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UNIVERSITÀ DEGLI STUDI DI PADOVA
Dipartimento di Scienze Economiche “Marco Fanno”
ASSESSING DIFFERENT DRIVERS OF THE
GREATMODERATION IN THE U.S.
EFREM CASTELNUOVO
Università di Padova
August 2006
“MARCO FANNO” WORKING PAPER N.25
Assessin g Different Drivers of the
GreatModerationintheU.S.
∗
Efrem Casteln uovo
University of Padua
August 2006
Abstract
This paper employs a calibrated new-Keynesian DSGE model to assess the
relative importance of two different, potentially important drivers of the Great
Moderation in the U.S., namely ’good policy’ vs. ’good luck’. The calibrated
model is capable to replicate the actual standard deviations of inflation and out-
put. Factual and counterfactual simulations are run in order to gauge the relative
importance of the systematic m o netary policy vs. the stochastic shocks hitting
the economic system in shaping some macroeconomic volatilities. Importantly,
under the bad policy scenario sunspots may influence the equilibrium values of the
macroeconomic variables of interest, and distortions in the transmission mecha-
nism going from the structural shocks to the variables of interest are allo wed for.
Our findings support the relevance of both drivers in causing inflation volatil-
ity. By contrast, output volatility can hardly be explained by a monetary policy
switch like the one occurred in the U.S. at the end of the ’70s.
JEL classification: E30, E52.
Keywords: Great Moderation, indeterminacy, good policy, good luck, coun-
terfactual simulations.
∗
First version: June 2006. Paper presented at the Society for Computational Economics 2006 (Li-
massol) and the Far Eastern Meeting of the Econometric Society 2006 (Beijing). We thank Margherita
Fort, Francesco Lisi, Giovanni Lombardo, Fabio Milani, Mic hael Reiter, and Paolo Surico for help-
ful discussions. All remaining errors are ours. Address for correspondence: Efrem Castelnuovo,
Department of Economics, Univ ersity of Padua, Via del Santo 33, I-35123 Padova (PD). E-mail:
efrem.castelnuo vo@unipd.it.
1 Introduction
One of the most debated topics in modern macroeconomics is undiscussably the ’Great
Moderation’, i.e. th e strik ing r edu ctio n o f inflation and ou tput volatilities occurred
in the last two decades in sev eral industrialized economies. This fact, common across
several countries, is surely a feature of the U.S. economy. Table 1 displays the boot-
strapped v ola tilities of annualized G D P inflation and detrended output in two differen t
samples, i.e. 1960Q1-1979Q3 , 1984Q1-1999Q4.
1
,
2
Quite evidently, there is some insta-
bility regarding these vo latilities. For instance, let’s take the statistics regarding the
subsamples reported in the firstrowoftheTable.
3
Notab ly, the median value of the
volatility of the inflation rate falls from 2.48 to 0.95, while its 90% confidence interval
evidently shrinks, and its standard deviation (not shown in th e Table) d rop s from 0.65
to 0.14. As far as detrended output is concerned, the median value of its bootstrapped
volatilit y lo wers from 1.69 to 0.92, while also its confidence interval tightens, and its
standard deviation mo v es from 0.5072 do wn to 0.3435. Table 1 shows that this ten-
dency finds empirical support also when employing the CBO output trend. Ov era ll,
these figures suggest that since the beginning of the ’80s the U.S. econom y has show n
a m u ch calmer behavior, a conclusion supported b y several recen t studies (Kim and
Nelson [1999], McConnel and P erez-Quiros [2000], Blanchard and Simon [2001], Stock
and Watson [2003], and Kim, Nelson, and Piger [2004] for o utput; Mumtaz and Surico
[2006] for the inflation rate).
[Table 1 about her e]
1
The beginning of the second subsample is suggested by several studies on the Great Moderation
(see references cited later). The exclusion of the period 1979Q4-1983Q4 is due to the ’experiment’
conducted by the Fed in that period. The choice of cutting the sample in 1999Q4 is justified by the
apparen t ’disconnect’ between inflation and output volatilities that has been observed in the U.S. since
the beginning of the current century (Gordon, 2005). However, our results are robust to the extension
of the second subsample to 2005Q3, or when starting our inv estigation in 1982Q4.
2
These distributions were computed with a semiparametric bootstrap procedure. First, we esti-
mated sub-sample specific AR(3) processes for the series under investigation (first subsample: 1960Q1-
1979Q3; second subsample: 1984Q1-1999Q4), specifically x
t
= c
x
+
3
X
j=1
α
x
j
x
t−j
+ ε
x
t
(with x standing
either for inflation or for detrended output). Next, the bootstrapped distributions were computed by
simulating 10,000 pseudo-series with the estimated models, keeping fixed the estimated autoregres-
sive parameters. The errors w ere sampled with replacement from the urns of the estimated residuals.
Following Davidson and MacKinnon (2006, eq. 23.11 page 821) the latter were rescaled to make the
variance of the sampled errors equal to that of the estimated autoregressive process. AR(2) models for
the t ime-series at hand delivered very similar results.
3
We mainly concentrate on annualized inflation - 4 times the quarterly inflation computed on the
PGDP c hain-weighted price index - and detrended real output - HP-filtered real log-GDP (1 decimal).
Following Lubik and Schorfheide (2004), w e computed the HP filter by considering as initial observation
the quarter 1955Q1. As an alternative measure of stochastic trend for the real GDP, we employ the
potential output computed by the Congressional Budget Office. In this study we also consider a
measure of the short-term interest rate, i.e. the federal funds rate (quarterly averages). The data
used in our analysis were downloaded on January 2006 from the Federal Reserve Bank of St. Louis’
web-site, i.e. http://research.stlouisfed.org/fred2/.
2
If this decline in inflation and output volatilities is mainly due to ’good policy’ ac-
tions (say a better monetary policy managem ent), then the low volatilities scenario
w e ha ve been observing for about two decades now could be maintained such b y keep
fighting inflation wit h the ’righ t’ systematic monetary policy. E v idence of a rem arkable
policyswitchattheendofthe’70sisprovided - among the others - by Judd and Rude-
busch (1998), Clarida, Galí, and Gertler (2000), Lubik and Sc horfheide (2004), Boivin
and Giannoni (2005), and C ogley and Sargent (20 05). B y constrast, if the Great M od-
eration is mo stly d ue to ’good luc k ’ (to be inter pre ted as m o re benign macroeconom ic
shocks), then nothing in principle can prev ent the U.S. economy to return to the high
volatilities scenario already liv ed in the ’60s and ’70s. Supporters of th e ’good lu ck’
view include Stock and Watson (2003), Primiceri (2005), Canova and Gam betti (2005),
Hansen (2005), C anova, Gambetti, and Pa ppa (2006), Sims and Zha (2006), Gordon
(2005), Arias, Hansen, and Ohanian (2006), and Ju stiniano and Prim iceri (2006).
4
Most o f the above cited studies concentrate on the estim ation of VAR-type or bac k-
ward looking models, as we ll as on their emplo ym ent for running factual and coun t erfac-
tual exercises. Th ese models un derscore the role pla yed by in flation expectations in in-
flue ncing the realizations of the variables of in terest, an aspect that is of k ey-im portance
when performing coun terfactual experiments. Moreo ver, the VAR -ba sed empirical evi-
dence on Great Moderation is challenged by Benati and Surico (2006), who show that
model-misspecification may lead to a severe up ward bias in the assessment of the merits
of the ’good luc k ’ hypothesis.
Of course, structural models in whic h agents ar e rational allow for the study of pol-
icy changes. Unfortunately, the few counterfactual experimen ts conducted with modern
DSGE monetary-policy models (e.g. Stock and Watson [2003] and Justiniano and Prim-
iceri [2006], who employ a Smets and Wouters (2003)-type model in their inv estiga tions)
ha v e neglected the role potentially pla yed by indeterminacy / sunspot shocks in the ’60s
and ’70s in the United States, a role whose statistical significance has been certified b y
Lubik and Schorfheide (2004).
5
,
6
If the mo ve from a ’passiv e’ to an ’activ e’ moneta ry
policy (implying the m ove from a multiple-equilibria scenario to a uniqu e equilibrium )
dram atic ally reduces the volatility of inflation expectations (and, con sequently, of infla-
tion and outp ut), then the om ission of the ’passive’ monetary policy h ypothesis might
lead a researc her to underestima te the role that systematic mon etary policy has possibly
4
Bernanke (2004) - among the others - also discusses the role potentially played by changes in the
(non-policy) economic s tructure for the reduction of the observed volatilities. In this paper we line up
with most of the literature and concentrate on the ’good policy’ vs. ’good luck’ drivers. We leave the
study of the structural change issue to future research.
5
For a contribution pointing towards the role of indeterminacy in explaining the dynamics of inflation
in reaction to a monetary policy shock in VAR models, see Castelnuovo and Surico (2006). Beyer and
Farmer (2005) point out how the support of the indeterminacy hypothesis in the ’60s and ’70s in the
U.S. might be driven by the imposition of untestable restrictions on the structure of the new-Keynesian
economic model employed by Lubik and Schorfheide (2004). For a reply, see Lubik and Schorfheide
(2006).
6
A notable exception is provided by Boivin and Giannoni (2005), that allow for indeterminacy in
their simulations but miss to gauge the role played by exogenous volatilities’ shifts in shaping the
macroeconomic scenario.
3
played as a driv er of the Great M oderation.
This paper works with a calibrated standard DSGE new-Keyn esian model to per-
form factual and coun ter factual sim ulation s in order to assess the relativ e importance
of ’policy’ vs’ luc k ’ in explaining the Grea t Moderation. Importantly, in modeling the
mon etary policy break, w e allow for sunspots and distortions in the moneta ry trans-
missio n mechanism under ’passive’ monetary policy, i.e. whe n the Taylor principle is
not met. To our know ledge, this is the first con tributio n that allows for indeterm inacy
when running counterfactua ls aimed at assessing the relativ e role of ’good policy’ vs.
’good luck’ in the U nited States.
Our results suggest that systematic monetary policy is lik ely to have played an
important role in stabilizing inflation in the ’80s and ’90s. Ho wever, it turns out that
outpu t stabilit y is hardly linked to an improv em ent in the monetar y policy managem ent.
Moreov er, the relative importance of the role pla yed by mor e benign macroeconom ic
shock s in influencing both inflation and the business cycle is likely to be higher than
the one pla yed by the monetary policy switch occurred at the end of the ’70s.
The p aper is structured as follo w s. Section 2 describes the m odel employ ed as DGP
for our factual and counter factual exercises, and describes our calibration strategy.
Section 3 explains the alternative scenarios w e concentra te on, and presents our results,
whose robustness is discussed in Section 4. Section 5 conclud es, and R eferen ces follo w .
2 M acroeconomic fram ework
As poin ted out in the Introduction, in performing our sim ulations w e emplo y the stan-
dard new-Keynesian framew ork survey ed by Clarida et al (1999). A key-element for
thechoiceofthismodelisthefactthatitistheonlynew-Keynesianmonetarypolicy
model estimated by allo wing for indeterminacy (see the w ork by Lubik and Sc horfheide
[2004]).Themodelreadsasfollows:
7
π
t
= βE
t
π
t+1
+ κ(x
t
− z
t
) (1)
x
t
= E
t
x
t+1
− τ(R
t
− E
t
π
t+1
)+g
t
(2)
R
t
=(1− ρ)[ρ
π
π
t
+ ρ
x
(x
t
− z
t
)] + ρR
t−1
+ ε
MP
t
(3)
z
t
= ρ
z
z
t−1
+ ε
z
t
,g
t
= ρ
g
g
t−1
+ ε
g
t
(4)
where x stands for real output, π represen ts inflation, R is the short term nominal
interest rate, z captures exogenous shifts of the marginal costs of production, g is a
7
The variables of the model are expressed in percentage deviation with respect to their steady state
values, or in the case of output from a trend path.
4
demand disturbance,
8
and ε
MP
is a mon etary policy shock . The random variables z
and g follow AR(1) processes whose roots are - respectively - ρ
z
and ρ
g
.Theshocks
ε
z
,ε
g
, and ε
MP
are white noise stoc hastic elements whose variance is, respectively,
σ
2
ε
z
,σ
2
ε
g
,andσ
2
ε
MP
.
Eq. (1) is the Euler equation maximizing the profit of the representativ e, m on opo-
listically competitiv e firm whose discoun t factor is iden tified by the p ar am eter β.Prices
are sticky due to a Calv o-t ype rigidity that allow s only a fraction of firms to r eoptimize
their prices or to quad ratic adjustment costs. The slo pe coefficient κ relates output and
the m arginal costs to the inflation rate.
Eq. (2) is a log-linearized IS curv e stemm ing from the household’s in tertem poral
problem in which consumption and bond holdings are the control variables. Con tem-
poraneou s output is ca used both by expectations on future realiza tion s of the b usiness
cycle and by the ex-ante real interest rate, the impa ct of the latter being regulated b y
the intertemporal elasticit y of substitution τ .
Eq. (3) is an interest rate rule accordin g to whic h th e central bank adjusts th e
policy rate in response to fluctuations in inflation and output. We in terpret the random
variable ε
MP
t
as the monetary policy shock.
It is well kno w n that this linear rational expectations model can be associated
to a unique solution as long as the Taylor principle is satisfied, i.e. the condition
ρ
π
> 1 −
(1−β)
κ
ρ
x
is met (Clarida et al, 2000; Woodford, 2003). If this condition does
not hold, m onetary authorities are unable to uniquely pin down private sector’s expec-
tations. Follo w ing Lubik and Schorfheide (2003,200 4), under indeterminacy w e allow
both for i) a zero-mean i.i.d. sunspot shock ζ
t
- who se variance is σ
2
ζ
-toinfluence the
equilibrium values of the variables of interes t, and for ii) a distortion in the transmissio n
mechanism going from the v ecto r of structural shocks to the endogen ous variables under
consider ation.
9
Notice that, giv en its simplicity, this model is lik ely to give systema tic
monetary policy a more importan t role than the one that other framew orks - sa y Sm ets
and Wouters (2003)’s - acknowledge it.
Model calibration
We emplo y the new-Keynesian model (1)-(4) to run simulations in order to compute
the v olatilities of inflation and output. To do so, we need to calibrate the model. We
divide the vector θ of the parameters of the model in three groups: policy parameters
8
Since the underlying model has no investment, output is proportional to consumption up to an
exogenous process that can be interpreted as time-varying government spending or, more broadly, as
preference change.
9
Technically, s uch a distortion is implemented by considering a vector
f
M affecting the transmission
from the structural shocks ε
t
to the endogenous forecast errors η
t
=[(x
t
− E
t−1
x
t
), (π
t
− E
t−1
π
t
)]
0
.
Lubik and Schorfheide (2003,2004) propose to compute the vector
f
M so to minimize the distance
between the on-impact reactions of the endogenous variables s
t
to the shocks ε
t
under indeterminacy
and those computed at the frontier dividing the parameter space into determinacy and indeterminacy.
We adopt this identification strategy, labeled as ’continuity’, as also done by Castelnuovo and Surico
(2006) and Benati and Surico (2006). A Technical Appendix containing further details is available
upon request.
5
θ
pol
= {ρ
π
,ρ
x
,ρ}, vo latilities θ
vol
= {σ
ε
z
,σ
ε
g
,σ
ε
MP
,σ
ζ
}, and non-policy, ’structural’
parameters θ
npol
=
©
τ,κ,ρ
z
,ρ
g
ª
. For calibrating the Taylor rules, we employ the poin t
estimates b y Clarida , Galí, and Gertler (2000, Table 2 p. 157). These estimates are
fairly in line with those provided by other authors on the U.S. monetary policy conduct
(see e.g. Judd and Ru debusch [1998], Lubik and S chorfheide [2004]). As far as th e
macr oeconomic shock s are concerned, in our bench m ark calibration we employ Lubik
and Schorfheide (2004)’s estimated volatilities. Interestin gly, the magn itudes of the
supply shocks are v ery similar to those one may obtain by estimating a VAR -type
model a la Rudebusch and Sv en sson (1999), w hile those of the dem an d shock s seem to
be slightly underestimated.
10
For calibrating the remaining ’non-policy’ parameters, we
follo w Canova (1994) and employ some (independen t) ’prior’ distributions for each of
the parameters of such vector. A recent study by Fuhrer and Rudebusch (2004) points
to wards a relatively small value of the in tertem poral elasticity of substitution τ for the
U.S. economy (spanning from 0.002 to 0.081 when the HP detrended real log-GDP is
considered), slightly smaller than the one provided by Rudebusch (2002). Follo wing
Lubik and Schorfheide (2004), w e c hoose (for τ
−1
) a gamma distribu tion h aving mean
19.94 (corresponding to
τ =0.05) and a fairly large standard, deviation, i.e. 14.07,in
order to allow for flat tails and a fairly wide range of drawn values in the calibration
exercise. To take into a ccount the large uncertain ty surround ing the value of the slope
coefficient κ (w hich, acco rd ing to Lu bik and Sch orfheide [2004]’s posterior me ans, may
span from .27 u p to 1.1 2), we co n sid er a gamma distribution ha vin g mean 0.75 and
standard deviation 0.31. Finally, we sample the values of the autoregressive parameters
ρ
z
and ρ
g
by imposing - respectively- a beta prior with mean 0.95 and standard deviation
0.05 and a beta prior with mean 0.50 and standard deviation 0.09.
11
Fixed the priors,
w e implement the following algorithm:
1. we sam p le a tuple j :
©
τ
j
,κ
j
,ρ
j
z
,ρ
j
g
ª
from the giv en distributions;
2. giv en the remaining parameters of the model (calibr ated as discussed abov e) and
the tuple j coming from step 1, w e simulate 500 times the new-Keynesian model
(1)-(4), and compute the medians of the sim ulated distributions of the endogenou s
variables of interest;
12
3. given the medians computed at step 2, we compute the following measure of
10
In particular, we OLS estimated the supply curve π
t
=
P
4
i=1
α
i
π
t−i
+α
x
x
t−1
+ε
π
t
and the demand
curve x
t
=
P
2
i=1
β
i
x
t−i
− β
r
(i
t−1
− π
t−1
)+ε
x
t
(where the upper-barred variables identify backward-
looking MA(4) processes) for the two subsamples 1960Q1-1979Q3, 1984Q1-1999Q4, and obtained:
dσ
ε
π
=1.18, dσ
ε
x
=0.89 (first sample); dσ
ε
π
=0.71, dσ
ε
x
=0.43 (second sample). We concentrate on
these point-estimates in our Robustness check.
11
The means of the beta distributions were selected on the basis of a preliminary grid-search we
performed by considering the following discrete domains (step-length: 0.05): τ[0 − 0.35],κ[0.55 −
1.0],ρ
z
[0.4 − 0.95],ρ
g
[0.4 − 0.95].
12
For all the model simulations we consider, we produce 1,000 pseudo-subsamples of a lenght compa-
rable to the historical one, i.e. 78 observations for the first subsample, and 65 for the second one. The
model simulations are stochastically initialized, and the first 100 pseudo-observations are discarded.
6
distance:
D
j
(ξ
nkm,j
(θ
j
),ξ
act
,V)=
h
¡
ξ
nkm,j
(θ
j
) − ξ
act
¢
0
V
−1
¡
ξ
nkm,j
(θ
j
) − ξ
act
¢
i
(5)
where ξ
nkm,j
=
£
σ
nkm,j
π
σ
nkm,j
y
¤
0
and ξ
act
=
£
σ
act
π
σ
act
y
¤
0
are (2x1) vectors con-
taining the medians of the distributions of the standard deviations of time-series of
interest, ’nkm’and’act’ stand respectively for ’new-K eynesian’ (simulated) an d ’ac-
tual’ (bootstrapped), and V is a (2x2) diagonal matrix whose non-zero elements are
represented by the standard deviations of the bootstrapped distributions of the actual
time-series in the sample under analysis;
4. we store the so computed distance D
j
and the tuple j, and go bac k to step 1.
We repeat steps 1-4 1,000 times. A t the end of the loop, we pin do w n the tuple
j
∗
that minimizes the distance (5) in the subsamp le at hand. In order to have a
calibration robust to sample uncertainty, w e consider the best (in terms of minimum
distance) 5% tuples and co m pute a weighted average of their elements, the weights
being the inv erse of the distance (5) for all the considered j
s
.Noticethatourmeasure
of distance is sample-specific, i.e. we perform t he minimum-distance searc h for each of
the two subsamp les.
The results of our calib r at ion are displaye d in Table 2.
13
It turns out that to match
the vola tilities of inflation and output one m ust impose a lo w degree of in tertemporal
elasticit y of substitution τ, very muc h in line with the above m entioned literature.
The slope coefficient κ is higher than the one suggested b y the posterior means b y
Lubik and Schorfheide (2004), and it is in terestingly outside the 90%-in terval of our
prior, so suggesting that the d ata (and not the prior) is actually driving the result.
The autoregressive coefficient of the demand shock is very similar to the posterior m ean
(firstsubsample)providedbyLubikandSchorfheide (2004), while the one of the shocks
to mar gin al costs is low er. Given the similarity of the two sets of subsamp le ’estim a tes’
we obtained, we constraint the vector
©
τ,κ,ρ
z
,ρ
g
ª
to assume the sam e values in both
the subsam ples of our in terest.
14
We summarize our calibration choices for the whole
vector of pa ram eters identifying the structure of the model (1)-(4) in Table 3.
15
[Tables 2 and 3 about here ]
13
Thesameexerciseperfomedwiththevectorξ
x,j
=
£
σ
x,j
π
σ
x,j
y
σ
x,j
i
¤
0
,withx ∈ {nkm, act},
delivered very similar results, i.e. τ=0.0488, κ=1.3335, ρ
g
=0.9479, ρ
z
=0.4988 for the first subsample,
and τ=0.0328, κ=1.0563, ρ
g
=0.9500, ρ
z
=0.5374 for the second one.
14
We employ the battery of the point-estimates obtained for the second subsample. The alternative
choice implies very similar results.
15
Notice that in terms of number of structural shocks there is an asymmetry between the first and
the second subsample due to the presence of the sunspot shock in the former but not in the latter.
Nevertheless, our results are robust to the ’elimination’ of the sunspot shock from the picture.
7
3 ’Good polic y ’ or ’good luck’? Co u nterfa ct ua l sim-
ulations
Once calibrated, the model is ready for perform ing factual and coun terfactual simula-
tions. In pa rticular, we first w ant to understand if this framework is able to deliver
’factual’ sim u lated volatilities wh ich are in line with the ’a ctu al’ bootstrapped ones.
For ’factual’ we mea n the volat ilities compute d with the m odel calibrat ed as de scribed
in the previous Section.
Table 4 reports the results of our factual sim ulations. The new-Keynesian model
at hand seems to offer a fairly good fit of the facts; in particular, all the medians
of the sim u lated vo latilities fall inside the bootstrapped interval d efined by the [5th;
95th] percen tiles. We tak e this calibration and the implied factual simulations as our
benchmark against which to confront the outcome of our conterfactual simulations.
We simulate four different coun ter factual scenarios: i) ’Good P olicy’, implemented
by ’plan ting’ the Volcker-Greenspan monetary policy conduct in the ’60s and ’70s; ii)
’Good Luc k’, featured by the presence of the more benign shoc ks of the ’80s and ’90s also
in the t wo earlier decades; iii) ’Bad Po licy’, ch a racte rized b y a ’passive’ monetary policy
in both the simula ted subsamples; and iv) ’Bad Luck ’, a scenario in wh ich the econ omy
is h it by highly vo latile shoc k s all time long. What w e expect is a better e con om ic
outcome - i.e. lower medians and tigh ter intervals - under the ’Good’ scenarios, and a
worse one - i.e. higher media ns and v o latilities - under the ’Bad’ ones. But are these
c h anges quan titatively importan t?
Figure 1 displays the benc hm ark vs. coun terfactual distributions of inflation and
output. A ll the distributions are tilted in the expected directions, but the magnitudes of
the shifts are different from eac h other. In particular, plan ting a good m on etary policy
in the ’60s and ’70s does exert a re m ar kable im pa ct on the distribu tion of inflation
in terms of median and standard deviation (both significantly lower); by contrast, the
distribution of output is basically unaffected by su ch a regim e shift.
16
This result finds
its confirmation in the somewhat ’symm etr ic’ scenario, when the bad policy is im posed
in the second subsample. In fact, according to our sim ula tion s under a passive mon etary
policy we would have observ ed a w orse outcome in terms of inflation volatility in the
second subsample (though the impact of the ’bad policy’ on the v olatilit y of inflation
seems to be lower than the one of the ’good policy’ in abso lute terms), but not so m u ch
in terms of output v olatility. Th is is confirmed by the figures collected in Table 3 , that
sho w how and ho w m uc h the distributions vary when different systema tic monetary
policies are imp lem ented.
Differently, the role of (either good or bad) luck seems to be relevant for both volat il-
ities. In fact, a ccordin g to our simulations milder shocks in the ’6 0s and ’70s would hav e
implied a m uch calmer beha v ior of the U.S. economy, both in terms of inflation and
16
Notice that, according to the Kolmogorov-Smirnov 2-sided test, all the ’counterfactual’ distribu-
tions plotted in Figure 1 are statistically different (at the 10% significance level) with respect to the
’factual’ ones. However, in this paper we are concerned with the economic relevance of the ’policy’ vs
’luck’ drivers.
8
in term s of output. ’Simmetric ally’, big macroeconom ic shoc ks in the las t two decades
would have triggered a large ma croeconomic instabilit y also under good m oneta ry pol-
icy.
Table 5 offers a synthetic sum mary o f our resu lts. First, systematic monetary policy
does influence the volatility of inflatio n. Suc h volatility w ould hav e been about 32%
smaller (or 30% bigger) if monetary policy had been tighter (or less agg ressive). These
figu res support the role played by the Fed in the ’80s and ’90s in stabilizing the v o latility
of the inflation rate, as also found b y Cogley an d Sargent (2005) and Mumtaz and
Surico (2006). Hence, indeterminacy triggered by a ’passiv e’ monetary policy is likely
to ha ve played an important role in forming the macroeconomic scenario of the pre-
Volc ker era, as previously poin ted out b y Clarid a, Galí, and Gertler (2000) and Lubik
and Sc horfheide (2004). Neverth eless, monetary policy has har d time in expla ining the
low e r o utp ut volatility of the ’80s and ’90s. Indeed, the historic ally re levan t policy
switch does not trigger much of a reaction in the business cycle. Th is finding supports
the resu lts com ing from Stock and Watson (2003)’s coun terfactual simulations. Third,
the role played b y good luck seems to be relativ e ly more important than the one of good
policy in both subsam ples. In this sense, our results corroborate those by Canova and
Gambetti (2005), Canova, Gam betti, and Pappa (2006), Primiceri (2005), Sims and
Zha (2006), Arias, Hansen, and Ohanian (2006), and Justiniano and Primiceri (2006).
[Table 4, Figure 1, Table 5 about here]
4 Robustness c heck
We perform a robustness ch eck along three dimensions: Th e magnitude of the supply
shocks, that of the in tertemporal elasticit y of substitution τ in the IS curv e, and that
of the slope parameter κ in th e Phillips cu rve.
Magnitude of the Supply Shocks
So far the analysis has m ainly relied upon Lubik and Sc horfheide (2004)’s estimates
of the vo la tilitie s of inflation, output, and the structural shocks. As a check, we estimate
an alternative model - i.e. th e Rudebusch and Svensson (1999) model - and concentrate
on the estim ated stand ard deviatio ns of the errors. We find cσ
ε
π
=1.18, cσ
ε
x
=0.89 for
the first sample, and cσ
ε
π
=0.71, cσ
ε
x
=0.43 for the second one.
17
Theremarkabledrop
in the supply shoc k - estimated b y Lubik and Schorfheide (2004) - seems to be confirm ed,
but the relativ e magnitude of the demand shoc k with respect to the supply shock is
much higher. By con ditionin g on these new values of the volatilities of the d em and and
supply shocks, and ke eping the vector of policy pa ram eters θ
pol
and t he volat ility of th e
monetary policy shock σ
ε
MP
and that of the sunspot shoc ks σ
ζ
unchanged, w e recalibrate
the vector
©
τ,κ,ρ
z
,ρ
g
ª
in order to matc h the medians of the actual v olatilities.
17
Newey-West correction for the VCV matrix (3 lags). A check with the CBO potential output (as
to substitute the HP measure for the output trend) delivered very similar estimates. The whole set of
estimates of the Rudebusch and Svensson (1999)’s model is available u pon request.
9
The results of our calibration, reported in Table 6, poin t to wards ’estimates’ that are
fairly similar to those previously obtaine d, with a sligh tly higher in terte m poral elasticity
of sustitution and a slighly lower slope of the Phillips curve. As previously done, we
emp loy the same set of calibrated parameter values for both the subsa m ples: O ur new
calibration is a vailable in Table 7.
18
[Tables 6 and 7 about here ]
The factual simulations confirm that the model is able to fit the data with a fair
precision (see Table 8 ). We then run new coun ter factual sim ulations. As far as the
conclusions about the role of systematic monetary policy vs. structural shoc ks is con-
cerned, also these sim ulation s lead us to remark the relevance of systematic mon etary
policy on inflation volatility, and that of the differen t magnitude of the supply and
demand shoc ks on both the volatilities under in vestigation. Figure 2 and Tables 8 and
9 give support to this statem ent.
[Table 8, Figure 2, Table 9 about here]
Higher Intertem poral Elasticity of Substitution τ
Our benchmark calibration delivers a value of τ of about 0.06, fairly in line with
recen t estimates of the IS curv e for the U.S. by Fuhrer and Rud ebu sch (2004). How ever,
alternative, higher estimates may be found in the literature. To assess the ro bu sness of
our fin din gs, we perform our simulations by raising its value up to 0.09, as in Rudebusc h
(2002). All the other parameters are calibrated as in Table 3.
Factual simulations rev eal that this parameterization delivers (median) v o latilities
that lay within the (or close to the) 90% bootstrapped confidence intervals. As far as
our qualitative results are concerned, Tables 10 and 11, as well as Figure 3, testify that
this para m eter perturbation does not affect our qualitativ e conclusions.
[Table 10, Figure 3, Table 11 about here]
Lower Slope of the Phillips curve κ
Given the importance of the parameter κ in the Phillips curve, we perturb it in order
to perform a further robustness c h eck. We assign to κ a new value, i.e. 0.58,avalue
in line with the posterior mean obtained by Lubik and Schorfheide (2004). We notice
that the factual simulations deliv e r median values for inflatio n that are m uch smaller
than the actual ones. How ever, when running ou r simulations with such a small value
for the parameter κ (the other parameter estimates are as those reported in Table 3),
our q ua litative results turn out to be confirmed (see Tables 12 and 13 and Figure 4).
[Table 12, Figure 4, Table 13 about here]
18
As before, we employ the battery of the point-estimates obtained for t he second subsample. The
alternative choice implies very similar results.
10
5Conclusions
In this paper w e calibrated a new-Keynesian model to perform factual and coun t erfac-
tual sim ulations relative to the U.S. macroeconomic behavior in order to understand
the relative merits of the ’Good (M on etary) Policy’ vs. ’Good Luck’ h ypotheses for
explaining the Great Moderation. Importantly, in performing our simulations under
the ’bad’ policy scenario, we allowed for sunspot shocks and distortions in the trans-
mission mechanism from the structural shocks to the endogenous variables to affect the
equilibrium values o f the variables of in terest.
Our results show that both an aggressive policy against inflation fluctuations and
benign macroeconomic shoc ks are likely to haveplayedabigroleinshapingthepath
of inflation and output. In particular, systematic moneta ry policy mov es turn out to
hav e been important in stabilizing the inflation rate, but have not been as effectiv e in
stabilizing the business c ycle. By co ntrast, less volatile ma croeconom ic shoc k s are q u ite
importan t for explaining the behavior of both variables.
All in all, while supporting the role of system atic monetary policy in influencing
inflation fluctuations, this paper supports the importance of the relative role pla y ed
by structura l shoc ks in the determ inatio n of the U.S. m acr oeconomic volatilities. This
find ing corroborates some recent con tribu tions by Stock and Watson (2003), Arias,
Hansen, and Ohanian (2006), and Justiniano and Primiceri (2006), who argue that
variations regarding - respectively - pure supply shocks, total factor productivity, and
inv estm e nt-specific technological shock s migh t be the cau ses of the red u ced vo latility of
the business cycle. R egard ing the fluctuations of the inflation rate, ou r results seem to
offer some support to Mankiw (2006, p. 184) who recently wrote: "I wonder whether
weexaggeratetheroleofpolicydecisionsandunderstatetheroleofluck. Onereasonis
that the ba d inflation performance of the 1970s and the good inflation performan ce of
the 1990 s were not limited to the Un ited States. If there was policy failure in the 1970s
and success in the 1990s, the blame and credit go to the world community of central
bankers, not to the single person leading the Federal Reserv e. I suspect, ho wev er, that
the differenc e cannot be fu lly explained by policy at all. [...] The favora ble supply-sid e
dev elopments of the 1990s w ere not caused by monetary policy, but they did make the
job of policymak e rs a lot easier. Luc k pla ys a large role in how history judges central
bankers."
Howev er, it m ust be recognized that a more cautious measurement of such ’exoge-
nous’ shocks is w arranted. In fact, what w e label as ’exogenous’ migh t be (at least
in part) the product of economic policies. Citing Krueger (2003, p. 64), "The [shock]
that leap s to mind imm ediately is the oil price increase in 1973 -74, whic h I think of as
ha ving come at the end of a commodity price boom - itself a result of the dollar inflation
and, for that matter, labor union strik es and things like this, which I think w ere partly
because of uncertain ty about relative prices. If so, trea ting those a s macroeconomic
shocks that are quite exogenous ma y understate quite significantly the ro le o f im p roved
monetary policy".
To take Krueger’s consideration up, one should work with more sophisticated models
11
able to take into accoun t e xchange rate fluctuations, imperfections in the labor market,
price heterogeneity, and so on, features that are just lack ing in the simplified view of
the w orld that the simple 3 equation new-Keynesian monetary policy model offers us.
We plan to pursue further research along this a ven ue in the future.
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14
1960Q1-1979Q3 19 84Q1-1999Q4
Outp u t trend σ
π
σ
x
σ
π
σ
x
HP
2.48
[1.68 ; 3.84 ]
1.69
[1.24; 2.2 5 ]
0.95
[0.77 ; 1.23 ]
0.92
[0.65; 1.33]
CBO
2.48
[1.68; 3.84 ]
2.33
[1.62; 3 .28]
0.95
[0.77; 1.23 ]
1.40
[.95; 2.20 ]
Table 1: INFLATION AND OUTPUT, BOOTSTRAPPED VO LATILITES. The Ta-
ble displa ys the 50th [5h; 95th] percen tiles of simulated distributions computed with
a semiparam etric bootstrap procedure. First, w e estimated su b-sam p le specificAR(3)
processes for the series under investigation. Next, the bootsrapped distributions w ere
comp uted b y simulating 10,000 pseudo-series with the estimated AR models, keeping
fixed the estimated autoregressiv e parameters. The errors w ere sampled with replace-
ment from the urns of the estim a ted residuals. Follow ing Davidson and MacK inno n
(2006, eq. 23 .11 page 82 1) the latter were rescaled to make the variance of the sampled
errors equal to the autogressive processes’s estimated one. In itial conditions for the AR
processes: Hystorical values. ARCH -La grang e Multiplier test (3 lags) supported the
assump tion of hom oschedasticity of the estim ated errors. First observation for the HP
trend comp utatio n: 1955Q1; last observation: 2005Q3. CBO : Output trend computed
b y the Congression al B udget Office.
’P rio r’ distr ib u tio ns Calibrated values
Parameters Type Mean Std 90%-interv al 1st su bsample 2nd s ubsample
τ
−1
Gam ma 19.94 14.07 [3.56; 47.02] 0.0551 (τ) 0.0594 (τ)
κ Gamma 0.75 0.31 [0.33; 1.3 1] 1.3581 1.3641
ρ
g
Beta 0.95 0.05 [0.85; 0.99] 0.9438 0.9407
ρ
z
Beta 0.50 0.09 [0.35; 0.65] 0.4655 0.4603
Table 2: CA L IB R ATED PARAMET E R VAL U E S. Calibration of the no n-policy pa ra-
meter s performed by minimizin g a distance function that tak e s into account the gaps
bet ween the model consistent vs. a ctual standa rd deviatio ns (m edians) of the variables
in the model. The momen ts are w eigh ted via the variance of the standard deviations
of the actual data. The ’poin t estimates’ of the non-policy parameters are a weigh ted
a verage of the elemen ts of the best 5 percent tuples. 1st subsam ple: 1960Q1-19 79Q 3,
2nd subsam ple: 1984Q 1-1999Q 4.
15
Par ameters 1st subsample 2nd subsample
ρ
π
0.83 2.15
ρ
x
0.27 0.93
ρ 0.68 0.79
σ
ε
z
1.13 0.64
σ
ε
g
0.27 0.18
σ
ε
MP
0.23 0.18
σ
ζ
0.20 −
τ 0.0594
κ 1.3641
ρ
g
0.9407
ρ
z
0.4603
Table 3: CALIBRATION OF THE DGP NEW-KEYNESIAN MODEL. Parameter
values borrow e d from the literature (see main text) / calibrated via a minimim distance
estimation. 1st subsample: 1960Q1-1979Q3, 2nd subsample: 1984Q1-1999Q4.
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
σ
x
σ
π
σ
x
Actual
(bootstr.)
2.48
[1.68; 3.84 ]
1.69
[1.24; 2.25]
0.95
[0.77; 1.23 ]
0.92
[0.65; 1.33]
Factual
(simulat.)
1.92
[1.66 ; 2.24 ]
2.15
[1.84; 2.52]
0.79
[0.67 ; 0.93 ]
1.14
[0.95; 1.3 6 ]
’Good Policy’
(simulat.)
1.40
[1.21; 1.62 ]
2.02
[1.72; 2.36]
Factual Factual
’Good Luck’
(simulat.)
1.14
[0.97; 1.34 ]
1.23
[1.05; 1.45]
Factual Factual
’Bad P olicy ’
(simulat.)
Factual Factual
1.06
[0.89; 1.25 ]
1.21
[1.00; 1.45]
’Bad Luc k ’
(simulat.)
Factual Factual
1.40
[1.19; 1.63 ]
2.03
[1.70; 2.38]
Table 4: V OLATILITIES COMPUTED WITH FA CTUAL AND COUNTERFAC-
TUAL SIMULATIONS, BENCHMARK CALIBRATION. The Table displays the 50th
[5th; 95th] percen tiles of the simulated distributions based on 10,000 repetitions. 1st
subsample: 1960Q1-1979Q3; 2nd subsample: 1984Q1-1999Q4.
16
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std output
Figure 1: FACTUAL VS. COUNTERFA CTUAL SIMULATIONS. Solid line: Factual
distributions; ’S quared ’ line: Coun terfa ctu al distributions. Calibra tion of the model:
Benchmark (see the explanation in the text). N u mber of repetitions: 10,000.
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
,%var σ
x
,%var σ
π
,%var σ
x
,%var
’Good Policy’
(simulat.)
-31.70% -6.48% --
’Good Lu ck’
(simulat.)
-52.02% -56.37% --
’Bad P olicy ’
(simulat.)
--29.53% 6.14%
’Bad Luc k ’
(simulat.)
--57.23% 57.51%
Table 5: STANDARD DEVIATION, PER CENTAGE VARIATION. The percentage
variations w ere compu ted on the medians of the simulated vola tilities with respect to
the benchm a rk factual scenario based on 10,000 repetitions. 1st subsample: 1960Q1-
1979Q3; 2n d subsample: 1984Q 1-1999Q4 .
17
’P rio r’ distr ib u tio ns Calibrated values
Parameters Type Mean Std 90%-interv al 1st su bsample 2nd s ubsample
τ
−1
Gam ma 19.94 14.07 [3.56; 47.02] 0.0627 (τ) 0.0771 (τ)
κ Gamma 0.75 0.31 [0.33; 1.3 1] 1.2946 1.2841
ρ
g
Beta 0.95 0.05 [0.85; 0.99] 0.9435 0.9436
ρ
z
Beta 0.50 0.09 [0.35; 0.65] 0.4486 0.4516
Table 6: CALIBRATED PARAMETER VALUES, R UDEBUSCH AND SVENSSON
(1999)’S DEM A N D A N D S U PP LY SHOC K S. Calibration of the ’non-policy’ parame-
ters performed by minimizin g a distance fun ction that take s into accoun t the gaps
bet ween the model consistent vs. a ctual standa rd deviatio ns (m edians) of the variables
in the model. The momen ts are w eigh ted via the variance of the standard deviations
of the actual data. The ’poin t estimates’ of the non-policy parameters are a weigh ted
a verage of the elemen ts of the best 5 percent tuples. 1st subsam ple: 1960Q1-19 79Q 3,
2nd subsam ple: 1984Q 1-1999Q 4.
Par ameters 1st subsample 2nd subsample
ρ
π
0.83 2.15
ρ
x
0.27 0.93
ρ 0.68 0.79
σ
ε
z
1.13 0.64
σ
ε
g
0.27 0.18
σ
ε
MP
0.23 0.18
σ
ζ
0.20 −
τ 0.0771
κ 1.2841
ρ
g
0.9436
ρ
z
0.4516
Table 7: CALIBRATION OF THE DGP NEW-KEYNESIAN MODEL, R UDEBUSCH
AND SVENSSON (1999)’S DEMAND AND SUPPLY SHOCKS. Rest of the calibration:
See the main text. 1st subsample: 1960Q1-1979Q3, 2nd subsample: 1984Q1-1999Q4.
18
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
σ
x
σ
π
σ
x
Actual
(bootstr.)
2.48
[1.68; 3.84 ]
1.69
[1.24; 2.25]
0.95
[0.77; 1.23 ]
0.92
[.065; 1.33]
Factual
(simulat.)
1.73
[1.46 ; 1.97 ]
2.20
[1.85; 2.54]
0.74
[0.64 ; 0.86 ]
1.22
[1.04; 1.4 5 ]
’Good Policy’
(simulat.)
1.22
[1.05; 1.42]
2.04
[1.74; 2.39]
Factual Factual
’Good Luck’
(simulat.)
1.09
[0.92; 1.28]
1.32
[1.11; 1.55]
Factual Factual
’Bad P olicy ’
(simulat.)
Factual Factual
1.01
[0.85; 1.18]
1.31
[1.09; 1.55]
’Bad Luc k ’
(simulat.)
Factual Factual
1.22
[1.03; 1.43]
2.03
[1.70; 2.44]
Table 8: V OLATILITIES COMPUTED WITH FA CTUAL AND COUNTERFAC-
TUAL SIMULATIONS, DEMAND AND SUPPLY SHOCKS A LA R UDEBUSCH
AND SVENSSON (1999). The Table displa ys the 50th [5th; 95th] percentiles of the
sim ulated distributions based on 10,000 repetitions. 1st subsample: 1960Q1-1979Q3;
2nd subsam ple: 1984Q 1-1999Q 4.
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
,%var σ
x
,%var σ
π
,%var σ
x
,%var
’Good Policy’
(simulat.)
-34.32% -7.28% --
’Good Lu ck’
(simulat.)
-45.76% -51.21% --
’Bad P olicy ’
(simulat.)
--30.97% 7.27%
’Bad Luc k ’
(simulat.)
--50.33% 50.96%
Table 9: V OLATILITIES COMPUTED WITH FA CTUAL AND COUNTERFAC-
TUAL SIMULATIONS. SHOCKS A LA RUDEBUSCH AND SVENSSON (1999). The
percentage variations were compu ted o n th e m ed ia ns of the simulated volatilities with
respect to the benchm a rk factu al scenario based on 10,000 repetitions. 1st subsample:
1960Q1-1979Q3; 2nd subsample: 1984Q1-1999Q4.
19
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std output
Figure 2: FACTUAL VS. COUNTERFACTUAL SIMULATIONS, SHOCKS A LA
RUDE B U SC H AND SVENSSON (1999). Calibration of th e model d riven by the shoc ks
estimated with the Rudebusch and S vensson (19 99)’s model. Solid line: Factual distri-
butions; ’Squared’ line: Coun terfac tual distributions. Number of repetitions: 10,000.
20
’60Q1-’79Q3 ’84Q1-’99Q 4
σ
π
σ
x
σ
π
σ
x
Actual
(bootstr.)
2.48
[1.66; 3.83]
1.68
[1.24; 2.24]
0.97
[0.78; 1.29 ]
1.23
[1.08; 1.43 ]
Factual
(simulat.)
1.74
[1.48 ; 2.02 ]
2.11
[1.79 ; 2.46]
0.70
[0.59 ; 0.81 ]
1.10
[0.93 ; 1.30 ]
’Good Policy’
(simulat.)
1.21
[1.05; 1.38]
1.95
[1.67; 2.27]
Factual Factual
’Good Luck’
(simulat.)
1.04
[0.89; 1.23]
1.21
[1.02; 1.40]
Factual Factual
’Bad Policy’
(simulat.)
Factual Factual
0.96
[0.81; 1.13]
1.19
[0.99; 1.41]
’Bad L uck’
(simulat.)
Factual Factual
1.22
[1.03; 1.43]
1.96
[1.64; 2.31]
Table 10: VOLATILITIES COMPUTED WITH FACTUAL AND COUNTERFAC-
TUAL SIMULATIONS, H IGH IES. The Table displays the 50th [2.5th; 97.5th] per-
centiles of the sim ula ted distributions based on 10,000 repetitions. 1st sub sam ple:
1960Q1-1979Q3; 2nd subsample: 1984Q1-1999Q4.
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
,%var σ
x
,%var σ
π
,%var σ
x
,%var
’Good Policy’
(simulat.)
-36.07% -7.89% --
’Good Lu ck’
(simulat.)
-51.27% -55.89% --
’Bad P olicy ’
(simulat.)
--32.32% 7.19%
’Bad Luc k ’
(simulat.)
--55.62% 57.29%
Table 11: VOLATILITIES COMPUTED WITH FACTUAL AND COUNTERFAC-
TU A L SIMULATION S , HIGH IES. The percentage variations w ere computed on the
medians of the simulated volatilities with respect to the bench m ark factual scenario
based on 10,000 repetitions. 1st subsample: 1960Q1-1979Q3; 2nd subsample: 1984Q1-
1999Q4.
21
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std output
Figure 3: FA CTUAL VS. COUNTERFA CTUAL SIMULATIONS, HIGH IES τ.High
in ter temporal elasticity of substitution, benc hm ark calibration for the rest of the model.
Solid line: Factual distributions; ’Squared’ line: Counterfactual distributions. Num ber
of repetitions: 10,000.
22
’60Q1-’79Q3 ’84Q1-’99Q4
σ
π
σ
x
σ
π
σ
x
Actual
(bootstr.)
2.48
[1.66; 3.83 ]
1.68
[1.24; 2.24]
0.97
[0.78; 1.29 ]
1.23
[1.08; 1.43]
Factual
(sim u la t.)
1.02
[0.85 ; 1.24 ]
2.23
[1.86; 2.61]
0.40
[0.34 ; 0.48 ]
1.19
[0.99; 1.4 3 ]
’Good Policy’
(simulat.)
0.71
[0.61; 0.81]
2.10
[1.80; 2.47]
Factual Factual
’Good Luck’
(simulat.)
0.69
[0.54; 0.95]
1.26
[1.05; 1.48]
Factual Factual
’Bad P olicy ’
(simulat.)
Factual Factual
0.52
[0.43; 0.61]
1.25
[1.04; 1.48]
’Bad Luc k ’
(simulat.)
Factual Factual
0.71
[0.59; 0.82]
2.10
[1.75; 2.49]
Table 12: VOLATILITIES COMPUTED WITH FACTUAL AND COUNTERFAC-
TUAL SIMULATIONS, LOW SLOPE k. The Table displays the 50th [5th; 95th]
percentiles of the sim ulated distributions based on 10,000 repetitions. 1st subsam ple:
1960Q1-1979Q3; 2nd subsample: 1984Q1-1999Q4.
’60Q1-’79Q3 ’82Q4-’98Q4
σ
π
,%var σ
x
,%var σ
π
,%var σ
x
,%var
’Good Policy’
(simulat.)
-36.82% -5.93% --
’Good Lu ck’
(simulat.)
-39.29% -57.31% --
’Bad P olicy ’
(simulat.)
--25.40% 4.64%
’Bad Luc k ’
(simulat.)
--55.99% 56.71%
Table 13: VOLATILITIES COMPUTED WITH FACTUAL AND COUNTERFAC-
TUA L SIMUL ATIONS , L OW SL O PE k. The percentage variations were compu ted
on the m edian s of the simulated volatilities w ith respect to the benc hm a rk factual sce-
nario based on 10,000 repetitions. 1st subsample: 1960Q1-1979Q3; 2nd subsample:
1984Q1-1999Q4.
23
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev inflation
Probability
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std inflation
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. good luck
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad policy
stdev output
0 2 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
factual vs. bad luck
std output
Figure 4: FA C T UA L V S. COU N TE R FACTUAL SIMULATIONS, LOW SLOPE κ.
Lo w slope coeffic ient in the P hillips curv e, benc hm ar k calibration for the rest of the
model. Solid line: Factual distributio ns; ’Squared’ line: Co unterfactual distributions.
Number of repetitions: 10,000.
24
Tec hnical Appendix: Solution of the LRE Model
Let’s consider a linear rational expectations model as the follow ing one:
π
t
= β[φ
π
E
t
π
t+1
+(1− φ
π
)π
t−1
]+κ(x
t
− z
t
)
x
t
= φ
x
E
t
x
t+1
+(1− φ
x
)x
t−1
− τ(R
t
− E
t
π
t+1
)+g
t
R
t
=(1− ρ)[ρ
π
π
t
+ ρ
x
(x
t
− z
t
)] + ρR
t−1
+ ε
MP
t
z
t
= ρ
z
z
t−1
+ ε
z
t
,g
t
= ρ
g
g
t−1
+ ε
g
t
This model can be cast in the follo w in g canonical form :
Γ
0
(θ)s
t
= Γ
1
(θ)s
t−1
+ Ψ(θ)ε
t
+ Π(θ)η
t
(A1)
where the vector s
t
=[x
t
,π
t
,R
t
,E
t
x
t+1
,E
t
π
t+1
,g
t
,z
t
]
0
collects the n variables of the
system, ε
t
=[ε
MP
t
,ε
π
t
,ε
x
t
] is the vecto r of l fundamen tal shoc ks, η
t
=[(x
t
−E
t−1
x
t
), (π
t
−
E
t−1
π
t
)]
0
collects the k rational expectations forecast errors, and θ is the v ector of the
param eters of the model outlined in th e previous section. The matrices of th e canonical
form are presen ted belo w:
Γ
0
=
100 0 0 0 0
010 0 0 0 0
001 0 0 0(1− ρ)ρ
x
00τ −φ
x
−τ 01
000 0 −βφ
π
0 κ
000 0 0 1 0
000 0 0 0 1
Γ
1
=
0001000
0000100
00ρ (1 − ρ)ρ
x
(1 − ρ)ρ
π
00
(1 − φ
x
)00−1000
0 β(1 − φ
π
)0 κ −100
00000ρ
g
0
000000ρ
z
1
Ψ =
000
000
100
000
000
010
001
, Π =
10
01
(1 − ρ)ρ
x
(1 − ρ)ρ
π
−10
κ −1
00
00
In th e exercises proposed in the paper, w e set φ
x
= φ
π
=1.
In order to transform the canonical form and solve the model, we follow Sims (2001)
and exploit the generalized complex Sc hur decom position (QZ) of the m atrices Γ
0
and
Γ
1
. This corresponds to computing the matrices Q, Z, Λ and ∆ such that QQ
0
= ZZ
0
=
I
n
, Λ and ∆ are upper triangular, Γ
0
= Q
0
ΛZ and Γ
1
= Q
0
∆Z.Defining w
t
= Z
0
s
t
and
pre-multiplying (A1) by Q, we obtain:
·
Λ
11
Λ
12
0 Λ
22
¸·
w
1,t
w
2,t
¸
=
·
∆
11
∆
12
0 ∆
22
¸·
w
1,t−1
w
2,t−1
¸
+
·
Q
1.
Q
2.
¸
(Ψε
t
+ Πη
t
) (A2)
where, without loss of generalit y, the vector of generalized eigenvalues λ,whichisthe
vector of the ratios between the diagonal elemen ts of ∆ and Λ, has been partitioned
such th at the lower block collects all the explosive eigenvalues. The matrices ∆, Λ and
Q hav e been partitioned accordingly, and therefore Q
j.
collects the bloc ks of rows that
correspond to the stable (j =1)and unstable (j =2)eigenvalues respectiv ely.
The explosive block of (A2) can be rewritten as:
1
w
2,t
= Λ
−1
22
∆
22
w
2,t−1
+ Λ
−1
22
Q
2.
(Ψε
t
+ Πη
t
) (A3)
Given the set of m equations (A3), a non-explosive solution of the linear rational ex-
pectations model (A1) for s
t
requires w
2,t
=0∀t ≥ 0. This can be obtained by settin g
w
2,0
=0and choosing for ev ery vector ε
t
the endogenous forecast error η
t
that satisfies
the following condition
Q
2.
(Ψε
t
+ Πη
t
)=0 (A4)
A general stable solution for the endogenous forecast error can be comp uted through
a singu lar value decomposition of Q
2.
Π
|
{z}
mxk
= U
|{z}
mxm
D
|{z}
mxk
V
0
|{z}
kxk
= U
.1
|{z}
mxr
D
11
|{z}
rxr
V
0
.1
|{z}
rxk
,whereD
11
is
a diagonal matrix and D and U are orthonorm al matrices. Using this decomposition,
1
It is possible to have some zero-elements on the main diagonal of Λ
22
. In this case, the latter matrix
is not inv ertible. The ’solving-forward’ solution proposed by Sims (2001) and extended by Lubik and
Schorfheide (2003) overcomes this problem. A Technical Appendix with a more detailed discussion of
the s olution strategy is available from the authors upon request.
2
Lubik and Schorfheide (2003) show that in equilibrium the vector of endogenous forecast
errors reads as follows:
η
t
=(−V
0
.1
D
−1
11
U
.1
Q
2.
Ψ + V
.2
f
M)ε
t
+ V
.2
ζ
t
(A5)
where
f
M is the (k − r)xl matrix governin g the influence of the sunspot shock on the
model dynamics.
Assum ing that Γ
−1
0
exists, th e solution (A5) can be combined w ith (A1) to yield the
follo w ing law of motion for the state vector:
s
t
= Γ
∗
1
s
t−1
+
h
Ψ
∗
− Π
∗
V
.1
D
−1
11
U
0
.1
Q
2.
Ψ + Π
∗
V
.2
f
M
i
ε
t
+ Π
∗
V
.2
ζ
t
(A6)
where a generic X
∗
= Γ
−1
0
X.
In general, w e can be confronted with three cases. If the num ber o f endogenous
forecast errors k is equal to the number of non zero singular values r, the system is
determ ined and the stabilit y condition (A4) uniquely determines η
t
. Insuchacase,
V
.2
=0, then the last t wo addends of (A6) drop out. This implies that the dynamics of
s
t
is purely a function of the structural p arameters θ.
If the number of endogenous forecast errors k exceeds the number of nonzero singular
values r, the system is indeterminate and sunspot fluctuations can a rise. Lubik and
Sc horfheide (2003) sho w that this can influence the solution along two dimensions. First,
sunspot fluctuations ζ
t
can affect the equilibrium dynamics. Second, the transmissio n
of fundamen tal shoc ks ε
t
is no longe r un ique ly ident ified as the elements of
f
M are not
pinned dow n by the structure of the linear ra tional expectations model.
Alternativ ely, the number of endogenous forecast errors k can be smaller than the
number of nonzero singular values r, and then the system has no solutions. These
three conditions generalize the procedure in B lan chard and Kahn (1980) of counting
the n umber of unstable roots and predetermined variables.
2
In order to compu te
f
M and then the solutions of the model under indeterminacy,
it is necessary to im pose some additional restrictions on the endogenous forecast er-
rors. Follo wing Lubik and Schorfheide (2004), we c hoose
f
M such that the impulse
responses
∂s
t
∂ε
0
t
associated with the system (A6) are con tin uous at the boundary between
the determin acy and the indeterm ina cy region. This solution is labelled ’con t inuity ’.
In particular, let Θ
I
and Θ
D
be the sets of all possible vectors of parameters θ
0
s in the
indeterm inacy and d eterm ina cy region r espectiv ely. For every vector θ ∈ Θ
I
we identify
2
The solution method proposed by Sims (2001) has the advantage that it does not require the
separation of predetermined variables from ’jump’ variables. Rather, it recognizes that in equilibrium
models expectational residuals are attached to equations and that the structure of the coefficient
matrices in the canonical form implicitly selects the linear combination of variables that needs to be
predetermined for a s olution to exist.
3
a corresponding vector
∼
θ ∈ Θ
D
that lies on the boundary of the two regions and c hoose
f
M suc h that the response of s
t
to ε
t
conditiona l on θ mimics the response conditional
on
∼
θ. This c orresponds to requiring that the condition
∂s
t
∂ε
0
t
(θ)=B
1
(θ)+B
2
(θ)=Ψ
∗
− Π
∗
V
.1
D
−1
11
U
0
.1
Q
2.
Ψ + Π
∗
V
.2
f
M (A7)
beascloseaspossibletothecondition
∂s
t
∂ε
0
t
(
e
θ)=B
1
(
e
θ) (A8)
Applyin g a least-square criterion, we can then com pu te
f
M =[B
0
2
(θ)B
2
(θ)]
−1
B
0
2
(θ)
h
B
1
(
e
θ) − B
1
(θ)
i
(A9)
and use (A9) to calculate the so lution of the m odel in (A5) and (A 6).
The new vector
∼
θ is obtained from θ by replacing ρ
π
with the co ndition that marks
the boundary between th e determinacy and indeterminacy region. Woodford (200 3)
shows that this condition corresponds to the follo w ing interest rate reaction to inflation
eρ
π
=1−
(1 − β)
κ
ρ
x
(A10)
As an alternative to the ’con tinuit y’ solution, we also compute the solution of the
model under indeterminacy by imposing
f
M =0
(k−r)xl
,i.e. theeffects of the sunspot
shocks are orthogonal to the effects of the structural shocks. This solution is dubbed
’orthogona lity’.
Contributions cited in this Tec hn ical Appendix
Lubik, T.A., and F. Schorfheide, 2003, Computing Sunspot Equilibria in Linear Ra-
tional Expectations Models, Journal of Economic Dynamics and Contr ol,28(2),
273-285.
Lubik, T.A., and F. Schorfheide, 2004, Testing for Indeterminacy: An Application to
US Monetary P olicy, The A m erican Economic Review,94(1),190-217.
Sims, C.A., 2001, Solving Linear Rational Expectations Models, Computational Ec o-
nomics, 20, 1-20.
4
