The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
orKmg raper series
Small Sample Properties of Generalized
Method of Moments Based Wald Tests
Craig Burnside and Martin Eichenbaum
3
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
August (W P-94-12)
FEDERAL RESERVE BANK
OF CHICAGO
S m a ll S a m p le P r o p e r tie s o f G e n e r a liz e d
M e th o d
o f M o m e n ts B a s e d
W a ld
T e s ts *
Craig Burnside
University of Pittsburgh
Martin Eichenbaum
Northwestern University, NBER
and Federal Reserve Bank of Chicago
May 1994
*We th a n k Law rence C hristiano, Lars Hansen and N arayana K o cherlakota for th eir
helpful com m ents and suggestions.
A bstract
This paper assesses the small sample properties of Generalized Method of Moments (GMM)
based Wald statistics. The analysis is conducted assuming that the data generating pro
cess corresponds to (i) a simple vector white noise process and (ii) an equilibrium business
cycle model. Our key findings are that the small sample size of the Wald tests exceeds
their asymptotic size, and that their size increases uniformly with the dimensionality of
joint hypotheses. For tests involving even moderate numbers of moment restrictions, the
small sample size of the tests greatly exceeds their asymptotic size. Relying on asymptotic
distribution theory leads one to reject joint hypothesis tests far too often. We argue that
the source of the problem is the difficulty of estimating the spectral density matrix of the
GMM residuals, which is needed to conduct inference in a GMM environment. Imposing
restrictions implied by the underlying economic model being investigated or the null hy
pothesis being tested on this spectral density matrix can lead to substantial improvements
in the small sample properties of the Wald tests.
Craig Burnside
Department of Economics
University of Pittsburgh
Pittsburgh, PA 15260
Martin Eichenbaum
Department of Economics
Northwestern University
Evanston, IL 60208
and NBER
1
Introduction
This paper assesses the small sample properties of Generalized Method of Moments (GMM)
based Wald statistics. The analysis is conducted assuming that the data generating pro
cess corresponds to (i) a simple vector white noise process and (ii) the equilibrium business
cycle model considered in Burnside and Eichenbaum (1994). Our key findings are that
the small sample size of the Wald tests exceeds their asymptotic size, and th at their size
increases uniformly with the dimensionality of joint hypotheses. For tests involving even
moderate numbers of moment restrictions, the small sample size of the tests greatly ex
ceeds their asymptotic size. Relying on asymptotic distribution theory leads one to reject
joint hypothesis tests far too often. We argue that the source of the problem is the diffi
culty of estimating the spectral density matrix of the GMM residuals, which is needed to
conduct inference in a GMM environment. Imposing restrictions implied by the underly
ing economic model being investigated or the null hypothesis being tested on this spectral
density matrix can lead to substantial improvements in the small sample properties of the
Wald tests.
A common approach to evaluating quantitative equilibrium business cycle models is to
compare model and non-model based estimates of the second moments of aggregate time
series. No uniform method for making these comparisons has emerged. Many authors in
the Real Business Cycle (RBC) literature make these comparisons in a way th at abstracts
from sampling uncertainty in estimates of models’ structural parameters (see for example
Kydland and Prescott (1982) or Hansen (1985)). Other authors have estimated and tested
RBC models using full information maximum likelihood methods (see for example Altug
(1989), Christiano (1988), McGratten, Rogerson and Wright (1993) and Leeper and Sims
(1994)).
An intermediate strategy is to simultaneously estimate model parameters and second
moments of the data using a variant of Hansen’s (1982) Generalized Method of Moments
(GMM) procedure. Christiano and Eichenbaum (1992) show how, in this framework,
simple Wald-type tests can be used to test models’ implications for second moments of
the data. Three advantages of this approach are that (i) at the estimation stage of the
analysis one need not completely specify agents’ environments, (ii) it is easy to specify
which aspects of the data one wishes to concentrate on for diagnostic purposes, and (iii)
1
it is substantially less demanding from a computational point of view than maximum
likelihood approaches. Use of this procedure has become more widespread. However its
properties in small samples are not well understood. This is disturbing in light of recent
results in the literature casting doubt on the extent to which asymptotic distribution
theory provides a good approximation to various aspects of the small sample behavior of
GMM based estim ators.1
In this paper we address four basic questions concerning the performance of GMM
based Wald statistics. First, does the small sample size of these tests closely approximate
their asymptotic size? Second, do joint tests of several restrictions perform as well or worse
than tests of simple hypotheses? Third, how can modeling assumptions, or restrictions
imposed by hypotheses themselves, be used to improve the performance of these tests?
Fourth, what practical advice, if any, can be given to the practitioner?
We answer these questions under two assumptions about the data generating process.
First, we assume th at the true process generating the macro time series is the equilibrium
business cycle model developed in Burnside and Eichenbaum (1994). This case is of interest
for two reasons: (i) the model generates time series that in several respects resemble U.S.
data, and (ii) we can study issues of size and inference in an applied context. Second,
we assume th at the data generating process corresponds to Gaussian vector white noise.
Working with such a simple process allows us to assess whether the findings th at emerge
with the more complicated data process also arise in simpler environments. In addition
we find it easier to build intuition about our results in the simpler environment.
Our main findings can be summarized as follows. First, there is a strong tendency
for GMM based Wald tests to over-reject. Second, the small sample size of these tests
increases uniformly as the dimension of joint tests increases. For even moderate number
of restrictions, the small sample size is dramatically larger than the asymptotic size of the
test. Indeed correcting for the small sample properties of the Wald test turns out to have a
substantive impact on inference about the empirical performance of the equilibrium busi
ness cycle model th at is being analyzed. Third, the basic problem is difficulty in accurately
estimating the spectral density matrix of the GMM error terms. We investigate various
nonparametric estimators of this matrix that have been suggested in the literature. While
1See for Tauchen (1986), Kocherlakota (1990), Ferson and Foerster (1991), Burnside (1992), Fuhrer,
Moore and Schuh (1993), Neely (1993), Christiano and den Haan (1994) and West and Wilcox (1994).
2
there is there is some sensitivity to which nonparametric estimator is used, these differ
ences do not affect our basic conclusions. Fourth, we argue that the size characteristics of
the Wald tests can be improved if the analyst imposes restrictions that emerge from the
model or the hypothesis being tested when estimating the covariance matrix component
of the Wald statistic. Not only does such information improve the size of simple tests, it
significantly ameliorates the problems associated with tests of joint hypotheses.
The remainder of this paper is organized as follows. Section 2 considers the case of
the Gaussian white noise generating process. In Section 3 we discuss the case where the
data are generated from an equilibrium business cycle model. Section 4 contains some
concluding remarks.
2
G a u s s ia n W h ite N o is e D a t a G e n e r a tin g P r o c e s s e s
In this section we consider the small sample properties of GMM based Wald statistics
within the confines of a very simple statistical environment. In particular we suppose
that data generating process is a mean zero, unit variance Gaussian white noise process.
There are several advantages to working with such a simple process. First, we are able to
document that the basic problems which arise in the more complex environment considered
in section 3 also arise here. Second, developing intuition for the results is easier in a
simpler environment. Third, we can examine the effects of imposing various assumptions
about the data generating processes on our procedures. Fourth, we can compute all
relevant population moments exactly. Fifth, simulation is straightforward and the number
of replications can be increased to gain accuracy in our Monte Carlo experiments.
The remainder of this section is organized as follows. Subsection 2.1 describes the
data generating process. In subsections 2.2 and 2.3 we discuss the hypothesis tests and
different experiments that we conducted. Finally, we report the results of our Monte Carlo
experiments in subsection 2.4.
2.1 The D ata G enerating Process
We suppose that am econometrician has time series data on J = 20 random variables X ,t,
i = 1, . . . , J , each of which are i.i.d. N(0,1) and mutually independent.2 The econometri
2 We also conducted experiments in which the data were independent MA(1 ) processes with Gaussian
innovations, and which were either positively or negatively serially correlated. In both cases our results were
3
cian has T = 100 observations on Xu, i — 1,
J . To simplify the analysis we assume
th at the econometrician knows that E X u = 0, for all i and t. The econometrician is
interested in estimating and testing hypotheses about the standard deviations, a,, of X it,
i = 1, 2, . . . , J . To estimate tr,- he uses a simple exactly identified GMM estimator based
on the moment restriction
E (X ft - o ? ) = 0 ,
i = 1 ,2 ,..., J.
(1)
This leads to the GMM estimators
2.2
H ypoth esis Testing
The econometrician estimates a,- in order to conduct inference. The hypotheses of interest
pertain to the variability of the series Xu- The specific hypotheses to be tested are of the
form
HM : <Ji = o2 = ■■■= oM = 1,
M < J.
We consider this hypothesis because of its similarity to a diagnostic procedure th at is
often used to evaluate RBC (and other) models. The basic idea is to see whether a model
can ‘account’ for various second moments of the data. In practice this amounts to test
ing whether the second moments of some series estimated in a nonparametric manner
equal the analogous second moment implications of a particular RBC model (see section
3). Early work on RBC models tended to concentrate on the volatility of different eco
nomic aggregates (see for example Hansen (1985)). Here there is no ‘model’. But we can
test sample moments against their true value (M = 1) and test whether various second
moments are equal to each other in population using similar statistical procedures.
The specific Wald statistic that we use to test H u is given by
W ? = T (a - \)'A!{AVTA ')-l A{o - 1).
Here A = ( I u
(3)
) and Vj> denotes a generic estimator of the asymptotic variance-
covariance matrix of s/T (d —<7o), where <r<) is the true value of the parameter vector
a = ( ai a2 • • • Oj j . Given well behaved estimators a and Vj>, “Wt
qualitatively similar to the white noise case.
4
X2{M).
We consider several questions that arise in testing H m - First, how does the choice of
estimator Vp affect inference? We are particularly interested in assessing the small sample
implications of using non-parametric estimators of Vr and understanding the gains to im
posing different types of restrictions on Vp. Particularly important sources of restrictions
are the economic theory being investigated and the null hypothesis being tested: For ex
ample, intertemporal consumption based asset pricing models typically imply restrictions
on the degree of serial correlation in the error terms that define Vp. (See for example
Hansen and Singleton (1982) or Eichenbaum and Hansen (1990)). A different example is
provided in section 3 where we can use the structural model itself to generate an estimate
of Vp. Since imposing restrictions on Vp can often be computationally burdensome, and
asymptotic inference is not affected, it is important to understand the nature of the small
sample gains to doing so.
Second, how does the dimension of the test, i.e. the degrees of freedom M , affect the
size of the test? This question is important because, in many applications, the model
gives rise to a large number of over-identifying restrictions. The issue is what trade-offs
are involved in simultaneously testing more or less of these moment restrictions.
Third, how are the small sample properties of the Wald statistic affected by reparame
terizing the example?3 An asymptotically equivalent way of assessing hypothesis H m is to
proceed as follows. Suppose that we estimate <7i, along with 0, = cr,/ Oi for * = 2, 3 , . . . , J.
To estimate 0, we utilize the following moment restrictions.
E M -a f)
= 0
£ (X S -« JX J)
= 0.
. = 2....... J
This leads to the estimators
The analogous hypothesis to H m is
H m : 0\ = Oj = • • • = $m — 1*
M < J.
3Gregory and Veall (1985) study the effects of reparameterizng Wald tests in a regression context.
5
(4)
The corresponding Wald test statistic for this hypothesis is
= T{0 - 1y m A V T A 'Y 'A tf - 1),
(5)
where 0 = ( <j\ 02 • • • Oj ) and Vt is some estimator for the asymptotic varianceA
covariance m atrix of 0. There is no a priori reason to suppose th at the small sample
properties of Wj? will be the same as those of W jf.
This example is of interest because it can shed light on the common practice in the RBC
literature of testing whether a model matches the volatility of output and the volatility
of various aggregates relative to output. One could simply test whether a model matches
the absolute volatility of all the relevant variables. Asymptotically this choice should not
m atter. But the small sample properties of the Wald tests in the two cases could be quite
different.
2.3
A ltern ative Covariance M atrix Estim ators
In this section we discuss our estimators of the asymptotic variance-covariance matrix of
a and 0. To be concrete we concentrate on the case of a. The case of 0 is discussed in
Appendix A. The moment conditions used to estimate <r, (1), can be written in the form
E[g{Xt,o)] = 0. Here g(', •) is the J x 1 vector valued function whose *th element is given
by (Xft — of). Denoting the true value of a by a0, the asymptotic covariance matrix of
V T (o — cr0) is given by
Vo = ( D & 'D o ) - 1,
where
r, _ v,9g(Xu ao)
D° ~ E
a✓
and
OO
So= J2
j=-oo
The corresponding estimator of Vo is given by
VT = (D't S ^ D t ) - \
where D t and S t are consistent estimators for D0 and S q.
6
We consider several estimators of V0. Each is defined in terms of some estimators D j
and S t . The different estimators impose varying amounts of information at the econome
trician’s disposal. Some of this information is in the nature of the maintained assumptions
concerning the serial and mutual independence properties of X u and Gaussianity. Other
information derives from the null hypothesis being tested.
Initially we consider estimators of So which do not exploit any of this information.
Instead we estimate So using versions of the nonparametric estimator proposed by Newey
and West (1987).4 A general version of this estimator can be written as
St =
E
y— ( T - i)
k(jr)n „
Bt
where
a
_ J ( i / T ) Y , l , t l g { X „ c ) g ( X t. i , a Y
for j > 0
'
1 ( l / r ) £ i l - y +i
»)»(*.. *)' for j < 0
and
k(x)
|S
—|x| for |x| < 1
otherwise
Here B t is a scalar that determines the bandwidth of the lag window k(-). We consider
three variants of this estimator,
•
uses bandwidth B t = 4,
• S} uses B t = 2,
• S^ has B t chosen automatically using a procedure suggested by Andrews (1991)
which is described in more detail in Appendix C.
The next group of estimators that we consider utilizes additional amounts of infor
mation about the underlying data generating process. The estimator Sj. exploits the
assumption th at the Xu are serially uncorrelated. This implies that
• Sj. has t/th element given by jr[S^.1(AT?t - d?)(X?t —a*)].
The estimator
imposes the mutual independence of the X u's as well as their serial
independence. This implies that
4 There are several alternative estimators which could be used at this stage. We have found our results
to be relatively insensitive to the choice of procedure in the RBC context so we present results based only
on the Newey and West (1987) method.
7
• Sj- is a diagonal matrix with n th element given by jr[X£=i(-X?< —d 2)2].
Our next estimator, Sy, also exploits the fact that the Xu are Gaussian. Since Gaussianity implies th at E(X*t) = Zaf,
• Sj. is a diagonal matrix with **th element given by 2af.
Our next two estimators impose additional restrictions derived from the null hypothesis
being tested. Under hypothesis H m , a, = 1 for * = 1, . . . , M , while <r,- is unrestricted for
* = M + 1 , . . . , J . This suggests the estimator
• Sj> which is a diagonal matrix with **th element 2 for t < M , and 2a f for i =
M H-1, . . . , J .
Corresponding to each estimator of So discussed above, there is am estimator for V0
given by,
v t = W ( ^ ) - 1^ ) - 1,
k = 1, 2, . . . , 7, where D? is a diagonal matrix with n th element - 2d,-. Since the null
hypothesis can also be imposed on
we also consider the estimator
v i = ( D ^ s i r ' D i r 1,
where D \ is a diagonal matrix with n th element —2 for t < Af, and - 2d,- for i = M +
1, . . . , J .5 Here the W statistic reduces to ]C^x(d,- —l ) 2/ 2.
We use the same differential information assumptions to define eight estimators for the
A
,
variance-covariance m atrix of 6 that axe analogous to V j, k = 1, 2 , . . . , 8 (see Appendix
A).
2.4
M onte Carlo Experim ents
Our experiments were conducted as follows. We generated 10,000 sets of synthetic time
series on {X k , X u , • • •, X j t}J=l, each of length 100. On each artificial data set, we esti
mated the parameter vector a, the different estimators of the variance covariance matrix
5If we imposed <r< = 1 for all t in the computation of ^ and
we would get numerically identical
results for our test statistics because all the matrices involved in the calculation are diagonal.
8
and then calculated the Wald test statistic,
that is relevant for testing hypothesis H m ,
M G {1,2,5,10,20}. This allowed us to generate an empirical distribution function for
W™ under the null hypothesis that if m is true, corresponding to the different estimators
of V0.
Our results are summarized in-Table 1, the columns of which correspond to different
specifications of M (which also equals the degrees of freedom of the test). The rows
correspond to fixed asymptotic sizes of the test while the entries in the table are the
percentages, out of 10,000 draws, for which the HI statistic exceeded the relevant critical
value of the chi-squared distribution.
A number of interesting results emerge here. Consider first the distribution of the test
statistics generated using V^, Vp, V? and V? (see Panels A-D of Table 1). First, even for
M = 1, the small sample size of the tests exceeds their asymptotic size. This result is
similar to th at obtained by Christiano and den Haan (1994) and Newey and West (1993).
Second, the small sample size of the tests rises uniformly with M . Indeed when we use the
estimator Vy, the HI statistic for hypothesis fTjo exceeds its asymptotic (1%, 5%, 10%)
critical values (59%, 73%, 80%) of the time. For even moderate sizes of M , relying on
asymptotic distribution theory leads one to reject H m fax more often than is warranted in
small samples. It is true that as the bandwidth decreases, the small sample performance
of the Wald test improves uniformly. But as panel D indicates, even when we impose the
white noise assumption (i.e. we use V^), the small sample performance of the large joint
tests is dismal. For example, with M = 20, tests with asymptotic size (1% , 5%, 10%),
lead to rejection (17%, 33%, 43%) of the time in samples of 100 observations.
The results generated using V? (which exploits the assumption that the X u are mu
tually independent) are presented in Panel E of Table 1. Comparing Panel E to Panels
A-D, we see th at the impact of imposing the independence assumption is to move the
small sample sizes of the tests substantially closer to their asymptotic values. Not surpris
ingly, the impact of this restriction becomes larger as M increases since there are more
off-diagonal elements being set to their population values. (In the case of M = 1 the two
panels are identical). With M = 20, the HI statistic for H m exceeds its asymptotic (1%,
5%, 10%) critical values (4.7%, 13.4%, 21.2%) of the time. This represents a substantial
improvement relative to the situation when we do not impose the zero off-diagonal ele
ment restriction. Even so, the Wald test still rejects too often in small samples. Panel F,
9
which reports results based on Vy, indicates that imposing the Gaussianity assumption
improves the small performance of
even further. To the extent th at fourth moments
are less accurately estimated than second moments for Gaussian processes this result is
not surprising.
Recall that the estimator Vy exploits information from the null hypothesis regarding
Of in constructing Sy. The results generated using Vy are reported in Panel G of Table
1. Comparing Panels F and G we see that the net effect of imposing these additional
restrictions is to move the small sample size of the test even closer to its asymptotic size
(except for the 10% critical value for M = 1). For example, with M = 20, the W statistic
for H m exceeds its asymptotic (1%, 5%, 10%) critical values (2.1%, 7.3%, 12.1%) of the
time.
Panel H of Table 1 reports results based on Vy where we impose the null hypothesis on
D t as well as on
S y . N ow
all of the anomalies associated with the small sample distribution
of the W statistic disappear. First, the degree to which the small sample sizes match their
asymptotic sizes is not affected by M . Second, the small sample size of the test statistic is
extremely close to the corresponding asymptotic size. Indeed, this is true even when we fix
the asymptotic size of the test at 1%. So, at least for the present example, the parameter
estimates appear to have a small sample distribution which is very well approximated by
their large sample distribution. The problem with the small sample distribution of the W
statistic seems to be closely related to the small sample distribution of Sy and to a much
smaller extent D t • The more information the econometrician imposes on S t and D t , the
better the performance of the tests appears to be in this example.
/x/
Table 2 presents results pertaining to the Wt statistic that is relevant for our alternative
parameterization of the problem in terms of relative standard deviations.6 In many ways
these results are qualitatively similar to those obtained with the original parameterization.
Broadly speaking, the second set of tests leads to slightly more rejections, although only
to a modest extent. However, unlike the previous parameterization, when we impose all
of the available information on Sy and DT (Panel H of Table 2), there is still a noticeable
tendency of joint tests with many degrees of freedom to reject more frequently than tests
of single hypotheses.
6 The column in Table 2 headed M = la is for tests of the hypothesis C\ = 1 , while the column headed
M = 16 is for tests of the hypothesis o^/cri = 1 . Both tests have one degree of freedom.
10
These results suggest that simply reparameterizing the problem will not dramatically
improve the performance of tests constructed using nonparametric estimators for S t and
D t • The key problem with inference seems to arise from difficulties in estimating the
spectral density matrix of the GMM error terms, So. Imposing as much information as
possible when estimating So and Do leads to significant improvements in the size properties
of the Wald tests. In the next section we investigate the extent to which these conclusions
continue to hold in a more complex statistical environment.
3
A R e a l B u s i n e s s C y c le M o d e l A s a D a t a G e n e r a t i n g P r o c e s s
In this section we consider the small sample properties of GMM based Wald statistics
assuming that the data generating process is given by the business cycle model developed
in Burnside and Eichenbaum (1994). The model is briefly summarized in subsection
3.1. Subsection 3.2 describes the way the model’s structural parameters were estimated.
Subsection 3.3 discusses the hypothesis tests we investigated. In subsection 3.4 we present
the results of our Monte Carlo experiments.
3.1 The M odel
The model economy is populated by a large number of infinitely lived individuals. To go
to work an individual must incur a fixed cost of f hours. Once at work, an individual
stays for a fixed shift length of / hours. The time t instantaneous utility of such a person
is given by
ln{Ct) + 6 l n { T - < - W tf )
(6)
Here T denotes the individual’s time endowment, Ct denotes time t privately purchased
consumption, 6 > 0, and Wt denotes the time t level of effort. The time t instantaneous
utility of a person who does not go to work is given by ln(Ct) + 6 ln(T).
Time t output, Yt, is produced via the Cobb-Douglas production function
Vt = (KtUty-INtfWtXt)*
(7)
where 0 < a < 1, Kt denotes the beginning of time t capital stock, Ut represents the
capital utilization rate, Nt denotes the number of individuals at work during time t, and
11
X t represents the time t level of technology. We assume that the time t depreciation rate
of capital, 6t, is given by
St = SUf
(8)
where 0 < 6 < 1 and <f>> 1. The stock of capital evolves according to
Kt+1 = (1 ~ St)K t + I t
(9)
where It denotes time t gross investment.
The level of technology, X t, evolves according to
X t = X t-1 exp (7 + vt)
where vt is a serially uncorrelated process with mean 0 and standard deviation av. The
aggregate resource constraint is given by
Ct + It + Gt < Yt
(10)
where Gt denotes the time t level of government consumption. We assume that Gt evolves
according to
Gt = Xtgl
Here
(11)
is the stationary component of government consumption and gt = ln(yt*) evolves
according to
gt = fi + pgt- i + et
(12)
where n is a scalar, |p| < 1 and et is a serially uncorrelated process with mean 0 and
standard deviation ae.
In the presence of complete markets the competitive equilibrium of this economy cor
responds to the solution of the social planning problem:
E0 f > ‘ (ln(Ct) + 9Nt ln(T - £ - Wtf ) + 0(1 - JV,) ln(T)]
t=0
(13)
subject to (7) - (12) by choice of contingency plans for {Ct,K t+i,N t, Ut,W t : t > 0}.
We obtain an approximate solution to this problem using King, Plosser and Rebelo’s
(1988) log-linear solution procedure.7 Let kt = \n(Kt/ X t-i), ht = ln(Ht), ct = In(Ct/ X t),
7See Burnside (1993) for details.
12
Wt = ln(Wt), «* =
Vt
= H Y t/ X t), at = ki(Yt/N tX t), it = \n[It/ X t), h°t = ln(tf®),
and a® = ln(Y*/H fX t). Here Ht and H f denote actual and observed time t hours of work.
As in Prescott (1986), we assume that
ln(JJ?) = ln(Ht) + &
where
(14)
is an i.i.d. random variable with mean zero and variance <r|. The time t state of
the system is given by
st = ( 1 kt ht vt gt 6 y
Define the vector of time t endogenous variables f t as
f t = ( c t wt ut yt at it h°t a°t
and the vector of time t shocks
it - ( 0 0 0 vt et 6 ) '.
Our assumptions about the exogenous variables and the log-linear approximation to the
model imply th at the evolution of the system can be summarized as
ft = n *
(15)
where M and n are functions of the model’s underlying structural parameters. We take
(15) to be the data generating mechanism in our Monte Carlo experiments.8
3.2 E stim ation
With certain exceptions, the parameters of the model were estimated using a variant of the
GMM procedure described in Christiano and Eichenbaum (1992). We did not estimate
/3, T, f and / . Instead we set /? = 1.03-1/4, T = 1369 hours per quarter, f = 60 and
chose / so that the nonstochastic steady state value of Wt is 1. Rather than estimating
the parameter 5, we estimated 6 = 6U*, where U is the nonstochastic steady state value
of Ut. To obtain a value of <f>we use the fact that in nonstochastic steady state,
^ = r 1e x p h ) - l + 1
0
8See Burnside (1993) for details.
13
In the data, the series gt displays a time trend, so this series was detrended using a
linear time trend. To simplify matters we did not include the time trend in the Monte
Carlo experiments. In addition, we chose to estimate the nonstcchastic steady state value
of Gt/ Y t, as the parameter g/y, rather than the mean of the process gt = ln(Gt/ X t).9
In light of these decisions, the vector of model parameters to be estimated, denoted by
¥ 1, is given by
^1 = ( 0 a 6 7 a„ g /y p a( ai )*.
The hypotheses th at we investigate involve various second moments of the data. Since
many of the relevant series exhibit marked trends, some stationary inducing transformation
of the data must be applied. To facilitate comparisons with the RBC literature, we chose
to process the data using the Hodrick and Prescott (1980) filter. Consequently, the second
moments to be discussed pertain to those of Hodrick and Prescott (HP) filtered data.10
We focus on a set of second moments that have received a great deal of attention
in the RBC literature: the standard deviation of output, c y, the standard deviations of
consumption, investment and hours relative to the standard deviation of output, <Te/crv,
Oi/oy and a ^/a y and the standard deviation of hours worked relative to the standard devi
ation of average productivity, <7*/<?<,- We also consider the dynamic correlations between
average productivity and hours, p'ah = Corr( A PI*, !?*+,•)> * = ± 1, ± 2, ±3, ±4, and the
dynamic correlations between average productivity and output p*ay = Corr( A PI*, 3^+,),
i — —4, —3, —2, —l .11 We denote the vector of diagnostic moments th a t must be estimated
in ways not involving the model by
*2 =
(°y
° c l° v
Oi/Oy ah/ c v ah/o a
p~£
p~*
p~*
p~£
Pah Pah Pah Pah Pay Pay Pay Pay ) •
9The mean of gt would matter in the linearised solution only in determining the steady state share of
government expenditure in output, which we parameterise directly.
10We have redone all of the experiments in this paper with first differenced data. For a comparison of
some of the small sample properties of GMM with HP-filtered and first differenced data see Christiano and
den Haan (1994).
11 The contemporaneous correlations between these variables and p%
ay9 » = 1 , 2 ,3,4, can be deduced from
the other moments that we consider.
14
3.2.1
M o m e n t C o n d itio n s U nd erly in g th e E s tim a to r of \&i
As discussed in Burnside and Eichenbaum (1994), our estimator of 'Pi is based on the
following moment conditions:
£ L | - i [ A l n W ) ] I + iA ln (fl;)A ln (fl'i’)}
= 0
(16)
£ [ln W f) - 1» W ) ]
= 0
(17)
£[ln(£t) - ln(S)]
= 0
(18)
+
= °
(19)
^ [in W J -ln ^ J -T f]
= 0
(20)
£[ln(X ") - l n ( X t°_i)]2 - a 2 ~ 2(p\a\
= 0
(21)
E[ln(Gt) - l n ( F ()-ln (ff/y )]
= 0
(22)
~ pg°t- i ) ln(^f_i)] + P<fi\o\
= 0
(23)
- (1 + P2)<p W (
= 0
(24)
E
E[{g°
E[(g°t ~ P9°t-xY\
In equation (16), H° and H° refer to our two measures of hours worked (see Appendix D).12
The variables N , representing the nonstochastic steady state value of N t, and <pz, a reduced
form parameter, are functions of the underlying parameter vector, 'Pi. Furthermore, X°
represents a measurement-error corrupted signal of the level of technology which can be
constructed given the data and a vector of parameters 'Pi. Similarly, g° is a signal of gt
based on the error-ridden measure of technology ATf.13
3 .2.2
M o m e n t C o n d itio n s U n d erly in g th e E s tim a to r o f $2
Our estimator of *P2 is based on the following moment conditions:
e
E [c\
[v ? - t f ]
= 0
-(*«/*,)V ] = 0
E [i2 - {o i/o y fy 2] = 0
12 Unlike Burnside and Eichenbaum (1994), we abstract from issues concerning the observability of &t and
K t . In particular, we assume, for the purposes of our Monte Carlo experiments, that the econometrician
observes these series directly.
13See Burnside and Eichenbaum (1994) for details.
15
E\h] - [oH/crfy]}
=0
E[h% - {phlo a)2a]\
=0
E [atht+i - p \ k ^ j
a \!
= 0, * = ± 1 , . . . , ±4
E l a t y t + i - p ^ ^ o l /(? £ )]
= 0, t = 1 ,... ,4,
where a lower case variable, e.g. Zt, is the cyclical component of ln(Z«) as defined by the
HP filter.
To define our joint estimator of ’J'i and \&2 consider the following generic representation
of our moment conditions:
E [ut(tf0)] = 0
t=
where 'fr° is the true value of ( \E,,1 ^2 ) and u< is a vector valued function of dimension
equal to the dimension of ^ °. Let
S rW =
1 <=i
The GMM estimator, \&r» minimizes
JT = TgTW ? TgT(9).
(25)
where T r is a symmetric positive definite weighting matrix of dimension equal to the
dimension of
Since our GMM estimator is exactly identified, ’J'r is independent of
T r- We simply set T r equal to the identity matrix in (25).
A consistent estimator of the variance-covariance matrix of y /T i^ r — ^o) is given by
V* = ( d ^ D
t)
'1
where D t = d U r C ^ r ) /^ ' and 5 r is a consistent estimate of S q, 2?r times the spectral
density matrix of ut(^°) at frequency zero.
3.3
H y p o th e sis T estin g
Suppose we wish to assess the empirical plausibility of the model’s implications for a q x 1
subset of \p2 given by u>. Let $('&) denote the value of u> implied by the model, given the
16
structural parameters t&i. Here $ denotes the (nonlinear) mapping between the model’s
structural parameters and the relevant population moments. Denote the nonparametric
estimate of u obtained without imposing restrictions from the model by T (^ ).u
The
hypotheses that we investigate are of the form
H0 : F(<V°) =
- r(tf°) = 0
(26)
Christiano and Eichenbaum (1992) show that a consistent estimate of the asymptotic
variance-covariance matrix of \/T [F (^ r) “ -F('J'o)] is
'a F ( ^ r ) ] '
d F {* r )
V*
aw
d*'
VP
and th at the test statistic
Wt = T F ( * Ty V f l F (¥ T)
(27)
is asymptotically distributed as a x 2 random variable with q degrees of freedom.
We consider two types of hypothesis tests. The first type involve tests of individual
moments of the data. The test numbers and corresponding moments being tested are
summarized in the following table.
Test #
i
2
3
4
5
Moment
Oy
Oc/Oy
Oi/Oy
Ohl<Jy
Oh/Oa
T e s t#
6
7
8
9
10
11
12
13
14
Moment
. —4
Pah
Plh
Pah
Pah
Pah
Pah
Plh
Pah
Pah
Test #
15
16
17
18
19
20
21
22
23
Moment
Pay
Pay
Pay
Pay
ray
Pay
Ply
Ply
Pay
The second type of tests involve joint moment restrictions.
Hypothesis H i states
that the values of erv, <re/crv, <7,/<rv, Oh/ay and Oh/oa implied by the model are the same,
in population, as the corresponding moments of the data generating process. Hypothe
ses H2, H3 and H4 are similar to hypothesis H i but pertain to the moments {p\h, i =14
14 Often the mapping T is linear. In particular, T is often a conformable matrix of zeros and ones that
selects the vector u from '&2 -
17
0, i l , i2 , i3 , i4 } , {pj^, $ — 0, i l , i2 , i3 , i4 } and {(Ty, o c/ o y ,
* = ±1,±2, ±3, ±4,
p*a v,
t
C i J o v , 0 h / G y i & h l& a i P*ahi
= —4 ,—3 ,-2 ,-1 } , respectively.15 The test statistics for Hi,
H2, H3 and H4 have 5, 9, 9, and 17 degrees of freedom, respectively.
To implement our hypotheses tests we require an estimator, Sr, of So. As in section
2 ,our estimators are of the form
r-i
£
St =
*
where
A. = f ( 1 / m S . y + i
for y > 0
’
1 (l/r )ES=-i+i4 it)“! for J < 0
The kernel function fc varies depending on the estimator, Hr is the bandwidth and
ut
=
Ut('f'r)- Our baseline results are generated using the Bartlett kernel function
‘" M
(0
k (x ) = {
v
1
« * N S 1
otherwise
and Andrews’ (1991) automatic selection procedure for Hr. Appendix C discusses the
other estimators of So that we consider. As it turns out, our basic results are robust
across these different estimators of So.
The bandwidth selection procedure that we used can be described as follows. Andrews
(1991) provides an expression for the optimal bandwidth corresponding to a given kernel,
a process u(, and a set of weights on the different elements of So. The bandwidth is
optimal in the sense that it leads to minimum M S E estimates of a weighted inner product
of the elements of So. Andrews’ (1991) procedure simplifies the dependence of the optimal
bandwidth on the entire spectral density of u t by assuming a simple parametric model for
the error term. The choice of model does not affect the consistency of Sr- The model
which we use corresponds to the simplest example in Andrews (1991). Specifically, we
treat the elements of u< as independent AR(1) scalar processes. No weight is given to
the off-diagonal elements of So. Under these circumstances, the bandwidth selected will
depend on the sample size, T, the weights, and coefficient estimates obtained by fitting
AR(l) processes to the elements of Ut(¥r). Roughly speaking, the more persistent the
errors, the greater the bandwidth.
lsThe last set of moments contains the nonredundant elements from among the moments involved in tests
1-23.
18
In the standard case, equal weight is placed on all of the error terms. However we
found that doing this led to test statistics with very poor small sample properties. (See
Appendix C). Instead we placed zero weight on (17), (18) and (22) along with unit weight
on the other error terms. The resulting median bandwidth across the different Monte
Carlo draws was 2.78.
3.4
Parameter Estimates and S o m e Results Based on Asymptotic Theory
Table 3 reports our point estimates of 'i'x along with corresponding standard errors. The
data set used to generate these estimates is described in Appendix D. Table 4 presents
the non-model and model based estimates of { a y ,
o J o y, 0i f o
y,O h/cry ,
C h / o a}.
Numbers
in parentheses are the standard errors of the corresponding point estimates. Numbers in
brackets are the asymptotic probability values of the
statistics for testing whether the
individual model and data population moment are the same. Notice that we cannot reject
any of the individual hypotheses in question.
Figures 1 and 2 summarize the model’s implications for the dynamic correlations be
tween hours worked and average productivity as well as the dynamic correlations between
average productivity and output. The dotted lines in row 1 correspond to the non-model
based estimates of {p^,, t = 0, ±1, ±2, ±3, ± 4 } , and {p *
ay,i
=
0, ± 1,±2, ±3, ± 4 } , while the
solid lines denote the moments implied by the model. The solid lines in row 2 graph the
differences between the model and non-model based estimates while the dotted lines de
pict an asymptotic two standard error band for the differences. According to these figures,
the model does quite well at accounting for the individual dynamic correlations between
average productivity and output as well as average productivity and hours worked.
W e now turn to our joint hypotheses. Columns 1 and 2 of Table 5 report the
W
statistics for hypotheses {Hi, H 2, H3, H4} and the corresponding asymptotic probability
values. Notice that hypotheses H 2, H3 and H4 are all rejected at very low significance
levels. To us the strength of these rejections seems at variance with the results of testing
the individual components of these hypotheses. One way to reconcile these results is to
invoke the pattern of covariances in question. However, in light of the results in section
2, these strong rejections may simply reflect the small sample properties of G M M based
Wald statistics as applied to hypotheses involving joint moment restrictions. With this as
19
motivation we turn to the Monte Carlo experiments
3.5
M o n t e Carlo Experiments
To generate data for our Monte Carlo experiments we proceeded as follows. Given the
estimated value of 'J'i,we generated artificial time series according to the following rules:
C t =
H t
exp(ct)Xt, Y t = exp(yt)Xt, K
= /exp(nt) and
variables X
t
St = 6 exp(<f>ut) .
t
= exp(A;t)Xt_1, G
Here
t
ct, y t, k t, u t, n t
= exp(y,)Xt, I t = exp(tt)Xt,
and
it
are given by (15). The
and gt were generated according to the laws of motion specified in section 3.1.
One thousand artificial time series data sets, each of length 113, were generated, assuming
that the stochastic elements of et were normally distributed.16
W e begin by reporting the small sample behavior of the W statistics for hypotheses HI,
H 2,H3 and H4. Column 3 of Table 5 reports the percentage of times (out of 1000 Monte
Carlo trials) that the
“W
statistics for these hypotheses were greater than or equal to the
corresponding W statistic obtained using U.S. data (see column 1). W e refer to this fraction
as the Monte Carlo probability. For hypothesis Hi, H 2 and H3 the asymptotic and Monte
Carlo probabilities are reasonably similar. However for hypothesis H4 the Monte Carlo
probability is much larger than the asymptotic probability (.06 versus .00). According to
asymptotic standard distribution theory, the W statistic which we obtained for hypothesis
H4 would be very unlikely ifthe model were specified correctly. But according to the small
sample results, one would obtain a W statistic this large or larger roughly 6% of the time.
A complementary way to assess the small sample properties of the Wald tests is to
consider the fraction of the time that the
TV
statistics emerging from the Monte Carlos
exceed the 1%, 5 % and 10% critical values of the relevant chi-squared distributions. These
axe displayed in columns 4, 5 and 6 of Table 5. Notice that the small sample sizes of the test
statistics for hypotheses Hi and H4 greatly exceed their asymptotic size. This tendency
is particularly dramatic in the case of H4, where the
W
statistics exceed their asymptotic
1%, 5% and 10% critical values 37%, 51% and 58% of the time.
16 With one exception all the moment conditions underlying our estimator of
hold exactly for the
artificial data generating process. The exception is the planner’s Euler equation for K t+ 1 , equation (19),
discussed in Appendix B. To deal with this problem, we computed the expectation in equation (19) for the
true log-linearized model. As it turns out, at these parameter values the error is approximately equal to
2 X 10-6. To correct for possible bias we implemented our Monte Carlos centering equation (19) around
2 X 10-6 rather than 0.
20
Before analyzing this finding, we briefly discuss the size of the test statistics applied to
the individual moments that make up joint hypotheses Hi, H 2 , H3 and H4. Our results
are displayed in Figure 3. The height of each bar graph in Panels A, B and C denotes
the percentage of times (out of 1000 trials) that the
W
statistic for a given hypothesis
exceeded the 10%, 5% and 1% critical values, of the asymptotic chi-squared distribution.
According to Figure 3, the small sample sizes of the test statistics for hypotheses 1 and
4-25 are moderately higher than their asymptotic sizes. The small sample sizes of the test
statistics associated with
o c/ a v
and
O i/ o y
are substantially larger than their asymptotic
sizes. This is consistent with our finding that Wald tests of hypotheses Hi and H4 overreject in small samples. However, these effects do not seem large enough to explain the
e x te n t
to which the Wald test over-rejects HI and H4.
Viewed overall, the outstanding feature of our experiments is the large (small sample)
size of the Wald test of hypothesis H4. Inference based on the asymptotic distribution of
the
Ml
statistic leads to a grossly overly critical assessment of the model’s performance.
In Appendix C we show that this conclusion is robust to various perturbations. First,
we consider the effects of different bandwidths when constructing Sx. These were chosen
both on an a
p r io r i
basis and using the Newey and West (1993) automatic bandwidth
procedure. Second, we consider different estimators of S o that correspond to different lag
windows. Third, we discuss the impact of using a small sample correction suggested by
Andrews (1991).
A different dimension along which our results could be sensitive is how we parameterize
the elements of ^ 2- Specifically, we could include the moments
{ o c / o y , O i/ o y , O h / 0 V, O h / o a }.
cr,-,<7* o a } rather them
Under these circumstances the moment conditions defining
our estimator of ¥2 are given by:
{ o e,
w ,- < z ]
= 0
= 0
= 0
£[/>?- » l ]
= 0
= 0
= 0, i
21
* — 1>•••»4.
Consistent with this reparameterization we redefined tests 2 through 5 so that they
pertained to
a c,
o,-,
and
oa
respectively and adjusted the definitions of HI and H 4
accordingly.17
Figure 4 reports the small sample size of the Wald tests with asymptotic size equal to
10% (Panel A), 5% (Panel B) and 1% (Panel C) for the reparameterized system. Notice
that in most cases, small sample size increases. This is true for hypotheses H1-H4, except
for the test of Hi at the 1% level. For the tests based on correlations (hypotheses 623) this is true for 51 out of 54 cases. Notice, however, that small sample performance
improves dramatically for the tests based on
O i/ O y
ae
and
a,
over those based on
a el a y
and
(hypotheses 2 & 3). Interestingly, this improvement does not translate into improved
performance for the test of hypothesis HI. So while the reparameterization appears to lead
to more uniform performance across the different moments, it does not solve the overall
excessive small sample size of the Wald tests. And it certainly does not account for the
dramatic problems associated with tests of hypothesis H4.
In the remainder of this section we discuss the factors underlying the large (small
sample) size of the Wald test of hypothesis H4. For this purpose we return to the original
parameterization of
distribution of the
W
and focus on the role played by the matrix S t in the small sample
statistic. To this end, we redid our Monte Carlo experiments using
the population value of
S t , S o,
that is implied by the parameters governing the data
generating process. Specifically, on each of the one thousand data sets, we estimated the
parameters of the model but formed the
W
statistic using the fixed matrix
S 0.
W e found
that the W statistics for H4 exceed their asymptotic (1%, 5%, 10%) critical values (4%, 8%,
11%) of the time. This contrasts with our baseline findings that the
W
statistic exceeds
its asymptotic (1%, 5%, 10%) critical values (37%, 51%, 58%) of the time.18 Evidently,
the fact that we must estimate
even when
So
So
accounts for a substantial part of the problem. But
is known, relying on asymptotic distribution theory would still lead us to
17The reparameterisation indirectly affected all of the other test statistics because of the covariance
between the GMM error terms.
18 We also found that the “W statistics for Hi, H2 and H3 exceeded their asymptotic (1%, 5%, 10%) critical
values (3%, 7%, 11%), (0%, 2%, 4%) and (2%, 5%, 7%) of the time, respectively. The analogous numbers in
the baseline Monte Carlo where we use S t rather than S q are (12%, 23%, 32%), (8%, 17%, 24%) and (7%,
13%, 20%).
22
reject hypothesis H4 too often.
A natural question arises as to whether the small sample distribution of the
W
statistic
for H4 would coincide even more closely to its asymptotic distribution if we imposed the
population values of D
t
and
F(^
t
)
as well as
St
in the Monte Carlo experiments. For
our data generating process the answer is no. Indeed, we found that the small sample size
of Wald test for H4 actually moved substantially farther away from its asymptotic size
under these circumstances. Specifically, the
W
statistic for H4 exceeded its asymptotic
(1%, 5 % 10%) critical values (16%, 25%, 32%) of the time. While this does not represent
a logical problem, we are surprised by the result.
Overall our results suggest that sampling error in S t plays a substantial role in the large
(small sample) sizes of Wald tests involving multiple moment conditions. This suggests
an alternative way to estimate 5j. Specifically, the econometrician could calculate the
implied population value of S t for any given set of parameter estimates when estimating
the model. The obvious drawback to this procedure, it that, for nontrivial models, it is
computationally quite burdensome.
4
C o n c lu s io n
This paper examined the small sample properties of Generalized Method of Moments
(GMM) based Wald statistics. For the data generating processes considered we found that
the small sample size of these tests exceeded their asymptotic size. The problem became
dramatically worse as the dimensionality of the joint tests being considered increased. We
offered evidence that the basic problem has to do with the difficulty of estimating the
spectral density matrix of the G M M residuals that is needed to conduct inference. Our
results lead us to be very skeptical that the problem can be resolved by losing any of
the alternative nonparametric estimators of this matrix that have been discussed in the
literature. Instead we advocate using estimators which impose as much a priori information
as possible. T wo important sources of such information are the economic theory being
investigated and the null hypothesis being tested. There are two costs associated with
pursuing this strategy. The first is computational. The second is that to pursue it the
analyst will often be required to make stronger distributional assumptions about the nature
of the unobservable shocks impacting on agents’ environments. But, in this case, two of
23
the prime reasons for using a G M M strategy, as opposed to maximum likelihood methods,
disappear.
R e fe re n c e s
1. Altug, Sumru (1989) “Time to Build and Aggregate Fluctuations: Some New Evi
dence,” I n t e r n a t i o n a l E c o n o m i c R e v i e w , Vol. 30, 889-920.
2. Andrews, Donald W.K. (1991) “Heteroskedasticity and Autocorrelation Consistent
Covariance Matrix Estimation,” E c o n o m e t r i c a , Vol. 59, 817-858.
3. Andrews, Donald W.K. and J. Christopher Monahan (1992) “An Improved Het
eroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator,” E c o n o m e t r i c a , Vol. 60, 953-966.
4. Brayton, F. and E. Mauskopf (1985) “The M P S Model of the United States Econ
omy,” Board of Governors of the Federal Reserve Bank of Minneapolis Working
Paper 425.
5. Burnside, Craig (1992) “Small Sample Properties of 2-Step Method of Moments
Estimators,” manuscript, University of Pittsburgh.
6 . Burnside, Craig (1993) “Notes on the Linearization and C M M Estimation of Real
Business Cycle Models,” manuscript, University of Pittsburgh.
7. Burnside, Craig and Martin Eichenbaum (1994) “Factor Hoarding and the Propa
gation of Business Cycle Shocks,” N B E R Working Paper 4675.
8. Christiano, Lawrence J. (1988) “W h y Does Inventory Investment Fluctuate So Much?,”
J o u r n a l o f M o n e t a r y E c o n o m i c s , Vol. 21, (March/May), 247-280.
9. Christiano, Lawrence J. and Wouter den Haan (1994) “Small Sample Properties of
G M M for Business Cycle Analysis,” manuscript, Northwestern University.
10. Christiano, L. and Eichenbaum, M. (1992) “Current Real Business Cycle Theories
and Aggregate Labor Market Fluctuations,” A m e r i c a n E c o n o m i c R e v i e w , Vol. 82,
(June), 430-450.
11. Eichenbaum, M.S. and L.P. Hansen (1990) “Estimating Models with Intertemporal
Substitution Using Aggregate Time Series Data,” J o u r n a l o f B u s i n e s s a n d E c o n o m i c
S t a t i s t i c s , 8 , 53-69.
12. Ferson, Wayne and Stephen E. Foerster (1991) “Finite Sample Properties of the
Generalized Method of Moments in Tests of Conditional Asset Pricing Models,”
manuscript, University of Chicago.
13. Fuhrer, Jeffrey, George Moore and Scott Schuh (1993) “Estimating the LinearQuadratic Inventory Model: Maximum Likelihood Versus Generalized Method of
Moments,” Finance and Economics Discussion Series 93-11, Board of Governors of
the Federal Reserve System.
14. Gallant, A.R. (1987)
N o n lin e a r S t a t is t ic a l M o d e ls .
24
New York: Wiley.
15. Gregory, Allan W. and Michael R. Veall (1985) “Formulating Wald Tests of Nonlinear
Restrictions,” E c o n o m e t r i c a, Vol. 53, 1465-70.
16. Hansen, Gary D. (1985) “Indivisible Labor and the Business Cycle,”
M o n e t a r y E c o n o m i c s , Vol. 16, (November), 309-328.
Jo u rn a l o f
17. Hansen, Lars P. (1982) “Large Sample Properties of Generalized Method of Moments
Estimators,” E c o n o m e t r i c a , Vol. 50, 1029-1054.
18. Hansen, L.P. and K.J. Singleton (1982) “Generalized Instrumental Variables Esti
mation of Nonlinear Rational Expectations Models,” E c o n o m e t r i c a , 50, 1269-1286.
19. Hodrick, Robert J. and Edward C. Prescott (1980) “Post-War Business Cycle: An
Empirical Investigation,” manuscript, Carnegie Mellon University.
20. King, Robert G., Charles I. Plosser and Sergio Rebelo (1988) “Production, Growth
and Business Cycles,” J o u r n a l o f M o n e t a r y E c o n o m i c s , Vol. 21, (March/May) 195232.
21. Kocherlakota, N. (1990) “On Tests of Representative Consumer Asset Pricing M o d
els,” J o u r n a l o f M o n e t a r y E c o n o m i c s , 26, 285-304.
22. Kydland, Finn E. and Edward C. Prescott (1982) “Time to Build and Aggregate
Fluctuations,” E c o n o m e t r i c a , Vol. 50, 1435-70.
23. Leeper, Eric and Christopher S. Sims (1994) “???” manuscript, Yale University.
24. McGrattan, Ellen, Richard Rogerson and Randall Wright (1993) “Household Pro
duction and Taxation in the Stochastic Growth Model,” Federal Reserve Bank of
Minneapolis, Research Department, Staff Report No. 166.
25. Neely, Christopher J. (1993) “A Reconsideration of Representative Consumer Asset
Pricing Models,” manuscript, University of Iowa.
26. Newey, Whitney K. and Kenneth D. West (1987) “A Simple, Positive Semi-definite,
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” E c o n o m e t
r i c a , Vol. 55, 703-708.
27. Newey, Whitney K. and Kenneth D. West (1993) “Automatic Lag Selection in Covariance Matrix Estimation,” manuscript, M.I.T.
28. Prescott, Edward C. (1986) “Theory Ahead of Business Cycle Measurement,”
e r a l R e s e r v e B a n k o f M i n n e a p o l i s Q u a r t e r l y R e v ie w , Vol. 10, Fall.
Fed
29. Tauchen, George (1986) “Statistical Properties of Generalized Method-of-Moments
Estimators of Structural Parameters Obtained from Financial Market Data,” J o u r n a l
o f B u s i n e s s a n d E c o n o m i c S t a t is t ic s , Vol. 4, 397-416.
30. West, Kenneth D. and David W. Wilcox (1994) “Some Evidence on Finite Sample
Distributions of Instrumental Variables Estimators of a Linear Quadratic Inventory
Model,” manuscript, Board of Governors of the Federal Reserve System.
25
TABLE 1
Small Sample Performance of Joint Tests
Using Normally Distributed White Noise Data
Estimating Standard Deviations
A. Estimated
*
II
h*
Asymptotic Size
2.69
7.49
12.65
1%
5%
10%
St , B ? = 4
Small
Af = 2
3.41
9.25
14.93
B. Estimated
*
II
Asymptotic Size
2.31
6.90
12.03
1%
5%
10%
1%
5%
10%
Af = 1
2.27
6.94
11.98
t
1%
5%
10%
Af = 1
2.15
6.74
11.79
t
M
= 20
58.68
73.37
80.29
= 2
M
= 20
28.88
45.62
55.88
by Andrews Procedure
Small
Af = 2
2.91
8.27
13.50
D. Estimated
Asymptotic Size
St , B
Small Sample Size (%)
M — 2
M = 5
Af = 10
2.87
4.83
9.17
8.26
12.22
19.91
13.62
19.32
28.55
C. Estimated Sr, B
Asymptotic Size
Sample Size (%)
Af = 5 Af = 10
6.99
16.98
15.61
30.92
23.32
40.10
S
t
Small
Af = 2
2.73
7.94
13.22
26
Sample Size (%)
Af = 5 Af = 10
4.71
9.06
11.94
19.27
19.04
27.87
,
Af = 20
26.64
43.43
53.83
No Lags
Sample Size (%)
Af = 5 Af = 10
6.67
4.17
10.82
16.23
24.10
17.43
Af = 20
17.31
32.87
42.51
Asymptotic Size
10 %
o
1%
5%
Small Sample Size (%)
M = 1 M = 2
M = 5 M = 10
2.15
2.67
3.33
3.88
6.74
7.58
9.32
11.04
11.79
13.04
15.50
17.56
II
E. Estimated Diagonal Sr, No Lags
4.71
13.39
21.20
F. Gaussianity Applied to E
Asymptotic Size
M
Small Sample Size (%)
M = 2 M = 5 M = 10
2.40
1.82
2.22
6.08
7.20
7.72
11.30
12.50
13.25
— 1
1.67
5.94
10.60
G. Impose
II
1.46
4.61
9.34
1%
5%
10 %
H. Impose
H
M
= 20
2.10
7.26
12.05
M
= 20
0.92
4.99
9.99
on Sr in F, and on Dr
1
0.96
5.16
10.14
0.97
4.90
10.13
27
0.99
5.08
10.20
s
II
o
=
*
II
to
M
1%
2.58
8.53
14.45
Small Sample Size (%)
Asymptotic Size
5%
10%
q
= 20
on Sr in F
Small Sample Size (%)
M = 2
M = 5 M = 10
1.67
2.03
2.10
5.33
5.97
6.58
9.55
10.47
11.70
h-*
Asymptotic Size
Ho
II
10 %
Cn
1%
5%
M
0.96
5.01
10.11
TABLE 2
Small Sample Performance of Joint Tests
Using Normally Distributed White Noise Data
Estimating Relative Standard Deviations
A. Estimated
Asymptotic Size
1%
5%
10%
= la
2.28
6.88
11.84
s
II
S r, B
t
D. Estimated
S
t
,
= 20
31.09
47.21
56.65
M
28
AT = 20
27.60
43.72
53.46
N o Lags
Small Sample Size (%)
M = 2
M = 5 M = 10
2.95
4.88
8.18
1.84
11.90
17.92
6.42
8.09
25.80
18.41
11.15
13.54
rH
II
1%
5%
10%
= la
2.15
6.74
11.79
*
M
= 20
59.25
73.11
79.98
by Andrews Procedure
Small Sample Size (%)
M = 10
M = 16
M = 5
9.80
1.90
3.12
5.45
8.46
6.57
12.87
20.65
11.40
13.93
19.98
28.84
Asymptotic Size
M
2
N
II
M
=
oH
r
II
C. Estimated
Asymptotic Size
t
S
e
II
2.31
6.90
12.03
10%
St , B
Small Sample Size (%)
M = 2
M = 5 M = 10
1.95
3.14
5.65
10.61
6.63
8.58
13.08
21.73
11.46
14.13
20.32
29.98
Asymptotic Size
1%
to
II
B. Estimated
5%
= 4
II
— la
2.59
7.49
12.65
1%
5%
10%
t
Small Sample Size (%)
M = 16
2.26
3.65
7.88
18.55
7.09
9.55
16.62
32.30
12.12
15.35
24.17
40.99
$
M
S r, B
M
= 20
20.29
34.91
44.46
E. Impose Mutual Independence
rH
II
2.15
6.74
11.79
1%
5%
10 %
II
h-‘
O
Small Sample Size (% )
Af = 2 Af = 5
1.79
2.87
4.07
5.37
6.24
7.88
10.43
12.44
11.07
13.29
16.60
19.24
II
t
—‘
&
Asymptotic Size
Af = 20
7.36
16.51
23.72
F. Gaussianity Applied to E
Small Sample Size (% )
Af = l b Af = 2 Af = 5 Af = 10
2.10
1.44
2.81
3.53
5.45
6.45
8.10
9.41
10.25
11.68
13.48
14.81
Asymptotic Size
M
1%
5%
10 %
= la
1.67
5.94
10.60
G. Impose
Asymptotic Size
5%
10 %
Af = la
1.46
4.61
9.34
10%>
St
H
q
on
St
in F
in F, and on
D
29
Af = 20
3.86
9.51
15.19
t
Small Sample Size (%)
Af = 16
Af = 5 Af = 10
1.36
1.21
1.71
2.02
5.43
5.36
6.19
6.71
10.25
10.11
11.24
11.92
II
1%
5%
Af = la
0.96
5.16
10.14
on
Small Sample Size (% )
Af = 16 M = 2 M = 5
2.79
1.76
2.50
2.98
6.09
5.63
7.04
8.45
9.86
9.82
11.71
13.23
H. Impose
Asymptotic Size
q
oH
r
II
1%
H
Af = 20
4.51
11.28
17.33
Af = 20
2.67
7.76
13.40
TABLE 3
Model Parameters
Estimates and Standard Errors*
Parameter
e
a
6
av
g/y
9o
9i
P
Estimate
3.5955
0.6422
0.0208
0.0038
0.0088
0.1763
1.7885
-0.0019
0.9456
0.0152
0.0088
Std. Error
(0.0377)
(0.0193)
(0.0002)
(0 .0012)
(0.0007)
(0 .0022)
(0.0809)
(0.0003)
(0.0299)
(0.0012)
(0.0011)
* All standard errors shown in this table are based on estimates of S t computed using
the Bartlett window suggested by Newey and West (1987), and the automatic bandwidth
selection procedure suggested by Andrews (1991).
30
TABLE 4
Tests of the Models
Moment
av
O c lo y
O ijO y
O h /O y
O h io a
U.S. Data
0.0192
(0.0018)
0.437
(0.029)
2.224
(0.068)
0.859
(0.069)
1.221
(0.115)
Model
0.0167
(0.0013)
0.480
(0.009)
2.244
(0.072)
0.795
(0.051)
1.033
(0.037)
W
1.614
(0.204)
2.005
(0.157)
0.044
(0.835)
0.990
(0.320)
2.258
(0.133)
‘Numbers under the heading U.S. Data are second moments of HP-filtered U.S. data.
Numbers under the heading model, are the model’s implications for the corresponding
moments as functions of ¥ 1. Standard errors for each are in parentheses. The p-values
for the corresponding W statistics are in parentheses.
TABLE 5
Small Sample Performance of the Joint Tests
Hypothesis
HI
H2
H3
H4
Test Performed Using U.S. Data*
p-value
M C p-value
W
6.64
0.25
0.48
43.7
0.00
0.01
35.5
0.00
0.01
66.3
0.00
0.06
Size (%) of Tests*
10%
5%
1%
31.7 23.0 11.9
23.6 16.5
7.6
20.2 13.3 6.5
57.6 50.7 36.7
‘The numbers under the heading ‘p-value’are the p-values obtained when the W statistics
for HI, H 2 , H3 and H4 are compared to x2 distributions with 5, 9, 9 and 17 degrees
of freedom respectively. The numbers under the heading ‘M C p-value’ are obtained by
comparing these statistics to the distribution of the W statistics generated by our Monte
Carlo experiments.
tThe numbers on this side of the table indicate the frequency (in %) with which the W
statistics from our Monte Carlo experiments exceed the 10%, 5% and 1% critical values
of the relevant x2 distributions.
31
TABLE 6
The Form of the Lag Window and Small Sample Performance
Moment
B
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
16.1
25.5
25.4
13.2
12.4
15.5
14.4
17.1
18.2
16.3
7.8
10.6
13.0
12.8
17.2
18.9
19.9
19.1
12.6
7.1
11.8
14.2
14.4
31.7
23.6
20.2
57.6
10%
P
16.0
24.0
25.7
14.2
13.4
15.0
14.2
17.4
18.3
16.9
8.8
10.1
12.3
12.8
16.4
17.6
19.1
19.3
13.9
8.5
11.1
13.6
13.9
32.6
28.1
23.7
63.6
Q
B
15.8
25.4
25.5
13.3
12.0
14.9
14.3
17.6
17.7
16.4
7.9
9.3
12.4
12.5
16.1
17.9
19.2
18.6
12.2
7.4
10.1
12.4
13.7
30.4
26.1
21.1
59.0
10.1
17.9
18.8
7.3
6.7
9.0
8.3
9.7
10.5
10.5
3.9
4.0
6.8
6.9
10.5
10.7
11.8
11.1
6.2
4.8
6.3
7.2
8.8
23.0
16.5
13.3
50.7
5%
P
9.7
16.3
19.7
8.0
7.6
8.9
8.0
9.5
10.6
10.5
5.2
4.5
6.9
7.5
9.8
10.1
11.6
11.1
6.5
5.5
5.9
7.1
8.2
24.1
21.3
16.7
56.9
Q
9.5
17.5
19.0
7.3
6.7
8.8
7.8
8.6
9.9
10.1
4.4
4.0
6.4
6.7
9.8
10.2
10.9
10.4
5.4
4.3
5.2
6.9
8.0
22.7
18.8
14.9
52.4
1%
P
Q
3.0
3.1
3.0
8.2
8.0
8.3
8.2
9.0
8.2
2.0
2.2
1.8
1.2
1.6
1.3
3.2
3.1
2.6
2.7
2.8
2.6
3.0
2.6
3.0
2.5
3.1
2.7
3.8
4.4
4.0
0.5
1.1
0.6
1.3
1.4
1.2
2.0
2.2
2.2
2.8
2.8
2.4
4.0
3.9
3.8
3.6
3.7
3.3
3.5
3.7
3.6
4.0
3.8
3.5
1.6
1.9
1.5
0.5
0.9
0.7
1.3
1.6
1.2
2.2
2.4
2.3
2.2
2.4
2.4
11.9 13.2 11.5
7.6 10.6 8.3
6.5
8.7
7.3
36.7 43.4 38.3
B
‘The sets of columns labelled x % refer to the sizes of tests (in % ) with asymptotic size
equal to x % . The labels B , P, Q refer to the Bartlett, Parzen and Quadratic Spectral
windows. The Bartlett kernel columns are our baseline case, the others differ from that
case only in the lag window used, and consequently in the bandwidths chosen by the
Andrews (1991) procedure. Tests are numbered as described in the text.
32
TABLE 7
The Impact of Excluding Some Moment Restrictions
Moment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
10%
Ex
In
16.1 28.8
25.5 33.4
25.4 35.7
13.2 28.5
12.4 28.1
15.5 30.0
14.4 27.0
17.1 26.6
18.2 30.9
16.3 29.8
7.8
23.5
10.6 26.1
13.0 26.6
12.8 25.9
17.2 33.9
18.9 33.5
19.9 34.2
19.1 32.2
12.6 29.6
7.1
28.5
11.8 29.3
14.2 31.3
14.4 31.3
31.7 85.4
23.6 95.8
20.2 93.5
57.6 100.0
5%
Ex
In
10.1 22.4
17.9 25.5
18.8 29.4
7.3
22.2
6.7 20.7
9.0 22.0
8.3
20.2
9.7
20.0
10.5 22.7
10.5 23.1
3.9
17.0
4.0
18.8
6.8
19.4
6.9
19.3
10.5 25.4
10.7 25.3
11.8 25.4
11.1 24.1
6.2
23.3
4.8
20.5
6.3
21.8
7.2
24.5
8.8
25.0
23.0 82.3
16.5 93.7
13.3 91.6
50.7 100.0
1%
Ex
In
3.0 13.0
8.2 14.9
8.2 19.2
2.0 12.4
1.2 11.0
3.2 12.1
2.7 11.6
3.0 11.8
2.5 13.2
3.8 13.7
0.5
9.0
1.3 10.3
2.0 11.0
2.8 11.0
4.0 15.1
3.6 13.5
3.5 15.1
4.0 14.5
1.6 12.0
0.5 10.4
1.3 13.1
2.2 14.7
2.2 14.8
11.9 71.7
7.6 88.2
6.5 85.9
36.7 99.9
‘The columns labelled ‘Ex’ correspond to our baseline case, while those labelled ‘In’cor
respond to experiments in which the three moment restrictions, excluded in our baseline
case, are not excluded in computing the automatic bandwidths.
33
TABLE 8
The Newey and West Automatic Bandwidth Procedure'
Moment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
A
16.1
25.5
25.4
13.2
12.4
15.5
14.4
17.1
18.2
16.3
7.8
10.6
13.0
12.8
17.2
18.9
19.9
19.1
12.6
7.1
11.8
14.2
14.4
31.7
23.6
20.2
57.6
10%
NW(4)
16.5
27.0
25.1
12.5
12.1
16.0
14.5
16.9
17.5
17.2
7.7
11.1
13.1
13.8
17.7
18.5
19.6
19.8
11.3
7.5
11.8
13.7
14.9
31.8
24.8
20.7
55.9
N W ( 12)
19.0
24.7
26.8
16.5
15.8
18.6
16.1
18.1
19.0
18.4
11.0
14.0
14.7
15.3
21.0
21.5
22.7
21.2
16.1
11.7
15.8
17.9
18.4
42.4
41.8
38.6
78.8
5%
A
NW(4)
10.1
9.8
17.9
18.9
18.8
17.6
7.1
7.3
6.7
6.7
9.0
9.4
8.3
8.7
9.7
9.4
10.5
9.6
10.5
10.0
3.9
3.8
4.0
5.0
6.8
7.2
6.9
7.6
10.5
10.5
10.7
11.3
11.8
11.8
11.1
11.2
6.2
6.1
4.8
4.5
6.3
6.6
7.2
8.1
8.8
9.4
23.0
21.8
16.5
17.2
13.3
14.9
50.7
49.8
N W ( 12) A
11.3
3.0
17.9
8.2
20.5
8.2
9.4
2.0
9.4
1.2
10.4
3.2
9.3
2.7
11.5
3.0
11.3
2.5
12.5
3.8
6.0
0.5
7.9
1.3
9.1
2.0
9.2
2.8
13.0
4.0
13.1
3.6
14.0
3.5
13.0
4.0
8.8
1.6
6.7
0.5
9.1
1.3
11.1
2.2
12.1
2.2
34.6
11.9
34.8
7.6
30.9
6.5
73.0
36.7
1%
NW(4)
3.3
9.2
7.5
2.0
1.3
3.0
3.4
2.4
2.7
3.9
0.5
1.5
2.3
2.9
4.0
4.1
4.2
3.8
1.3
0.6
1.4
2.6
2.9
12.1
8.7
8.4
36.4
N W ( 12)
4.0
9.2
10.0
2.9
2.8
4.2
4.2
3.2
3.8
5.6
1.8
2.2
3.3
4.2
4.7
5.3
5.3
5.0
2.6
1.7
2.8
4.6
4.4
20.8
23.9
20.7
63.0
‘The columns labelled ‘A ’ correspond to our baseline case, which uses the Andrews au
tomatic bandwidth procedure, while those labelled *NW(x)’ correspond to experiments
using the Newey and West procedure, with x being the value of n.
34
TABLE 9
Variable Versus Fixed Bandwidth'
Moment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
V
16.1
25.5
25.4
13.2
12.4
15.5
14.4
17.1
18.2
16.3
7.8
10.6
13.0
12.8
17.2
18.9
19.9
19.1
12.6
7.1
11.8
14.2
14.4
31.7
23.6
20.2
57.6
10%
2
16.4
28.3
24.2
12.0
10.9
15.5
14.8
17.6
17.8
15.8
6.1
9.2
12.8
13.1
17.1
18.7
19.8
18.8
9.5
5.8
9.7
13.1
14.2
28.6
20.0
16.7
49.6
4
16.6
23.9
25.6
13.6
13.6
15.3
14.4
17.1
17.9
17.4
8.7
11.4
12.7
13.0
16.5
18.4
20.1
19.5
13.6
8.7
12.7
14.4
14.2
32.3
25.6
21.6
62.7
V
10.1
17.9
18.8
7.3
6.7
9.0
8.3
9.7
10.5
10.5
3.9
4.0
6.8
6.9
10.5
10.7
11.8
11.1
6.2
4.8
6.3
7.2
8.8
23.0
16.5
13.3
50.7
5%
2
10.5
20.1
17.4
6.5
5.9
9.3
8.5
8.9
9.5
10.1
2.9
3.7
6.8
7.3
11.0
10.9
11.8
10.4
4.1
2.5
5.0
7.2
8.4
19.6
12.9
10.6
41.2
4
9.8
16.3
19.6
8.0
7.5
8.8
8.0
10.3
10.4
10.4
4.4
4.7
7.0
7.5
9.8
10.7
11.8
11.7
6.1
4.9
6.9
7.6
8.6
24.3
19.4
14.9
55.2
1%
V
2
4
3.0
3.0
3.1
8.2
9.6
7.9
8.2
6.4
8.7
2.0
1.1
2.1
1.2
1.1
1.6
3.2
2.6
3.2
2.7
2.4
2.9
3.2
3.0
2.2
2.5
2.2
2.9
4.3
3.8
3.5
0.5
0.3
0.9
1.3
1.2
1.4
2.0
2.4
2.2
2.8
3.1
2.5
3.9
3.6
4.0
3.6
3.9
3.9
3.5
3.4
4.2
4.0
3.2
3.8
1.6
1.0
1.7
0.5
0.5
0.2
1.3
1.5
0.9
2.7
2.2
2.4
2.3
2.2
2.4
11.9 10.7 12.8
8.7
7.6
5.8
7.2
6.5
5.3
36.7 27.5 41.6
’The sets of columns labelled x % refer to tests with asymptotic size equal to x % . The
labels V, 2 and 4 refer to variable bandwidths picked with the Andrews (1991) procedure,
a fixed bandwidth of 2 and a fixed bandwidth of 4 respectively. All results are based on
our other baseline choices.
35
TABLE 10
Variable Versus Fixed Bandwidth
Moment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
V
16.1
25.5
25.4
13.2
12.4
15.5
14.4
17.1
18.2
16.3
7.8
10.6
13.0
12.8
17.2
18.9
19.9
19.1
12.6
7.1
11.8
14.2
14.4
31.7
23.6
20.2
57.6
10%
6
17.8
23.1
26.7
16.0
15.1
16.3
15.5
17.3
18.0
18.6
11.0
13.3
13.6
14.2
18.2
20.9
21.0
21.0
15.1
10.9
15.2
17.3
16.8
37.8
34.7
29.8
76.3
8
19.0
23.2
26.5
16.9
16.6
17.8
15.9
17.6
19.0
19.5
12.2
14.6
14.4
15.2
20.2
23.0
22.7
21.8
17.2
12.6
17.1
18.8
19.2
42.9
43.2
39.7
86.6
V
10.1
17.9
18.8
7.3
6.7
9.0
8.3
9.7
10.5
10.5
3.9
4.0
6.8
6.9
10.5
10.7
11.8
11.1
6.2
4.8
6.3
7.2
8.8
23.0
16.5
13.3
50.7
5%
. 6
10.4
15.8
20.0
9.0
8.6
9.3
8.6
10.7
11.0
11.5
5.7
5.9
8.1
8.7
10.8
12.0
13.1
12.4
8.3
6.0
8.1
10.0
10.9
28.5
26.4
22.0
69.6
1%
8
V
6
8
11.6 3.0
3.6
3.9
16.6 8.2
7.6
8.0
20.8 8.2
9.8
9.9
10.3 2.0
2.5
2.8
9.6
1.2
2.2
2.8
9.9
3.2
4.2
5.0
9.9
2.7
3.4
3.6
11.2 3.0
3.5
4.0
11.6 2.5
3.4
4.0
12.5 3.8
5.0
5.3
7.0 0.5
1.7
1.4
7.4
1.3
2.1
2.3
9.1
2.0
3.0
3.3
9.4
2.8
3.7
4.1
12.6 4.0 4.0
4.9
14.0 3.6
5.3
4.4
14.8 3.5
4.8
5.0
13.9 4.0
5.1
5.2
10.1
1.6
2.4
3.3
7.3
0.5
1.6
2.1
9.3
1.3
2.2
2.7
12.4 2.2
3.5
4.8
12.5 2.2
5.1
3.6
34.1 11.9 16.3 20.3
35.6 7.6 14.8 21.6
30.8 6.5 11.8 17.8
81.9 36.7 55.9 72.0
‘The sets of columns labelled x % refer to tests with asymptotic size equal to x % . The
labels V, 6 and 8 refer to variable bandwidths picked with the Andrews (1991) procedure,
a fixed bandwidth of 6 and a fixed bandwidth of 8 respectively. All results are based on
our other baseline choices.
36
TABLE 11
First Order V A R Prewhitening'
Moment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
HI
H2
H3
H4
10%
Ex
In
7.1
11.5
8.5
14.1
15.2 21.6
6.6
12.7
7.3
11.9
4.6
11.0
5.1
9.7
5.5
10.6
7.0
12.3
7.0
11.6
2.8
6.7
6.9
3.6
3.0
6.5
3.6
6.8
6.6
12.4
8.5
15.2
8.4
13.8
8.6
14.5
7.3
13.3
7.7
4.1
3.7
6.9
2.6
6.4
4.6
7.5
39.8 68.5
49.5 89.7
49.1 88.8
91.1 100.0
5%
Ex
4.2
5.8
9.2
4.0
4.1
2.5
3.0
3.2
3.7
3.9
1.8
1.8
1.7
1.9
4.2
4.9
4.5
4.3
3.2
2.3
2.2
1.4
2.9
30.8
42.3
40.5
88.1
In
7.7
9.5
15.4
9.2
7.8
6.3
6.0
6.5
7.5
7.7
3.9
3.9
4.1
4.4
8.5
10.4
9.5
9.9
9.0
4.5
5.2
3.6
4.5
62.9
85.9
84.5
100.0
1%
Ex
In
1.2
3.4
2.0
4.6
3.2
6.9
1.0
3.3
1.3
3.7
1.1
2.9
1.2
3.3
0.5
2.7
1.0
2.7
1.4
4.2
0.3
1.8
0.6
1.8
0.4
1.1
0.7
2.5
1.5
3.2
1.7 4.2
1.4
4.6
1.6
4.3
0.9
3.3
0.8
2.4
0.6
2.0
0.5
1.8
0.7
2.4
19.7 51.2
28.9 79.9
28.1 77.2
81.8 99.7
’The columns labelled ‘Ex’ correspond to our baseline case, while those labelled ‘In’cor
respond to experiments in which the three moment restrictions, excluded in our baseline
case, are not excluded in computing the automatic bandwidths.
37
FIGURE 1
C o r r e l a t io n
of
APLt
w it h
F actor Hoarding Model
Difference
Correlation
Benchmark yodel
H t+i ( H P F il t e r )*
FIGURE 2
C o r r e l a t io n
of
APLt
w it h
F acto r Hoarding Model
Difference
Correlation
Benchmark Model
Yt+i ( H P F il t e r )*
*In the Correlation panels: solid line - model predicted correlations, dashed line - sample
correlations. In the Difference panels the dashed lines represent a 2-standard error band
around the difference.
38
FIGURE 3
S m a ll S a m p l e S iz e o f t h e W T e s t s
R B C Example
A.
A s y m p t o t ic S iz e = 10%
CO
o
.11
I I I
. . ■ I I I
. . I l l
■i i i i i i i i i n i i i i i i i n i i i i i i i
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
H1
H2
H3
H4
20
21
22
23
H1
H2
H3
H4
Test
B.
A s y m p t o t ic S iz e = 5 %
Test
C.
A s y m p t o t ic S iz e =
Wo
o
_______________________________________________
in
o
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Test
39
15
16
17
18
19
FIGURE 4
S m a ll S a m p l e S iz e
of the
W T ests
Reparameterized RBC Example
A.
A s y m p t o t ic S iz e = 10%
CO
o
in
i n
■ ■ i
■m
1
2
3
i m
4
5
6
7
_ . . ■
i m
8
9
10
11
m
12
13
14
..in
1 1 1
m
15
16
17
m
18
19
i m
i i i
20
21
22
23
HI
H2
H3
H4
20
21
22
23
H1
H2
H3
H4
Test
B. A s y m p t o t ic S iz e = 5
%
C. A s y m p t o t ic S iz e = 1%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Test
40
15
16
17
18
19
A
C o v a ria n c e M a tr ic e s fo r th e R e p a r a m e tr iz e d
C ase
W h it e N o is e
In this appendix we discuss our estimators of the asymptotic variance-covariance matrix of
0. The estimators Sy, S ? and S ? are defined as for a . When we allow the econometrician
to exploit the lack of serial correlation in the data generating mechanism, we obtain
• S£, with (1,1) element
- d?j2], (!»j) and (j» l) elements
° l ) { X ? t - 0 ? X \ t)) for j > 2, and (i,j) element given by £[£f=1(X?t - 0t
2X*t)(X?t0 ? X \ t)\,
for * > 2, and
j
> 2.
If the econometrician imposes the mutual independence restriction then we obtain
•
with (1,1) element £ 521=1
for j
>
(1»j) and
2, (*,*') element £ £f=1 X f t -
element given by
6?0?
(jr52f= i X
ft
—
Of o f + Of
b fj
(j\
1) elements 0?
52f= i X f t
(o f
—
£ 52j= i X f ^
- d*) for * > 2 and
( i,j)
for * > 2 and 2 < ; ^ *•
If in addition the econometrician exploits the fact that the X u axe Gaussian, we have
that E ( X f t) = 3<7i and E ( X f t) = 3erf = 3Of o f for j > 2. This restriction can be imposed
on Sy, which yields
•
with (1,1) element 2 o f , (l,j) and ( j , 1) elements — 20 ? b f for j > 2 , (t,t) element
4Of o f for t > 2 and (:,j ) element given by 26*0?o f for * > 2 and 2 < j / i .
S%,
W e consider two types of hypotheses when estimating 6 : (i) hypothesis H m :0,- = 1,
* = 1,..., M , and (ii) hypothesis H u ‘.62 = 1 . When we impose H m we obtain
with (1,1) element given by 2; (1,j) and ( j , 1) element —2 for 2 < j < M and
— 26? for j > M ; (t,t) element equal to 4 for 2 < t < M
and 4O f for * > M ; and
•
(t,j) element equal to 2 for 2 <
i , j < M , 26?
for 2 < * <
M , j
> M
and
2 0 ?6 ?
for
t ,j > M .
When we impose hypothesis
•
H u
we obtain
which is identical to
except in the second row and second column. The
diagonal (2 ,2) element is 4o f while the (2 ,j) and (j,2) elements are 2 0 ? o f , j > 2 .
S j- ,
For each of the estimators above we construct an estimator for the variance-covariance
matrix of the G M M estimator:
V-T* = ( R f tS j) - 1^ ] - 1,
is a diagonal matrix with (1,1) element — 2di and ($,*)) element — 20,0 *, for
j > 2 . Since the null hypothesis H m cam also be imposed on D \ , we also consider the
estimator
V-r = [ D » { S I ) - ' D ' t \~ \
where
D ?
where D \ is a diagonal matrix with nth element —2 for i < M , and — 20,- for i > M
the hypothesis H u we have D \ equal to D ? except that the (2,2) element is — 2 b \ .
41
.
For
B
T h e E u l e r E q u a t io n f o r C a p it a l in t h e R B C E x a m p le
In this appendix we discuss our procedure for ensuring that the Euler equation for capital
holds exactly in the data generating process underlying our Monte Carlo experiments.
The Euler equation for K t+1 does not hold exactly for our linearized representation of the
model. This equation is given by
(B.l)
E
As a result, when we estimate the model using artificially generated time series from
the linearized model it is important to adjust this moment restriction appropriately. W e
compute the expectation in equation (B.l) for our linearized model (it is approximately
2 x 10-5) evaluated at the parameter values we use to generate the artificial data. W e
then center the moment restriction around that value rather than 0. This expectation,
denoted by e , is computed as follows
e
=
E
=
E
1 -
P
exp(ct - cm )((1 - a) (1 - ^-1)exp(yt+1 -
k t+ i ) +
exp( - 7 - vt+1))
Let = ( 1 k t N t Vt gt & ) and st+i = ( sj+1
) . Any variable in the linearized
model, say Z t, determined at time t is given as a function 7r'st, for some vector tcx de
termined by the solution to the model. Therefore we can write the Euler equation error
simply as
e = E
1 -/?((!-
a)
(1 -
<f>
exp(/zist+i) + exp(-^) exp(A*'2st+i))J,
where
Hi
In our simulations we assume that the innovations to the exogenous variables are normally
distributed. In this case the properties of log-normal random variables can be exploited
to show that
e = 1 - #[(1 - a)(l -
<f>
x)e x p ( n i E s
,^i) + exp( ~ 7 +
+
n'2E s
-I-
where E s and T, are the mean and unconditional covariance matrix of
both computable as a by-product of the solution method.
42
,
These are
C
A lte r n a tiv e C o v a ria n c e M a t r ix E s tim a to r s
This appendix considers the robustness of the results presented in section 3 to alternative
estimators, Sr,of the the weighting matrix S 0.
W e consider various forms of S t which depend on
i. whether we include a small sample correction or not,
ii. the form of the lag window, &(•),
iii. the method for determining the bandwidth,
B
t
,
and
iv. whether we prewhiten the errors u t.
As described in the text, we take as our baseline case an estimator which
i. does not apply any small sample correction,
ii. uses the Bartlett lag window suggested by Newey and West (1987), and
iii. selects the bandwidth automatically using the method suggested by Andrews
(1991), and
iv. does not prewhiten the errors.
C.l
The Small Sample Correction
The large sample performance of the tests is unaffected by the inclusion of a small sample
degrees of freedom correction in the estimator S t - In a regression context, Andrews (1991)
suggests the small sample correction of multiplying S t by a factor of T / ( T — d ) where
d is the dimension of the coefficient vector. In our simulated samples, the sample size
is T = 113, while the length of the parameter vector ^ is d = 27. Therefore, applying
Andrews’small sample correction increases the magnitude of the elements of S t by a factor
of 31%. Since the effect is uniform, applying the correction unambigously decreases all test
statistics by 31%. This decreases the small sample size of the tests, although this effect will
not be uniform. W e did not apply the small sample correction in our baseline experiments.
Although applying the correction would have improved our results somewhat, we found
that this was special to results based on the H P filter, as opposed to results based on first
differenced data.
C.2
The Lag W i n d o w
W e consider three forms of lag window,
Bartlett kernel given by
k(X
k (-).
Newey and West (1987) suggest using the
1 —1*1 for |x| < 1
0
otherwise.
43
Gallant (1987) proposes using the Parzen kernel given by
( 1 — 6xJ + 6 |x|3 for 0 < |x| < 1/2,
= < 2(1 — |x|)3
for 1 /2 < |x| < 1,
(0
otherwise.
k p (x)
Andrews (1991) examines the properties of the Quadratic Spectral (QS) kernel given by
,
,
25
\
(
sin(67rx/5)
.
(-6 ^ 5 —
.
\
cos(6,rl/5)J '
Andrews (1991) shows that within a certain class of estimators, which includes each of
these kernels, the QS kernel is optimal in the sense that it minimizes the asymptotic M S E
of Sr*
W e use the Bartlett kernel as our baseline lag window. Holding the other elements of
the baseline estimator fixed we examine the small sample performance of our Wald tests
using the Parzen and QS kernels. The results are summarized in Table 6 which shows the
small sample size of the tests using the different lag windows. The results indicate that
the small sample size of the tests is insensitive to the choice of lag window, at least in our
example.
C.3
The Choice of Bandwidth
An issue which arises immediately with these estimators is how to choose the bandwidth
parameter B t for a given kernel. Andrews (1991) shows that the optimal bandwidths (in
an M S E sense) for the three kernels are
1.1447[a(l)T]1/s Bartlett kernel
Bf = < 2.6614[a(2)7’]1/5 Parzen kernel
( 1.322l[a(2)T]1/5 QS kernel,
(
where a ( q ) is a function which depends on the unknown spectral density matrix of
given by
/(A ) = i
£
ut
n ,e -« \
j= -o o
where fly =
Since S p is a matrix estimator, its M S E is typically measured with respect to some
weighting scheme such as (following Andrews 1991)
M S E ( T / B r,Sr,W ) = -^-£vec(Sr
Bt
-
S0)'Wvec(Sr - S0),
where W is some d? x d? positive definite matrix. The measure of M S E depends on the
choice of the matrix W . Given a particular matrix W the optimal bandwidth formulas
can be made operational since
.._
2(vec/^)Wvec/to)
= trW(/ + K p P) /( 0 ) ® /( 0 ) ’
44
where
Kpp
is defined so that vec(A') =
K ppv e c ( A ) ,
/<” = s
t
J=-oo
and
u rn # .
Automated bandwidth selection procedures provide a means of estimating the a’s in the
above formulas.
C.3.1
A n d r e w s ’ (1991) Automated Bandwidth Procedures
Andrews (1991) proposes various automatic bandwidth estimators. These are data-based
procedures which implement the above formulas for estimates of a(l) and a(2). There
are many possible procedures, both parametric and nonparametric, that can be used
to estimate a(l) and a(2). Parametric estimators require choosing an approximating
parametric model for the errors ut. Typical choices are parsimoniously parameterized and
may model the errors individually. They further require the choice of a weighting matrix
W .
Since the possibilities were numerous, we chose perhaps the simplest approach which
is to choose a weighting matrix which only puts weight on the diagonal elements of S T
and to model each error term as an AR(1). Of course, the errors do not follow AR(l)
processes but this does not affect consistency of the estimators for S o . Rather it generates
a bias in estimates of the optimal bandwidth. In the AR(1) case
d(2)
*
1=1
a (l)
z >
4 6 2ct4
k
(i - A ) 1 ,=1
IPi& i
(1 k
P i) 4
ot
i= 1
where tu* is the weight given to error term i in computing the estimator, and (p,,^«) are
standard estimates of the parameters of the A R model obtained from residual t. The
simplest estimator sets to,- = 1 for all i .
Andrews suggests setting tVi to zero for any error terms corresponding to a constant re
gressor in a regression model. Presumably, this ismotivated by the fact that the covariance
properties of those error terms are qualitatively dissimilar to the covariance properties of
the error terms corresponding to nonconstant regressors. In our examples, we placed no
weight on the error terms corresponding to (17), (18) and (22) and unit weight on all other
error terms, as these error terms behave very differently than the others. This constitutes
our baseline method. To assess the impact of excluding these three moment restrictions
we compared our baseline results to experiments where they were not excluded. In our
baseline experiments the median bandwidth from the 1000 draws was 2.78 for the Bartlett
kernel. With equal weight given to all moment conditions the median bandwidth rose sig
nificantly to 40.1. Furthermore, as can be seen in Table 7, the small sample performance
of some of our tests deteriorated significantly.
45
C.3.2
N e w e y and West’s (1993) Automated Bandwidth Procedure
Newey and West’s (1993) procedure is related to the procedures outlined above but is
nonparametric in the sense that no pseudo-model of the residuals is specified in order
to estimate the a ’s. Newey and West note that when the M S E criterion is rewritten as
w ' ( S t — 5o)t2; for some d x 1 vector to, the formula for a ( q ) can be rewritten as
«(?) = £ |iPw'ftiw] / f £
./=- oo
J /
L/=—oo
w% w
.
In order to estimate 0 (9), Newey and West (1993) suggest the approximation
d(9) =
J2
J=—n
J /
/ [Ly=-n
£ w'fy*5
»
where n is chosen a p r i o r i in order to be consistent with n —*■ 00 and n / T 2/9 — ► 00 (for
the Bartlett window) as T —► 00. Newey and West cite evidence that S t is less sensitive
to arbitrary n than it is to arbitrary choices of B t W e present results for choices of n = 4 and n = 12. The weight vector we use puts
zero weight on the same moment restrictions we excluded from our baseline Andrews
method. The results are summarized in Table 8 . When we chose n = 4 we obtained
very similar results to when we used the Andrews procedure. This is not surprising:
the median bandwidth chosen by the Newey and West procedure was 2.74 while the
median bandwidth chosen by the Andrews procedure was 2.78. W h e n we used n = 12
this increased the median bandwidth of the Newey and West procedure to 7.39. This
led to massive overrejections of the joint hypotheses similar in scale to when we used a
fixed bandwidth of 8 (see the next subsection). Overall our results indicate that automated
procedures may perform similarly but only ifthey are ‘timed’in a way that happens to lead
to similar bandwidths. W e suspect that the Andrews procedure, while it has no parameter
like n to be chosen, would be analagously sensitive to the choice of pseudo-model for the
error terms.
C.3.3
Arbitrary Fixed Bandwidth
It is difficult to compare results obtained with fixed bandwidths to those obtained using
variable bandwidths, since any results we find may not be interpretable beyond the confines
of our example. However, in Table 9 and Table 10 we compare our baseline results with
results obtained using fixed bandwidths of 2, 4, 6 and 8 in repeated samples. While the
results are mixed for small bandwidths, the results indicate a deterioration of small sample
performance (especially for joint tests) for bandwidths of 6 and 8 . What is also clear from
these tables is that the bandwidth affects the various tests differentially.
C.4
Prewhitening of the Errors
Andrews and Monahan (1992) suggest a procedure which prewhitens the error terms as
a first step prior to the computation of Sy. Prewhitening is motivated by the apparent
46
problem in estimating S t when the nature of the persistence in the errors is unknown. A
particular bandwidth in tandem with a particular lag window may not adequately capture
the nature of the persistence in the errors in small samples. The whiter are the error
terms the less important will be the choice of lag window and bandwidth. A prewhitening
procedure uses an arbitrary procedure to whiten the error terms, computes the equivalent
of S t for those whiter errors, then recolors the estimated matrix.
As an example, suppose that a first-order V A R is fit to process ut,
ut
= nut-i +
fjt -
Suppose that n converges to II asymptotically so that the errors ut('Jro) have the repre
sentation
Wt(^o) = Ilu t-i^ o ) +
Define
fit-
^
^
SS = £
m n ',-,) = £
j=—oo
;=-oo
n;.
Then notice that
s0 = {i - n ) - lsz[i - n ' ) - 1.
An analagous estimator
St
is
s T = ( i - i t ) - l s } ( i - i t ' ) - 1,
where S £ is an estimator of the variety described in previous sections applied to f\t . For
higher order VARs represented as [ I — II(L)]ut = T)t , the corresponding estimator would
be
sT = [i-fi{i)]~ls } [ i - i i ( i y ] ~ \
W e conducted experiments using lst-order VARs for the error terms. The results of
these experiments are presented in Table 11. Given that we prewhitened the errors we
thought that comparisons should be made with both the ‘Ex’and the ‘In’columns in Table
7. Notice that small sample sample performance of the tests changes dramatically. Some
of the tests reject less often, but the joint tests perform terribly. The test of H4 almost
always rejects in the ‘In’case where we include all the error terms in our bandwidth calcu
lations. These results are somewhat surprising. W e might expect prewhitening to improve
performance. However, the median bandwidth chosen by the automated procedures rises
to 7.72 in the ‘Ex’case and drops to 29.7 in the ‘In’case. The rise in the ‘Ex’case is not
surprising since the included errors have been projected onto lags of the excluded errors.
D
D a ta
In this appendix we describe the data that was used to estimate the R B C model of section
3. Private consumption, C t , was measured as the sum of private sector expenditures on
nondurable goods plus services plus the imputed service flow from the stock of durable
47
goods. The first two measures were obtained from the Survey of Current Business. The
third measure was obtained from Brayton and Mauskopf (1985). Government consump
tion, G t , was measured by real government (federal, state and local) purchases of goods
minus real government investment. The government data was provided to us by John Musgrave at the Bureau of Economic Analysis. The official capital stock, K t , was measured as
the sum of consumer durables, producer structures and equipment, and government and
private residential capital plus government non-residential capital. Data on gross invest
ment, J(,are the flow data that conceptually match the capital stock data. Gross output,
Y u was measured as ( C t + G t + /*) plus time t inventory investment. Our basic measure of
hours worked is the quarterly time series constructed by Hansen (1985), which we refer to
as household hours. The data cover the period 1955:3-1984:1 and were converted to per
capita terms using an efficiency weighted measure of the population.19 W e use Prescott’s
(1986) model of measurement error in hours worked. In particular we assume that the
log of measured hours worked differs from the log of actual hours worked by an i.i.d.
random variable that has mean zero and standard deviation
To estimate
we need
two measures of hours worked. The first is Hansen’s measure of hours worked which is
based on the household survey conducted by the Bureau of the Census. The second is the
establishment survey conducted by the Bureau of Labor Statistics.
19See Christiano (1988, appendix) for further details.
48
Working Paper Series
A seriesofresearchstudiesonregionaleconomicissuesrelatingtotheSeventhFederal
ReserveDistrict,andonfinancialandeconomictopics.
REGIONAL ECONOMIC ISSUES
EstimatingMonthlyRegionalValueAdded byCombiningRegionalInput
With NationalProductionData
PhilipR.IsrailevichandKennethN.Kuttner
WP-92-8
LocalImpactofForeignTradeZone
WP-92-9
DavidD.Weiss
TrendsandProspectsforRuralManufacturing
WP-92-12
StateandLocalGovernmentSpending-The Balance
Between InvestmentandConsumption
W P-92-14
WilliamA.Testa
RichardH.Mattoon
ForecastingwithRegionalInput-OutputTables
Pi?.I
srailevich,R.Mahidhara,andGJD.Hewings
WP-92-20
A PrimeronGlobalAuto Markets
WP-93-1
IndustryApproaches toEnvironmentalPolicy
intheGreatLakesRegion
W P-93-8
The MidwestStockPriceIndex-LeadingIndicator
ofRegionalEconomic Activity
WP-93-9
Lean ManufacturingandtheDecisiontoVerticallyIntegrate
Some EmpiricalEvidenceFrom theU.S.AutomobileIndustry
WP-94-1
DomesticConsumptionPatternsandtheMidwestEconomy
WP-94-4
PaulD.BallewandRobertH.Schnorbus
DavidR.AUardice,RichardH.MattoonandWilliamA.Testa
WilliamA.Strauss
ThomasH.Klier
RobertSchnorbusandPaulBallew
1
W orking paper series continued
To Trade orNot otTrade: Who Participates inRECLAIM?
Thomas H. Klier and Richard H. Mattoon
WP-94-11
ISSUES IN FINANCIAL REGULATION
Incentive Conflictin Deposit-Institution Regulation: Evidence from Australia
Edward J . Kane and George G. Kaufman
WP-92-5
Capital Adequacy and the Growth ofU.S. Banks
Herbert Baer and John McElravey
WP-92-11
Bank Contagion: Theory and Evidence
George G. Kaufman
WP-92-13
Trading Activity, Progarm Trading and the Volatilityof Stock Returns
James T. Moser
WP-92-16
Preferred Sources ofMarket Discipline: Depositors vs.
Subordinated Debt Holders
Douglas D. Evanoff
WP-92-21
An Investigation ofReturns Conditional
on Trading Performance
James T. Moser and Jacky C. So
The Effect of Capital on PortfolioRisk atLife Insurance Companies
Elijah Brewer III , Thomas H. Mondschean, and Philip E. Strahan
A Framework forEstimating the Value and
InterestRate Risk ofRetail Bank Deposits
David E. Hutchison, George G. Pennacchi
WP-92-24
W P-92-29
WP-92-30
Capital Shocks and Bank Growth-1973 to 1991
Herbert L . Baer and John N . McElravey
WP-92-31
The Impact of S&L Failures and Regulatory Changes
on the CD Market 1987-1991
Elijah Brewer and Thomas H . Mondschean
WP-92-33
2
W orking p aper series continued
Junk Bond Holdings, Premium Tax Offsets, and Risk
Exposure atLife Insurance Companies
Elijah Brewer III and Thomas H. Mondschean
WP-93-3
Stock Margins and the Conditional Probability ofPrice Reversals
Paul Kojman and James T. Moser
WP-93-5
IsThere Lif(f)e After DTB?
Competitive Aspects ofCross Listed Futures
Contracts on Synchronous Markets
Paul Kojman, Tony Bouwman and James T. Moser
Opportunity Cost and Prudentiality: A RepresentativeAgent Model of Futures Clearinghouse Behavior
Herbert L. Baerf Virginia G. France and James T. Moser
The Ownership Structure ofJapanese Financial Institutions
Hesna Genay
Origins of the Modem Exchange Clearinghouse: A History ofEarly
Clearing and Settlement Methods atFutures Exchanges
James T. Moser
The Effect of Bank-Held Derivatives on Credit Accessibility
Elijah Brewer ///, Bernadette A . Minton and James T. Moser
Small Business Investment Companies:
Financial Characteristics and Investments
Elijah Brewer III and Hesna Genay
WP-93-11
WP-93-18
WP-93-19
WP-94-3
WP-94-5
WP-94-10
MACROECONOMIC ISSUES
An Examination ofChange inEnergy Dependence and Efficiency
in the Six Largest Energy Using Countries-1970-1988
JackL. Hervey
WP-92-2
Does theFederal Reserve Affect Asset Prices?
Vefa Tarhan
WP-92-3
Investment and Market Imperfections in theU.S. Manufacturing Sector
Paula R. Worthington
W P-92-4
3
W orking paper series continued
Business Cycle Durations and Postwar Stabilizationof theU.S. Economy
Mark W. Watson
A Procedure forPredicting Recessions with Leading Indicators: Econometric Issues
and Recent Performance
James H. Stock and Mark W. Watson
Production and Inventory Control atthe General Motors Corporation
During the 1920s and 1930s
Anil K. Kashyap and David W. Wilcox
Liquidity Effects,Monetary Policy and the Business Cycle
Lawrence J. Christiano and Martin Eichenbaum
Monetary Policy and External Finance: Interpreting the
Behavior ofFinancial Flows and InterestRate Spreads
Kenneth N. Kuttner
WP-92-6
WP-92-7
WP-92-10
WP-92-15
WP-92-17
Testing Long Run Neutrality
Robert G. King and Mark W. Watson
WP-92-18
A Policymaker’sGuide to Indicators ofEconomic Activity
Charles Evans, Steven Strongin, and Francesca Eugeni
WP-92-19
Barriers toTrade and Union Wage Dynamics
Ellen R. Rissman
WP-92-22
Wage Growth and Sectoral Shifts: PhillipsCurve Redux
Ellen R. Rissman
WP-92-23
Excess Volatility and The Smoothing of InterestRates:
An Application Using Money Announcements
Steven Strongin
Market Structure, Technology and the Cyclicality ofOutput
Bruce Petersen and Steven Strongin
The Identificationof Monetary Policy Disturbances:
Explaining theLiquidity Puzzle
Steven Strongin
WP-92-25
WP-92-26
WP-92-27
4
W orking paper series continued
Earnings Losses and Displaced Workers
Louis S. Jacobson, Robert J. LaLonde, and Daniel G. Sullivan
Some Empirical Evidence of the Effects on Monetary Policy
Shocks on Exchange Rates
Martin Eichenbaum and Charles Evans
WP-92-28
WP-92-32
An Unobserved-Components Model of
Constant-Inflation Potential Output
Kenneth N. Kuttner
WP-93-2
Investment, Cash Flow, and Sunk Costs
Paula R. Worthington
W P-93-4
Lessons from theJapanese Main Bank System
forFinancial System Reform inPoland
Takeo Hoshi, Anil Kashyap, and Gary Loveman
Credit Conditions and the Cyclical Behavior ofInventories
Anil K . Kashyap, Owen A . Lament and Jeremy C. Stein
Labor Productivity During the Great Depression
Michael D. Bordo and Charles L. Evans
Monetary Policy Shocks and Productivity Measures
in the G-7 Countries
Charles L. Evans and Fernando Santos
WP-93-6
WP-93-7
WP-93-10
WP-93-12
Consumer Confidence and Economic Fluctuations
John G. Matsusaka and Argia M. Sbordone
WP-93-13
Vector Autoregressions and Cointegration
Mark W. Watson
WP-93-14
Testing for Cointegration When Some of the
Cointegrating Vectors Are Known
Michael T. K. Horvath and Mark W. Watson
Technical Change, Diffusion, and Productivity
Jeffrey R. Campbell
WP-93-15
WP-93-16
5
W orking p aper series continued
Economic Activity and the Short-Term Credit Markets:
An Analysis ofPrices and Quantities
Benjamin M. Friedman and Kenneth N. Kuttner
Cyclical Productivity in a Model ofLabor Hoarding
Argia M. Sbordone
W P-93-17
WP-93-20
The Effects of Monetary Policy Shocks: Evidence from the Flow ofFunds
Lawrence J. Christiano, Martin Eichenbaum and Charles Evans
WP-94-2
Algorithms forSolving Dynamic Models with Occasionally Binding Constraints
Lawrence J. Christiano and Jonas D M . Fisher
WP-94-6
Identification and the Effects of Monetary Policy Shocks
Lawrence J . Christiano, Martin Eichenbaum and Charles L. Evans
WP-94-7
Small Sample Bias inG M M Estimation of Covariance Structures
Joseph G. Altonji and Lewis M. Segal
WP-94-8
Interpreting theProcyclical Productivity ofManufacturing Sectors:
External Effects ofLabor Hoarding?
Argia M . Sbordone
Small Sample Properties of Generalized
Method of Moments Based Wald Tests
Craig Burnside and Martin Eichenbaum
WP-94-9
WP-94-12
6
@FedFRASER