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Central Bank Losses and Monetary Policy
Rules: A DSGE Investigation
Jonathan Benchimol†and André Fourçans‡
May 2019
Abstract
Central banks’ monetary policy rules being consistent with policy objec-
tives are a fundamental of applied monetary economics. We seek to deter-
mine, first, which of the central bank’s rules are most in line with the historical
data for the US economy and, second, what policy rule would work best to
assist the central bank in reaching its objectives via several loss function mea-
sures. We use Bayesian estimations to evaluate twelve monetary policy rules
from 1955 to 2017 and over three different sub-periods. We find that when
considering the central bank’s loss functions, the estimates often indicate the
superiority of NGDP level targeting rules, though Taylor-type rules lead to
nearly identical implications. However, the results suggest that various cen-
tral bank empirical rules, be they NGDP or Taylor type, are more appropriate
to achieve the central bank’s objectives for each type of period (stable, crisis,
recovery).
Keywords: Monetary policy, Monetary rule, Central bank loss, Nominal in-
come targeting.
JEL Codes: E52, E58, E32.
This paper does not necessarily reflect the views of the Bank of Israel. We thank the referees,
Lahcen Bounader, Makram El-Shagi, Johannes Pfeifer, Guy Segal, Volker Wieland, participants at
the CEPR Network on Macroeconomic Modelling and Model Comparison (MMCN) and Western
Economic Association International conferences, and participants at the Tel Aviv University Macro
Workshop, Henan University, Bank of England, and National Bank of Romania research seminars
for their valuable comments.
†Bank of Israel, Jerusalem, Israel. Corresponding author. Email: jonathan.benchimol@boi.org.il
‡ESSEC Business School and THEMA, Cergy Pontoise, France.
1
1 Introduction
Monetary economists generally contend that central bankers should follow policy
rules rather than use their own discretion when devising monetary policy. Debates
held during the 1970s and 1980s suggested nominal income targeting concepts,
even if they were not always presented as such.1The consensus on Taylor (1993)
rules increased during the last two decades.2However, criticism of such mone-
tary policy rules also increased,3especially during and after the Global Financial
Crisis4(GFC/ZLB), arguing that nominal income targeting could be a better way
to achieve the central banks’ objectives.
An interesting way to compare and evaluate different monetary policy pro-
posals and rules is to introduce them within the framework of a macroeconomic
Dynamic Stochastic General Equilibrium (DSGE) model. Because the dynamics
are so important and difficult to work through intuitively, such empirical models
can provide invaluable clarification of the matter (Taylor, 2013).
Our aim in this paper is to use the Smets and Wouters (2007) framework, the
well-known baseline DSGE model fitted for the US, to evaluate different monetary
policy rules and their consequences in terms of current and forecasted central bank
objectives.
These objectives may differ for various reasons, hence the need to analyze sev-
eral hypotheses regarding the current and forecasted preferences of the central
banker. Such an approach implies an analysis of the impact of policies on central
bank loss functions (Taylor and Wieland, 2012; Walsh, 2015).
Two main research strategies may be used to deal with these issues: a rather
common one using optimal monetary policy theory and an empirical one based
on historical economic dynamics. The latter appears better suited to capture the
real economic behavior of a central bank. Indeed, commitment, discretionary or
optimal monetary rules imply theoretically optimal behavior but do not neces-
sarily represent the actual behavior of central bankers. In addition, this theoretical
framework does not permit the analysis of empirical central bank losses over time.
Since central banks do not necessarily behave optimally, our empirical exercise
offers a more realistic framework regarding central bank behavior than the the-
oretical one. Finally, the implications of ad-hoc monetary policy rules are rarely
analyzed in terms of various central bank losses, be they current or forecasted, es-
pecially during different sample periods, which we consider in our analysis. Our
approach allows for such an analysis through a medium-scale DSGE model.
1See Friedman (1971), Meade (1978), and McCallum (1973, 1987).
2See Bernanke and Mishkin (1997), Svensson (1999), and Taylor (1999).
3See for instance Hall and Mankiw (1994), Frankel and Chinn (1995), McCallum and Nelson
(1999), and Rudebusch (2002a).
4Hendrickson (2012), Woodford (2012), Frankel (2014), Sumner (2014, 2015), Belongia and Ire-
land (2015), and McCallum (2015) for example.
3
The monetary policy rules we examine are of three types: Taylor-type rules,
nominal income growth rules, and nominal income level rules. There are four
Taylor-type rules, following (1) a structure à la Smets and Wouters (2007), where
the nominal interest rate responds to an inflation gap, an output gap and out-
put gap growth; (2) a structure à la Taylor (1993), where the nominal interest rate
responds to an inflation gap and an output gap; (3) a structure à la Galí (2015),
where the nominal interest rate responds to an inflation gap, an output gap and a
natural interest rate defined as the interest rate in the flexible-price economy; and
(4) a structure à la Garín et al. (2016), where the nominal interest rate responds to
an inflation gap and to output growth. There are also four nominal GDP (NGDP)
growth rules that replace the core functions of the Taylor-type rules with an NGDP
growth targeting function. Finally, our last four rules replace the core functions of
the Taylor-type rules with an NGDP level rule.
We apply Bayesian techniques to estimate our twelve DSGE models (each type
is composed of 4 structures) using US data. Note that this approach goes further
than the literature generally does. First, we consider a large set of monetary policy
rules. Second, our models are studied over different time periods. Third, we
add the analysis of central bank losses, current and forecasted, over these models
and periods. Fourth, the model structure we use (Smets and Wouters, 2007) is a
sophisticated medium-scale model.5We believe that our analysis and estimates
enrich the literature in an informative and innovative way.
Specifically, we estimate all of the parameters over several sample periods: the
overall available sample (1955-2017) and three sub-samples, each with different
economic environments and monetary policy styles, running from 1955 to 1985,
from 1985 to 2007 and from 2007 to 2017.
Monetary policy during the GFC/ZLB period can hardly be described by a
monetary policy rule in which the monetary policy shock is assumed to be nor-
mally distributed. To overcome this statistical problem, while taking into consid-
eration most of such unconventional monetary policies (credit easing, quantita-
tive easing, and forward guidance), we use the shadow rate6(Kim and Singleton,
2012; Krippner, 2013). The shadow rate is a version of the federal funds rate that
can take negative values; it is also consistent with a term structure of interest rates.
Thus, it allows for meaningful monetary policy analysis and interpretation during
low interest rate regimes, without ignoring data from high interest rate periods.
From the estimations and simulations of our models, we analyze, among other
factors, the monetary policy rules’ parameters, in-sample fits (which monetary
5Models like this have been extensively used for policy analysis at various central banks.
6Wu and Xia (2016) devised a shadow Fed funds rate that can be negative, reflecting the Fed’s
unconventional policies. When quantitative easing or forward guidance is pursued, the Fed’s
current rate is zero (ZLB), while the shadow rate changes. When rates are above the ZLB, the
shadow rate is identical to the Fed funds rate. Once the ZLB is reached, the Wu and Xia (2016) rate
uses a Gaussian affine term structure model to generate an effective rate.
4
rule is most in line with historical data) and the central bank’s loss functions,
current or forecasted. Estimated parameters, estimated shocks, impulse response
functions, and variance decompositions are presented in the online appendix.
We find that when considering the central bank’s loss functions, the estimates
often indicate the superiority of NGDP level rules, although Taylor-type rules
have nearly identical implications. However, this being given, the results sug-
gest that historical fitting and the central bank’s objectives cannot be achieved
by one single rule over all time frames. For each type of period (more or less
stable, crisis, recovery), a specific rule performs better than others. Policy institu-
tions, which base their forecasts and policy recommendations on such models and
rules, should refresh their estimates regularly because the parameter estimates of
the rule vary over time.
The remainder of the paper is organized as follows. Section 2 describes the
theoretical setup. Section 3 describes the empirical methodology. Monetary rule
parameters estimates as well as in-sample fit results and analysis are presented in
Section 4. Central bank loss measures are presented in Section 5. Our results are
interpreted in Section 6. Section 7 draws some policy implications, and Section 8
concludes the paper. The online appendix presents additional empirical results.
2 The models
The Smets and Wouters (2007) model is the core model used in this paper. How-
ever, in their article and other working paper versions, those authors do not de-
scribe a flexible-price economy. We perform this work in the detailed description
of the log-linearized sticky- and flexible-price economies in our online appendix.
This (generic) model, also detailed in the online appendix, needs to be com-
pleted by adding an ad hoc monetary policy reaction function (Table 1). Despite
their different formulations, all of these functions include a smoothing process
that captures the degree of rule-specific smoothing.
Taylor-type rules
Model 1is the original Smets and Wouters (2007) monetary policy rule, which
gradually responds to deviations of inflation (πt) from an inflation objec-
tive (normalized to zero), the output gap, defined as the difference between
sticky-price (yt) and flexible-price (yp
t) outputs (see the online appendix), and
deviations of the output gap from the previous period (4yt 4yp
t).
Model 2is based on the Taylor (1993) monetary policy rule, which gradually
responds to deviations of inflation from an inflation objective (normalized to
5
zero) and of the output gap, as previously defined.7
Model 3is the Galí (2015) monetary policy rule, which gradually responds to
the natural interest rate (r
t), as defined in Galí (2015), deviations of inflation
from an inflation objective (normalized to zero) and of the output gap, as
previously defined.
Model 4gradually responds to the deviations of inflation from an inflation
objective (normalized to zero) and output growth (Iacoviello and Neri, 2010;
Garín et al., 2016). It assumes that the natural output (yp
t), as well as the
natural interest rate, are not observable in real time.
Nominal GDP growth rules
Model 5is the Adapted NGDP Growth targeting monetary policy rule, which
gradually responds to deviations of nominal output growth (πt+4yt) from
an objective, as in McCallum and Nelson (1999), and deviations of the output
gap from the previous period (output gap growth, as in model 1).
Model 6is the NGDP Growth targeting monetary policy rule, which gradu-
ally responds to deviations of nominal output growth from its flexible-price
counterpart (FPC).
Model 7is the NGDP Growth targeting monetary policy rule including a nat-
ural interest rate (NIR) component, where the policy gradually responds to
the NIR, as in Rudebusch (2002a), and deviations of nominal output growth
from its flexible-price counterpart.
Model 8is the NGDP Growth targeting monetary policy rule where the pol-
icy gradually responds to the deviations of nominal output growth.
Nominal GDP level rules
Model 9is the Adapted NGDP Level targeting monetary policy rule, which
gradually responds to nominal output level (pt+yt) deviations from its
flexible-price counterpart,8as suggested by McCallum (2015), and devia-
tions of the output gap from the previous period (output gap growth, as
in model 1).
7In the original Taylor rule, the natural interest rate is constant (Taylor, 1993). Log-linearization
around the steady state eliminates this (constant) natural interest rate. Note that rule 1 also (Smets
and Wouters, 2007) does not include the natural interest rate.
8The level of nominal output is pt+yt, where prices ptare deducted from the definition of
inflation πt=ptpt1.
6
Model 10 is the NGDP Level targeting monetary policy rule, which gradu-
ally responds to nominal output level deviations from its flexible-price coun-
terpart (FPC).
Model 11 is the NGDP Level targeting monetary policy rule including an
NIR component, where the policy gradually responds to the NIR and to de-
viations of the nominal output level from its flexible-price counterpart.
Model 12 is the NGDP Level targeting monetary policy rule where the policy
gradually responds to the nominal output level.
As indicated above, there are three categories of rules. The first four (1 to 4)
are of the Taylor-type. Rules 5 to 8 are nominal GDP rules targeting nominal GDP
growth. Rules 9 to 12 target the level of nominal GDP.
Rules 5 and 9 include output gap growth, as in rule 1 (Smets and Wouters,
2003, 2007). Rules 7 and 11 include the natural interest rate, as in rule 3 (Galí,
2015). Including these variables allows us to compare the various rules with their
standard versions as presented by the above-cited authors.
These three categories of rules represent the main policy rules in the contem-
porary literature.
As these rules are all ad hoc, they do not require changes in the specification of
the core model. The unique differentiating feature of the twelve models therefore
comes from their respective monetary policy rule. Concerning NGDP Level tar-
geting rules (models 9 to 12), we add to the core model and the monetary policy
rule the definition of prices, derived from (in log form) πt=ptpt1, where pt
represents the log-price index at time t.
In addition, the inflation rate in the flexible-price economy at time tis πp
t=0,
as in Smets and Wouters (2007). Then, the flexible-price nominal income is only
defined by 4yp
t(growth) or yp
t(level). These assumptions are used in rules 5 to 7
(NGDP Growth rules) and 9 to 11 (NGDP Level rules) in Table 1.
3 Methodology
3.1 Data
The models, with various monetary policy rules, are estimated between 1955 and
2017 and over three different periods within this time interval: from 1955Q1 to
1985Q1, a period when the economy was rather unstable and featured ups and
downs and monetary policy could be characterized as discretionary; from 1985Q1
to 2007Q1, the Great Moderation era (GM), when the economy was rather stable and
monetary policy more predictable; and from 2007Q1 to 2017Q1, the GFC/ZLB era,
7
Models Sources Monetary policy rules
1Smets and Wouters (2007) rt=ρrt1+(1ρ)rππt+ryytyp
t+r4y4yt 4yp
t+εr
t
2Taylor (1993) rt=ρrt1+(1ρ)rππt+ryytyp
t+εr
t
3Galí (2015) rt=ρrt1+(1ρ)r
t+rππt+ryytyp
t+εr
t
4Garín et al. (2016) rt=ρrt1+(1ρ)rππt+ry4yt+εr
t
5Adapted NGDP Growth Targeting rt=ρrt1+(1ρ)rnπt+4yt 4yp
t+r4y4yt 4yp
t+εr
t
6NGDP Growth + FPC Targeting rt=ρrt1+(1ρ)rnπt+4yt 4yp
t+εr
t
7NGDP Growth + NIR Targeting rt=ρrt1+(1ρ)r
t+rnπt+4yt 4yp
t+εr
t
8NGDP Growth Targeting rt=ρrt1+(1ρ) [rn(πt+4yt)] +εr
t
9Adapted NGDP Level Targeting rt=ρrt1+(1ρ)rnpt+ytyp
t+r4y4yt 4yp
t+εr
t
10 NGDP Level + FPC Targeting rt=ρrt1+(1ρ)rnpt+ytyp
t+εr
t
11 NGDP Level + NIR Targeting rt=ρrt1+(1ρ)r
t+rnpt+ytyp
t+εr
t
12 NGDP Level Targeting rt=ρrt1+(1ρ) [rn(pt+yt)] +εr
t
NIR and FPC stand for the natural interest rate (r
t) and the flexible-price counterpart à la Galí (2015), respectively.
Table 1: Summary of monetary policy rules used in this study
8
the crisis and recovery period when monetary policy followed an unusual ZLB
track.
During our first sub-sample (1955-1985), monetary policy was rather discre-
tionary and severely criticized in the literature (Friedman, 1982). Since the 1980s,
the predictability and stability of monetary policy has improved, with many re-
searchers currently recommending rule-based rather than discretionary monetary
policy decisions (Kydland and Prescott, 1977; Taylor, 1986, 1987; Friedman, 1982;
Taylor, 1993). Notice that monetary policies occurring during our first sub-sample
(1955-1985) were often modeled by a rule in the literature (Smets and Wouters,
2007; Nikolsko-Rzhevskyy and Papell, 2012; Nikolsko-Rzhevskyy et al., 2014).
Our second sub-sample (1985-2007) is inspired by Clarida (2010), describing
the period 1985-2007 as the GM. Although our second sub-sample is in line with
the literature (Clarida, 2010; Meltzer, 2012; Taylor, 2012; Nikolsko-Rzhevskyy et al.,
2014), we extend it until 2007, to define a sub-sample with a relatively stable econ-
omy (despite the dot-com crisis beginning in the 2000s) that can be compared with
the crisis period starting in 2007.
Our third sub-sample (2007-2017) is well documented in the crisis and recov-
ery period literature (Gorton, 2009; Cúrdia and Woodford, 2011; Benchimol and
Fourçans, 2017).
The series are quarterly, and data transformations, data sources9and measure-
ment equations10 are exactly the same as in Smets and Wouters (2007).
We estimate our models over the third sub-sample (2007-2017) by using the
shadow rate11 data for the US pursuant to Wu and Xia (2016).
3.2 Calibration
To maintain consistency across models for comparison purposes, we specify and
calibrate prior distributions for all model parameters as in Smets and Wouters
(2007). A detailed description of these parameters, and their calibrations, is pro-
vided in the online appendix.
Except for NGDP targeting rules, monetary policy rule parameters in Table 2
have the same calibration as in Smets and Wouters (2007).
Of course, r4yequals zero in models 2 to 4, 6 to 8, and 10 to 12. rπand ryare
not used in models 5 to 12, and rnis not used in models 1 to 4.
As explained in Rudebusch (2002a), rnis higher than one for NGDP growth
targeting rules and positive and smaller than one for NGDP level targeting rules.
9Detailed data sources, measurement equations and data transformations are available in the
online appendix.
10Measurement equations are presented in the online appendix.
11From December 16, 2008, to December 15, 2015, the effective federal funds rate was in the 0 to
1/4 percent range. In this zero lower bound environment, shadow rate models are used (Kim and
Singleton, 2012; Krippner, 2013).
9
Law Mean Std.
ρBeta 0.75 0.10
rπNormal 1.50 0.25
ryNormal 0.125 0.05
r∆yNormal 0.125 0.05
rnNormal 1.5()/0.5()0.25
Table 2: Prior distribution of monetary policy rule parameters. ()stands for
NGDP growth targeting (rules 5 to 8). ()stands for NGDP level targeting (rules
9 to 12).
3.3 Estimation
As in Smets and Wouters (2007), we apply Bayesian techniques to estimate our
DSGE models with different specifications of monetary policy rules. We estimate
all the parameters presented above over the four different periods defined in Sec-
tion 3.1.
To achieve draw acceptance rates between 20% and 40%, we calibrate the tun-
ing parameter on the covariance matrix for each model and each period. Our
results, for each model and each period, are based on the standard Monte Carlo
Markov Chain (MCMC) algorithm with 6 000 000 draws of 2 parallel chains (where
3 000 000 draws are used for burn-in).
To avoid undue complexity, we do not present all the estimates. We prefer to
concentrate on the analysis of the parameters of the different monetary rules. All
the estimation results12 are available in the online appendix. These results confirm
well-identified parameters and shock estimates in line with the literature. The
comparison of prior and posterior distributions for approximately 35 estimated
parameters, for all 48 estimations (12 rules for each 4 periods), does not highlight
any identification issues.
4 Monetary rule parameters and in-sample fit
Parameter estimates are detailed in the online appendix with all impulse response
functions and variance decompositions. To draw policy conclusions from our
models, we assess monetary policy rule parameters (estimated values) in Section
4.1 and the models’ in-sample fit in Section 4.2.
12Estimated parameters (mean), estimated standard errors (std), highest posterior density inter-
vals (HPDi) and estimated shocks are presented in the online appendix. Other detailed results are
available upon request.
10
4.1 Monetary rule parameters
Fig. 1 presents the estimates of the smoothing parameter (ρ), the inflation coeffi-
cient (rπ), the output gap coefficient (ry), the output gap growth coefficient (r4y)
and the nominal income coefficient (rn).
As Fig. 1 shows, the smoothing parameter is in line with the literature (Jus-
tiniano and Preston, 2010), at approximately 0.8, and rather stable over time, al-
though it appears somewhat smaller for rules 9, 10 and 12, a result in accordance
with Rudebusch (2002a,b).
The inflation coefficient (for rules 1 to 4) remains between 1.5 and 2, also in line
with the literature (Smets and Wouters, 2007; Adolfson et al., 2011). Note that it
is somewhat smaller during the GFC/ZLB, suggesting less reaction by the Fed to
inflation developments than during more stable periods, notably than during the
GM, from 1985 to 2007.
The value of the coefficient of the output gap varies across the periods. It ap-
pears to be higher during the GFC/ZLB period (it remains between 0.15 and 0.20)
than between 1955 and 1985 (its value ranges from 0.10 to 0.15, except for rule 4).
This difference is not as significant when we compare the crisis period with the
1985-2007 period (except for rule 3, to some extent).
These estimates of the Taylor-type rules (rules 1 to 4) imply a Fed that placed
greater emphasis (on the margin) on the output gap during the crisis than during
the previous, stabler period.
The output gap growth coefficient varies somewhat across periods and rules
(between 0.10 and 0.23). At least for rule 9, this coefficient appears to be somewhat
higher during the GFC/ZLB than during the GM, implying a larger reaction to
output growth during the crisis than during the previous, stabler period. For rule
1, this coefficient is the highest during the sub-period 1955-1985, yet with rule 5 it
becomes the smallest, notably during the GFC/ZLB.
The nominal income coefficient associated with the NGDP rules is higher for
the growth rules than the level rules, over all periods, a result that echoes Rude-
busch (2002a). For the growth and level rules, this coefficient is lower during the
GFC/ZLB than otherwise, especially during the GM. The coefficient for the NGDP
level rules changes (with time and rule) but is lower during the GFC/ZLB period
than during the other periods.
4.2 In-sample fit
Which monetary rule best fits the historical data is significant for understanding
and analyzing the behavior of a central bank and drawing implications about pol-
icy debates. It does not mean that the central banker always followed monetary
rules, but at least implicitly, he (generally) behaved “as if” he followed some kind
11
1955-2017
1 2 3 4 5 6 7 8 9 101112
0
0.5
12007-2017
1 2 3 4 5 6 7 8 9 101112
0
0.5
11985-2007
1 2 3 4 5 6 7 8 9101112
0
0.5
11955-1985
1 2 3 4 5 6 7 8 9101112
0
0.5
1
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9101112
0
1
2
1 2 3 4 5 6 7 8 9101112
0
1
2
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9101112
0
0.1
0.2
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9101112
0
1
2
1 2 3 4 5 6 7 8 9101112
0
1
2
Figure 1: Monetary policy rule parameter values for each model (1 to 12).
12
of rule. Unveiling such rules, which may vary with the state of the economy, may
clarify the background of monetary policy over time.
Furthermore, assessing in-sample fit is important to determine whether histor-
ical data (sample) are more or less in line with data generated by the estimated
model. Table 3 shows the Laplace approximation13 around the posterior mode
(based on a normal distribution), i.e., log marginal densities, for each model and
for each sample.
Sample Rule
123456789101112
1955-2017 -1491 -1515 -1512 -1510 -1464 -1481 -1488 -1514 -1563 -1602 -1556 -1548
2007-2017 -269 -270 -285 -285 -307 -308 -283 -302 -258 -262 -278 -254
1985-2007 -386 -428 -408 -406 -406 -404 -396 -410 -393 -395 -405 -383
1955-1985 -817 -824 -835 -840 -840 -855 -837 -842 -844 -846 -863 -853
Table 3: Log marginal data densities for each rule and each period (Laplace ap-
proximation). Best values for each period in gray.
Table 3 suggests that the last NGDP rule in levels (rule 12), the pure NGDP
level targeting without flexible-price output, best fits the historical data during
the GFC/ZLB, yet rule 9 comes close. Rule 12 also performs best during the GM
period, while rule 1, the Smets and Wouters (2007) rule, comes close. This result
suggests that the Fed may have changed strategy during the GM compared to
what it did before 1985. It may have switched from a Taylor type framework to
an NGDP Level targeting framework, and then maintained, and even reinforced,
this targeting type once the federal funds rate hit the zero lower bound.
Finally, rule 1 dominates the other rules over the period 1955-1985, whereas
rule 5 ranks first over the whole sample.
For each period, a different monetary policy rule best fits the historical data,
except for rule 12 that places first twice. Note that standard Taylor-type rules (rules
2 to 4) and NGDP growth targeting rules (rules 5 to 8) are generally inferior to the
other rules in explaining historical data, at least over the various sub-periods.
However, this result does not imply that models with lower log marginal data
densities should be discarded. Whatever the log marginal data density function,
it may be argued that each model is designed to capture only certain characteris-
tics of the data. Whether the marginal likelihood is a good measure to evaluate
how well the model accounts for particular aspects of the data is an open ques-
tion (Koop, 2003; Fernández-Villaverde and Rubio-Ramírez, 2004; Del Negro et al.,
2007; Benchimol and Fourçans, 2017).
13The Geweke (1999) mean harmonic estimator provides a similar ranking of models.
13
5 Central bank losses
It is traditional to assume that central banks seek to minimize a loss function based
on the historical variances of the variables of interest to the bank. Generally, the
current values of these variables are considered. However, the decision maker
could also use forecasted values to determine which monetary policy is best as far
as economic dynamics are concerned, hence our decision to analyze the minimiza-
tion of two types of loss functions, one based on current (and past) outcomes in
Section 5.1 and the other on forecasted ones in Section 5.2.
5.1 Current loss function
As noted above, the preferences of the central banker are generally represented
by a loss function that he seeks to minimize. This minimization process is also
supposed to represent the objectives of society.
In this section, we present current loss measures based on the historical vari-
ances of the variables of interest from the central bank’s perspective. These vari-
ances are estimated for each model and for each period.
Many ad hoc central bank loss functions appear in the literature (Svensson and
Williams, 2009; Taylor and Wieland, 2012; Adolfson et al., 2014). Our methodology
intends to summarize all standard possibilities. For various sets of weights defin-
ing these functions, we compute the ex post loss functions consistent with the
estimated DSGE model. This approach is used in the literature to investigate em-
pirical monetary policy rules (Taylor, 1979; Fair and Howrey, 1996; Taylor, 1999)
and is different from the optimal monetary policy literature (Schmitt-Grohé and
Uribe, 2007; Billi, 2017).
Non-separability between consumption and labor (worked hours) in the Smets
and Wouters (2007) household utility function (see the online appendix) intro-
duces labor-related variables into the inflation and output equations. By mini-
mizing its loss function with respect to these two equations, the central bank must
also consider labor-related variables, such as wages (the price of worked hours).
Our general central bank loss function, Lt, is defined in a traditional way as14
Lt=var (πt)+λyvar ytyp
t+λrvar (∆rt)+λwvar (wt)(2)
14See Galí (2015) for further details. Another loss measure based on the squared distance of
variables generated by the models can be defined as
Lt=π2
t+λyytyp
t2+λr(∆rt)2+λww2
t(1)
By the definition of the variance operator, this type of formulation leads to a ranking similar to
those given by Eq. 2.
14
where var (.)is the variance operator, λythe weight on output gap variances, λr
the weight on nominal interest rate differential variance, and λwthe weight on
wage inflation variance. The weight on price inflation variance is normalized to
unity. πtis price inflation, ytyp
tthe output gap, ∆rtthe nominal interest rate
differential, and wtwage inflation.15
First, in Fig. 2, we present the estimated variances of each variable (inflation,
output gap, nominal interest rate differential, and wage inflation) entering the
central bank loss functions.
The variances of all variables under consideration are significantly higher be-
fore 1985 and over the full sample. Even during the 2007-2017 period, these vari-
ances were lower than before 1985 and little different from those during the GM
period. The fact that estimated variances over the GFC/ZLB period are compara-
ble across the models with those of the GM period does not mean that the vari-
ances of historical data during the GFC/ZLB and GM are comparable. Indeed, the
variances presented in Fig. 2 are estimated from the models while assuming that
the Fed followed various rules and the US economy behaved as in the Smets and
Wouters (2007) model. The high inflation period cum various significant ups and
downs in economic activity and interest rates explain the high values observed
between 1955 and 1985.
However, changes in the Fed’s monetary policy and the stabilization period
that occurred during the 1990s explain the low variance of the GM period rela-
tive to the 1955-1985 period. Output variances are a somewhat higher during the
GFC/ZLB period than during the GM period, while those of the inflation rate
come close. The low interest rates of the GFC/ZLB period lead to lower variances
of the shadow interest rate differentials during the GFC/ZLB than during the GM
period, although the difference is not large. The variances of wages were also
smaller during the GFC/ZLB period than during the GM period.
Interestingly, the output gap exhibits a low level of volatility during
theGFC/ZLB, just a bit higher than during the GM (Fig. 2). The low variances
of the output gap over the GM period are due to the fact that, as in most DSGE
models, the potential output covaries in general with the current output16 (Kiley,
2013; Coibion et al., 2018).
During the GFC/ZLB, the correlation between the current output (historical)
and the potential output (unobservable, based on our estimations) is rather low
compared to the one during the GM. The low output gap variances during the
GFC/ZLB are essentially due to the low variances exhibited by both the current
and potential outputs over most of the sample after 2009Q2, if not from 2008Q1 to
2009Q1.
15See the online appendix for further details on the variables in the models.
16The same type of result comes from some from non-DSGE models, for instance Fernald (2015).
15
1955-2017
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
0.3
0.4
2007-2017
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
0.3
0.4
1985-2007
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
0.3
0.4
1955-1985
1 2 3 4 5 6 7 8 9 101112
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
1 2 3 4 5 6 7 8 9 101112
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 101112
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 101112
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 101112
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 9 101112
0
0.2
0.4
0.6
Figure 2: Estimated variances of central bank loss function variables, for each pe-
riod and each rule.
Second, we compute ad hoc loss functions based on Eq. 2. The following
heatmaps (Tables 4 and 5) present the best (white shading) to the worst (black
shading) loss functions in percentage variance for each line.
The loss increases for all rules and for all periods when the weight on the vari-
ance of one variable included in the loss functions increases. This is directly re-
lated to the linear quadratic functional form of the central bank loss function. The
ranking of the monetary policy rules follows from the value of the loss function
given by each line, and this ranking changes with respect to the weighting scheme
allowed by the central banker’s preferences.
When considering the full sample (Table 4, left panel), there is no clear result,
with the best rule (rules 9 and 12) being especially sensitive to the values of λw.
Rules 9 and 10 (Table 4, right panel) lead to the lowest losses over the GFC/ZLB
period, but rule 12 leads to nearly identical results.
Over the GM period (Table 5, left panel) rule 2 often dominates, but rules 1, 9
and 10 come close.
16
1 2 3 4 5 6 7 8 9 10 11 12
25.1
34.1
43.0
27.8
36.7
45.6
30.5
39.4
48.3
37.1
46.0
54.9
39.8
48.7
57.6
42.5
51.4
60.3
49.1
58.0
66.9
51.7
60.6
69.5
54.4
63.3
72.2
24.1
28.4
32.7
27.4
31.7
36.0
30.7
35.0
39.3
65.2
68.5
71.8
94.2
98.2
34.8
43.0
38.3
46.4
41.7
49.9
71.1
74.5
78.0
74.8
62.7
77.8
50.5
65.6
80.7
60.8
76.0
91.1
63.8
78.9
94.0
66.7
81.9
97.0
74.6
62.5
77.5
50.5
65.5
80.5
60.7
75.7
90.7
63.6
78.6
93.6
66.6
81.5
96.5
15.6
22.1
28.5
19.0
25.4
31.8
22.3
28.7
35.2
38.2
44.6
51.1
41.5
48.0
54.4
44.8
51.3
57.7
60.7
67.2
73.6
64.1
70.5
77.0
67.4
73.8
80.3
25.3
31.3
37.3
29.4
35.4
41.4
39.5
45.5
42.8
48.9
54.9
46.9
53.0
59.0
51.0
57.0
63.1
60.4
66.4
72.5
64.5
70.5
76.5
68.6
74.6
80.6
16.6
26.1
35.5
20.8
30.2
39.6
24.9
34.3
43.7
31.1
40.5
49.9
35.2
44.6
54.0
39.3
48.7
58.1
45.5
54.9
64.3
49.6
59.0
68.5
53.7
63.2
72.6
16.5
35.4
54.3
20.7
39.6
58.5
24.9
43.8
62.7
35.6
54.5
73.4
39.8
58.7
77.6
44.0
62.9
81.8
54.7
73.6
92.5
58.9
77.8
96.7
63.1
82.0
100.9
56.6
60.9
59.9
64.2
63.2
67.5
89.0
93.3
97.6
92.3
96.6
100.9
95.6
99.9
104.2
36.4
49.4
62.4
40.4
53.4
66.4
44.4
57.4
70.4
52.3
65.3
78.3
56.3
69.3
82.3
60.3
73.3
86.3
68.2
81.2
72.3
85.2
76.3
89.3
102.3
26.7
30.1
33.5
54.8
62.9
58.2
66.4
61.6
69.8
82.8
91.0
99.2
86.3
94.4
102.6
89.7
97.9
106.0
28.3
43.5
58.6
31.3
46.4
61.5
34.2
49.4
64.5
44.6
59.7
47.5
28.6
43.6
58.5
31.5
46.5
61.5
34.4
49.4
64.4
44.6
59.6
47.6
28.1
47.9
67.8
31.1
50.9
70.8
34.1
54.0
73.8
48.6
68.5
88.4
51.7
71.5
91.4
54.7
74.6
94.4
69.2
89.1
108.9
72.3
92.1
112.0
75.3
95.1
115.0
31.0
57.2
83.4
34.5
60.7
86.9
38.1
64.3
90.5
51.6
77.7
103.9
55.1
81.3
107.5
58.7
84.8
111.0
72.1
98.3
124.5
75.7
101.9
128.1
79.2
105.4
131.6
33.4
1 2 3 4 5 6 7 8 9 10 11 12
3.9
6.1
8.2
4.7
6.8
9.0
5.5
7.6
9.8
4.9
7.1
9.2
5.7
7.9
10.0
6.5
8.6
10.8
5.9
8.1
10.2
6.7
8.9
11.0
7.5
9.7
11.8
3.9
5.9
7.9
4.8
6.8
8.8
5.7
7.7
9.7
4.8
6.8
8.9
5.7
7.7
9.7
6.6
8.6
10.6
5.7
7.7
9.8
6.6
8.6
10.7
7.5
9.5
11.5
4.1
7.6
11.1
5.1
8.6
12.1
6.2
9.7
13.2
6.4
9.9
13.4
7.5
11.0
14.5
8.5
12.0
15.5
8.8
12.3
15.8
9.8
13.3
16.8
10.8
14.3
17.8
4.5
7.5
10.5
5.4
8.4
11.4
6.3
9.3
12.3
5.6
8.6
11.6
6.6
9.6
12.5
7.5
10.5
13.5
6.8
9.8
12.8
7.7
10.7
13.7
8.6
11.6
14.6
15.9
16.8
13.3
17.7
13.6
17.9
14.5
18.8
15.4
19.7
18.3
19.2
20.1
9.0
12.5
9.6
13.1
10.3
13.8
11.3
14.8
11.9
15.4
12.6
16.1
13.6
17.1
14.3
17.8
14.9
18.4
3.9
5.8
7.6
4.7
6.5
8.4
5.5
7.3
9.2
4.7
6.6
8.5
5.5
7.4
9.3
6.3
8.2
10.0
5.6
7.5
9.3
6.4
8.2
10.1
7.1
9.0
10.9
3.8
5.7
7.5
4.6
6.5
8.3
5.5
7.3
9.1
4.7
6.5
8.4
5.5
7.3
9.2
6.3
8.1
10.0
5.5
7.3
9.2
6.3
8.2
10.0
7.1
9.0
10.8
4.4
7.6
10.7
5.4
8.6
11.8
6.4
9.6
12.8
6.1
9.3
12.5
7.1
10.3
13.5
8.1
11.3
14.5
7.9
11.0
14.2
8.9
12.0
15.2
9.9
13.1
16.2
3.8
6.4
9.0
4.5
7.1
9.7
5.3
7.9
10.5
4.4
7.0
9.6
5.2
7.8
10.4
6.0
8.6
11.2
5.1
7.7
10.3
5.9
8.5
11.1
6.7
9.3
11.9
5.1
9.5
13.8
6.0
10.4
14.8
6.9
11.3
15.7
7.2
11.5
8.1
12.4
9.0
9.2
10.1
11.0
5.7
10.0
14.3
6.6
10.9
15.2
7.5
11.8
16.1
7.7
12.0
16.3
8.6
12.9
17.2
9.5
13.8
18.1
9.7
14.0
10.6
14.9
11.5
15.8
5.5
6.1
6.8
7.8
8.5
9.1
10.1
10.8
11.4
5.3
12.5
19.7
6.4
13.6
20.8
7.5
14.7
22.0
9.1
16.4
23.6
10.3
17.5
24.7
11.4
18.6
25.8
13.0
20.2
27.4
14.1
21.3
28.5
15.2
22.4
29.6
Table 4: Central bank losses, for each rule (1 to 12), between 1955 and 2017 (left panel) and 2007 and 2017 (right panel). The
shading scheme is defined separately in relation to each line. The lighter the shading is, the smaller the loss.
1 2 3 4 5 6 7 8 9 10 11 12
4.2
6.2
8.2
4.8
6.8
8.8
5.4
7.4
9.5
8.8
10.8
12.8
9.4
11.4
13.5
10.0
12.0
14.1
13.4
15.4
17.4
14.0
16.0
18.1
14.6
16.7
18.7
5.2
6.6
8.0
5.9
7.3
8.7
6.6
8.0
9.4
8.5
9.9
11.3
9.3
10.7
12.1
10.0
11.4
12.8
11.9
13.3
14.7
12.6
14.0
15.4
13.3
14.7
16.1
7.3
9.2
8.2
10.0
9.0
10.8
9.4
11.2
13.0
10.2
12.1
13.9
11.1
12.9
14.7
13.3
15.1
16.9
14.1
15.9
17.8
15.0
16.8
18.6
10.5
14.0
17.5
11.2
14.7
18.2
12.0
15.5
19.0
14.4
17.9
21.4
15.1
18.6
22.1
15.9
19.4
22.9
13.5
5.0
6.4
7.8
5.7
7.1
8.5
6.4
7.8
9.2
9.7
11.1
12.4
10.4
11.8
13.2
11.1
12.5
13.9
14.3
15.7
17.1
15.1
16.4
17.8
15.8
17.2
18.5
4.7
6.1
7.4
5.5
6.8
8.2
6.2
7.5
8.9
9.5
10.9
12.2
10.2
11.6
13.0
11.0
12.3
13.7
14.3
15.7
17.0
15.0
16.4
17.8
15.8
17.1
18.5
5.0
6.6
8.1
5.8
7.4
9.0
6.6
8.2
9.8
9.7
11.3
12.9
10.6
12.1
13.7
11.4
12.9
14.5
14.5
16.1
17.6
15.3
16.9
18.4
16.1
17.7
19.2
4.3
8.2
5.1
8.9
12.7
5.8
9.6
13.4
10.4
14.2
18.1
11.1
14.9
18.8
11.8
15.7
19.5
16.5
20.3
17.2
21.0
24.8
17.9
21.7
25.6
5.5
6.3
7.2
6.6
10.1
13.6
7.3
10.8
14.3
8.1
11.6
15.1
5.8
9.1
12.5
6.5
9.8
13.2
7.1
10.5
13.8
12.1
15.5
18.8
12.8
16.1
19.5
13.4
16.8
20.2
18.4
21.8
25.1
19.1
22.4
25.8
19.8
23.1
26.5
6.0
9.5
13.1
6.7
10.2
13.7
7.3
10.9
14.4
13.9
17.4
21.0
14.6
18.1
21.6
15.2
18.8
22.3
21.8
25.3
28.9
22.5
26.0
29.5
23.2
26.7
30.2
6.5
9.4
12.3
7.1
10.0
12.9
7.7
10.6
14.2
17.1
19.9
14.7
17.6
20.5
15.3
18.2
21.1
21.8
24.7
27.6
22.4
25.3
28.1
22.9
25.8
28.7
5.5
11.0
16.5
6.5
12.0
17.5
7.5
13.0
18.5
14.1
19.5
25.0
15.0
20.5
26.0
16.0
21.5
27.0
22.6
28.1
33.5
23.6
29.0
34.5
24.6
30.0
35.5
12.0
24.1
1 2 3 4 5 6 7 8 9 10 11 12
34.2
38.0
41.9
39.4
43.3
47.1
44.7
48.6
52.4
55.2
52.8
56.6
60.5
58.0
61.9
65.7
60.8
64.7
68.5
66.1
69.9
73.8
71.4
75.2
79.0
41.4
44.8
47.3
50.8
53.3
56.7
56.2
61.1
66.0
64.1
69.0
73.9
72.1
77.0
81.9
60.1
66.1
72.2
60.3
66.3
72.3
58.3
64.0
69.7
45.3
52.9
51.0
62.6
58.6
70.3
66.3
77.9
25.8
28.1
30.4
32.4
34.7
37.0
38.9
41.3
43.6
36.1
38.4
40.7
42.6
45.0
47.3
49.2
51.5
53.8
46.3
48.7
51.0
52.9
55.2
57.5
59.5
61.8
64.1
47.3
53.5
59.6
65.1
68.8
67.6
71.3
75.0
73.7
77.4
81.1
46.4
53.0
48.3
60.1
55.0
66.8
61.7
73.5
47.5
51.3
37.9
43.9
49.8
52.2
55.6
59.1
58.1
61.6
65.0
64.1
67.5
70.9
66.4
69.9
73.3
72.4
75.8
79.2
78.3
81.7
85.2
43.2
48.1
53.0
51.1
56.0
60.9
59.1
64.0
68.9
49.7
54.6
59.5
57.6
62.5
67.4
65.6
70.5
75.4
38.9
44.5
50.1
45.2
50.8
56.4
51.5
57.1
62.7
57.5
63.1
68.7
63.7
69.3
74.9
70.0
75.6
81.2
76.0
81.6
87.2
82.3
87.9
93.5
88.6
94.2
99.8
42.6
51.8
61.1
48.7
57.9
67.1
54.7
63.9
73.1
51.4
60.6
69.8
57.4
66.6
75.8
63.4
72.7
81.9
69.3
78.5
75.4
84.6
81.4
90.6
42.2
51.7
61.2
48.2
57.7
67.2
54.2
63.7
73.2
51.2
60.7
70.2
57.2
66.7
76.2
63.3
72.8
82.3
69.8
79.3
75.8
85.3
81.8
91.3
41.2
53.2
65.1
46.9
58.9
70.8
52.6
64.6
76.5
49.7
61.7
73.6
55.5
67.4
79.3
61.2
73.1
85.1
70.2
82.2
75.9
87.9
81.7
93.6
39.6
51.2
62.8
47.2
58.9
70.5
54.9
66.5
78.1
56.9
68.5
64.6
76.2
60.6
72.2
83.8
74.2
81.9
89.5
45.3
47.8
50.2
52.1
54.6
57.1
59.0
61.4
63.9
56.5
59.0
61.5
63.4
65.8
68.3
70.2
72.7
75.1
67.7
70.2
72.7
74.6
77.1
79.5
81.4
83.9
86.4
39.9
43.6
46.1
49.8
52.2
55.9
50.7
54.4
58.1
56.8
60.5
64.2
63.0
66.7
70.4
61.4
44.4
56.2
68.0
51.1
62.9
74.7
57.7
69.5
81.3
58.2
70.0
64.8
76.6
59.7
71.5
83.3
71.9
78.6
85.3
Table 5: Central bank losses, for each rule (1 to 12), between 1985 and 2007 (left panel) and 1955 and 1985 (right panel). The
shading scheme is defined separately in relation to each line. The lighter the shading is, the smaller the loss.
Over the GFC/ZLB period, such sensitivity to wage inflation is low, and the
ranking of rules does not particularly depend on taking wages into consideration.
The sensitivity of the results is also low with respect to the values of λyand λr.
The same can be said during the 1955-1985 period and even during the GM pe-
riod, albeit to a somewhat lesser extent. Interestingly, in all periods, the change in
the loss is minor for a given λywhen λrchanges, compared to the change in the
loss for a given λrwhen λychanges. One can interpret this result in light of the
interest rate smoothing assumption. Most of the monetary policy rules used in the
literature assume interest rate smoothing, as we do. This smoothing implies that
the central bank already minimizes the variances in the interest rate differential
over time, hence the small gain generated by changing the interest rate differen-
tial coefficient in the central bank loss function for a given λyor λw.
From all these observations, it can be inferred that during the exceptional-
GFC/ZLB period, the Fed would have minimized its loss by following an NGDP
rule in levels, especially rules 9 and 10. During this period, rule 12 performs better
under a less credible configuration (λy=0). However, had it employed Taylor-
type rules 1 and 2, the difference in terms of loss would have been minor. Over
more stable periods such as the GM period, the central bank would have mini-
mized its losses with a Taylor-type rule, especially rules 1 and 2, but NGDP in
level rules (rules 9 and 10) would have led to nearly identical results.
5.2 Forecasted loss function
As noted above, a central banker may want to minimize a forecasted loss function
based on the dynamics of the model of the economy he uses.
The Bayesian estimation procedure we use allows us to compute the distrib-
ution of out-of-sample forecasts while taking into account the uncertainty about
parameters and shocks. We use these point forecasts (3 years ahead) to draw the
price inflation, output-gap, nominal interest rate differential and wage inflation
posterior variances to compute the various forecasted loss functions.
The out-of-sample forecasted losses over a three-year out-of-sample period are
presented in Tables 6 and 7. They are based on the estimation of the model with
the various monetary rules over the full sample period and over each sub-period.
20
1 2 3 4 5 6 7 8 9 10 11 12
15.1
19.1
23.2
17.4
21.4
25.4
19.6
23.6
27.7
44.8
20.9
23.1
23.7
25.9
26.5
28.8
38.5
41.3
44.1
24.7
27.7
30.6
37.0
37.0
39.9
39.9
42.8
49.3
52.2
55.1
17.0
27.3
19.6
29.8
22.2
27.3
32.4
34.9
40.0
45.1
30.4
23.1
32.9
16.4
19.0
21.6
27.4
33.1
27.5
33.3
39.1
30.2
35.9
41.7
32.8
38.5
44.3
38.7
44.5
50.2
41.3
47.1
52.8
43.9
49.7
55.4
26.7
29.8
32.8
35.7
44.7
38.7
47.7
41.8
50.8
13.4
16.4
19.3
16.3
19.2
22.1
19.1
22.0
24.9
25.9
28.8
31.7
28.7
31.6
34.5
31.5
34.4
37.3
38.3
41.2
44.1
41.1
44.0
47.0
43.9
46.9
49.8
15.7
18.3
21.0
19.0
21.7
24.3
22.3
25.0
27.6
33.6
36.3
36.9
39.6
40.2
42.9
14.1
17.9
21.7
17.5
21.3
25.1
20.9
24.7
28.5
13.8
19.7
25.6
17.0
22.9
28.8
20.2
26.1
32.0
17.6
23.5
29.4
20.8
26.7
32.6
24.0
29.9
35.8
21.4
27.3
33.2
24.6
30.5
36.4
27.8
33.7
39.6
32.3
36.3
40.3
34.5
38.5
42.6
36.8
40.8
49.4
53.4
57.5
51.7
55.7
59.7
53.9
57.9
62.0
18.7
21.5
24.3
34.0
36.3
36.8
39.1
39.6
41.9
49.4
51.6
53.9
52.2
54.4
56.7
55.0
57.3
59.5
21.3
26.0
30.8
24.7
29.4
34.1
28.0
32.8
37.5
39.5
44.2
48.9
42.9
47.6
52.3
46.2
50.9
55.7
57.7
62.4
67.1
61.1
65.8
70.5
64.4
69.1
73.8
18.9
21.8
21.8
24.7
24.7
27.6
31.1
34.1
34.0
37.0
43.4
46.3
46.3
49.2
49.2
52.2
22.2
24.7
29.8
34.9
40.0
32.4
37.5
42.6
42.5
47.6
52.7
45.1
50.2
55.3
47.7
52.8
57.9
18.1
23.0
27.9
20.6
25.5
28.0
32.0
36.9
41.7
34.5
39.3
44.2
37.0
41.8
46.7
45.8
50.7
55.6
48.3
53.2
58.1
50.8
55.7
60.6
22.1
27.9
24.8
30.5
17.7
26.7
35.7
20.8
29.8
38.8
23.8
32.9
41.9
35.7
44.7
38.8
47.8
41.8
50.8
53.7
56.7
59.8
30.9
34.3
37.6
46.2
48.9
51.5
49.5
52.2
54.8
52.8
55.5
58.2
37.1
40.9
44.7
40.5
44.3
48.1
43.8
47.7
51.5
60.1
63.9
67.7
63.4
67.3
71.1
66.8
70.6
74.5
1 2 3 4 5 6 7 8 9 10 11 12
3.7
5.3
6.9
4.3
5.9
7.5
5.0
6.6
8.2
5.0
6.6
8.2
5.7
7.3
8.8
6.3
7.9
9.5
6.4
7.9
9.5
7.0
8.6
10.2
7.6
9.2
10.8
3.9
5.3
6.8
4.6
6.0
7.5
6.8
8.2
5.1
6.6
8.0
5.8
7.3
8.7
6.6
8.0
9.4
6.3
7.8
9.2
7.1
8.5
10.0
7.8
9.2
10.7
3.6
5.5
7.4
4.4
6.3
8.2
5.3
7.2
9.0
7.3
9.2
8.2
10.1
9.0
10.9
9.2
11.1
10.1
11.9
10.9
12.8
6.1
8.1
6.9
8.9
7.7
9.7
7.5
9.5
8.3
10.3
9.1
11.1
8.8
10.8
9.6
11.6
10.4
12.4
6.6
8.9
7.2
9.6
7.9
10.3
8.3
10.7
9.0
11.4
9.7
12.0
10.0
12.4
10.7
13.1
11.4
13.8
7.0
9.5
7.7
10.2
8.4
10.9
8.7
11.2
9.5
11.9
10.2
12.6
10.5
13.0
11.2
13.7
11.9
14.4
3.5
5.3
7.0
4.1
5.8
7.6
4.6
6.4
8.2
5.0
6.7
8.5
5.5
7.3
9.1
6.1
7.9
9.6
6.4
8.2
10.0
7.0
8.8
10.5
7.6
9.3
11.1
3.7 3.6
4.9
6.3
4.2
5.6
6.9
4.8
6.2
7.5
4.7
6.1
7.5
5.4
6.7
8.1
6.0
7.4
8.7
5.9
7.3
8.6
6.6
7.9
9.3
7.2
8.5
9.9
3.6
5.1
6.5
4.3
5.7
7.2
5.0
6.4
7.8
4.8
6.2
7.6
5.4
6.9
8.3
6.1
7.5
9.0
5.9
7.3
8.8
6.6
8.0
9.4
7.2
8.7
10.1
3.6
5.5
7.4
4.4
6.3
8.2
5.1
7.1
9.0
5.1
7.0
8.9
5.9
7.8
9.7
8.6
10.5
8.6
10.5
9.4
11.3
10.2
12.1
3.6
5.6
7.5
4.2
6.2
8.1
4.9
6.8
8.8
4.3
6.2
8.2
4.9
6.9
8.8
5.6
7.5
9.5
5.0
6.9
8.9
5.6
7.6
9.5
6.3
8.2
10.2
5.3
5.5
6.3
7.1
7.3
8.2
9.0
4.1
4.9
5.7
5.5
6.3
7.1
6.8
7.6
8.4
4.2
4.9
5.6
5.9
6.6
7.3
7.7
8.4
9.0
4.5
5.2
5.9
6.3
7.0
7.7
8.0
8.8
9.5
10.3
17.0
4.6
11.3
17.9
5.6
12.2
18.8
5.8
12.4
19.1
6.8
13.4
20.0
7.7
14.3
20.9
7.9
14.6
21.2
8.9
15.5
22.1
9.8
16.4
23.1
6.7
6.7
7.5
8.3
Table 6: Forecasted central bank losses, for each rule (1 to 12), between 2017 and 2019 based on full sample estimates (left panel)
and based on 2007-20017 estimates (right panel). The shading scheme is defined separately in relation to each line. The lighter
the shading is, the smaller the loss.
1 2 3 4 5 6 7 8 9 10 11 12
7.1
6.4
7.7
6.9
8.2
8.7
10.0
7.9
9.2
10.6
8.5
9.8
11.1
10.2
11.5
12.9
10.8
12.1
13.4
11.3
12.6
14.0
6.9
6.4
7.5
7.0
8.1
7.3
8.4
9.5
7.9
9.0
10.1
8.4
9.5
10.6
9.9
11.0
12.1
10.4
11.5
12.6
11.0
12.1
13.2
5.6
6.6
6.2
7.2
6.9
7.9
6.9
7.9
8.9
7.6
8.6
9.6
8.3
9.3
10.3
9.2
10.3
11.3
9.9
10.9
11.9
10.6
11.6
12.6
7.3
7.9
8.5
9.7
11.6
13.5
10.3
12.2
14.1
10.9
12.8
14.7
5.4 8.2
7.2
8.7
12.1
3.8
4.9
5.9
4.4
5.4
6.5
4.9
6.0
7.0
6.6
7.6
8.7
7.2
8.2
9.2
7.7
8.8
9.8
9.4
10.4
11.5
10.0
11.0
12.0
10.5
11.6
12.6
3.5
4.5
5.6
4.1
5.1
6.1
4.7
5.7
6.7
6.3
7.3
8.3
6.9
7.9
8.9
7.4
8.4
9.4
9.0
10.0
11.0
9.6
10.6
11.6
10.2
11.2
12.2
3.7
4.6
5.5
4.3
5.2
6.1
4.9
5.8
6.7
6.5
7.4
8.3
7.1
8.0
8.9
7.7
8.6
9.5
9.3
10.2
11.1
9.9
10.8
11.7
10.5
11.4
12.3
3.6
4.2
4.8
7.1
6.8
7.3
7.9
10.2
9.9
10.5
12.8
11.0
13.4
4.5
5.8
5.0
5.6
7.4
4.7
5.8
5.3
5.9
4.6
5.2
5.9
4.8
6.7
8.6
5.5
7.4
9.2
6.1
8.0
9.9
9.2
11.0
9.8
11.7
10.4
12.3
4.2
6.3
8.4
4.8
6.9
9.0
7.5
9.6
7.7
9.7
11.8
8.2
10.3
12.4
8.8
10.9
13.0
11.1
13.2
15.2
11.6
13.7
15.8
12.2
14.3
16.4
4.4
6.4
8.4
5.0
7.0
9.0
5.6
7.6
9.6
8.4
10.4
12.5
9.0
11.0
13.0
9.6
11.6
13.6
12.4
14.5
16.5
13.0
15.1
17.1
13.6
15.6
17.7
4.7
6.2
7.7
5.1
6.7
5.6
8.1
9.6
11.1
8.6
10.1
11.6
9.0
10.6
11.5
13.0
14.5
12.0
13.5
15.0
12.4
14.0
15.5
4.2
6.9
9.6
5.1
7.8
10.5
6.0
8.6
11.3
7.9
10.6
13.3
8.8
11.4
14.1
9.6
12.3
15.0
11.6
14.2
16.9
12.4
15.1
17.8
13.3
16.0
18.7
6.0
8.3
6.5
8.9
9.4
9.1
11.4
9.7
12.0
12.6
12.2
14.6
15.1
15.7
1 2 3 4 5 6 7 8 9 10 11 12
41.1
43.2
37.6
33.6
38.6
43.7
39.6
43.7
44.7
48.7
49.8
53.8
26.1
30.9
35.7
40.4
31.4
36.2
41.0
41.0
45.8
36.8
41.5
46.2
41.6
46.3
51.0
46.4
51.1
55.8
26.4
32.1
38.7
37.8
43.5
44.4
26.9
28.7
30.4
32.4
34.2
35.9
38.0
39.7
41.5
33.0
34.8
36.5
38.5
40.3
42.1
44.1
45.8
47.6
39.1
40.9
42.7
44.7
46.4
48.2
50.2
52.0
53.7
21.2
23.0
24.7
26.9
28.6
30.3
32.5
34.2
36.0
27.4
29.2
30.9
33.1
34.8
36.5
38.7
40.4
42.2
33.6
35.4
37.1
39.3
41.0
42.7
44.9
46.6
48.4
26.5
32.1
37.7
27.8
33.8
33.4
39.4
39.0
45.0
29.1
35.1
41.1
34.7
40.7
46.7
40.3
46.3
52.3
29.8
32.0
34.1
34.4
36.5
38.7
39.0
40.3
42.4
44.5
44.8
47.0
49.1
49.4
51.5
53.6
50.7
52.8
54.9
55.3
57.4
59.5
59.8
61.9
64.1
30.3
32.3
34.4
35.6
37.6
39.6
40.8
42.8
44.8
39.4
41.4
43.4
44.6
46.6
48.6
49.9
51.9
53.9
48.4
50.4
52.4
53.6
55.6
57.6
58.9
60.9
62.9
31.6
34.8
38.0
38.2
41.4
44.6
44.8
48.0
51.2
36.9
40.1
43.3
43.5
46.7
49.9
50.1
53.3
56.5
42.3
45.5
48.7
48.9
52.1
55.3
55.5
58.7
61.9
28.5
31.5
34.6
33.9
36.9
39.9
39.2
42.2
45.2
39.1
42.1
45.1
44.4
47.4
50.5
49.8
52.8
55.8
49.7
52.7
55.7
55.0
58.0
61.0
60.3
63.3
66.4
28.6
32.8
37.0
33.8
38.0
42.2
38.9
43.1
47.3
34.7
38.9
43.1
39.9
44.1
48.3
45.0
49.2
53.4
40.8
45.0
49.2
46.0
50.2
54.3
51.1
55.3
59.5
27.5
31.5
35.5
32.5
36.6
40.6
41.6
45.6
37.6
41.6
42.6
46.7
47.7
51.7
47.7
52.8
57.8
30.8
35.5
35.6
40.3
45.1
36.1
40.9
45.7
50.5
32.1
37.8
33.0
38.8
44.5
39.7
45.4
51.1
37.8
43.5
44.4
50.2
45.3
51.1
56.8
49.2
50.1
55.9
51.0
56.8
62.5
32.6
34.6
36.7
37.7
39.8
41.8
42.9
44.9
46.9
39.0
41.0
43.0
44.1
46.1
48.1
49.2
51.2
53.3
45.3
47.3
49.4
50.4
52.5
54.5
55.6
57.6
59.6
32.5
38.6
38.1
44.2
43.7
49.7
39.8
45.4
51.0
Table 7: Forecasted central bank losses, for each rule (1 to 12), between 2007 and 2009, based on 1985-2007 estimates (left panel)
and between 1985 and 1987, based on 1955-1985 estimates (right panel). The shading scheme is defined separately in relation to
each line. The lighter the shading is, the smaller the loss.
The objective here is not to compare to what extent these ex ante forecasted
losses diverge from the ex post actual losses over the different sub-periods. A cen-
tral banker interested in these forecasted values, and who decides today which
monetary rule to use, cannot know the ex post values of these losses. He is inter-
ested only in minimizing the forecasted values given his model of the economy.
Table 6, left panel, presents the forecasted loss function for 2017-2019 using the
full sample. This table shows that rule 9 dominates the other rules when wage
inflation is not taken into consideration. Otherwise, rule 12 gives the best results.
Table 6, right panel, presents the forecasted loss function for 2017-2019 using
the GFC/ZLB data. Although rule 12 performs well, rules 9 and 10 appear to be
optimal if the central banker is interested in realistic forecasted central bank losses
(λy>0).
Table 7, left panel, presents the forecasted loss function for 2007-2009 using the
GM period data. Rule 10 (closely followed by 11) is recommended for minimizing
central bank losses in the following years (2007-2009).
Table 7, right panel, presents the forecasted loss function for 1985-1987 using
the 1955-1985 data. This table clearly shows that rule 10 (again) should be fol-
lowed to minimize the central bank’s loss in the next period. Rule 12 appears to
be optimal for less realistic central bank losses (λw=0 and λy=0).
What is remarkable from all these results is that whatever the period used to
establish the forecasts, rule 10 (closely followed by 9 and 12) dominates in terms
of minimizing the forecasted losses. This is generally the case regardless of the
values of the different weights assigned to each variable (inflation, output, wages
and interest rate differential).
From this exercise, one can therefore state that the NGDP in level rules clearly
dominate the other rules.
If the central bank seeks to minimize such a forecasted loss function in deter-
mining its monetary policy, it should choose this type of monetary policy rule.
6 Interpretation
6.1 Essential facts
Table 8 summarizes our results to capture the essential facts of our exercise.
In terms of fitting the data, the marginal density values show that rule 12 per-
forms better than all others during the GM and GFC/ZLB periods. Rule 1 is best
over 1955-1985, while rule 5 dominates from the full sample estimates.
However, for reasons explained in Section 4.2, the values of the marginal den-
sities are not definitive proof that we have the correct ranking of rules. These
values constitute an indication as to which rules were more or less followed dur-
23
1955-2017 2007-2017 1985-2007 1955-1985
Fitting
Marginal density 5 12 12 1
Central bank loss
Current 9,11 9,10 (1,2,12) 2,10 (1,9) 10
Forecasted 9,12 9,12 (10) 10 (11) 10 (12)
Table 8: Summary of the best rule(s) for each criterion. Rules close to the best
one(s) are in parentheses.
ing the various periods, assuming that the Fed followed a policy rule and that the
economy behaved as in the Smets and Wouters (2007) model.
Note that during the GFC/ZLB and the GM periods, the pure NGDP level rule
best fits the data but the Smets and Wouters (2007) monetary policy rule is very
close during the GM, and even dominates over 1955-1985.
An analysis of the current losses of the central bank generally indicates the
superiority of NGDP level rules for all periods, even if the Taylor rule performs as
well during the GM.
From Table 8, it can be inferred that during the GFC/ZLB, in-sample fitting
and current central bank loss functions indicate that the best performance can be
obtained using some NGDP rules. This is not always the case during the other
periods.
These results are not intended to prove that the Fed followed any given type
of rule in a given period. An explicit rule is only a model that attempts to cap-
ture some monetary policy parameters and explain the methodology whereby the
central bank determines its interest rate.
The estimates show that an NGDP level rule would be best to minimize the
current loss function of the central bank over the various periods (with the Taylor
rule performing somewhat better during the GM period).
Table 8 shows that during almost all sub-periods, minimizing the current cen-
tral bank loss functions does not necessarily lead to one and only one specific
preferable monetary policy rule, even if some NGDP in level rules appear to often
be ahead of or at approximately the same ranking as some other rules.
Importantly, the implications that can be drawn from the forecasted losses are
illuminating in that respect, showing that the choice of NGDP in level rules would
clearly be optimal for minimizing the forecasted losses, whatever the period.
24
6.2 Role of price stickiness and indexation
We examine the coefficients of two important variables of the core model, the de-
gree of price stickiness (ξp) and the degree of indexation to past prices (ιp), to better
understand why some monetary rules perform better than others over the differ-
ent periods.17 Of course, all the coefficients of the models impact the empirical
results (approximately 20 parameters, in addition to those in the monetary rules),
but it would be particularly cumbersome to deal with all of them.
We choose these two parameters because of their significance in terms of mon-
etary policy effectiveness. As noted by Schmitt-Grohé and Uribe (2004), price-
related parameters, such as the degrees of price stickiness and indexation, af-
fect the reaction of monetary policy after a cost-push shock. They also affect the
transmission channel of a monetary policy shock to price dynamics. According
to Woodford (2010), the variance of such a reaction is determinant in assessing
optimal monetary policy.
Let us first examine whether a pattern exists between the different rules during
the same period with respect to fitting, central bank losses and forecasted central
bank losses with the help of Table 9.
In terms of fitting, over the full sample period, no obvious pattern appears
with respect to the degree of price stickiness for rule 5 (the best performer), but
the value of the degree of price indexation is the second-highest value in this case.
This may mean that during the full period, this last variable played a more signif-
icant role than under the other rules in explaining the best fit.
Now, and through the end of this section, we will focus on what is the most
interesting exercise: the analysis of the coefficients concerning the GFC/ZLB and
the GM periods and the comparison of the results between those periods.
In terms of fit, as far as the GFC/ZLB is concerned, and for the best fitting rule
(rule 12), the degree of price stickiness does not exhibit any particular characteris-
tics, but the degree of indexation to past prices is among the highest values (close
to rules 9 and 10).
Over the GM period, the degree of price stickiness shows the highest value,
whereas the degree of indexation to past prices is among the highest.
The analysis of the coefficients concerning current central bank losses during
the GFC/ZLB and GM periods leads to some interesting observations. For the
2007-2017 period the coefficients of the price stickiness variable are almost the low-
est for rules 9 and 10 (with the coefficients of rule 2 being close), whereas those of
the price indexation are the highest, hence the implication that the second variable
played a more important role at the margin than the first one. The coefficients of
price stickiness and price indexation are the second highest for rule 10 over the
17For further details about these coefficients and the core model, please refer to our online ap-
pendix.
25
Sample Rule
1 2 3 4 5 6 7 8 9 10 11 12
ξp
1955-2017 0.803 0.691 0.608 0.646 0.747 0.737 0.737 0.762 0.909 0.873 0.730 0.914
(0.031) (0.034) (0.029) (0.029) (0.028) (0.023) (0.024) (0.023) (0.011) (0.014) (0.013) (0.013)
C,F C F
2007-2017 0.879 0.861 0.902 0.841 0.896 0.893 0.908 0.897 0.866 0.863 0.895 0.875
(0.023) (0.025) (0.018) (0.029) (0.021) (0.007) (0.017) (0.012) (0.032) (0.029) (0.021) (0.026)
C,F C F
1985-2007 0.875 0.859 0.869 0.734 0.871 0.877 0.872 0.878 0.905 0.907 0.905 0.909
(0.031) (0.029) (0.026) (0.025) (0.020) (0.023) (0.020) (0.020) (0.016) (0.017) (0.015) (0.017)
C C,F
1955-1985 0.588 0.566 0.569 0.529 0.568 0.567 0.576 0.575 0.695 0.750 0.632 0.716
(0.037) (0.049) (0.043) (0.021) (0.030) (0.033) (0.039) (0.039) (0.033) (0.038) (0.036) (0.051)
C,F
ιp
1955-2017 0.247 0.206 0.151 0.216 0.275 0.291 0.197 0.269 0.148 0.211 0.243 0.166
(0.055) (0.042) (0.067) (0.077) (0.052) (0.041) (0.051) (0.057) (0.035) (0.033) (0.052) (0.030)
C,F C F
2007-2017 0.288 0.292 0.248 0.278 0.274 0.267 0.234 0.254 0.305 0.305 0.291 0.303
(0.069) (0.081) (0.066) (0.068) (0.050) (0.049) (0.107) (0.053) (0.082) (0.093) (0.119) (0.080)
C,F C F
1985-2007 0.275 0.243 0.244 0.210 0.233 0.239 0.246 0.246 0.347 0.335 0.305 0.320
(0.086) (0.043) (0.053) (0.161) (0.061) (0.051) (0.076) (0.049) (0.073) (0.128) (0.063) (0.054)
C C,F
1955-1985 0.235 0.221 0.228 0.238 0.344 0.333 0.270 0.224 0.207 0.249 0.206 0.227
(0.047) (0.075) (0.058) (0.086) (0.057) (0.120) (0.083) (0.068) (0.065) (0.055) (0.069) (0.072)
C,F
Table 9: Degree of price stickiness (ξp) and indexation (ιp) posterior estimates for
each period and each rule. The corresponding estimated (posterior) standard de-
viation is in parentheses. The best in-sample fit for each period is marked in gray
(Section 4.2). C and F denote the best current and forecasted loss function, respec-
tively.
26
GM period, meaning that, again at the margin, these two variables contribute to
explaining the superiority of rule 10 over the other rules.
Regarding the comparison of the GFC/ZLB and GM periods, note that the
coefficients of price stickiness and price indexation are lower over 2007-2017 than
over 1985-2007, meaning that these two parameters played a less significant role,
at the margin, over the GFC/ZLB period than over the GM period.
If we follow the same type of analysis with the forecasted losses, the implica-
tions are the same as with the current losses for the GFC/ZLB and GM periods.
Nevertheless, when comparing the two periods, the marginal impact of both
variables on forecasted losses appear to be somewhat stronger during the GM
period than the GFC/ZLB period.
Ultimately, the marginal roles of price stickiness and price indexation vary
across periods. The role of both these parameters appears to be higher over the
GM than over the GFC/ZLB periods. The changes in these structural parameters
over time support the implication that there is a need to regularly renew model
estimations to capture changing policy effects.
7 Policy implications
Irrespective of the period in question, central bank’s objectives are not achieved
by one single monetary policy rule with the same weights given to each variable
entering a rule. For each period, there is a preferred monetary policy reaction
function. In other words, for each type of period (more or less stable, crisis, recov-
ery), a given type of reaction function performs better than others. However, if we
consider the current and forecasted loss functions of the central bank, the results
indicate the superiority of NGDP rules in levels, except during the Great Modera-
tion where the Taylor rule works better with the current loss functions (but some
NGDP Level rules are close). The forecasted losses yield non-ambiguous results
on the matter: whatever the period and the specific loss function used, it is opti-
mal to use the NGDP level-type rules. However, there is no specific empirical rule,
i.e., a rule with fixed parameters, that must always be used whatever the period.
Parameter estimates change with respect to the period considered, for any
given monetary policy rule. Policy institutions, which base their forecasts and
policy recommendations on such models and rules, should refresh their estimates
regularly to avoid inaccurate policy conclusions.
In line with Wieland et al. (2012), our analysis demonstrates that central banks
should compare several monetary policy rules to base their policy on a broader
scope of results than those obtained by only one model or monetary policy rule.
Most of central banks’ DSGE models use ad hoc monetary policy rules, such as
Taylor-type rules. It is also standard practice to assume that a central bank seeks
27
to minimize a loss function that includes, at least, inflation and output variances.
Would this minimization process necessarily lead to a standard Taylor rule? Our
results show that this is not necessarily the case. NGDP in level rules are often
superior in terms of minimizing a loss function (current or forecasted). However,
both of these types of rules may still sometimes be (or close to) compatible.
In fine, what is significant is to use a rule, the NGDP level type being probably
the most frequently indicated, especially during crisis and unstable periods, but a
Taylor-type rule would also perform well, especially during more stable periods.
Furthermore, it is necessary to regularly re-estimate the model, and therefore the
monetary policy rule parameters, to better fit to the dynamics of the economy.
8 Conclusion
The purpose of this paper is to shed light on the effects of different monetary pol-
icy rules on the macroeconomic equilibrium. Specifically, we seek to determine,
first, which of the various monetary policy rules is most in line with the historical
data for the US economy and, second, what policy rule would work best to as-
sist the central bank in achieving its objectives via several loss function measures,
current and forecasted.
To conduct this type of analysis, we compare Taylor-type and nominal income
rules through the well-known Smets and Wouters (2007) DSGE model.
We consider twelve monetary policy rules. Four are of the Taylor-type, and
eight are of the nominal income targeting type (NGDP), either in growth or levels.
We test the model with these various rules through Bayesian estimations from
1955 to 2017, over three different periods: 1955-1985, 1985-2007, and 2007-2017.
These sub-periods are selected to capture the impact of policy rules given different
economic environments (more or less stable periods, crisis and recovery).
In terms of fit with historical data, the marginal density values suggest that
one NGDP level targeting rule exhibits the best fit during the GFC/ZLB and the
GM–and an NGDP growth targeting rule is the best fit over the whole sample. A
Taylor-type rule is best during the 1955-1985 period.
The results regarding the current losses of the central bank suggest the superi-
ority of NGDP level targeting rules, over all periods except the Great Moderation,
when a Taylor-type rule performs better. However, during the GFC/ZLB period,
Taylor-type rules yield results that come close to those of NGDP in level rules, and
during the GM period, NGDP in level rules lead also to results that come close to
those of Taylor-type rules.
The results are even more clearly in favor of NGDP in level rules, whatever the
period, when the minimization of a forecasted loss function is used as an instru-
mental goal of the central banker.
28
Several policy implications can be drawn.
First, although a central bank’s objectives are best achieved by using monetary
rules in the decision-making process, these objectives are not achieved by one sin-
gle empirical rule with the same weights given to each variable entering the rule.
For each type of period (more or less stable, crisis, recovery), a specific reaction
function performs better than others.
Second, central banks, which base their forecasts and policy recommendations
on such models and rules, should refresh their estimates regularly to avoid inac-
curate policy decisions.
Third, policy makers should estimate central bank losses (current or fore-
casted) through several empirical monetary policy rules and models to better as-
sess their interest rate decisions.
References
Adolfson, M., Laséen, S., Lindé, J., Svensson, L. E. O., 2011. Optimal monetary
policy in an operational medium-sized DSGE model. Journal of Money, Credit
and Banking 43 (7), 1287–1331.
Adolfson, M., Laséen, S., Lindé, J., Svensson, L. E. O., 2014. Monetary policy trade-
offs in an estimated open-economy DSGE model. Journal of Economic Dynam-
ics and Control 42, 33–49.
Belongia, M. T., Ireland, P. N., 2015. A "working" solution to the question of nomi-
nal GDP targeting. Macroeconomic Dynamics 19 (03), 508–534.
Benchimol, J., Fourçans, A., 2017. Money and monetary policy in the Eurozone: an
empirical analysis during crises. Macroeconomic Dynamics 21 (3), 677–707.
Bernanke, B. S., Mishkin, F., 1997. Inflation targeting: a new framework for mone-
tary policy ? Journal of Economic Perspectives 11 (2), 97–116.
Billi, R. M., 2017. A note on nominal GDP targeting and the zero lower bound.
Macroeconomic Dynamics 21 (08), 2138–2157.
Clarida, R., 2010. The mean of the new normal is an observation rarely realized:
focus also on the tails. Global perspectives, PIMCO.
Coibion, O., Gorodnichenko, Y., Ulate, M., 2018. The cyclical sensitivity in esti-
mates of potential output. Brookings Papers on Economic Activity 49 (2 (Fall)),
343–441.
Cúrdia, V., Woodford, M., 2011. The central-bank balance sheet as an instrument
of monetary policy. Journal of Monetary Economics 58 (1), 54–79.
29
Del Negro, M., Schorfheide, F., Smets, F., Wouters, R., 2007. On the fit of New
Keynesian models. Journal of Business & Economic Statistics 25, 123–143.
Fair, R. C., Howrey, E. P., 1996. Evaluating alternative monetary policy rules. Jour-
nal of Monetary Economics 38 (2), 173–193.
Fernald, J. G., 2015. Productivity and potential output before, during, and after the
Great Recession. NBER Macroeconomics Annual 29 (1), 1–51.
Fernández-Villaverde, J., Rubio-Ramírez, J. F., 2004. Comparing dynamic equilib-
rium models to data: a Bayesian approach. Journal of Econometrics 123 (1), 153–
187.
Frankel, J., 2014. Nominal GDP targeting for middle-income countries. Central
Bank Review 14 (3), 1–14.
Frankel, J., Chinn, M., 1995. The stabilizing properties of a nominal GNP rule.
Journal of Money, Credit and Banking 27 (2), 318–334.
Friedman, M., 1971. A monetary theory of nominal income. Journal of Political
Economy 79 (2), 323–337.
Friedman, M., 1982. Monetary policy: theory and practice. Journal of Money,
Credit and Banking 14 (1), 98–118.
Galí, J., 2015. Monetary policy, inflation and the business cycle: an introduction to
the New Keynesian framework, 2nd Edition. Princeton, NJ: Princeton Univer-
sity Press.
Garín, J., Lester, R., Sims, E., 2016. On the desirability of nominal GDP targeting.
Journal of Economic Dynamics and Control 69, 21–44.
Geweke, J., 1999. Using simulation methods for Bayesian econometric models: in-
ference, development, and communication. Econometric Reviews 18 (1), 1–73.
Gorton, G. B., 2009. Information, liquidity, and the (ongoing) panic of 2007. Amer-
ican Economic Review 99 (2), 567–72.
Hall, R., Mankiw, N., 1994. Nominal income targeting. Working Papers 4439, Na-
tional Bureau of Economic Research.
Hendrickson, J. R., 2012. An overhaul of Federal Reserve doctrine: nominal in-
come and the Great Moderation. Journal of Macroeconomics 34 (2), 304–317.
Iacoviello, M., Neri, S., 2010. Housing market spillovers: evidence from an esti-
mated DSGE model. American Economic Journal: Macroeconomics 2 (2), 125–
164.
30
Justiniano, A., Preston, B., 2010. Can structural small open-economy models ac-
count for the influence of foreign disturbances ? Journal of International Eco-
nomics 81 (1), 61–74.
Kiley, M. T., 2013. Output gaps. Journal of Macroeconomics 37 (C), 1–18.
Kim, D. H., Singleton, K. J., 2012. Term structure models and the zero bound: an
empirical investigation of Japanese yields. Journal of Econometrics 170 (1), 32–
49.
Koop, G., 2003. Bayesian Econometrics. New York, NY: John Wiley & Sons.
Krippner, L., 2013. Measuring the stance of monetary policy in zero lower bound
environments. Economics Letters 118 (1), 135–138.
Kydland, F. E., Prescott, E. C., 1977. Rules rather than discretion: the inconsistency
of optimal plans. Journal of Political Economy 85 (3), 473–91.
McCallum, B. T., 1973. Friedman’s missing equation: another approach.
Manchester School 41 (3), 311–328.
McCallum, B. T., 1987. The case for rules in the conduct of monetary policy: a
concrete example. Review of World Economics 123 (3), 415–429.
McCallum, B. T., 2015. Nominal GDP targeting: Policy rule or discretionary
splurge ? Journal of Financial Stability 17, 76–80.
McCallum, B. T., Nelson, E., 1999. Nominal income targeting in an open-economy
optimizing model. Journal of Monetary Economics 43 (3), 553–578.
Meade, J. E., 1978. The meaning of internal balance. Economic Journal 88 (351),
423–435.
Meltzer, A. H., 2012. Federal Reserve policy in the Great Recession. Cato Journal
32 (2), 255–263.
Nikolsko-Rzhevskyy, A., Papell, D., 2012. Taylor rules and the Great Inflation.
Journal of Macroeconomics 34 (4), 903–918.
Nikolsko-Rzhevskyy, A., Papell, D. H., Prodan, R., 2014. Deviations from rules-
based policy and their effects. Journal of Economic Dynamics and Control 49,
4–17.
Rudebusch, G., 2002a. Assessing nominal income rules for monetary policy with
model and data uncertainty. Economic Journal 112 (479), 402–432.
31
Rudebusch, G. D., 2002b. Term structure evidence on interest rate smoothing and
monetary policy inertia. Journal of Monetary Economics 49 (6), 1161–1187.
Schmitt-Grohé, S., Uribe, M., 2004. Optimal fiscal and monetary policy under
sticky prices. Journal of Economic Theory 114 (2), 198–230.
Schmitt-Grohé, S., Uribe, M., 2007. Optimal simple and implementable monetary
and fiscal rules. Journal of Monetary Economics 54 (6), 1702–1725.
Smets, F., Wouters, R., 2003. An estimated dynamic stochastic general equilibrium
model of the Euro area. Journal of the European Economic Association 1 (5),
1123–1175.
Smets, F., Wouters, R., 2007. Shocks and frictions in US business cycles: a Bayesian
DSGE approach. American Economic Review 97 (3), 586–606.
Sumner, S., 2014. Nominal GDP targeting: a simple rule to improve Fed perfor-
mance. Cato Journal 34 (2), 315–337.
Sumner, S., 2015. Nominal GDP futures targeting. Journal of Financial Stability 17,
65–75.
Svensson, L. E. O., 1999. Inflation targeting as a monetary policy rule. Journal of
Monetary Economics 43 (3), 607–654.
Svensson, L. E. O., Williams, N., 2009. Optimal monetary policy under uncertainty
in DSGE models: a Markov jump-linear-quadratic approach. In: Central Bank-
ing, Analysis, and Economic Policies Book Series. Vol. 13. Central Bank of Chile,
pp. 077–114.
Taylor, J. B., 1979. Staggered wage setting in a macro model. American Economic
Review 69 (2), 108–13.
Taylor, J. B., 1986. An appeal for rationality in the policy activism debate. In: R.W.
Hafer, ed. The monetary versus fiscal policy debate: lessons from two decades.
Littlefield, NJ: Rowman and Allenheld, pp. 151–163.
Taylor, J. B., 1987. Externalities associated with nominal price and wage rigidities.
In: Barnett, W., Singleton, K. (Eds.), New approaches to monetary economics.
Cambridge, UK: Cambridge University Press, pp. 350–367.
Taylor, J. B., 1993. Discretion versus policy rules in practice. Carnegie-Rochester
Conference Series on Public Policy 39 (1), 195–214.
Taylor, J. B., 1999. A historical analysis of monetary policy rules. In: NBER Chap-
ters. National Bureau of Economic Research, pp. 319–348.
32
Taylor, J. B., 2012. Monetary policy rules work and discretion doesn’t: a tale of two
eras. Journal of Money, Credit and Banking 44 (6), 1017–1032.
Taylor, J. B., 2013. MacroMania on nominal GDP targeting and the Taylor rule.
Blog post, EconomicsOne.com.
Taylor, J. B., Wieland, V., 2012. Surprising comparative properties of monetary
models: results from a new model database. Review of Economics and Statistics
94 (3), 800–816.
Walsh, C. E., 2015. Goals and rules in central bank design. International Journal of
Central Banking 11 (S1), 295–352.
Wieland, V., Cwik, T., Müller, G. J., Schmidt, S., Wolters, M., 2012. A new com-
parative approach to macroeconomic modeling and policy analysis. Journal of
Economic Behavior & Organization 83 (3), 523–541.
Woodford, M., 2010. Optimal monetary stabilization policy. In: Friedman, B. M.,
Woodford, M. (Eds.), Handbook of Monetary Economics. Vol. 3 of Handbook of
Monetary Economics. Elsevier, Ch. 14, pp. 723–828.
Woodford, M., 2012. Methods of policy accommodation at the interest-rate lower
bound. Proceedings - Economic Policy Symposium - Jackson Hole, 185–288.
Wu, J. C., Xia, F. D., 2016. Measuring the macroeconomic impact of monetary pol-
icy at the zero lower bound. Journal of Money, Credit and Banking 48 (2-3),
253–291.
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