Efficient Estimation of Dynamic Panel Data Models:
Alternative Assumptions and Simplified Estimation
Seung C. Ahn
Arizona State University, Tempe, AZ 85187, USA
Peter Schmidt*
Michigan State University, E. Lansing, MI 48824, USA
Abstract
This paper considers the estimation of dynamic models for panel data. It shows how
to count and express the moment conditions implied by a variety of covariance restrictions.
These conditions can be imposed in a GMM framework. Many of the moment conditions are
nonlinear in the parameters. We derive a simple linearized estimator that is asymptotically as
efficient as the nonlinear GMM estimator, and convenient tests of the validity of the nonlinear
restrictions.
Key Words: Panel data; Dynamic models; Stationarity; GMM estimation; Conditional
moment tests
JEL Classification: C23
August, 1993
Revised July, 1994
Revised July, 1995
*
Corresponding author, Department of Economics, Michigan State University, E. Lansing, MI
48824, USA. Phone: (517) 355-8385; FAX: (517) 336-1068.
References
Ahn, S.C., 1990, Three essays on share contracts, labor supply, and the estimation of models
for dynamic panel data, Unpublished Ph.D. dissertation (Michigan State University, E.
Lansing, MI).
Ahn, S.C., 1995, Model specification testing based on root-T consistent estimators,
Unpublished manuscript (Arizona State University, Tempe, AZ).
Ahn, S.C. and P. Schmidt, 1995a, A separability result for GMM estimation, with applications
to GLS prediction and conditional moment tests, Econometric Reviews 14, 19-34.
Ahn, S.C. and P. Schmidt, 1995b, Efficient estimation of models for dynamic panel data,
Journal of Econometrics 68, 5-27.
Anderson, T.W. and C. Hsiao, 1981, Estimation of dynamic models with error components,
Journal of the American Statistical Association 76, 598-606.
Arellano, M. and S. Bond, 1991, Tests of specification for panel data: Monte Carlo evidence
and an application to employment equations, Review of Economic Studies 58, 277-97.
Arellano, M. and O. Bover, 1990, Another look at the instrumental variables estimation of
error-component Models, Unpublished manuscript (London School of Economics,
London).
Arellano, M. and O. Bover, 1995, Another look at the instrumental variables estimation of
error-component models, Journal of Econometrics 68, 29-51.
Blundell, R. and S. Bond, 1994, The role of initial conditions in GMM and ML estimators for
dynamic panel data models, Unpublished manuscript (University College, London).
Breusch, T.S., G.E. Mizon and P. Schmidt, 1989, Efficient estimation using panel data,
Econometrica 57, 695-701.
Holtz-Eakin, D., 1988, Testing for individual effects in autoregressive models, Journal of
Econometrics 39, 297-308.
Holtz-Eakin, D., W. Newey and H.S. Rosen, 1988, Estimating vector autoregressions with
panel data, Econometrica 56, 1371-1396.
Hsiao, C., 1986, Analysis of panel data (Cambridge University Press, New York, NY).
Newey, W., 1985, Generalized method of moments specification testing, Journal of
Econometrics 29, 229-256.
Schmidt, P., S.C. Ahn and D. Wyhowski, 1992, Comment, Journal of Business and Economic
Statistics 10, 10-14.
Wooldridge, J.M., 1995, Estimating systems of equations with different instruments for
different equations, Journal of Econometrics, forthcoming.
1. INTRODUCTION*
A standard approach to the estimation of the dynamic panel data regression model [e.g.,
Anderson and Hsiao (1981), Hsiao (1986), Holtz-Eakin (1988), Holtz-Eakin, Newey and
Rosen (1988), Arellano and Bond (1991)] is to first difference the equation, and then use
instrumental variables (IV). More recent papers [Ahn and Schmidt (1995b), Arellano and
Bover (1990, 1995)] have shown that, under certain assumptions, there are additional moment
conditions that are not exploited by these IV estimators, and have suggested that the
additional moment conditions be imposed in a generalized method of moments (GMM)
framework.
In this paper, we extend these analyses in three ways. First, we count and express the
moment conditions available under alternative sets of assumptions about initial conditions and
the errors. Second, we propose a linearized GMM estimator, starting from an initial
consistent estimator such as a linear estimator based on a subset of the moment conditions,
and we derive simple moment tests of the nonlinear moment conditions. Third, for the case
that only linear moment conditions are used, we investigate the circumstances under which
the optimal GMM estimator is equivalent to a linear IV estimator. These circumstances are
quite restrictive, and go beyond uncorrelatedness and homoskedasticity of the errors.
The plan of the paper is as follows. Section 2 shows how to count and express the
moment conditions available under alternative sets of assumptions. Section 3 considers some
*
The first author gratefully acknowledges the financial support of the College of Business,
Arizona State University. The second author gratefully acknowledges the financial support of
the National Science Foundation. Both authors are grateful for the helpful comments of the
editor and two referees.
2
implications of the assumption of stationarity. Section 4 discusses linearized estimation and
conditional moment tests. Finally, Section 5 contains some concluding remarks.
2. Moment Conditions under Alternative Assumptions
In this section, we count and express the moment conditions implied by various sets of
assumptions. We consider the simple dynamic panel data model:
(1)
Here i = 1, ..., N denotes cross sectional unit (individual) and t = 1, ..., T denotes time.
Throughout the paper, we consider the case that N is large and T is small, so that asymptotics
will be derived under the assumption that N → ∞ with T fixed. We can also write the T
observations on (1) for person i as
(2)
where yi′ = (yi1,...,yiT) and yi,-1 and ui are defined similarly. See Ahn and Schmidt (1995b) for
more detail on the notation.
This model does not contain any additional regressors beyond the lagged dependent
variable, but this is only an expository point. The moment conditions implied by exogeneity
assumptions on additional regressors have been identified by Schmidt, Ahn and Wyhowski
(1992), Ahn and Schmidt (1995b) and Arellano and Bover (1995); they can just be added to
the sets of moment conditions identified in this section.
We assume independence of all variables across individuals. We will consider various
subsets of the following assumptions.
3
For all i, εit is uncorrelated with yi0 for all t.
(A.1)
For all i, εit is uncorrelated with αi for all t.
(A.2)
For all i, the εit are mutually uncorrelated.
(A.3)
For all i, var(εit) is the same for all t.
(A.4)
Given the four assumptions (A.1)-(A.4), there are 16 possible cases corresponding to
imposing or not imposing each of them. One is the degenerate case of no assumptions.
Furthermore, although we have deliberately avoided assumptions about the generation of yi0, it
is reasonable to view it as being a linear function of αi and of εi0, εi,-1, ... . Therefore we will
not impose (A.1) (uncorrelatedness of yi0 and εit, t=1,...,T) unless we also impose both (A.2)
(uncorrelatedness of αi and the εit) and (A.3) (mutual uncorrelatedness of the εit). This leaves
nine subsets of assumptions to consider.
We begin with the case in which we assume all of (A.1)-(A.4). We will call this CASE
A, and it is the only case that we will consider in detail. The method by which the moment
conditions implied by (A.1)-(A.4) are identified is as follows. Let Σ be the covariance matrix
of the things we make assumptions about, namely (εi1,...,εiT,yi0,αi); see Ahn and Schmidt
(1995b, equation (6)). Let Λ be the covariance matrix of the "observables" that can be
written in terms of data and parameters, namely (u1i,...,uiT,yi0); see Ahn and Schmidt (1995b,
equation (7)). Assumptions (A.1)-(A.4) impose restrictions on Σ and imply restrictions on Λ.
For example, under (A.1)-(A.4), we have σαt ≡ cov(εit,αi) = 0 for all t, σ0t ≡ cov(yi0,εit) = 0
for all t, σts ≡ cov(εit,εis) = 0 for all t ≠ s, and σtt (≡ var(εit)) the same (≡ σεε) for all t.
This implies three types of restrictions on Λ. First, λ0t ≡ cov(yi0,uit) is the same for t = 1,...,T.
Second, λtt ≡ var(uit) is the same for t = 1,...,T. Third, λts ≡ cov(uit,uis) is the same for t,s =
4
1,...,T, t ≠ s. The total number of restrictions is (T-1) + (T-1) + [T(T-1)/2 - 1] = T(T-1)/2 +
2T - 3. The number of restrictions may also be obtained simply by noting that the
(T+1)(T+2)/2 distinct elements of Λ depend on the four parameters σαα (≡ var(αi)), σ0α (≡
cov(yi0,αi)), σ00 (≡ var(yi0)) and σεε. These restrictions correspond to the moment conditions:
(3A)
(3B)
(3C)
Tedious but straightforward algebra shows that the conditions (3A)-(3C) are equivalent to
(4A)
(4B)
(4C)
where ūi = T-1Σtuit. The T(T-1)/2 moment conditions in (4A) can be motivated by firstdifferencing (1) to get
(5)
Holtz-Eakin (1988), Holtz-Eakin, Newey and Rosen (1988) and Arellano and Bond (1991)
have noted that the set of available instruments for (5) is [yi0,...,yi,t-2], and the moment
conditions (4A) express the validity of these instruments. Given assumptions (A.1)-(A.4),
Ahn and Schmidt (1995b) show that conditions in (4B) can be replaced by
which are linear in δ.
5
(6)
We now consider the moment conditions that arise under various subsets of (A.1)-(A.4),
as discussed above.
CASE B. In this case we impose (A.1), (A.2) and (A.3); that is, we impose all of the
assumptions except that we do not assume homoskedasticity of the εit. This is the set of
assumptions that was identified as the "standard assumptions" (SA) in Ahn and Schmidt
(1995b). We have the T-1 restrictions that λ0t is the same for all t, and the T(T-1)/2 - 1
restrictions that λts is the same for all t ≠ s, just as we did in CASE A. These restrictions can
be expressed as the linear moment conditions in (4A), plus the T-2 quadratic moment
conditions in (4B).
CASE C. In this case we impose (A.2), (A.3), and (A.4). That is, we impose all
assumptions except that we do not assume that yi0 is uncorrelated with the εit. This may be
arguably a poorly motivated case; however, it has been discussed in the literature as a "fixed
effects" treatment for the initial condition yi0, since it is allowed to be correlated with both αi
and the εit. The parameters σ0α and the σ0t are not separately identified. The elements of Λ
depend on the T+3 identified parameters σαα, σ00, σεε, (σ0α+σ01), ..., (σ0α+σ0T), so that the
number of restrictions is (T+1)(T+2)/2 - (T+3) = T(T-1)/2 + T - 2. We have the T-1
restrictions that λtt is the same for all t, and the T(T-1)/2 - 1 restrictions that λts is the same
for all t ≠ s, just as we did in CASE A. These correspond to the moment conditions in (3B)
and (3C).
CASE D. In this case we impose (A.2) and (A.3). Thus we assume that the εit are
mutually uncorrelated and uncorrelated with αi. The elements of Λ depend on the 2T + 2
6
identified parameters σαα, σ00, (σ0α+σ01), ..., (σ0α+σ0T), σ11, ..., σTT, and the total number of
moment conditions is T(T-1)/2 - 1. Specifically, λts is the same for all t ≠ s, just as in CASE
A or CASE C. These restrictions correspond to the moment conditions given in equation
(3C) above.
CASE E. In this case we impose (A.2) and (A.4). Thus we assume only that the εit are
homoskedastic and that they are uncorrelated with αi. This may be plausible if the εit are
autocorrelated. In this case Λ depends on the T(T-1)/2 + T + 2 identified parameters σ00,
(σαα+σεε), (σ0α+σ01), ..., (σ0α+σ0T) and (σαα+σst), s ≠ t. We have only the T-1 restrictions
that λtt is the same for all t. They can be expressed as in equation (3B) above.
CASE F. In this case we impose only (A.3) and (A.4); that is, we assume that the εit are
homoskedastic and mutually uncorrelated, but we make no assumptions about their
relationship to yi0 or αi. The elements of Λ depend on the 2T+2 identified parameters σ00,
σεε, (σ0α+σ01), ..., (σ0α+σ0T), (σαα+2σα1), ..., (σαα+2σαT). Thus we have T(T-1)/2 - 1
moment conditions. Specifically, we have the restrictions that λtt + λss - 2λts is the same for
all t, s = 1, ... , T, t ≠ s. These restrictions correspond to the moment conditions that E[(uituis)2] is the same for all t ≠ s; that is,
(7)
There has always been some question in the dynamic model about what constitutes a
"fixed effects" treatment. So-called "random effects" treatments of the model have typically
imposed stronger assumptions than ours about the generation of yi0, and some treatments have
assumed that yi0 is uncorrelated with αi [see, e.g., Hsiao (1986, section 4.3.2)]. The
treatments based on the moment conditions (4A) above have sometimes been called fixed-
7
effects treatments, even though they clearly depend on assumptions about yi0 and αi, such as
lack of correlation of these quantities with the εit. The case just considered is arguably as
close in spirit as we can get to a fixed effects treatment of this model without literally
assuming αi and yi0 to be fixed.
CASE G. In this case we assume only (A.2), that αi is uncorrelated with the εit. This
implies no moment conditions.
CASE H. In this case we assume only (A.3), that the εit are mutually uncorrelated. For
T ≥ 4, there are restrictions implied by the fact that the T(T-1)/2 off-diagonal elements of Λ
corresponding to covariances of the uit depend only on the T identified parameters
(σαα+2σα1), ..., (σαα+2σαT). Thus we have T(T-1)/2 - T = T(T-3)/2 restrictions, of the form
λqr + λst = λqs + λrt for distinct q,r,s,t = 1,...,T. The corresponding moment conditions are
(8)
CASE I. In this case we assume only (A.4), homoskedasticity of the εit. This implies no
moment conditions.
3. Assumptions about Stationarity
In this section, we consider the implications of two different types of stationarity
assumptions. We will first consider the following assumption:
The series yi0, ..., yiT is covariance stationary.
(S.1)
This assumption is made in addition to assumptions (A.1)-(A.4) of the previous section.
Arellano and Bover (1990) discuss the following condition:
cov(αi,yit) is the same for t = 0, 1, ..., T .
(9)
8
This is an assumption of the type made by Breusch, Mizon and Schmidt (1989); it requires
equal covariance between the effects and the variables with which they are correlated. Ahn
and Schmidt (1995b) show that, given assumptions (A.1)-(A.4), the condition in (9)
corresponds to the restriction that
(10)
and implies one additional moment restriction. Furthermore, they show that it also allows the
entire set of available moment conditions to be written linearly; see their equations (12A)(12B). An alternative, equivalent set of moment conditions for this case is given by Blundell
and Bond (1994).
To see the connection between (S.1) and (9), we use the solution
(11)
to calculate
(12)
where the calculation assumes (A.1)-(A.4). (S1) implies that var(yit) = σ00 for all t, which
occurs if and only if (10) holds and also
(13)
Thus (S.1) implies (10), which in turn implies (9). However, it also implies the restriction
9
(13) on the variance of the initial observation yi0. Imposing (13) as well as (A.1)-(A.4) and
(9) yields one additional, nonlinear moment condition:
(14)
The assumption (S.1) can be compared to the weaker stationarity assumption:
Conditional on (αi,yi0), (εi1, ..., εiT) is stationary.
(S.2)
This assumption implies the following restrictions on the covariance matrix Λ. First, λtt is the
same for t = 1, 2, ..., T. Second, λ0t is the same for t = 1, 2, ..., T. Third, for j = 1, 2, ...,
T-2, λt,t+j is the same for t = 1, 2, ..., T-j. The total number of moment conditions under (S.2)
is (T-1) + (T-1) + (T-2)(T-1)/2 = (T+2)(T-1)/2. The moment conditions can be expressed as
in (3A)-(3B) above, plus
(15)
4. Estimation and Specification Testing
In this section we provide some econometric detail on GMM estimation and specification
tests. We provide a simple linearized GMM estimator and convenient moment tests of the
validity of the moment conditions. We also discuss the relationship between GMM based on
the linear moment conditions and IV estimation. Our discussion will proceed under
assumptions (A.1)-(A.4), but can easily be modified to accommodate the other cases.
4.1. Notation and General Results
We consider the model
(16)
10
as given by Ahn and Schmidt (1995b), equation (24). The model may contain time-varying
explanatory variables (X) and time-invariant explanatory variables (Z) in addition to the
lagged dependent variable. For purposes of GMM it is convenient to focus on the T
observations for person i, and we will write
(17)
to emphasize the dependence of ui on ξ.
Exogeneity assumptions on X and Z generate linear moment conditions of the form
(18)
where Ri is a function of the exogenous variables. See Ahn and Schmidt (1995b), equations
(33) and (34), for definitions of Ri under different exogeneity assumptions. In addition, the
moment conditions given by (4A), (6) and (4C) above are valid. The moment conditions in
(4A) above are linear in ξ and can be written as E[Ai′ui(ξ)] = 0, where Ai is the T × T(T-1)/2
matrix
(19)
Similarly, the moment conditions in (6) above are also linear in ξ and can be written as
E[B1i′ui(ξ)] = 0, where B1i is the T × (T-2) matrix defined by
11
(20)
However, the moment conditions in (4C) above are quadratic in ξ.
We will discuss GMM estimation based on all of the available moment conditions and
GMM based on a subset (possibly all) of the linear moment conditions. Suppose that
is the
total number of moment conditions given in (4A), (6), (4C) and (18). Let Si (T × 1) be
made up of columns of Ri, Ai and B1i, so that it represents some or all of the available linear
instruments. The corresponding linear moment conditions are E[fi(ξ)] = 0, with
(21)
If the dimension of ξ is k, we assume
remaining
2
=
-
1
1
≥ k so that the conditions fi can identify ξ. The
moment conditions will be written as E[gi(ξ)] = 0. Since they are at
most quadratic, we can write
(22)
where g1i, g2i and g3i are
2
× 1,
2
× k and 2k × k matrices of functions of data,
respectively, and the dimension of the identity matrix is 2. An efficient estimator of ξ can
be obtained by GMM based on all of the moment conditions:
12
(23)
Define mN = N-1Σimi(ξ), with fN, f1N, f2N, gN, g1N, g2N and g3N defined similarly; and define MN
= ∂mN/∂ξ′ = [∂fN′/∂ξ,∂gN′/∂ξ]′ = [FN′,GN′]′, where GN(ξ) = g2N + 2(I
ξ′)g3N and FN = f2N.
Let M = plim MN, with F and G defined similarly. Define the optimal weighting matrix:
(24)
Let Ω̂ be a consistent estimate of Ω of the form
(25)
where ξ̂ is an initial consistent estimate of ξ (perhaps based on the linear moment conditions
fi, as discussed below); partition it similarly to Ω.
In this notation, the efficient GMM estimator ξ̃GMM minimizes NmN(ξ)′Ω̂-1mN(ξ). Using
standard results, the asymptotic covariance matrix of N½(ξ̃GMM-ξ) is [M′Ω-1M]-1. We can also
test the validity of the moment conditions E[mi(ξ)] = 0 using the usual overidentification test
statistic JN = NmN(ξ̃GMM)′Ω̂-1mN(ξ̃GMM). This statistic is asymptotically chi-squared with ( -k)
degrees of freedom under the joint hypothesis that all the moment conditions are legitimate.
4.2. Linear Moment Conditions and IV
Some interesting questions arise when we consider GMM based on the linear moment
conditions fi(ξ) only. The optimal GMM estimator based on these conditions is
(26)
13
This GMM estimator can be compared to the linear IV estimator that is of the same form,
but with Ω̂ff replaced by (ΣiSi′Si). The GMM estimator is generally more efficient than the
linear IV estimator. They are asymptotically equivalent in the case that Ωff is proportional to
E(Si′Si), which occurs if E(Si′uiui′Si) is proportional to E(Si′Si). For the case that Si consists
only of columns of Ri, so that only the moment conditions (18) based on exogeneity of X and
Z are imposed, this equivalence will generally hold. Arellano and Bond (1991) considered
the moment conditions (4A), so that Si also contains Ai, and noted that asymptotic
equivalence between the IV and GMM estimates fails if we relax the homoskedasticity
assumption (A.4), even though the moment conditions (4A) are still valid under only
assumptions (A.1)-(A.3) (our CASE B). In fact, even the full set of assumptions (A.1)-(A.4)
is not sufficient to imply the asymptotic equivalence of the IV and GMM estimates when the
moment conditions (4A) are used. Assumptions (A.1)-(A.4) deal only with second moments,
whereas asymptotic equivalence of IV and GMM involves restrictions on fourth moments
(e.g., cov(yi02,εit2) = 0). Ahn (1990) proved the asymptotic equivalence of the IV and GMM
estimators based on the moment conditions (4A) for the case that (A.4) is maintained and
(A.1)-(A.3) are strengthened by replacing uncorrelatedness with independence. Wooldridge
(1995) provides a more general treatment of cases in which IV and GMM are asymptotically
equivalent. In the present case, his results indicate that asymptotic equivalence would hold if
we rewrite (A.1)-(A.4) in terms of conditional expectations instead of uncorrelatedness; that
is, if we assume
(A.5)
(A.6)
14
A more novel observation is that the asymptotic equivalence of IV and GMM fails
whenever we use the additional linear moment conditions (6). This is so even if assumptions
(A.1)-(A.4) are strengthened by replacing uncorrelatedness with independence. When
uncorrelatedness in (A.1)-(A.3) is replaced by independence, Ahn (1990, Chapter 3, Appendix
3) shows that, while E(Ai′uiui′Ai) = σεεE(Ai′Ai) and E(Ai′uiui′B1i) = σεεE(Ai′B1i),
(27)
where d = E(ε4) - 3σεε2 and C is a square matrix of dimension (T-2) with 2 on the diagonal,
-1 one position off the diagonal, and 0 elsewhere. Under normality d = 0 but the term σεεC
remains.
4.3. Linearized GMM and Specification Tests
We now consider a linearized GMM estimator. Suppose that ξ̃ is any consistent estimator
of ξ; for example, ξ̃f. Following Newey (1985, p. 238), the linearized GMM estimator is of
the form
(28)
This estimator is consistent and has the same asymptotic distribution as ξ̃GMM.
When the LGMM estimator is based on the initial estimator ξ̃f, some further
simplification is possible. Applying the usual matrix inversion rule to Ω̂ and using the fact
that f2N′Ω̂ff-1fN(ξ̃f) = 0, we can write the LGMM estimator as follows:
(29)
Here Γ̂-1 = [f2N′Ω̂ff-1f2N]-1 denotes the usual estimate of the asymptotic covariance matrix of
15
N½(ξ̃f-ξ); Ω̂bb = Ω̂gg - Ω̂gfΩ̂ff-1Ω̂fg; bN(ξ) = gN(ξ) - Ω̂gfΩ̂ff-1fN(ξ); BN(ξ) = ∂bN/∂ξ′ = GN(ξ) - Ω̂gfΩ̂fff2N ; and B̃N and b̃N are shorthand for BN(ξ̃f) and bN(ξ̃f), respectively.
1
We now consider alternative tests of the validity of the moment conditions. One can of
course use JN to test the joint hypothesis that all the moment conditions are legitimate. We
consider an alternative procedure in which we maintain the validity of the moment conditions
E[fi(ξ)] = 0 that yield the linear estimate ξ̃f, and we test the additional moment conditions
E[gi(ξ)] = 0. This may be a reasonable empirical strategy because the moment conditions
based on fi(ξ) are linear and therefore easy to impose; we can test the validity of the hard-toimpose (nonlinear) restrictions before we impose them. Alternatively, we might choose fi(ξ)
to be only the set of moment conditions based on the exogeneity assumptions (18), and we
can test the validity of the further moment conditions based on second-moment assumptions
on the errors. Following Newey (1985, p. 243), we can construct the test statistic
(30)
This statistic is asymptotically chi-squared with
2
degrees of freedom under the null
hypothesis that the moment conditions based on gi are legitimate.
We can also construct a modified overidentification test that is closely related to CN.
Letting m̃N = mN(ξ̃f) and M̃N = MN(ξ̃f), we define:
(31)
where JNf = NfN(ξ̃f)′Ω̂ff-1fN(ξ̃f) is the statistic for testing E[fi(ξ)] = 0, and the second equality
results from the fact that f2N′Ω̂ff-1fN(ξ̃f) = 0. Following Ahn (1995), we can show that MJN and
JN are asymptotically identical under the hypothesis that all the moment conditions are valid.
In fact, MJN and JN are asymptotically equivalent under local alternative hypotheses H* =
{HT* }∞T=1 where HT*: T½E[mN(ξ)] = O(1). See Ahn (1995, Proposition 1).
16
An alternative and perhaps more intuitive approach to the estimation and testing problem
is as follows. Define ρ = E[gi(ξ)], an
2
× 1 vector of auxiliary parameters; we wish to test
the hypothesis ρ = 0. Define θ = (ξ′,ρ′)′, and consider the following GMM problem:
(32)
for which the solution is θ̈ = (ξ̈′,ρ̈′)′. This problem satisfies the conditions for the
"separability" result of Ahn and Schmidt (1995a) and its solution is simply ξ̈ = ξ̃f and ρ̈ = b̃N.
By standard GMM results, the asymptotic covariance matrix of N½(θ̈-θ) is given by Ξ-1 =
[(M,J)′Ω-1(M,J)]-1, where J′ = [0,-I], with 0 2× 1 and I of dimension 2. It is consistently
estimated by
(33)
which is a standard calculation since Ξ̃ is just the Hessian of the GMM minimand in (32).
Following Ahn and Schmidt (1995a), the conventional Wald statistic (based on ρ̈ and Ξ̃-1) for
the hypothesis ρ = 0 is numerically identical to CN, provided that the same weighting matrix
is used. The statistic MJN can be then obtained by simply adding this Wald statistic and JNf .
Once ξ̈ (= ξ̃f), ρ̈ (= b̃N) and Ξ̃-1 are computed, it is natural to obtain an efficient minimum
distance (MD) estimator of ξ. Suppose that ξ̃MD solves:
(34)
Then it is straightforward to show that ξ̃MD = ξ̈ + Ξ̃11-1Ξ̃12ρ̈, where Ξ̃11 = MN(ξ̃f)′Ω̂-1MN(ξ̃f) and
Ξ̃12 = MN(ξ̃f)′Ω̂-1J are submatrices of Ξ̃. With these substitutions, and substituting ξ̈ = ξ̃f and
ρ̈ = b̃N = gN(ξ̃f) - Ω̂gfΩ̂ff-1fN(ξ̃f) as given above, we obtain ξ̃MD = ξ̃LGMM. Unsurprisingly, there
17
is a close link between linearization of the moment conditions gi(ξ) in estimation and tests of
their validity.
5. Concluding Remarks
In this paper, we have considered the estimation of dynamic models for panel data. We
have counted and expressed the moment conditions implied by a wide variety of different
assumptions. These conditions can be imposed in a GMM framework. While many of the
moment conditions are nonlinear in the parameters, we derive a very simple linearized
estimator that is asymptotically as efficient as the nonlinear GMM estimator, and simple
moment tests of the validity of the nonlinear restrictions.
Some evidence on the efficiency gains from using nonlinear moment conditions is given
in Ahn and Schmidt (1995b), for one of the sets of assumptions considered in this paper.
These efficiency gains appear to be large enough to justify the effort needed to impose the
nonlinear moment conditions. Thus the methods developed in this paper to test and impose
these conditions should be of practical use.
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