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Central banks' monetary policy rules being consistent with policy objectives are a fundamental of applied monetary economics. We seek to determine, 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 measures. 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 central 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).
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Central bank losses and monetary
policy rules: a DSGE investigation
Jonathan Benchimolyand André Fourçansz
March 15, 2019
Abstract
Central banks’monetary policy rules being consistent with policy
objectives are a fundamental of applied monetary economics. We seek
to determine, …rst, 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 objec-
tives via several loss function measures. We use Bayesian estimations
to evaluate twelve monetary policy rules from 1955 to 2017 and over
three di¤erent sub-periods. We …nd 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.
JEL Classi…cation: E52, E58, E32.
.
This paper does not necessarily re‡ect 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 Eng-
land, and National Bank of Romania research seminars for their valuable comments.
yBank of Israel, POB 780, 91007 Jerusalem, Israel. Phone: +972-2-6552641. Fax:
+972-2-6669407. Corresponding author. Email: jonathan.benchimol@boi.org.il
zESSEC Business School and THEMA, Cergy Pontoise, France.
1
Please cite this paper as:
Benchimol, J., and Fourçans, A., 2019. Central bank losses and
monetary policy rules: a DSGE investigation. International Review
of Economics & Finance, 61, 289-303.
2
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.2
However, criticism of such monetary policy rules also increased,3especially
during and after the Global Financial Crisis4(GFC/ZLB), arguing that nom-
inal income targeting could be a better way to achieve the central banks’
objectives.
An interesting way to compare and evaluate di¤erent monetary policy
proposals and rules is to introduce them within the framework of a macro-
economic Dynamic Stochastic General Equilibrium (DSGE) model. Because
the dynamics are so important and di¢ cult to work through intuitively, such
empirical models can provide invaluable clari…cation 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 …tted for the US, to evaluate di¤er-
ent monetary policy rules and their consequences in terms of current and
forecasted central bank objectives.
These objectives may di¤er for various reasons, hence the need to analyze
several 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 empiri-
cal one based on historical economic dynamics. The latter appears better
suited to capture the real economic behavior of a central bank. Indeed, com-
mitment, discretionary or optimal monetary rules imply theoretically opti-
mal behavior but do not necessarily 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 o¤ers a more realistic framework regarding central bank behavior
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 Ireland (2015), and McCallum (2015) for example.
3
than the theoretical 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, especially during di¤erent sample periods, which we
consider in our analysis. Our approach allows for such an analysis through
a medium-scale DSGE model.
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 in‡ation gap, an
output gap and output gap growth; (2) a structure à la Taylor (1993), where
the nominal interest rate responds to an ination gap and an output gap;
(3) a structure à la Galí (2015), where the nominal interest rate responds
to an in‡ation gap, an output gap and a natural interest rate de…ned as the
interest rate in the ‡exible-price economy; and (4) a structure à la Garín et al.
(2016), where the nominal interest rate responds to an in‡ation 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 di¤erent
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.
Speci…cally, we estimate all of the parameters over several sample periods:
the overall available sample (1955-2017) and three sub-samples, each with
di¤erent 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
normally distributed. To overcome this statistical problem, while taking into
consideration most of such unconventional monetary policies (credit easing,
quantitative easing, and forward guidance), we use the shadow rate6(Kim
and Singleton,2012;Krippner,2013). The shadow rate is a version of the
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, re‡ecting the
Fed’s unconventional policies. When quantitative easing or forward guidance is pursued,
4
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 …ts (which
monetary 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 …nd that when considering the central bank’s loss functions, the es-
timates often indicate the superiority of NGDP level rules, although Taylor-
type rules have nearly identical implications. However, this being given, the
results suggest that historical …tting 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 speci…c rule performs better than
others. Policy institutions, which base their forecasts and policy recommen-
dations 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 2describes
the theoretical setup. Section 3describes the empirical methodology. Mone-
tary rule parameters estimates as well as in-sample …t 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 7draws some policy im-
plications, and Section 8concludes 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.
However, in their article and other working paper versions, those authors do
not describe a ‡exible-price economy. We perform this work in the detailed
description of the log-linearized sticky- and ‡exible-price economies in our
online appendix.
This (generic) model, also detailed in the online appendix, needs to be
completed by adding an ad hoc monetary policy reaction function (Table 1).
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 a¢ ne term structure model to generate an
ective rate.
5
Despite their di¤erent formulations, all of these functions include a smoothing
process that captures the degree of rule-speci…c smoothing.
Taylor-type rules
Model 1is the original Smets and Wouters (2007) monetary policy
rule, which gradually responds to deviations of in‡ation (t) from an
in‡ation objective (normalized to zero), the output gap, de…ned as
the di¤erence between sticky-price (yt) and ‡exible-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 in‡ation from an in‡ation objective
(normalized to zero) and of the output gap, as previously de…ned.7
Model 3is the Galí (2015) monetary policy rule, which gradually re-
sponds to the natural interest rate (r
t), as de…ned in Galí (2015), devi-
ations of ination from an in‡ation objective (normalized to zero) and
of the output gap, as previously de…ned.
Model 4gradually responds to the deviations of in‡ation from an in‡a-
tion 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
(t+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
gradually responds to deviations of nominal output growth from its
exible-price counterpart (FPC).
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.
6
Model 7is the NGDP Growth targeting monetary policy rule including
a natural interest rate (NIR) component, where the policy gradually re-
sponds to the NIR, as in Rudebusch (2002a), and deviations of nominal
output growth from its ‡exible-price counterpart.
Model 8is the NGDP Growth targeting monetary policy rule where the
policy 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 ‡exible-price counterpart,8as suggested by McCallum (2015),
and deviations of the output gap from the previous period (output gap
growth, as in model 1).
Model 10 is the NGDP Level targeting monetary policy rule, which
gradually responds to nominal output level deviations from its ‡exible-
price counterpart (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 deviations of the nominal output level from its ‡exible-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 …rst 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
contemporary literature.
As these rules are all ad hoc, they do not require changes in the speci-
cation of the core model. The unique di¤erentiating feature of the twelve
8The level of nominal output is pt+yt, where prices ptare deducted from the de…nition
of in‡ation t=ptpt1.
7
Models Sources Monetary policy rules
1Smets and Wouters (2007)rt=rt1+ (1 ) [rt+ry(ytyp
t)] + r4y(4yt 4yp
t) + "r
t
2Taylor (1993)rt=rt1+ (1 ) [rt+ry(ytyp
t)] + "r
t
3Galí (2015)rt=rt1+ (1 ) [r
t+rt+ry(ytyp
t)] + "r
t
4Garín et al. (2016)rt=rt1+ (1 ) [rt+ry4yt] + "r
t
5Adapted NGDP Growth Targeting rt=rt1+ (1 ) [rn(t+4yt 4yp
t)] + r4y(4yt 4yp
t) + "r
t
6NGDP Growth + FPC Targeting rt=rt1+ (1 ) [rn(t+4yt 4yp
t)] + "r
t
7NGDP Growth + NIR Targeting rt=rt1+ (1 ) [r
t+rn(t+4yt 4yp
t)] + "r
t
8NGDP Growth Targeting rt=rt1+ (1 ) [rn(t+4yt)] + "r
t
9Adapted NGDP Level Targeting rt=rt1+ (1 ) [rn(pt+ytyp
t)] + r4y(4yt 4yp
t) + "r
t
10 NGDP Level + FPC Targeting rt=rt1+ (1 ) [rn(pt+ytyp
t)] + "r
t
11 NGDP Level + NIR Targeting rt=rt1+ (1 ) [r
t+rn(pt+ytyp
t)] + "r
t
12 NGDP Level Targeting rt=rt1+ (1 ) [rn(pt+yt)] + "r
t
NIR and FPC stand for the natural interest rate (r
t) and the ‡exible-price counterpart à la Galí (2015), respectively.
Table 1: Summary of monetary policy rules used in this study
8
models therefore comes from their respective monetary policy rule. Concern-
ing NGDP Level targeting rules (models 9 to 12), we add to the core model
and the monetary policy rule the de…nition of prices, derived from (in log
form) t=ptpt1, where ptrepresents the log-price index at time t.
In addition, the in‡ation rate in the ‡exible-price economy at time tis
p
t= 0, as in Smets and Wouters (2007). Then, the ‡exible-price nominal
income is only de…ned 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 di¤erent 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 dis-
cretionary; 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, the crisis and recovery period
when monetary policy followed an unusual ZLB track.
During our …rst sub-sample (1955-1985), monetary policy was rather dis-
cretionary and severely criticized in the literature (Friedman,1982). Since
the 1980s, the predictability and stability of monetary policy has improved,
with many researchers currently recommending rule-based rather than dis-
cretionary monetary policy decisions (Kydland and Prescott,1977;Taylor,
1986,1987;Friedman,1982;Taylor,1993). Notice that monetary policies oc-
curring during our …rst 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), describ-
ing 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 de…ne a sub-sample with a
relatively stable economy (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 re-
covery period literature (Gorton,2009;Cúrdia and Woodford,2011;Benchi-
mol and Fourçans,2017).
9
The series are quarterly, and data transformations, data sources9and
measurement 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 spec-
ify and calibrate prior distributions for all model parameters as in Smets
and Wouters (2007). A detailed description of these parameters, and their
calibrations, is provided in the online appendix.
Except for NGDP targeting rules, monetary policy rule parameters in
Table 2have the same calibration as in Smets and Wouters (2007).
Law Mean Std.
Beta 0.75 0.10
rNormal 1.50 0.25
ryNormal 0.125 0.05
ryNormal 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).
Of course, r4yequals zero in models 2 to 4, 6 to 8, and 10 to 12. rand
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.
3.3 Estimation
As in Smets and Wouters (2007), we apply Bayesian techniques to estimate
our DSGE models with derent speci…cations of monetary policy rules. We
9Detailed data sources, measurement equations and data transformations are available
in the online appendix.
10 Measurement equations are presented in the online appendix.
11 From December 16, 2008, to December 15, 2015, the e¤ective 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).
10
estimate all the parameters presented above over the four di¤erent periods
de…ned in Section 3.1.
To achieve draw acceptance rates between 20% and 40%, we calibrate the
tuning 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 di¤erent mon-
etary rules. All the estimation results12 are available in the online appendix.
These results con…rm well-identi…ed 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 identication issues.
4 Monetary rule parameters and in-sample
t
Parameter estimates are detailed in the online appendix with all impulse re-
sponse functions and variance decompositions. To draw policy conclusions
from our models, we assess monetary policy rule parameters (estimated val-
ues) in Section 4.1 and the models’in-sample …t in Section 4.2.
4.1 Monetary rule parameters
Fig. 1presents the estimates of the smoothing parameter (), the in‡a-
tion coe¢ cient (r), the output gap coe¢ cient (ry), the output gap growth
coe¢ cient (r4y) and the nominal income coe¢ cient (rn).
As Fig. 1shows, the smoothing parameter is in line with the literature
(Justiniano and Preston,2010), at approximately 0.8, and rather stable over
time, although it appears somewhat smaller for rules 9, 10 and 12, a result
in accordance with Rudebusch (2002a,b).
The in‡ation coe¢ cient (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 in‡ation developments than during more stable
periods, notably than during the GM, from 1985 to 2007.
12 Estimated parameters (mean), estimated standard errors (std), highest posterior den-
sity intervals (HPDi) and estimated shocks are presented in the online appendix. Other
detailed results are available upon request.
11
The value of the coe¢ cient of the output gap varies across the periods. It
appears 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 di¤erence is not as signi…cant when we compare the
crisis period with the 1985-2007 period (except for rule 3, to some extent).
1955-2017
1 2 3 4 5 6 7 8 910 1112
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 9 1011 12
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 8 910 1112
0
1
2
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9 1011 12
0
1
2
1 2 3 4 5 6 7 8 9101112
0
1
2
1 2 3 4 5 6 7 8 910 1112
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 1011 12
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 8 910 1112
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 1011 12
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 8 910 1112
0
1
2
1 2 3 4 5 6 7 8 9 101112
0
1
2
1 2 3 4 5 6 7 8 9 1011 12
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).
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 coe¢ cient varies somewhat across periods and
rules (between 0.10 and 0.23). At least for rule 9, this coe¢ cient appears to
be somewhat higher during the GFC/ZLB than during the GM, implying a
12
larger reaction to output growth during the crisis than during the previous,
stabler period. For rule 1, this coe¢ cient 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 co cient associated with the NGDP rules is higher
for the growth rules than the level rules, over all periods, a result that echoes
Rudebusch (2002a). For the growth and level rules, this coe¢ cient is lower
during the GFC/ZLB than otherwise, especially during the GM. The coef-
cient 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 …t
Which monetary rule best …ts the historical data is signi…cant for understand-
ing and analyzing the behavior of a central bank and drawing implications
about policy debates. It does not mean that the central banker always fol-
lowed monetary rules, but at least implicitly, he (generally) behaved “as if”
he followed some kind 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 …t is important to determine whether
historical data (sample) are more or less in line with data generated by the
estimated model. Table 3shows 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
1 2 3 4 5 6 7 8 9 10 11 12
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
approximation). Best values for each period in gray.
Table 3suggests that the last NGDP rule in levels (rule 12), the pure
NGDP level targeting without ‡exible-price output, best …ts 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,
13 The Geweke (1999) mean harmonic estimator provides a similar ranking of models.
13
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 …rst over the whole sample.
For each period, a di¤erent monetary policy rule best …ts the historical
data, except for rule 12 that places …rst 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 cer-
tain characteristics 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 question (Koop,2003;Fernández-Villaverde and Rubio-
Ramírez,2004;Del Negro et al.,2007;Benchimol and Fourçans,2017).
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 minimization 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 repre-
sented 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
variances of the variables of interest from the central bank’s perspective.
These variances are estimated for each model and for each period.
14
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 de…ning these functions, we compute the ex post loss functions
consistent with the estimated DSGE model. This approach is used in the
literature to investigate empirical monetary policy rules (Taylor,1979;Fair
and Howrey,1996;Taylor,1999) and is di¤erent 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 appen-
dix) introduces labor-related variables into the in‡ation and output equa-
tions. By minimizing 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 de…ned in a traditional way
as14
Lt=var (t) + yvar (ytyp
t) + rvar (rt) + wvar (wt)(2)
where var (:)is the variance operator, ythe weight on output gap vari-
ances, rthe weight on nominal interest rate di¤erential variance, and w
the weight on wage in‡ation variance. The weight on price in‡ation variance
is normalized to unity. tis price in‡ation, ytyp
tthe output gap, rtthe
nominal interest rate di¤erential, and wtwage in‡ation.15
First, in Fig. 2, we present the estimated variances of each variable
(in‡ation, output gap, nominal interest rate derential, and wage in‡ation)
entering the central bank loss functions.
The variances of all variables under consideration are signi…cantly higher
before 1985 and over the full sample. Even during the 2007-2017 period,
these variances were lower than before 1985 and little di¤erent from those
during the GM period. The fact that estimated variances over the GFC/ZLB
period are comparable across the models with those of the GM period does
not mean that the variances of historical data during the GFC/ZLB and GM
are comparable. Indeed, the variances presented in Fig. 2are estimated
from the models while assuming that the Fed followed various rules and the
14 See Galí (2015) for further details. Another loss measure based on the squared distance
of variables generated by the models can be de…ned as
Lt=2
t+y(ytyp
t)2+r(rt)2+ww2
t(1)
By the de…nition of the variance operator, this type of formulation leads to a ranking
similar to those given by Eq. 2.
15 See the online appendix for further details on the variables in the models.
15
US economy behaved as in the Smets and Wouters (2007) model. The high
in‡ation period cum various signi…cant ups and downs in economic activity
and interest rates explain the high values observed between 1955 and 1985.
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 10 1112
0
0.1
0.2
0.3
0.4
1985-2007
1 2 3 4 5 6 7 8 91011 12
0
0.1
0.2
0.3
0.4
1955-1985
1 2 3 4 5 6 7 8 91011 12
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 10 1112
0
0.2
0.4
1 2 3 4 5 6 7 8 91011 12
0
0.2
0.4
1 2 3 4 5 6 7 8 91011 12
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 10 1112
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 91011 12
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 91011 12
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 10 1112
0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 91011 12
0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 91011 12
0
0.2
0.4
0.6
Figure 2: Estimated variances of central bank loss function variables, for
each period and each rule.
However, changes in the Fed’s monetary policy and the stabilization pe-
riod that occurred during the 1990s explain the low variance of the GM
period relative 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 in‡ation rate come close. The low interest rates of the GFC/ZLB
period lead to lower variances of the shadow interest rate di¤erentials during
the GFC/ZLB than during the GM period, although the di¤erence is not
16
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 the
GFC/ZLB, just a bit higher than during the GM (Fig. 2). The low vari-
ances 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.,2019).
During the GFC/ZLB, the correlation between the current output (his-
torical) 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.
Second, we compute ad hoc loss functions based on Eq. 2. The following
heatmaps (Tables 4and 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 variance of one variable included in the loss functions increases. This is
directly related 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 w.
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.
The results vary somewhat when considering the 1955-1985 period (Table
5, right panel), where rule 10 clearly dominates.
Over the 1955-2017 period, generally rule 11 dominates. Yet if the central
bank does not pay attention to wage in‡ation, rule 9 leads to the best results.
16 The same type of result comes from some from non-DSGE models, for instance Fernald
(2015).
17
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 de…ned 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
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 de…ned separately in relation to each line. The lighter the shading is, the smaller the
loss.
Over the GFC/ZLB period, such sensitivity to wage in‡ation is low, and
the ranking of rules does not particularly depend on taking wages into con-
sideration. The sensitivity of the results is also low with respect to the values
of yand r. The same can be said during the 1955-1985 period and even
during the GM period, albeit to a somewhat lesser extent. Interestingly, in
all periods, the change in the loss is minor for a given ywhen rchanges,
compared to the change in the loss for a given rwhen ychanges. 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 di¤erential over time, hence the
small gain generated by changing the interest rate di¤erential coe¢ cient in
the central bank loss function for a given yor w.
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 con…guration (y= 0). However, had it
employed Taylor-type rules 1 and 2, the di¤erence in terms of loss would have
been minor. Over more stable periods such as the GM period, the central
bank would have minimized 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 dis-
tribution 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 in‡ation, output-gap, nominal interest rate di¤erential
and wage in‡ation 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 6and 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
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 de…ned 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
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 de…ned
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 fore-
casted losses diverge from the ex post actual losses over the di¤erent sub-
periods. A central 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 interested 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 in‡ation 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 (y>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 followed to minimize the central bank’s loss in the next period.
Rule 12 appears to be optimal for less realistic central bank losses (w= 0
and y= 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 di¤erent weights assigned to each variable (in‡ation,
output, wages and interest rate di¤erential).
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
determining its monetary policy, it should choose this type of monetary policy
rule.
6 Interpretation
6.1 Essential facts
Table 8summarizes our results to capture the essential facts of our exercise.
In terms of …tting the data, the marginal density values show that rule
12 performs better than all others during the GM and GFC/ZLB periods.
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.
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
densities are not denitive proof that we have the correct ranking of rules.
These values constitute an indication as to which rules were more or less
followed during 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 …ts 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
tting and current central bank loss functions indicate that the best perfor-
mance 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 capture 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 8shows that during almost all sub-periods, minimizing the current
central bank loss functions does not necessarily lead to one and only one
speci…c preferable monetary policy rule, even if some NGDP in level rules
24
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.
6.2 Role of price stickiness and indexation
We examine the coe¢ cients of two important variables of the core model, the
degree of price stickiness (p) and the degree of indexation to past prices (p),
to better understand why some monetary rules perform better than others
over the di¤erent periods.17 Of course, all the coe¢ cients 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 signi…cance in terms
of monetary policy e¤ectiveness. As noted by Schmitt-Grohe and Uribe
(2004), price-related parameters, such as the degrees of price stickiness and
indexation, a¤ect the reaction of monetary policy after a cost-push shock.
They also a¤ect 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 …rst examine whether a pattern exists between the di¤erent rules
during the same period with respect to …tting, central bank losses and fore-
casted central bank losses with the help of Table 9.
In terms of …tting, 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 signi…cant role than under the other rules in explaining the
best …t.
Now, and through the end of this section, we will focus on what is the most
interesting exercise: the analysis of the coe¢ cients concerning the GFC/ZLB
and the GM periods and the comparison of the results between those periods.
In terms of …t, as far as the GFC/ZLB is concerned, and for the best …tting
rule (rule 12), the degree of price stickiness does not exhibit any particular
characteristics, but the degree of indexation to past prices is among the
17 For further details about these coe¢ cients and the core model, please refer to our
online appendix.
25
Sample Rule
1 2 3 4 5 6 7 8 9 10 11 12
p
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) (0.0 11) (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) (0.0 32) (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) (0.0 16) (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) (0.0 33) (0.038) (0.036) (0.051)
C,F
p
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) (0.0 35) (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) (0.0 82) (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) (0.0 73) (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) (0.0 65) (0.055) (0.069) (0.072)
C,F
Table 9: Degree of price stickiness (p) and indexation (p) posterior estimates
for each period and each rule. The corresponding estimated (posterior) stan-
dard deviation is in parentheses. The best in-sample …t for each period is
marked in gray (Section 4.2). C and F denote the best current and forecasted
loss function, respectively.
26
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 coe¢ cients concerning current central bank losses
during the GFC/ZLB and GM periods leads to some interesting observations.
For the 2007-2017 period the coe¢ cients of the price stickiness variable are
almost the lowest for rules 9 and 10 (with the coe¢ cients 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 …rst one. The coe¢ cients of price stickiness and price indexation are the
second highest for rule 10 over the 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 coe¢ cients of price stickiness and price indexation are lower over 2007-
2017 than over 1985-2007, meaning that these two parameters played a less
signi…cant 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
implications 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 e¤ects.
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 vari-
able 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 sta-
ble, crisis, recovery), 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 lev-
els, except during the Great Moderation where the Taylor rule works better
27
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 speci…c loss function used, it is optimal to use the NGDP
level-type rules. However, there is no speci…c empirical rule, i.e., a rule with
xed 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 to minimize a loss function that includes, at least, in‡ation
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 …ne, what is signi…cant 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 …t
to the dynamics of the economy.
8 Conclusion
The purpose of this paper is to shed light on the e¤ects of di¤erent monetary
policy rules on the macroeconomic equilibrium. Speci…cally, we seek to de-
termine, …rst, 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 assist 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 in-
come 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
28
or levels. We test the model with these various rules through Bayesian estima-
tions from 1955 to 2017, over three di¤erent periods: 1955-1985, 1985-2007,
and 2007-2017. These sub-periods are selected to capture the impact of pol-
icy rules given derent economic environments (more or less stable periods,
crisis and recovery).
In terms of …t with historical data, the marginal density values suggest
that one NGDP level targeting rule exhibits the best …t during the GFC/ZLB
and the GMand an NGDP growth targeting rule is the best …t 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
superiority 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
instrumental goal of the central banker.
Several policy implications can be drawn.
First, although a central bank’s objectives are best achieved by using mon-
etary rules in the decision-making process, these objectives are not achieved
by one single empirical rule with the same weights given to each variable en-
tering the rule. For each type of period (more or less stable, crisis, recovery),
a speci…c reaction function performs better than others.
Second, central banks, which base their forecasts and policy recommen-
dations on such models and rules, should refresh their estimates regularly to
avoid inaccurate policy decisions.
Third, policy makers should estimate central bank losses (current or fore-
casted) through several empirical monetary policy rules and models to better
assess their interest rate decisions.
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... New Deal policies) played an important role in the weak recovery during the Great Depression. 6 The estimated price markup shock identifies the oil crisis as the main driving force of the Stagflation. Nevertheless, the overall increase in wages may also played an important role in this recession. ...
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We revisit three major US recessions through the lens of a standard medium-scale DSGE model (Smets and Wouters, 2007) augmented with financial frictions. We first estimate the DSGE model using a Bayesian approach for three alternative periods, each containing a major US recession: the Great Depression, the Stagflation and the Great Recession. Then, we assess the stability of structural parameters, and analyze what frictions were particularly important and what shocks were the main drivers of aggregate fluctuations in each historical period. This exercise can be understood as a test of the standard New-Keynesian DSGE model with financial accelerator in closed economies. We find that the estimated DSGE model is able to provide a sound explanation of all three recessions by closely relating both estimated structural shocks and frictions with well known economic events. JEL classification: E30, E44
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We revisit three major US recessions through the lens of a standard medium-scale DSGE model (Smets, F., and R. Wouters. 2007. “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach.” The American Economic Review 97: 586–606.) augmented with financial frictions. We first estimate the DSGE model using a Bayesian approach for three alternative periods, each containing a major US recession: the Great Depression, the Stagflation and the Great Recession. Then, we assess the stability of structural parameters, and analyze what frictions were particularly important and what shocks were the main drivers of aggregate fluctuations in each historical period. This exercise can be understood as a test of the standard New-Keynesian DSGE model with financial accelerator in closed economies. We find that the estimated DSGE model is able to provide a sound explanation of all three recessions by closely relating both estimated structural shocks and frictions with well known economic events.