commodity
prices be regarded as early warning signals of inflation changes?
To answer these questions, we measured inflation as the monthly change in the Consumer Price
Index (in short CPI) in percentage points. We considered the period ranging from September 2009
up until April 2022, hence covering a period of roughly 13 years. We downloaded it from FRED and
considered the CPI for all urban consumers indexed for 1982 −1984 = 100. We used the seasonally
adjusted version of this CPI index to remove the effects of seasonal change, including production
12
cycles, weather, etc. 11 For a graphical representation of how it evolves over time, please refer to
the line plot in Figure Ain Appendix AInflation will be our macroeconomic low-frequency variable
since it is available once per month.
Instead, we use futures on commodities as our high-frequency variable. Indeed, various studies un-
derscore that commodity price futures, compared to commodity spot prices, better reflect movements
in inflation in the medium-long run and can be fully informative about future economic developments
(Chinn and Coibion (2014), Gospodinov and Ng (2013), Hong and Yogo (2012) and Deschamps et al.
(2021)). We downloaded a selection of future continuous contracts from Kibot and we picked one
commodity for each main category. That is, we selected gold for the precious metals, crude oil for
energy, and corn for agriculture. In light of the recent Ukrainian-Russia war and its repercussions
on the economy, we decided to look at corn, alongside wheat as the main agriculture commodity, to
rule out any bias coming from this extreme event.
Then, we calculated the logarithmic returns of each of the given commodities, and we relied on
these figures for our analysis. We considered continuous contracts for every commodity. Similarly to
inflation, we considered the period ranging from September 2009 to April 2022. Commodity future
prices are available daily, thus we set m= 21, setting that there are 21 high-frequency observations
for one low-frequency observation, namely for a monthly value of inflation.
The Pseudo-R Square suggests that the optimal autoregressive structure for inflation has two
lags. Accordingly, we consider a Quantile-MIDAS model with two lags for all the applications. That
is, we set n= 2 in Equation (1) and (12).
3.1 Inflation & Gold Futures Contracts
We begin our empirical analysis, by investigating the relationship between gold and inflation. Gold
is one of the most important commodities, both for governments and private investors, with trading
taking place 24 hours a day worldwide and daily transactions comprising billions of dollars occur-
ring on a regular basis. Gold has historically been regarded as one of the most sage inflation hedge
11U.S. Bureau of Labor Statistics, Consumer Price Index for All Urban Consumers: All Items in U.S. City Average
[CPIAUCSL], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/CPIAUCSL,
January 29, 2023.
13
commodities, with its prices rising considerably at times of high inflation, such as those preceding
the great financial crisis Dempster and Artigas (2010). Indeed, Junttila et al. (2018) have found
that gold is among the commodities providing the safest hedge against economic and financial risks.
Specifically, gold has proved to be a safe haven during periods when investors are wary of a recession
or concerned about credit markets, target rate cuts, and uncertain about changing inflation rates, as
highlighted by Tuysuz (2013), Baur and McDermott (2010). Accordingly, while gold as a commodity
bears a monetary nature and is very sensitive to inflation and interest rate changes (Batten et al.,
2014), a formal causal gold-inflation linkage has not been established yet. We believe that this is a
relevant gap to fill, to be able to better hedge and foresee inflation.
Preliminary Quantile-MIDAS Estimation. Following our methodology in Section 2, we begin
estimating the Quantile-MIDAS model in Equation 1. The estimation results in Table 1suggests
that θ1∈[−0.006; 0.2] and θ2∈[−0.008; 0.15]. To obtain a reasonable range of values of polyno-
mial weights, we select a step of 0.01 for both, thus considering 21x16=336 possible combinations of
(θ1, θ2). These are reported in the first two columns of Table B1 in Appendix B
Mixed-Frequency Granger Causality Test in Quantiles. To establish if gold Granger causes infla-
tion, we compute the test statistic JTin Equation (10) at each and every one of these combinations
and we extrapolate the maximum value of JTat every quantile, collecting which combinations of
(θ1, θ2) yield given values of JT. These combinations are reported in Table 2. Table B1 in Appendix
Breports all the values of JTfor every pairs of (θ1, θ2) in the range indicated above. Then, to eval-
uate the causal relation between Gold and Inflation throughout the whole distribution, we plot JT
at various quantiles. Figure 3suggests that JTexceeds the critical value 2.57 when 0 ≤τ≤0.26,
0.39 ≤τ≤0.64, and 0.84 ≤τ≤0.97. Therefore, we derive that returns on gold futures are a
prima facie cause for inflation in these intervals and they are not in the remaining intervals. JT
indicates that causality is strongest in the left tail, suggesting that low returns impact inflation the
most. Accordingly, looking at the results in Figure 3, we find that returns on gold futures have a
significant predictive power for extreme changes in inflation. Therefore, we provide strong evidence
justifying the usage of gold futures to hedge inflation movements. To rule out any concern arising
14
Table 1: Quantile-MIDAS Preliminary Analysis: Estimating Weights for Gold Returns Collapse
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0:intercept -0.263***
[0.000]
-0.032***
[0.000]
0.014***
[0.000]
0.093***
[0.000]
0.307***
[0.000]
β1:yt−1
1.161***
[0.000]
0.404***
[0.000]
0.572***
[0.000]
0.546***
[0.004]
0.559***
[0.000]
β2:yt−2
-0.944***
[0.000]
-0.248***
[0.000]
0.227***
[0.000]
0.202***
[0.003]
0.312***
[0.000]
β3:|yt−1|-0.217***
[0.000]
-0.198***
[0.001]
-0.335***
[0.000]
-0.204***
[0.000]
-0.176***
[0.000]
β4:|yt−2|0.567***
[0.001]
0.053***
[0.001]
0.289***
[0.000]
0.370***
[0.001]
0.073***
[0.000]
β5: Gold Log Returns 11.675***
[0.000]
3.711***
[0.001]
3.665***
[0.008]
3.783***
[0.008]
7.998***
[0.000]
θ1
0.045***
[0.000]
0.002***
[0.001]
0.004
[0.065]
0.002***
[0.000]
0.199***
[0.000]
θ2
0.017***
[0.001]
0.061***
[0.000]
0.132***
[0.002]
0.147***
[0.001]
-0.008***
[0.000]
Pseudo R Squared 0.17 0.18 0.21 0.25 0.24
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the inde-
pendent variables used in our model along the rows. Each cell contains the estimated coeffi-
cients, with their respective p-values presented in square brackets underneath. These p-values
are to be interpreted using the significance codes provided in the table. The polynomial weights,
which are pivotal at this stage, have been highlighted in grey.
Table 2: Table showing the maximum values of JTat every quantile along with the combinations of
(θ1, θ2)
τJTθ1θ2
0.05 40.89 0.194 0.142
0.25 2.97 -0.006 0.002
0.50 4.90 0.194 0.012
0.75 1.36 -0.006 -0.008
0.95 6.45 -0.006 0.032
from simultaneously computing several inferences, we apply the Bonferroni correction. By setting
the significance cut-off at α
n, where αis the aggregated significance level and nis the number of
tests performed, we compensate for the probability of running into a type I error with the confidence
intervals being larger (Dunn,1961).
Causal Inference. To investigate the relation between gold futures and inflation in a more precise
way, we estimate the Quantile-MIDAS model again, fixing (θ1, θ2) at the values indicated in Table
2, which returns the results depicted in Table 3. Looking at the coefficient β5, we conclude that
15
Table 3: Table showing Quantile-MIDAS regression results with fixed polynomial weights.
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.230***
[0.000]
-0.031***
[0.000]
0.022***
[0.000]
0.119***
[0.000]
0.349***
[0.000]
β1:yt−11.258***
[0.000]
0.552***
[0.000]
0.552***
[0.000]
0.416***
[0.004]
0.629***
[0.000]
β2:yt−2-1.035***
[0.000]
0.074***
[0.000]
0.197***
[0.000]
0.323***
[0.003]
0.379***
[0.000]
beta3:|yt−1|-0.225***
[0.000]
-0.168***
[0.001]
-0.330***
[0.000]
-0.183***
[0.000]
-0.227***
[0.000]
β4:|yt−2|0.540***
[0.000]
0.101***
[0.001]
0.299***
[0.000]
0.295***
[0.001]
-0.043***
[0.000]
β5:Gold Log Returns 6.833***
[0.000]
10.739***
[0.001]
3.920***
[0.001]
0.064***
[0.001]
−2.708***
[0.003]
Pseudo R Squared 0.27 0.17 0.22 0.25 0.27
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the independent
variables used in our model along the rows. Each cell contains the estimated coefficients, with their
respective p-values presented in square brackets underneath. These p-values are to be interpreted
using the significance codes provided in the table. The coefficient of the MIDAS polynomial function
is reported in bold, as it is the key element in this stage.
the impact of gold futures log returns on inflation is larger in the left tail than in the center of the
inflation distribution and in the right tail. Accordingly, when inflation is very low, returns on gold
futures have the highest positive impact on inflation, increasing it by 0.07 units in the 5th quantile
and by 0.11 units 25th. In the upper quantile of the inflation distribution (i.e. at τ= 0.95), the
coefficient takes a negative sign. This means that if inflation is very high, log returns on gold futures
decrease it by 0.03 units. Figure 4below shows the estimated conditional quantiles against the actual
data, attesting that our model captures the observations fairly well.
As a robustness check, we estimate an unrestricted Quantile-MIDAS model as well and our causal-
ity test accordingly. The estimation results and the test statistic across the whole distribution are
reported in Appendix C2 in Table C2 and he results of the Granger causality quantiles at all quantiles
is reported in Figure C1. The estimation results show that the significance is fading, with almost no
coefficient being statistically different from zero. This is the case since, using this model specification,
our time series become very short and there are too many parameters. In any case, the causality test
is very similar to the one performed with the polynomial weights and confirm the results discussed
above.
16
Significance of the Weighting Function. The combinations of weights that return the maximum
value of test statistics across the distribution, indicated in Figure 3, assume the polynomial shape
depicted in Figure 5. These plots suggest that in the tails and in the center of the distribution, the
last days of the months matter the most in explaining variations in inflation.
To further investigate the pieces of information revealed by the graphs in Figure 5, we check if the
underlying weighting function is significant at every quantile and how its significance evolves through-
out the monthly time series. For the baseline case, we evaluate the significance of the Almong Lag
polynomial function. To this aim, we resort to the δ-Method firstly introduced by Dorfman (1938)
to approximate the variance of the weighting function and to ultimately evaluate the significance of
the MIDAS expansion dictated by the Almon Lag polynomial.
We begin by computing the gradient of the weighting function relative to θ1and θ2analytically.
Then, we compute the gradient at those values of θ1and θ2reported in Table 2, that is those we used
for the causal inference. Next, we retrieve the variance-covariance matrix at every quantile from the
estimation results in table 3and we extract only the grids relative to the polynomial weights, that
is (θ1, θ2). Finally, we approximate the standard error of the MIDAS expansion (i.e. of the Almon
Lag polynomial as a weighting function) multiplying the gradient of the function at the (θ1, θ2) max-
imizing our test statistic, times the variance-covariance matrix, times the same gradient transposed.
Results are reported in Figure 6below. Interestingly, we can see that in the tails of the distribution,
almost every day is important to predict inflation. Instead, at the center of the distribution, only
the last days of the month are in fact relevant.
Nowcasting Inflation at Quantiles: Causality from Gold. To complete the analysis of the rela-
tionship between gold and inflation, we perform a nowcasting exercise following Adrian et al. (2019),
Ba´nbura et al. (2013) and Giannone et al. (2008). We analyze and compare 3 different models, that
is our Quantile-MIDAS model (QMIDAS), a Quantile model where the high-frequency part is simply
collapsed to the average (QAVG), and a Quantile Autoregressive model with 2 lags (QAR(2)), which
we refer to as the benchmark. For all them we consider 19 quantiles, ranging from 0.05 to 0.95,
equally spaced with a step of 0.05. In line with the rest of our intuition, we take advantage of the
17
high-frequency nature of returns on gold futures to produce real-time forecasts. Indeed, we use a
one-step expanding window updating both the inflation information and the daily return on gold
futures, meaning that for each monthly observation, we can produce 21 forecasts. Since we have 149
monthly inflation observations, we set the length of the in-sample period to 96 months, corresponding
to observations until December 2017. Accordingly, the out-of-sample periods starts at January 2018,
corresponding to the latest 53 months in the sample, thus ending in April 2022. For each of these
53 months, though, we have daily information regarding returns on gold futures, and accordingly, we
are able to produce 21 forecast for each of these 53 months.
To begin our nowcasting analysis, we compare the three models with quantile scores. Results
are reported in Table 4below, where the different quantiles are detailed on the columns, and our
three different models on the rows. For the full selection of quantiles, please refer to Table D1 in
Appendix D. Table 4also reports the outcome of the Diebold-Mariano test against the benchmark
(the univariate and autoregressive model with two lags of inflation only) and the model confidence
set at 5% with respect to all the models. These results show that both our model and the QAVG,
that is the model with the average monthly return on gold futures, perform better than QAR(2).
This implies that returns on gold futures are relevant when predicting inflation.
Table 4: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). ***, ** and * indicate that ratios are significantly different from 1 at 1%, 5% and
10%, according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.25 0.50 0.75 0.90 0.95
QAR(2) 1.971 3.342 4.911 3.819 1.495 0.313 0.085
QMIDAS 0.954*** 0.964*** 0.985*** 0.971*** 0.943*** 0.981*** 0.978***
QAVG 0.968*** 0.988*** 0.999*** 0.995*** 0.996*** 0.982*** 0.966***
To understand which one performs better, we look at the cumulative sum of the differences be-
tween the quantile scores of our model (QMIDAS) and the model with the average (QAVG) over
the out-of-sample period. Results are plotted in Figure 7. For the sake of readability, we reported
a selection of quantiles only. Please refer to Appendix Dfor the complete distribution. While both
quantile scores are above 0, the blue line, hence the one corresponding to our QMIDAS model, is
consistently above the red line, corresponding to the QAVG model. This indicates that our QMIDAS
model is consistently outperforming the QAVG model over the out-of-sample period.
18
3.2 Inflation & Crude Oil Futures Contracts
As a second application, we look at another rather impactful commodity, that is crude oil. Various
studies highlight that oil prices move along with a country’s economic growth and that they mirror
a country’s economic conditions (Darby (1982) and Kilian and Vigfusson (2017)). Indeed, under
ordinary times, as the economy grows, so does the demand for oil, and when a country is in a
recession or is under distress, oil prices are likely to shrink, as happened after the global crisis in
2020 and the subsequent lockdowns in fairly every country worldwide. More recently, the growing
tensions between Russia and Ukraine, and the rest of the world are fomenting fears about a crude
oil supply shortage. In turn, these events considerably contribute to rising inflation, among other
important economic indicators. Indeed, some works have highlighted that crude oil price fluctuations
are a major driver of inflation variability, but there is a strong asymmetric relationship between crude
oil prices and inflation, both for oil-importing and -exporting countries (Chen (2009), Raheem et al.
(2020) and ´
Alvarez et al. (2011)). We gather that while it is clear that this relationship exist and
has crucial relevance, a formal link between the two variables hasn’t been established yet. With this
application, we aim to fill this gap and shed light onto this important, yet still rather fuzzy relation.
Preliminary Quantile-MIDAS Estimation Similarly to the previous section, we begin our empirical
analysis by estimating our Quantile-MIDAS model in Equation 1. Estimations results are reported
in Table 5and suggest that θ1∈[−0.1; 0.25] and θ2∈[−0.006; 0.02], where each value represent the
minimum and maximum value that resulted statistically significant in the estimation. To obtain a
reasonable value grid of polynomial weights we select a step of 0.02 for θ1and of 0.002 θ2, ending up
with 18x14=252 combinations. These are reported in the first two columns of Table B2 in Appendix B
Mixed-Frequency Granger Causality Test in Quantiles At this point, to establish a formal link
between crude oil and inflation, that is to investigate whether crude oil prices Granger cause infla-
tion, we compute the test statistic JTin Equation (10) at various quantiles. We extrapolate the
maximum value of JTat every quantile considered and we mark which combination of polynomial
weights return given values of JT. We report them in Table 6. Table B2 in Appendix Breports all
the values of JTfor every pairs of (θ1, θ2) in the range indicated above. In Figure 8we show how
19
Table 5: Quantile-MIDAS Preliminary Analysis: Estimating Weights for Crude Oil Returns Collapse
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: Intercept -0.208
[9.815]
0.008***
[0.000]
0.062***
[0.000]
0.117***
[0.000]
0.281***
[0.000]
β1:yt−10.929
[21987]
0.645***
[0.000]
0.387***
[0.000]
0.355***
[0.000]
0.283***
[0.000]
β2:yt−2-0.305
[30401]
-0.073***
[0.001]
0.232***
[0.000]
0.384***
[0.000]
0.372***
[0.000]
β3:|yt−1|-0.282
[44220]
-0.046**
[0.015]
0.002***
[0.000]
-0.067***
[0.000]
0.145***
[0.000]
β4:|yt−2|0.091
[48460]
-0.108***
[0.015]
-0.021***
[0.000]
0.113***
[0.001]
-0.070***
[0.000]
β5: Crude Oil Log Returns 19.786
[68161]
25.506***
[0.024]
19.978***
[0.008]
21.533***
[0.000]
19.272***
[0.003]
θ1
-0.195
[107.43]
0.104***
[0.000]
0.074
[0.065]
-0.022***
[0.000]
0.232***
[0.000]
θ2
0.014***
[0.001]
-0.001***
[0.000]
-0.003***
[0.002]
0.02***
[0.000]
-0.006***
[0.000]
Pseudo R Squared 0.30 0.29 0.29 0.34 0.43
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the independent
variables used in our model along the rows. Each cell contains the estimated coefficients, with their
respective p-values presented in square brackets underneath. These p-values are to be interpreted
using the significance codes provided in the table. The polynomial weights, which are pivotal at this
stage, have been highlighted in grey.
Granger causality between crude oil prices and inflation evolves throughout the whole distribution.
We can see that the test statistic JTis always above the critical value 2.57, hence indicating that
crude oil prices are a prima facie cause for inflation at any price level. JTonly drops below 2.57
at the very end of the distribution (i.e. at τ= 0.96), hence indicating that extremely high returns
on crude oil futures do not Granger cause inflation to abruptly change. Consequently, returns on
crude oil futures have significant predictive power for extreme, as well as non-extreme, movements
in inflation and could reliably be used as hedging instruments against inflation. As a robustness,
similar to the gold application, we applied the Bonferroni correction to rule out concerns arising from
multiple testing.
Causal Inference. Then, to further investigate the relationship between log returns on crude
oil futures and inflation, we estimate our Quantile-MIDAS model again fixing (θ1, θ2) at the values
20
Table 6: Table showing the maximum values of JTat every quantile along with the combinations of
(θ1, θ2)
τJTθ1θ2
0.05 45.79 0.10 -0.006
0.25 5.99 0.04 -0.002
0.50 8.82 -0.10 0.004
0.75 2.94 0.20 0.018
0.95 10.65 0.22 0.018
indicated in Table 6. Table 7reports the estimation of our Quantile-MIDAS model with fixed poly-
nomial weights. The coefficient β5relative to Crude Oil Log Returns takes a positive sign throughout
the whole distribution and is significant at every quantile considered. We can notice that, while it
remains rather large at any inflation status, it takes on more sizable values in the left tail of the
distribution, the magnitude of which then gradually decreases as the right tail is approached. This
is in line with the causality test depicted in Figure 8, where it appeared that the causal relation was
strongest on the left tail, that is up until τ= 0.20, while it was weaker, yet still present, in the rest
of the distribution. It follows that, when inflation is very low, log returns on crude oil futures, do
have the biggest impact on inflation, increasing it by 0.32 units at τ= 0.05 and by 0.24 units at
τ= 0.25. The effect is still fairly large around the median, where if crude oil futures increase by 1%,
inflation increases by 0.18 units. Toward the end of the distribution, the effect shrinks but is still
significant and positive. At τ= 0.75, inflation increases by 0.05 units if crude oil futures increase by
1%, whereas at τ= 0.95, by 0.06 units. In short, we gather that at times of low inflation, crude oil
futures are powerful instruments to foresee how inflation will move in the future.
To have a graphical intuition of our model fit, we plot the conditional quantiles of our estimated
model, against the actual data. The model captures the actual data (the blue dots) pretty well.
Moreover, we observe that conditional quantiles do not cross, implying that the conditional quantile
function satisfied the imposed monotonicity constraint.
Similarly to the section above, as robustness, we estimate an unrestricted Quantile-MIDAS model
as well and perform our mixed-frequency Granger causality test accordingly. Results hold. Please
consult Appendix C.2 for the estimation and test results. The same reasoning explained in the pre-
vious sub-section applies.
21
Table 7: Table showing Quantile-MIDAS regression results with fixed polynomial weights.
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.116***
[0.000]
0.012***
[0.000]
0.054***
[0.005]
0.173***
[0.000]
0.318***
[0.000]
β1:yt−10.354***
[0.000]
0.661***
[0.001]
0.330
[0.280]
0.494***
[0.000]
0.709***
[0.000]
β2:yt−2-0.258***
[0.000]
-0.143***
[0.001]
0.266
[0.283]
0.164***
[0.000]
0.299***
[0.000]
β3:|yt−1|-0.102***
[0.000]
0.009***
[0.001]
0.037
[0.286]
-0.119***
[0.001]
-0.123***
[0.000]
β4:|yt −2|0.080***
[0.000]
-0.064***
[0.001]
0.008
[0.283]
0.095***
[0.000]
-0.121***
[0.000]
β5: Crude Oil Log Returns 31.979***
[0.000]
24.428***
[0.001]
18.457***
[0.001]
5.531***
[0.000]
6.423***
[0.000]
Pseudo R Squared 0.29 0.23 0.25 0.32 0.42
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the independent variables
used in our model along the rows. Each cell contains the estimated coefficients, with their respective p-values
presented in square brackets underneath. These p-values are to be interpreted using the significance codes
provided in the table. The coefficient of the MIDAS polynomial function is reported in bold, as it is the key
element in this stage.
Significance of the Weighting Function. Figure 10 plots the exponential Almon lag polynomial
function of Equation (2) using given polynomial weights as inputs. This function takes a concave
shape in the left tail of the distribution (i.e. at τ= 0.05 and τ= 0.25), and a convex shape around
the median, suggesting that in times of low inflation, the first 10 days have more predictive power
than the last 11 days, whereas at τ= 0.5, the first and the last 5 days of the month are those that
matter the most. Instead, when inflation is very high (i.e. at τ= 0.75 and τ= 0.95), the last 5 days
of the month are those with the largest predictive power.
Similar to the previous section, we evaluate the significance of the MIDAS expansion according
to the Almon Lag Polynomial specification. As shown in the plots in Figure 11, when evaluating the
relevance of crude oil in predicting inflation, throughout nearly the whole distribution, every day of
the month is relevant.
Nowcasting Inflation at Quantiles: Causality from Crude Oil. As a last step of our empirical
analysis on crude oil, we perform the nowcasting exercise explained in the previous Subsection 3.1.
22
We compare our model (QMIDAS) and an average model (QAVG) model against the benchmark of
the AR(2) model for inflation using quantile scores. A snapshot of results is reported in Table 8; for
the whole selection of quantiles, please refer to Table D2. Results show that both models perform
significantly better than the benchmark at all quantiles, suggesting that when predicting inflation,
returns on crude oil futures are indeed relevant.
Table 8: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). ***, ** and * indicate that ratios are significantly different from 1 at 1%, 5% and
10%, according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.25 0.50 0.75 0.90 0.95
QAR(2) 1.971 3.342 4.911 3.819 1.495 0.313 0.085
QMIDAS 0.799*** 0.861*** 0.892*** 0.864*** 0.887*** 0.910*** 0.891***
QAVG 0.815*** 0.827*** 0.847*** 0.866*** 0.891*** 0.936*** 0.955***
To understand which model perform better, we look at the cumulative sum of the differences
between the quantile scores of our model and the QAVG model. Results are shown in Figure 12.
While there aren’t significant differences in the performance of the two models in the center and in
the left tail of the distribution, in the most right tail of the distribution, our model performs better
than the QAVG model.
23
3.3 Inflation & Corn Future Contracts
Table 9: Quantile-MIDAS Preliminary Analysis: Estimating Weights for Crude Oil Collapse
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.149***
[0.000]
-0.010***
[0.000]
0.018***
[0.000]
0.119***
[0.000]
0.317***
[0.000]
β1:yt−11.195***
[0.000]
0.507***
[0.001]
0.569***
[0.000]
0.417***
[0.000]
0.565***
[0.000]
β2:yt−2-0.816***
[0.000]
0.169***
[0.001]
0.221***
[0.000]
0.324***
[0.000]
0.388***
[0.000]
β3:|yt−1|-0.247***
[0.000]
-0.176**
[0.000]
-0.336***
[0.000]
-0.183***
[0.000]
-0.204***
[0.000]
β4:|yt−2|0.259***
[0.000]
-0.044***
[0.000]
0.286***
[0.000]
0.295***
[0.000]
0.030**
[0.000]
β5: Crude Oil Log Returns 0.118***
[68161]
1.177***
[0.001]
0.017***
[0.000]
0.040***
[0.000]
0.045***
[0.003]
θ1
0.000
[0.001]
0.222***
[0.000]
0.003***
[0.000]
0.004***
[0.000]
0.016***
[0.000]
θ2
0.022***
[0.000]
-0.008***
[0.000]
0.125***
[0.000]
0.140***
[0.000]
0.012***
[0.000]
Pseudo R Squared 0.14 0.17 0.21 0.25 0.18
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the independent
variables used in our model along the rows. Each cell contains the estimated coefficients, with their
respective p-values presented in square brackets underneath. These p-values are to be interpreted
using the significance codes provided in the table. The polynomial weights, which are pivotal at this
stage, have been highlighted in grey.
For the last empirical application, we look at two main food commodities, namely corn and wheat
future contracts. While corn is the most produced and consumed crop worldwide with nearly 1.1
billion tons harvested every year, wheat comes near, especially in the region of Europe and Central
Asia, as opposed to the U.S., with nearly 800 million tons produced yearly. Since the market for
wheat endured major disruptions following the Russia-Ukraine war, we decided to focus on corn for
our empirical application. There is a general consensus that food commodity prices are relevant to
predict inflation and, more generally, that they could be powerful instruments to foresee economic
trends. In fact, some countries are predominantly driven by the exports of essential crops, such
as corn, wheat, soybean and, accordingly, changes in the market and prices for these commodities
would yield a major impact on the underlying economy and especially inflation. Nevertheless, very
24
few studies have focused on establishing a formal link between inflation and food commodities. Cec-
chetti and Moessner (2008) suggested that suggests that rising food commodity prices have generally
not spawned strong second-round effects on inflation. Furceri et al. (2016) showed that global food
price shocks have a great impact on domestic inflation in a large group of countries, increasing it
on average by 0.5 percentage points after a year. Peersman (2022) corroborated these results and
found that exogenous shifts in food prices indirectly trigger inflationary effects via rising wages and
that they explain almost 30% of the euro-area inflation volatility. None of these studies, though,
approached the issue dealing with the mismatch in the frequency of inflation and commodity prices.
We aim to fill this gap and establish a formal link between preserving the original frequency to obtain
real-time indications about inflation starting from futures on food commodities.
Quanile-MIDAS Estimation. Similarly to the previous sections, we begin estimating our Quantile-
MIDAS model of Equation (1). The estimation results are reported in Table 9and suggest that
θ1∈[−0.07; 0.32] and θ2∈[−0.18; 0.14], where each value represent the minimum, and maximum
value that proved to be statistically significant. To obtain a value grid out of these boundary values,
we selected a step of 0.03 for both θ1and θ2, for a total of 14x11=154 combinations. These are
reported in the first two columns of Table B3 in Appendix B
Mixed-Frequency Granger Causality Test in Quantiles. At this point, having gained insights into
the polynomial weights (θ1, θ2), we can proceed with the Granger causality test. We compute the
test statistic JTin Equation (10) for every combination and at various quantiles. We extrapolate the
maximum value of the test statistic at every quantile considered and we mark which combination of
(θ1;θ2) return given values of JT. We report them in Table 10. Table B3 in Appendix Breports all
the values of JTfor every pairs of (θ1, θ2) in the range indicated above. Figure 13 shows how Granger
causality between futures on corn and inflation evolves across the whole distribution. JTis beyond
the critical value 2.57 when 0 ≤τ≤0.27, 0.38 ≤τ≤0.64, 0.79 ≤τ≤0.96. Therefore, e gather
that returns on corn futures are a prima facie cause for inflation in these intervals, whilw they are
not when 0.28 ≤τ≤0.37 and 0.80 ≤τ≤0.95, and at th last three percentiles of the distribu-
tion. Figure 13 suggests that causality is stronger in the left tail, rather than in the center of the
25
Table 10: Table showing the maximum values of JTat every quantile along with the combinations of
(θ1, θ2)
τJTθ1θ2
0.05 65.36 0.32 0.12
0.25 3.82 0.29 0.12
0.50 5.49 0.32 0.03
0.75 0.61 0.32 0
0.95 9.96 -0.04 0
distribution since the value of JTpeaks at the lower quantiles and slowly decreases at the end of the
distribution. Thus, low returns on corn futures cause inflation to abruptly change. More generally,
returns on corn futures have significant predictive power for extreme, as well as non-extreme changes
in inflation and could be used as hedging instruments against inflation. As a robustness, similar to
the gold application, we applied the Bonferroni correction to rule out concerns arising from multiple
testing.
Table 11: Table showing Quantile-MIDAS regression results with fixed polynomial weights.
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.157***
[0.000]
-0.022***
[0.000]
0.019***
[0.000]
0.105***
[0.000]
0.320***
[0.000]
β1:yt−11.173
[8713]
0.517
[1321]
0.573***
[0.000]
0.384***
[0.000]
0.686***
[0.000]
β2:yt−2-0.856
[8713]
0.118
[1321]
0.225***
[0.000]
0.332***
[0.000]
0.394***
[0.000]
beta3:|yt−1|-0.213***
[0.000]
-0.175***
[0.002]
-0.339***
[0.000]
-0.151***
[0.000]
-0.230***
[0.000]
β4:|yt−2|0.332***
[0.000]
0.033***
[0.002]
0.282***
[0.000]
0.327***
[0.000]
0.010***
[0.000]
β5:Corn Log Returns 0.055***
[0.000]
0.275***
[0.000]
0.022***
[0.000]
0.176***
[0.000]
-0.405***
[0.000]
Pseudo R Squared 0.12 0.17 0.21 0.25 0.29
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
Note: This table displays the estimated conditional quantiles along the columns and the independent
variables used in our model along the rows. Each cell contains the estimated coefficients, with their
respective p-values presented in square brackets underneath. These p-values are to be interpreted
using the significance codes provided in the table. The coefficient of the MIDAS polynomial function
is reported in bold, as it is the key element in this stage.
Causal Inference. Having established that there is indeed causality, namely that returns on corn
26
futures Granger cause inflation at various quantiles, we now examine this linkage in more depth.
To this aim, we estimate our Quantile-MIDAS model of Equation (12) fixing (θ1, θ2) at the values
reported in Table 10. Table 11 reports the estimation result of our Quantile-MIDAS model with
fixed polynomial weights. The coefficient β5relative to Corn log returns is significant throughout
the whole distribution. It always returns a positive sign, except on the left tail (i.e. at τ= 0.95),
where the coefficient is negative. Compared to the other two commodities considered, that is gold
and crude oil, the relationship between inflation and corn futures is weaker. Indeed, in Table 11,
we can observe that the coefficients on corn log returns are rather small, yet strongly significant.
Accordingly, while futures on corn Granger cause inflation to change as shown in Figure 13, the
impact that the former has on the latter is minor. Indeed, when inflation is low (i.e. at τ= 0.05
and τ= 0.25), if log returns on corn futures increase by 1%, inflation increases by 0.0005 units and
0.003 units respectively. When inflation lies at a median level (i.e. at τ= 0.5), a 1% increase in
corn futures returns leads to a 0.0002 units increase in inflation. When inflation is rather high (i.e.
at τ= 0.75), a 1% increase in corn futures leads to a 0.002 units increase in inflation. Instead, if
inflation is very high (i.e. at τ= 0.95), a 1% increase in corn futures returns will bring down inflation
by 0.004 units. We conclude that while food commodities can relate about inflation movements, gold
and crude oil are more powerful and leads to larger changes in inflation.
To have a graphical intuition of our model fit, in Figure 14 we plot the conditional quantiles of
our estimated model, against the actual data. The model captures the actual data (the blue dots)
pretty well. We observe that conditional quantiles do not cross, implying that the conditional quan-
tile function satisfied the imposed monotonicity constraint.
As a robustness check, we estimate an unrestricted Quantile-MIDAS model as well and our causal-
ity test accordingly. Please refer to Appendix C.3 for the estimation results in Table C3 and the test
statistic development throughout the whole distribution in Figure C6.
Significance of the Weighting Function. Figure 15 plots the exponential Almon lag polynomial
function in Equation 2using as input the combinations of (θ1, θ2) that maximize JT, that is those
reported in Table 10. This function is increasing up until τ= 0.75 and decreasing at τ= 0.95.
These shapes suggest that when inflation is very low and when it lies around the median value, it at
27
τ= 0.05 through τ= 0.75, the last 5 days of the month entail the largest predictive power, compared
to the first days. At τ= 0.95, that is at left tail of the distribution, the function becomes decreasing,
suggesting that the first days of the month matter the most to foresee inflation, while as the end of
the month approaches, information becomes gradually less insightful.
To better understand this trend, we evaluate the significance of the MIDAS expansion function,
that is of the Almon Lag Polynomial for our baseline case. The plots in Figure 16 show that around
the median and at τ= 0.75, only the last day of the month (i.e. m= 21), is relevant to predict infla-
tion. Nevertheless, when the left tail of the distribution is considered, as well as τ= 0.95, every day
in the month is statistically significant. This result predominantly demonstrates the importance of
analyzing the whole distribution and a MIDAS specification when trying to evaluate the relationship
between a macroeconomic and a financial variable such as inflation and returns on commodities.
Table 12: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). ***, ** and * indicate that ratios are significantly different from 1 at 1%, 5% and
10%, according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.25 0.50 0.75 0.90 0.95
QAR(2) 1.971 3.342 4.911 3.819 1.495 0.313 0.085
QMIDAS 0.961*** 0.960*** 0.973*** 0.992*** 0.978*** 0.981*** 0.943***
QAVG 0.903*** 0.923*** 0.961*** 0.996*** 0.999*** 0.986*** 0.987***
Nowcasting Inflation at Quantiles: Causality from Corn. As a last step of our empirical analysis
on corn, we perform a nowcasting exercise as discussed and described in the previous subsections.
We compare our model (QMIDAS) and the QAVG model against the usual benchmark of AR(2)
for inflation with compare quantile scores. A snapshot of results is reported in Table 12, while the
full selection of quantiles is reported in Table D3 in Appendix D. It shows that both QMIDAS and
QAVG are significantly better then the benchmark across the whole distribution, suggesting that,
once again, returns on corn futures are relevant to predict inflationary trends. To understand which
model specification works better, we plot the cumulative sum of the differences between the quantile
scores of our model and the QAVG model. Results are displayed in Figure 17. We conclude that at
the center of the distribution and up until τ= 0.75, our model performs better. Nevertheless, in the
28
left tail of the distribution QAVG performs better. Instead, at the utmost right tail, the two models
perform equally accurately.
4 Concluding Remarks
By extending the nonparametric causality test in quantiles initially proposed by Jeong et al. (2012),
we introduce a modified version of this test capable of accommodating mixed-frequency data. The
appealing feature of our test statistic lies in its ability to explore Granger causality between two
variables sampled at different frequencies across various conditional quantiles.
We estimate the polynomial weights to accurately collapse the high-frequency variable into a low-
frequency one via a Quantile-MIDAS model, hence placing the causality testing within the framework
of Jeong et al. (2012). We then use the so-estimated weights to evaluate the magnitude of the relation-
ship between two mixed-frequency variables. Our first contribution is thus on the methodological
side, as we introduce a practical approach to detect Granger causality at quantiles, and as a bi-
product, to measure the intensity of the causality.
We then apply our proposed methodology to investigate Granger causality between inflation and
returns on futures on commodities, that is gold, crude oil, and corn continuous contracts. We find
that logarithmic returns on commodity futures are a prima facie cause for inflation in the lower quan-
tiles of the distribution and marginally around the median. Particularly, starting with the future
contract on gold, we find that the effect is larger in the left tail than in the center of the inflation
distribution. This means that when inflation is very low (i.e. in the 5th ) and 25th quantile of the
distribution, if returns on gold futures increase by 1%, inflation increases by 0.06 units and 0.11
units, respectively. The application to future contracts on oil yields similar results. In the 5th and
25th quantile of the inflation distribution, if returns on oil futures increase by 1%, inflation increases
by 0.31 units and 0.24 units. While there is causality between futures on agricultural commodities
and inflation, the magnitude of the relationship is rather small. We conclude that precious metals
and energy commodities have the most prominent impact on inflation.
Our proposed framework can be favorably used by investors and policymakers to anticipate infla-
tion movements through the lens of financial market perception and, as an example, through futures
29
on commodities. This is particularly relevant since financial data are available at a very high fre-
quency and the information they entail could offer a head start on inflation movements, which would
otherwise be known only after the end of the month or quarter.
30
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Figures
Figure 1: Stylized methodology
Figure 2: Plot of the daily gold (solid blue line), crude oil (dash-dotted orange line), corn (dotted
green line), and wheat (dashed purple line) prices from 2009 to 2022
Figure 3: Test statistic JTwith respect to various quantiles. The blue solid line represents the test
statistic based on the Almon Lag polynomial. The red dashed line marks the aggregated critical value
at α= 0.01 and the associated standard normal critical value of 2.57. The green dotted line marks
the significance level increased as per the Bonferroni correction (i.e. at α= 0.00003 and an associated
critical value of 3.98).
Figure 4: Conditional Quantiles and actual data.
Figure 5: Almon Polynomial Shape for the combination of (θ1, θ2) that maximize JTat various quantiles
τindicated above each plot.
Figure 6: These plots show the significance of the Almon Lag Polynomial as a weighting function for
the MIDAS component at different quantiles. The y-axis reports the z-scores and the x-axis each day
min a month. The blue squares indicate the z-scores within the [−1.96,1.96] range, that is the non-
significant days. The green dots indicate the z-scores that are above or below the confidence interval,
that is the significant days.
Figure 7: Quantile Scores of QMIDAS vs QAVG model in the out-of-sample period. The blue line
refers to the QMIDAS model; the red one to the QAVG model.
Figure 8: Test statistic JTwith respect to various quantiles. The red dashed line marks the aggregated
critical value at α= 0.01 and the associated standard normal critical value of 2.57. The green dotted
line marks the significance level increased as per the Bonferroni correction (i.e. at α= 0.00003 and an
associated critical value of 3.86).
Figure 9: Conditional Quantiles and actual data.
Figure 10: Almon Polynomial Shape for the combination of (θ1, θ2) that maximize JTat various
quantiles τindicated above each plot.
Figure 11: These plots show the significance of the Almon Lag Polynomial as a weighting function for
the MIDAS component at different quantiles. The y-axis reports the z-scores and the x-axis each day
min a month. The blue squares indicate the z-scores within the [−1.96,1.96, that is the non-significant
days. The green dots indicate the z-scores that are above or below the confidence interval, that is the
significant days.
Figure 12: Quantile Scores of QMIDAS vs QAVG model in the out-of-sample period. The blue line
refers to the QMIDAS model; the red one to the QAVG model.
Figure 13: Test statistic JTwith respect to various quantiles. The red dashed line marks the aggregated
critical value at α= 0.01 and the associated standard normal critical value of 2.57. The green dotted
line marks the significance level increased as per the Bonferroni correction (i.e. at α= 0.00003 and an
associated critical value of 3.82).
Figure 14: Conditional Quantiles and actual data.
Figure 15: Almon Polynomial Shape for the combination of (θ1, θ2) that maximize JTat various
quantiles τindicated above each plot.
Figure 16: These plots show the significance of the Almon Lag Polynomial as a weighting function for
the MIDAS component at different quantiles. The y-axis reports the z-scores and the x-axis each day
min a month. The blue squares indicate the z-scores within the [−1.96,1.96, that is the non-significant
days. The green dots indicate the z-scores that are above or below the confidence interval, that is the
significant days.
Figure 17: Quantile Scores of QMIDAS vs QAVG model in the out-of-sample period. The blue solid
line refers to the QMIDAS model; the red dotted one to the QAVG model.
A Inflation
Figure A1: Line plot showing how inflation evolves over time
B Values Grids for JT
Table B1: Table showing all possible outcomes of the test statistic JTfor the causal relation between futures on gold and inflation at various
quantiles (on the columns) for all plausible pairs of the polynomial weights (θ1, θ2) (on the rows). The pairs yielding the maximum values of
JTat every quantile considered are indicated in bold.
Quantiles
θ1θ20.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
-0.006 -0.008 34.68 17.69 37.98 17.12 1.18 0.30 -0.12 1.88 3.35 2.34 3.67 2.01 -0.74 -0.65 1.36 2.76 2.35 2.41 4.09
-0.006 0.002 37.67 19.84 33.17 17.03 2.98 1.48 1.24 2.02 3.19 3.87 4.91 2.80 0.21 0.75 0.95 1.68 0.31 1.66 2.15
-0.006 0.012 30.07 16.24 32.37 15.01 1.68 1.31 0.95 2.70 4.14 4.08 5.27 2.81 0.46 -0.19 -0.03 1.18 2.07 2.93 3.42
-0.006 0.022 32.59 16.85 34.77 17.06 2.31 1.86 1.92 3.73 5.32 4.88 6.28 4.34 2.27 1.60 0.83 2.91 3.15 3.56 5.52
-0.006 0.032 34.49 17.74 36.50 18.89 2.03 1.31 1.27 3.25 4.85 4.38 5.58 4.06 1.52 1.38 0.52 2.80 3.51 3.46 6.45
-0.006 0.042 37.95 20.41 37.64 20.12 2.13 1.18 1.13 2.95 4.51 4.13 5.30 3.94 1.04 1.09 0.39 2.65 3.04 3.14 5.77
-0.006 0.052 38.92 21.45 37.34 20.28 2.42 1.08 1.22 2.90 4.45 4.12 5.37 4.13 0.94 1.13 0.53 2.43 2.54 2.74 4.82
-0.006 0.062 39.50 21.74 36.95 20.28 2.50 0.89 1.16 2.80 4.35 3.98 5.31 4.20 0.87 1.14 0.57 2.26 2.20 2.48 4.27
-0.006 0.072 39.96 21.84 36.72 20.29 2.46 0.71 1.03 2.70 4.24 3.82 5.18 4.16 0.77 1.07 0.49 2.12 1.97 2.33 4.03
-0.006 0.082 40.29 21.89 36.59 20.29 2.41 0.58 0.92 2.61 4.15 3.70 5.06 4.10 0.68 0.99 0.40 2.01 1.81 2.25 3.92
-0.006 0.092 40.51 21.91 36.51 20.30 2.36 0.49 0.84 2.55 4.08 3.61 4.97 4.05 0.61 0.92 0.32 1.93 1.70 2.20 3.88
-0.006 0.102 40.65 21.93 36.47 20.30 2.33 0.43 0.79 2.50 4.03 3.55 4.91 4.01 0.56 0.87 0.26 1.88 1.63 2.17 3.86
-0.006 0.112 40.75 21.94 36.44 20.30 2.31 0.39 0.75 2.48 4.00 3.52 4.87 3.98 0.53 0.84 0.22 1.84 1.59 2.16 3.85
-0.006 0.122 40.81 21.94 36.42 20.30 2.29 0.36 0.73 2.46 3.98 3.49 4.84 3.96 0.51 0.81 0.20 1.81 1.55 2.15 3.85
-0.006 0.132 40.85 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.97 3.47 4.83 3.94 0.49 0.80 0.18 1.80 1.53 2.14 3.85
-0.006 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.004 -0.008 34.61 17.59 37.93 17.07 1.14 0.36 -0.12 1.90 3.38 2.35 3.66 1.97 -0.80 -0.70 1.26 2.55 2.16 2.25 3.85
0.004 0.002 37.16 19.78 32.67 16.67 2.81 1.25 1.11 1.88 3.05 3.90 4.69 2.83 0.36 0.74 0.86 1.83 0.35 2.06 2.37
0.004 0.012 30.32 16.33 32.47 15.09 1.67 1.34 0.96 2.76 4.20 4.10 5.32 2.88 0.53 -0.15 0.00 1.22 2.14 3.09 3.56
0.004 0.022 32.56 16.81 34.78 17.09 2.32 1.85 1.93 3.73 5.32 4.87 6.26 4.33 2.27 1.62 0.83 2.91 3.16 3.55 5.55
0.004 0.032 34.60 17.81 36.55 18.94 2.02 1.30 1.26 3.23 4.84 4.36 5.56 4.06 1.50 1.37 0.51 2.80 3.50 3.45 6.45
0.004 0.042 37.99 20.45 37.64 20.13 2.14 1.18 1.13 2.95 4.51 4.13 5.30 3.94 1.04 1.09 0.39 2.64 3.03 3.13 5.75
0.004 0.052 38.94 21.47 37.33 20.28 2.42 1.08 1.22 2.90 4.45 4.11 5.37 4.13 0.94 1.13 0.53 2.43 2.53 2.73 4.80
0.004 0.062 39.52 21.75 36.94 20.28 2.50 0.89 1.15 2.80 4.35 3.98 5.30 4.20 0.87 1.14 0.57 2.25 2.19 2.47 4.27
0.004 0.072 39.97 21.84 36.71 20.29 2.46 0.71 1.03 2.69 4.24 3.82 5.17 4.16 0.77 1.07 0.49 2.12 1.96 2.33 4.02
0.004 0.082 40.29 21.89 36.59 20.29 2.41 0.58 0.92 2.61 4.14 3.70 5.06 4.10 0.68 0.99 0.40 2.01 1.81 2.25 3.92
0.004 0.092 40.51 21.91 36.51 20.30 2.36 0.49 0.84 2.55 4.08 3.61 4.97 4.05 0.61 0.92 0.32 1.93 1.70 2.20 3.88
0.004 0.102 40.66 21.93 36.47 20.30 2.33 0.43 0.79 2.50 4.03 3.55 4.91 4.01 0.56 0.87 0.26 1.88 1.63 2.17 3.86
0.004 0.112 40.75 21.94 36.44 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.98 0.53 0.84 0.22 1.84 1.58 2.16 3.85
0.004 0.122 40.81 21.94 36.42 20.30 2.29 0.36 0.73 2.46 3.98 3.49 4.84 3.96 0.50 0.81 0.20 1.81 1.55 2.15 3.85
0.004 0.132 40.85 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.97 3.47 4.83 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.004 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.014 -0.008 34.57 17.50 37.95 17.05 1.09 0.39 -0.12 1.92 3.40 2.35 3.63 1.88 -0.86 -0.75 1.14 2.31 1.92 2.07 3.60
0.014 0.002 35.98 19.43 32.01 16.19 2.54 0.97 0.94 1.68 2.86 3.79 4.39 2.71 0.42 0.58 0.64 1.85 0.36 2.38 2.40
0.014 0.012 30.59 16.45 32.58 15.17 1.66 1.36 0.97 2.81 4.26 4.13 5.38 2.95 0.59 -0.10 0.03 1.28 2.22 3.25 3.68
0.014 0.022 32.53 16.78 34.80 17.13 2.33 1.85 1.93 3.73 5.33 4.85 6.24 4.32 2.26 1.63 0.83 2.92 3.18 3.54 5.58
0.014 0.032 34.71 17.88 36.59 18.98 2.02 1.29 1.25 3.22 4.82 4.35 5.55 4.05 1.49 1.36 0.50 2.79 3.49 3.45 6.45
0.014 0.042 38.03 20.50 37.65 20.14 2.15 1.18 1.13 2.95 4.51 4.13 5.30 3.94 1.03 1.09 0.39 2.64 3.01 3.12 5.72
0.014 0.052 38.95 21.48 37.32 20.28 2.43 1.08 1.22 2.90 4.45 4.11 5.37 4.14 0.94 1.13 0.54 2.42 2.52 2.72 4.78
0.014 0.062 39.53 21.75 36.93 20.28 2.50 0.88 1.15 2.80 4.35 3.98 5.30 4.20 0.86 1.14 0.57 2.25 2.18 2.47 4.26
0.014 0.072 39.98 21.84 36.71 20.29 2.46 0.70 1.03 2.69 4.23 3.82 5.17 4.16 0.77 1.07 0.49 2.11 1.96 2.33 4.02
0.014 0.082 40.30 21.89 36.59 20.30 2.41 0.57 0.92 2.61 4.14 3.69 5.05 4.10 0.68 0.99 0.40 2.01 1.81 2.24 3.92
0.014 0.092 40.52 21.91 36.51 20.30 2.36 0.49 0.84 2.54 4.08 3.61 4.97 4.05 0.61 0.92 0.32 1.93 1.70 2.20 3.88
0.014 0.102 40.66 21.93 36.47 20.30 2.33 0.43 0.79 2.50 4.03 3.55 4.91 4.01 0.56 0.87 0.26 1.87 1.63 2.17 3.86
0.014 0.112 40.75 21.94 36.44 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.98 0.53 0.84 0.22 1.84 1.58 2.16 3.85
0.014 0.122 40.81 21.94 36.42 20.30 2.29 0.36 0.73 2.45 3.98 3.49 4.84 3.96 0.50 0.81 0.20 1.81 1.55 2.15 3.85
0.014 0.132 40.85 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.83 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.014 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.024 -0.008 34.56 17.45 38.06 17.08 1.03 0.41 -0.13 1.95 3.44 2.34 3.59 1.77 -0.92 -0.78 1.00 2.03 1.65 1.87 3.38
0.024 0.002 34.40 18.84 31.14 15.54 2.16 0.68 0.70 1.42 2.61 3.58 4.08 2.50 0.41 0.32 0.39 1.68 0.31 2.53 2.26
0.024 0.012 30.87 16.58 32.68 15.25 1.66 1.38 0.99 2.86 4.32 4.17 5.44 3.03 0.66 -0.05 0.07 1.33 2.29 3.38 3.80
0.024 0.022 32.50 16.75 34.82 17.16 2.34 1.85 1.93 3.74 5.33 4.84 6.22 4.32 2.25 1.64 0.83 2.92 3.20 3.53 5.62
0.024 0.032 34.83 17.95 36.64 19.03 2.01 1.28 1.23 3.20 4.81 4.34 5.54 4.04 1.47 1.35 0.50 2.79 3.49 3.44 6.44
0.024 0.042 38.07 20.54 37.65 20.15 2.16 1.18 1.13 2.95 4.50 4.13 5.30 3.95 1.03 1.09 0.40 2.63 3.00 3.11 5.69
0.024 0.052 38.97 21.49 37.30 20.28 2.43 1.07 1.22 2.89 4.45 4.11 5.37 4.14 0.93 1.13 0.54 2.42 2.51 2.71 4.77
0.024 0.062 39.54 21.75 36.92 20.28 2.50 0.88 1.15 2.80 4.34 3.97 5.30 4.20 0.86 1.14 0.57 2.24 2.18 2.46 4.25
0.024 0.072 39.99 21.84 36.70 20.29 2.46 0.70 1.02 2.69 4.23 3.81 5.17 4.16 0.76 1.07 0.49 2.11 1.95 2.32 4.02
0.024 0.082 40.31 21.89 36.58 20.30 2.41 0.57 0.92 2.60 4.14 3.69 5.05 4.10 0.67 0.99 0.39 2.01 1.80 2.24 3.92
0.024 0.092 40.52 21.91 36.51 20.30 2.36 0.48 0.84 2.54 4.07 3.61 4.97 4.05 0.61 0.92 0.32 1.93 1.70 2.20 3.88
0.024 0.102 40.66 21.93 36.46 20.30 2.33 0.43 0.79 2.50 4.03 3.55 4.91 4.00 0.56 0.87 0.26 1.87 1.63 2.17 3.86
0.024 0.112 40.75 21.94 36.43 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.98 0.53 0.84 0.22 1.84 1.58 2.16 3.85
0.024 0.122 40.81 21.94 36.41 20.30 2.29 0.36 0.73 2.45 3.98 3.49 4.84 3.96 0.50 0.81 0.20 1.81 1.55 2.15 3.85
0.024 0.132 40.85 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.83 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.024 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.034 -0.008 34.60 17.45 38.25 17.17 1.00 0.41 -0.12 1.99 3.48 2.33 3.56 1.64 -0.95 -0.78 0.86 1.76 1.36 1.68 3.25
0.034 0.002 33.01 18.29 30.38 14.89 1.77 0.41 0.48 1.22 2.41 3.37 3.85 2.31 0.40 0.09 0.20 1.46 0.30 2.62 2.08
0.034 0.012 31.16 16.72 32.79 15.33 1.67 1.41 1.02 2.92 4.38 4.21 5.50 3.11 0.73 0.01 0.10 1.39 2.36 3.51 3.91
0.034 0.022 32.49 16.73 34.85 17.20 2.35 1.84 1.92 3.74 5.33 4.83 6.21 4.31 2.24 1.65 0.83 2.92 3.22 3.53 5.65
0.034 0.032 34.94 18.03 36.69 19.07 2.01 1.27 1.22 3.19 4.79 4.33 5.52 4.03 1.45 1.34 0.49 2.79 3.48 3.44 6.44
0.034 0.042 38.11 20.58 37.65 20.16 2.16 1.18 1.14 2.95 4.50 4.13 5.30 3.95 1.02 1.08 0.40 2.63 2.98 3.09 5.67
0.034 0.052 38.98 21.50 37.29 20.28 2.44 1.07 1.22 2.89 4.45 4.11 5.37 4.15 0.93 1.13 0.54 2.41 2.50 2.70 4.75
0.034 0.062 39.55 21.76 36.92 20.28 2.50 0.87 1.14 2.79 4.34 3.97 5.29 4.20 0.86 1.13 0.56 2.24 2.17 2.46 4.24
0.034 0.072 40.00 21.84 36.70 20.29 2.46 0.70 1.02 2.69 4.23 3.81 5.16 4.16 0.76 1.06 0.48 2.11 1.95 2.32 4.01
0.034 0.082 40.31 21.89 36.58 20.30 2.40 0.57 0.91 2.60 4.14 3.69 5.05 4.10 0.67 0.98 0.39 2.00 1.80 2.24 3.92
0.034 0.092 40.53 21.91 36.51 20.30 2.36 0.48 0.84 2.54 4.07 3.60 4.97 4.04 0.61 0.92 0.31 1.93 1.70 2.20 3.88
0.034 0.102 40.66 21.93 36.46 20.30 2.33 0.43 0.78 2.50 4.03 3.55 4.91 4.00 0.56 0.87 0.26 1.87 1.63 2.17 3.86
0.034 0.112 40.76 21.94 36.43 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.98 0.53 0.84 0.22 1.84 1.58 2.15 3.85
0.034 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.96 0.50 0.81 0.20 1.81 1.55 2.15 3.85
0.034 0.132 40.86 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.83 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.034 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.044 -0.008 34.67 17.50 38.48 17.28 1.01 0.41 -0.09 2.04 3.53 2.34 3.54 1.53 -0.94 -0.74 0.74 1.51 1.09 1.53 3.24
0.044 0.002 32.07 17.88 29.86 14.34 1.44 0.21 0.30 1.10 2.31 3.21 3.75 2.18 0.37 -0.11 0.06 1.29 0.37 2.72 1.94
0.044 0.012 31.46 16.87 32.91 15.42 1.68 1.43 1.05 2.97 4.44 4.26 5.56 3.19 0.81 0.06 0.14 1.46 2.43 3.62 4.02
0.044 0.022 32.47 16.71 34.87 17.24 2.35 1.83 1.92 3.73 5.33 4.81 6.19 4.30 2.23 1.66 0.82 2.92 3.24 3.52 5.69
0.044 0.032 35.05 18.10 36.74 19.12 2.00 1.27 1.21 3.18 4.78 4.32 5.51 4.02 1.43 1.33 0.48 2.78 3.47 3.43 6.44
0.044 0.042 38.14 20.61 37.65 20.17 2.17 1.18 1.14 2.94 4.50 4.13 5.31 3.96 1.02 1.08 0.40 2.62 2.97 3.08 5.64
0.044 0.052 39.00 21.51 37.28 20.28 2.44 1.06 1.22 2.89 4.44 4.11 5.37 4.15 0.93 1.13 0.54 2.41 2.49 2.70 4.73
0.044 0.062 39.57 21.76 36.91 20.28 2.50 0.87 1.14 2.79 4.34 3.96 5.29 4.20 0.86 1.13 0.56 2.24 2.16 2.45 4.23
0.044 0.072 40.01 21.85 36.70 20.29 2.46 0.69 1.02 2.68 4.22 3.81 5.16 4.16 0.76 1.06 0.48 2.10 1.95 2.32 4.01
0.044 0.082 40.32 21.89 36.58 20.30 2.40 0.57 0.91 2.60 4.14 3.69 5.05 4.10 0.67 0.98 0.39 2.00 1.80 2.24 3.91
0.044 0.092 40.53 21.92 36.51 20.30 2.36 0.48 0.84 2.54 4.07 3.60 4.96 4.04 0.60 0.92 0.31 1.93 1.69 2.20 3.88
0.044 0.102 40.67 21.93 36.46 20.30 2.33 0.42 0.78 2.50 4.03 3.55 4.91 4.00 0.56 0.87 0.26 1.87 1.63 2.17 3.86
0.044 0.112 40.76 21.94 36.43 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.97 0.52 0.83 0.22 1.84 1.58 2.15 3.85
0.044 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.96 0.50 0.81 0.19 1.81 1.55 2.15 3.85
0.044 0.132 40.86 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.044 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.96 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.054 -0.008 34.72 17.57 38.67 17.39 1.03 0.40 -0.05 2.09 3.58 2.37 3.54 1.44 -0.88 -0.65 0.65 1.32 0.88 1.44 3.37
0.054 0.002 31.51 17.59 29.59 13.96 1.18 0.08 0.18 1.09 2.31 3.14 3.76 2.10 0.30 -0.31 -0.09 1.19 0.46 2.81 1.87
0.054 0.012 31.75 17.02 33.04 15.51 1.70 1.46 1.08 3.02 4.49 4.31 5.63 3.27 0.88 0.12 0.18 1.52 2.49 3.72 4.12
0.054 0.022 32.46 16.69 34.90 17.28 2.36 1.83 1.92 3.73 5.33 4.80 6.17 4.30 2.22 1.66 0.82 2.92 3.26 3.51 5.73
0.054 0.032 35.16 18.17 36.78 19.16 2.00 1.26 1.20 3.17 4.76 4.31 5.49 4.01 1.42 1.32 0.48 2.78 3.46 3.43 6.43
0.054 0.042 38.17 20.65 37.64 20.18 2.18 1.17 1.14 2.94 4.50 4.13 5.31 3.96 1.01 1.08 0.40 2.62 2.96 3.07 5.62
0.054 0.052 39.01 21.52 37.27 20.28 2.44 1.06 1.22 2.89 4.44 4.10 5.37 4.15 0.93 1.13 0.55 2.40 2.48 2.69 4.71
0.054 0.062 39.58 21.76 36.90 20.28 2.50 0.86 1.14 2.79 4.34 3.96 5.29 4.20 0.85 1.13 0.56 2.23 2.16 2.45 4.22
0.054 0.072 40.02 21.85 36.69 20.29 2.46 0.69 1.01 2.68 4.22 3.80 5.16 4.16 0.76 1.06 0.48 2.10 1.94 2.32 4.01
0.054 0.082 40.33 21.89 36.58 20.30 2.40 0.56 0.91 2.60 4.13 3.68 5.05 4.09 0.67 0.98 0.39 2.00 1.79 2.24 3.91
0.054 0.092 40.53 21.92 36.51 20.30 2.36 0.48 0.83 2.54 4.07 3.60 4.96 4.04 0.60 0.91 0.31 1.92 1.69 2.19 3.87
0.054 0.102 40.67 21.93 36.46 20.30 2.33 0.42 0.78 2.50 4.03 3.55 4.91 4.00 0.56 0.87 0.26 1.87 1.62 2.17 3.86
0.054 0.112 40.76 21.94 36.43 20.30 2.31 0.39 0.75 2.47 4.00 3.51 4.87 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.054 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.96 0.50 0.81 0.19 1.81 1.55 2.15 3.85
0.054 0.132 40.86 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.054 0.142 40.88 21.95 36.39 20.30 2.27 0.34 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.064 -0.008 34.69 17.62 38.78 17.43 1.06 0.37 -0.02 2.13 3.61 2.42 3.56 1.38 -0.77 -0.52 0.60 1.22 0.76 1.43 3.65
0.064 0.002 31.22 17.40 29.55 13.72 1.00 0.02 0.12 1.17 2.40 3.18 3.85 2.05 0.20 -0.51 -0.27 1.11 0.53 2.83 1.87
0.064 0.012 32.05 17.16 33.17 15.60 1.72 1.48 1.11 3.07 4.54 4.36 5.69 3.35 0.96 0.17 0.22 1.59 2.56 3.81 4.22
0.064 0.022 32.46 16.68 34.93 17.32 2.36 1.82 1.91 3.73 5.33 4.79 6.15 4.29 2.20 1.67 0.81 2.92 3.27 3.51 5.76
0.064 0.032 35.28 18.25 36.83 19.21 1.99 1.25 1.19 3.15 4.75 4.30 5.48 4.01 1.40 1.31 0.47 2.77 3.45 3.42 6.42
0.064 0.042 38.21 20.69 37.64 20.19 2.19 1.17 1.14 2.94 4.50 4.13 5.31 3.97 1.01 1.08 0.41 2.61 2.94 3.06 5.59
0.064 0.052 39.03 21.53 37.26 20.28 2.45 1.05 1.22 2.89 4.44 4.10 5.37 4.15 0.93 1.14 0.55 2.40 2.47 2.68 4.70
0.064 0.062 39.59 21.77 36.90 20.28 2.50 0.86 1.14 2.79 4.33 3.96 5.29 4.20 0.85 1.13 0.56 2.23 2.15 2.45 4.22
0.064 0.072 40.02 21.85 36.69 20.29 2.45 0.68 1.01 2.68 4.22 3.80 5.15 4.15 0.75 1.06 0.48 2.10 1.94 2.31 4.00
0.064 0.082 40.33 21.89 36.57 20.30 2.40 0.56 0.91 2.60 4.13 3.68 5.04 4.09 0.67 0.98 0.38 2.00 1.79 2.24 3.91
0.064 0.092 40.54 21.92 36.51 20.30 2.36 0.48 0.83 2.54 4.07 3.60 4.96 4.04 0.60 0.91 0.31 1.92 1.69 2.19 3.87
0.064 0.102 40.67 21.93 36.46 20.30 2.33 0.42 0.78 2.50 4.03 3.55 4.90 4.00 0.56 0.87 0.26 1.87 1.62 2.17 3.86
0.064 0.112 40.76 21.94 36.43 20.30 2.30 0.39 0.75 2.47 4.00 3.51 4.87 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.064 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.064 0.132 40.86 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.064 0.142 40.88 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.074 -0.008 34.48 17.59 38.75 17.39 1.06 0.29 -0.01 2.14 3.61 2.46 3.59 1.33 -0.64 -0.37 0.59 1.21 0.75 1.52 4.08
0.074 0.002 31.06 17.27 29.55 13.54 0.91 0.02 0.12 1.29 2.55 3.30 3.97 2.01 0.11 -0.68 -0.44 1.06 0.56 2.80 1.91
0.074 0.012 32.33 17.30 33.31 15.70 1.74 1.51 1.15 3.11 4.59 4.41 5.76 3.43 1.04 0.23 0.26 1.66 2.61 3.89 4.31
0.074 0.022 32.46 16.66 34.96 17.36 2.37 1.81 1.90 3.73 5.33 4.77 6.13 4.29 2.19 1.67 0.81 2.92 3.29 3.51 5.80
0.074 0.032 35.39 18.32 36.87 19.25 1.99 1.25 1.18 3.14 4.74 4.29 5.47 4.00 1.39 1.30 0.46 2.77 3.44 3.42 6.41
0.074 0.042 38.24 20.72 37.64 20.19 2.20 1.17 1.15 2.94 4.50 4.13 5.31 3.97 1.01 1.08 0.41 2.61 2.93 3.05 5.57
0.074 0.052 39.05 21.54 37.25 20.28 2.45 1.05 1.22 2.89 4.44 4.10 5.37 4.16 0.93 1.14 0.55 2.39 2.46 2.67 4.68
0.074 0.062 39.60 21.77 36.89 20.28 2.50 0.85 1.13 2.78 4.33 3.95 5.28 4.20 0.85 1.13 0.56 2.23 2.14 2.44 4.21
0.074 0.072 40.03 21.85 36.69 20.29 2.45 0.68 1.01 2.68 4.22 3.80 5.15 4.15 0.75 1.06 0.48 2.10 1.93 2.31 4.00
0.074 0.082 40.34 21.89 36.57 20.30 2.40 0.56 0.91 2.60 4.13 3.68 5.04 4.09 0.67 0.98 0.38 1.99 1.79 2.24 3.91
0.074 0.092 40.54 21.92 36.50 20.30 2.36 0.48 0.83 2.54 4.07 3.60 4.96 4.04 0.60 0.91 0.31 1.92 1.69 2.19 3.87
0.074 0.102 40.68 21.93 36.46 20.30 2.33 0.42 0.78 2.50 4.03 3.54 4.90 4.00 0.55 0.87 0.26 1.87 1.62 2.17 3.86
0.074 0.112 40.76 21.94 36.43 20.30 2.30 0.38 0.75 2.47 4.00 3.51 4.87 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.074 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.074 0.132 40.86 21.95 36.40 20.30 2.28 0.35 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.80 0.18 1.79 1.53 2.14 3.85
0.074 0.142 40.88 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.084 -0.008 34.01 17.44 38.48 17.23 0.99 0.17 -0.07 2.08 3.55 2.47 3.60 1.28 -0.51 -0.25 0.61 1.26 0.81 1.69 4.59
0.084 0.002 31.00 17.18 29.47 13.37 0.90 0.05 0.15 1.42 2.70 3.43 4.11 2.01 0.10 -0.77 -0.51 1.05 0.63 2.80 1.97
0.084 0.012 32.60 17.44 33.46 15.80 1.77 1.53 1.19 3.16 4.64 4.46 5.83 3.51 1.12 0.29 0.30 1.73 2.67 3.95 4.39
0.084 0.022 32.46 16.66 35.00 17.41 2.37 1.80 1.89 3.72 5.33 4.76 6.12 4.28 2.17 1.67 0.80 2.92 3.31 3.50 5.83
0.084 0.032 35.50 18.40 36.92 19.29 1.99 1.24 1.17 3.13 4.72 4.28 5.46 3.99 1.37 1.29 0.46 2.77 3.43 3.41 6.41
0.084 0.042 38.27 20.76 37.63 20.20 2.20 1.17 1.15 2.94 4.49 4.13 5.31 3.98 1.00 1.08 0.41 2.60 2.92 3.04 5.54
0.084 0.052 39.06 21.55 37.24 20.28 2.45 1.05 1.22 2.88 4.44 4.10 5.37 4.16 0.93 1.14 0.55 2.39 2.45 2.67 4.66
0.084 0.062 39.62 21.77 36.88 20.28 2.50 0.85 1.13 2.78 4.33 3.95 5.28 4.20 0.85 1.13 0.56 2.22 2.14 2.44 4.20
0.084 0.072 40.04 21.85 36.68 20.29 2.45 0.68 1.01 2.67 4.22 3.79 5.15 4.15 0.75 1.05 0.47 2.09 1.93 2.31 4.00
0.084 0.082 40.34 21.89 36.57 20.30 2.40 0.56 0.90 2.59 4.13 3.68 5.04 4.09 0.66 0.97 0.38 1.99 1.78 2.23 3.91
0.084 0.092 40.55 21.92 36.50 20.30 2.36 0.47 0.83 2.54 4.07 3.60 4.96 4.04 0.60 0.91 0.31 1.92 1.69 2.19 3.87
0.084 0.102 40.68 21.93 36.46 20.30 2.33 0.42 0.78 2.50 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.87 1.62 2.17 3.86
0.084 0.112 40.76 21.94 36.43 20.30 2.30 0.38 0.75 2.47 4.00 3.51 4.87 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.084 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.084 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.18 1.79 1.53 2.14 3.85
0.084 0.142 40.88 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.79 0.17 1.78 1.52 2.14 3.85
0.094 -0.008 33.25 17.17 37.96 17.00 0.87 0.01 -0.16 1.97 3.44 2.46 3.61 1.23 -0.41 -0.17 0.60 1.33 0.89 1.89 5.09
0.094 0.002 31.15 17.18 29.44 13.30 0.96 0.12 0.22 1.55 2.83 3.56 4.28 2.08 0.18 -0.76 -0.45 1.14 0.79 2.90 2.01
0.094 0.012 32.86 17.56 33.60 15.90 1.79 1.56 1.23 3.20 4.69 4.52 5.89 3.59 1.19 0.35 0.34 1.80 2.72 4.01 4.47
0.094 0.022 32.47 16.65 35.04 17.45 2.37 1.78 1.88 3.72 5.32 4.75 6.10 4.28 2.15 1.67 0.79 2.92 3.33 3.50 5.87
0.094 0.032 35.61 18.47 36.96 19.33 1.99 1.24 1.16 3.12 4.71 4.27 5.44 3.98 1.36 1.28 0.45 2.76 3.42 3.41 6.40
0.094 0.042 38.30 20.79 37.63 20.21 2.21 1.17 1.15 2.94 4.49 4.13 5.32 3.98 1.00 1.08 0.42 2.60 2.90 3.03 5.51
0.094 0.052 39.08 21.56 37.23 20.28 2.46 1.04 1.21 2.88 4.43 4.09 5.37 4.16 0.93 1.14 0.55 2.38 2.44 2.66 4.65
0.094 0.062 39.63 21.77 36.88 20.28 2.50 0.84 1.13 2.78 4.32 3.94 5.28 4.20 0.85 1.13 0.56 2.22 2.13 2.43 4.19
0.094 0.072 40.05 21.85 36.68 20.29 2.45 0.67 1.00 2.67 4.21 3.79 5.15 4.15 0.75 1.05 0.47 2.09 1.92 2.31 3.99
0.094 0.082 40.35 21.89 36.57 20.30 2.40 0.55 0.90 2.59 4.13 3.67 5.04 4.09 0.66 0.97 0.38 1.99 1.78 2.23 3.91
0.094 0.092 40.55 21.92 36.50 20.30 2.36 0.47 0.83 2.54 4.07 3.59 4.96 4.04 0.60 0.91 0.31 1.92 1.68 2.19 3.87
0.094 0.102 40.68 21.93 36.46 20.30 2.32 0.42 0.78 2.50 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.87 1.62 2.17 3.86
0.094 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 4.00 3.51 4.86 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.094 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.49 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.094 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.18 1.79 1.53 2.14 3.85
0.094 0.142 40.88 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.104 -0.008 32.34 16.85 37.25 16.74 0.74 -0.14 -0.25 1.84 3.31 2.45 3.62 1.18 -0.33 -0.17 0.55 1.36 0.90 2.03 5.45
0.104 0.002 31.57 17.33 29.62 13.40 1.10 0.23 0.32 1.66 2.95 3.69 4.51 2.21 0.35 -0.69 -0.30 1.31 1.04 3.09 2.03
0.104 0.012 33.10 17.67 33.75 16.00 1.82 1.58 1.27 3.24 4.73 4.57 5.95 3.66 1.27 0.41 0.38 1.87 2.77 4.05 4.55
0.104 0.022 32.49 16.65 35.08 17.49 2.36 1.77 1.87 3.71 5.32 4.74 6.08 4.27 2.13 1.67 0.79 2.91 3.35 3.50 5.90
0.104 0.032 35.72 18.55 37.00 19.37 1.99 1.24 1.16 3.11 4.70 4.26 5.43 3.98 1.34 1.27 0.45 2.76 3.41 3.40 6.39
0.104 0.042 38.33 20.82 37.63 20.21 2.22 1.17 1.16 2.94 4.49 4.13 5.32 3.98 1.00 1.08 0.42 2.59 2.89 3.02 5.49
0.104 0.052 39.09 21.57 37.22 20.28 2.46 1.04 1.21 2.88 4.43 4.09 5.37 4.17 0.92 1.14 0.56 2.38 2.43 2.65 4.63
0.104 0.062 39.64 21.78 36.87 20.28 2.49 0.84 1.12 2.77 4.32 3.94 5.27 4.20 0.84 1.13 0.55 2.22 2.13 2.43 4.19
0.104 0.072 40.06 21.85 36.68 20.29 2.45 0.67 1.00 2.67 4.21 3.79 5.14 4.15 0.74 1.05 0.47 2.09 1.92 2.30 3.99
0.104 0.082 40.36 21.89 36.57 20.30 2.40 0.55 0.90 2.59 4.13 3.67 5.03 4.09 0.66 0.97 0.38 1.99 1.78 2.23 3.91
0.104 0.092 40.55 21.92 36.50 20.30 2.35 0.47 0.83 2.53 4.06 3.59 4.95 4.04 0.60 0.91 0.30 1.92 1.68 2.19 3.87
0.104 0.102 40.68 21.93 36.46 20.30 2.32 0.42 0.78 2.50 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.87 1.62 2.17 3.86
0.104 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.51 4.86 3.97 0.52 0.83 0.22 1.83 1.58 2.15 3.85
0.104 0.122 40.82 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.104 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.18 1.79 1.53 2.14 3.85
0.104 0.142 40.88 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.114 -0.008 31.55 16.65 36.52 16.57 0.68 -0.23 -0.30 1.73 3.18 2.47 3.65 1.18 -0.26 -0.25 0.43 1.31 0.80 2.07 5.58
0.114 0.002 32.21 17.60 29.99 13.64 1.30 0.36 0.46 1.77 3.06 3.82 4.76 2.38 0.54 -0.58 -0.10 1.53 1.34 3.32 2.03
0.114 0.012 33.31 17.77 33.90 16.10 1.84 1.60 1.31 3.27 4.77 4.62 6.01 3.73 1.34 0.46 0.41 1.94 2.81 4.08 4.62
0.114 0.022 32.50 16.65 35.12 17.54 2.36 1.76 1.85 3.70 5.31 4.72 6.06 4.27 2.12 1.66 0.78 2.91 3.37 3.50 5.93
0.114 0.032 35.83 18.63 37.04 19.41 1.99 1.23 1.15 3.10 4.69 4.25 5.42 3.97 1.33 1.26 0.44 2.76 3.40 3.39 6.37
0.114 0.042 38.35 20.86 37.62 20.22 2.23 1.16 1.16 2.93 4.49 4.13 5.32 3.99 0.99 1.08 0.42 2.59 2.88 3.01 5.46
0.114 0.052 39.11 21.58 37.21 20.28 2.46 1.03 1.21 2.88 4.43 4.09 5.37 4.17 0.92 1.14 0.56 2.37 2.42 2.64 4.62
0.114 0.062 39.65 21.78 36.86 20.28 2.49 0.83 1.12 2.77 4.32 3.94 5.27 4.20 0.84 1.12 0.55 2.21 2.12 2.43 4.18
0.114 0.072 40.07 21.85 36.67 20.29 2.45 0.67 1.00 2.67 4.21 3.78 5.14 4.15 0.74 1.05 0.47 2.08 1.92 2.30 3.99
0.114 0.082 40.36 21.90 36.56 20.30 2.39 0.55 0.90 2.59 4.12 3.67 5.03 4.09 0.66 0.97 0.37 1.99 1.78 2.23 3.91
0.114 0.092 40.56 21.92 36.50 20.30 2.35 0.47 0.83 2.53 4.06 3.59 4.95 4.04 0.60 0.91 0.30 1.91 1.68 2.19 3.87
0.114 0.102 40.69 21.93 36.46 20.30 2.32 0.42 0.78 2.49 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.86 1.62 2.17 3.86
0.114 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.51 4.86 3.97 0.52 0.83 0.22 1.83 1.57 2.15 3.85
0.114 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.98 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.114 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.18 1.79 1.53 2.14 3.85
0.114 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.124 -0.008 31.18 16.70 35.97 16.59 0.72 -0.23 -0.28 1.65 3.10 2.53 3.71 1.21 -0.21 -0.36 0.26 1.17 0.60 1.97 5.41
0.124 0.002 32.93 17.92 30.38 13.93 1.54 0.50 0.61 1.87 3.15 3.94 4.99 2.52 0.68 -0.47 0.09 1.73 1.61 3.51 2.02
0.124 0.012 33.51 17.86 34.03 16.19 1.87 1.63 1.35 3.31 4.81 4.66 6.07 3.80 1.42 0.52 0.45 2.01 2.85 4.10 4.68
0.124 0.022 32.52 16.65 35.16 17.59 2.36 1.74 1.84 3.69 5.31 4.71 6.05 4.26 2.10 1.66 0.77 2.91 3.38 3.50 5.97
0.124 0.032 35.93 18.70 37.08 19.45 1.99 1.23 1.15 3.09 4.68 4.24 5.41 3.96 1.31 1.25 0.43 2.75 3.39 3.39 6.36
0.124 0.042 38.38 20.89 37.61 20.22 2.24 1.16 1.16 2.93 4.49 4.13 5.32 3.99 0.99 1.08 0.43 2.58 2.86 3.00 5.44
0.124 0.052 39.12 21.59 37.20 20.28 2.47 1.03 1.21 2.88 4.43 4.08 5.37 4.17 0.92 1.14 0.56 2.37 2.41 2.64 4.60
0.124 0.062 39.66 21.78 36.86 20.28 2.49 0.83 1.12 2.77 4.31 3.93 5.27 4.20 0.84 1.12 0.55 2.21 2.11 2.42 4.17
0.124 0.072 40.08 21.86 36.67 20.29 2.45 0.66 0.99 2.67 4.21 3.78 5.14 4.15 0.74 1.05 0.46 2.08 1.91 2.30 3.98
0.124 0.082 40.37 21.90 36.56 20.30 2.39 0.55 0.90 2.59 4.12 3.67 5.03 4.08 0.66 0.97 0.37 1.98 1.77 2.23 3.90
0.124 0.092 40.56 21.92 36.50 20.30 2.35 0.47 0.82 2.53 4.06 3.59 4.95 4.03 0.59 0.91 0.30 1.91 1.68 2.19 3.87
0.124 0.102 40.69 21.93 36.46 20.30 2.32 0.42 0.78 2.49 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.86 1.62 2.17 3.86
0.124 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.51 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.124 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.124 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.17 1.79 1.53 2.14 3.85
0.124 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.134 -0.008 31.33 17.04 35.65 16.78 0.87 -0.16 -0.18 1.62 3.05 2.65 3.81 1.29 -0.17 -0.48 0.07 0.98 0.34 1.74 4.97
0.134 0.002 33.62 18.21 30.68 14.17 1.78 0.64 0.77 1.95 3.23 4.06 5.17 2.59 0.73 -0.41 0.22 1.85 1.81 3.59 2.01
0.134 0.012 33.67 17.92 34.16 16.28 1.89 1.65 1.39 3.34 4.85 4.71 6.12 3.86 1.49 0.57 0.48 2.07 2.88 4.12 4.74
0.134 0.022 32.55 16.65 35.21 17.63 2.35 1.73 1.82 3.69 5.30 4.70 6.03 4.26 2.08 1.65 0.76 2.90 3.40 3.50 6.00
0.134 0.032 36.04 18.77 37.12 19.49 1.99 1.23 1.14 3.08 4.67 4.24 5.40 3.96 1.30 1.24 0.43 2.75 3.38 3.38 6.35
0.134 0.042 38.40 20.92 37.61 20.23 2.24 1.16 1.16 2.93 4.49 4.13 5.32 4.00 0.99 1.08 0.43 2.57 2.85 2.99 5.41
0.134 0.052 39.13 21.60 37.19 20.28 2.47 1.02 1.21 2.87 4.42 4.08 5.36 4.17 0.92 1.14 0.56 2.37 2.40 2.63 4.59
0.134 0.062 39.67 21.79 36.85 20.28 2.49 0.83 1.11 2.77 4.31 3.93 5.26 4.20 0.84 1.12 0.55 2.20 2.11 2.42 4.17
0.134 0.072 40.09 21.86 36.67 20.29 2.44 0.66 0.99 2.66 4.20 3.78 5.13 4.14 0.74 1.04 0.46 2.08 1.91 2.30 3.98
0.134 0.082 40.37 21.90 36.56 20.30 2.39 0.54 0.89 2.59 4.12 3.66 5.03 4.08 0.65 0.97 0.37 1.98 1.77 2.23 3.90
0.134 0.092 40.56 21.92 36.50 20.30 2.35 0.47 0.82 2.53 4.06 3.59 4.95 4.03 0.59 0.90 0.30 1.91 1.68 2.19 3.87
0.134 0.102 40.69 21.93 36.46 20.30 2.32 0.41 0.77 2.49 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.86 1.61 2.17 3.86
0.134 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.51 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.134 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.134 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.17 1.79 1.53 2.14 3.85
0.134 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.144 -0.008 31.82 17.48 35.41 17.04 1.12 -0.03 -0.03 1.62 3.02 2.79 3.92 1.39 -0.14 -0.58 -0.13 0.77 0.06 1.41 4.36
0.144 0.002 34.22 18.46 30.85 14.32 1.99 0.77 0.93 2.01 3.28 4.16 5.27 2.58 0.70 -0.40 0.29 1.90 1.90 3.53 2.01
0.144 0.012 33.81 17.98 34.28 16.35 1.92 1.67 1.42 3.37 4.89 4.75 6.17 3.92 1.55 0.63 0.51 2.14 2.91 4.12 4.79
0.144 0.022 32.58 16.66 35.25 17.68 2.34 1.71 1.80 3.68 5.29 4.69 6.01 4.25 2.05 1.65 0.75 2.90 3.42 3.49 6.03
0.144 0.032 36.14 18.85 37.16 19.53 1.99 1.22 1.13 3.08 4.66 4.23 5.39 3.95 1.29 1.23 0.43 2.75 3.37 3.37 6.33
0.144 0.042 38.43 20.95 37.60 20.23 2.25 1.16 1.17 2.93 4.49 4.13 5.33 4.01 0.98 1.09 0.43 2.57 2.84 2.98 5.39
0.144 0.052 39.15 21.60 37.18 20.28 2.47 1.02 1.21 2.87 4.42 4.08 5.36 4.18 0.92 1.14 0.56 2.36 2.39 2.62 4.57
0.144 0.062 39.69 21.79 36.85 20.28 2.49 0.82 1.11 2.76 4.31 3.92 5.26 4.19 0.83 1.12 0.55 2.20 2.10 2.41 4.16
0.144 0.072 40.09 21.86 36.66 20.29 2.44 0.66 0.99 2.66 4.20 3.77 5.13 4.14 0.74 1.04 0.46 2.08 1.90 2.29 3.98
0.144 0.082 40.38 21.90 36.56 20.30 2.39 0.54 0.89 2.58 4.12 3.66 5.02 4.08 0.65 0.96 0.37 1.98 1.77 2.23 3.90
0.144 0.092 40.57 21.92 36.49 20.30 2.35 0.46 0.82 2.53 4.06 3.59 4.95 4.03 0.59 0.90 0.30 1.91 1.67 2.19 3.87
0.144 0.102 40.69 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.54 4.90 4.00 0.55 0.86 0.25 1.86 1.61 2.17 3.86
0.144 0.112 40.77 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.144 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.55 2.14 3.85
0.144 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.17 1.79 1.53 2.14 3.85
0.144 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.154 -0.008 32.37 17.82 35.10 17.24 1.41 0.14 0.16 1.65 3.01 2.95 4.05 1.50 -0.11 -0.64 -0.29 0.57 -0.16 1.04 3.70
0.154 0.002 34.79 18.68 30.89 14.41 2.19 0.91 1.08 2.07 3.32 4.23 5.30 2.53 0.60 -0.44 0.28 1.86 1.90 3.34 2.03
0.154 0.012 33.92 18.02 34.39 16.43 1.94 1.69 1.46 3.40 4.92 4.79 6.21 3.97 1.61 0.68 0.53 2.20 2.94 4.12 4.84
0.154 0.022 32.61 16.67 35.30 17.73 2.34 1.70 1.79 3.66 5.28 4.67 6.00 4.25 2.03 1.64 0.74 2.90 3.43 3.49 6.06
0.154 0.032 36.24 18.92 37.19 19.56 1.99 1.22 1.13 3.07 4.65 4.22 5.38 3.95 1.27 1.22 0.42 2.74 3.36 3.37 6.32
0.154 0.042 38.45 20.97 37.59 20.24 2.26 1.16 1.17 2.93 4.48 4.13 5.33 4.01 0.98 1.09 0.44 2.56 2.82 2.97 5.36
0.154 0.052 39.16 21.61 37.17 20.28 2.47 1.01 1.21 2.87 4.42 4.08 5.36 4.18 0.92 1.14 0.56 2.36 2.38 2.62 4.56
0.154 0.062 39.70 21.79 36.84 20.28 2.49 0.82 1.11 2.76 4.31 3.92 5.26 4.19 0.83 1.12 0.55 2.20 2.10 2.41 4.15
0.154 0.072 40.10 21.86 36.66 20.29 2.44 0.65 0.99 2.66 4.20 3.77 5.13 4.14 0.73 1.04 0.46 2.07 1.90 2.29 3.98
0.154 0.082 40.38 21.90 36.56 20.30 2.39 0.54 0.89 2.58 4.12 3.66 5.02 4.08 0.65 0.96 0.37 1.98 1.76 2.23 3.90
0.154 0.092 40.57 21.92 36.49 20.30 2.35 0.46 0.82 2.53 4.06 3.59 4.95 4.03 0.59 0.90 0.30 1.91 1.67 2.19 3.87
0.154 0.102 40.70 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.54 4.90 3.99 0.55 0.86 0.25 1.86 1.61 2.16 3.86
0.154 0.112 40.78 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.154 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.81 1.54 2.14 3.85
0.154 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.17 1.79 1.53 2.14 3.85
0.154 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.164 -0.008 32.80 17.97 34.65 17.33 1.74 0.33 0.40 1.69 3.03 3.13 4.21 1.65 -0.08 -0.67 -0.40 0.40 -0.32 0.70 3.15
0.164 0.002 35.25 18.84 30.88 14.47 2.36 1.06 1.20 2.14 3.37 4.27 5.30 2.48 0.50 -0.47 0.23 1.77 1.85 3.06 2.04
0.164 0.012 34.00 18.04 34.49 16.49 1.96 1.71 1.49 3.43 4.95 4.82 6.26 4.02 1.67 0.73 0.56 2.26 2.96 4.11 4.88
0.164 0.022 32.64 16.69 35.35 17.77 2.33 1.68 1.77 3.65 5.27 4.66 5.98 4.25 2.01 1.64 0.73 2.89 3.45 3.49 6.10
0.164 0.032 36.34 19.00 37.23 19.60 1.99 1.22 1.13 3.06 4.64 4.21 5.37 3.94 1.26 1.21 0.42 2.74 3.35 3.36 6.30
0.164 0.042 38.48 21.00 37.59 20.24 2.27 1.15 1.17 2.93 4.48 4.13 5.33 4.02 0.98 1.09 0.44 2.56 2.81 2.96 5.34
0.164 0.052 39.18 21.62 37.16 20.28 2.48 1.01 1.21 2.87 4.42 4.07 5.36 4.18 0.91 1.14 0.57 2.35 2.38 2.61 4.54
0.164 0.062 39.71 21.79 36.83 20.28 2.49 0.81 1.11 2.76 4.30 3.92 5.25 4.19 0.83 1.12 0.54 2.19 2.09 2.41 4.15
0.164 0.072 40.11 21.86 36.66 20.29 2.44 0.65 0.98 2.66 4.20 3.77 5.12 4.14 0.73 1.04 0.45 2.07 1.90 2.29 3.97
0.164 0.082 40.39 21.90 36.55 20.30 2.39 0.54 0.89 2.58 4.12 3.66 5.02 4.08 0.65 0.96 0.36 1.98 1.76 2.22 3.90
0.164 0.092 40.58 21.92 36.49 20.30 2.35 0.46 0.82 2.53 4.06 3.58 4.95 4.03 0.59 0.90 0.30 1.91 1.67 2.19 3.87
0.164 0.102 40.70 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.54 4.89 3.99 0.55 0.86 0.25 1.86 1.61 2.16 3.86
0.164 0.112 40.78 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.164 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.80 1.54 2.14 3.85
0.164 0.132 40.86 21.95 36.40 20.30 2.28 0.34 0.71 2.44 3.96 3.47 4.82 3.94 0.49 0.79 0.17 1.79 1.53 2.14 3.85
0.164 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.52 2.14 3.85
0.174 -0.008 33.09 17.96 34.01 17.28 2.09 0.58 0.68 1.75 3.07 3.31 4.38 1.81 -0.05 -0.66 -0.46 0.25 -0.41 0.43 2.78
0.174 0.002 35.36 18.86 30.87 14.51 2.46 1.17 1.27 2.19 3.41 4.29 5.27 2.46 0.42 -0.49 0.14 1.65 1.78 2.74 2.04
0.174 0.012 34.05 18.04 34.57 16.55 1.98 1.72 1.53 3.46 4.98 4.85 6.29 4.07 1.73 0.78 0.58 2.31 2.97 4.10 4.92
0.174 0.022 32.68 16.70 35.39 17.82 2.32 1.66 1.75 3.64 5.26 4.65 5.96 4.24 1.99 1.63 0.72 2.89 3.46 3.49 6.12
0.174 0.032 36.44 19.07 37.26 19.63 2.00 1.21 1.12 3.05 4.63 4.21 5.37 3.94 1.25 1.20 0.41 2.74 3.33 3.35 6.29
0.174 0.042 38.50 21.03 37.58 20.24 2.27 1.15 1.18 2.93 4.48 4.13 5.33 4.02 0.98 1.09 0.44 2.55 2.80 2.95 5.31
0.174 0.052 39.19 21.63 37.15 20.28 2.48 1.00 1.21 2.86 4.41 4.07 5.36 4.18 0.91 1.14 0.57 2.35 2.37 2.60 4.53
0.174 0.062 39.72 21.79 36.83 20.28 2.49 0.81 1.10 2.76 4.30 3.91 5.25 4.19 0.83 1.11 0.54 2.19 2.09 2.40 4.14
0.174 0.072 40.12 21.86 36.65 20.29 2.44 0.65 0.98 2.65 4.19 3.76 5.12 4.14 0.73 1.04 0.45 2.07 1.89 2.29 3.97
0.174 0.082 40.40 21.90 36.55 20.30 2.39 0.53 0.89 2.58 4.11 3.66 5.02 4.08 0.65 0.96 0.36 1.97 1.76 2.22 3.90
0.174 0.092 40.58 21.92 36.49 20.30 2.35 0.46 0.82 2.53 4.06 3.58 4.94 4.03 0.59 0.90 0.29 1.91 1.67 2.19 3.87
0.174 0.102 40.70 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.53 4.89 3.99 0.55 0.86 0.24 1.86 1.61 2.16 3.86
0.174 0.112 40.78 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.174 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.80 1.54 2.14 3.85
0.174 0.132 40.87 21.95 36.40 20.30 2.28 0.34 0.70 2.44 3.96 3.47 4.82 3.94 0.48 0.79 0.17 1.79 1.53 2.14 3.85
0.174 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.51 2.14 3.85
0.184 -0.008 33.22 17.86 33.17 17.09 2.42 0.84 0.97 1.81 3.11 3.47 4.55 1.98 -0.01 -0.62 -0.49 0.12 -0.45 0.27 2.62
0.184 0.002 34.97 18.66 30.85 14.53 2.45 1.23 1.26 2.21 3.42 4.26 5.22 2.46 0.37 -0.49 0.04 1.52 1.70 2.41 2.02
0.184 0.012 34.08 18.03 34.65 16.60 2.00 1.74 1.56 3.48 5.01 4.88 6.33 4.11 1.79 0.83 0.60 2.36 2.99 4.08 4.95
0.184 0.022 32.73 16.72 35.44 17.87 2.31 1.64 1.72 3.62 5.24 4.64 5.94 4.24 1.97 1.62 0.71 2.88 3.47 3.49 6.15
0.184 0.032 36.53 19.14 37.30 19.67 2.00 1.21 1.12 3.05 4.62 4.20 5.36 3.94 1.24 1.20 0.41 2.73 3.32 3.34 6.27
0.184 0.042 38.52 21.05 37.57 20.25 2.28 1.15 1.18 2.93 4.48 4.13 5.34 4.03 0.97 1.09 0.45 2.55 2.79 2.94 5.29
0.184 0.052 39.21 21.63 37.14 20.28 2.48 1.00 1.20 2.86 4.41 4.07 5.36 4.18 0.91 1.15 0.57 2.34 2.36 2.60 4.52
0.184 0.062 39.73 21.80 36.82 20.28 2.49 0.80 1.10 2.75 4.30 3.91 5.25 4.19 0.82 1.11 0.54 2.19 2.08 2.40 4.13
0.184 0.072 40.13 21.86 36.65 20.29 2.44 0.64 0.98 2.65 4.19 3.76 5.12 4.14 0.73 1.03 0.45 2.07 1.89 2.29 3.97
0.184 0.082 40.40 21.90 36.55 20.30 2.39 0.53 0.88 2.58 4.11 3.65 5.02 4.08 0.65 0.96 0.36 1.97 1.76 2.22 3.90
0.184 0.092 40.58 21.92 36.49 20.30 2.35 0.46 0.82 2.53 4.06 3.58 4.94 4.03 0.59 0.90 0.29 1.90 1.67 2.19 3.87
0.184 0.102 40.70 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.53 4.89 3.99 0.54 0.86 0.24 1.86 1.61 2.16 3.86
0.184 0.112 40.78 21.94 36.43 20.30 2.30 0.38 0.74 2.47 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.83 1.57 2.15 3.85
0.184 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.84 3.95 0.50 0.81 0.19 1.80 1.54 2.14 3.85
0.184 0.132 40.87 21.95 36.40 20.30 2.28 0.34 0.70 2.44 3.96 3.47 4.82 3.94 0.48 0.79 0.17 1.79 1.53 2.14 3.85
0.184 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.51 2.13 3.85
0.194 -0.008 33.16 17.67 32.18 16.77 2.65 1.07 1.19 1.86 3.14 3.61 4.72 2.15 0.02 -0.58 -0.51 0.02 -0.45 0.20 2.65
0.194 0.002 34.20 18.30 30.81 14.53 2.36 1.22 1.19 2.19 3.40 4.19 5.15 2.47 0.33 -0.49 -0.07 1.39 1.62 2.10 2.00
0.194 0.012 34.08 18.01 34.70 16.64 2.02 1.75 1.59 3.51 5.04 4.90 6.35 4.15 1.84 0.88 0.62 2.42 2.99 4.05 4.98
0.194 0.022 32.77 16.74 35.49 17.91 2.29 1.63 1.70 3.61 5.23 4.63 5.93 4.23 1.95 1.61 0.70 2.88 3.48 3.49 6.18
0.194 0.032 36.62 19.21 37.33 19.70 2.00 1.21 1.12 3.04 4.61 4.19 5.35 3.93 1.22 1.19 0.41 2.73 3.31 3.33 6.25
0.194 0.042 38.54 21.08 37.56 20.25 2.29 1.15 1.18 2.93 4.48 4.13 5.34 4.03 0.97 1.09 0.45 2.54 2.77 2.93 5.27
0.194 0.052 39.22 21.64 37.13 20.28 2.48 1.00 1.20 2.86 4.41 4.06 5.36 4.19 0.91 1.15 0.57 2.34 2.35 2.59 4.50
0.194 0.062 39.74 21.80 36.82 20.28 2.49 0.80 1.10 2.75 4.29 3.90 5.24 4.19 0.82 1.11 0.54 2.18 2.07 2.40 4.13
0.194 0.072 40.13 21.86 36.65 20.29 2.44 0.64 0.98 2.65 4.19 3.76 5.12 4.13 0.72 1.03 0.45 2.06 1.88 2.28 3.96
0.194 0.082 40.41 21.90 36.55 20.30 2.39 0.53 0.88 2.58 4.11 3.65 5.01 4.07 0.64 0.96 0.36 1.97 1.75 2.22 3.90
0.194 0.092 40.59 21.92 36.49 20.30 2.35 0.46 0.81 2.52 4.05 3.58 4.94 4.03 0.59 0.90 0.29 1.90 1.67 2.18 3.87
0.194 0.102 40.71 21.93 36.45 20.30 2.32 0.41 0.77 2.49 4.02 3.53 4.89 3.99 0.54 0.85 0.24 1.86 1.61 2.16 3.86
0.194 0.112 40.78 21.94 36.43 20.30 2.30 0.38 0.74 2.46 3.99 3.50 4.86 3.97 0.52 0.83 0.21 1.82 1.57 2.15 3.85
0.194 0.122 40.83 21.94 36.41 20.30 2.29 0.36 0.72 2.45 3.97 3.48 4.83 3.95 0.50 0.81 0.19 1.80 1.54 2.14 3.85
0.194 0.132 40.87 21.95 36.40 20.30 2.28 0.34 0.70 2.44 3.96 3.47 4.82 3.94 0.48 0.79 0.17 1.79 1.53 2.14 3.85
0.194 0.142 40.89 21.95 36.39 20.30 2.27 0.33 0.70 2.43 3.95 3.46 4.81 3.93 0.48 0.78 0.16 1.78 1.51 2.13 3.85
Table B2: Table showing all possible outcomes of the test statistic JTfor the causal relation between futures on crude oil and inflation at
various quantiles (on the columns) for all plausible pairs of the polynomial weights (θ1, θ2) (on the rows). The pairs yielding the maximum
values of JTat every quantile considered are indicated in bold.
Quantiles
θ1θ20.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
-0.1 -0.006 44.66 22.28 35.67 17.11 1.85 0.93 1.10 4.26 6.33 4.80 7.32 4.57 1.17 0.74 -0.07 2.05 2.69 1.53 5.97
-0.1 -0.004 43.68 22.03 35.80 17.08 1.93 1.04 1.31 4.42 6.51 4.92 7.53 4.87 1.17 0.80 0.03 2.21 2.91 1.58 5.95
-0.1 -0.002 41.72 21.12 35.43 16.68 1.93 1.12 1.47 4.54 6.63 4.99 7.70 5.12 1.22 0.88 0.13 2.41 3.19 1.82 6.16
-0.1 0 40.32 20.17 34.80 16.38 2.28 1.37 1.94 4.61 6.66 5.26 8.04 5.45 1.64 1.20 0.18 2.59 3.26 2.33 6.37
-0.1 0.002 42.22 20.92 34.41 16.91 4.20 2.49 3.69 4.79 6.73 6.92 9.70 6.91 3.25 2.38 0.56 3.08 3.41 3.56 6.25
-0.1 0.004 38.89 20.47 32.28 16.29 5.24 3.17 4.93 4.91 6.76 8.82 10.86 8.52 5.07 3.90 1.74 4.40 4.19 5.02 6.44
-0.1 0.006 33.45 18.63 30.35 14.24 3.72 2.50 3.27 3.35 5.03 7.23 9.56 7.90 3.19 2.65 1.70 4.20 5.16 4.53 6.08
-0.1 0.008 32.03 18.50 30.29 14.59 2.36 2.17 2.51 3.14 4.90 7.50 10.17 8.49 3.10 2.98 2.18 5.22 6.44 6.33 6.31
-0.1 0.01 34.02 18.71 30.49 14.79 2.33 1.96 2.54 3.07 4.87 7.07 10.01 8.16 3.39 2.47 1.31 4.18 5.04 5.89 7.25
-0.1 0.012 35.94 18.62 30.69 14.58 1.78 1.41 2.28 2.89 4.74 6.55 9.64 7.34 3.21 2.20 0.66 3.74 4.19 5.30 6.85
-0.1 0.014 38.17 19.07 31.13 14.90 1.59 1.35 2.28 2.94 4.80 6.41 9.45 6.86 3.14 2.11 0.57 3.88 3.71 4.88 7.25
-0.1 0.016 40.12 19.84 32.21 15.78 1.84 1.61 2.55 3.19 5.02 6.43 9.35 6.76 3.27 2.32 1.04 4.53 3.81 4.75 8.11
-0.1 0.018 39.88 19.81 33.12 16.26 1.71 1.54 2.55 3.21 5.08 6.34 9.22 6.84 3.43 2.65 1.74 5.49 4.34 4.87 9.04
-0.1 0.02 38.47 19.29 33.58 16.26 1.31 1.20 2.31 3.03 4.96 6.18 9.04 6.95 3.55 2.88 2.33 6.43 4.88 5.01 9.76
-0.08 -0.006 43.73 22.03 35.92 17.15 1.93 1.08 1.34 4.47 6.56 4.99 7.59 4.89 1.17 0.81 0.01 2.17 2.88 1.62 6.00
-0.08 -0.004 42.35 21.42 35.84 16.94 1.93 1.18 1.48 4.61 6.71 5.04 7.72 5.09 1.18 0.84 0.09 2.34 3.15 1.82 6.17
-0.08 -0.002 40.75 20.44 35.36 16.53 1.96 1.28 1.66 4.67 6.75 5.11 7.89 5.29 1.43 1.06 0.18 2.59 3.42 2.28 6.61
-0.08 0 41.30 20.42 35.07 16.85 3.11 1.82 2.66 4.73 6.73 5.91 8.74 6.02 2.42 1.77 0.31 2.82 3.33 2.97 6.62
-0.08 0.002 42.69 21.19 34.10 17.05 5.35 3.13 4.71 4.98 6.88 8.24 10.88 8.16 4.64 3.42 1.26 3.93 4.16 4.97 6.72
-0.08 0.004 35.92 19.78 31.70 15.63 4.74 2.90 4.29 4.40 6.18 8.25 10.14 8.15 4.76 3.75 2.15 4.76 4.87 5.42 6.55
-0.08 0.006 32.58 18.29 29.65 13.27 2.30 1.70 2.20 2.54 4.23 6.59 9.15 7.55 2.27 2.10 1.30 4.04 4.89 4.64 5.81
-0.08 0.008 33.01 18.91 30.89 15.29 2.67 2.37 2.70 3.26 5.02 7.58 10.32 8.58 3.38 2.86 2.04 5.02 6.20 6.54 7.01
-0.08 0.01 35.00 18.75 30.63 14.69 2.22 1.76 2.47 3.01 4.82 6.89 9.90 7.89 3.31 2.29 1.03 3.88 4.67 5.67 7.06
-0.08 0.012 36.32 18.53 30.70 14.55 1.64 1.30 2.21 2.85 4.72 6.47 9.57 7.13 3.13 2.10 0.54 3.66 3.97 5.15 6.84
-0.08 0.014 38.82 19.28 31.36 15.08 1.64 1.40 2.33 3.00 4.85 6.41 9.41 6.79 3.15 2.11 0.63 3.97 3.65 4.79 7.44
-0.08 0.016 40.29 19.92 32.47 15.97 1.86 1.63 2.58 3.23 5.06 6.41 9.31 6.76 3.31 2.40 1.20 4.74 3.91 4.75 8.35
-0.08 0.018 39.55 19.68 33.28 16.28 1.60 1.45 2.49 3.17 5.05 6.30 9.17 6.86 3.46 2.72 1.91 5.74 4.49 4.90 9.25
-0.08 0.02 38.17 19.18 33.65 16.25 1.23 1.12 2.24 2.98 4.93 6.13 8.99 6.98 3.57 2.92 2.44 6.63 4.99 5.04 9.90
-0.06 -0.006 42.74 21.55 36.07 17.09 1.94 1.24 1.51 4.68 6.78 5.11 7.77 5.09 1.15 0.83 0.07 2.29 3.12 1.84 6.18
-0.06 -0.004 41.45 20.79 35.86 16.80 1.89 1.31 1.59 4.75 6.85 5.10 7.84 5.22 1.30 0.97 0.16 2.53 3.43 2.21 6.59
-0.06 -0.002 40.92 20.24 35.53 16.71 2.32 1.53 2.02 4.77 6.84 5.42 8.24 5.58 1.94 1.42 0.24 2.79 3.55 2.83 7.06
-0.06 0 43.09 21.17 35.35 17.36 4.43 2.45 3.68 4.80 6.75 6.98 9.88 7.05 3.59 2.61 0.75 3.40 3.70 3.97 7.02
-0.06 0.002 40.73 20.93 33.77 16.99 5.59 3.21 4.98 5.00 6.85 8.80 11.08 8.71 5.45 4.17 1.97 4.71 4.63 5.64 6.99
-0.06 0.004 35.54 19.63 31.18 15.01 4.27 2.62 3.61 3.61 5.29 7.56 9.72 7.77 3.75 3.03 1.93 4.27 5.29 4.85 6.30
-0.06 0.006 32.76 18.71 30.06 13.96 2.14 1.81 2.16 2.70 4.38 6.92 9.48 7.56 2.33 2.23 1.55 4.42 5.48 5.35 5.62
-0.06 0.008 33.62 18.74 30.64 15.05 2.58 2.17 2.59 3.11 4.87 7.34 10.17 8.37 3.26 2.42 1.59 4.39 5.49 6.29 7.32
-0.06 0.01 35.57 18.67 30.72 14.63 2.08 1.59 2.40 2.97 4.79 6.74 9.80 7.63 3.24 2.18 0.82 3.70 4.41 5.50 6.88
-0.06 0.012 36.86 18.59 30.77 14.59 1.56 1.25 2.19 2.86 4.73 6.42 9.51 6.97 3.08 2.03 0.48 3.65 3.79 5.01 6.94
-0.06 0.014 39.38 19.49 31.62 15.29 1.70 1.46 2.40 3.07 4.91 6.40 9.37 6.74 3.17 2.14 0.73 4.10 3.64 4.73 7.65
-0.06 0.016 40.30 19.94 32.72 16.11 1.84 1.63 2.59 3.25 5.08 6.38 9.26 6.77 3.34 2.48 1.38 4.97 4.04 4.77 8.60
-0.06 0.018 39.19 19.55 33.40 16.28 1.50 1.36 2.43 3.12 5.02 6.25 9.11 6.89 3.49 2.78 2.07 5.98 4.63 4.94 9.45
-0.06 0.02 37.91 19.08 33.70 16.25 1.15 1.04 2.18 2.94 4.89 6.09 8.94 7.01 3.60 2.95 2.54 6.82 5.09 5.06 10.03
-0.04 -0.006 41.97 20.99 36.19 17.01 1.89 1.37 1.59 4.83 6.94 5.15 7.85 5.19 1.22 0.91 0.13 2.46 3.40 2.17 6.55
-0.04 -0.004 41.15 20.35 35.97 16.82 1.98 1.46 1.74 4.85 6.95 5.23 8.02 5.38 1.66 1.23 0.23 2.77 3.69 2.73 7.18
-0.04 -0.002 41.85 20.50 35.75 17.15 3.21 1.88 2.70 4.79 6.82 6.07 8.94 6.19 2.76 1.97 0.41 3.10 3.64 3.46 7.39
-0.04 0 44.11 21.64 35.63 17.72 5.53 3.04 4.56 4.93 6.82 8.04 10.87 8.23 4.87 3.56 1.46 4.24 4.44 5.29 7.36
-0.04 0.002 37.80 20.55 33.32 16.46 5.18 3.06 4.61 4.72 6.51 8.51 10.50 8.37 5.30 4.21 2.43 5.08 4.90 5.67 6.90
-0.04 0.004 34.85 19.43 31.07 14.19 3.18 2.17 2.76 2.90 4.55 6.82 9.29 7.36 2.67 2.31 1.47 3.97 4.83 4.10 5.70
-0.04 0.006 33.87 19.46 31.28 15.35 2.71 2.36 2.64 3.16 4.86 7.50 10.09 8.10 2.85 2.47 1.85 4.76 5.98 5.93 5.87
-0.04 0.008 34.33 18.62 30.56 14.80 2.42 1.92 2.48 3.01 4.78 7.11 10.04 8.15 3.18 2.16 1.24 3.94 4.95 5.94 7.23
-0.04 0.01 35.85 18.53 30.74 14.59 1.92 1.44 2.32 2.93 4.76 6.61 9.71 7.38 3.16 2.09 0.65 3.58 4.17 5.33 6.75
-0.04 0.012 37.53 18.77 30.91 14.69 1.54 1.26 2.21 2.89 4.76 6.39 9.46 6.85 3.07 2.00 0.48 3.68 3.66 4.88 7.08
-0.04 0.014 39.83 19.67 31.90 15.52 1.76 1.53 2.47 3.13 4.96 6.39 9.33 6.72 3.20 2.20 0.86 4.27 3.68 4.70 7.88
-0.04 0.016 40.17 19.89 32.94 16.21 1.78 1.59 2.57 3.24 5.08 6.35 9.22 6.79 3.38 2.56 1.57 5.22 4.18 4.81 8.84
-0.04 0.018 38.83 19.42 33.51 16.28 1.40 1.28 2.36 3.08 4.99 6.20 9.06 6.92 3.52 2.84 2.21 6.21 4.77 4.98 9.62
-0.04 0.02 37.68 19.00 33.75 16.24 1.07 0.96 2.12 2.89 4.86 6.05 8.90 7.03 3.62 2.97 2.62 6.99 5.18 5.09 10.15
-0.02 -0.006 41.52 20.48 36.35 17.01 1.87 1.47 1.65 4.93 7.04 5.19 7.94 5.29 1.47 1.10 0.21 2.70 3.70 2.62 7.09
-0.02 -0.004 41.24 20.14 36.15 17.01 2.37 1.63 2.08 4.87 6.97 5.57 8.42 5.75 2.25 1.60 0.32 3.04 3.83 3.28 7.71
-0.02 -0.002 43.69 21.29 36.23 17.73 4.44 2.31 3.55 4.72 6.68 6.92 9.86 7.13 3.83 2.75 0.88 3.70 4.02 4.39 7.75
-0.02 0 42.59 21.39 35.45 17.74 5.86 3.20 4.90 4.98 6.81 8.59 11.13 8.86 5.72 4.31 2.10 4.94 4.95 6.10 7.38
-0.02 0.002 37.40 20.52 32.59 15.89 4.72 2.70 3.92 3.93 5.63 7.74 9.78 7.71 4.26 3.37 2.11 4.39 5.12 5.07 6.43
-0.02 0.004 34.30 19.35 30.63 13.84 2.22 1.68 2.05 2.43 4.05 6.42 8.92 6.72 1.89 1.71 1.12 3.68 4.52 4.28 5.36
-0.02 0.006 34.20 19.28 31.31 15.54 2.85 2.41 2.70 3.17 4.88 7.54 10.22 8.28 2.97 2.26 1.73 4.46 5.71 6.20 6.56
-0.02 0.008 34.97 18.57 30.66 14.69 2.29 1.73 2.43 2.98 4.77 6.93 9.95 7.90 3.16 2.06 1.00 3.70 4.62 5.69 7.04
-0.02 0.01 36.06 18.41 30.73 14.55 1.75 1.31 2.25 2.89 4.74 6.52 9.62 7.17 3.09 2.02 0.52 3.51 3.96 5.16 6.72
-0.02 0.012 38.23 19.00 31.11 14.85 1.56 1.30 2.25 2.94 4.81 6.37 9.42 6.77 3.07 2.00 0.51 3.76 3.58 4.78 7.26
-0.02 0.014 40.16 19.81 32.19 15.75 1.81 1.58 2.52 3.19 5.01 6.38 9.29 6.71 3.24 2.28 1.02 4.47 3.76 4.70 8.12
-0.02 0.016 39.92 19.80 33.13 16.27 1.70 1.53 2.53 3.21 5.07 6.31 9.17 6.81 3.42 2.64 1.75 5.48 4.34 4.85 9.06
-0.02 0.018 38.50 19.29 33.60 16.27 1.31 1.19 2.30 3.03 4.95 6.15 9.01 6.94 3.54 2.88 2.34 6.43 4.90 5.01 9.78
-0.02 0.02 37.48 18.94 33.81 16.23 1.00 0.89 2.06 2.86 4.84 6.03 8.87 7.06 3.65 2.99 2.69 7.13 5.25 5.11 10.25
0 -0.006 41.28 20.09 36.59 17.13 2.00 1.57 1.80 4.98 7.11 5.37 8.16 5.52 1.92 1.40 0.31 3.00 3.94 3.13 7.72
0 -0.004 42.03 20.40 36.48 17.44 3.19 1.82 2.64 4.76 6.81 6.13 9.01 6.34 3.05 2.13 0.52 3.42 3.95 3.91 8.05
0 -0.002 45.14 21.96 37.02 18.28 5.41 2.75 4.22 4.71 6.60 7.67 10.60 8.16 4.93 3.61 1.54 4.46 4.66 5.54 7.98
0 0 39.90 21.23 35.21 17.48 5.68 3.20 4.83 4.91 6.69 8.62 10.79 8.70 5.75 4.46 2.44 5.11 4.86 5.77 7.16
0 0.002 37.10 20.48 32.29 15.25 3.92 2.44 3.22 3.24 4.86 7.13 9.42 7.28 3.09 2.57 1.64 3.96 4.82 3.85 5.56
0 0.004 34.84 19.97 31.31 14.89 2.47 2.02 2.31 2.75 4.38 6.98 9.45 7.08 2.16 1.84 1.38 4.08 5.14 4.96 5.31
0 0.006 34.29 18.83 30.87 15.16 2.68 2.16 2.55 3.02 4.75 7.33 10.14 8.22 2.94 1.99 1.46 4.05 5.25 6.17 7.10
0 0.008 35.43 18.52 30.77 14.66 2.18 1.59 2.40 2.98 4.78 6.79 9.85 7.66 3.14 2.02 0.82 3.55 4.37 5.49 6.84
0 0.01 36.40 18.39 30.73 14.53 1.62 1.24 2.19 2.87 4.73 6.44 9.55 7.00 3.04 1.96 0.44 3.49 3.78 5.02 6.79
0 0.012 38.88 19.24 31.35 15.04 1.60 1.37 2.31 3.00 4.85 6.37 9.37 6.71 3.09 2.03 0.59 3.88 3.55 4.71 7.45
0 0.014 40.34 19.91 32.47 15.95 1.83 1.62 2.56 3.22 5.05 6.36 9.25 6.72 3.28 2.37 1.20 4.70 3.88 4.72 8.37
0 0.016 39.59 19.68 33.29 16.29 1.60 1.45 2.48 3.17 5.05 6.27 9.13 6.84 3.45 2.72 1.92 5.73 4.50 4.90 9.28
0 0.018 38.19 19.18 33.67 16.27 1.23 1.11 2.24 2.98 4.92 6.11 8.97 6.97 3.57 2.92 2.45 6.64 5.01 5.04 9.92
0 0.02 37.31 18.88 33.87 16.24 0.94 0.82 2.01 2.82 4.82 6.00 8.84 7.09 3.67 3.01 2.76 7.26 5.31 5.13 10.35
0.02 -0.006 41.43 20.00 36.93 17.40 2.40 1.65 2.10 4.91 7.03 5.70 8.53 5.92 2.53 1.79 0.43 3.34 4.09 3.68 8.25
0.02 -0.004 43.84 21.22 37.22 18.10 4.23 2.05 3.27 4.55 6.53 6.73 9.65 7.14 3.98 2.84 0.99 4.00 4.33 4.78 8.35
0.02 -0.002 44.32 21.89 37.13 18.40 5.88 3.01 4.62 4.81 6.62 8.21 10.95 8.85 5.80 4.33 2.08 5.05 5.09 6.31 7.82
0.02 0 38.89 21.15 34.57 16.98 5.25 2.92 4.29 4.33 6.03 7.98 9.99 7.93 4.86 3.76 2.27 4.60 4.80 5.09 6.53
0.02 0.002 36.31 20.27 31.99 14.56 2.81 1.96 2.37 2.59 4.16 6.42 8.85 6.48 2.01 1.71 1.15 3.41 4.17 3.44 5.07
0.02 0.004 35.25 20.02 31.85 15.69 2.86 2.37 2.63 3.02 4.67 7.39 9.92 7.62 2.46 1.87 1.55 4.19 5.45 5.48 5.52
0.02 0.006 34.53 18.54 30.67 14.85 2.47 1.89 2.43 2.94 4.69 7.13 10.06 8.09 2.95 1.87 1.21 3.74 4.86 5.93 7.19
0.02 0.008 35.74 18.46 30.84 14.65 2.06 1.48 2.36 2.97 4.78 6.67 9.76 7.43 3.10 1.99 0.66 3.45 4.15 5.31 6.68
0.02 0.01 36.93 18.50 30.79 14.57 1.54 1.22 2.18 2.87 4.74 6.39 9.49 6.87 3.01 1.92 0.41 3.52 3.64 4.90 6.92
0.02 0.012 39.45 19.47 31.62 15.26 1.66 1.44 2.38 3.06 4.90 6.36 9.33 6.68 3.13 2.09 0.71 4.03 3.57 4.67 7.67
0.02 0.014 40.36 19.93 32.72 16.11 1.82 1.62 2.58 3.24 5.07 6.34 9.22 6.74 3.32 2.46 1.39 4.95 4.02 4.75 8.62
0.02 0.016 39.22 19.55 33.42 16.30 1.50 1.36 2.42 3.12 5.02 6.22 9.08 6.87 3.48 2.78 2.08 5.98 4.65 4.94 9.47
0.02 0.018 37.93 19.09 33.72 16.26 1.15 1.03 2.17 2.93 4.89 6.07 8.92 7.00 3.59 2.95 2.55 6.83 5.10 5.07 10.05
0.02 0.02 37.15 18.83 33.94 16.25 0.88 0.77 1.96 2.79 4.80 5.98 8.81 7.11 3.69 3.02 2.81 7.37 5.35 5.14 10.43
0.04 -0.006 42.50 20.47 37.58 17.94 3.11 1.72 2.53 4.70 6.77 6.12 8.97 6.46 3.26 2.29 0.68 3.77 4.26 4.30 8.55
0.04 -0.004 45.33 21.94 38.15 18.65 5.05 2.34 3.75 4.42 6.32 7.21 10.17 7.95 4.86 3.58 1.52 4.59 4.83 5.72 8.52
0.04 -0.002 42.12 21.82 36.97 18.35 5.99 3.19 4.84 4.92 6.68 8.51 10.93 8.92 6.04 4.57 2.33 5.07 4.85 5.93 7.44
0.04 0 39.12 21.26 33.92 16.35 4.53 2.55 3.56 3.54 5.15 7.24 9.34 7.12 3.52 2.74 1.69 3.91 4.57 3.79 5.53
0.04 0.002 36.22 20.65 31.99 14.82 2.43 1.83 2.15 2.48 4.05 6.52 8.93 6.31 1.74 1.40 1.03 3.39 4.24 3.94 5.03
0.04 0.004 35.12 19.51 31.64 15.68 2.90 2.36 2.64 3.00 4.68 7.39 10.05 7.89 2.57 1.73 1.49 3.97 5.29 5.84 6.14
0.04 0.006 34.84 18.39 30.70 14.72 2.31 1.70 2.37 2.93 4.70 6.96 9.97 7.89 2.99 1.84 1.00 3.54 4.56 5.68 7.06
0.04 0.008 35.95 18.38 30.84 14.62 1.91 1.37 2.30 2.94 4.76 6.58 9.68 7.23 3.06 1.95 0.53 3.39 3.95 5.16 6.62
0.04 0.01 37.61 18.72 30.93 14.67 1.51 1.24 2.20 2.90 4.77 6.36 9.43 6.76 3.01 1.92 0.42 3.58 3.54 4.79 7.08
0.04 0.012 39.91 19.66 31.91 15.51 1.73 1.51 2.45 3.13 4.96 6.35 9.28 6.67 3.17 2.16 0.85 4.22 3.63 4.66 7.90
0.04 0.014 40.22 19.90 32.95 16.22 1.77 1.58 2.56 3.23 5.08 6.31 9.18 6.76 3.37 2.55 1.58 5.21 4.18 4.80 8.86
0.04 0.016 38.86 19.42 33.53 16.30 1.40 1.27 2.35 3.07 4.98 6.18 9.03 6.90 3.51 2.84 2.23 6.22 4.78 4.98 9.64
0.04 0.018 37.70 19.01 33.77 16.25 1.07 0.96 2.11 2.89 4.86 6.04 8.88 7.03 3.62 2.98 2.63 7.00 5.19 5.09 10.17
0.04 0.02 37.00 18.78 34.02 16.27 0.83 0.71 1.91 2.77 4.78 5.96 8.79 7.12 3.71 3.03 2.86 7.46 5.39 5.15 10.50
0.06 -0.006 44.11 21.26 38.41 18.59 3.98 1.81 2.97 4.41 6.39 6.50 9.36 7.07 4.02 2.92 1.10 4.26 4.58 5.06 8.75
0.06 -0.004 45.30 22.16 38.53 18.83 5.61 2.66 4.17 4.51 6.34 7.73 10.60 8.59 5.62 4.21 1.92 4.99 5.09 6.32 8.36
0.06 -0.002 40.38 21.64 36.63 18.14 5.76 3.16 4.61 4.70 6.37 8.17 10.29 8.25 5.34 4.08 2.29 4.72 4.47 5.02 6.73
0.06 0 38.37 21.08 33.56 15.74 3.56 2.27 2.82 2.92 4.46 6.68 8.95 6.55 2.43 1.97 1.38 3.47 4.16 2.99 4.81
0.06 0.002 36.13 20.65 32.01 15.34 2.58 2.02 2.31 2.64 4.23 6.91 9.32 6.65 1.87 1.38 1.20 3.62 4.75 4.58 5.07
0.06 0.004 34.92 18.97 31.15 15.30 2.73 2.13 2.51 2.91 4.62 7.26 10.05 7.99 2.64 1.63 1.37 3.75 5.03 5.99 6.81
0.06 0.006 35.15 18.33 30.81 14.69 2.21 1.57 2.36 2.96 4.74 6.82 9.88 7.68 3.02 1.86 0.82 3.42 4.32 5.46 6.85
0.06 0.008 36.16 18.31 30.80 14.57 1.75 1.28 2.24 2.90 4.75 6.49 9.60 7.05 3.01 1.90 0.43 3.37 3.79 5.04 6.66
0.06 0.01 38.32 18.98 31.13 14.83 1.53 1.28 2.24 2.95 4.81 6.34 9.38 6.69 3.02 1.94 0.48 3.68 3.49 4.71 7.26
0.06 0.012 40.24 19.82 32.20 15.75 1.78 1.57 2.51 3.18 5.00 6.34 9.25 6.67 3.22 2.25 1.02 4.45 3.74 4.67 8.14
0.06 0.014 39.97 19.81 33.15 16.28 1.69 1.52 2.52 3.21 5.06 6.28 9.14 6.79 3.41 2.64 1.76 5.48 4.35 4.85 9.09
0.06 0.016 38.52 19.29 33.62 16.29 1.31 1.19 2.29 3.02 4.95 6.13 8.99 6.93 3.54 2.89 2.35 6.44 4.91 5.02 9.80
0.06 0.018 37.50 18.94 33.82 16.24 1.00 0.89 2.06 2.85 4.83 6.01 8.85 7.06 3.64 3.00 2.70 7.14 5.26 5.11 10.27
0.06 0.02 36.86 18.73 34.11 16.29 0.79 0.66 1.87 2.75 4.77 5.94 8.76 7.13 3.72 3.03 2.89 7.52 5.41 5.16 10.56
0.08 -0.006 45.32 21.90 39.20 19.02 4.69 2.01 3.34 4.22 6.12 6.82 9.74 7.67 4.70 3.52 1.51 4.62 4.91 5.79 8.89
0.08 -0.004 43.91 22.24 38.41 18.90 5.98 2.99 4.60 4.75 6.50 8.23 10.84 8.85 6.03 4.54 2.15 5.06 4.86 6.14 7.87
0.08 -0.002 40.28 21.47 35.72 17.42 5.13 2.72 3.90 3.87 5.44 7.36 9.33 7.09 4.02 2.99 1.73 3.96 4.17 3.82 5.65
0.08 0 37.85 21.29 33.42 15.49 2.83 1.96 2.31 2.52 4.05 6.41 8.75 6.06 1.77 1.35 1.00 3.07 3.80 3.03 4.67
0.08 0.002 36.11 20.32 32.16 15.73 2.79 2.21 2.50 2.77 4.40 7.13 9.64 7.09 2.03 1.34 1.29 3.64 4.98 5.07 5.23
0.08 0.004 34.84 18.59 30.84 14.96 2.50 1.88 2.39 2.86 4.60 7.10 10.02 7.97 2.72 1.61 1.20 3.57 4.77 5.89 7.11
0.08 0.006 35.50 18.33 30.92 14.70 2.13 1.49 2.36 2.98 4.77 6.72 9.80 7.47 3.03 1.87 0.66 3.33 4.12 5.28 6.65
0.08 0.008 36.50 18.33 30.77 14.54 1.61 1.22 2.19 2.88 4.74 6.42 9.52 6.90 2.97 1.87 0.38 3.39 3.64 4.92 6.77
0.08 0.01 38.98 19.24 31.38 15.03 1.57 1.35 2.30 3.00 4.85 6.33 9.33 6.65 3.06 1.99 0.57 3.82 3.48 4.65 7.47
0.08 0.012 40.42 19.92 32.49 15.96 1.81 1.60 2.55 3.22 5.04 6.33 9.21 6.69 3.26 2.35 1.21 4.69 3.87 4.70 8.39
0.08 0.014 39.63 19.69 33.31 16.31 1.60 1.44 2.47 3.16 5.04 6.24 9.09 6.83 3.44 2.72 1.94 5.74 4.51 4.90 9.30
0.08 0.016 38.22 19.18 33.69 16.28 1.23 1.11 2.23 2.98 4.91 6.09 8.94 6.96 3.56 2.92 2.46 6.65 5.02 5.05 9.94
0.08 0.018 37.32 18.88 33.88 16.25 0.94 0.82 2.01 2.82 4.81 5.99 8.82 7.08 3.67 3.02 2.77 7.27 5.32 5.13 10.37
0.08 0.02 36.72 18.68 34.20 16.32 0.74 0.62 1.83 2.73 4.76 5.92 8.73 7.12 3.73 3.03 2.92 7.57 5.43 5.16 10.60
0.1 -0.006 45.79 22.30 39.69 19.17 5.24 2.28 3.72 4.24 6.07 7.27 10.16 8.17 5.28 4.00 1.73 4.80 5.00 6.18 8.85
0.1 -0.004 41.99 22.07 38.24 18.97 5.94 3.16 4.65 4.80 6.48 8.17 10.46 8.38 5.55 4.21 2.14 4.73 4.29 5.11 7.16
0.1 -0.002 40.13 21.50 35.30 16.89 4.29 2.48 3.25 3.28 4.80 6.86 8.90 6.43 2.89 2.22 1.52 3.53 4.05 2.87 4.80
0.1 0 37.30 21.26 32.91 15.42 2.51 1.79 2.13 2.38 3.93 6.48 8.81 6.02 1.56 1.08 0.94 3.05 3.98 3.64 4.77
0.1 0.002 35.91 19.77 31.98 15.74 2.85 2.22 2.53 2.79 4.45 7.16 9.81 7.45 2.18 1.30 1.30 3.55 4.92 5.48 5.75
0.1 0.004 34.86 18.36 30.79 14.78 2.31 1.68 2.32 2.87 4.63 6.96 9.97 7.85 2.81 1.64 1.01 3.42 4.52 5.68 7.08
0.1 0.006 35.82 18.34 30.97 14.71 2.04 1.42 2.35 2.98 4.78 6.63 9.72 7.28 3.02 1.87 0.54 3.29 3.94 5.14 6.54
0.1 0.008 37.04 18.47 30.83 14.57 1.52 1.20 2.18 2.89 4.75 6.36 9.45 6.78 2.96 1.85 0.36 3.43 3.53 4.82 6.92
0.1 0.01 39.55 19.48 31.65 15.26 1.64 1.42 2.37 3.06 4.90 6.32 9.28 6.63 3.10 2.06 0.70 4.00 3.52 4.63 7.68
0.1 0.012 40.43 19.95 32.75 16.13 1.81 1.61 2.57 3.24 5.06 6.31 9.17 6.71 3.31 2.45 1.40 4.95 4.02 4.74 8.64
0.1 0.014 39.26 19.55 33.45 16.32 1.50 1.36 2.41 3.12 5.01 6.20 9.05 6.86 3.48 2.79 2.10 5.99 4.66 4.95 9.49
0.1 0.016 37.95 19.09 33.74 16.27 1.15 1.03 2.17 2.93 4.88 6.06 8.90 6.99 3.59 2.96 2.56 6.84 5.12 5.08 10.07
0.1 0.018 37.16 18.83 33.95 16.26 0.88 0.77 1.96 2.79 4.80 5.97 8.80 7.10 3.69 3.03 2.82 7.38 5.37 5.15 10.45
0.1 0.02 36.58 18.63 34.30 16.35 0.70 0.57 1.78 2.70 4.74 5.89 8.69 7.11 3.72 3.02 2.93 7.59 5.43 5.16 10.62
0.12 -0.006 45.18 22.52 39.64 19.27 5.70 2.61 4.17 4.43 6.19 7.79 10.50 8.45 5.69 4.35 1.91 4.94 4.81 6.18 8.45
0.12 -0.004 41.21 21.62 37.42 18.45 5.61 2.91 4.22 4.21 5.75 7.51 9.50 7.24 4.43 3.26 1.78 4.08 3.86 3.94 6.03
0.12 -0.002 39.20 21.50 35.01 16.58 3.52 2.24 2.67 2.77 4.24 6.53 8.70 6.06 2.12 1.61 1.22 3.13 3.72 2.49 4.40
0.12 0 36.84 20.91 32.43 15.40 2.45 1.78 2.15 2.38 3.96 6.65 9.03 6.18 1.53 0.95 0.99 3.15 4.37 4.24 4.86
0.12 0.002 35.59 19.23 31.52 15.44 2.72 2.08 2.46 2.77 4.47 7.11 9.90 7.68 2.33 1.32 1.27 3.48 4.81 5.77 6.47
0.12 0.004 34.94 18.22 30.86 14.72 2.19 1.55 2.30 2.90 4.68 6.84 9.90 7.68 2.88 1.69 0.83 3.31 4.29 5.45 6.89
0.12 0.006 36.07 18.32 30.94 14.67 1.91 1.34 2.31 2.96 4.78 6.55 9.65 7.11 2.99 1.86 0.44 3.28 3.79 5.03 6.55
0.12 0.008 37.72 18.71 30.97 14.67 1.49 1.22 2.19 2.91 4.78 6.33 9.39 6.69 2.97 1.86 0.39 3.52 3.46 4.72 7.09
0.12 0.01 40.01 19.68 31.94 15.51 1.71 1.50 2.44 3.12 4.95 6.31 9.24 6.63 3.15 2.15 0.85 4.20 3.61 4.63 7.92
0.12 0.012 40.29 19.91 32.98 16.24 1.76 1.58 2.55 3.23 5.07 6.28 9.14 6.74 3.36 2.55 1.59 5.22 4.19 4.80 8.88
0.12 0.014 38.89 19.42 33.56 16.31 1.40 1.27 2.35 3.07 4.98 6.15 9.01 6.89 3.51 2.84 2.24 6.23 4.80 4.99 9.67
0.12 0.016 37.72 19.01 33.79 16.26 1.07 0.96 2.11 2.89 4.85 6.02 8.87 7.02 3.62 2.98 2.64 7.01 5.20 5.10 10.18
0.12 0.018 37.01 18.78 34.03 16.27 0.83 0.71 1.91 2.77 4.78 5.95 8.77 7.12 3.71 3.04 2.87 7.47 5.40 5.16 10.51
0.12 0.02 36.45 18.58 34.40 16.38 0.66 0.53 1.74 2.68 4.72 5.86 8.65 7.08 3.71 3.00 2.94 7.60 5.43 5.15 10.64
0.14 -0.006 43.38 22.38 39.32 19.35 5.73 2.86 4.37 4.62 6.31 7.94 10.39 8.23 5.47 4.19 1.96 4.77 4.33 5.49 7.81
0.14 -0.004 40.99 21.44 36.74 17.72 4.84 2.55 3.56 3.53 5.01 6.93 8.80 6.23 3.29 2.38 1.49 3.48 3.71 2.93 5.08
0.14 -0.002 38.57 21.58 34.55 16.17 2.85 1.84 2.23 2.37 3.86 6.31 8.55 5.78 1.61 1.09 0.91 2.77 3.56 2.77 4.40
0.14 0 36.74 20.53 32.43 15.61 2.57 1.91 2.27 2.46 4.09 6.78 9.30 6.55 1.63 0.92 1.05 3.19 4.59 4.72 5.00
0.14 0.002 35.33 18.80 31.11 15.10 2.52 1.88 2.36 2.76 4.49 7.02 9.94 7.79 2.48 1.36 1.18 3.42 4.67 5.82 6.96
0.14 0.004 35.16 18.18 30.97 14.73 2.14 1.48 2.32 2.96 4.73 6.74 9.82 7.50 2.93 1.74 0.68 3.23 4.10 5.26 6.67
0.14 0.006 36.29 18.29 30.87 14.60 1.75 1.27 2.25 2.93 4.77 6.47 9.57 6.96 2.96 1.83 0.37 3.28 3.66 4.94 6.63
0.14 0.008 38.44 18.99 31.17 14.83 1.51 1.27 2.24 2.96 4.81 6.30 9.33 6.63 2.99 1.90 0.46 3.64 3.43 4.65 7.28
0.14 0.01 40.34 19.84 32.24 15.76 1.77 1.56 2.50 3.18 5.00 6.30 9.20 6.64 3.20 2.25 1.03 4.44 3.72 4.65 8.17
0.14 0.012 40.02 19.82 33.17 16.30 1.69 1.52 2.52 3.20 5.05 6.25 9.10 6.77 3.40 2.64 1.78 5.48 4.36 4.85 9.11
0.14 0.014 38.55 19.30 33.64 16.31 1.31 1.19 2.29 3.02 4.94 6.11 8.96 6.92 3.53 2.89 2.37 6.45 4.92 5.02 9.82
0.14 0.016 37.52 18.94 33.83 16.25 1.00 0.89 2.05 2.85 4.83 6.00 8.84 7.05 3.64 3.00 2.71 7.15 5.27 5.12 10.29
0.14 0.018 36.87 18.73 34.12 16.30 0.79 0.66 1.87 2.75 4.77 5.93 8.75 7.13 3.72 3.04 2.90 7.53 5.42 5.17 10.57
0.14 0.02 36.32 18.53 34.49 16.40 0.62 0.48 1.69 2.66 4.70 5.83 8.61 7.04 3.69 2.97 2.93 7.59 5.42 5.14 10.64
0.16 -0.006 42.40 21.97 38.70 19.15 5.66 2.85 4.26 4.34 5.89 7.54 9.63 7.32 4.59 3.44 1.78 4.18 3.76 4.26 6.77
0.16 -0.004 40.30 21.50 36.66 17.61 4.22 2.46 3.10 3.11 4.54 6.64 8.57 5.92 2.57 1.98 1.45 3.33 3.68 2.38 4.45
0.16 -0.002 37.80 21.29 33.56 15.61 2.36 1.50 1.93 2.12 3.66 6.23 8.52 5.65 1.32 0.77 0.80 2.67 3.71 3.37 4.54
0.16 0 36.56 20.05 32.35 15.73 2.69 2.01 2.38 2.55 4.21 6.87 9.52 6.98 1.83 0.96 1.11 3.20 4.61 5.15 5.42
0.16 0.002 35.16 18.50 30.96 14.88 2.33 1.69 2.29 2.78 4.54 6.93 9.94 7.77 2.62 1.44 1.03 3.33 4.49 5.69 7.08
0.16 0.004 35.54 18.24 31.07 14.76 2.10 1.44 2.35 3.00 4.78 6.66 9.76 7.33 2.96 1.78 0.56 3.19 3.93 5.12 6.51
0.16 0.006 36.63 18.33 30.82 14.55 1.61 1.21 2.20 2.90 4.76 6.39 9.49 6.82 2.93 1.81 0.33 3.32 3.54 4.85 6.76
0.16 0.008 39.11 19.26 31.42 15.04 1.55 1.34 2.29 3.01 4.85 6.29 9.28 6.60 3.04 1.96 0.56 3.79 3.44 4.61 7.48
0.16 0.01 40.51 19.94 32.52 15.97 1.80 1.60 2.54 3.21 5.03 6.29 9.16 6.66 3.25 2.35 1.22 4.69 3.87 4.69 8.42
0.16 0.012 39.67 19.70 33.34 16.33 1.59 1.44 2.46 3.16 5.03 6.21 9.06 6.81 3.44 2.72 1.95 5.75 4.52 4.90 9.32
0.16 0.014 38.24 19.19 33.71 16.29 1.23 1.11 2.22 2.97 4.91 6.07 8.92 6.95 3.56 2.93 2.47 6.66 5.04 5.06 9.96
0.16 0.016 37.34 18.89 33.89 16.25 0.94 0.82 2.00 2.82 4.81 5.98 8.81 7.08 3.66 3.02 2.78 7.28 5.33 5.14 10.38
0.16 0.018 36.73 18.68 34.21 16.33 0.75 0.62 1.83 2.73 4.75 5.91 8.72 7.12 3.72 3.03 2.92 7.57 5.44 5.17 10.61
0.16 0.02 36.19 18.47 34.58 16.43 0.57 0.43 1.64 2.63 4.68 5.79 8.56 7.00 3.67 2.94 2.92 7.57 5.40 5.12 10.64
0.18 -0.006 41.73 21.52 37.84 18.38 5.15 2.49 3.74 3.70 5.15 6.96 8.79 6.21 3.54 2.52 1.43 3.44 3.43 3.13 5.69
0.18 -0.004 39.20 21.36 36.12 17.19 3.53 2.10 2.56 2.59 4.00 6.36 8.41 5.70 1.94 1.42 1.13 2.88 3.46 2.25 4.18
0.18 -0.002 37.30 20.94 32.83 15.29 2.16 1.40 1.87 2.06 3.65 6.28 8.67 5.71 1.19 0.61 0.79 2.74 4.04 3.96 4.68
0.18 0 36.20 19.53 31.93 15.56 2.67 1.98 2.39 2.61 4.30 6.91 9.69 7.33 2.04 1.05 1.16 3.24 4.60 5.52 6.12
0.18 0.002 35.03 18.28 30.97 14.78 2.18 1.55 2.26 2.83 4.61 6.83 9.89 7.67 2.73 1.52 0.86 3.23 4.28 5.47 6.95
0.18 0.004 35.93 18.30 31.09 14.77 2.04 1.40 2.35 3.01 4.81 6.60 9.69 7.18 2.97 1.80 0.46 3.19 3.79 5.02 6.46
0.18 0.006 37.17 18.48 30.86 14.58 1.51 1.19 2.18 2.90 4.76 6.34 9.42 6.71 2.92 1.81 0.33 3.38 3.45 4.75 6.92
0.18 0.008 39.67 19.50 31.69 15.27 1.62 1.42 2.36 3.07 4.90 6.28 9.23 6.59 3.09 2.04 0.70 3.98 3.50 4.60 7.71
0.18 0.01 40.51 19.97 32.78 16.14 1.80 1.60 2.56 3.23 5.06 6.27 9.13 6.69 3.30 2.46 1.41 4.95 4.03 4.74 8.67
0.18 0.012 39.30 19.56 33.47 16.33 1.50 1.36 2.40 3.11 5.00 6.17 9.02 6.84 3.47 2.79 2.11 6.00 4.67 4.95 9.51
0.18 0.014 37.97 19.09 33.76 16.28 1.15 1.03 2.16 2.93 4.88 6.04 8.88 6.99 3.59 2.96 2.57 6.85 5.13 5.08 10.09
0.18 0.016 37.18 18.84 33.96 16.26 0.89 0.76 1.95 2.79 4.79 5.96 8.79 7.10 3.69 3.03 2.83 7.39 5.38 5.15 10.46
0.18 0.018 36.59 18.63 34.30 16.35 0.71 0.57 1.78 2.70 4.74 5.89 8.68 7.11 3.72 3.02 2.94 7.60 5.44 5.16 10.64
0.18 0.02 36.07 18.42 34.67 16.45 0.53 0.39 1.59 2.60 4.65 5.75 8.50 6.94 3.64 2.91 2.91 7.53 5.38 5.10 10.62
0.2 -0.006 41.08 21.50 37.76 18.12 4.59 2.37 3.31 3.28 4.70 6.66 8.40 5.72 2.90 2.18 1.46 3.35 3.49 2.51 4.86
0.2 -0.004 38.62 21.28 35.35 16.47 2.75 1.56 2.05 2.14 3.61 6.08 8.23 5.46 1.44 0.90 0.82 2.50 3.36 2.57 4.18
0.2 -0.002 37.15 20.63 32.70 15.40 2.26 1.52 1.99 2.14 3.78 6.40 8.93 6.04 1.28 0.61 0.85 2.83 4.28 4.43 4.82
0.2 0 35.85 19.08 31.45 15.25 2.53 1.86 2.34 2.65 4.37 6.90 9.80 7.55 2.24 1.14 1.14 3.26 4.56 5.71 6.74
0.2 0.002 34.99 18.13 31.03 14.76 2.11 1.47 2.26 2.90 4.68 6.74 9.83 7.52 2.82 1.60 0.70 3.15 4.08 5.26 6.72
0.2 0.004 36.23 18.32 31.03 14.71 1.92 1.34 2.32 2.99 4.80 6.53 9.62 7.03 2.95 1.80 0.38 3.20 3.67 4.94 6.51
0.2 0.006 37.85 18.73 31.00 14.68 1.48 1.22 2.19 2.92 4.78 6.30 9.35 6.63 2.94 1.83 0.37 3.48 3.39 4.67 7.10
0.2 0.008 40.13 19.71 31.98 15.52 1.69 1.49 2.43 3.12 4.95 6.28 9.19 6.59 3.14 2.14 0.86 4.20 3.59 4.61 7.95
0.2 0.01 40.36 19.92 33.00 16.26 1.76 1.58 2.55 3.22 5.06 6.25 9.10 6.72 3.35 2.55 1.61 5.22 4.20 4.80 8.91
0.2 0.012 38.93 19.43 33.58 16.33 1.40 1.27 2.34 3.06 4.97 6.13 8.98 6.88 3.50 2.85 2.25 6.24 4.81 5.00 9.69
0.2 0.014 37.74 19.02 33.80 16.27 1.08 0.96 2.10 2.89 4.85 6.01 8.85 7.02 3.61 2.99 2.65 7.02 5.22 5.11 10.20
0.2 0.016 37.03 18.79 34.04 16.28 0.84 0.71 1.91 2.77 4.78 5.94 8.76 7.12 3.70 3.04 2.87 7.47 5.41 5.16 10.53
0.2 0.018 36.46 18.58 34.40 16.38 0.66 0.53 1.74 2.68 4.72 5.86 8.64 7.08 3.71 3.00 2.94 7.61 5.44 5.16 10.65
0.2 0.02 35.95 18.37 34.75 16.46 0.48 0.34 1.53 2.58 4.63 5.70 8.45 6.88 3.60 2.87 2.88 7.48 5.35 5.07 10.59
0.22 -0.006 39.81 21.24 37.47 17.96 4.08 2.23 2.91 2.86 4.23 6.42 8.23 5.55 2.36 1.86 1.39 3.20 3.47 2.17 4.31
0.22 -0.004 38.06 21.05 34.21 15.66 2.15 1.14 1.70 1.85 3.39 5.91 8.17 5.28 1.10 0.55 0.65 2.37 3.46 3.15 4.34
0.22 -0.002 36.99 20.27 32.70 15.64 2.47 1.72 2.17 2.30 3.96 6.57 9.22 6.53 1.51 0.71 0.96 2.91 4.36 4.86 5.16
0.22 0 35.60 18.75 31.19 15.01 2.36 1.71 2.28 2.69 4.44 6.86 9.85 7.64 2.42 1.25 1.05 3.24 4.46 5.69 7.03
0.22 0.002 35.19 18.11 31.12 14.79 2.09 1.43 2.30 2.97 4.75 6.67 9.77 7.37 2.88 1.68 0.58 3.11 3.92 5.10 6.52
0.22 0.004 36.46 18.30 30.93 14.62 1.76 1.27 2.26 2.96 4.79 6.45 9.54 6.88 2.92 1.79 0.33 3.22 3.55 4.87 6.62
0.22 0.006 38.58 19.01 31.21 14.84 1.49 1.27 2.23 2.96 4.81 6.27 9.29 6.58 2.97 1.88 0.44 3.61 3.38 4.61 7.30
0.22 0.008 40.45 19.87 32.28 15.77 1.75 1.55 2.49 3.17 4.99 6.27 9.16 6.61 3.19 2.25 1.04 4.44 3.72 4.65 8.20
0.22 0.01 40.08 19.83 33.20 16.32 1.69 1.52 2.51 3.20 5.05 6.22 9.06 6.75 3.39 2.65 1.79 5.49 4.37 4.85 9.14
0.22 0.012 38.58 19.30 33.67 16.32 1.31 1.19 2.28 3.02 4.94 6.09 8.94 6.91 3.53 2.90 2.38 6.47 4.94 5.03 9.84
0.22 0.014 37.54 18.95 33.85 16.26 1.01 0.89 2.05 2.85 4.83 5.99 8.82 7.05 3.64 3.01 2.72 7.16 5.29 5.13 10.30
0.22 0.016 36.88 18.74 34.12 16.30 0.79 0.66 1.87 2.75 4.77 5.92 8.74 7.12 3.72 3.04 2.91 7.54 5.43 5.17 10.58
0.22 0.018 36.33 18.53 34.49 16.41 0.62 0.48 1.69 2.66 4.70 5.82 8.60 7.04 3.69 2.98 2.94 7.60 5.43 5.14 10.65
0.22 0.02 35.83 18.31 34.82 16.47 0.43 0.29 1.48 2.55 4.60 5.66 8.39 6.81 3.56 2.83 2.85 7.42 5.32 5.04 10.56
0.24 -0.006 38.82 20.94 36.74 17.33 3.36 1.76 2.36 2.33 3.72 6.15 8.08 5.39 1.79 1.30 1.06 2.72 3.31 2.14 4.06
0.24 -0.004 37.58 20.77 33.28 15.15 1.87 0.99 1.58 1.76 3.36 5.91 8.31 5.30 0.93 0.38 0.61 2.43 3.77 3.74 4.53
0.24 -0.002 36.65 19.81 32.36 15.62 2.58 1.82 2.28 2.44 4.12 6.69 9.47 6.97 1.77 0.83 1.06 3.02 4.41 5.27 5.78
0.24 0 35.38 18.49 31.14 14.88 2.20 1.58 2.24 2.75 4.53 6.79 9.86 7.62 2.57 1.36 0.90 3.17 4.29 5.51 7.00
0.24 0.002 35.62 18.20 31.19 14.82 2.09 1.42 2.35 3.03 4.80 6.63 9.72 7.23 2.92 1.73 0.48 3.10 3.78 5.00 6.42
0.24 0.004 36.78 18.35 30.86 14.57 1.61 1.21 2.21 2.92 4.77 6.37 9.46 6.75 2.90 1.77 0.30 3.27 3.45 4.79 6.77
0.24 0.006 39.24 19.29 31.46 15.05 1.54 1.34 2.29 3.01 4.85 6.26 9.23 6.56 3.02 1.95 0.56 3.78 3.41 4.58 7.51
0.24 0.008 40.61 19.97 32.56 15.99 1.79 1.59 2.53 3.21 5.03 6.26 9.12 6.63 3.25 2.35 1.23 4.69 3.87 4.69 8.45
0.24 0.01 39.72 19.71 33.36 16.35 1.59 1.44 2.46 3.16 5.02 6.19 9.03 6.79 3.43 2.73 1.97 5.76 4.53 4.91 9.35
0.24 0.012 38.27 19.19 33.73 16.31 1.23 1.11 2.22 2.97 4.90 6.06 8.90 6.95 3.56 2.94 2.49 6.67 5.05 5.06 9.98
0.24 0.014 37.36 18.89 33.90 16.26 0.94 0.82 2.00 2.82 4.81 5.97 8.80 7.08 3.66 3.03 2.79 7.29 5.34 5.15 10.40
0.24 0.016 36.74 18.69 34.21 16.33 0.75 0.62 1.82 2.72 4.75 5.90 8.71 7.12 3.72 3.04 2.93 7.58 5.45 5.17 10.62
0.24 0.018 36.20 18.48 34.58 16.43 0.58 0.44 1.64 2.63 4.68 5.79 8.55 7.00 3.67 2.95 2.93 7.57 5.41 5.12 10.65
0.24 0.02 35.72 18.26 34.89 16.48 0.38 0.24 1.42 2.52 4.57 5.61 8.33 6.74 3.52 2.78 2.82 7.35 5.29 5.00 10.52
Table B3: Table showing all possible outcomes of the test statistic JTfor the causal relation between futures on corn and inflation at various
quantiles (on the columns) for all plausible pairs of the polynomial weights (θ1, θ2) (on the rows). The pairs yielding the maximum values of
JTat every quantile considered are indicated in bold.
Quantiles
θ1θ20.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
-0.07 -0.18 48.63 25.79 42.37 19.37 2.23 -0.16 -0.28 0.57 1.75 1.91 3.38 1.10 0.02 -0.34 0.07 1.13 3.02 3.94 8.33
-0.07 -0.15 48.71 25.76 42.63 19.52 2.22 -0.17 -0.29 0.55 1.72 1.83 3.25 1.00 -0.06 -0.36 -0.01 0.98 2.78 3.55 7.67
-0.07 -0.12 47.76 25.11 42.17 19.18 2.13 -0.15 -0.30 0.62 1.81 1.87 3.26 1.00 -0.14 -0.41 -0.16 0.76 2.44 3.06 7.14
-0.07 -0.09 47.19 24.60 41.46 18.71 2.11 -0.08 -0.26 0.78 1.99 1.99 3.35 1.00 -0.27 -0.42 -0.34 0.55 1.96 2.39 6.57
-0.07 -0.06 49.78 25.87 41.86 19.09 2.40 0.10 -0.07 1.06 2.28 2.17 3.41 0.98 -0.46 -0.36 -0.52 0.52 1.38 1.61 5.77
-0.07 -0.03 56.77 28.10 43.48 19.91 3.17 0.12 0.28 1.56 2.85 2.53 3.63 1.30 -0.33 -0.52 -0.74 0.79 1.19 1.88 5.46
-0.07 0 49.44 23.90 38.40 17.28 3.16 0.93 1.38 3.78 5.50 4.95 6.05 3.91 0.77 -0.28 -0.49 2.05 2.00 4.44 9.15
-0.07 0.03 51.59 23.46 39.00 18.20 1.38 0.26 1.10 2.34 4.29 5.13 6.84 4.92 0.97 0.42 0.40 2.05 1.94 1.79 6.32
-0.07 0.06 61.51 28.04 48.26 22.97 2.00 0.35 0.92 2.79 4.95 5.21 7.19 4.61 0.88 0.09 -0.02 2.41 2.41 2.29 6.72
-0.07 0.09 64.98 30.74 52.01 25.12 2.21 0.07 0.34 2.50 4.64 4.17 6.09 3.58 0.39 -0.34 -0.36 2.04 2.18 2.16 6.28
-0.07 0.12 65.33 31.16 52.45 25.41 2.26 0.03 0.26 2.44 4.58 4.00 5.90 3.41 0.30 -0.41 -0.41 1.96 2.12 2.10 6.19
-0.04 -0.18 48.82 25.87 42.52 19.47 2.24 -0.17 -0.29 0.55 1.72 1.87 3.33 1.07 0.00 -0.34 0.06 1.09 2.96 3.84 8.13
-0.04 -0.15 48.57 25.66 42.59 19.49 2.21 -0.18 -0.30 0.54 1.71 1.81 3.22 0.99 -0.09 -0.37 -0.03 0.93 2.70 3.43 7.49
-0.04 -0.12 47.36 24.86 41.94 19.03 2.10 -0.16 -0.31 0.63 1.83 1.88 3.27 1.00 -0.16 -0.41 -0.20 0.70 2.32 2.90 6.99
-0.04 -0.09 47.40 24.70 41.44 18.71 2.12 -0.06 -0.25 0.81 2.03 2.01 3.37 1.00 -0.32 -0.40 -0.36 0.53 1.82 2.18 6.39
-0.04 -0.06 50.97 26.40 42.18 19.36 2.51 0.15 -0.01 1.14 2.36 2.23 3.45 1.01 -0.49 -0.34 -0.53 0.60 1.30 1.51 5.61
-0.04 -0.03 58.45 28.54 43.46 19.82 3.39 0.14 0.40 1.72 3.04 2.65 3.73 1.43 -0.24 -0.53 -0.74 1.00 1.26 2.10 5.50
-0.04 0 47.28 22.91 37.56 16.60 2.47 0.53 0.82 3.21 5.05 4.56 5.87 3.94 0.77 -0.28 -0.52 2.05 2.18 4.12 9.96
-0.04 0.03 51.87 23.60 39.18 18.29 1.40 0.27 1.12 2.36 4.31 5.16 6.88 4.94 0.98 0.43 0.40 2.07 1.94 1.79 6.34
-0.04 0.06 61.70 28.16 48.45 23.08 2.01 0.34 0.90 2.78 4.94 5.17 7.15 4.57 0.86 0.07 -0.03 2.40 2.40 2.29 6.71
-0.04 0.09 65.00 30.76 52.03 25.14 2.22 0.07 0.33 2.49 4.63 4.16 6.08 3.57 0.38 -0.34 -0.36 2.04 2.18 2.15 6.27
-0.04 0.12 65.34 31.16 52.46 25.41 2.26 0.03 0.26 2.44 4.57 4.00 5.90 3.41 0.30 -0.41 -0.41 1.96 2.12 2.10 6.19
-0.01 -0.18 48.93 25.91 42.63 19.55 2.26 -0.18 -0.29 0.53 1.70 1.83 3.28 1.03 -0.03 -0.34 0.05 1.05 2.89 3.73 7.92
-0.01 -0.15 48.29 25.49 42.47 19.41 2.18 -0.19 -0.32 0.54 1.72 1.80 3.21 0.98 -0.11 -0.38 -0.07 0.87 2.61 3.30 7.33
-0.01 -0.12 47.01 24.65 41.72 18.88 2.06 -0.16 -0.32 0.65 1.86 1.89 3.28 1.01 -0.19 -0.41 -0.24 0.63 2.20 2.74 6.86
-0.01 -0.09 47.89 24.97 41.56 18.80 2.16 -0.02 -0.23 0.85 2.07 2.04 3.40 0.99 -0.37 -0.37 -0.37 0.55 1.69 1.98 6.22
-0.01 -0.06 52.28 26.90 42.45 19.58 2.64 0.21 0.06 1.23 2.46 2.30 3.50 1.06 -0.50 -0.36 -0.52 0.71 1.28 1.52 5.51
-0.01 -0.03 59.59 28.78 43.17 19.57 3.57 0.16 0.51 1.91 3.26 2.82 3.86 1.62 -0.09 -0.44 -0.71 1.30 1.39 2.38 5.59
-0.01 0 52.52 25.30 39.02 17.45 2.31 0.49 0.55 2.71 4.63 4.41 5.98 4.17 0.73 -0.14 -0.24 2.29 2.87 3.94 9.67
-0.01 0.03 52.13 23.73 39.35 18.38 1.43 0.28 1.13 2.37 4.33 5.20 6.92 4.96 0.99 0.44 0.40 2.08 1.95 1.79 6.37
-0.01 0.06 61.88 28.28 48.63 23.18 2.02 0.33 0.88 2.77 4.93 5.13 7.11 4.53 0.84 0.05 -0.05 2.39 2.40 2.29 6.70
-0.01 0.09 65.02 30.78 52.06 25.15 2.22 0.07 0.33 2.49 4.63 4.15 6.07 3.56 0.38 -0.35 -0.36 2.03 2.18 2.15 6.27
-0.01 0.12 65.34 31.16 52.46 25.41 2.26 0.03 0.26 2.44 4.57 4.00 5.90 3.40 0.30 -0.41 -0.41 1.96 2.12 2.10 6.19
0.02 -0.18 48.92 25.90 42.68 19.58 2.26 -0.19 -0.30 0.51 1.67 1.80 3.23 1.00 -0.05 -0.35 0.03 1.01 2.82 3.61 7.72
0.02 -0.15 47.88 25.25 42.28 19.27 2.14 -0.19 -0.33 0.55 1.73 1.80 3.21 0.99 -0.13 -0.38 -0.11 0.80 2.51 3.16 7.19
0.02 -0.12 46.81 24.52 41.58 18.78 2.03 -0.14 -0.33 0.68 1.89 1.91 3.31 1.01 -0.24 -0.40 -0.26 0.59 2.08 2.56 6.72
0.02 -0.09 48.69 25.39 41.80 19.00 2.23 0.04 -0.19 0.91 2.14 2.09 3.45 1.01 -0.41 -0.33 -0.37 0.61 1.59 1.81 6.06
0.02 -0.06 53.61 27.29 42.56 19.67 2.80 0.26 0.14 1.34 2.58 2.39 3.57 1.11 -0.51 -0.42 -0.51 0.81 1.30 1.58 5.49
0.02 -0.03 59.65 28.57 42.46 19.13 3.66 0.22 0.64 2.15 3.53 3.09 4.08 1.91 0.13 -0.22 -0.62 1.73 1.62 2.78 5.77
0.02 0 50.57 24.38 38.01 16.78 1.93 0.18 0.31 2.12 3.90 4.34 5.67 4.11 0.72 -0.33 -0.41 1.74 2.36 3.57 8.89
0.02 0.03 52.38 23.85 39.53 18.47 1.45 0.29 1.15 2.39 4.35 5.23 6.96 4.98 1.00 0.45 0.40 2.10 1.96 1.79 6.39
0.02 0.06 62.06 28.39 48.81 23.27 2.03 0.32 0.85 2.77 4.93 5.10 7.07 4.49 0.83 0.03 -0.06 2.38 2.40 2.29 6.68
0.02 0.09 65.04 30.81 52.08 25.17 2.22 0.07 0.33 2.49 4.63 4.14 6.06 3.55 0.37 -0.35 -0.37 2.03 2.17 2.15 6.27
0.02 0.12 65.34 31.16 52.46 25.42 2.26 0.03 0.26 2.44 4.57 3.99 5.90 3.40 0.30 -0.41 -0.41 1.96 2.12 2.10 6.19
0.05 -0.18 48.77 25.81 42.65 19.56 2.24 -0.20 -0.32 0.50 1.66 1.77 3.20 0.98 -0.08 -0.36 0.01 0.96 2.75 3.49 7.53
0.05 -0.15 47.39 24.98 42.05 19.11 2.09 -0.20 -0.35 0.56 1.75 1.81 3.22 1.00 -0.15 -0.38 -0.15 0.74 2.41 3.02 7.07
0.05 -0.12 46.83 24.53 41.55 18.75 2.02 -0.12 -0.34 0.72 1.93 1.93 3.33 1.01 -0.29 -0.39 -0.28 0.57 1.97 2.38 6.58
0.05 -0.09 49.74 25.92 42.12 19.26 2.32 0.11 -0.13 1.00 2.22 2.16 3.52 1.05 -0.43 -0.28 -0.36 0.72 1.53 1.71 5.94
0.05 -0.06 54.85 27.54 42.45 19.59 2.99 0.29 0.23 1.45 2.70 2.48 3.66 1.15 -0.51 -0.53 -0.49 0.88 1.34 1.64 5.56
0.05 -0.03 58.28 27.77 41.22 18.47 3.66 0.32 0.78 2.40 3.82 3.50 4.44 2.33 0.43 0.09 -0.46 2.22 1.94 3.33 6.14
0.05 0 46.77 22.82 36.42 16.29 2.18 0.53 0.83 2.81 4.50 4.83 5.81 3.90 0.74 -0.48 -0.55 1.22 1.65 3.21 7.94
0.05 0.03 52.61 23.96 39.70 18.56 1.47 0.31 1.16 2.40 4.37 5.25 6.99 4.99 1.01 0.45 0.40 2.12 1.96 1.79 6.42
0.05 0.06 62.23 28.50 48.98 23.37 2.04 0.31 0.83 2.76 4.92 5.06 7.04 4.46 0.81 0.02 -0.07 2.37 2.39 2.29 6.67
0.05 0.09 65.06 30.83 52.11 25.18 2.22 0.06 0.32 2.48 4.62 4.14 6.05 3.54 0.37 -0.35 -0.37 2.03 2.17 2.15 6.26
0.05 0.12 65.34 31.17 52.46 25.42 2.26 0.03 0.26 2.44 4.57 3.99 5.90 3.40 0.30 -0.41 -0.41 1.96 2.12 2.10 6.19
0.08 -0.18 48.46 25.63 42.55 19.48 2.21 -0.21 -0.33 0.49 1.66 1.76 3.18 0.97 -0.10 -0.36 -0.02 0.90 2.67 3.37 7.37
0.08 -0.15 46.93 24.72 41.83 18.96 2.04 -0.20 -0.37 0.58 1.78 1.83 3.24 1.01 -0.17 -0.38 -0.18 0.67 2.30 2.88 6.96
0.08 -0.12 47.14 24.69 41.65 18.82 2.02 -0.08 -0.33 0.76 1.98 1.96 3.37 1.01 -0.34 -0.37 -0.28 0.59 1.86 2.21 6.44
0.08 -0.09 50.96 26.46 42.42 19.53 2.43 0.20 -0.05 1.09 2.33 2.25 3.59 1.10 -0.43 -0.25 -0.33 0.87 1.54 1.72 5.88
0.08 -0.06 55.89 27.63 42.11 19.34 3.18 0.31 0.32 1.55 2.81 2.56 3.76 1.20 -0.47 -0.60 -0.45 0.99 1.44 1.73 5.65
0.08 -0.03 55.71 26.47 39.57 17.67 3.56 0.45 0.91 2.64 4.07 3.99 4.90 2.80 0.75 0.44 -0.33 2.55 2.23 3.87 6.74
0.08 0 43.96 21.08 37.39 16.41 1.63 0.43 0.82 3.04 4.87 5.02 6.11 3.95 0.58 -0.58 -0.39 1.50 1.47 2.50 6.74
0.08 0.03 52.84 24.06 39.88 18.65 1.49 0.32 1.18 2.42 4.40 5.28 7.03 5.01 1.02 0.46 0.39 2.13 1.97 1.79 6.44
0.08 0.06 62.39 28.60 49.15 23.46 2.04 0.30 0.81 2.75 4.91 5.02 7.00 4.42 0.79 0.00 -0.09 2.36 2.39 2.29 6.65
0.08 0.09 65.08 30.85 52.13 25.20 2.22 0.06 0.32 2.48 4.62 4.13 6.05 3.54 0.36 -0.36 -0.37 2.02 2.17 2.15 6.26
0.08 0.12 65.34 31.17 52.47 25.42 2.26 0.03 0.26 2.44 4.57 3.99 5.90 3.40 0.30 -0.41 -0.41 1.95 2.11 2.10 6.19
0.11 -0.18 48.00 25.39 42.37 19.35 2.17 -0.22 -0.35 0.50 1.67 1.76 3.17 0.98 -0.11 -0.36 -0.06 0.84 2.58 3.25 7.25
0.11 -0.15 46.60 24.54 41.69 18.85 1.99 -0.18 -0.38 0.61 1.81 1.85 3.26 1.02 -0.21 -0.38 -0.21 0.63 2.20 2.73 6.85
0.11 -0.12 47.75 25.02 41.89 18.99 2.06 -0.02 -0.30 0.82 2.04 2.01 3.43 1.02 -0.37 -0.33 -0.27 0.67 1.78 2.06 6.33
0.11 -0.09 52.17 26.92 42.59 19.70 2.55 0.29 0.05 1.20 2.44 2.36 3.67 1.16 -0.45 -0.25 -0.28 1.01 1.59 1.78 5.89
0.11 -0.06 56.71 27.60 41.66 18.98 3.36 0.29 0.37 1.61 2.89 2.62 3.85 1.27 -0.39 -0.54 -0.37 1.21 1.61 1.91 5.71
0.11 -0.03 52.85 25.07 37.99 16.93 3.43 0.59 1.00 2.80 4.22 4.37 5.31 3.16 0.91 0.67 -0.29 2.56 2.40 4.23 7.49
0.11 0 43.87 21.09 38.24 16.76 1.30 0.27 0.70 2.94 4.84 5.13 6.45 4.16 0.87 -0.30 0.09 2.27 2.54 2.06 6.12
0.11 0.03 53.06 24.16 40.06 18.74 1.51 0.33 1.19 2.44 4.42 5.31 7.07 5.02 1.03 0.46 0.39 2.15 1.98 1.80 6.47
0.11 0.06 62.54 28.71 49.31 23.55 2.05 0.29 0.78 2.74 4.90 4.98 6.96 4.38 0.77 -0.01 -0.10 2.35 2.38 2.29 6.64
0.11 0.09 65.10 30.86 52.15 25.21 2.23 0.06 0.31 2.48 4.62 4.12 6.04 3.53 0.36 -0.36 -0.37 2.02 2.17 2.14 6.25
0.11 0.12 65.35 31.17 52.47 25.42 2.26 0.02 0.26 2.44 4.57 3.99 5.89 3.40 0.30 -0.41 -0.41 1.95 2.11 2.10 6.19
0.14 -0.18 47.46 25.10 42.14 19.19 2.11 -0.22 -0.37 0.51 1.68 1.77 3.18 1.00 -0.13 -0.36 -0.10 0.78 2.49 3.13 7.14
0.14 -0.15 46.49 24.48 41.67 18.82 1.96 -0.16 -0.39 0.65 1.86 1.87 3.29 1.02 -0.26 -0.38 -0.22 0.61 2.10 2.57 6.74
0.14 -0.12 48.65 25.49 42.22 19.25 2.12 0.06 -0.25 0.90 2.13 2.08 3.52 1.07 -0.38 -0.27 -0.24 0.80 1.73 1.96 6.26
0.14 -0.09 53.19 27.19 42.50 19.70 2.69 0.37 0.14 1.32 2.57 2.47 3.76 1.21 -0.47 -0.30 -0.23 1.12 1.66 1.83 5.99
0.14 -0.06 57.29 27.47 41.19 18.56 3.48 0.22 0.36 1.63 2.93 2.65 3.90 1.36 -0.30 -0.38 -0.31 1.48 1.77 2.12 5.69
0.14 -0.03 50.28 23.81 36.74 16.33 3.27 0.63 0.99 2.78 4.18 4.46 5.46 3.23 0.86 0.62 -0.30 2.37 2.56 4.44 8.16
0.14 0 43.72 21.02 37.63 16.46 0.98 0.12 0.63 2.74 4.64 5.06 6.55 4.14 1.07 -0.11 0.29 2.54 3.25 2.17 6.04
0.14 0.03 53.28 24.25 40.25 18.83 1.52 0.34 1.20 2.46 4.45 5.34 7.11 5.04 1.04 0.46 0.38 2.17 2.00 1.81 6.49
0.14 0.06 62.69 28.81 49.46 23.64 2.06 0.28 0.76 2.73 4.89 4.95 6.92 4.34 0.75 -0.03 -0.11 2.34 2.37 2.29 6.63
0.14 0.09 65.11 30.88 52.17 25.22 2.23 0.06 0.31 2.48 4.61 4.11 6.03 3.52 0.36 -0.36 -0.37 2.01 2.16 2.14 6.25
0.14 0.12 65.35 31.17 52.47 25.42 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.30 -0.41 -0.41 1.95 2.11 2.10 6.19
0.17 -0.18 46.93 24.82 41.93 19.04 2.05 -0.22 -0.39 0.53 1.71 1.78 3.20 1.02 -0.15 -0.35 -0.13 0.72 2.39 3.01 7.06
0.17 -0.15 46.66 24.58 41.78 18.88 1.94 -0.12 -0.38 0.69 1.91 1.90 3.33 1.02 -0.31 -0.37 -0.21 0.63 2.01 2.42 6.64
0.17 -0.12 49.77 26.02 42.57 19.53 2.21 0.16 -0.17 1.00 2.24 2.19 3.64 1.15 -0.36 -0.20 -0.19 0.98 1.76 1.97 6.25
0.17 -0.09 53.80 27.19 42.07 19.48 2.83 0.45 0.25 1.43 2.70 2.57 3.87 1.27 -0.46 -0.35 -0.14 1.21 1.79 1.90 6.15
0.17 -0.06 57.46 27.23 40.70 18.08 3.50 0.13 0.30 1.63 2.95 2.61 3.87 1.44 -0.25 -0.24 -0.31 1.65 1.78 2.25 5.59
0.17 -0.03 47.62 22.50 35.59 15.78 3.06 0.56 0.87 2.61 3.96 4.29 5.36 3.04 0.71 0.39 -0.27 2.21 2.80 4.59 8.57
0.17 0 43.66 20.74 36.92 16.10 0.79 -0.04 0.54 2.50 4.38 4.88 6.46 3.93 1.12 0.03 0.35 2.58 3.53 2.52 6.22
0.17 0.03 53.50 24.34 40.44 18.93 1.54 0.35 1.21 2.48 4.48 5.37 7.15 5.05 1.05 0.47 0.38 2.18 2.01 1.82 6.52
0.17 0.06 62.83 28.91 49.61 23.72 2.06 0.27 0.74 2.72 4.88 4.91 6.88 4.31 0.74 -0.05 -0.13 2.32 2.37 2.28 6.61
0.17 0.09 65.13 30.90 52.18 25.23 2.23 0.06 0.31 2.47 4.61 4.11 6.02 3.51 0.35 -0.36 -0.38 2.01 2.16 2.14 6.25
0.17 0.12 65.35 31.18 52.47 25.42 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.29 -0.41 -0.41 1.95 2.11 2.10 6.19
0.2 -0.18 46.52 24.61 41.80 18.93 1.99 -0.20 -0.40 0.56 1.75 1.81 3.22 1.03 -0.18 -0.35 -0.16 0.67 2.30 2.88 6.97
0.2 -0.15 47.14 24.85 42.02 19.03 1.95 -0.06 -0.36 0.75 1.97 1.95 3.39 1.02 -0.35 -0.34 -0.20 0.71 1.94 2.29 6.57
0.2 -0.12 50.93 26.50 42.81 19.75 2.32 0.27 -0.06 1.12 2.37 2.33 3.77 1.25 -0.33 -0.14 -0.13 1.19 1.85 2.06 6.31
0.2 -0.09 53.96 26.95 41.35 19.05 2.96 0.51 0.33 1.53 2.80 2.66 4.01 1.35 -0.41 -0.34 0.00 1.36 2.02 2.05 6.30
0.2 -0.06 57.09 26.85 40.20 17.58 3.45 0.03 0.20 1.61 2.96 2.52 3.73 1.45 -0.28 -0.21 -0.37 1.64 1.67 2.29 5.52
0.2 -0.03 45.64 21.56 34.93 15.64 3.02 0.53 0.82 2.52 3.81 4.12 5.18 2.73 0.49 0.21 -0.21 2.04 2.85 4.26 8.65
0.2 0 43.96 20.52 36.40 15.91 0.74 -0.12 0.49 2.33 4.17 4.73 6.35 3.76 1.11 0.16 0.41 2.61 3.69 2.85 6.44
0.2 0.03 53.71 24.42 40.64 19.03 1.55 0.36 1.22 2.50 4.51 5.39 7.18 5.06 1.06 0.47 0.37 2.20 2.03 1.83 6.54
0.2 0.06 62.97 29.00 49.75 23.80 2.07 0.26 0.72 2.71 4.87 4.87 6.85 4.27 0.72 -0.06 -0.14 2.31 2.36 2.28 6.60
0.2 0.09 65.14 30.91 52.20 25.24 2.23 0.05 0.30 2.47 4.61 4.10 6.01 3.51 0.35 -0.37 -0.38 2.01 2.16 2.14 6.24
0.2 0.12 65.35 31.18 52.48 25.43 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.29 -0.41 -0.41 1.95 2.11 2.09 6.19
0.23 -0.18 46.34 24.53 41.79 18.90 1.94 -0.18 -0.41 0.59 1.79 1.83 3.25 1.03 -0.22 -0.34 -0.17 0.65 2.22 2.75 6.89
0.23 -0.15 47.92 25.26 42.36 19.27 1.98 0.02 -0.32 0.83 2.06 2.03 3.48 1.07 -0.36 -0.29 -0.17 0.84 1.91 2.20 6.53
0.23 -0.12 51.96 26.81 42.81 19.83 2.44 0.38 0.05 1.24 2.50 2.47 3.90 1.36 -0.31 -0.11 -0.04 1.37 1.97 2.16 6.44
0.23 -0.09 53.89 26.56 40.57 18.53 3.08 0.52 0.37 1.56 2.85 2.71 4.13 1.42 -0.36 -0.27 0.12 1.53 2.21 2.16 6.43
0.23 -0.06 56.14 26.35 39.74 17.13 3.33 -0.05 0.11 1.57 2.94 2.43 3.58 1.42 -0.34 -0.26 -0.44 1.45 1.55 2.33 5.60
0.23 -0.03 44.65 21.17 35.04 16.06 3.18 0.68 0.87 2.57 3.77 3.90 4.87 2.28 0.23 -0.10 -0.44 1.49 2.38 3.24 8.08
0.23 0 44.58 20.44 36.17 15.88 0.77 -0.13 0.48 2.21 4.03 4.61 6.25 3.66 1.09 0.27 0.49 2.64 3.79 3.09 6.61
0.23 0.03 53.93 24.51 40.84 19.13 1.57 0.37 1.23 2.52 4.53 5.42 7.22 5.08 1.07 0.46 0.37 2.22 2.04 1.85 6.56
0.23 0.06 63.10 29.09 49.89 23.87 2.08 0.25 0.70 2.70 4.86 4.84 6.81 4.24 0.70 -0.08 -0.15 2.30 2.36 2.28 6.58
0.23 0.09 65.15 30.93 52.22 25.25 2.23 0.05 0.30 2.47 4.61 4.09 6.01 3.50 0.35 -0.37 -0.38 2.01 2.16 2.13 6.24
0.23 0.12 65.35 31.18 52.48 25.43 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.29 -0.41 -0.41 1.95 2.11 2.09 6.19
0.26 -0.18 46.46 24.61 41.92 18.96 1.91 -0.14 -0.41 0.64 1.84 1.87 3.29 1.03 -0.27 -0.34 -0.16 0.67 2.15 2.62 6.81
0.26 -0.15 48.94 25.75 42.73 19.55 2.05 0.12 -0.25 0.94 2.18 2.14 3.61 1.15 -0.33 -0.20 -0.12 1.05 1.93 2.19 6.55
0.26 -0.12 52.63 26.87 42.47 19.68 2.56 0.48 0.17 1.35 2.63 2.60 4.04 1.47 -0.29 -0.09 0.07 1.52 2.14 2.26 6.62
0.26 -0.09 53.69 26.07 39.82 17.98 3.17 0.45 0.35 1.53 2.82 2.70 4.19 1.47 -0.36 -0.20 0.11 1.63 2.20 2.10 6.50
0.26 -0.06 54.59 25.69 39.30 16.71 3.12 -0.13 0.01 1.48 2.83 2.43 3.52 1.41 -0.36 -0.27 -0.57 1.14 1.46 2.42 5.88
0.26 -0.03 44.15 21.17 35.63 16.70 3.59 0.99 1.15 2.75 3.85 3.92 4.77 2.43 0.48 -0.22 -0.66 0.98 1.78 2.27 6.97
0.26 0 45.29 20.46 36.14 15.94 0.83 -0.10 0.50 2.14 3.93 4.53 6.16 3.60 1.06 0.36 0.55 2.67 3.82 3.23 6.69
0.26 0.03 54.14 24.59 41.05 19.23 1.59 0.38 1.24 2.54 4.56 5.45 7.25 5.09 1.07 0.46 0.36 2.24 2.06 1.86 6.59
0.26 0.06 63.22 29.18 50.02 23.95 2.08 0.24 0.68 2.69 4.85 4.81 6.77 4.20 0.69 -0.09 -0.16 2.29 2.35 2.27 6.57
0.26 0.09 65.16 30.94 52.23 25.26 2.24 0.05 0.30 2.47 4.60 4.09 6.00 3.50 0.34 -0.37 -0.38 2.00 2.15 2.13 6.24
0.26 0.12 65.35 31.18 52.48 25.43 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.29 -0.41 -0.41 1.95 2.11 2.09 6.18
0.29 -0.18 46.90 24.86 42.19 19.12 1.90 -0.08 -0.39 0.69 1.91 1.91 3.34 1.03 -0.31 -0.33 -0.14 0.74 2.09 2.50 6.76
0.29 -0.15 50.05 26.23 43.03 19.80 2.14 0.24 -0.14 1.06 2.32 2.29 3.78 1.28 -0.27 -0.11 -0.04 1.29 2.03 2.28 6.63
0.29 -0.12 52.76 26.63 41.73 19.27 2.68 0.59 0.29 1.46 2.76 2.73 4.21 1.59 -0.24 -0.05 0.24 1.68 2.39 2.40 6.79
0.29 -0.09 53.21 25.42 39.01 17.34 3.19 0.31 0.27 1.46 2.75 2.66 4.18 1.49 -0.38 -0.16 -0.07 1.66 1.96 1.86 6.45
0.29 -0.06 52.37 24.76 38.60 16.22 2.81 -0.17 -0.11 1.34 2.66 2.47 3.57 1.44 -0.36 -0.28 -0.68 0.93 1.54 2.65 6.34
0.29 -0.03 42.41 20.55 35.93 17.10 3.82 1.21 1.45 2.95 4.02 4.09 4.91 2.97 0.92 -0.27 -0.77 0.53 1.16 1.46 5.72
0.29 0 45.88 20.52 36.19 16.03 0.89 -0.07 0.52 2.09 3.86 4.48 6.10 3.59 1.04 0.42 0.59 2.68 3.80 3.30 6.70
0.29 0.03 54.37 24.68 41.27 19.33 1.60 0.39 1.24 2.57 4.59 5.47 7.29 5.10 1.08 0.46 0.35 2.26 2.08 1.88 6.61
0.29 0.06 63.33 29.27 50.14 24.02 2.09 0.23 0.66 2.68 4.84 4.77 6.74 4.17 0.67 -0.10 -0.17 2.28 2.34 2.27 6.55
0.29 0.09 65.18 30.96 52.25 25.27 2.24 0.05 0.30 2.47 4.60 4.08 5.99 3.49 0.34 -0.37 -0.38 2.00 2.15 2.13 6.23
0.29 0.12 65.36 31.18 52.48 25.43 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.40 0.29 -0.41 -0.41 1.95 2.11 2.09 6.18
0.32 -0.18 47.65 25.26 42.56 19.36 1.93 0.00 -0.35 0.77 2.00 1.98 3.43 1.06 -0.34 -0.29 -0.11 0.87 2.07 2.41 6.75
0.32 -0.15 51.07 26.56 43.12 19.93 2.25 0.36 -0.02 1.19 2.47 2.46 3.96 1.43 -0.21 -0.02 0.06 1.53 2.18 2.42 6.77
0.32 -0.12 52.39 26.15 40.73 18.68 2.79 0.68 0.40 1.55 2.85 2.84 4.36 1.69 -0.20 -0.01 0.40 1.79 2.62 2.50 6.93
0.32 -0.09 52.62 24.73 38.28 16.73 3.19 0.14 0.17 1.40 2.71 2.59 4.10 1.48 -0.40 -0.16 -0.34 1.62 1.64 1.58 6.28
0.32 -0.06 49.81 23.63 37.53 15.60 2.49 -0.14 -0.17 1.25 2.53 2.50 3.69 1.54 -0.33 -0.29 -0.66 0.96 1.89 3.13 6.82
0.32 -0.03 39.26 19.09 36.09 17.33 3.54 1.23 1.50 3.10 4.22 4.22 5.09 3.21 1.13 -0.39 -0.92 0.11 0.64 0.77 4.54
0.32 0 46.27 20.56 36.26 16.13 0.93 -0.05 0.55 2.05 3.80 4.46 6.06 3.62 1.03 0.46 0.61 2.68 3.75 3.33 6.65
0.32 0.03 54.59 24.77 41.49 19.44 1.62 0.39 1.25 2.59 4.62 5.49 7.32 5.11 1.09 0.46 0.34 2.28 2.10 1.90 6.63
0.32 0.06 63.45 29.35 50.26 24.09 2.09 0.23 0.64 2.67 4.83 4.74 6.70 4.14 0.65 -0.12 -0.18 2.27 2.33 2.27 6.54
0.32 0.09 65.19 30.97 52.26 25.28 2.24 0.05 0.29 2.47 4.60 4.08 5.99 3.49 0.34 -0.38 -0.39 2.00 2.15 2.13 6.23
0.32 0.12 65.36 31.19 52.48 25.43 2.26 0.02 0.25 2.44 4.57 3.99 5.89 3.39 0.29 -0.41 -0.41 1.95 2.11 2.09 6.18
C Unrestricted Quantile-MIDAS
C.1 Gold
Figure C1: Test statistic JTwith respect to various quantiles. The red dashed line marks the aggre-
gated critical value at α= 0.01 and the associated standard normal critical value of 2.57.
Figure C2: Unrestricted Quantile MIDAS Conditional Quantiles and actual data
Table C1: Table showing Unrestricted Quantile-MIDAS Estimation Results
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.164
[1.000]
-0.002
[0.122]
0.009***
[0.000]
0.108***
[0.000]
0.256***
[0.000]
β1:yt−10.642
[1.000]
0.638***
[0.000]
0.617***
[0.000]
0.528
[0.999]
0.515
[1.000]
β2:yt−2-0.436
[1.000]
0.031***
[0.000]
0.203***
[0.000]
0.301
[0.999]
0.686
[1.000]
β3:|yt−1|0.107
[1.000]
-0.141***
[0.000]
-0.341
[0.999]
-0.195***
[0.000]
-0.229
[1.000]
β4:|yt−2|0.178
[1.000]
-0.018***
[0.000]
0.314
[1.000]
0.271***
[0.000]
0.024
[1.000]
β5:m= 1 0.809
[1.000]
1.826***
[0.000]
1.179
[1.000]
2.025
[0.999]
-3.077
[0.999]
β5:m= 2 1.622
[1.000]
-1.202 .
[0.015]
-0.137
[0.999]
-1.727
[0.999]
-2.131
[1.000]
β5:m= 3 -0.472
[1.000]
-0.117
[0.438]
-0.371
[0.999]
-0.091
[1.000]
-3.281
[0.999]
β5:m= 4 -4.838
[1.000]
-6.040***
[0.000]
-4.346
[0.999]
-3.219
[0.999]
1.272
[1.000]
β5:m= 5 -2.458
[1.000]
0.876***
[0.000]
1.956
[0.999]
-1.403
[1.000]
0.496
[1.000]
β5:m= 6 2.578
[1.000]
4.535***
[0.000]
2.283
[0.999]
0.044
[1.000]
1.735
[0.999]
β5:m= 7 1.723
[1.000]
1.042***
[0.000]
1.759
[0.999]
4.036
[0.999]
4.778
[0.999]
β5:m= 8 -0.344
[1.000]
-2.520***
[0.000]
-0.121
[1.000]
-0.265
[1.000]
3.354
[1.000]
β5:m= 9 2.569
[1.000]
-1.160***
[0.000]
0.949
[0.999]
-2.274
[0.999]
-3.894
[0.999]
β5:m= 10 5.150
[0.999]
3.365***
[0.000]
1.806
[0.999]
2.184
[0.999]
-1.364
1.000]
β5:m= 11 3.441
[1.000]
0.695***
[0.000]
1.889
[0.999]
4.540
[0.999]
10.083
[0.999]
β5:m= 12 -1.960
[1.000]
2.019***
[0.000]
0.841
[0.999]
-0.974
[0.999]
4.531
[0.999]
β5:m= 13 2.903
[1.000]
-0.065
[0.640]
0.474
[0.999]
-0.118
[1.000]
0.277
[1.000]
β5:m= 14 -4.065
[1.000]
1.542***
[0.000]
-1.341
[0.999]
-0.487
[1.000]
-0.546
[1.000]
β5:m= 15 -3.869
[1.000]
1.781***
[0.000]
-0.017
[1.000]
-0.593
[1.000]
-1.061
[1.000]
β5:m= 16 -1.072
[1.000]
-0.887 .
[0.041]
1.336
[0.999]
2.308
[0.999]
6.461
[0.999]
β5:m= 17 -1.849
[1.000]
-2.288***
[0.000]
-1.511
[0.999]
1.875
[0.999]
2.561
[0.999]
β5:m= 18 -0.044
[1.000]
1.969***
[0.000]
-1.407
[0.999]
-1.588
[0.999]
-0.393
[1.000]
β5:m= 19 3.731
[0.999]
-1.159***
[0.000]
-1.142
[0.999]
-0.848
[0.999]
-0.884
[1.000]
β5:m= 20 3.986
[0.997]
0.213 .
[0.042]
-2.249
[0.999]
-2.578
[0.999]
-2.974
[0.999]
β5:m= 21 2.729
[0.999]
2.572***
[0.000]
2.228
[0.999]
1.223
[0.999]
-2.078
[0.999]
Pseudo R Squared 0.29 0.27 0.30 0.31 0.38
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
C.2 Crude Oil
Figure C3: Unrestricted Quantile MIDAS Conditional Quantiles and actual data
Figure C4: Test statistic JTwith respect to various quantiles. The red dashed line marks the aggre-
gated critical value at α= 0.01 and the associated standard normal critical value of 2.57.
Table C2: Table showing Unrestricted Quantile-MIDAS Estimation Results
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.126***
[0.000]
-0.008***
[0.000]
0.036***
[0.000]
0.139***
[0.000]
0.321
[1.000]
β1:yt−10.362
[1.000]
0.446***
[0.000]
0.245***
[0.000]
0.213***
[0.000]
0.258
[1.000]
β2:yt−2-0.039
[1.000]
0.194***
[0.000]
0.442***
[0.000]
0.472***
[0.000]
0.241
[1.000]
beta3:|yt−1|0.124
[1.000]
-0.041***
[0.000]
-0.066***
[0.000]
0.146***
[0.000]
0.327
[1.000]
β4:|yt−2|-0.155
[1.000]
-0.152***
[0.000]
0.056***
[1.000]
-0.178***
[0.000]
-0.404
[1.000]
β5:m= 1 2.621
[0.999]
1.865***
[0.000]
1.276***
[0.000]
1.610
[0.999]
0.275
[0.999]
β5:m= 2 0.865
[0.999]
0.117***
[0.000]
1.229***
[0.000]
0.467
[0.999]
-0.257
[1.000]
β5:m= 3 2.573
[0.999]
1.512***
[0.000]
0.017
[0.884]
0.482 .
[0.078]
1.009
[0.999]
β5:m= 4 0.299
[1.000]
-0.265**
[0.001]
0.061
[0.656]
-1.221
[0.999]
-2.029
[1.000]
β5:m= 5 0.186
[1.000]
1.174***
[0.000]
1.207***
[0.000]
2.176
[0.999]
2.610
[1.000]
β5:m= 6 2.906
[0.999]
0.518***
[0.000]
-0.590***
[0.000]
0.779***
[0.000]
0.517
[0.999]
β5:m= 7 -0.249
[1.000]
0.869***
[0.000]
0.476***
[0.000]
-0.012
[0.271]
0.579
[0.999]
β5:m= 8 1.069
[1.000]
2.037***
[0.000]
1.031***
[0.000]
0.357
[1.000]
0.689
[1.000]
β5:m= 9 1.425
[1.000]
1.461***
[0.000]
1.890***
[0.000]
2.228***
[0.000]
2.429
[0.999]
β5:m= 10 3.074
[0.999]
2.561***
[0.000]
0.210 .
[0.074]
0.799
[0.999]
2.517
1.000]
β5:m= 11 2.270
[1.000]
1.841***
[0.000]
1.803***
[0.000]
1.689
[0.999]
2.656
[0.999]
β5:m= 12 -1.349
[0.999]
0.689***
[0.000]
1.401***
[0.000]
1.363
[0.999]
-0.092
[0.999]
β5:m= 13 0.463
[1.000]
1.821***
[0.640]
1.239***
[0.000]
0.934
[0.999]
2.476
[1.000]
β5:m= 14 0.475
[1.000]
1.189***
[0.000]
1.333***
[0.000]
0.884***
[0.000]
0.797
[1.000]
β5:m= 15 3.389
[0.999]
3.467***
[0.000]
1.587***
[0.000]
1.178
[0.999]
-1.214
[1.000]
β5:m= 16 2.518
[1.000]
1.962***
[0.041]
2.373***
[0.000]
2.059***
[0.000]
2.059
[0.999]
β5:m= 17 -0.269
[1.000]
1.360***
[0.000]
1.001***
[0.000]
1.631
[0.999]
1.170
[0.999]
β5:m= 18 0.384
[1.000]
-1.302***
[0.000]
-1.423***
[0.000]
0.064
[0.999]
2.517
[1.000]
β5:m= 19 0.259
[1.000]
1.065***
[0.000]
1.056***
[0.000]
0.894
[0.999]
3.929
[1.000]
β5:m= 20 2.484
[1.000]
2.786***
[0.042]
2.586***
[0.000]
2.849
[0.999]
2.101
[0.999]
β5:m= 21 1.935
[0.999]
0.005
[0.776]
0.269
[0.156]
1.615
[0.999]
3.822
[0.999]
Pseudo R Squared 0.46 0.40 0.36 0.42 0.50
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
C.3 Corn
Figure C5: Unrestricted Quantile MIDAS Conditional Quantiles and actual data
Figure C6: Test statistic JTwith respect to various quantiles. The red dashed line marks the aggre-
gated critical value at α= 0.01 and the associated standard normal critical value of 2.57.
Table C3: Table showing Unrestricted Quantile-MIDAS Estimation Results
Quantiles
yt: Inflation 0.05 0.25 0.50 0.75 0.95
β0: intercept -0.176
[0.983]
0.013
[1.000]
0.035
[0.999]
0.157
[0.643]
0.285
[0.253]
β1:yt−10.420
[1.000]
0.586
[1.000]
0.552
[1.000]
0.591
[1.000]
0.337
[1.000]
β2:yt−2-0.007
[1.000]
-0.036
[1.000]
0.365
[1.000]
0.248
[1.000]
0.225
[1.000]
beta3:|yt−1|0.016
[1.000]
-0.166
[1.000]
-0.155
[0.998]
-0.222
[1.000]
0.037
[1.000]
β4:|yt−2|0.076
[1.000]
-0.092
[1.000]
0.005
[0.999]
0.174
[1.000]
0.196
[0.987]
β5:m= 1 -0.036
[1.000]
-0.011
[1.000]
-0.025
[0.999]
-0.025
[1.000]
-0.058
[0.929]
β5:m= 2 -0.012
[0.999]
-0.009
[1.000]
-0.018
[0.999]
-0.029
[1.000]
-0.071
[1.000]
β5:m= 3 0.069
[0.999]
0.018
[1.000]
0.033
[1.000]
0.021
[1.000]
0.022
[1.000]
β5:m= 4 0.141
[0.999]
0.114
[1.000]
0.080
[1.000]
0.035
[1.000]
0.045
[1.000]
β5:m= 5 0.008
[1.000]
0.025
[1.000]
0.036
[1.000]
0.008
[1.000]
0.030
[1.000]
β5:m= 6 0.029
[0.999]
-0.008
[1.000]
0.003
[0.990]
-0.027
[0.999]
0.052
[1.000]
β5:m= 7 0.073
[1.000]
0.043
[1.000]
0.027
[1.000]
-0.007
[1.000]
0.039
[1.000]
β5:m= 8 0.091
[1.000]
0.075
[1.000]
0.044
[1.000]
-0.012
[0.999]
0.046
[1.000]
β5:m= 9 0.032
[1.000]
0.038
[1.000]
-0.002
[0.998]
-0.041
[1.000]
-0.035
[1.000]
β5:m= 10 0.054
[0.999]
0.086
[1.000]
0.042
[1.000]
-0.013
[1.000]
0.020
[1.000]
β5:m= 11 0.064
[1.000]
0.118
[1.000]
0.039
[1.000]
0.009
[1.000]
0.044
[1.000]
β5:m= 12 0.092
[0.999]
0.133
[1.000]
0.031
[1.000]
-0.031
[1.000]
-0.005
[1.000]
β5:m= 13 0.077
[0.999]
0.127
[1.000]
0.007
[1.000]
-0.049
[1.000]
-0.014
[1.000]
β5:m= 14 -0.036
[1.000]
0.111
[1.000]
0.016
[1.000]
-0.054
[1.000]
-0.029
[1.000]
β5:m= 15 -0.034
[0.999]
0.102
[1.000]
0.016
[1.000]
-0.078
[1.000]
-0.037
[1.000]
β5:m= 16 -0.033
[1.000]
0.068
[1.000]
0.022
[1.000]
-0.076
[1.000]
0.018
[1.000]
β5:m= 17 0.051
[1.000]
0.099
[1.000]
0.054
[1.000]
-0.054
[1.000]
0.017
[1.000]
β5:m= 18 -0.009
[1.000]
0.080
[1.000]
0.055
[1.000]
-0.036
[1.000]
0.056
[1.000]
β5:m= 19 0.039
[1.000]
0.094
[1.000]
0.055
[1.000]
0.006
[1.000]
0.087
[1.000]
β5:m= 20 0.059
[1.000]
0.080
[1.000]
0.033
[1.000]
-0.005
[0.999]
0.109
[1.000]
β5:m= 21 0.082
[0.999]
0.078
[1.000]
0.037
[1.000]
0.025
[1.000]
0.075
[1.000]
Pseudo R Squared 0.38 0.40 0.28 0.33 0.48
Significance Codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
D Nowcasting: Full Selection of Quantiles
D.1 Nowcasting at Quantiles: Causality from Gold
Table D1: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). , and indicate that ratios are significantly different from 1 at 1%, 5% and 10%,
according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
QAR(2) 1.971 3.342 4.219 4.743 4.911 4.888 4.746 5.501 4.196 3.819
QMIDAS 0.954*** 0.964*** 0.985*** 0.960*** 0.985*** 0.991*** 0.985*** 0.980*** 0.975*** 0.971***
QAVG 0.968*** 0.988*** 0.999*** 0.998*** 0.999*** 0.998*** 0.997*** 0.998*** 0.997*** 0.995***
τ0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
QAR(2) 3.389 2.928 2.453 1.969 1.495 1.051 0.650 0.313 0.085
QMIDAS 0.969*** 0.963** 0.952*** 0.946*** 0.943*** 0.942*** 0.944*** 0.981*** 0.978***
QAVG 0.990*** 0.989*** 0.987*** 0.988** 0.996*** 0.997*** 0.989*** 0.982*** 0.966***
Figure D1: Quantile Scores of QMDIDAS (solid blue line) vs QAVG model (dashed red line) over the
out-of-sample period.
D.2 Nowcasting at Quantiles: Causality from Crude Oil
Table D2: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). , and indicate that ratios are significantly different from 1 at 1%, 5% and 10%,
according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
QAR(2) 1.971 3.342 4.219 4.743 4.911 4.888 4.746 5.501 4.196 3.819
QMIDAS 0.799*** 0.861*** 0.831*** 0.852*** 0.892*** 0.851*** 0.885*** 0.857*** 0.864*** 0.864***
QAVG 0.815*** 0.827*** 0.838*** 0.836*** 0.847*** 0.860*** 0.863*** 0.865*** 0.864*** 0.866***
τ0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
QAR(2) 3.389 2.928 2.453 1.969 1.495 1.051 0.650 0.313 0.085
QMIDAS 0.868*** 0.892** 0.892*** 0.903*** 0.887*** 0.921*** 0.907*** 0.910*** 0.891***
QAVG 0.866*** 0.866*** 0.869*** 0.878** 0.891*** 0.901*** 0.913*** 0.936*** 0.955***
Figure D2: Quantile Scores of QMDIDAS (solid blue line) vs QAVG model (dashed red line) over the
out-of-sample period.
D.3 Nowcasting at Quantiles: Causality from Corn
Table D3: Table showing quantile scores for different quantiles (on the columns) and for three different
models (on the rows). , and indicate that ratios are significantly different from 1 at 1%, 5% and 10%,
according to the Diebold-Mariano test. The model in italic is our benchmark model.
τ0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
QAR(2) 1.971 3.342 4.219 4.743 4.911 4.888 4.746 5.501 4.196 3.819
QMIDAS 0.961*** 0.960*** 0.953*** 0.965*** 0.973*** 0.982*** 0.988*** 0.947*** 0.993*** 0.992***
QAVG 0.903*** 0.923*** 0.934*** 0.948*** 0.961*** 0.972*** 0.983*** 0.990*** 0.995*** 0.996***
τ0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
QAR(2) 3.389 2.928 2.453 1.969 1.495 1.051 0.650 0.313 0.085
QMIDAS 0.994*** 0.995** 0.992*** 0.983*** 0.978*** 0.979*** 0.972*** 0.981*** 0.943***
QAVG 0.998*** 0.999*** 0.999*** 0.999** 0.999*** 0.996*** 0.988*** 0.986*** 0.987***
Figure D3: Quantile Scores of QMDIDAS (solid blue line) vs QAVG model (dashed red line) over the
out-of-sample period.