Panel Data Models with Nonadditive Unobserved Heterogeneity:
Estimation and Inference
by
Joonhwan Lee
B.A. Economics
Seoul National University, 2001
SUBMITTED TO THE DEPARTMENT OF ECONOMICS IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MASSAHU
I
OF TECHNOLOGY
FEBRUARY 2014
MAR 2 4 2014
@Joonhwan Lee. All rights reserved.
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Department of Economics
January 10, 2014
Certified by:
N
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Victor Chernozhukov
Professor of Economics
Thesis Supervisor
Accepted by_
. N
Michael Greenstone
3M Professor of Environmental Economics
Chairman, Departmental Committee on Graduate Studies
PANEL DATA MODELS WITH NONADDITIVE UNOBSERVED
HETEROGENEITY: ESTIMATION AND INFERENCE
IVAN FERNANDEZ-VAL§
JOONHWAH LEEt
ABSTRACT. This paper considers fixed effects estimation and inference in linear and nonlin-
ear panel data models with random coefficients and endogenous regressors. The quantities
of interest - means, variances, and other moments of the random coefficients - are estimated
by cross sectional sample moments of GMM estimators applied separately to the time series of each individual. To deal with the incidental parameter problem introduced by the
noise of the within-individual estimators in short panels, we develop bias corrections. These
corrections are based on higher-order asymptotic expansions of the GMM estimators and
produce improved point and interval estimates in moderately long panels. Under asymptotic
sequences where the cross sectional and time series dimensions of the panel pass to infinity
at the same rate, the uncorrected estimator has an asymptotic bias of the same order as
the asymptotic variance. The bias corrections remove the bias without increasing variance.
An empirical example on cigarette demand based on Becker, Grossman and Murphy (1994)
shows significant heterogeneity in the price effect across U.S. states.
JEL Classification: C23; J31; J51.
Keywords: Correlated Random Coefficient Model; Panel Data; Instrumental Variables;
GMM; Fixed Effects; Bias; Incidental Parameter Problem; Cigarette demand.
Date: This version of October 15, 2013. First version of April 2004. This paper is based in part on the
second chapter of Ferndndez-Val (2005)'s MIT PhD dissertation. We wish to thank Josh Angrist, Victor
Chernozhukov and Whitney Newey for encouragement and advice. For suggestions and comments, we are
grateful to Manuel Arellano, Mingli Chen, the editor Elie Tamer, three anonymous referees and the participants to the Brown and Harvard-MIT Econometrics seminar. We thank Aju Fenn for providing us the data
for the empirical example. All remaining errors are ours. FernAndez-Val gratefully acknowledges financial
support from Fundaci6n Caja Madrid, Fundaci6n Ram6n Areces, and the National Science Foundation.
Please send comments or suggestions to ivanffbu.edu (Ivan) or jhlee82@mit.edu (Joonhwan).
§ Boston University, Department of Economics, 270 Bay State Road,Boston, MA 02215, ivanf@bu.edu.
t Department of Economics, MIT, 50 Memorial Drive, Cambridge, MA 02142, jhlee82@mit.edu.
1
2
1. INTRODUCTION
This paper considers estimation and inference in linear and nonlinear panel data models
with random coefficients and endogenous regressors. The quantities of interest are means,
variances, and other moments of the distribution of the random coefficients. In a state level
panel model of rational addiction, for example, we might be interested in the mean and variance of the distribution of the price effect on cigarette consumption across states, controlling
for endogenous past and future consumptions. These models pose important challenges in
estimation and inference if the relation between the regressors and random coefficients is
left unrestricted. Fixed effects methods based on GMM estimators applied separately to
the time series of each individual can be severely biased due to the incidental parameter
problem. The source of the bias is the finite-sample bias of GMM if some of the regressors
is endogenous or the model is nonlinear in parameters, or nonlinearities if the parameter of
interest is the variance or other high order moment of the random coefficients. Neglecting the
heterogeneity and imposing fixed coefficients does not solve the problem, because the resulting estimators are generally inconsistent for the mean of the random coefficients (Yitzhaki,
1996, and Angrist, Graddy and Imbens, 2000).1 Moreover, imposing fixed coefficients does
not allow us to estimate other moments of the distribution of the random coefficients.
We introduce a class of bias-corrected panel fixed effects GMM estimators. Thus, instead
of imposing fixed coefficients, we estimate different coefficients for each individual using the
time series observations and correct for the resulting incidental parameter bias. For linear
models, in addition to the bias correction, these estimators differ from the standard fixed
effects estimators in that both the intercept and the slopes are different for each individual.
Moreover, unlike for the classical random coefficient estimators, they do not rely on any
restriction in the relationship between the regressors and random coefficients; see Hsiao and
Pesaran (2004) for a recent survey on random coefficient models. This flexibility allows us
to account for Roy (1951) type selection where the regressors are decision variables with
levels determined by their returns. Linear models with Roy selection are commonly referred
to as correlated random coefficient models in the panel data literature. In the presence of
endogenous regressors, treating the random coefficients as fixed effects is also convenient to
overcome the identification problems in these models pointed out by Kelejian (1974).
The most general models we consider are semiparametric in the sense that the distribution of the random coefficients is unspecified and the parameters are identified from moment
conditions. These conditions can be nonlinear functions in parameters and variables, accommodating both linear and nonlinear random coefficient models, and allowing for the presence
of time varying endogeneity in the regressors not captured by the random coefficients. We
'Heckman and Vytlacil (2000) and Angrist (2004) find sufficient conditions for fixed coefficient OLS and IV
estimators to be consistent for the average coefficient.
3
use the moment conditions to estimate the model parameters and other quantities of interest
via GMM methods applied separately to the time series of each individual. The resulting
estimates can be severely biased in short panels due to the incidental parameters problem,
which in this case is a consequence of the finite-sample bias of GMM (Newey and Smith,
2004) and/or the nonlinearity of the quantities of interest in the random coefficients. We
develop analytical corrections to reduce the bias.
To derive the bias corrections, we use higher-order expansions of the GMM estimators,
extending the analysis in Newey and Smith (2004) for cross sectional estimators to panel data
estimators with fixed effects and serial dependence. If n and T denote the cross sectional
and time series dimensions of the panel, the corrections remove the leading term of the bias
of order O(T 1 ), and center the asymptotic distribution at the true parameter value under
sequences where n and T grow at the same rate. This approach is aimed to perform well in
econometric applications that use moderately long panels, where the most important part
of the bias is captured by the first term of the expansion. Other previous studies that used
a similar approach for the analysis of linear and nonlinear fixed effects estimators in panel
data include, among others, Kiviet (1995), Phillips and Moon (1999), Alvarez and Arellano
(2003), Hahn and Kuersteiner (2002), Lancaster (2002), Woutersen (2002), Hahn and Newey
(2004), and Hahn and Kuersteiner (2011). See Arellano and Hahn (2007) for a survey of this
literature and additional references.
A first distinctive feature of our corrections is that they can be used in overidentified models where the number of moment restrictions is greater than the dimension of the parameter
vector. This situation is common in economic applications such as rational expectation models. Overidentification complicates the analysis by introducing an initial stage for estimating
optimal weighting matrices to combine the moment conditions, and precludes the use of
the existing methods. For example, Hahn and Newey's (2004) and Hahn and Kuersteiner's
(2011) general bias reduction methods for nonlinear panel data models do not cover optimal
two-step GMM estimators. A second distinctive feature is that our results are specifically
developed for models with multidimensional nonadditive heterogeneity, whereas the previous studies focused mostly on models with additive heterogeneity captured by an scalar
individual effect. Exceptions include Arellano and Hahn (2006) and Bester and Hansen
(2008), which also considered multidimensional heterogeneity, but they focus on parametric
likelihood-based panel models with exogenous regressors. Bai (2009) analyzed related linear
panel models with exogenous regressors and multidimensional interactive individual effects.
Bai's nonadditive heterogeneity allows for interaction between individual effects and unobserved factors, whereas the nonadditive heterogeneity that we consider allows for interaction
4
between individual effects and observed regressors. A third distinctive feature of our analysis is the focus on moments of the distribution of the individual effects as one of the main
quantities of interest.
We illustrate the applicability of our methods with empirical and numerical examples
based on the cigarette demand application of Becker, Grossman and Murphy (1994). Here,
we estimate a linear rational addictive demand model with state-specific coefficients for price
and common parameters for the other regressors using a panel data set of U.S. states. We find
that standard estimators that do not account for non-additive heterogeneity by imposing a
constant coefficient for price can have important biases for the common parameters, mean of
the price coefficient and demand elasticities. The analytical bias corrections are effective in
removing the bias of the estimates of the mean and standard deviation of the price coefficient.
Figure 1 gives a preview of the empirical results. It plots a normal approximation to the
distribution of the price effect based on uncorrected and bias corrected estimates of the
mean and standard deviation of the distribution of the price coefficient. The figure shows
that there is important heterogeneity in the price effect across states. The bias correction
reduces by more than 15% the absolute value of the estimate of the mean effect and by 30%
the estimate of the standard deviation.
Some of the results for the linear model are related to the recent literature on correlated
random coefficient panel models with fixed T. Graham and Powell (2008) gave identification
and estimation results for average effects. Arellano and Bonhomme (2010) studied identification of the distributional characteristics of the random coefficients in exogenous linear
models. None of these papers considered the case where some of the regressors have time
varying endogeneity not captured by the random coefficients or the model is nonlinear. For
nonlinear models, Chernozhukov, FernAndez-Val, Hahn and Newey (2010) considered identification and estimation of average and quantile treatment effects. Their nonparametric and
semiparametric bounds do not require large-T, but they do not cover models with continuous
regressors and time varying endogeneity.
The rest of the paper is organized as follows. Section 2 illustrates the type of models
considered and discusses the nature of the bias in two examples. Section 3 introduces the
general model and fixed effects GMM estimators. Section 4 derives the asymptotic properties
of the estimators. The bias corrections and their asymptotic properties are given in Section
5. Section 6 describes the empirical and numerical examples. Section 7 concludes with a
summary of the main results. Additional numerical examples, proofs and other technical
details are given in the online supplementary appendix Ferndndez-Val and Lee (2012).
5
2. MOTIVATING EXAMPLES
In this section we describe in detail two simple examples to illustrate the nature of the bias
problem. The first example is a linear correlated random coefficient model with endogenous
regressors. We show that averaging IV estimators applied separately to the time series of each
individual is biased for the mean of the random coefficients because of the finite-sample bias
of IV. The second example considers estimation of the variance of the individual coefficients
in a simple setting without endogeneity. Here the sample variance of the estimators of the
individual coefficients is biased because of the non-linearity of the variance operator in the
individual coefficients. The discussion in this section is heuristic leaving to Section 4 the
specification of precise regularity conditions for the validity of the asymptotic expansions
used.
2.1. Correlated random coefficient model with endogenous regressors. Consider
the following panel model:
(2.1)
Yit = a0j + 0ixit + cjt, (i = 1, ..., n;
=
1, ...,T);
where yit is a response variable, xit is an observable regressor, eit is an unobservable error
term, and i and t usually index individual and time period, respectively. 2 This is a linear random coefficient model where the effect of the regressor is heterogenous across individuals, but
no restriction is imposed on the distribution of the individual effect vector ac := (aoi, aii)'.
The regressor can be correlated with the error term and a valid instrument (1, zit) is available
for (1, xit), that is E[cit I ai] = 0, E[ziteit I a] = 0 and Cov[zitxit I ai] f 0. An important
example of this model is the panel version of the treatment-effect model (Wooldridge, 2002
Chapter 10.2.3, and Angrist and Hahn, 2004). Here, the objective is to evaluate the effect
of a treatment (D) on an outcome variable (Y). The average causal effect for each level
of treatment is defined as the difference between the potential outcome that the individual
would obtain with and without the treatment, Yd - Yo. If individuals can choose the level
of treatment, potential outcomes and levels of treatment are generally correlated. An instrumental variable Z can be used to identify the causal effect. If potential outcomes are
represented as the sum of permanent individual components and transitory individual-time
specific shocks, that is Yt = Yi + cjit for j E {0, 1}, then we can write this model as a
special case of (2.1) with yit = (1 - Dit)Yot + DitY1it, ao = Yo, ali = Y1i - Yoi, xit = Dit,
zit = Zit, and cit = (1 - Dit)oit + Dirtcit.
Suppose that we are ultimately interested in a, := Eaii], the mean of the random slope
coefficient. We could neglect the heterogeneity and run fixed effects OLS and IV regressions
2
More generally, i denotes a group index and t indexes the observations within the group. Examples of
groups include individuals, states, households, schools, or twins.
6
in
Yit = aOi + aixit + Uit,
where uit
=
xit(aii - ai) + cit in terms of the model (2.1). In this case, OLS and IV estimate
weighted means of the random coefficients in the population; see, for example, Yitzhaki
(1996) and Angrist and Krueger (1999) for OLS, and Angrist, Graddy and Imbens (2000)
for IV. OLS puts more weight on individuals with higher variances of the regressor because
they give more information about the slope; whereas IV weighs individuals in proportion to
the variance of the first stage fitted values because these variances reflect the amount of information that the individuals convey about the part of the slope affected by the instrument.
These weighted means are generally different from the mean effect because the weights can
be correlated with the individual effects.
To see how these implicit OLS and IV weighting schemes affect the estimand of the fixedcoefficient estimators, assume for simplicity that the relationship between xit and zit is linear,
that is xit = 7roi + 7rizt + vit, (Eit, vit) is normal conditional on (zit, caj, 7i), zit is independent
of (ai, 7ri), and (ai, -ri) is normal, for 7ri := (7roi, iris)'. Then, the probability limits of the OLS
and IV estimators are3
aOLS
ZIV
=
a +Cov[t,vit]
=
a, + Cov[aii, iris]/E[iri].
+2E[7ri]Var[zit]Cov[ai,
ri]}/Var[xit],
These expressions show that the OLS estimand differs from the average coefficient in presence
of endogeneity, i.e. non zero correlation between the individual-time specific error terms, or
whenever the random coefficients are correlated; while the IV estimand differs from the
average coefficient only in the latter case. 4
In the treatment-effects model, there exists
correlation between the error terms in presence of endogeneity bias and correlation between
the individual effects arises under Roy-type selection, i.e., when individuals who experience
a higher permanent effect of the treatment are relatively more prone to accept the offer
of treatment.
Wooldridge (2005) and Murtazashvile and Wooldridge (2005) give sufficient
conditions for consistency of standard OLS and IV fixed effects estimators. These conditions
amount to Cov[Et , vit] = 0 and Cov[xit, a1i|acio] = 0.
Our proposal is to estimate the mean coefficient from separate time series estimators
for each individual. This strategy consists of running OLS or IV for each individual, and
then estimating the population moment of interest by the corresponding sample moment
3
The limit of the IV estimator is obtained from a first stage equation that imposes also fixed coefficients,
that is Xit = 7r0 i + 7rizit + wit, where wit = Zit(7rij - 7ri) + vit. When the first stage equation is different for
each individual, the limit of the IV estimator is
2
a = ai + 2E[7ri]Cov[ai, 7ris]/{E[risJ
+ Var [ri]}.
See Theorems 2 and 3 in Angrist and Imbens (1995) for a related discussion.
4
This feature of the IV estimator is also pointed out in Angrist, Graddy and Imbens (1999), p. 507.
7
of the individual estimators. For example, the mean of the random slope coefficient in the
population is estimated by the sample average of the OLS or IV slopes. These sample
moments converge to the population moments of interest as number of individuals n and
time periods T grow. However, since a different coefficient is estimated for each individual,
the asymptotic distribution of the sample moments can have asymptotic bias due to the
incidental parameter problem (Neyman and Scott, 1948).
To illustrate the nature of this bias, consider the estimator of the mean coefficient a,
constructed from individual time series IV estimators. In this case the incidental parameter
problem is caused by the finite-sample bias of IV. This can be explained using some expansions. Thus, assuming independence across t, standard higher-order asymptotics gives (e.g.
Rilstone et. al., 1996), as T -+ oc
ali) =
VIT(-Iv
where
it
=
TT
+
#+o(T1/2),
E[citzit I ai,7ri]-izitct is the influence function of IV, Oi = -E[zit.it I a,, r,]-2
E[zige,,it I ai, Ti] is the higher-order bias of IV (see, e.g., Nagar, 1959, and Buse, 1992), and
the variables with tilde are in deviation from their individual means, e.g., zit = zit - E[zit I
ai, 7i]. In the previous expression the first order asymptotic distribution of the individual
estimator is centered at the truth since \/T(-{f - aij) ->d N(0, ou) as T -+ oc, where
o2 =
E[itzit
,
] 2[2
2 I
r] 1,
a, 7rit.
Let a' = n- 1 E_1&{, the sample average of the IV estimators. The asymptotic distribution of a' is not centered around al in short panels or more precisely under asymptotic
sequences where T/y -+ 0. To see this, consider the expansion for ai
v (a1
- al) =
(i
- a).
-ai)+(
i=1
=
The first term is the standard influence function for a sample mean of known elements. The
second term comes from the estimation of the individual elements inside the sample mean.
Assuming independence across i and combining the previous expansions,
nn
i(&i-cai)= }
(ai - a)+
V/n-2
i=1
=OP (1)
T
Tn()it_+ T n Y.1+O
i=1
v/_/nT .=1 1t=1
=OP(l1//T)=O//T
This expression shows that the bias term dominates the asymptotic distribution of &1 in
short panels under sequences where T/# -+ 0. Averaging reduces the order of the variance
of a'G, without affecting the order of its bias. In this case the estimation of the random
coefficients has no first order effect in the asymptotic variance of 81 because the second term
is of smaller order than the first term.
8
A potential drawback of the individual by individual time series estimation is that it might
more be sensitive to weak identification problems than fixed coefficient pooled estimation. 5
In the random coefficient model, for example, we require that Ezitzit | aj, fri] = 7rig = 0 with
probability one, i.e., for all the individuals, whereas fixed coefficient IV only requires that this
condition holds on average, i.e., E[7ri] 4 0. The individual estimators are therefore more
sensitive than traditional pooled estimators to weak instruments problems. On the other
hand, individual by individual estimation relaxes the exogeneity condition by conditioning on
additive and non-additive time invariant heterogeneity, i.e, E[itcit I ai, 1ri] = 0. Traditional
fixed effects estimators only condition on additive time invariant heterogeneity. A formal
treatment of these identification issues is beyond the scope of this paper.
2.2. Variance of individual coefficients. Consider the panel model:
Yit
=
ci + Et, Eit | I
~
(0, of), ci ~ (a, oQ), (t
=
1,
... ,
T; i = 1, ... , n);
where yit is an outcome variable of interest, which can be decomposed in an individual effect
ao with mean a and variance a , and an error term cit with zero mean and variance o
conditional on ac. The parameter of interest is o = Var[ai] and its fixed effects estimator
is
n
i2
(n-)-1
-)2
i=1
where &i= T
t=1 yit
and & =
-1 Z= 1 ai.
Let , = (a, - a)2 _ o and Pei =
- o . Assuming independence across i and t, a
standard asymptotic expansion gives, as n, T -+ oc,
S
V/n .
T
z=1yT
=0P (1)
'/T i=1 t=1
=0P(1/s/T)
+n
T
=(l-T
The first term corresponds to the influence function of the sample variance if the ai's were
known. The second term comes from the estimation of the ai's. The third term is a bias
term that comes from the nonlinearity of the variance in O'i. The bias term dominates the
expansion in short panels under sequences where T/w/n -+ 0. As in the previous example,
the estimation of the ai's has no first order affect in the asymptotic variance since the second
term is of smaller order than the first term.
5
We thank a referee for pointing out this issue.
9
3. THE MODEL AND ESTIMATORS
We consider a general model with a finite number of moment conditions d.. To describe it,
let the data be denoted by zit (i = 1, ... , n; t = 1, . . ., T). We assume that zit is independent
over i and stationary and strongly mixing over t. Also, let 0 be a do-vector of common
parameters, {ai
1 < i < n} be a sequence of de-vectors with the realizations of the
individual effects, and g(z; 6, aj) be an dg-vector of functions, where dg ;> do + dQ.' The
model has true parameters 60 and {ajo: 1 < i < n}, satisfying the moment conditions
E [g (zit; Oo, ajo) ] =0, (t = 1, ... , T;-i = I1, ...
,I
n),
where E[-] denotes conditional expectation with respect to the distribution of zit conditional
on the individual effects.
Let E[-] denote the expectation taken with respect to the distribution of the individual
effects. In the previous model, the ultimate quantities of interest are smooth functions of
parameters and observations, which in some cases could be the parameters themselves,
= EE[(i(zit; 6o, ajo)],
if EEI(i(zit; 0o, aeo)| < oo, or moments or other smooth functions of the individual effects
[t = EP[p(ajo)],
if EIp(ajo)| < oo. In the correlated random coefficient example, g(zit; 60, ajo) = Zit(Yit - ceo aioxit), 6 = 0, do = 0, da = 2, and p(ajo) = a 1 0 . In the variance of the random coefficients
example, g(zit; 60, aio) = (yit - aojo), 6 = 0, do = 0, da = 1 , and p(ajo) = (
- E[aigo])2
Some more notation, which will be extensively used in the definition of the estimators and
in the analysis of their asymptotic properties, is the following
Qjj(0, ac)
:=
E[g(zit; 0, aj)g(z,tj; 6, ce)'], j C {0, 1, 2,
Go,(6, a)
:=
E[Go(zit;0, a )
G,(0, aij)
=
...}
E [ag(zit; 0,a)/ 0'],
E[G,(zit; 0, ai)] = E [g(zit; 6, ac)/ i],
where superscript ' denotes transpose and higher-order derivatives will be denoted by adding
subscripts. Here Qjj is the covariance matrix between the moment conditions for individual
i at times t and t - j, and Go, and Go, are time series average derivatives of these conditions.
6
We impose that some of the parameters are common for all the individuals to help preserve degrees of
freedom in estimation of short panels with many regressors. An order condition for this model is that the
number of individual specific parameters d, has to be less than the time dimension T.
10
Analogously, for sample moments
T
Uj
(O, ac)
T
1
g(zit; 6, cei)g(zi,&tj; 6, Ci)',
j E {0, 1, ..., T
-1,
t=j+1
T
T
Go (0, ac)
T
1-
Go(zit; 6, ac)
=
T- [
T
T
T- 1
Ga.(O, ca)
ag(zit; 6, ac)/D6',
t=1
t=1
GQ(zit; 6, ac)
=
&g(zit; 6, ai)/c9.
T- l
t=1
t=1
In the sequel, the arguments of the expressions will be omitted when the functions are
evaluated at the true parameter values (60, a'o)', e.g., g(zit) means g(zit; Oo, aco).
In cross-section and time series models, parameters defined from moment conditions are
usually estimated using the two-step GMM estimator of Hansen (1982).
To describe how
to adapt this method to panel models with fixed effects, let $j(6,ai) := T - _1 g(zit; 6, ai),
and let (6', {'} 1)' be some preliminary one-step FE-GMM estimator, given by (6', {&'}U3' =
arginfI(ot, )/rE
1 En
space, and {Wi
41
cs)'
11(6,
-(6, aj), where T C Rd+d denotes the parameter
1 < i < n} is a sequence of positive definite symmetric dg x dg weighting
matrices. The two-step FE-GMM estimator is the solution to the following program
n
(6', {a'}7_)'
=
arg
3
inf
(6 , ac)'Q(0,
di)-
1
(6, ai),
where Qj(6, di) is an estimator of the optimal weighting matrix for individual i
cc
Qi=Q 0 i + Z(Q.ii + Qvi).
j=1
To facilitate the asymptotic analysis, in the estimation of the optimal weighting matrix
we assume that g(zit; 6o, ajo) is a martingale difference sequence with respect to the sigma
algebra o-(a, zi,t_1, Zi,t-2,
... ),
so that Qj = Q0 j and Qj(O, di)
holds in rational expectation models.
=
Qoi(6, di). This assumption
We do not impose this assumption to derive the
limiting distribution of the one-step FE-GMM estimator.
For the subsequent analysis of the asymptotic properties of the estimator, it is convenient
to consider the concentrated or profile problem. This problem is a two-step procedure. In
the first step the program is solved for the individual effects, given the value of the common
parameter 6. The First Order Conditions (FOC) for this stage, reparametrized conveniently
as in Newey and Smith (2004), are the following
ti(0, p6)) -
( a,(0, ^(6))'Ai(6)
C
( -
~ a)-
gi( , ai (0)) + Qj (0, as) Ai (0))
= 0, (i=1,...,n),
11
where Ai is a d4-vector of individual Lagrange multipliers for the moment conditions, and
yi:= (a/, A')' is an extended (dQ + dg)-vector of individual effects. Then, the solutions to
the previous equations are plugged into the original problem, leading to the following first
order conditions for 0, 1(0)
F(O) = n
=
0, where
S
(()) = - -1
i=1
5 G(0,(0, -
i(())'A3(0),
=
is the profile score function for 0.7
Fixed effects estimators of smooth functions of parameters and observations are constructed using the plug-in principle, i.e.
(= (0) where
n
(0)
=
(nT)
T
? S ((zit; 0, &^(0)).
i=1 t=1
Similarly, moments of the individual effects are estimated by -= -(0),
^(0) = n-
where
/(a (0)).
i=1
4. ASYMPTOTIC THEORY FOR FE-GMM ESTIMATORS
In this section we analyze the properties of one-step and two-step FE-GMM estimators in
large samples. We show consistency and derive the asymptotic distributions for estimators
of individual effects, common parameters and other quantities of interest under sequences
where both n and T pass to infinity with the sample size. We establish results separately
for one-step and two-step estimators because the former are derived under less restrictive
assumptions.
We make the following assumptions to show uniform consistency of the FE-GMM one-step
estimator:
Condition 1 (Sampling and asymptotics). (i) For each i, conditional on aj, zi := {zit : 1 < t < T}
is a stationary mixing sequence of random vectors with strong mixing coefficients ai(l)
sup supAEAP,DE
IP(A n D) - P(A)P(D)I, where A' = o-(ai, zit, zi,t_1,
such that supi Iai(l)i1
where 0 <
and'D = -(az, zit, zi,t+,,
Cal for some 0 < a < 1 and some C > 0; (ii) {(zi, aei) :1 < i < n}
are independent and identically distributed across i; (iii) n, T -+
2
=
< oo; and (iv) dim [g(.; 0, ai)] = dg < oo.
7
1n the original parametrization, the FOC can be written as
n-1 Z
i (w, ai())'i(6, &i)-(,
where the superscript - denotes a generalized inverse.
ai(O)) = 0,
oo such that n/T -+ 01
12
For a matrix or vector A, let JAl denote the Euclidean norm, that is JAl 2 = trace[AA'].
Condition 2 (Regularity and identification). (i) The vector of moment functions g(.; 6, a) =
(gi (-; 6, a) , ... , gd, (.; 0, a))' is continuous in (0, a) E T; (ii) the parameter space T is a
compact, convex subset of Rdo+d-; (iii) dim (0, a) = do + dc
function M (zit) such that Ik (zit; 0, aj)|I
dg; (iv) there exists a
M (zit), |ag9k (zit; 0, ai) /a (6, ai)I 5 M (zit), for
k = 1, ... , dg, and supi E [M (zit)4+] < oc for some 6 > 0; and (v) there exists a deterministic sequence of symmetric finite positive definite matrices {W : 1 < i < n} such that
sup 1<i<n
W - W -+p 0, and, for each r1 > 0
inf QW (A, aio) inf [Q1~ (0,~)
Q" (6, a) > 0,
sup
{(0,a):I(O,a)-((0,cio)1>})
where
(0, aj)' W i'gi (0, aj) , gi (0, aj) :=E [- (0, aj)]
QW (0, aj) :-gi
Conditions 1(i)-(ii) impose cross sectional independence, but allow for weak time series
dependence as in Hahn and Kuersteiner (2011). Conditions 1(iii)-(iv) describe the asymptotic
sequences that we consider where T and n grow at the same rate with the sample size, whereas
the number of moments d9 is fixed. Condition 2 adapts standard assumptions of the GMM
literature to guarantee the identification of the parameters based on time series variation for
all the individuals, see Newey and McFadden (1994). The dominance and moment conditions
in 2(iv) are used to establish uniform consistency of the estimators of the individual effects.
Theorem 1 (Uniform consistency of one-step estimators). Suppose that Conditions 1 and
2 hold. Then, for any q > 0
) = o(T1),
Pr (O-o
Z=
n Q (0, cei) and QW (0, ai) :=
1).r}&
where 0 = arg max{(O,
-g
(6, at)' W-Ig (6, al).
Also, for any q > 0
Pr
SUP jdj - aeoj > 7=
o (T- 1 ) and Pr (sup
where di = arg max, QW"(6, a) and j = -W
Let Ej := (G' W
= 0 (T-1)
\l<i<n/
(I<i<n
G4)1, H
G
:
Ji(6, i).
W4,
EwG',W
P
:= W-1 -W
GcHr, Jj:
G'. P Gi and .J := E[J~f]. We use the following additional assumptions to derive the
limiting distribution of the one-step estimator:
Condition 3 (Regularity). (i) For each i, (0o, ajo) E int [T]; and (ii) Jr
definite, and {G', W71 G,
is finite positive
: 1 < i < n} is a sequence of finite positive definite matrices,
where {W : 1 < i < n} is the sequence of matrices of Condition 2(v).
13
Condition 4 (Smoothness). (i) There exists a function M (zit) such that, for k = 1,..., dq,
d,+d2 gA
and supi E [M
(zit; 0, ai) /ad1&
2a
< M (zit)
0 < di + d2 < 1,..
,
,
5,
< 00, for some 6 > 0 and 0 < v < 1/10; and (ii)
(Zit)5(dO+da+6)/(1-10V)+6]
I 1 (t zit) /T + R W/T, where maxi|RWI = op(T1
there exists j (zit) such that Wi = W+ E
E[ j(zjt)] = 0, and supi E[Igi(zit)120/(1-10o)+5] < oo, for some 6 > 0 and 0 < v < 1/10.
Condition 3 is the panel data analog to the standard asymptotic normality condition for
GMM with cross sectional data, see Newey and McFadden (1994). Condition 4 is similar to
Condition 4 in Hahn and Kuersteiner (2011), and guarantees the existence of higher order
expansions for the GMM estimators and the uniform convergence of their remainder terms.
Let Gea := (G'c, ,,.. ., G'a)', where Ga,,j = E [G,(zit)/Iaajj],and Go,:
(G', 1 , . . . G'jq)',
where Goai, = E[aGO,(zjt)/8aci,]. The symbol 9 denotes kronecker product of matrices, Id
a d, x dc, identity matrix, ej a unitary d.-vector with 1 in row j, and PgV the j-th column
of Pr. Recall that the extended individual effect is y4 = (a', A')'.
Lemma 1 (Asymptotic expansion for one-step estimators of individual effects). Under Conditions 1, 2, 3, and 4,
'/T_( 'o
(4.1)
- yjo) = i
+ T-1/2QW + T-'RK,
where Yio := ~(00),
\
(Hw
"W
-
T
T 1/2 E
_
n-1/2 I
sup 1 <i
n R2
Vw
N(0,
Bw,1
'I
En
n1
QW
4
g[BW], B
= Bw, + BwG + Bw,1s,
op(V J), for
g'
((P1
,HHZ
Br)=
BwG
Bw,G
Ai
tiBws
~~Bw,s
(,i
d
7
=
W
VW]),
g (Zit)
t=1
cii
Bs
Ai
P
) (H P ),
(
(
w
_
=j
j'
gw
as
-Hv,
EG E [Gi(zt)'Pg(z,tj)
(
-oo
aa
j=1
*~
PZ
-
Ga H0H
e)a~~k~/)
j=-oo
a
+
F(Gc (z )Hwgg(zt_.)]
Ec
00
i
'nPG~'2(
a
E [Wizit)Pzg(zi't _j)]
j=-o0
,
j=1
j=1
'mIaOiHn~~|
2,
14
Theorem 2 (Limit distribution of one-step estimators of common parameters). Under Conditions 1, 2, 3 and 4,
nT(O
-
Oo)
4
-(Jv)-'N (KBW, V W)
where
J4
Go] ,Vw = E [G' P
[GI P S~~~L
=
[BB + BWC + BlW'V
=
,B
PGo]
a
Zi
si
±i
s0
Jj
and
Bw + BWG + Bw,1)
BB = -G'.
B''V =
-_
'c
The expressions for Bw,1, BG
,
oo__
E,
G'1
Go(Id 0 e )Hi'Q Ps,,i2.
PWQ.H1/2 -
G'
=
Bc
and Bw'1s are given in Lemma 1.
The source of the bias is the non-zero expectation of the profile score of 6 at the true
parameter value, due to the substitution of the unobserved individual effects by sample estimators.
These estimators converge to their true parameter value at a rate yI, which
the rate of convergence of the estimator of the common parameter.
Intuitively, the rate for jo is v'7 ' because only the T observations for individual i convey
is slower than \iT,
In nonlinear and dynamic models, the slow convergence of the es-
information about 7-y.
timator of the individual effect introduces bias in the estimators of the rest of parameters.
The expression of this bias can be explained with an expansion of the score around the true
value of the individual effects 8
E [s"(60o, ~5'o)]
=
E [-"'] + E [
]' E [Yio - -yio] + E [(s[ - E [sl])'(jo - yjo)]
-d, +dg
+
([doj - 'yio,j)E ['i]
E
(~yjo -
Yio)] /2 + o(T- 1 )
.j=1
-
0 + B,B/T + BW/C/T + B'V/T + o(T- 1).
This expression shows that the bias has the same three components as in the MLE case, see
Hahn and Newey (2004). The first component, BB,
comes from the higher-order bias of the
estimator of the individual effects. The second component, BCC, is a correlation term and
is present because individual effects and common parameters are estimated using the same
8
Using the notation introduced in Section 3, the score is
n
n
R/o)
=0,
=
= --
i=1
where ijo = (&'0,
do (Oo, 6io)/io,
j
i=1
'O) is the solution to
Pi, (0
io)
-i
-(G
=0.
(o, dio)'io
( i (00, dio) +Wi
o )=
15
observations. The third component, Br,v, is a variance term. The bias of the individual
effects, BWB can be further decomposed in three terms corresponding to the asymptotic
bias for a GMM estimator with the optimal score, BW"", when W is used as the weighting
function; the bias arising from estimation of Gas, BW'G; and the bias arising from not using
an optimal weighting matrix, Brw"S
We use the following condition to show the consistency of the two-step FE-GMM estimator:
Condition 5 (Smoothness, regularity, and martingale). (i) There exists a function M (zit)
such that Igk (zit; 0, ai)j
M (zit), I gk (zit; 0, a ) /a (0, aj) I
M (zit), for k = 1, ..., dg,
and supi E [Al (zit)10(do+dQ+6)/(1-10v)+6I < o, for some 6 > 0 and 0 < v < 1/10; (ii)
{Ri : 1 < i < n} is a sequence of finite positive definite matrices; and (iii) for each i,
g(zit; Oo, ao) is a martingale difference sequence with respect to o-(ai, zi,t-1, Zi,t-2, . . .).
Conditions 5(i)-(ii) are used to establish the uniform consistency of the estimators of the
individual weighting matrices. Condition 5(iii) is convenient to simplify the expressions of
the optimal weighting matrices. It holds, for example, in rational expectation models that
commonly arise in economic applications.
Theorem 3 (Uniform consistency of two-step estimators). Suppose that Conditions 1, 2, 3
and 5 hold. Then, for any q > 0
Pr (0 - 00 > ) = o (T- 1 ),
where 0 = arg max(o, ,),,U
er
Qn(0, ci) and Q (0, ac)
:=
-
(, i cv)'
Ui(6,
di)
9i (0, ai).
Also, for any rj > 0
Pr
where
&i
sup
i~
--
ao| >
= o (T- 1) and Pr (sup
= arg max, Q (0, o) and '(0,&S)
+ Qi(O, &j)A
=
Ai >7
o (T-1) ,
0.
We replace Condition 4 by the following condition to obtain the limit distribution of the
two-step estimator:
Condition 6 (Smoothness). There exists some M (zit) such that, for k = 1,..., dg
ad1+d2g
(zit; 0, ai) /laod,
1
;2
Al (zit)
0 < d1 + d 2
1,..., 5,
and supi E IM (zo )10(do+da+6)/(1-10v)+] < 0c, for some ( > 0 and 0 < v < 1/10.
Condition 6 guarantees the existence of higher order expansions for the estimators of the
weighting matrices and uniform convergence of their remainder terms. Conditions 5 and 6
are stronger versions of conditions 2(iv), 2(v) and 4. They are presented separately because
they are only needed when there is a first stage where the weighting matrices are estimated.
16
Let Z
:= (GIQjGa)
1,
Ha,
,i:=
G'.Q 1, and Pa,:= Q-
1
- Q71 G, Hat.
Lemma 2 (Asymptotic expansion for two-step estimators of individual effects). Under the
Conditions 1, 2, 3, 4, and 5,
(4.2)
/T(9
where '0
-
'yio)
+ T-1/ 2 B, + T-R
2i,
g(zit) -4
T-1/2
N(O, Vt),
-+
4 N(0, E[V]), B = Bi + B7 + BQ + Br, sup1 <i<n R 2i =
2
Qaiu
4
:= i(0),
-
n-1
=
oP (\/T ),
with, for
zj,
= 0Qaig
Vi
diag
Yi
=
P
-
B
BG
BG
H a
(B
B=
H::
1 E [G ,(z
i)'P, g (zi,tj)],
j=0
BBQ
EE[g(zjt)g(zjt)'Pasg(zj,t_j)],
BwO
B
r /2 + E [Ga, (zit)Haig(zitj)])
GaaG ,
H
B
)
a
,j(H
Hij.
Theorem 4 (Limit distribution for two-step estimators of common parameters). Under the
Conditions 1, 2, 3,
4,
5 and 6,
SnT(-- 0o) -4 -J-N (KB,, J,) ,
where J, =
[G',PatGo] , B,
=0 E [Go (zit)'Pajg(zi,tj)].
E [BB + Bq] , BB= -G'
[B', + B G+
B9 + B],
B
=
The expressions for B,, B , B" and B[ are given in Lemma
2.
Theorem 4 establishes that one iteration of the GMM procedure not only improves asymptotic efficiency by reducing the variance of the influence function, but also removes the
variance and non-optimal weighting matrices components from the bias. The higher-order
bias of the estimator of the individual effects, B , now has four components, as in Newey and
Smith (2004). These components correspond to the asymptotic bias for a GMM estimator
with the optimal score, B(; the bias arising from estimation of G,,
BG; the bias arising
from estimation of Qj, BO; and the bias arising from the choice of the preliminary first step
estimator, BT. An additional iteration of the GMM estimator removes the term BT.
17
The general procedure for deriving the asymptotic distribution of the FE-GMM estimators
consists of several expansions. First, we derive higher-order asymptotic expansions for the
estimators of the individual effects, with the common parameter fixed at its true value 00.
Next, we obtain the asymptotic distribution for the profile score of the common parameter
at 0 using the expansions of the estimators of the individual effects. Finally, we derive the
asymptotic distribution of estimator for the common parameter multiplying the asymptotic
distribution of the score by the limit profile Jacobian matrix. This procedure is detailed
in the online appendix FernAndez-Val and Lee (2012). Here we characterize the asymptotic
bias in a linear correlated random coefficient model with endogenous regressors. Motivated
by the numerical and empirical examples that follow, we consider a model where only the
variables with common parameter are endogenous and allow for the moment conditions not
to be martingale difference sequences.
Example: Correlated random coefficient model with endogenous regressors. We
consider a simplified version of the models in the empirical and numerical examples. The
notation is the same as in the theorems discussed above. The moment condition is
g(zit; 0, cai) = wit(Yit - x-Tiai where wi = (xli, w'it)' and zit = (x'i, '2i)W
mon coefficients are endogenous. Let
it =
T'2it),
, yit)'. That is, only the regressors with com-
Yit - x'taio - x't0o. To simplify the expressions
for the bias, we assume that Eit wi, ai - i.i.d.(0, o) and E[x2iti,t-j wi, ai] = E[x2its,-j],
for wi = (wil, ---, wiT)' and j E {0, ±1,.. .}. Under these conditions, the optimal weighted
matrices are proportional to E[witwi], which do not depend on 0 and azo. We can therefore
obtain the optimal GMM estimator in one step using the sample averages T
1 it
to estimate the optimal weighting matrices.
ZEt
In this model, it is straightforward to see that the estimators of the individual effects have
W,1S - 0. By linearity of the first order conditions in 0 and
no bias, that is BW = BW
ai, Bsv = 0. The only source of bias is the correlation between the estimators of 0 and ai.
After some straightforward but tedious algebra, this bias simplifies to
-I
00
B'c
= -(dg -
d,)
S
E[xsutitj]
For the limit Jacobian, we find
Jw = g { E[z 2it'
]Et[f2iz'b2i
2it ] 1 E]
}
where variables with tilde indicate residuals of population linear projections of the corresponding variable on x 1 t, for example -2it = X2it - EIX2aia]E[1xli]-lxit. The expression
18
of the bias is
'3(0)
(4.3)
(dg - do)(Jw)- 1 E
E[z2it( i,t-j -
-/
0
j=-oo
In random coefficient models the ultimate quantities of interest are often functions of
the data, model parameters and individual effects. The following corollaries characterize
the asymptotic distributions of the fixed effects estimators of these quantities. The first
corollary applies to averages of functions of the data and individual effects such as average
partial effects and average derivatives in nonlinear models, and average elasticities in linear
models with variables in levels. Section 6 gives an example of these elasticities. The second
corollary applies to averages of smooth functions of the individual effects including means,
variances and other moments of the distribution of these effects. Sections 2 and 6 give
examples of these functions. We state the results only for estimators constructed from twostep estimators of the common parameters and individual effects. Similar results apply to
estimators constructed from one-step estimators. Both corollaries follow from Lemma 2 and
Theorem 4 by the delta method.
Corollary 1 (Asymptotic distribution for fixed effects averages). Let ((z; 0, a) be a twice
continuously differentiable function in its second and third argument, such that infi Var[((zjt)] >
0, EE[((zte) 2 ] < oo, E|c(zjt)| 2 < oo, and E |EI(zit)| 2 < oc, where the subscripts on
denote partial derivatives. Then, under the conditions of Theorem 4, for some deterministic
sequence rnT -+ o such that rT = O nT),
7'nT (
where
=
B =
F
E
-
- B( T)
-4 N(0, V),
EF [((zit)],
d,
o0
-(;c,(zit)'Hcg(zitj)
+ (c, , ((
'
V=
=
zit) Z' /2 -
3
zit'J-Bs] 7i
j=1
j=0
for B,,
-
+
B' + BG + BQ + Br, and for r 2 = lim2,T4O0riT/(nT),
1
r2E [(,, (zjt)'Fas C., (zit) + (0 (zit)'J,- o (zit)] +
lim
nT-4oo
ST E
[/2
1 T
((zt)
( Tt=1
-
)()
21
.
Corollary 2 (Asymptotic distribution for smooth functions of individual effects). Let p(ai)
be a twice differentiable function such that E [p(ajo) 2 ] < oo and EIpAi(a 0 )| 2 < 0c, where the
subscripts on t denote partial derivatives. Then, under the conditions of Theorem 4
P
-4
N(rBit, V),
19
where M =P [p(aio)],
d,,
=y
p[iaaio'Baj
+
ipanii(aio)'Ea /2
j=1
for Ba, = B' + BG + B + B', and V,=
[(/_(a,0) - P)2
The convergence rate rnT in Corollary 1 depends on the function ((z; 0, cei). For example,
rrT = VnT for functions that do not depend on aj such as ((z; 0, cii) = c'0, where c is
a known do vector. In general, rnT = V/n for functions that depend on oz. In this case
r2 = 0 and the first two terms of V drop out. Corollary 2 is an important special case
of Corollary 1. We present it separately because the asymptotic bias and variance have
simplified expressions.
5. BIAS CORRECTIONS
The FE-GMM estimators of common parameters, while consistent, have bias in the asymptotic distributions under sequences where n and T grow at the same rate. These sequences
provide a good approximation to the finite sample behavior of the estimators in empirical
applications where the time dimension is moderately large. The presence of bias invalidates
any asymptotic inference because the bias is of the same order as the variance. In this section
we describe bias correction methods to adjust the asymptotic distribution of the FE-GMM
estimators of the common parameter and smooth functions of the data, model parameters
and individual effects. All the corrections considered are analytical. Alternative corrections
based on variations of Jackknife can be implemented using the approaches described in Hahn
and Newey (2004) and Dhaene and Jochmans (2010).9
We consider three analytical methods that differ in whether the bias is corrected from the
estimator or from the first order conditions, and in whether the correction is one-step or
iterated for methods that correct the bias from the estimator. All these methods reduce the
order of the asymptotic bias without increasing the asymptotic variance. They are based on
analytical estimators of the bias of the profile score B, and the profile Jacobian matrix J'.
Since these quantities include cross sectional and time series means E and E evaluated at the
true parameter values for the common parameter and individual effects, they are estimated
by the corresponding cross sectional and time series averages evaluated at the FE-GMM
estimates. Thus, for any function of the data, common parameter and individual effects
z2=
fit(0, ai), let fai() = fit(0, a(0)), fi(0) = E[fit(O)] = T- 1
1fit(0) and f (6) = E[fi(0)] =
1
n- 1 Z fi(0). Next, define Za,() = [G,,()'Q-TG,()]- , Hai(0) = Zai (0)Oa (0)'Q ,
9
Hahn, Kuersteiner and Newey (2004) show that analytical, Bootstrap, and Jackknife bias corrections methods are asymptotically equivalent up to third order for MLE. We conjecture that the same result applies to
GMM estimators, but the proof is beyond the scope of this paper.
20
and P(6) =
U1
() H,, (0). To simplify the presentation, we only give explicit formulas
for FE-GMM three-step estimators in the main text. We give the expressions for one and
two-step estimators in the Supplementary Appendix. Let
'3(0) = -Js()- Bs(0), Bs(6) = E[Bs3(6) + Bs(0)], Js(6) = E[Go,(0)'Paj(0)Go,(0)|,
where Bq(6) = -Go (6)'[B,(.() + B%(0)+
B(0) + B (0),
B
R
d,
=
-P2,(0)
Et (6)
=
Ha (6)'
Z (0)
=
Pcj(6) ET-
j=0
T-
j=0
(0) = T-1 Q
T
5
(5aas,(6)Za.()/2+
Pai(6)ST- 1
Gc(6)Ha,(6)g,tj(6),
t=j+1
j=0
j=1
BI(6)
1
1
E Gaa (O)'P'i( 6 )i,t-j(0),
t=j+1
>
it(0)4it(0)'Pa (6)i,t-j(6),
t=j+1
T=+
In the previous expressions, the spec(-j().
1
tral time series averages that involve an infinite number of terms are trimmed. The trimming
0 and /T -+ 0
parameter f is a positive bandwidth that need to be chosen such that -+ oo
and
as T -+ oo (Hahn and Kuersteiner, 2011)
The one-step correction of the estimator subtracts an estimator of the expression of the
asymptotic bias from the estimator of the common parameter. Using the expressions defined
above evaluated at 0, the bias-corrected estimator is
(5.1)
BC
=
(9T
6-
This bias correction is straightforward to implement because it only requires one optimization. The iterated correction is equivalent to solving the nonlinear equation
(5.2)
IBC
When 0 + '(0)
equation.
10
^_
=
B^(g
T
BC)/T
is invertible in 0, it is possible to obtain a closed-form solution to the previous
Otherwise, an iterative procedure is needed. The score bias-corrected estimator
is the solution to the following estimating equation
(5.3)
s(^SBC)
-
B
(SBC
This procedure, while computationally more intensive, has the attractive feature that both
estimator and bias are obtained simultaneously. Hahn and Newey (2004) show that fully
iterated bias-corrected estimators solve approximated bias-corrected first order conditions.
IBC and SBC are equivalent if the first order conditions are linear in 0.
10
See MacKinnon and Smith (1998) for a comparison of one-step and iterated bias correction methods.
21
Example: Correlated random coefficient model with endogenous regressors. The
previous methods can be illustrated in the correlated random coefficient model example in
Section 4. Here, the fixed effects GMM estimators have closed forms:
S(0) (T
i
and
andn
_
iW2itE
2
~=1 [ZT
it
2
)
w2j
E
T
'2ii
-1
t=1
t=1
t=1
(2:T
2i
-
it(T
1
i=1
-
6t=1~ litlit3
1it yit
t=1
T
0 =W(JW
where JT~V
-=
22it], and variables with tilde now
1_
indicate residuals of sample linear projections of the corresponding variable on x~i,
example
X2it = X2it -
zt=1 X2it'
t
'
for
1
We can estimate the bias of 0 from the analytic formula in expression (4.3) replacing
population by sample moments and 0 by 0, and trimming the number of terms in the
spectral expectation,
n
B(O) ==7 -(dg - dc,)(8
J(6))1
-
t
3
min(T,T+j)
E
3
~
E
E
i=1
j=-t=max(1,j+1)
(itj-z24,-Vitj)
2it(Yi,t-.j
The one-step bias corrected estimates of the common parameter 0 and the average of the
individual parameter a := Elai] are
BC
I
BC
f(BC
-1
i=1
The iterated bias correction estimator can be derived analytically by solving
IBC
_
_ 3(BC)
which has closed-form solution
dIBC =
Idne (r - 0.
12iti2i,t-j
n J
i=1 j= -
min(T,T+j)
2
t=max(1,J+a)
S(dg
-- d,)4v
W_1 Y
E
E
-2it9,t-j/(n
2)
i=1 j=-t t=max(1,J+1)
The score bias correction is the same as the iterated correction because the first order conditions are linear in 0.
The bias correction methods described above yield normal asymptotic distributions centered at the true parameter value for panels where n and T grow at the same rate with
22
the sample size. This result is formally stated in Theorem 5, which establishes that all the
methods are asymptotically equivalent, up to first order.
Theorem 5 (Limit distribution of bias-corrected FE-GMM). Assume that vx/7(B8 (O) Bs)/T 24 0 and v/n~T(J(O)-J,)/T 2*0, for some 6 = 6o+Op((nT)1 '/ 2 ). Under Conditions
1, 2, 3, 4, 5 and 6, for C E {BC, SBC, IBC}
viT(OC
(5.4)
where
UBC,
jIBC
and
9
SBC
Qo) 4 N (0, J, 1 ),
-
are defined in (5.1), (5.2) and (5.3), and J, = E [G'.,PGo,].
The convergence condition for the estimators of B, and J, holds for sample analogs evaluated at the initial FE-GMM one-step or two-step estimators if the trimming sequence is
chosen such that f -+ oo and f/T -+ 0 as T -+ oc. Theorem 5 also shows that all the bias-
corrected estimators considered are first-order asymptotically efficient, since their variances
achieve the semiparametric efficiency bound for the common parameters in this model, see
Chamberlain (1992).
The following corollaries give bias corrected estimators for averages of the data and individual effects and for moments of the individual effects, together with the limit distributions
of these estimators and consistent estimators of their asymptotic variances. To construct
the corrections, we use bias corrected estimators of the common parameter. The corollaries
then follow from Lemma 2 and Theorem 5 by the delta method. We use the same notation
as in the estimation of the bias of the common parameters above to denote the estimators
of the components of the bias and variance.
Corollary 3 (Bias correction for fixed effects averages). Let c(z; 0, ac) be a twice continuously differentiable function in its second and third argument, such that infi Var[((zit)] > 0,
EE[C(zjt)2 ] < oc, E E[(,(z t) 2 ] < o0, and E|o(zjt)|2 < oo. For C e { BC, SBC, IBC}, let
)
;=
B_(c )/T where
1T
B (0)
T
=
+dj=
(,i(O)'t_ (0)+< a(0)'B,(0) +
where f is a positive bandwidth such that f
-+
oc and f/T
-.
>
c(i
(0)'Ea(0)/2 ,
0 as T -+ oo. Then, under the
conditions of Theorem 5
rnT
(
4-)N(0,
-
V),
where rnT, (, and V are defined in Corollary 1. Also, for any 0 =
++
+ Op((nT)-1 1 2 ) and
+o
(p(r(-[(
)
rnT
23
is a consistent estimatorfor V.
Corollary 4 (Bias correction for smooth functions of individual effects). Let p(aj) be a
twice differentiable function such that E[pu(ao) 2] < oo and
{BC, SBC, IBC}, let p^
Pas
(0)'B,(0) +
Z
= E[p(6c)]
-
t
= 1
=
F [p(ajo)1 and V,
B (6)/T, where ' (0)
=
For C E
p(a(0)), and b,(0)
=
(0)/2]. Then, under the conditions of Theorem 5
,,(0)'Z0
1
Vn0where u
Epa(ajo)| 2 < oo.
=
[(bp(ao)
d N(0, Vyt),
_p_)
-
P)2]. Also, for any 0 = Oo
+ Op((nT>-1 / 2 ) and
+ Op(n- 1 / 2 ),
(5.5)
[{j,(5)
2
+
- f}
V = Ea(O)'Zai(6)ia
is a consistent estimator for V1.
()/T
,
The second term in (5.5) is included to improve the finite
sample properties of the estimator in short panels.
6. EMPIRICAL EXAMPLE
We illustrate the new estimators with an empirical example based on the classical cigarette
demand study of Becker, Grossman and Murphy (1994) (BGM hereafter). Cigarettes are addictive goods. To account for this addictive nature, early cigarette demand studies included
lagged consumption as explanatory variables (e.g., Baltagi and Levin, 1986). This approach,
however, ignores that rational or forward-looking consumers take into account the effect of
today's consumption decision on future consumption decisions. Becker and Murphy (1988)
developed a model of rational addiction where expected changes in future prices affect the
current consumption. BGM empirically tested this model using a linear structural demand
function based on quadratic utility assumptions. The demand function includes both future
and past consumptions as determinants of current demand, and the future price affects the
current demand only through the future consumption. They found that the effect of future
consumption on current consumption is significant, what they took as evidence in favor of
the rational model.
Most of the empirical studies in this literature use yearly state-level panel data sets. They
include fixed effects to control for additive heterogeneity at the state-level and use leads and
lags of cigarette prices and taxes as instruments for leads and lags of consumption. These
studies, however, do not consider possible non-additive heterogeneity in price elasticities or
sensitivities across states. There are multiple reasons why there may be heterogeneity in the
price effects across states correlated with the price level. First, the considerable differences
in income, industrial, ethnic and religious composition at inter-state level can translate into
different tastes and policies toward cigarettes. Second, from the perspective of the theoretical
model developed by Becker and Murphy (1988), the price effect is a function of the marginal
24
utility of wealth that varies across states and depends on cigarette prices.
If the price
effect is heterogenous and correlated with the price level, a fixed coefficient specification
may produce substantial bias in estimating the average elasticity of cigarette consumption
because the between variation of price is much larger than the within variation. Wangen
(2004) gives additional theoretical reasons against a fixed coefficient specification for the
demand function in this application.
We consider the following linear specification for the demand function
(6.1)
Cit =
+ C1liPit
cOi +
0
1Ci,t-1
+ 02Ci,t+1 + Xi'tt + Eit,
where Cit is cigarette consumption in state i at time t measured by per capita sales in packs;
ae0 is an additive state effect; ali is a state specific price coefficient; Pit is the price in 1982-
1984 dollars; and Xit is a vector of covariates which includes income, various measures of
incentive for smuggling across states, and year dummies. We estimate the model parameters
using OLS and IV methods with both fixed coefficient for price and random coefficient for
price. The data set, consisting of an unbalanced panel of 51 U.S. states over the years 1957
to 1994, is the same as in Fenn, Antonovitz and Schroeter (2001). The set of instruments for
Cs,t_1 and Ci,t+1 in the IV estimators is the same as in specification 3 of BGM and includes
Xit, Pit, Pj,t_1 , Pi,t+, Taxit, Taxi,t_1, and Taxi,t+i, where Taxit is the state excise tax for
cigarettes in 1982-1984 dollars.
Table 1 reports estimates of coefficients and demand elasticities. We focus on the coefficients of the key variables, namely Pit, Ct-1 and C,t+1.
Throughout the table, FC refers
to the fixed coefficient specification with aci = ai and RC refers to the random coefficient
specification in equation (6.1). BC and IBC refer to estimates after bias correction and iterated bias correction, respectively. Demand elasticities are calculated using the expressions in
Appendix A of BGM. They are functions of Cit,Pt, ce1i, 61 and
02,
linear in a1i. For random
coefficient estimators, we report the mean of individual elasticities, i.e.
1
n
(h= 7 T(h(zit;0,
ai)
i=1 t=1
where (h(zit; 0, ij) = alog Cit(h)/a log Pit(h) are price elasticities at different time horizons
h.
Standard errors for the elasticities are obtained by the delta method as described in
Corollaries 3 and 4. For bias-corrected RC estimators the standard errors use bias-corrected
estimates of 6 and oi.
As BGM, we find that OLS estimates substantially differ from their IV counterparts.
IV-FC underestimates the elasticities relative to IV-RC. For example, the long-run elasticity estimate is -0.70 with IV-FC, whereas it is -0.88 with IV-RC. This difference is also
pronounced for short-run elasticities, where the IV-RC estimates are more than 25 percent
25
larger than the IV-FC estimates. We observe the same pattern throughout the table for
every elasticity. The bias comes from both the estimation of the common parameter 62
and the mean of the individual specific parameter Eaii]. The bias corrections increase the
coefficient of future consumption Ci,t±1 and reduce the absolute value of the mean of the
price coefficient. Moreover, they have significant impact on the estimator of dispersion of
the price coefficient. The uncorrected estimates of the standard deviation are more than
20% larger than the bias corrected counterparts. In the online appendix Ferndndez-Val and
Lee (2012), we show through a Monte-Carlo experiment calibrated to this empirical example,
that the bias is generally large for dispersion parameters and the bias corrections are effective
in reducing this bias. As a consequence of shrinking the estimates of the dispersion of 0 1i,
we obtain smaller standard errors for the estimates of E[aii] throughout the table. In the
Monte-Carlo experiment, we also find that this correction in the standard errors provides
improved inference.
7. CONCLUSION
This paper introduces a new class of fixed effects GMM estimators for panel data models with unrestricted nonadditive heterogeneity and endogenous regressors. Bias correction
methods are developed because these estimators suffer from the incidental parameters problem. Other estimators based on moment conditions, like the class of GEL estimators, can be
analyzed using a similar methodology. An attractive alternative framework for estimation
and inference in random coefficient models is a flexible Bayesian approach. It would be interesting to explore whether there are connections between moments of posterior distributions
in the Bayesian approach and the fixed effects estimators considered in the paper. Another
interesting extension would be to find bias reducing priors in the GMM framework similar
to the ones characterized by Arellano and Bonhomme (2009) in the MLE framework. We
leave these extensions to future research.
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-I
0
--
Uncorrected/
Bias Corrected
40%
/
CV~)
0
CN
0
0
di
I
-60
-50
-40
-30
-20
-10
Price effect
Normal approximation to the distribution of price effects using
FIGURE 1.
uncorrected (solid line) and bias corrected (dashed line) estimates of the mean
and standard deviation of the distribution of price effects. Uncorrected estimates of the mean and standard deviation are -36 and 13, bias corrected
estimates are -31 and 10.
29
Table 1: Estimates of Rational Addiction Model for Cigarette Demand
OLS-FC IV-FC
OLS-RC
IV-RC
NBC
BC
IBC
NBC
BC
IBC
-13.49
(3.55)
-13.58
(3.55)
-13.26
(3.55)
-36.39 -31.26
(4.85) (4.62)
-31.26
(4.64)
4.35
(0.98)
4.22
(1.02)
4.07
(1.03)
12.86
(2.35)
10.45
(2.13)
10.60
(2.15)
0.44
0.44
Coefficients
(Mean) Pt
-9.58
(1.86)
-34.10
(4.10)
(Std. Dev.) Pt
Ct_1
0.49
0.45
0.48
0.48
0.48
(0.01)
(0.06)
(0.04)
(0.04)
(0.04)
(0.04) (0.04)
0.44
0.45
(0.04)
0.44
0.17
0.44
0.43
0.23
0.29
0.27
(0.01)
(0.07)
(0.04)
(0.04)
(0.04) (0.05)
(0.05)
(0.05)
-1.05
-0.70
-1.30
-1.31
-1.28
-0.88
-0.91
-0.90
(0.24)
(0.12)
(0.28)
(0.28)
(0.28)
(0.09)
(0.10)
(0.10)
Own Price
(Anticipated)
-0.20
(0.04)
-0.32
(0.04)
-0.27
(0.06)
-0.27
(0.06)
-0.27
(0.06)
-0.38
(0.04)
-0.35
(0.04)
-0.35
(0.04)
Own Price
(Unanticipated)
-0.11
(0.02)
-0.29
(0.03)
-0.15
(0.04)
-0.16
(0.04)
-0.15
(0.04)
-0.33
(0.04)
-0.29
(0.04)
-0.29
(0.04)
Future Price
(Unanticipated)
-0.07
(0.01)
-0.05
(0.03)
-0.10
(0.02)
-0.10
(0.02)
-0.09
(0.02)
-0.09
(0.02)
-0.10
(0.02)
-0.09
(0.02)
Past Price
(Unanticipated)
-0.08
(0.01)
-0.14
(0.02)
-0.11
(0.03)
-0.11
(0.02)
-0.10
(0.03)
-0.16
(0.02)
-0.15
(0.02)
-0.15
(0.02)
Short-Run
-0.30
-0.35
-0.41
-0.41
-0.40
-0.44
-0.44
-0.43
(0.05)
(0.06)
(0.12)
(0.12)
(0.12)
(0.06)
(0.06)
(0.06)
Ct+1
Price elasticities
Long-run
RC/FC refers to random/fixed coefficient model. NBC/BC/IBC refers to no bias-correction/bias
correction/iterated bias correction estimates.
Note: Standard errors are in parenthesis.
1
Supplementary Appendix to Panel Data Models with Nonadditive Unobserved
Heterogeneity: Estimation and Inference
Ivin Ferndndez-Val and Joonhwan Lee
October 15, 2013
This supplement to the paper "Panel Data Models with Nonadditive Unobserved Heterogeneity: Estimation and Inference" provides additional numerical examples and the proofs of the main results. It is organized
in seven appendices. Appendix A contains a Monte Carlo simulation calibrated to the empirical example of
the paper. Appendix B gives the proofs of the consistency of the one-step and two-step FE-GMM estimators. Appendix C includes the derivations of the asymptotic distribution of one-step and two-step FE-GMM
estimators. Appendix D provides the derivations of the asymptotic distribution of bias corrected FE-GMM
estimators. Appendix E and Appendix F contain the characterization of the stochastic expansions for the
estimators of the individual effects and the scores. Appendix G includes the expressions for the scores and
their derivatives.
Throughout the appendices
Oup and onp will denote uniform orders in probability. For example, for a
sequence of random variables {(i : 1 < i < n},
j = Oup(l) means supi<j<,
j = Op(1) as n -+ oo, and
j = op(l) as n -+ oo. It can be shown that the usual algebraic properties for
j = osp(1) means supl<ig<
Op and op orders also apply to the uniform orders Oup and ovp. Let ej denote a 1 x dg unitary vector with
a one in position
j.
For a matrix A, JAl denotes Euclidean norm, that is JAl 2 = trace[AA']. HK refers to
Hahn and Kuersteiner (2011).
APPENDIX
A.
NUMERICAL EXAMPLE
We design a Monte Carlo experiment to closely match the cigarette demand empirical example in the
paper. In particular, we consider the following linear model with common and individual specific parameters:
C
=
Pit=
aoi
+ ai Pit + OlCi,t-1 +
rj
+ 7Taxjt +ujt,
where {(acjivji) : 1 < i < n} is i.i.d.
correlation p3 , for
0
2Ci,t+1 +
?Eit,
(i = 1,2,...,n, t = 1,2,...,T);
bivariate normal with mean (py, tj), variances ( ? a'),
j E {0, 1}, independent across j; {uit : 1 < t
and
< T, 1 < i < n} is i.i.d N(0, C2); and
{Eit : 1 < t < T, 1 < i < n} is i.i.d. standard normal. We fix the values of Taxit to the values in the data
set. All the parameters other than p, and V are calibrated to the data set. Since the panel is balanced for
only 1972 to 1994, we set T = 23 and generate balanced panels for the simulations. Specifically, we consider
n = 51, T = 23; po = 72.86, p1 = -31.26, pq0 = 0.81, pu, = 0.13, oo = 18.54, ', = 10.60, cr, = 0.1 4 ,
cg7 = 2.05, o, = 0.15, 01 = 0.45, 02 = 0.27,
po = -0.17, pi E {0, 0.3, 0.6, 0.9},
In the empirical example, the estimated values of p1 and
b
bE
{2, 4, 6}.
are close to 0.3 and 5, respectively.
Since the model is dynamic with leads and lags of the dependent variable on the right hand side, we
construct the series of Cit by solving the difference equation following BGM. The stationary part of the
solution is
Ct = 0012
-
1q81(#2 -
hi(t + s) + 0122-1)
)
=1(
hi(t - s)
=)
2
where
hi(t) = aeo + aiiP,t-1 + ci,t-1, 01
In our specification, these values are
experiments are pi and V.
#1
1
= 0.31 and
--
(1
#2
-
=
1+ (1
46102)1/2
261
'
=
-
46102)1/2
202
1.91. The parameters that we vary across the
The parameter p, controls the degree of correlation between
aih and Pt and
determines the bias caused by using fixed coefficient estimators. The parameter V) controls the degree of
endogeneity in Ci,t_1 and Ci,t+1, which determines the bias of OLS and the incidental parameter bias of
random coefficient IV estimators. Although V is not an ideal experimental parameter because it is the
variance of the error, it is the only free parameter that affects the endogeneity of Cit-i and Ci,t+1. In this
design we cannot fully remove the endogeneity of Ci,t_ 1 and
Ci,t+1 because of the dynamics.
In each simulation, we estimate the parameters with standard fixed coefficient OLS and IV with additive
individual effects (FC) , and the FE-GMM OLS and IV estimators with the individual specific coefficients
(RC). For IV, we use the same set of instruments as in the empirical example. We report results only for
the common coefficient 62, and the mean and standard deviation of the individual-specific coefficient aii.
Throughout the tables, Bias refers to the mean of the bias across simulations; SD refers to the standard
deviation of the estimates; SE/SD denotes the ratio of the average standard error to the standard deviation;
and p; .05 is the rejection frequency of a two-sided test with nominal level of 0.05 that the parameter is equal
to its true value. For bias-corrected RC estimators the standard errors are calculated using bias corrected
estimates of the common parameter and individual effects.
Table A.1 reports the results for the estimators of 62. We find significant biases in all the OLS estimators
relative to the standard deviations of these estimators. The bias of OLS grows with V. The IV-RC estimator
has bias unless p, = 0, that is unless there is no correlation between aii and P, and its test shows size
distortions due to the bias and underestimation in the standard errors. IV-RC estimators have no bias in
every configuration and their tests display much smaller size distortions than for the other estimators. The
bias corrections preserve the bias and inference properties of the RC-IV estimator.
Table A2 reports similar results for the estimators of the mean of the individual specific coefficient pi
=
E[aii]. We find substantial biases for OLS and IV-FC estimators. RC-IV displays some bias, which is
removed by the corrections in some configurations. The bias corrections provide significant improvements
in the estimation of standard errors. IV-RC standard errors overestimate the dispersion by more than 15%
when b is greater than 2, whereas IV-BC or IV-IBC estimators have SE/SD ratios close to 1. As a result
bias corrected estimators show smaller size distortions. This improvement comes from the bias correction in
the estimates of the dispersion of ali that we use to construct the standard errors. The bias of the estimator
of the dispersion is generally large, and is effectively removed by the correction. We can see more evidence
on this phenomenon in Table A3.
Table A3 shows the results for the estimators of the standard deviation of the individual specific coefficient
E[(ii - pl)2]1/2. As noted above, the bias corrections are relevant in this case. As 'b increases, the
C-1 =
bias grows in orders of 0. Most of bias is removed by the correction even when V is large. For example,
when V) = 6, the bias of IV-RC estimator is about 4 which is larger than two times its standard deviation.
The correction reduces the bias to about 0.5, which is small relative to the standard deviation. Moreover,
despite the overestimation in the standard errors, there are important size distortions for IV-RC estimators
for tests on -1 when V) is large. The bias corrections bring the rejection frequencies close to their nominal
levels.
3
Overall, the calibrated Monte-Carlo experiment confirms that the IV-RC estimator with bias correction
provides improved estimation and inference for all the parameters of interest for the model considered in the
empirical example.
APPENDIX B. CONSISTENCY OF ONE-STEP AND Two-STEP FE-GMM ESTIMATOR
Lemma 3. Suppose that the Conditions 1 and 2 hold. Then, for every 77 > 0
Pr Isup
1<i<n
(
sup E
a ( a)
j (0,
Q,a)C
z
QW (0 0a)>n
=o(T -- ),
and
sup Q '(6, a) - Q '(O',a)| 5 C - E[M(zit)]2 6- 'l
for some constant C > 0.
Proof. First, note that
Igi(6, a)'W-' (6, a) - gj(6, a)'W7'1gj(6, a)I + jg (0, a)'(WJ- - W-' )g(6, a)
Q '(6, a) - Qf (6, a) <
|[ (6, a) - gj(0, a)]'W;4[ j(0, a) - gj(6, a)]I + 2 -gj(6, a)'W-l[
1
+ [gi(6, a) - gj(0, a)]'(Wi
- Wl- )[9i(6, a) - gi(6,a)] + 2 [ j(6, a) - gj(6,a)]'(HWi1 - Wil 1 )g (6, a)
+ gi(6,a)'(W-' - Wi_
W- )gi(0, a)
+ 2d2 sup E[M(zi)] Wilj
W.
where we use that sup<i<,n
1
gk,i(0, a) -
gk,i(0,
(6, a) - gj(6, a)]
1
d2 max I ,i(6, a) - gk,i(6, a)2 IWi|
91 k<dg
max fgk,j(,a) -gk,i(6,a) + op (max
- W
,j(,a)-9k,i(0,a),
= op(l). Then, by Condition 2, we can apply Lemma 4 of HK to
a)I to obtain the first part.
The second part follows from
|Q '(6,a) - Qf(6',a)j
5
gi(6,a)'Wi~'[gi(6,a)- gi(6',a)]I + [gi(6,a) -g(6',a)]'Wi~lg9(6',a)I
2 d 2 E[M(zit)]2 |Wi|
|61-
6'|
.
B.1. Proof of Theorem 1.
Proof. Part I: Consistency of 6. Foranyq > 0, letE:= infi[QW (60, ao)-sup{(,o.):1(0,a)-(-0,a0o)I>.} QW'(6,a)] >
0 as defined in Condition 2. Using the standard argument for consistency of extremum estimator, as in Newey
and McFadden (1994), with probability 1 - o(T-1)
max
n~-IIi
10-00 1>7),Q1,.. .,an
(6,aj) <
by definition of c and Lemma 3. Thus, by continuity of
we conclude that Pr
- , >7]
o (T-1).
Q"
n-ZQ'
(60, ao)-
and the definition of the lefthand side above,
Part II: Consistency of di. By Part I and Lemma 3,
(B.1)
Pr
sup sup Qj
L1<i<i a
(o, a)
3 6)
- Qf (0o, a) >
1
=
j
o(T-1)
4
for any q > 0. Let
:=inf Q (0o, ao)T I
{ai: Iai -
sup sup I
(&, a) - Q
QW (0, ao)
sup
ajo
> 0.
}
Condition on the event
1<i<n a3
e ,
(00, a) I
which has a probability equal to 1 - o (T-') by (B.1). Then
max
|ais-ao
Q1" (6,
QW (0o, ai) + -e < QZW (0,
max
3
ai-Q >a
ai) <
is2
hissinconsistent with
Q' (0, i)
;>
Q" (,
ceio) -
_E < Q
3
-0.
ao
z
), and therefore, I5i - aco| 17 with probability 1-o(T-)
for every i.
First, note that
Part III: Consistency of Ai.
-1(9,
i
Ai
<
dW
+
dg I4Wi
&i)
dg
Wi_
max
Wi
sup
1<k<d 9 (6,ai)ET
M(zi)
max
1 <k~dg
(gk,i(,
1 k,i(0,ai)--g,i(0,
d- 0 + dg W
Then, the result follows because sup<i%<,
&i) - g,i(d, &i) +
gk,i(0,
)
a)
M(zit) I - aol.
Wi - Wil = op(l) and {W: 1
i < n} are positive definite
by Condition 2, maxliksdg sup(e,,i)cT 1 k,d(,ct) - gk,i(O, ai) = op(l) by Lemma 4 in HK, and d - D0
op(l) and sup 1 <i<
i
=
E
- ciol = op(l) by Parts I and II.
B.2. Proof of Theorem 3.
Proof. First, assume that Conditions 1, 2, 3 and 5 hold. The proofs are exactly the same as that of Theorem
1
using the uniform convergence of the criterion function.
To establish the uniform convergence of the criterion function as in Lemma 3, we need
sup Qii(,5&)-Qi(60, aio)
1<i<n
= op(l),
along with an extended version of the continuous mapping theorem for op. This can be shown by noting
that
Ni(d, &i) - Flj(0o, aio) I
QA(,
&i) -
6i(6, &i) - Qi(5, &i) + IQi(,
&i) - flj(0o, aio)
&i(,
&,) + dE [M(zit) 21 (, &i) - (0o,
Qjo)
.
The convergence follows by the consistency of 0 and &i's, and the application of Lemma 2 of HK to
gk (zit; 0, ai)gt (zit; 0, ao)
2
using that Igk (zit; 0, ai)g, (zit; 0, a,)I < M(zit)
APPENDIX C. ASYMPTOTIC DISTRIBUTION OF ONE-STEP AND Two-STEP FE-GMM ESTIMATOR
C.1. Some Lemmas.
Lemma 4. Assume that Condition 1 holds. Let h(zit; 0, ai) be a function such that (i) h(zit; 6, ai) is
continuously differentiable in (0, ai) - T C R d+da; (ii) T is convex; (iii) there exists a function M(zit) such
oc
that Jh(zit; 6, ai)l 5 M(zit) and IJh(zit; 6, ai)/3(6, cei)j 5 M(zit) with E [M(zit)5(do+d+6)/(1-lov)+6I
5
for some J > 0 and 0 < v < 1/10. Define fA(O, ai) := T-'
, h(zit; 0, a), and Hi(0, cj) :E
E
Hi (0, ai)].
Let
a* = arg maxQi (0*,
ai),
such that a any 6 between
= O p(Tas ) and 6* - 00 = op(T ao), with -2/5 < a < 0, for a = max(a 0 , ao). Then, for
6* and 0 0, and T, between a and ajo,
oup (T'1 1 0 ),
[Ii ( , a) - Hi=(, Zi)]
v
H(6, Zi)
Hi(60,
-
o) = osp (Ta).
Proof. The first statement follows from Lemma 2 in HK. The second statement follows by the first statement
and the conditions of the Lemma by a mean value expansion since
T
li (-0, -di) - Hi (0, ajo)
5
11T
-0
M (zit) + -di - aeio
1;
=oUp(Ta)
=OUp(l)
=Oup(l)
Hi (0o, a o) - Hi (0,
+
EM (zit)
=o.p(Ta)
_
oUP(Ta.
jo)
=oUp(T-2/5)
0
Lemma 5. Assume that Conditions 1, 2, 3 and
4 hold. Let iwi(6, yi) denote the first stage GMM score of
)
the fixed effects, that is
=iw
(0(
(6, ac)'A
+ Wi Ai
aj)
'g' (0,
where -yj = (a',A')', 91V (0, -yi) denote the one-step GMM score for the common parameter, that is
91(0, -y)- -ei(0,
ai)'Ai,
and j'(0) be such that iF (6,
j(6)) = 0.
Let TW(6, -y)
denote
8i7(O, 7y)/a7y8o9,; and
dg + da, where yyj is the jth element of -yj and
dt7(O,-y)/D0' and Sr(,-yi) denote
(0, gi) denote a'9(0,7y)/y7j,, for some 0 < j
= 0 denotes no second derivative. Let NRr(0,-y) denote
M
j
N'(0,7j)/a0'.
n
Let (O,~y1 ,...
be the one-step GMM estimator.
Then, for any 6 between 6 and 00, and 7i between i and Yio,
N, Ti-T
op(1
MTi)-7"
(0i 1
=
)-
Also, for any ;yj between 'yiO and jo = ~1(0
v'7T7(Oo,
VYM
1
),
io) = oup (T/10) ,\T (Tu(
(0,;Yio)
- MW)
oup (1) , §r(5,74)-3r = OUP (1).
K(,T)N
60 ,
yjo)
- TW) = oup
(T1/la)
= oup (T1/10)
Proof. The first set of results follows by inspection of the scores and their derivatives (the expressions are
given in Appendix G), uniform consistency of j by Theorem 1 and application of the first part of Lemma 4
to
6* = 6 and a!
=
di with a = 0.
The following steps are used to prove the second set of result. By Lemma 4,
VFIW = oUp
(T1/1l), If(
60 ,0TiO)
- Tr
= oup (1)
6
where
T7 io
is between 7io and
7yjO.
Then, a mean value expansion of the FOC of ijo,
i-4'(0,
ijo) = 0, around
Yio = 7YiO gives
VI- o-
-YiO)
=
-
-(TW
=O.(1
=OUP(T111)
oup (T1/1 0)
=
1
(w(oo
- (TW)-
/
=0"(1)
+ oup (V(io -
0 io)
-
Tw)V7(i'Yio)
=O "r(1)
io),
by Condition 3 and the previous result. Therefore,
(1 + oup(1)) VT(~jo - yio) = oup(T 1 / 10 ) =>
v'i(jio - 7o0) = oup(T111 L
Given this uniform rate for jo, the desired result can be obtained by applying the second part of Lemma
4 to 0* =0o and a* = dio with a = -2/5.
C.2. Proof of Theorem 2.
Proof. By a mean value expansion of the FOC for 0 around
0=gw(6) =w(0)+
=
0 D0,
dO'
(6-C0),
where C lies between 0 and 00.
Part I: Asymptotic limit of dsw()/d0'. Note that
dgw(6)
dO'
I
-
(C.1)
dsj?( , is 5)
d'
n
=
,i
9 (D~
dO'
)) +(0)a77(5, i (0))
+
17
0'
By Lemma 5,
OP'A
S
Then, differentiation of the FOC for 'i(0), i' (0,
Tr
=Mr +oUp(1).
+ OSP(1),
~ A)C9
(0)) = 0, with respect to 0 and
'i
gives
) + Rw ( , ~73(A) = 0,
By repeated application of Lemma 5 and Condition 3,
n~(0)
=-(TW)
1
NrW +
oup(1).
Finally, replacing the expressions for the components in (C. 1) and using the formulae for the derivatives,
which are provided in the Appendix G,
d~G'w(,
!'~GpGe
dO'
nL~
(C.2)
)J"-+hop
+ op (1)
(1),
jJ = $[G',PZGo].
0
Part II: Asymptotic Expansion for 0- o. By (C.2) and Lemma 22, which states the stochastic expansion
of V'
sw(Oo),
0
=
V'nThW(0o) +
Op(1
d- (0)
dO'
Op (1)
nT(-0o).
7
Therefore, v'/nT(6 - Oo) = Op(l), and by part I, Lemma 22 and Condition 3,
/fn(
-
(rBw, VyW)
00) + -(Jr)-'N
C.3. Proof of Theorem 4.
Proof. Applying Lemma 4 with a minor modification, along with Condition 4, we can prove an exact counterpart to Lemma 5 for the two-step GMM score for the fixed effects
Ti (, 1 Yi) = Pi (, 1 Yi) + Fi (0, -yi),
iF
where the expressions of Pi and
are given in the Appendix G, and for the two-step score of the common
parameter
9i (0, i (0)) = -Gi (0, ai (0))'Ii(0),
The only difference arises due to the term it(0, -yi), which involves Q(0, &j) - Qj. Lemma 8 shows that
V(U(O, i ) - Qj) = oup(T 1/' 0 ), so that a result similar to Lemma 5 holds for the two-step scores.
Thus, we can make the same argument as in the proof of Theorem 2 using the stochastic expansion of
1
/_nT(60o) given in Lemma 23.
APPENDIx D.
ASYMPTOTIC
DISTRIBUTION OF BIAS-CORRECTED TWO-STEP
GMM
ESTIMATOR
D.1. Some Lemmas.
Lemma 6. Assume that Conditions 1, 2, 3,
4 and 5
hold. Let j(O, yi) denote the two-step GMM score for
the fixed effects, 9j(0, yi) denote the two-step GMM score for the common parameter,and
1(6)
be such that
for some 0 < j 5 d9 + da, where -y,j is the jth
0. Let Tj,(0, -y) denote 8t9(0, y) ly8-1,,
component of -yj and j = 0 denotes no second derivative. Let Ni(0, -Y) denote Di (6, -yi)/DO'. Let Mi,3 (0, 'yi)
d9 + da. Let S§(0, -yi) denote aO3(0, yi)/ 90 '. Let (0, {g}tL) be
denote D ?j(0, -y)/Oa4D'y,, , for some 0 < j
(0,
(0))
=
the two-step GMM estimators.
Then, for any 6 between 6 and 60, and y1 between
(ti,d(
'/
,;Yi)
- Ti,d)
=
oup
- Nj)
=
oup (T1/10) ,
\/T (&ij(,;j)
Proof. Let
'/f(T7 -
i
=
9(0) and io
Yio) =
D
(T1/10)
i
(
=
(-
,
and -yi,
\/7
(M,,j
v'
(
- Mij,)
= oup
(Ti/l1)
) - Si) = oup (T1/10)
ij (0o). First, note that
Oo)
=
-(T") - N V(
=0.(1)
- Oo) + oup (VI/(
- 00)) = Op(n-1/2).
=Op(,n-1/2)
where the second equality follows from the proof of Theorem 2 and 4. Thus, by the same argument used in
the proof of Lemma 5,
-V/Tj(i7 - 'io) = .IT(;, - ijo) + /T( jo - 'yio) = oup(T1/10
Given this result and inspection of the scores and their derivatives (see the Appendix G), the proof is similar
to the proof of the second part of Lemma 5.
8
1 holds. Let hj(zit; 6, ai), j = 1, 2 be two functions such that (i)
hj(zit; 6, ai) is continuously differentiable in (6, a%) E T C R d+d- ; (ii) T is convex; (iii) there exists a function M(zit) such that Ihj(zit; 6, aI)< M(zit) and hj(zit; 6, at)/O(0, ai) 5 M(zit) with E [M(zit)10(do+d,+ 6 )/(1-10v)+6
Lemma 7. Assume that Condition
oo for some 6 > 0 and 0 < v
Fi(6, aj) := E [F4(0, a)]. Let
< 1/10.
Define
i(6,ai) := T-1 ET
a* = arg sup
h(zit;6,a)h
2 (zit;6,ai), and
(0*, a),
such that at - ajo = oup(Ta- ) and 0* - 0= op(Ta, ), with -2/5 < a < 0, for a = max(aa,ao). Then, for
any 6 between 0* and 60, and Zi between a* and a&o,
F(0, iii) - F (60 , ajo) = oup(T),
[(,i)
= oP(T110).
F(,)
-
Proof. Same as for Lemma 4, replacing Hi by F, and M(zit) by M(zat) 2 .
Lemma 8. Assume that Conditions 1, 2, 3, 4, 5, and 6 hold. Let 6j(0, Ti) = T-' t=1 g(zit; #, zi)g(zit; 0, zi)'
be an estimator of the covariance function Qj = E[g(z t)g(zjt)'], where 6 = Go + op(T- 2 /") and ai =
aco + oup(T- 2 /1). Let
Q
6d 1 d 2
(0,
v
)
=
ad1+d2Qi(Oy,)/adiatad2o,for 0
(
(5,
adOd 2
)-
< d 1 + d 2 ! 2. Then,
ou, (T1/10) .
=
Proof. Note that
Ig(zit; 6,-)g(ztt;
, i)' - E [g(zit; 6, Ti)g(zat; 6, Zi)']
< d2
Then we can apply Lemma 7 to h,
max
Igk(zit;6,ii)gI(zit;#,Ni)' - E [9k(Zit; , i)g1(zit; ,ai)
and h 2
= 9A
=
gi with a = -2/5. A similar argument applies to the
derivatives, since they are sums of products of elements that satisfy the assumption of Lemma 7.
Lemma 9. Assume that Conditions 1, 2, 3,
4, 5, and 6 hold, and f -> oc such that t/T
any
(),
6
between 0 and 0, let
-2
1Q
)Z, (6) = [ac
(5) ft, (s), I,
RV
(i)
(R)' W
=
do, (6) Na, (6) 00 (6), Pq(6) = T-1
Bi(0)+
Q
-
z-L
G
(0)
=
=
-a(0)
H (6)'
Y,
=
P0 i(0) ZTj=0
()
,Ha (0) = Ea, (0) Ga, (#)'wj, .i (5) =
1 Goi (0)'Pai(0)
Ai (0)
=
(
=
Pc. (0) L
j=1
G, (5)'Pa (O)gi,t _ (0),
it(0)t(0)'ai(6) it-j(6),
[Hal (0) - H(
(0)],
T
,(0)ZT-1
t=j+1
t=j+1
S,
d
,aj
i,t-j(0), and B$(0) = -Goi (0)'[B,(0)+
j=0
1 1
(m)=
15,
(j6)Zi(0)/2
+
T
(m)
, Rai
e
ZScfc
j=1
j=0
B$(0)
Pai
-+ 0 as T -+ oo. For
(#)
d
BIU(6)
()
0
d
'((0)], where
Ai (0 )+
.
((),
t=j+1
<
9
be estimators of)ZQ,
F
Ha,, Pai, EW, Hr, J,, BC and BP. Let Fadlod2 i(O, &(0))
and Fd10d 2 i(0,aj), with
E{ , H, P,Ew , Hw, Jj, Bq, B-P} denote their derivatives for 0 < d1 + d 2
VT
where
Fadi od2 i
(
(
adod2i
1. Then,
p,&i(T)) - Fadiod2 i) = oup (T1/1)
:= F if d, + d2 = 0.
Proof. The results follow by Theorem 3 and Lemma 6, using the algebraic properties of the oup orders and
Lemma 12 of HK to show the properties of the estimators of the spectral expectations.
3, 4, 5, and 6 hold. Then, for any 0 between 0 and 00,
Lemma 10. Assume that Conditions 1, 2,
(b) = J, +
J,
op(T-2/ 5 ).
Proof. Note that
VT
[,
-
Go, P, Go, =oup(T110),
by Theorem 3 and Lemmas 6 and 9, using the algebraic properties of the o,,p orders. The result then follows
by a CLT for independent sequences since
ZJ(0) - J = E[G 0 ,()'Pa,(0)Go,(0)] - E[GoPciGoj = n
- E[G'.PceGi]) + onp(T-2 )
3, 4, 5, and 6 hold. Then, for any 6 between 0 and 00,
Lemma 11. Assume that Conditions 1, 2,
(o)
B3
(G'.PoG
5
= B, + op(T -2/ ).
Proof. Analogous to the proof of Lemma 10 replacing J, by B,.
Lemma 12. Assume that Conditions 1, 2,
'B= -Jj'B
8
3, 4, 5, and 6 hold. Then, for any 0 between 0 and 00, and
,
3(1) =
1
-Z
(W)
= 3 + op(T -2/
5
).
Proof. The result follows from Lemmas 10 and 11, using a Taylor expansion argument.
D
D.2. Proof of Theorem 5.
Proof. Case I: C = BC. By Lemmas 10 and 25
/-nT(W-
0)
=
-!, ( )'1(0o)
= -J;
(0)
+op(T
2
/O
_ -J.(0
0
) +
p(l).
Then, by Lemmas 12 and 25
SnT
(jBC
- 00)
=
=
VOo (-J[~
oo)
- v'nH
E
si+
(0) = -J ~-9(0
0 )+
aBS-
B,
+op(1)
J,-1B, + op(1)
N(0, J-).
Case II: C = SBC. First, note that since the correction of the score is of order Op(T-1), OsBC _
Op(T
1
). Then, by a Taylor expansion of the corrected FOC around SBC = 0 0
0
=
(SBC)
-
T-B8
(OSBC)
= g(0O)
+ J8 (6) (PsBC -
Oo)
-
T-1B, + op(T~ 2 ),
-
10
where 0 lies between
SBC
V/T
00
(jSBC
and Oo. Then by Lemma 25
=
-J_)
[VZTs(60o) - n1/2T-1/2B,] + op(1)
- Z (5)~
[7
s~~v
i+
Bs
Bs-
+ op (1)4N(0, J-).
Case III: C = IBC. A similar argument applies to the estimating equation (5.2), since
1
O(T- ) neighborhood of
0
IBC is in a
Go.
APPENDIX E. STOCHASTIC EXPANSION FOR 'io =
'i(Oo) AND
io =
i(6o)
We characterize the stochastic expansions up to second order for one-step and two-step estimators of the
individual effects given the true common parameter. We only provide detailed proofs of the results for the
io, because the proofs the one-step estimator ~io follow by similar arguments. Lemmas 1
and 2 in the main text are corollaries of these expansions. The expressions for the scores and their derivatives
two-step estimator
in the components of the expansions are given in Appendix G.
Lemma 13. Suppose that Conditions 1, 2, 3, and 4 hold. Then
+ T-1 2 RK
VT_(io - 7io) =
4 N(0, VW),
where
1
Tw= -
y/7T"
=
oup(T 11 10 ), RW
=
oup(T 1 /)
yw
= E[?
('].
Also
SOp(1).
7=1
Proof. We just show the part of the remainder term because the rest of the proof is similar to the proof of
Lemma 16. By the proof of Lemma 5, rT/(io - yio) = oUp(T'/10) and
i=
(TW) 1 (w(o 0 ,. 0) - Tw) 'VT(io -7io)
=0uPk-1
10)
=O2(1)
= oup(T 115 )
Lemma 14. Suppose that Conditions 1, 2, 3, and 4 hold. Then,
V
iT(~yo
-
YiO) = gi + T-1/2Q
+
where
d ,+d
QW =
=
-[(Tr)
v/T(Sw -
A
TW)
+
=
(iz
oup(T11 10 ), R
Also,
Qi = Op (1).
i=1
Tyw
=
= oup(T1/5
3/10).
11
Proof. Similar to the proof of Lemma 18.
Lemma 15. Suppose that Conditions 1, 2, 3, and 4 hold. Then,
in
d
+~ N(O, E[VW])
i
n
QIi=14
=: B
+ Bw,1s
g[Bw, + Bw,G
-iY
where
H
Vw
B,
B,YG
=
(
Qi HW',ai
)
E [Ga, (zit)H
BW,1s
Br/ is
Bw,1S
)
)
(
(
w
HW
+ al
H
~1Ga,)
da
g(zi,tj)] - L Gaae, H[QiiHf
j=1
d,
d9
Z G'aiPc"Q H
'/2 + L
G'a (kdn
(9 ej)HWi Pj
j=1
j=1
00
N
j=-oo
for EW = (G' W
2
(j=-o
00
EB [Gaj (zit)'P, "g(zi,t-j)]
HW'
a,
H |
,G
Bw,~G
B
=
P' ) ,
PZ )
E [ i(zi)P0g(zi,t-j)]
G', W-1, and P 7 = W 1 - WJ 1Ga, H,1 .
= E
Proof. The results follow from Lemmas 13 and 14, noting that
(TW
1
=
K
Hw )
Hw'
ai
(
H
E[A WV"J
E
00
)
)
Ar
(H pW
)
g(zit),
H',P
(E
[G
(zit)'P!/g(zi t-j)]
(Gazt)'Hwg(zit-j)] + E [,i(zit)P,g(zit-j)]
E
E
i
Pli
)
]
_
HG' Hw' ) )
G/aIn
e
Lemma 16. Suppose that Conditions 1, 2, 3,
4,
d,,;
j
> da.
if
0 ej-d.) Ha.QPj,
GI,,j(Id
if j
H aP
5, and 6 hold. Then,
(io - yio) = Vj + T-1 2 Ri
d
N(0, V),
where
= lki =
(Ti')
VT
= oup (T1/10
,) R1
Also
;~
=
Op(1)
=
oup (T1/5 )
Vi = E[
|2,
12
Proof. The statements about
follow by the proof of Lemma 5 applied to the second stage, and the CLT
/i
in Lemma 3 of HK. From a similar argument to the proof of Lemma 5,
Rij
=
(i(O0,
-(Ti)
=O (1)
=
Ti) - Ti') vT(iO - 7Yo) - (Ti[)
=OUP(T1/10)
=OUP(TI/10)
VT(f(60,;7) - TR) V(Mio - -YiO)
=OILP(T1/10)
=OU(1)
=O.P(Tl/10)
oup(T1/5),
by Conditions 3 and 4.
Lemma 17. Assume that Conditions 1, 2, 3,
4
Qi(O, &)=
and 5 hold. Then,
i+T-
a +T-1R
where
d"
W
=
(
'
Qi
-
+
oUP(T'1
=
10
),
R
= oup(T1/
5
),
j=1
and
i4
is the jth element of
.
Proof. By a mean value expansion around (0, aio),
da
i (0, &i)
do
6 + L
=
- cei~j) + ZL
j=1
(ii)(&ij
j=1
oj(P,(,)( j -
00j),
&i) and (0, ajo). The expressions for 4i can be obtained using the expansions
for 'io in Lemma 13 since 7-i - io = oup(T- 3 /1 0 ). The order of this term follows from Lemma 13 and the
where (0, ai) lies between (b,
CLT for independent sequences. The remainder term is
d
do
Rii=
[ Qas Ri'
/(Q
+
(P, d) - Qa )T(&ij
o oj)1
-
+ E
j=1
di)T(6 - Oo,j).
6(6,
j=1
The uniform rate of convergence then follows by Lemmas 8 and 13, and Theorem 1.
Lemma 18. Suppose that Conditions 1, 2, 3, 4, and 5 hold. Then,
(E.1)
+ T-1 2 Q1
VTT(io - 7io) =
+ T
R2i,
where
dg+d
Qi
i,
ad
) =
-
(T"
T2"
ji+
1
A
=
V(
--
T"I)
=
(Ti/),
o
diag[OAW
=
oUP
T/
R 2 = OuP (T3/10
Also,
n
Qi= Op(1.
i=1
Proof. By a second order Taylor expansion of the FOC for io, we have
dg+d,
0t
(o,
is0 ) = t
+ T(Mo - g'o) +
E
j=1
(iO,j - 70,5) fi'j (00,;7i)(MO -
YiO),
13
where yi is between io and
7yiO.
The expression for Qji can be obtained in a similar fashion as in Lemma
A4 in Newey and Smith (2004). The rest of the properties for Qji follow by Lemma 5 applied to the second
stage, Lemma 16, and an argument similar to the proof of Theorem 1 in HK that uses Corollary A.2 of Hall
and Heide (1980, p. 278) and Lemma 1 of Andrews (1991). The remainder term is
dg+d.
-
=
(T ) -1 A9Ri +
i, Ti Ri] /2
- -yio) +
d9 +d .
j/1
dq +da
- -yio)/2
VI(920,j - YiO,j) )d-(T" (0, 77i) -- TPn ) -_(io
-(Till)
j=1
-
]Rji.
(?io- -Yio) + diag[0,
diag[ , R1IV
(Ti")
0
The uniform rate of convergence then follows by Lemmas 5 and 16, and Conditions 3 and 4.
Lemma 19. Suppose that Conditions 1, 2,
3, 4, 5, and 6 hold. Then,
EZ i + N(0, E[Vi,
Q1 -
[B
+B
+BO
+B]
=: B,,
where
diag(E,, Pi)
V=
B'
-
( )
B
BG
__
BG
HI
(
B
-i
H
,
)
a:
E[g(zi)g(zi)'Pa g(zi,t _)],
H
- H
GaHi.
C GI Q-', and P, = Q-1 - Q
)G, H
(zitj)
/2+
2 + E [Ga,(zit)H
(zi)'Pasg(zitj),
E[G
Qci
X Lk = (G=1G
,,
"
B
for
Gac
Proof. The resu lts follow by Lemmas 16 and 18, noting that
--
EE~
*
0
E
T
E
H
H'(
P
) g(zit),
P
0 ) , E
Pi
G( '
0,
Ha
i
Zc,
E [Ge, (zit)'Pig(zi,tj)]
A
j=
if j
E [GQ, (zit)'Hig(zi,tj)]
d;
if j >d,.
E [diag[O,,Wjk]
= E[g(zit)g(zit)'Paig(zit+j)]+
a
H
- Ha
)
14
APPENDIx
F.
'(00, oy)
STOCHASTIC EXPANSION FOR
AND 9i(0o,
io)
We characterize stochastic expansions up to second order for one-step and two-step profile scores of the
common parameter evaluated at the true value of the common parameter. The expressions for the scores
and their derivatives in the components of the expansions are given in Appendix G.
Lemma 20. Suppose that Conditions 1, 2,
3, and 4 hold. Then,
=T 1 /2
(0, ~)
+ T-'Q
+T
R
where
=M>w"Vi
=
oup(T 1 0 ),
QW
= Mw
+
w
+ 1d
M 0i
=oup(T
j=1
V
d(Mv - MW)
=
=
osp(T 1/ 10), R
= oup (T 2 /5).
Also,
Qi = Op (1).
Op(1),
Vn i.1
ni=1
Proof. By a second order Taylor expansion of
o
s
o)=
s0
+
O
i
1 (0o, ijo) around ijo =
Y iO
-iO) - -
'Yio,
7i) (io - 'Yio),
-j Yij)Mij(0,
j=1
where
7y is between ijo and yio. Noting that 1 (0, yio)
14, we can obtain the expressions for
terms follow by the properties of
RC
=
iW and QW , after some algebra.
QW. The remainder term is
jo in Lemma
The rest of the properties for these
W and
1d9 +d~,W
MWRR
= 0 and using the expansion for
+ 0w RW + -
R
2Mi - Yio) +
AlyM
M R
1
j=1
dg+d .'i)
- E
+
- -Yio).
(~7;N0j - 74i0, )VIT(M iW(0,7Z) - MM)"T-No
j=1
The uniform order of R
follows by the properties of the components in the expansion of ~jo, Lemma 5,
E
and Conditions 3 and 4.
Lemma 21. Suppose that Conditions 1, 2, 3, and 4 hold. We then have
yw = R[G'
-+ N(0,V7),
EQW
where B
BW'
=
,B =
-G'
Z~
G'
(G',W-1Gr)~,
B'
= -GI
-
E[Q
=
+ B'
+B.'|)
=: BW
Bw,' +BWG + Bw1S) , BX"c = Eoo_
PQ H W'/2 and P~7
] =[B,
PWQ.PWGo ],
l1 G', (Id . 9 ej) H"QiP
W-1 - W--'G,2 H
.
./|2, HP
E [Goe(zi)'Pjg(zi,t-j)],
=
W G2,WJ-=,
=
15
Proof. The results follow by Lemmas 20 and 15, noting that
Mw( PWQ.HW'
Qi
a,
E
E [
pW Qj pW
)
MW'
E [Go (zit'Pa g(zitj)]
=-oo
E [.WM
P',
_Qj
H
, RHI
.P'
0 ej-dJH $QiPW j
1a
G.(Id,,
-G
v]
if j < d,;
if j > d,.
0
Lemma 22. Suppose that Conditions 1, 2, 3, and 4 hold. Then, for 9W( 0o) = n- 1
-+ N (rB',
\5W(00)
V
where B'
_1'
,(Oo,),
VW),
and V7w are defined in Lemma 21.
Proof. By Lemma 20,
nTgw(0o)
1:
-Ca
T~>RW
'Si
i=1
i=1
=Op(1)
7=
=oP(1)
=0P(i)
ZQ
4n
1)
+op(1).
Tni=1 18
Then, the result follows by Lemma 21.
Lemma 23. Suppose that Conditions 1, 2, 3, 4, 5, and 6 hold. Then,
9i(0o, 4o)
= T-
1/2 '
+T-'
Ql,
+T-3/2R2si,
where all the terms are identical to that of Lemma 20 after replacing W by Q. Also, the properties of all the
terms of the expansion are the analogous to those of Lemma 20.
Proof. The proof is similar to the proof of Lemma 20.
Lemma 24. Suppose that Conditions 1, 2, 3, 4, 5, and 6 hold. Then,
I
d
n
=
Ha, = E3aG'
+ B" + BW) , B 0
-G'.o (B, + B
,-1 and E,
-
J
=
E[GPjGoj]
$E [Q1,i = R[BB + BC] =: B,,
Q1t
where B
N(0, J5 ),
-
0 E [Got
j'Pg(z_,
Pt , =
0-7GHO,
'G
(G'$,
Proof. The results follow by Lemmas 16, 18, 19 and 23, noting that
E [?,P'[
= M
0
)
Ml
E
aCE
[Go, (zit)'Pig(zi,t._j)],
E [jijM
] = 0.
j=O
0
16
Lemma 25. Suppose that Conditions 1, 2, 3, 5, and 4 hold. Then, or i(6 0 ) = n-,
z si +
-nTh'(60 ) = IE
-B
T
V_ i=1
8
± op(l)
~Nh
Zi=1
'i(00, Yio),
J)
8
where ,qi and B, are defined in Lemmas 23 and 24, respectively.
Proof. Using the expansion form obtained in Lemma 23, we can get the result by examining each term with
D
Lemma 24.
APPENDIx
G.
SCORES AND DERIVATIVES
G.1. One-Step Score and Derivatives: Individual Effects. We denote dimensions of g(zit), aj, and 6
by dg, da and do. The symbol 0 denotes kronecker product of matrices, and Id. denotes a da-order identity
matrix. Let Ga, (zit; 6, ai) := (G00 i (zit; 6, al)', ... , GOfiad (zit; 6, al)')', where
We denote derivatives of G,. (zit; 0,
scripts for higher order derivatives.
Gti (zit; 0, ai)
=
(zt;
Gaaij
(zit; 6,, ai
ail) =
Ge
ac) with respect to aij by G 0 a,.
3
ac), and use additional sub-
(zit; 0,
G.1.1. Score.
Gai (zit; 0, aj)'Ai
g(zit; 6, oe) + WA8
I1
T
(5ai(0,
ai)'Ai
g (0, aj) + W AJ
G.1.2. Derivatives with respect to the fixed effects.
First Derivatives
t'(7Y, 1 )
^wii )
T((,,o
do(0,
G ai)'(1d,
(& Ai)
Wi
T~W
TW
=
c))G]=6(o
E[T
-Q [T]
0i e
ci
Second Derivatives
Gacice1 j (6, ai)'(d~. 09 Aj)
Ti
(0, 7)
-
E2T
(0,
Gar,
(6, at)'(I
0 ed
)
0
=E-
(-yio; Oo)
I
GI
0
G'
0
0
if
,
\
if j < d,;
Gaaij
G0
(Id
if j > da;
0
G5aai (0, aj)
)J
0e
9 3 -dJ
0
0
0
if
j
> d,.
j > d,.
17
Third Derivatives
(0, a)'(I1d. 0 Ai) Go00ik (0, a%)'
0
Goaacjk (0, ac)
(0, a)'(d. 0 ekd-) 0
dcfala~aij
G,,j
8
'Y,k
0
'Yi,j 9-Y4
if
3,
3,
0
0
afiiv(0, Y )
,) j
Giacik (0 1ci)(d-
0
dj
0
0
0
0
Tijk-j( ~k
0
G'aaij
Gaeak
,k
0
)k I
0
G/cai, (Id- 0 ek-d)
=
0
0
E[TWj
0
GIa~k~(d
GQUik (Id
0 0 ejdj~
0
0
0
if j
dak > d 0 ;
if j
df),k> d,;
5 dc, k < d,;
if j 5 da, k > dc;
if j > da,k < da;
if j > da, k > da.
if j > dc,k > da-
0
G.1.3. Derivatives with respect to the common parameter.
First Derivatives
(0,t (y 0, )
=
]
E
N£=
7
ij
_
_
(
s (0, c i)'A
0GE[9
Goi '
G.2. One-Step Score and Derivatives: Common Parameters. Let Goa, (zit; 0,
(GeoI (zit; 0, oi)',
.
., G
d
Goa,,j (zit; 0, ac)
= 9Go(zit; 0, a)
We denote the derivatives of Goa, (zit; 0, aj) with respect to a , by Goa,c,,
(zit; 0, c), and use additional
subscripts for higher order derivatives.
G.2.1. Score.
T
o'i)
sr(0,
=
Go(zit; 0, aj)'A = -Go(0,
T
t=1
G.2.2. Derivatives with respect to the fixed effects.
First Derivatives
ao)
(zit; 0, ac)')', where
i)'Ai.
18
099
aj(0)yz)
=
Miw
=
_
= --
E [A=
_/
--
(Goj (6, a)'(Id. 0 Aj)
-
Goe(0, a)
).
0 G's.
-(
Second Derivatives
2
_
j (0,7Yi)
%W(0,y)
(
-
_
do,,,ij (0, ozi)'(Id. 0 Aj)
if j
E M I(0, 7 o) =
=
G'sa (Id.
-
0
ej-d.)
),
da;
if j > da.
0 ),
50i (0, Cei)'(Id. 0 ej-d)
a-yi~j
ay
4Y3
if j
0a, (0,
da;
if j > d,.
Third Derivatives
V
(, Y
g7
k"0
= Yi,ka7i,jayi
0
-
=
E
0 ekd~a)
)
G 1a,aik(, 1ai)'(d. 0 ej-d
-
M,
(0, )'(I.
G,,a,aij
0
G'a
if
i~j
0
if j
da, k > da;
0if
j > d, k < da
if j > da, k > da.
j < da,k
(Id
Ga aij
ek-d)
0
if j < da,k > dca;
-
G0I
ejdj
0
if
j
> da,k
da;
da;
[ik]
da;
if'j > da, k > da.
0 0 ),
G.2.3.
da, k
),
0
,
if j
Goaaijk(0,ai)'
- oa,aa, 3 k (0, a )'(Id . 0 Aj)
Derivatives with respect to the common parameters.
First Derivatives
( ,3)
(0
=
=
E[
=Yj)
-Goo
(0, aj)'Ai.
] =0.
G.3. Two-Step Score and Derivatives: Fixed Effects.
G.3.1. Score.
1
T t=_
G,(zit; 0, ai)'Ai
( g(zit; 6,ai) + 2i(0,&)Ai
Sa(0, cei)'j
g (o, aj) + Q Ai
Note that the formulae for the derivatives of Appendix G. 1 apply for
need to derive the derivatives for t .
(U2i - Qi)Ai
in, replacing W by Q. Hence, we only
19
G.3.2. Derivatives with respect to the fixed effects.
First Derivatives
(01 ai,
a(o,
T R Tf~(6'yB)
i-y) (0
ayi ys)
(
0
o
0 Qi(0, di) -- pi
0
E0
E[TR--(
1 fl=
-T
TR
-
[~-]
0 E [ 2j -- Qj
Second and Third Derivatives
Since TfR(-yi,
6) does not depend on ^yj, the derivatives (and its expectation) of order greater than one are
zero.
G.3.3. Derivatives with respect to the common parameters.
First Derivatives
a
a-
vR(0,
(, -Y') = 0.
G.4. Two-Step Score and Derivatives: Common Parameters.
G.4.1. Score.
T
s9 (6,'y)
=
Go (zit;6, a)'Ai = -do, (6, aj)'Ai.
T
t=1
Since this score does not depend explicitly on 6i(,
di), the formulae for the derivatives are the same as in
Appendix G.2.
REFERENCES
ANDREWS, D. W. K. (1991), "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation," Econometrica 59, 817-858.
FERNANDEZ-VAL,
I. AND, J. LEE (2012), "Panel Data Models with Nonadditive Unobserved Heterogene-
ity: Estimation and Inference," unpublished manuscript, Boston University.
HAHN, J., AND G. KUERSTEINER (2011), "Bias Reduction for Dynamic Nonlinear Panel Models with
Fixed Effects," Econometric Theory 27, 1152-1191.
HALL, P., AND C. HEIDE (1980), Martingale Limit Theory and Applications. Academic Press.
NEWEY, W.K., AND D. McFADDEN (1994), "Large Sample Estimation and Hypothesis Testing," in R.F.
ENGLE AND D.L. McFADDEN, eds., Handbook of Econometrics, Vol.
4. Elsevier Science. Amsterdam:
North-Holland.
NEWEY, W.K., AND R. SMITH (2004), "Higher Order Properties of GMM and Generalized Empirical
Likelihood Estimators," Econometrica 72, 219-255.
Table Al: Common Parameter 62
Pi =
0
p, =
0.3
Estimator
Bias
SD
SE/SD
p;.05 Bias
SD
SE/SD
OLS - FC
IV - FC
OLS - RC
0.06
0.00
0.04
0.04
0.01
0.01
0.01
0.84
0.90
0.97
1.00 0.06
0.08 -0.01
0.01
0.02
0.83
0.84
0.01
0.01
0.97
0.97
1.00
1.00
1.00
0.04
0.04
0.04
0.01
0.01
0.01
0.99
0.99
0.99
0.00 0.01
0.00 0.01
0.00 0.01
1.00
0.99
0.99
0.06
0.06
0.06
0.00 0.01
0.00
0.00
0.01
0.01
1.01
1.01
1.01
OLS - FC
IV - FC
OLS - RC
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
0.12
0.00
0.10
0.10
0.10
0.00
0.00
0.00
0.01
0.02
0.01
0.01
0.01
0.02
0.02
0.02
1.09
0.94
1.06
1.06
1.06
0.98
0.97
0.97
1.00 0.12
0.07 -0.01
1.00 0.10
1.00 0.10
1.00 0.10
0.06 0.00
0.05 0.00
0.05 0.00
0.01
0.02
0.01
0.01
0.01
0.02
0.02
0.02
1.03
0.89
1.05
1.05
1.05
0.96
0.95
0.95
OLS - FC
IV - FC
OLS - RC
BC - OLS
IBC - OLS
IV - RC
0.16
0.00
0.15
0.15
0.15
0.00
0.00
0.00
0.01
0.03
0.01
0.01
0.01
0.03
0.03
0.03
0.16
0.00
0.15
0.15
0.15
0.00
0.00
0.00
0.01
0.03
0.01
0.01
0.01
0.03
0.03
0.03
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
0.04
p, =
p;.05 Bias
V=2
1.00 0.06
0.11 -0.01
1.00 0.04
1.00 0.04
1.00 0.04
0.05 0.00
0.05 0.00
0.05 0.00
SD
0.6
pi =
SE/SD p;.05 Bias
SD
0.9
SE/SD p;.05
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.84
0.78
1.02
1.02
1.02
1.00
1.00
1.00
1.00
0.18
1.00
1.00
1.00
0.05
0.05
0.05
0.07
-0.01
0.04
0.04
0.04
0.00
0.00
0.00
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.71
0.63
0.96
0.96
0.96
1.01
1.00
1.00
1.00
0.28
1.00
1.00
1.00
0.05
0.05
0.05
0.01
0.02
0.01
0.01
0.01
0.02
0.02
0.02
1.10
0.92
1.08
1.08
1.08
1.01
1.00
1.00
1.00
0.09
1.00
1.00
1.00
0.05
0.05
0.05
0.12
-0.01
0.11
0.11
0.11
0.00
0.00
0.00
0.01
0.03
0.01
0.01
0.01
0.02
0.02
0.02
1.07
0.79
1.07
1.07
1.07
1.00
0.99
0.99
1.00
0.15
1.00
1.00
1.00
0.06
0.06
0.06
0.01
0.03
1.25
0.92
1.00
0.08
0.16
-0.01
0.01
0.04
1.34
0.92
1.00
0.08
10 = 4
BC - IV
IBC - IV
1.27
1.00
0.95
0.06
1.20
1.00
1.20
1.20
0.98
0.95
0.95
1.00
1.00
0.06
0.06
0.06
0.94
1.00 0.12
0.08 -0.01
1.00 0.11
1.00 0.11
1.00 0.11
0.06 0.00
0.06 0.00
0.06 0.00
0 =6
1.00 0.16
0.06 -0.01
1.21
1.00
0.15
0.01
1.21
1.00
0.15
0.01
1.26
1.00
1.21
1.21
1.00
0.97
0.97
1.00
1.00
0.04
0.15
0.15
0.00
0.01
0.01
0.03
1.21
1.21
1.01
1.00
1.00
0.05
0.15
0.15
0.00
0.01
0.01
0.03
1.26
1.26
1.05
1.00
1.00
0.04
0.04
0.00
0.03
0.98
0.05
0.00
0.03
1.02
0.04
0.04
0.00
0.03
0.98
0.05
0.00
0.03
1.02
0.04
1.22
RC/FC rcfcrs to randon/fixed coefficiinL model. BC/IBC refers to bias corrected/iterated bias corrected estimates.
Note: 1, 000 repetitions.
C-
Table A2: Mean of Individual Specific Parameter Pi =
pi =
0
p, =
0. 3
Estimator
Bias
SD
SE/SD
p;.05
Bias
SD
SE/SD
OLS - FC
IV - FC
2.33
0.08
OLS - RC
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
1.16
1.16
1.16
0.01
-0.01
-0.01
1.65
1.59
1.53
1.53
1.53
1.51
1.51
1.51
0.35
0.40
1.02
0.97
1.07
1.02
1.02
0.78
0.44
0.12
0.14
0.14
0.04
0.04
0.04
2.58
0.16
1.15
1.15
1.15
-0.01
-0.02
-0.03
1.91
1.72
1.65
1.65
1.65
1.62
1.62
1.62
0.31
0.37
0.96
0.92
0.92
1.00
0.96
0.96
OLS - FC
4.15
0.09
3.19
3.19
3.19
0.06
0.00
-0.01
1.84
1.85
1.76
1.76
1.76
1.78
1.78
1.78
0.52
0.59
1.06
0.93
0.93
1.15
1.02
1.02
0.90
0.25
0.41
0.50
0.50
0.03
0.05
0.05
4.43
0.21
3.12
3.12
3.12
-0.01
-0.08
-0.08
1.95
1.92
1.81
1.81
1.81
1.86
1.86
1.86
0.49
0.57
1.04
0.91
0.91
1.10
0.98
0.98
5.62
0.14
4.69
4.69
4.69
0.09
-0.05
-0.06
2.13
2.29
2.10
2.10
2.10
2.30
2.30
2.30
0.62
0.69
1.08
0.88
0.88
1.12
0.95
0.95
0.93
0.17
0.53
0.69
0.69
0.04
0.06
0.06
5.87
0.26
4.59
4.59
4.59
0.05
-0.10
-0.10
2.25
2.34
2.14
2.14
2.14
2.33
2.33
2.32
0.58
0.68
1.07
0.88
0.88
1.11
0.94
0.94
IV - FC
OLS - RC
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
OLS
- FC
IV - FC
OLS - RC
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
0.97
p, =
p;.05
E[a]
0. 6
p_ =
Bias
SD
SE/SD
p;.05
Bias
SD
SE/SD
p;.05
0.79 3.01
0.47 0.46
0.12 1.19
0.14 1.19
0.14 1.19
0.05 0.02
0.06 0.00
0.06 0.00
* = 4
0.90 4.90
0.27 0.56
0.38 3.12
0.48 3.12
0.48 3.12
0.03 0.03
0.05 -0.04
0.05 -0.04
V=6
0.92 6.19
0.19 0.53
0.51 4.52
0.67 4.52
0.67 4.52
0.03 0.03
0.06 -0.12
0.06 -0.13
1.92
1.66
1.59
1.59
1.59
1.56
1.56
1.56
0.30
0.39
0.99
0.95
0.95
1.04
1.00
1.00
0.84
0.46
0.12
0.14
0.14
0.05
0.06
0.06
3.68
0.96
1.25
1.25
1.25
0.08
0.06
0.06
2.20
1.80
1.62
1.62
1.62
1.59
1.59
1.59
0.27
0.37
0.97
0.93
0.93
1.03
0.98
0.98
0.89
0.53
2.08
1.89
1.76
1.76
1.76
1.78
1.78
1.78
0.46
0.58
1.07
0.94
0.94
1.15
1.02
1.02
0.93
0.27
0.38
0.47
0.47
0.03
0.05
0.05
5.45
1.03
3.18
3.18
3.18
0.10
0.03
0.03
2.22
1.94
1.78
1.78
1.78
1.78
1.78
1.78
0.43
0.57
1.06
0.93
0.93
1.15
1.02
1.02
0.95
0.32
0.38
0.47
0.47
0.03
0.05
0.05
2.31
2.31
2.11
2.11
2.11
2.23
2.23
2.23
0.57
0.69
1.09
0.89
0.89
1.16
0.97
0.97
0.93
0.19
0.50
0.64
0.64
0.02
0.06
0.06
6.35
0.80
4.30
4.30
4.30
-0.10
-0.26
-0.26
2.28
2.26
2.01
2.01
2.01
2.18
2.18
2.18
0.57
0.70
1.15
0.94
0.94
1.19
1.00
1.00
0.95
0.20
0.46
0.61
0.61
0.02
0.05
0.05
RC/FC refers to random/fixed coefficient model. BC/IBC refers to bias corrected/iterated bias corrected estimates.
Note: 1,000 repetitions.
0.9
0.13
0.15
0.15
0.05
0.06
0.06
Table A3: Standard Deviation of the Individual Specific Parameter
Estimator
Bias
P, = 0
SD SE/SD
p, =
p;.05
Bias
SD
0. 3
SE/SD
p, =
p;.05 Bias
=
0.01
BC - OLS -0.63
IBC - OLS -0.63
IV - RC
0.38
OLS - RC
1.06
1.10
1.10
1.08
1.02
1.04
1.04
1.03
0.05
0.10
0.10
0.05
0.06
0.06
0.15
-0.48
-0.48
0.47
-0.16
-0.16
1.06
1.11
1.11
1.10
1.14
1.14
1.02
0.98
-1.13
-1.13
1.83
-0.26
-0.26
1.20
1.44
1.44
1.29
1.52
1.52
1.17
2.60
-1.71
-1.71
3.87
-0.42
-0.41
1.40
2.14
2.14
1.55
2.23
2.23
BC - IV
IBC - IV
-0.25
-0.25
1.13
1.13
1.05
OLS - RC
0.89
BC - OLS
IBC - OLS
IV - RC
BC - IV
IBC - IV
-1.24
-1.24
1.84
-0.25
-0.25
1.21
1.46
1.46
1.28
1.52
1.52
1.17
1.19
1.19
1.20
0.04
0.08
0.08
0.17
0.03
0.03
OLS - RC
BC - OLS
IBC - OLS
2.35
-2.06
-2.06
1.33
2.04
2.04
IV - RC
3.79
1.52
BC - IV
-0.49
-0.49
2.13
2.13
1.38
1.41
1.41
1.31
1.37
1.37
0.14
0.00
0.00
0.46
0.00
0.00
1.05
1.04
1.04
1.02
1.04
1.04
SD
o1 = E[(ci -
0. 6
pi)2 1/2
P, =
SE/SD
p;.05
0.99
1.01
1.01
0. 9
Bias
SD
SE/SD
p;.05
0.06
0.17
1.06
0.10
-0.46
1.11
0.10 -0.46
0.06 0.46
0.07 -0.17
0.07 -0.17
1.11
1.11
1.16
1.16
0.99
1.01
1.01
0.99
1.00
1.00
0.06
0.09
0.09
0.06
0.06
0.06
0.06
0.05
0.05
1.09
-0.98
-0.98
1.05
1.23
1.24
1.23
1.24
1.22
0.18
1.87
1.18
1.17
0.03
0.03
-0.21
-0.21
1.37
1.38
1.18
1.16
0.08
0.03
0.03
0.20
0.02
0.02
0.21
0.00
0.00
0.47
2.69
-1.54
-1.54
3.87
1.31
1.78
1.80
1.48
1.28
1.28
1.26
1.23
0.00
-0.40 1.96
1.24
0.00
-0.39
1.22
2
0.05 0.11 1.08
0.09 -0.52 1.12
0.09 -0.52 1.12
0.06 0.41 1.13
0.06 -0.22 1.18
0.06 -0.22 1.18
0.98
1.00
1.00
,= 4
IBC - IV
1.17
1.20
1.20
1.20
0.05 1.08
0.08 -1.02
1.16
1.41
1.19
1.20
1.20
0.08
-1.02
1.41
1.16
1.19
1.19
0.16
0.02
0.02
1.85
-0.26
-0.26
1.26
1.51
1.51
1.18
1.30
1.30
0.21
0.01
V= 6
2.57
-1.75
1.37
2.06
1.31
1.35
1.30
1.28
1.29
1.29
0.01
-1.75
2.06
0.49
3.78
1.50
1.35
1.30
0.01
0.01
-0.55 2.14
-0.55 2.14
1.35
1.35
1.17
1.18
RC/FC refers to random/fixed coefficient model. BC/IBC refers to bias corrected/iterated bias corrected estimates.
Note: 1,000 repetitions.
1.97
0.38
0.00
0.00
0.60
0.01
0.02