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Working Paper Series

The Lucas Critique and the Stability of
Empirical Models

WP 06-05

Thomas A. Lubik
Federal Reserve Bank of Richmond
Paolo Surico
Bank of England and
University of Bari

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

The Lucas Critique and the Stability of Empirical Models∗
Thomas A. Lubik
Federal Reserve Bank of Richmond

Paolo Surico
Bank of England and
University of Bari

July 2006
Working Paper No. 06-05

Abstract
This paper re-considers the empirical relevance of the Lucas critique using a DSGE
sticky price model in which a weak central bank response to inflation generates equilibrium indeterminacy. The model is calibrated on the magnitude of the historical shift in
the Federal Reserve’s policy rule and is capable of generating the decline in the volatility
of inflation and real activity observed in U.S. data. Using Monte Carlo simulations and
a backward-looking model of aggregate supply and demand, we show that shifts in the
policy rule induce breaks in both the reduced-form coefficients and the reduced-form
error variances. The statistics of popular parameter stability tests are shown to have
low power if such heteroskedasticity is neglected. In contrast, when the instability of
the reduced-form error variances is accounted for, the Lucas critique is found to be
empirically relevant for both artificial and actual data.
JEL Classification: C52, E38, E52.
Key Words: Lucas critique, heteroskedasticity, parameter stability tests, rational
expectations, indeterminacy.

∗

We are grateful to Luca Benati, Andrew Blake, Jon Faust, Jan Groen, Haroon Mumtaz, Serena Ng,
Christoph Schleicher, Frank Schorfheide, Shaun Vahey and seminar participants at the Reserve Bank of
New Zealand, the Bank of England, the 2006 meeting of the Society of Computational Economics, the conference on “Macroeconometrics and Model Uncertainty” held on 27-28 June 2006 at the Reserve Bank of New
Zealand, and our discussant Timothy Kam for valuable comments and suggestions. The views expressed in
this paper are those of the authors, and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System, nor the Bank of England or the Monetary Policy Committee. Address
for correspondence: Thomas A. Lubik: Research Department, Federal Reserve Bank of Richmond, P.O. Box
27622, Richmond, VA 23261. Thomas.Lubik@rich.frb.org. Paolo Surico: Bank of England, Threadneedle
Street, London, EC2R 8AH. paolo.surico@bankofengland.co.uk.

1

1

Introduction

The Lucas critique comes as close to a natural law as seems possible in macroeconomics.
While its theoretical validity is largely uncontested, there is, however, a surprising lack of
empirical support in the literature. A host of studies have argued that the relevance of the
Lucas critique is limited in practice. In particular, shifts in policy seem not to have any
significant effects on the stability of backward representations of macroeconomic models for
various historical episodes. We argue in this paper that the evident inapplicability of the
Lucas critique is due to problems with the size and power of the econometric tests used.
We use a simple structural model of the U.S. economy as a data-generating process (DGP)
to illustrate these issues both conceptually and by means of a Monte Carlo analysis. Our
empirical findings confirm that the Lucas critique is relevant for the shift in U.S. monetary
policy behavior in the early 1980s.
It is difficult to underestimate the importance of Lucas (1976) in the development of
modern macroeconomic thought. The introduction of rational expectations in macroeconomics in the middle of the 1970s represented an intellectual revolution for the profession
and a serious challenge for large-scale, backward-looking econometric models that were used
for policy analysis. Lucas argued that changes in policy have an immediate effect on agents’
decision rules since they are inherently forward-looking and adapt to the effects of the new
policy regime. An important corollary of this argument is that any policy evaluation based
on backward-looking macroeconomic models is misleading whenever such policy shifts occur.
Empirical studies of the validity of the Lucas critique (e.g. Estrella and Fuhrer, 2003,
Rudebusch, 2005) suggest that it is unimportant in practice. Tests for parameter stability in
backward-looking specifications or reduced forms of macroeconomic relationships typically
fail to reject the null of structural stability in the presence of well-documented policy shifts.
This evidence would support the conclusion that policy changes are, in the terminology of
Leeper and Zha (2003), ‘modest’ enough not to alter the behavior of private agents in a
manner that is detectable by the econometrician. A further implication is that backwardlooking monetary models of the type advocated by Rudebusch and Svensson (1999), which
perform well empirically, are safe to use in policy experiments.
We argue in this paper that the failure of these studies to detect Lucas-critique effects
rests on the use of parameter stability tests such as the Chow break-point test and the
superexogeneity test. These tests implicitly assume equality of the variances of the reducedform innovations across sub-samples. It is, however, well known, at least since Toyoda

2

(1974), that even moderate heteroskedasticity has a considerable impact on the significance
levels of the null of parameter stability, which leads to an incorrect acceptance of parameter
invariance. Moreover, and similar to Lindé (2001), we also find evidence of a small-sample
bias which tends to hide the instability of backward-looking specifications.
At the heart of Lucas’ critique lies an emphasis on the use of fully-specified, optimisationbased dynamic stochastic general equilibrium (DSGE) models for policy analysis. We follow
this approach and work within the confines of a structural New Keynesian model that has
been widely used for monetary policy analysis. We treat this model as our DGP from
which we generate simulated time series for a Monte Carlo analysis. This allows us to
control the environment in which the policy change occurs. Otherwise, it might be difficult
to distinguish between actual policy shifts and changes in the economy’s driving processes.
Our main experiment is to model the monetary policy change in the U.S. that occurred
at the turn of the 1980s. We follow Clarida et al. (2000) and Lubik and Schorfheide (2004)
in assuming that policy during the 1970s responded only weakly to inflation, whereas with
the start of Volcker’s tenure policy became much more aggressively anti-inflationary. In
the context of our model, this implies that in the first sub-sample the rational expectations
(RE) equilibrium was indeterminate, and determinate later on. On the basis of the approach developed by Lubik and Schorfheide (2003) to describe indeterminate equilibria, we
show, first, that a policy shift of the magnitude observed in the Federal Reserve’s monetary
policy rule is capable of reproducing the historical decline in the volatility of real activity,
inflation, and the interest rate. Furthermore, we demonstrate that such a decline invalidates
the assumption of constant reduced-form innovation variances in a widely used backwardlooking model of aggregate supply and aggregate demand estimated over the sub-samples
associated with the regime shift.
Monte Carlo simulations show that the instability of the error variances severely affects
the power of the Chow test and the superexogeneity test to the effect that it prevents the
rejection of the incorrect null hypothesis of parameter stability. When the heteroskedasticity
in the estimated backward-looking specifications is accounted for, we find robust evidence
in favor of the empirical relevance of the Lucas critique. An application to U.S. data reveals
that controlling for the decline in the variance of the estimated error terms matters also in
practice. We show that this overturns the results from parameter stability tests which are
based on the (incorrect) assumption of equal error variance between sub-samples. Hence,
we conclude that the Lucas critique is alive and well.
To our knowledge, this is the first paper that investigates the effect of equilibrium inde-

3

terminacy on the stability of reduced-form models. The message of the paper is, however,
not predicated on policy changes that induce switches between determinacy and indeterminacy, but this assumption helps to sharpen our argument. We also report results from
simulations that preserve a specific regime.
The paper is organized as follows. Section 2 presents and discusses analytical results
in the context of simple examples on how policy changes in structural models affect their
reduced-form representations and how this relates to the debate about the empirical validity
of the Lucas critique. Section 3 introduces the structural model that we use as DGP for the
Monte Carlo study. This section also shows that the policy-induced shift from indeterminacy
to determinacy is capable of explaining the U.S. Great Moderation. Section 4 describes the
simulation strategy for testing the empirical relevance of the Lucas critique. In section 5,
we report the Monte Carlo evidence on the structural stability of a backward-looking model
of aggregate supply and demand, and provide a sensitivity analysis for variations in the
values of some parameters of the structural model. The sixth section contains empirical
results obtained on actual U.S. data, while the last section concludes.

2

Rational Expectations Equilibria and the Lucas Critique

We assume throughout our paper that the data are generated by a DSGE model. Analyzing
the effects of parameter changes thus requires an understanding of the reduced-form properties of structural linear rational expectations models. In this sense, our paper is similar to
Lindé (2001) and Rudebusch (2005). However, we go further than the earlier literature in
analyzing the effects of a policy break that changes the equilibrium properties of the economy, specifically, a change from indeterminacy to determinacy. In this section, we therefore
give a brief overview on indeterminacy in linear rational expectations models.

2.1

Determinate and Indeterminate Equilibria

Consider the simple expectational difference equation:
xt = aEt xt+1 + εt ,

(1)

where a is a parameter, εt is a white noise process with mean zero and variance σ 2 , and Et
is the rational expectations operator conditional on information at time t. It is well known
that the type of solution depends on the value of the parameter a. If |a| < 1 there is a
unique (‘determinate’) solution which is simply:

xt = εt .
4

On the other hand, if |a| > 1, there are multiple solutions and the rational expectations

equilibrium is indeterminate.

In order to derive the entire set of solutions we follow the approach developed by Lubik
and Schorfheide (2003). For this purpose it is often convenient to rewrite the model by
introducing endogenous forecast errors η t = xt − Et−1 xt . Define ξ t = Et xt+1 so that
equation (1) can be rewritten as:

ξt =

1
1
1
ξ t−1 − εt + η t .
a
a
a

(2)

Under indeterminacy this is a stable difference equation which imposes no further restrictions on the evolution of the endogenous forecast error η t .1 Hence, any covariance-stationary
stochastic process for η t is a solution for this linear rational expectations model. The forecast
error can then be expressed as a linear combination of the model’s fundamental disturbances
and extraneous sources of uncertainty, typically labeled ‘sunspots’. The coefficients on the
shocks in this decomposition generally depend on parameters of the model and on parameters that index a specific sunspot equilibrium. To wit, the forecast error can be written
as:
η t = mεt + ζ t ,
where the sunspot ζ t is a martingale-difference sequence, and m is an unrestricted parameter.2 Substituting this into Eq. (2) yields the full solution under indeterminacy:
xt =

1
1
xt−1 + mεt − εt−1 + ζ t .
a
a

The evolution of xt now depends on an additional (structural) parameter m which indexes
specific rational expectations equilibria.
Indeterminacy affects the behavior of the model in three main ways. First, indeterminate solutions exhibit a much richer lag structure and more persistence than the corresponding determinate solution. This feature has been exploited by Lubik and Schorfheide
(2004) in distinguishing between the two types of rational expectations equilibria in U.S.
data. In the simple example, this is strikingly evident: under determinacy the solution for
xt is white noise, while under indeterminacy the solution is described by an ARMA(1,1)
process. Specifically, the (composite) error term exhibits both serial correlation and a different variance when compared to the determinate solution. Second, under indeterminacy
1

In the case of determinacy, the restriction imposed is that ξ t = 0, ∀ t, which implies ηt = εt .
There is a technical subtlety in that ζ t is actually, in the terminology of Lubik and Schorfheide (2003), a
reduced-form sunspot shock, with ζ t = mζ ζ ∗t . Furthermore, in less simple models, there would be additional
restrictions on the coefficients which depend on other structural parameters.
2

5

sunspot shocks can affect equilibrium dynamics. Other things being equal, data generated
by sunspot equilibria are inherently more volatile than their determinate counterparts. The
third implication, especially emphasized by Lubik and Schorfheide (2003), is that indeterminacy affects the response of the model to fundamental shocks, whereas the response
to sunspot shocks is uniquely determined. In the example, innovations to εt could either
increase or decrease xt depending on the sign of m.

2.2

Indeterminacy and the Lucas Critique

We now try to provide some insight into the analytics of the Lucas critique by means
of a simple example. We consider the following two-equation model which describes the
evolution of an economic variable xt and a policy variable yt :
xt = aEt xt+1 + bEt yt+1 + ε1t ,

(3)

yt = cxt + ε2t .

(4)

ε1t , ε2t are white noise processes with variances σ 21 , σ 22 , respectively. a, b are structural
parameters, and c is a policy parameter. We assume for simplicity that all parameters are
positive. Eq. (4) is a feedback rule of the type often used in monetary policy models.
The model has a unique rational expectations equilibrium if 0 < c <

1−a
b ,

the solution

of which is:
xt = ε1t ,
yt = cε1t + ε2t .
As in the simple example above, both variables are white noise processes. If c >

1−a
b

the

solution is indeterminate. The laws of motion for the two variables are as follows:
xt =
yt =

1
1
xt−1 + mε1t −
ε1t−1 + ζ t ,
a + bc
a + bc
1
c
1
yt−1 + mcε1t −
ε1t−1 + ε2t −
ε2t−1 + cζ t .
a + bc
a + bc
a + bc

Again, the change in the stochastic properties of the variables, when moving across the
parameter space, is quite evident. If (3) - (4) is the DGP, then the reduced-form equations
above are the representations upon which tests for the empirical relevance of the Lucas
critique would have to be based.
The type of experiment we are interested in is an exogenous, unanticipated change in the
policy parameter c. We can distinguish four different scenarios: a shift from determinacy
6

to indeterminacy, from indeterminacy to determinacy, and changes that preserve previously
determinate and indeterminate equilibria. If the break in c is such that the solution stays
determinate, the behavior of xt is unaffected while the variance of yt changes. However, an
econometrician could not distinguish between a change in σ 22 and the policy coefficient by
observing yt . A change in the variance of ε1t , on the other hand, could be deduced from
observations on both variables.
The more interesting case is a change in policy behavior that moves c across the boundary
1−a
b

that separates the determinacy from the indeterminacy region. Suppose the economy

is initially in an indeterminate equilibrium associated with a policy parameter c0 . An
unexpected (and believed to be permanent) policy shift to c1 <

1−a
b

< c0 moves the economy

to a determinate equilibrium. This implies a dramatic change in the nature of the stochastic
processes for xt and yt , as they switch from persistence to white noise. Specifically, both
their variance and the degree of auto-correlation decline. Furthermore, sunspot shocks, and
the extra volatility associated with them, no longer affect the dynamics. It is this type of
scenario we have in mind in our empirical analysis. The standard example is the change
in monetary policy that occurred during Paul Volcker’s tenure at the Board of Governors.
Naturally, similar reasoning applies to the opposite case when a policy change moves the
economy from a determinate to an indeterminate equilibrium.
Yet, even if equilibrium properties are unaffected by shifts in policy, the properties of
the reduced-form model are not. Crucially, the variance of the error term in a regression
of the endogenous variables on their first lag would (i) change with the shift in policy,
and (ii) be correlated with the regressor. The latter issue is, of course, well known, and
methods to deal with this, such as IV-estimation, are now widely used.3 The former is not
as well appreciated, however. Consequently, the main thrust of our paper is directed at this
form of heteroskedasticity, which reduces the power of parameter stability tests based on
reduced-form regressions.4

3

The Structural Model

In this section, we lay out a small-scale sticky-price monetary model which will serve as DGP
for the Monte Carlo experiments. After describing the calibration, we use the structural
3
See, however, Lubik and Schorfheide (2005) for a set of examples in the context of DSGE-models where
IV-methods fail.
4
Structural estimation methods are immune against this problem since the structure of the reduced-form
error would be reflected in, say, the likelihood function. This is likely the main reason why the GMM-based
results of Collard et al. (2002) are an outlier in the literature in that they find strong evidence for the
empirical relevance of the Lucas critique.

7

model to assess the extent to which the ‘good policy’ and the ‘good luck’ explanations of
the U.S. Great Moderation are capable of replicating the facts.

3.1

Specification

Our simulation analysis is based on a log-linearized, microfounded New-Keynesian sticky
price model of the business cycle of the kind popularized by Clarida et al. (1999), King
(2000) and Woodford (2003) among others. The model consists of three aggregate relationships which describe the dynamic behavior of output yt , inflation π t , and the nominal
interest rate Rt :
yt = Et yt+1 − τ (Rt − Et π t+1 ) + gt ,

(5)

π t = βEt π t+1 + κ (yt − zt ) ,
¡
¢
Rt = ρR Rt−1 + (1 − ρR ) ψ π π t + ψ y (yt − zt ) + εR,t .

(6)
(7)

All variables are expressed in percentage deviations from the steady state.
Eq. (5) is a log-linearized IS-curve derived from a household’s intertemporal optimization problem in which consumption and nominal bond holdings are the control variables.
Since there is no physical capital in this economy, consumption is proportional to total
resources up to an exogenous process gt . The latter is typically interpreted as a government
spending shock or a shock to preferences.5 The parameter τ > 0 represents the intertemporal elasticity of substitution.
The Phillips-curve relationship (6) describes inflation dynamics as a function of output.
It captures the staggered feature of an economy with Calvo-type price setting in which firms
adjust their optimal price with a constant probability in any period, independently of the
time elapsed from the last adjustment. The discrete nature of price setting creates an incentive to adjust prices by more the higher expected future inflation is. The contemporaneous
rate of inflation is thus related to the difference between output and the stochastic marginal
cost of production zt via the parameter κ > 0, which can be interpreted as the inverse of
the sacrifice ratio. 0 < β < 1 is the agents’ discount factor.
The policy rule (7) characterizes the behavior of the monetary authorities according
to which the central bank adjusts the policy variable in response to current inflation and
the output gap (yt − zt ). ψ π , ψ y ≥ 0 are the policy coefficients. These adjustments are

implemented smoothly, with 0 < ρR < 1 capturing the degree of interest rate inertia. The
5

The IS curve can easily be reinterpreted as a schedule explaining the behavior of the ‘output gap’ defined
as the difference between actual output and the hypothetical flexible price level of output (see Clarida et al.
1999). In this case, the shock gt is also a source of potential output variations.

8

random variable εR,t stands for the monetary policy shock, which can be interpreted either
as unexpected deviations from the policy rule or as a policy implementation error. We
assume it to be a white noise process with mean zero and variance σ 2ε .
The model description is completed by specifying the stochastic properties of the exogenous shocks gt and zt . We assume they are first-order autoregressive processes with
lag-coefficients 0 ≤ ρg , ρz < 1. Their innovations are assumed to be i.i.d. with variances σ 2g

and σ 2z , respectively. The equation system (5) - (7), together with the two processes for the

exogenous shocks, describes a linear rational expectations model that can be solved using
the methods described in Sims (2002). We solve the model for both determinacy and indeterminacy. In the latter case, we follow the approach developed by Lubik and Schorfheide
(2003).

3.2

Calibration

We choose parameter values in line with the estimates in Lubik and Schorfheide (2004),
who analyzed a model similar to ours. The values of the structural parameters are reported
in Panel A of Table 1. We assign the same values in both subsamples to focus on changes in
the policy coefficients. Unlike Lubik and Schorfheide (2004), however, we set the variance
of sunspot shocks to zero in the case of indeterminacy. This is designed to facilitate the
comparison between our simulations and the previous literature, which does not take the
presence of sunspot fluctuations into account.
A large body of empirical literature has documented the dramatic change in the conduct
of U.S. monetary policy at the turn of the 1980s. The established consensus is that the
nominal interest rate response to inflation in estimated policy rules became substantially
more aggressive in the early 1980s. As the number of available estimates is quite large,
we set the coefficients in the policy rules for Periods I and II to values that are broadly
in line with the evidence presented by Clarida et al. (2000). Within the indeterminacy
and determinacy regions, the results are fairly robust to deviations from the baseline values
reported in Panel B of Table 1.
The interest rate response to inflation over the indeterminacy sample does not guarantee
a unique rational expectations equilibrium because ψ π = 0.4 violates the Taylor principle.6
On the other hand, the parameter constellation associated with Period II guarantees a
determinate equilibrium. As detailed above, the solution of the model under indeterminacy
6

The additional requirement is that the other policy coefficients in the rule are not ‘too large’ to compensate for the weak inflation response. The analytics and intuition of indeterminate equilibria in the New
Keynesian monetary model are discussed extensively by Lubik and Marzo (2006).

9

is affected along various dimensions. Most importantly, equilibrium dynamics can be driven
by sunspot fluctuations. Secondly, the transmission of fundamental shocks can be altered
relative to the unique rational expectations solution. Having set the variance of sunspot
disturbances to zero, we will thus focus on the transmission mechanism effect.

3.3

Replicating the Stylized Facts

Before evaluating the empirical relevance of the Lucas critique, we assess the extent to
which the magnitude of historical policy shifts can account for the decline in volatility and
persistence of inflation, output, and the interest rate observed in U.S. data at the beginning
of the 1980s. A large number of contributions, including Kim and Nelson (1999), McConnell
and Perez-Quiros (2000), and Blanchard and Simon (2001), document a sharp decline in
the volatility of many U.S. macroeconomic variables. The variance of inflation declined by
more than two thirds and the variance of output by just less than one half. Following Stock
and Watson (2002), this set of facts has come to be known as the Great Moderation.
There is, however, a simmering debate whether this was due to good luck (a decline
in the volatility of exogenous shocks) or good policy (a shift to a more stabilizing policy
regime). Empirical evidence by Stock and Watson (2002) and Kim et al. (2004) shows that
a sizeable part of such decline was driven by a reduction in the volatility of the reduced-form
innovations of estimated backward-looking specifications for inflation and output. This has
been invariably interpreted as prima facie evidence in favor of good luck.7
We thus ask the question whether the good luck and good policy scenarios are capable
of explaining this decline in volatilities in the context of our model. We present results for
two sets of model calibrations. In the first experiment, we change the policy parameters
such that the equilibrium switches from indeterminacy to determinacy. Differences in the
dynamics are therefore driven only by a shift in the monetary policy rule, as all non-policy
parameters of the model are kept fixed across regimes. In the second calibration, we simulate
a decline in the variance of the structural shocks such as to match the magnitude of the
decline in the volatility of inflation and output observed in the data. All other parameters
are fixed in the second simulation. In particular, the Taylor principle applies and the
equilibrium is therefore always determinate.
Simulation results for the good policy hypothesis are reported in Table 2. Panel A
displays the standard deviation of output, inflation and the interest rate. All variables are far
7

In work in progress, Benati and Surico show, however, that this interpretation is unwarranted since a
move from indeterminacy to determinacy is also consistent with a decline in the innovation variances of VAR
models.

10

less volatile in the post-1982 simulated sample. The volatility of output and the interest rate
declined by 42%, while the standard deviation of inflation was reduced by 67% when moving
from indeterminacy to determinacy. Panel B reports a measure of persistence obtained as
the sum of autoregressive coefficients from the estimates of an univariate AR(p) process.
In line with the evidence in Cogley and Sargent (2005), the most dramatic change occurs
for inflation whose inertia drops by 58%. The impact on the interest rate is considerably
smaller. This can be explained by the higher smoothing parameter under determinacy
which outweighs the impact of the ‘extra’ persistence induced by monetary policy under
indeterminacy.8
We contrast this with the statistics obtained under the good luck hypothesis, reported
in Table 3. While a decline in the variance of the structural shocks can replicate the
drop in volatility, it cannot match the observed decrease in persistence, in particular the
well-documented decline in inflation persistence. This strongly points toward a change
in the transmission mechanism. A similar conclusion has been reached by McConnell and
Perez-Quiros (2000), Boivin and Giannoni (2005) and Canova, Gambetti and Pappa (2005),
pinpointing monetary policy changes. Our specific explanation is the shift from an indeterminate to a determinate equilibrium induced by a change in the Federal Reserve’s policy
behavior. As we demonstrated in Section 2 such a shift changes the dynamic properties of
the reduced form of the model.
We also want to emphasize that the Lucas critique is economically relevant in our setting.
Two metrics that are typically considered for this assessment are the sum of the autoregressive coefficients of inflation and the value of a loss function that weighs the unconditional
variances of inflation and output. The results of Table 2 indicate that the critique is, indeed, economically important in that both the persistence of inflation and the sum of the
unconditional variances of inflation and output decline drastically in the move to a more
anti-inflationary policy.
Overall, we find that both explanations are capable of explaining the Great Moderation.
However, only the good policy hypothesis successfully reproduces the decline in inflation
persistence. A shift from passive to active monetary policy can account on its own for the
sharp decline in the volatility and persistence of inflation and output observed in the early
1980s. Furthermore, the decline in the volatility of the series is likely to contaminate the
8

On the other hand, output persistence actually increases, which is likely due to the behavior of the exante real interest rate. Across regimes, the persistence of the nominal interest rate is almost unchanged and
close to 0.8. In contrast, the persistence of expected inflation, whose behavior is similar to actual inflation,
declines remarkably, from 0.7 to 0.3. This implies that the persistence of the ex-ante real interest rate is
higher in the second sub-sample, which makes output more persistent.

11

properties of the error terms in estimated reduced-form models. As many widely used break
point tests like the Chow test are based on the assumption of constancy of the innovations
variance, we show below that the neglected heteroskedasticity leads to incorrect inference
on the empirical relevance of the Lucas critique.

4

Testing for the Lucas Critique

This section presents a simple algorithm to investigate the empirical relevance of the Lucas
critique. To focus on the importance of a change in monetary policy, we simulate a shift in
the coefficients of the policy rule while keeping the parameters describing the structure of
the economy fixed across simulations. For each simulation, we obtain two sets of artificial
data. The first set is generated from an indeterminate equilibrium and is associated with
the pre-1979 estimates of the policy rule typically found in the literature. The second
sample is generated under the assumption of determinacy and it corresponds to the post1979 description of policy behavior. Any difference in the estimates on the two sets of
artificial data is thus only attributable to changes in policy. As the Lucas critique is about
(in)stability of reduced-form parameters, we specify a backward-looking representation of
the economy and then assess the impact of a policy shift on the reduced-form representation.

4.1

A Backward-Looking Model

The backward-looking model of aggregate supply and demand is a quarterly version of the
specification in Svensson (1997), which has been used by Rudebusch and Svensson (1999)
and Rudebusch (2005) among many others:

π t = α1 π t−1 + α2 π t−2 + α3 π t−3 + α4 π t−4 + αy yt−1 + uπt ,

(8)

yt = β 1 yt−1 + β 2 yt−2 + β 3 yt−3 + β 4 yt−4 + β r (R − π)t−1 + uyt .

(9)

Inflation depends on its own past values within a year and on the lagged value of real
activity. Aggregate demand is characterized by an autoregressive structure with four lags
augmented by the lagged value of the real interest rate. The model is closed with a standard
Taylor-type rule with interest rate smoothing:
Rt = γ R Rt−1 + γ π π t + γ y yt + uR
t .

(10)

The model (8)-(10) is estimated on simulated data under the assumption that the DGP
is the structural model (5)-(7).
12

We want to emphasize that we do not restrict the backward-looking model to the
reduced-form representation of the DSGE model. In particular, the backward-looking specification is allowed, but not required, to have a richer lag structure relative to the DSGE
model. In other words, we let the data choose what lags of the dependent variables have
explanatory power. Our reasoning is twofold. First, earlier contributions have used very
similar backward-looking specifications and we wish to contrast our results with those of
the previous literature. Second, and more importantly, we have shown in Section 2 that
equilibrium indeterminacy is an additional source of serial correlation in the error terms.9
Augmenting the backward-looking specification with additional lags of the dependent variable is a simple way to make the residuals white noise and therefore control for serial
correlation.

4.2

Simulation Strategy

To assess the empirical relevance of the Lucas critique, we design a Monte Carlo experiment
in which we postulate a break of the magnitude of the historical shift in the Federal Reserve’s
policy rule. All other parameters of the structural model are kept fixed across policy regimes,
and we are thus isolating any Lucas-critique effects.
The procedure in the simulations is as follows:
1. Solve the structural model under both indeterminacy and determinacy, and generate
two samples of 82 and 61 observations10 for output, inflation and the interest rate.11
2. For each artificial sample, estimate the backward-looking equations for output, inflation, and the interest rate.
3. Perform tests for error variance constancy and parameter stability of the backwardlooking equations across the two periods; compute the relevant statistics and probability values.
4. Repeat steps 1. to 3. 20, 000 times; for each sub-period select the median values of
the statistics of interest.
9

We are grateful to Adrian Pagan for raising our awareness of this issue.
The number of observations has been chosen to match the quarterly data points which are typically used
in sub-sample analyses of US data (see Lubik and Schorfheide, 2004, and Clarida et al., 2000). The first
period ranges from 1960:1 to 1979:2 while the second corresponds to 1982:4 to 1997:4. In each simulated
sample, 100 extra observations are produced to provide us with a stochastic vector of initial conditions,
which are then discarded.
11
The solution of the model under indeterminacy is computed using the continuity assumption in Lubik
and Schorfheide (2004). Similar results, not reported but available upon request, are obtained by imposing
that the structural shocks are orthogonal to the sunspot shocks.
10

13

5. Prior to the simulations, Steps 1 - 4 are carried out under the assumption of no
policy regime shift, and small-sample critical values are computed from the actual
distributions of the relative statistics under such scenario. This ensures a correct size
of the tests for parameter instability.
If the probability of rejecting parameter stability in Steps 1 - 4 computed under the
assumption of a policy break is higher than the probability in Step 5 based on the assumption
of no change in the policy parameters, then the Lucas critique is judged empirically relevant.

5

Results on Simulated Data

This section presents the results of the Monte Carlo analysis. We present the relevant
statistics of the test for equal innovation variances and parameter stability together with
some robustness checks.

5.1

Testing for Equal Innovation Variances

Table 4 reports residual standard deviations from the estimated model (8) - (10) together
with the statistics of the Goldfeld-Quandt test of innovation variance constancy.12 The
variances of reduced-form innovations in the output and inflation equation exhibit a dramatic decline from the indeterminacy to the determinacy period of more than 50%. The
decline for the interest rate equation is less dramatic. The null hypothesis of stability across
samples is overwhelmingly rejected for the output and inflation equations.
This leads us to conclude that heteroskedasticity is present in the reduced-form equations
over the two sub-samples. From the point of view of a structural DSGE model this is not
surprising as we demonstrated in Section 2: it is precisely what would be expected in the
presence of a monetary policy shift. What makes this observation potentially relevant for
the empirical literature on the Lucas critique is that, so far, inference had been based on
parameter stability tests that neglected this feature of the data. More specifically, the
Chow test and the superexogeneity test, used by Favero and Hendry (1992), Lindé (2001),
and Rudebusch (2005) among many others, implicitly assume homoskedasticity of the error
variances between sample regimes and are therefore subject to our criticism.
Toyoda (1974) has demonstrated that the size and power of the Chow test can be
considerably affected by neglecting differences in reduced-form error variances. The problem
12

Estimates of the reduced-form coefficients are available from the authors upon request. We chose not
to report these since the actual estimates are immaterial to our discussion. What matters is the joint
significance, since it is well known that two sets of parameters can be jointly different from each other even
though the parameters are not statistically different individually.

14

is most serious when samples of similar size are used. When overall sample size is small
even moderate degrees of heteroskedasticity reduce the power of the test significantly. Both
scenarios apply to the historical episode of the monetary policy shift in the U.S. Perhaps not
surprisingly, the conclusions drawn in the literature almost always go against the statistical
importance of Lucas’ critique.13 We consequently ask the question whether the failure to
detect any effects is due to a lack of power of these tests.

5.2

Parameter Stability Tests

To quantify the importance of heteroskedasticity for the results of conventional parameter
stability tests, we follow the literature and use the Chow and superexogeneity tests on
the two simulated samples. The superexogeneity test measures the probability of rejecting
the null of parameter stability in the output and inflation equations conditional on having
rejected the null hypothesis of parameter stability in the interest rate equation.14
We present results for three versions of the parameter stability and superexogeneity
tests. The first version is based on the OLS residual sum of squares, RSS, of (8) and (9),
and therefore implicitly (and incorrectly) assumes homoskedasticity. A prominent example
of this class of tests is the Chow statistics used by Lindé (2001) and Rudebusch (2005):
Chow =

[RSSf ull − (RSS1 + RSS2 )] /m
,
(RSS1 + RSS2 ) / (T − k)

(11)

where m is the number of restrictions in an equation of k parameters and i = 1, 2 indexes
the sub-samples.
The second and third versions of the tests, in contrast, account for possible heteroskedasticity across sub-samples. In particular, we use a GLS version based on a two-step procedure.
In the first step, Eqs. (8) and (9) are estimated by OLS over the two sub-samples. In the
second step, the variables are normalized by the square root of the estimated innovation
variance-covariance matrix. Statistics of interest are then computed using the residuals
of the OLS estimates of (8) and (9) whereby the original variables are replaced with the
transformed variables.
The third version is based on the difference of the parameter estimates, θ̂i . The test
statistic is:

³
´0 ³
´−1 ³
´
W ald = θ̂1 − θ̂2
V̂1 + V̂2
θ̂1 − θ̂2 ,

(12)

13
An exception is Lindé (2001) who argues in favor of its empirical relevance on the basis of an empirical
money demand equation, showing that the superexogeneity tests suffers from a serious small-sample problem.
14
See Favero and Hendry (1992) for the strategy behind this test.

15

where V̂i is the estimated variance-covariance matrix of the parameters corrected for heteroskedasticity and autocorrelation in the error terms. Following Newey and West (1987),
we set the truncation lag in the autocorrelation function of the residuals, qi , to f loor(4 ∗
(Ti /100)ˆ(2/9)), which corresponds to 3 in our sub-samples.15

Parameter stability tests based on the RSS such as (11) and on the Wald form such
as (12) give the same algebraic results only in one special case: when the restrictions on
the parameters are linear and the error covariance matrix is homoskedastic (see Hamilton,
1994, Ch. 8, and Hansen, 2001). The Newey-West correction for heteroskedasticity and
autocorrelation, in fact, has no influence on the RSS, implying that only an expression
such as (12) is suited for dealing with heteroskedasticity.
Table 5 reports the results of the three versions of the parameter stability and superexogeneity tests of 5% empirical size. The size-corrected power of the respective tests, i.e., the
probability of rejecting the null hypothesis when it is false, is computed using the empirical
5% significance level and the simulation strategy described in Section 4.2. According to the
standard Chow test (labeled ‘OLS-based’) there is little evidence of parameter instability
in the output and inflation equations indicated by probabilities that are never larger than
0.10. Yet, the Chow test is capable of detecting instability in the interest rate equation.
The power of the test to detect parameter instability increases noticeably for our second
and third versions which take the instability of the reduced-form innovations variance into
account. When a GLS-based correction is used, the probability of (correctly) rejecting the
null hypothesis increases by a factor of five for both the output and the inflation equations.
While the power of the test for the inflation specification is good, it is still less than satisfactory in the case of output. This need not be surprising per se for two reasons. First, there
may still be a small-sample issue. We investigate this possibility further below. Second, it
may very well be the case that the effect of the policy parameter change on the behavior of
the output specification is statistically small. In other words, the policy intervention might
be modest in the Leeper-Zha sense. Additionally, the estimation of a similar model in Lubik
and Schorfheide (2004) reveals that output over the sample period (and conditional on the
structural model) is almost exclusively driven by technology shocks, and that the feedback
from the policy equation is minor. Similar conclusions can be drawn in the case of the
Newey-West correction.
The simulation evidence on the performance of the superexogeneity tests corroborates
15

We obtain similar results by setting qi = 0. Moreover, the residuals display no sign of serial correlation,
suggesting that four lags of the dependent variable in the inflation and output equations provide a reasonably
good approximation of the dynamics of the model.

16

the notion of a bias induced by neglecting heteroskedasticity. Incidentally, the probabilities
of rejecting the null of superexogeneity reported in Panel B of Table 5 are virtually identical.
We conclude that the test for parameter stability using either a GLS-based or a Newey-West
correction for the presence of heteroskedasticity and serial correlation always have higher
power than the tests based on OLS.

5.3

Sensitivity Analysis

We assess the robustness of our assessment along three dimensions. We first modify our
baseline calibration with respect to a few key parameters, while maintaining the regime
shift from indeterminacy to determinacy. Secondly, we keep the baseline specification for
the structural parameters, but simulate a policy shift between determinate regimes. Finally,
we address the small-sample problems emphasized by Lindé (2001).
The results for variations of our baseline calibration are reported in Table 6. We consider
three cases. The simulation using a smaller Phillips-curve slope coefficient κ = 0.2 reinforces
the neglected heteroskedasticity bias as the power of the tests on the stability of the output
and inflation equations become 0.18 and 0.60, respectively. Similar conclusions are reached
on the basis of the superexogeneity tests in the last two columns or using even smaller values
of κ. When aggregate demand is less sensitive to interest rate movements, τ −1 = 3, the
GLS-based probabilities of rejecting the null for the output and inflation equations are three
times as large as those neglecting the instability of the reduced-form innovations variance
in the column OLS-based.
Lastly, to appreciate fully the effect of a shift from indeterminacy to determinacy, the
bottom panel reports the results based on the value of 0.2 for σ ζ , the standard deviation
of sunspot shocks estimated by Lubik and Schorfheide (2004). This modification does
not vary the baseline results much and confirms the conclusion reached by Castelnuovo and
Surico (2006) that the passive monetary policy regime influenced U.S. aggregate fluctuations
through a change in the transmission mechanism, rather than through sunspot shocks.
The second robustness check analyzes to what extent our results are driven by the change
in the equilibrium properties of the model. As demonstrated in Section 2, the policy-induced
shift from indeterminacy to determinacy reduces both persistence and volatility of the
endogenous variables. Policy shifts that maintain a determinate equilibrium, on the other
hand, have a less dramatic effect on the reduced-form error variance.16 Our experiment is to
16

Rudebusch (2005) rules out the possibility of a shift from indeterminacy to determinacy by adjusting
the estimated inflation response coefficient upwards. This is done out of concern for avoiding instability in
the equation system. However, the issue with indeterminacy is that there is not enough instability in the

17

change the inflation coefficient from 1.5 to 2.5. The other structural and policy parameters
are as in Table 1. Two observations emerge: first, the policy shift within the determinacy
region is less successful in replicating the stylized facts, in particular the decline in output
volatility and persistence. While obviously far from conclusive, we regard this as a simple
plausibility check on the driving forces behind the Great Moderation. Secondly, tests for
error variance constancy, reported in Table 8, show that the null of homoskedasticity is not
rejected at typical confidence levels with the exception of inflation. Differences between
variances are in any case noticeably smaller than in the baseline case. Not surprisingly, this
affects the improvement in power of the heteroskedasticity-adjusted test statistics in Table
9. The size-adjusted probabilities of rejecting parameter stability and superexogeneity are
now more similar.
Small-sample problems as documented by Lindé (2001) in the case of the superexogeneity
test are also a possible explanation. We return to our baseline calibration and simulate a
shift from indeterminacy to determinacy, but this time impose a sub-sample size of 150
observations, twice as many as in the baseline case. The results can be found in Table 10.
Small-sample problems are clearly present for all three versions of the test for parameter
stability. Interestingly, a doubling of sample size has little effect on the probabilities of
rejecting the null hypothesis for the output equation, while the power of heteroskedasticityadjusted statistics improves by more than that of the simple OLS-based statistics. A similar
conclusion can be drawn for the superexogeneity test.

6

Detecting the Lucas Critique in Practice: The Volcker Policy Shift

We conclude our analysis by an application to historical data. In fact, many contributions,
for instance Stock and Watson (2002), Kim, Nelson and Piger (2004), and Cogley and Sargent (2005), have shown that post-war U.S. data are characterized by a substantial amount
of heteroskedasticity. The conceptual background for our simulation analysis is the break in
U.S. monetary policy behavior in the early 1980s, characterized by the tenure of Paul Volcker. We are interested in whether parameter stability tests adjusted for heteroskedasticity
detect this shift in the context of a reduced-form model.
The full sample spans the period 1960:1 - 2005:3. Quarterly data were collected from the
FRED database at the Federal Reserve Bank of St. Louis. We use data on output growth,
measured as the quarter-to-quarter change in real GDP; inflation as the quarter-to-quarter
system since there are infinitely many stable adjustment paths.

18

change in the GDP deflator; and the federal funds rate as policy instrument.
Figure 1 depicts the statistics of two recursive tests for parameter stability of the
reduced-form model (8)-(10). The minimum length of a sub-sample is eight years, implying that in the first recursion, Period I ends in 1968:4, whereas in the last recursion,
Period II begins in 1998:1. The left column reports the results of the Chow test (11) which
is based on the RSS and therefore implicitly assumes spherical disturbances. The right
column refers to the Wald test (12) which, in contrast, is corrected for heteroskedasticity
and autocorrelation in the error terms.
The top dashed horizontal line in each panel represents the 5% empirical critical value
based on 10, 000 bootstrap repetitions in which we impose the null of parameter stability. The dashed horizontal line in the middle stands for the 5% asymptotic critical value
computed by Andrews (1993) for tests of parameter instability with unknown change point
within the middle 70% of the sample. For sake of comparison, the critical values of the
same tests, but with known break date, are reported as dotted horizontal lines.17 The null
hypothesis of parameter stability is rejected if the maximum value of the statistics is above
the critical value.
In analogy with the Monte Carlo simulations in Section 5, using the Wald test dramatically changes the inference on parameter stability. The sup of the Wald statistics in the
right column are above the bootstrapped 5% critical values for all equations. In contrast,
the recursive Chow tests in the left column uniformly fail to reject the null hypothesis of
parameter stability. The only possible exception is the interest rate equation conditional on
a priori knowledge that a break occurred in 1980:2. However, even if the econometrician
were equipped with information, say, on the basis of the empirical literature on monetary
policy rules (Clarida et al., 2000), the statistics of the Chow tests would not detect any
structural change in the output and inflation equations, being far below the relevant 5%
critical values.
While the focus of the paper is on the policy shift associated with the beginning of
Volcker’s tenure, we cannot exclude, in principle, that the backward-looking model (8)-(10)
is in fact characterized by multiple structural breaks. To investigate this possibility, we use
the tests for (pure) multiple changes at unknown dates proposed by Bai and Perron (1998).
Following the practical recommendations in Bai and Perron (2003), we set the number of
maximum structural breaks, M , to 4, and consider a trimming ε of 0.15. In analogy with
17
Andrews and Fair (1988) show that under the null hypothesis of parameter stability, θ̂1 = θ̂2 , the Wald
statistic is asymptotically distributed as a χ2 random variable with k degrees of freedom.

19

the testing procedure on simulated data, different variances of the residuals are allowed
across segments. P-values are based on 10, 000 bootstrap repetitions obtained under the
null of parameter stability. The significance level is 5%.
The first two columns of Table (11) report the UDmax and WDmax double maximum
statistics for a test of no structural break against the alternative of an unknown number
of breaks given the upper bound M . According to both statistics, there is at least one
structural change in all three equations. To determine the number of breaks we then examine
the Sup-F( + 1| ) statistics for testing the null of

break(s) against the alternative that

an additional break exists. The third column of Table (11) shows that the hypothesis that
only one change occurred is never rejected against the alternative of at least two changes.
The estimates of the break date obtained by minimizing the sum of the RSS over the
sub-samples are 1983:2 for output, 1981:2 for inflation and 1980:3 for the policy rule.

7

Conclusions

We present two main arguments in this paper. First, we emphasize that within the framework of structural DSGE models, a change in policy parameters affects the stability of
both reduced-form coefficients and reduced-form error variances. We show that tests for
parameter instability based on the assumption of homoskedasticity have low power. We
consequently argue that this is behind the tendency in the literature to reject the empirical
relevance of the Lucas critique. We suggest adjustments for heteroskedasticity in commonly
used parameter stability tests, and show by means of a Monte Carlo analysis that the power
of these tests is improved and Lucas critique effects are detected in simulated data.
As an empirical example we test for the presence of Lucas critique effects on reducedform specifications of output and inflation using post-war U.S. data. We do, indeed, find
evidence of a break in the behavior of the Federal Reserve in 1980, and of parameter
instability across the sub-samples using our heteroskedasticity-adjusted test statistic. We
believe this is a new finding in the literature. Obviously, our results are model-dependent,
and further investigation in a richer framework is certainly warranted.
We want to conclude on a somewhat critical note. Given a fully-specified, structural
DSGE model as a DGP, the question arises why researchers should bother at all with
reduced-form specifications that are subject to the Lucas critique. One answer is certainly
ease of implementation. Furthermore, very little is known about structural break tests
within the context of estimated DSGE-models.18 However, a deeper issue is whether DSGE18

A notable exception is Ireland (2001). Initial steps in this direction have also been made by Canova

20

models that are used for policy analysis are not themselves subject to the Lucas critique.
Implicitly, Lucas’ argument rests on the notion that the information set of economic agents
and their decision problems were not fully specified in traditional macroeconometric models.
Yet, with the use of ad hoc monetary policy rules that very issue surely comes up in DSGE
models that do not include optimizing policy-makers.

(2005) and Justiniano and Primiceri (2006).

21

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[15] Favero, Carlo and David Hendry (1992): “Testing the Lucas Critique: A Review”.
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Macroeconomic Fluctuations”. Manuscript, Northwestern University
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Stable? A Bayesian Approach Based on a Markov-Switching Model of the Business
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Economy: A Bayesian Investigation of Timing, Breadth, and Potential Explanations”.
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[29] Lucas, Robert E. (1976): “Econometric Policy Evaluation: A Critique”. CarnegieRochester Conference Series on Public Policy, 1, 19-46.
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Targeting”. In: Monetary Policy Rules, ed. by John B. Taylor. Chicago: University of
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Policy. Princeton University Press. Princeton, NJ.

24

Table 1: Model Parameters

Panel A: Structural Parameters
β

κ

τ −1

ρg

ρz

σg

σz

0.99

0.7

1.4

0.7

0.8

0.3

1.1

Subsample

ψπ

ψy

ρR

σR

I: Indeterminacy
II: Determinacy

0.4
1.5

0.2
0.5

0.6
0.8

0.3
0.3

Subsample
I and II

Panel B: Monetary Policy Rule

25

Table 2: Good Policy Hypothesis: Descriptive Statistics

Panel A: Standard Deviations
Equation

I: Indeterminacy

II: Determinacy

% Change

2.52
2.17
0.81

1.47
0.72
0.47

-41.58
-66.91
-41.85

I: Indeterminacy

II: Determinacy

% Change

0.69
0.69
0.85

0.75
0.29
0.77

8.96
-58.07
- 9.33

Output
Inflation
Interest Rate
Panel B: Persistence
Equation
Output
Inflation
Interest Rate

26

Table 3: Good Luck Hypothesis: Descriptive Statistics

Panel A: Standard Deviations
Equation

I: High Variance

II: Low Variance

% Change

1.50
0.72
0.48

0.80
0.26
0.21

-46.57
-64.65
-57.65

I: High Variance

II: Low Variance

% Change

0.78
0.30
0.76

0.83
0.33
0.80

6.50
9.27
5.08

Output
Inflation
Interest Rate
Panel B: Persistence
Equation
Output
Inflation
Interest Rate

Table 4: Innovation Standard Errors: Descriptive Statistics and Stability Tests

Equation
Output
Inflation
Interest Rate

I: Indeterminacy

II: Determinacy

Statistics (p-value)

1.79
1.55
0.25

0.88
0.65
0.23

4.11 (0.00)
5.64 (0.00)
1.22 (0.23)

27

Table 5: Parameter Stability Tests

Panel A: Size-Adjusted Probability of Rejecting Parameter Stability
Equation
Output
Inflation
Interest Rate

OLS-based

GLS-based

Newey-West corrected

0.03
0.09
0.91

0.17
0.45
0.94

0.11
0.38
0.92

Panel B: Size-Adjusted Probability of Rejecting Super-Exogeneity
Equation

OLS-based

GLS-based

Newey-West corrected

Output
Inflation

0.02
0.09

0.11
0.46

0.11
0.37

Table 6: Parameter Stability Tests
-Sensitivity Analysis-

Parameter Stability
OLS-based GLS-based

Superexogeneity
OLS-based GLS-based

κ = 0.2
Output
Inflation
Interest Rate

0.03
0.08
0.30

0.18
0.60
0.44

0.03
0.22
-

0.13
0.61
-

τ −1 = 3
Output
Inflation
Interest Rate

0.05
0.18
0.97

0.20
0.53
0.99

0.04
0.16
-

0.14
0.54
-

σ ζ = 0.2
Output
Inflation
Interest Rate

0.03
0.08
0.84

0.16
0.45
0.89

0.03
0.08
-

0.11
0.45
-

28

Table 7: Good Policy Hypothesis: Descriptive Statistics
- from Determinacy to Determinacy -

Panel A: Standard Deviations
Equation

I: Determinacy

II: Determinacy

% Change

1.49
0.69
0.73

1.50
0.52
0.50

0.35
-24.57
-31.75

I: Determinacy

II: Determinacy

% Change

0.77
0.47
0.78

0.75
0.25
0.77

1.98
-45.66
- 1.38

Output
Inflation
Interest Rate
Panel B: Persistence
Equation
Output
Inflation
Interest Rate

Table 8: Innovation Standard Errors: Descriptive Statistics and Stability Tests
- from Determinacy to Determinacy Equation
Output
Inflation
Interest Rate

I: Determinacy

II: Determinacy

Statistics (p-value)

0.89
0.59
0.26

0.90
0.48
0.23

0.97 (0.55)
1.55 (0.05)
1.28 (0.18)

29

Table 9: Parameter Stability Tests
- from Determinacy to Determinacy -

Panel A: Size-Adjusted Probability of Rejecting Parameter Stability
Equation
Output
Inflation
Interest Rate

OLS-based

GLS-based

0.06
0.10
0.48

0.16
0.23
0.61

Newey-West corrected
0.10
0.20
0.61

Panel B: Size-Adjusted Probability of Rejecting Super-Exogeneity
Equation

OLS-based

GLS-based

Output
Inflation

0.06
0.21

0.11
0.23

Newey-West corrected
0.11
0.20

Table 10: Parameter Stability Tests
-from Indeterminacy to Determinacy: sub-samples of 150 obs-

Panel A: Size-Adjusted Probability of Rejecting Parameter Stability
Equation
Output
Inflation
Interest Rate

OLS-based

GLS-based

0.03
0.19
0.99

0.12
0.78
0.99

Newey-West corrected
0.12
0.75
0.99

Panel B: Size-Adjusted Probability of Rejecting Super-Exogeneity
Equation

OLS-based

GLS-based

Output
Inflation

0.03
0.15

0.11
0.76

30

Newey-West corrected
0.09
0.71

Table 11: Tests for Multiple Structural Changes at Unknown Dates

Equation
Output
Inflation
Interest Rate
a

UDmax (p-value)a

WDmax (p-value)a

SupF(2|1) (p-value)a

31.35 (0.010)
35.35 (0.010)
32.12 (0.035)

31.35 (0.015)
35.35 (0.015)
32.12 (0.063)

16.57 (0.117)
11.35 (0.588)
11.63 (0.443)

bootstrapped p-values, trimming = 0.15, maximum number of breaks = 4.

31

Break dates
1983Q2
1981Q2
1980Q3

Figure 1: Parameter Instability Tests at Unknown Break Date

Output Equation

Output Equation
40

bootstrapped 5% critical value
asymp. 5% critical value - unknow date
asymp. 5% critical value - know date

30

Wald Statistics

Chow Statistics

40

20

10

0

70

75
80
85
90
Range of possible break dates

30

20

10

0

95

70

40

30

30

20

10

0

70

75
80
85
90
Range of possible break dates

20

10

0

95

70

30

30

20

10

70

75
80
85
90
Range of possible break dates

75
80
85
90
Range of possible break dates

95

Interest Rate Equation
40

Wald Statistics

Chow Statistics

Interest Rate Equation
40

0

95

Inflation Equation

40

Wald Statistics

Chow Statistics

Inflation Equation

75
80
85
90
Range of possible break dates

20

10

0

95

32

70

75
80
85
90
Range of possible break dates

95