Efficient Estimation of Dynamic Panel Data Models: Alternative Assumptions and Simplified Estimation

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1 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) ; FAX: (517)

2 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Ahn, S.C., 1990, Three essays on share contracts, labor supply, and the estimation of models for dynamic panel data, Unpublished Ph.D. dissertation (Michigan State University, E. Lansing, MI). Ahn, S.C., 1995, Model specification testing based on root-t consistent estimators, Unpublished manuscript (Arizona State University, Tempe, AZ).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, Ahn, S.C. and P. Schmidt, 1995b, Efficient estimation of models for dynamic panel data, Journal of Econometrics 68, Anderson, T.W. and C. Hsiao, 1981, Estimation of dynamic models with error components, Journal of the American Statistical Association 76, 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58, Arellano, M. and O. Bover, 1990, Another look at the instrumental variables estimation of error-component Models, Unpublished manuscript (London School of Economics, London).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, 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,

3 Holtz-Eakin, D., 1988, Testing for individual effects in autoregressive models, Journal of Econometrics 39, Holtz-Eakin, D., W. Newey and H.S. Rosen, 1988, Estimating vector autoregressions with panel data, Econometrica 56, 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, Schmidt, P., S.C. Ahn and D. Wyhowski, 1992, Comment, Journal of Business and Economic Statistics 10, Wooldridge, J.M., 1995, Estimating systems of equations with different instruments for different equations, Journal of Econometrics, forthcoming.

4 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.

5 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) Herei = 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 y i =(y i1,...,y it ) and y i,-1 and u i 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.

6 3 For all i, ε it is uncorrelated with y i0 for all t. For all i, ε it is uncorrelated with α i for all t. For all i, the ε it are mutually uncorrelated. For all i, var(ε it ) is the same for all t. (A.1) (A.2) (A.3) (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 y i0,it is reasonable to view it as being a linear function of α i and of ε i0, ε i,-1,.... Therefore we will not impose (A.1) (uncorrelatedness of y i0 and ε it, t=1,...,t) unless we also impose both (A.2) (uncorrelatedness of α i and the ε it ) and (A.3) (mutual uncorrelatedness of the ε it ). 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 (ε i1,...,ε it,y i0,α i ); 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 (u 1i,...,u it,y i0 ); 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 σ αt cov(ε it,α i ) = 0 for all t, σ 0t cov(y i0,ε it )=0 for all t, σ ts cov(ε it,ε is ) = 0 for all t s, and σ tt ( var(ε it )) the same ( σ εε ) for all t. This implies three types of restrictions on Λ. First, λ 0t cov(y i0,u it ) is the same for t = 1,...,T. Second, λ tt var(u it ) is the same for t = 1,...,T. Third, λ ts cov(u it,u is ) is the same for t,s =

7 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 σ αα ( var(α i )), σ 0α ( cov(y i0,α i )), σ 00 ( var(y i0 )) and σ εε. 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 ū i =T -1 Σ t u it. The T(T-1)/2 moment conditions in (4A) can be motivated by first- differencing (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 [y i0,...,y i,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.

8 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 ε it. 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 λ 0t is the same for all t, and the T(T-1)/2-1 restrictions that λ ts 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 y i0 is uncorrelated with the ε it. 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 y i0, since it is allowed to be correlated with both α i and the ε it. The parameters σ 0α and the σ 0t are not separately identified. The elements of Λ depend on the T+3 identified parameters σ αα, σ 00, σ εε,(σ 0α +σ 01 ),..., (σ 0α +σ 0T ), so that the number of restrictions is (T+1)(T+2)/2 - (T+3) = T(T-1)/2 + T-2. Wehave the T-1 restrictions that λ tt is the same for all t, and the T(T-1)/2-1 restrictions that λ ts 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 ε it are mutually uncorrelated and uncorrelated with α i. The elements of Λ depend on the 2T + 2

9 6 identified parameters σ αα, σ 00,(σ 0α +σ 01 ),..., (σ 0α +σ 0T ), σ 11,..., σ TT, and the total number of moment conditions is T(T-1)/2-1. Specifically, λ ts 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 ε it are homoskedastic and that they are uncorrelated with α i. This may be plausible if the ε it are autocorrelated. In this case Λ depends on the T(T-1)/2 +T+2identified parameters σ 00, (σ αα +σ εε ), (σ 0α +σ 01 ),..., (σ 0α +σ 0T ) and (σ αα +σ st ), s t. We have only the T-1 restrictions that λ tt 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 ε it are homoskedastic and mutually uncorrelated, but we make no assumptions about their relationship to y i0 or α i. The elements of Λ depend on the 2T+2 identified parameters σ 00, σ εε,(σ 0α +σ 01 ),..., (σ 0α +σ 0T ), (σ αα +2σ α1 ),..., (σ αα +2σ αt ). Thus we have T(T-1)/2-1 moment conditions. Specifically, we have the restrictions that λ tt + λ ss -2λ ts is the same for allt,s=1,...,t,t s. These restrictions correspond to the moment conditions that E[(u it - u is ) 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 y i0, and some treatments have assumed that y i0 is uncorrelated with α i [see, e.g., Hsiao (1986, section 4.3.2)]. The treatments based on the moment conditions (4A) above have sometimes been called fixed-

10 7 effects treatments, even though they clearly depend on assumptions about y i0 and α i, such as lack of correlation of these quantities with the ε it. 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 α i and y i0 to be fixed. CASE G. In this case we assume only (A.2), that α i is uncorrelated with the ε it. This implies no moment conditions. CASE H. In this case we assume only (A.3), that the ε it 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 u it depend only on the T identified parameters (σ αα +2σ α1 ),..., (σ αα +2σ αt ). Thus we have T(T-1)/2 -T=T(T-3)/2 restrictions, of the form λ qr + λ st = λ qs + λ rt 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 ε it. 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 y i0,..., y it 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(α i,y it ) is the same for t=0,1,..., T. (9)

11 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(y it )=σ 00 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

12 9 (13) on the variance of the initial observation y i0. 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 (α i,y i0 ), (ε i1,..., ε it ) is stationary. (S.2) This assumption implies the following restrictions on the covariance matrix Λ. First, λ tt is the same for t=1,2,..., T. Second, λ 0t is the same for t=1,2,..., T. Third, for j=1,2,..., T-2, λ t,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 Notation and General Results We consider the model (16)

13 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 u i on ξ. Exogeneity assumptions on X and Z generate linear moment conditions of the form (18) where R i is a function of the exogenous variables. See Ahn and Schmidt (1995b), equations (33) and (34), for definitions of R i 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[A i u i (ξ)] = 0, where A i 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[B 1i u i (ξ)] = 0, where B 1i is the T (T-2) matrix defined by

14 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 S i (T 1 )be made up of columns of R i,a i and B 1i, so that it represents some or all of the available linear instruments. The corresponding linear moment conditions are E[f i (ξ)] = 0, with (21) If the dimension of ξ is k, we assume 1 k so that the conditions f i can identify ξ. The remaining 2 = - 1 moment conditions will be written as E[g i (ξ)] = 0. Since they are at most quadratic, we can write (22) where g 1i,g 2i and g 3i are 2 1, 2 k and 2 k 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:

15 12 (23) Define m N =N -1 Σ i m i (ξ), with f N,f 1N,f 2N,g N,g 1N,g 2N and g 3N defined similarly; and define M N = m N / ξ =[ f N / ξ, g N / ξ] =[F N,G N ], where G N (ξ) =g 2N + 2(I ξ )g 3N and F N =f 2N. Let M = plim M N, with F and G defined similarly. Define the optimal weighting matrix: (24) Let ˆΩ be a consistent estimate of Ω of the form (25) where ξˆ is an initial consistent estimate of ξ (perhaps based on the linear moment conditions f i, as discussed below); partition it similarly to Ω. In this notation, the efficient GMM estimator ξ GMM minimizes Nm N (ξ) ˆΩ -1 m N (ξ). Using standard results, the asymptotic covariance matrix of N ½ (ξ GMM -ξ) is[m Ω -1 M] -1. We can also test the validity of the moment conditions E[m i (ξ)] = 0 using the usual overidentification test statistic J N =Nm N ( ξ GMM ) Ωˆ -1 m N ( ξ GMM ). This statistic is asymptotically chi-squared with ( -k) degrees of freedom under the joint hypothesis that all the moment conditions are legitimate Linear Moment Conditions and IV Some interesting questions arise when we consider GMM based on the linear moment conditions f i (ξ) only. The optimal GMM estimator based on these conditions is (26)

16 13 This GMM estimator can be compared to the linear IV estimator that is of the same form, but with ˆΩ ff replaced by (Σ i S i S i ). The GMM estimator is generally more efficient than the linear IV estimator. They are asymptotically equivalent in the case that Ω ff is proportional to E(S i S i ), which occurs if E(S i u i u i S i ) is proportional to E(S i S i ). For the case that S i consists only of columns of R i, 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 S i also contains A i, 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(y 2 i0,ε 2 it ) = 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)

17 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(A i u i u i A i )=σ εε E(A i A i ) and E(A i u i u i B 1i )=σ εε E(A i B 1i ), (27) where d=e(ε 4 )-3σ εε 2 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=0buttheterm σ εε C remains Linearized GMM and Specification Tests We now consider a linearized GMM estimator. Suppose that ξ is any consistent estimator of ξ; for example, ξ f. 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 ξ GMM. When the LGMM estimator is based on the initial estimator ξ f, some further simplification is possible. Applying the usual matrix inversion rule to ˆΩ and using the fact that f 2N ˆΩ -1 ff f N ( ξ f ) = 0, we can write the LGMM estimator as follows: (29) Here ˆΓ -1 =[f 2N ˆΩ ff -1 f 2N ] -1 denotes the usual estimate of the asymptotic covariance matrix of

18 N ½ ( ξ f -ξ); ˆΩ bb = ˆΩ gg - ˆΩ gf ˆΩ ff -1 ˆΩ fg ;b N (ξ)=g N (ξ)- ˆΩ gf ˆΩ ff -1 f N (ξ); B N (ξ) = b N / ξ =G N (ξ)- ˆΩ gf ˆΩ ff - 1 f 2N ; and B N and b N are shorthand for B N ( ξ f ) and b N (ξ f), respectively. 15 We now consider alternative tests of the validity of the moment conditions. One can of course use J N 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[f i (ξ)] = 0 that yield the linear estimate ξ f, and we test the additional moment conditions E[g i (ξ)] = 0. This may be a reasonable empirical strategy because the moment conditions based on f i (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 f i (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 g i are legitimate. We can also construct a modified overidentification test that is closely related to C N. Letting m N =m N ( ξ f ) and M N =M N ( ξ f ), we define: (31) where J f N =Nf N ( ξ f ) ˆΩ -1 ff f N ( ξ f ) is the statistic for testing E[f i (ξ)] = 0, and the second equality results from the fact that f 2N ˆΩ -1 ff f N ( ξ f ) = 0. Following Ahn (1995), we can show that MJ N and J N are asymptotically identical under the hypothesis that all the moment conditions are valid. In fact, MJ N and J N are asymptotically equivalent under local alternative hypotheses H * = {H T} * T=1 where H T:T * ½ E[m N (ξ)] = O(1). See Ahn (1995, Proposition 1).

19 16 An alternative and perhaps more intuitive approach to the estimation and testing problem is as follows. Define ρ = E[g i (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 θ =(ξ, ρ ). This problem satisfies the conditions for the "separability" result of Ahn and Schmidt (1995a) and its solution is simply ξ = ξ f and ρ =b N. By standard GMM results, the asymptotic covariance matrix of N ½ ( θ-θ) is given by Ξ -1 = [(M,J) Ω -1 (M,J)] -1, where J = [0,-I], with and I of dimension 2. It is consistently estimated by (33) which is a standard calculation since Ξ is just the Hessian of the GMM minimand in (32). Following Ahn and Schmidt (1995a), the conventional Wald statistic (based on ρ and Ξ -1 ) for the hypothesis ρ = 0 is numerically identical to C N, provided that the same weighting matrix is used. The statistic MJ N can be then obtained by simply adding this Wald statistic and JN. f Once ξ (= ξ f), ρ (= b N) and Ξ -1 are computed, it is natural to obtain an efficient minimum distance (MD) estimator of ξ. Suppose that ξ MD solves: (34) Then it is straightforward to show that ξ MD = ξ + Ξ 11-1 Ξ 12 ρ, where Ξ 11 =M N (ξ f) ˆΩ -1 M N (ξ f) and Ξ 12 =M N (ξ f) ˆΩ -1 J are submatrices of Ξ. With these substitutions, and substituting ξ = ξ f and ρ =b N =g N ( ξ f )- ˆΩ gf ˆΩ ff -1 f N (ξ f) as given above, we obtain ξ MD = ξ LGMM. Unsurprisingly, there

20 is a close link between linearization of the moment conditions g i (ξ) in estimation and tests of their validity 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.