On the optimal weighting matrix for the GMM system estimator in dynamic panel data models

Transcription

1 Discussion Paper: 007/08 On the optimal weighting matrix for the GMM system estimator in dynamic panel data models Jan F Kiviet wwwfeeuvanl/ke/uva-econometrics Amsterdam School of Economics Department of Quantitative Economics Roetersstraat 08 WB AMSTERDAM The Netherlands

2 On the optimal weighting matrix for the GMM system estimator in dynamic panel data models Jan F Kiviet (Universiteit van Amsterdam & Tinbergen Institute) preliminary version: December 007 JEL-code: C3; C5; C3 Keywords: e ect stationarity, generalized method of moments, initial conditions, optimal weighting matrix, two-step estimation Abstract An analytical expression is obtained for the optimal weighting matrix for the generalized method of moments system estimator (GMMs) in the dynamic panel data model with predetermined regressors, individual e ect stationarity of all variables, homoskedasticity and cross-section independence As yet such an expression had not been obtained Therefore GMMs has always been applied by rst using some simple form of sub-optimal weighting matrix in -step GMMs, after which asymptotically e cient -step GMMs can be obtained in the usual way The optimal weighting matrix is found to depend on unobservable model and data parameters, so asymptotically e cient -step GMMs is not operational However, next to the few suboptimal naive -step GMMs weighting matrices presently in use, the expression for the optimal weighting matrix suggests alternative feasible more sophisticated GMMs implementations, including a point-optimal weighting matrix In Monte Carlo experiments we analyse and compare some of these and nd that -step GMMs is rather sensitive to the initial choice of weighting matrix in moderately large samples Some guidelines are given for choosing weights in order to reduce the mean (squared) estimation errors Department of Quantitative Economics, Amsterdam School of Economics, Universiteit van Amsterdam, Roetersstraat, 08 WB Amsterdam, The Netherlands (jfkiviet@uvanl)

3 Introduction To achieve asymptotic e ciency in GMM (generalized method of moments) estimation when there are more moment restrictions than parameters to be estimated, the sample moment conditions are minimized in a quadratic form that entails a weighting matrix that has a probability limit which is proportional to the inverse of the variance of the limiting distribution of the orthogonality condition vector In many circumstances this variance is hard to derive and it may depend on nuisance parameters Dynamic panel data models with unobserved individual e ects are often estimated by GMM, and when the cross-sections are independent and homoskedastic the optimal weighting matrix for the linear moment conditions is well-known and not determined by nuisance parameters, see Arellano and Bond (99) However, it has been observed that this estimator may su er from serious bias in moderately large samples and it has been suggested that exploiting additional moment conditions that build on stationarity assumptions in a so-called GMM system estimator (GMMs) yields much better results This estimator, put forward by Arellano and Bover (995) and Blundell and Bond (998), is called a system estimator because it simultaneously exploits moment conditions for the dynamic panel data model in levels and in its rst-di erenced form Up to date the optimal weighting matrix for this estimator had not been derived so that even under very strict assumptions GMMs has to be employed in a -step procedure in which an empirical assessment of the optimal weights is used that is based on consistent -step GMMs estimates that are obtained from an operational but suboptimal weighting matrix In this paper we derive the optimal GMMs weighting matrix for the dynamic panel data model with both a lagged dependent variable regressor and an arbitrary number of further predetermined regressors, where both the dependent variable and the regression variables are e ect stationary This means that their correlation with the unobserved heterogeneity (the individual e ects) is time-invariant For many elements of the GMMs orthogonality condition vector it is extremely di cult to derive their covariance, which complicates the assessment of the optimal weighting matrix However, we show that asymptotically equivalent results are obtained by deriving a conditional covariance, and that di erent conditioning sets may be used for the various elements of the variance matrix By choosing for all the various types of elements in the variance matrix of the orthogonality conditions a conditioning set that leads to rather straightforward expressions for their conditional variance we are able to obtain an explicit expression for the asymptotically optimal weighting matrix This matrix is shown to depend on all the observed instrumental variables and also on the unknown values of the covariance between the regressors and the individual e ects, as well as on the unknown covariance between current regressors and lagged disturbances, ie on the pattern in the feedback mechanism of current disturbances into future observations of the regressors Due to the dependence on nuisance parameters asymptotically e cient -step GMMs is unfeasible, although a better founded initial choice of weighting matrix is possible now At least a choice better than those currently in use, which are based either on the usually wrong assumption that the individual e ects have zero variance, or on taking as weighting matrix the inverse of the sample moment matrix of the instrumental variables, ie wrongly assuming that the disturbances of the system are iid and therefore applying GIV (generalized instrumental variables), which in this model is clearly far from optimal An alternative approach regarding improving the quality of weighting matrices can be found in Doran and Schmidt (005)

4 In Monte Carlo experiments [in the present version of the paper we just simulated the autoregressive model without further predetermined regressors] we compare the results obtained by employing the initial weighting matrices presently in use, and by exploiting some new operational forms inspired by the optimal matrix We nd that -step GMMs is rather sensitive to the initial choice of weighting matrix in moderately large samples As a yardstick we also present results obtained by the in practice unfeasible -step GMMs estimator that uses (or is inspired by) the optimal weights Some guidelines, including a point-optimal version of the weighting matrix, are given for choosing the weights in order to reduce the mean (squared) estimation errors The paper is structured as follows In Section we introduce our notation, state the model assumptions, review some general GMM results for panels, present the GMMs estimator and the various forms of suboptimal weighting matrix that have been suggested in the literature Section 3 contains our main analytical results regarding the derivation of the optimal weighting matrix Next, in Section 4, we present the Monte Carlo design that we used and describe our simulation ndings Finally, Section 5 concludes Standard and system GMM for dynamic panels In this section we introduce the linear rst-order dynamic panel data model with random individual e ects and an arbitrary number of further regressors All regressors are assumed to be predetermined with respect to the homoskedastic and serially and contemporaneously uncorrelated idiosyncratic disturbance terms From the various model assumptions the linear orthogonality conditions are derived that are exploited in what we will address as the standard GMM estimator, see Arellano and Bond (99) We pay extra attention to an additional model assumption that we will call e ect stationarity This, when valid, gives rise to additional linear orthogonality conditions, which can be exploited in the system GMM estimator (GMMs), see Blundell and Bond (998), who considered this estimator in the pure autoregressive case Generic results are obtained for -step and -step estimation that cover both the standard and the system GMM panel data situation This provides the setting from which we can embark on the derivation of the optimal weighting matrix for GMMs Model assumptions We consider the linear dynamic panel data model y it = y i;t + x 0 it + i + " it ; () where x it and are K vectors and i = ; :::; N refers to the cross-sections and t = ; :::; T to the time-periods We suppose that we have a balanced panel data set; hence, if x it happens to contain also the rst lag of all current explanatory variables then both the dependent and the other individual explanatory variables should have been observed for the time-indices f0; :::; T g for all cross-section units Below we shall use the notation y t i (y i;0 ; :::; y i;t ) Xi t (x 0 i; :::; x 0 t = ; :::; T; it) 3

5 where y t i is t and Xi t is tk: We also de ne 3 3 y t X Y t t 5 ; X t 6 4 y t N XN t 7 5 ; where is N ; Y t is N t and X t is N tk: Regarding the two random error components i and " it in model () we make the assumptions (i; j = ; :::; N; t; s = ; :::; T ): 9 E(" it j Y t ; X t ; ) = 0; 8i; t >= E(" it j Y t ; X t ; ) = " > 0; 8i; t E(" it " js j Y t ; X t () ; ) = 0; 8t < s; 8i; j and 8i 6= j; 8t s >; E( i ) = 0; E( i ) = 0; E( i j ) = 0 8i 6= j These assumptions entail that all regressors are predetermined with respect to the idiosyncratic disturbances " it ; which are homoskedastic and serially and contemporaneously uncorrelated In fact, we assume that all cross-section units are independent Additional assumptions that may be made and which are crucial when the GMMs estimator is employed involve 0 N C A ; E(y it i ) = y and E(x it i ) = x ; 8i; t: (3) Here y is a scalar and x is a K vector Note that both y and x are assumed to be time-invariant By multiplying the model equation () by i and taking expectations we nd that (3) implies y = y + 0 x + or y = 0 x + : (4) This condition we will call for obvious reasons e ect-stationarity Moment conditions By taking rst-di erences the model simpli es in the sense that only one unobservable error component remains Estimating by OLS would yield inconsistent estimators because y it = y i;t + (x it ) 0 + " it (5) E(y i;t " it ) = E(y i;t " i;t ) = " 6= 0: Unless " would be known this moment condition cannot directly be exploited in a method of moments estimator Note, however, that it easily follows from the model assumptions () that for i = ; :::; N we have E(Y t " it ) = O E(X t t = ; :::; T; " it ) = O Note that in the (habitual) case where x it contains a unit element and an intercept this model has no longer K + unknown coe cients In the analysis that follows we neglect that situation, but it is reasonably straigtforward to adapt the results (mainly the dimensions of matrices) for models with an intercept 4

6 which together provide (K + )NT (T )= moment conditions for estimating just K + coe cients Especially when the cross-section units are independent many of these conditions will produce weak or completely ine ective instruments Then it will be more appropriate to exploit just the (K + )T (T )= moment conditions E(y t i " it ) = 0 0 E(X t i " it ) = 0 0 t = ; :::; T; as is done in the Arellano-Bond (99) estimator Upon substituting (5) for " it it is obvious that these moment conditions are linear in the unknown coe cients and ; ie Efy t i [y it y i;t (x it ) 0 ]g = 0 0 EfX t i [y it y i;t (x it ) 0 ]g = 0 0 t = ; :::; T: (6) Blundell and Bond (998) argue that assumption (3), when valid, may yield relatively strong additional useful instruments for estimating the undi erenced equation () These additional instruments are the rst-di erenced variables De ning y t i (y i ; :::; y i;t ) X t i (x 0 i; :::; 0 x i;t ) t = ; :::; T; which are (t ) and (t )K respectively, it follows from (3) that (for i = ; :::; N) From () we nd E(y t i i ) = 0 0 and E(X t i i ) = 0 0 : E(y t i " it ) = 0 0 and E(X t i " it ) = 0 0 : Combining these and substituting i +" it = y it y i;t x 0 it yields the (K+)T (T )= linear moment conditions E[y t i (y it y i;t x 0 it)] = 0 0 E[Xi t (y it y i;t x 0 it)] = 0 0 t = ; :::; T: (7) These can be transformed linearly into two subsets of (K + )(T ) and (K + )(T )(T )= conditions respectively, viz E[y i;t (y it y i;t x 0 it)] = 0 0 E[x 0 it(y it y i;t x 0 it)] = 0 0 t = ; :::; T; (8) and Efy t i [y it y i;t (x it ) 0 ]g = 0 0 EfX t i [y it y i;t (x it ) 0 ]g = 0 0 t = 3; :::; T: (9) The second subset (9), though, can also be obtained by a simple linear transformation of (6) Hence, e ect stationarity only leads to the (K + )(T ) additional linear moment conditions (8) These involve estimation of the undi erenced model () by employing all the rst-di erenced regressor variables as instruments Due to the iid assumption regarding " it further (non-linear) moment conditions do hold in the present dynamic panel data model, see Ahn and Schmidt (995, 997), but below we will stick to the linear conditions mentioned above There would in fact be fewer unique moment conditions if the model includes an intercept, indicator or dymmy variables or particular regressors in combination with their lags In what follows we will not take this into account, so that in actual applications of our results adaptations will have to be made 5

7 3 Generic results for -step and -step panel GMM Both the standard GMM and the system estimator GMMs t into the following simple generic setup After appropriate manipulation (transformation and stacking) of the panel data observations one has y i = X i + " i ; i = ; :::; N; (0) where yi and " i are T vectors and the T K matrix Xi contains all regressor variables (including transformations of the lagged-dependent variable) The K vector of coe cients is estimated by employing L K moment conditions that hold for all i = ; :::; N; viz E[Zi 0 (yi Xi )] = 0; () where Zi is T L : The GMM estimator using the L L semi-positive de nite weighting matrix W is obtained by minimizing a quadratic form, viz ^ N P 0 N W = arg min Z 0 i (yi Xi P ) W Zi 0 (yi Xi ) : () i= i= Assuming that X 0 Z W Z 0 X has rank K with probability a unique minimum exists which yields ^ W = (X0 Z W Z 0 X ) X 0 Z W Z 0 y ; (3) where y = (y 0 ; :::; yn 0)0 ; X = (X 0 ; :::; XN 0)0 and Z = (Z 0 ; :::; ZN 0)0 : It is obvious that ^ W is invariant for the scale of W : Below, when useful, we will implicitly assume that scaling took place such that W = O p (): We only consider cases where convenient regularity conditions hold, including the existence with full column rank (for N! ) of plim N Z 0 X ; plim N Z 0 Z and plim N X 0 Z W Z 0 X : According to () the instruments are valid, thus P plim N i= N! N Z0 i " i = 0 and ^ W is consistent Since the cross-section units are assumed to be independent the CLT (central limit theorem) yields NP p N i= Z 0 i " i! N (0; V Z 0 " ) ; where V Z 0 " lim N! N NP i= Var(Z 0 i " i ); (4) from which we obtain p N(^ W 0)! N 0; ; with (5) V^ W V^ W A V Z 0 " A0 ; A plim[n(x 0 Z W Z 0 X ) X 0 Z W ]: The asymptotically e cient GMM estimator in the class of estimators exploiting instruments Z is obtained if W is chosen such 3 that, after appropriate scaling, it has probability limit proportional to the inverse of the variance matrix of the limiting distribution of N P = N i= Z0 i " i :Hence, optimality is attained by using a weighting matrix W opt such that P N W opt / Var(Zi 0 " i ) : (6) i= 3 see Hansen (98) and, in the context of dynamic panel data, Arellano (003) and Baltagi (005) 6

8 When W opt is known the asymptotically e cient GMM estimator is self-evidently given by ^ W opt = (X0 Z W opt Z 0 X ) X 0 Z W opt Z 0 y ; and p N(^ W opt 0)! N 0; ; with V^ W opt V^ [plim N (X 0 Z V W opt Z 0 " Z 0 X )] (7) For the special case " it j Zi t iid(0; " ); where Zt i contains the rst t rows of Zi ; one nds Var(Zi 0 " i ) = " E(Z0 i Zi ) and hence asymptotic e ciency is achieved by taking W opt idd = (Z0 Z ) ; which yields the familiar SLS or GIV result : ^ GIV = [X 0 P Z X ] X 0 P Z y ; (8) where P Z Z (Z 0 Z ) Z 0 : In case K = L this specializes to the simple instrumental variable estimator ^ IV = (Z 0 X ) Z 0 y : (9) If matrix W opt is not directly available then one may use some arbitrary initial weighting matrix W that produces a consistent (though ine cient) -step GMM estimator ^ W, and then exploit the -step residuals ^" i to construct the empirical weighting matrix c W ; where yielding the -step GMM estimator ^" i = yi Xi ^ W ; cw = ( P N i= Z0 i ^" i ^" 0 i Zi ) ; ^ c W = (X 0 Z c W Z 0 X ) X 0 Z c W Z 0 y : (0) Estimator ^ W c is asymptotically equivalent to ^ W opt; and hence it is e cient in the class of estimators exploiting instruments Z For inference purposes one uses the approximation ^ c W a N ; (X 0 Z c W Z 0 X ) : () Note that, provided the moment conditions are valid, ^ GIV is consistent thus could be employed as a -step GMM estimator When K = L using a weighting matrix for the orthogonality conditions is redundant, because the criterion function () will be zero for any W ; as all moment conditions can then be imposed on the sample observations 4 Implementation of ordinary and system GMM In case of the Arellano-Bond ordinary GMM estimator the above generic setup involves y i = y i y it C A ; " B i " i " it C A and Xi 6 = 4 y i x 0 i y i;t x 0 it 7 5 ; () hence T = T : When there is an intercept in and corresponding unit value in x it (which vanishes after rst-di erencing) we have K = K; otherwise K = K + and 7

9 = (; 0 ) 0 : From (6) it follows that L = (K + )T (T )= and the T L matrix Zi is taken to be 3 yi Xi Zi AB 6 = 4 0 O O 7 O 5 : (3) y T i X T i Since E(" it " i;t ) 6= 0 the " it are not iid, thus the GIV estimator is not e cient It can be derived that for the ordinary GMM estimator W opt AB / N P (Zi AB i= ) 0 HZi AB (4) with (T ) (T ) matrix H : (5) 7 5 So, under our strong assumptions regarding homoskedasticity and independence of the idiosyncratic disturbances " it -step ordinary GMM employing (4) is asymptotically e cient within the class of estimators exploiting the instruments Zi AB : In case of GMMs we have K = K + ; = (; 0 ) and T = (T ) with 0 yi = y i y it y i y it 0 " i ; " " i = it and X i = C B i + " i C 6 A 4 i + " it y i x 0 i y i;t x 0 it y i x 0 i y i;t x 0 it 3 ; (6) 7 5 and from (6) and (??) it follows that L = (K + )(T )(T + )= whereas the T L matrix Zi = Zi BB is 3 Zi AB 0 0 O O Zi BB 0 0 y i x 0 i = O 5 : (7) y i;t x 0 it Note that E[(" it ) ] = " di ers from E[(" it + i ) ] = " + when 6= "; and for t 6= s we nd E[(" it + i )(" is + i )] = 0: Thus, again it does not hold here that " it j Zi t iid(0; " ); so GIV is not e cient However, here an appropriate weighting matrix is not readily available due to the complexity of Var(Zi " i ): 8

10 5 Weighting matrices in use for -step GMMs To date the GMMs optimal weighting matrix has only been obtained for the no individual e ects case = 0 by Windmeijer (000), who presents with P N WBB F W / i= (Zi BB H D F W C = C 0 I T ) 0 D F W Zi BB (8) ; (9) where C is the (T ) (T ) matrix : : : 0 0 C = 0 0 : (30) : : : 0 Blundell, Bond and Windmeijer (000, footnote ), and Doornik et al (00, p9) in the computer program DPD, use in -step GMMs the operational weighting matrix W DP D BB P N _ (Zi BB i= ) 0 D DP D Zi BB ; (3) with H O D DP D = O I T : (3) The motivation for that is unclear The block diagonality of D DP D does not lead to an interesting reduction of computational requirements, and there is no special situation (not even = 0) for which these weights are optimal Blundell and Bond (998) did use (see page 30, 7 lines from bottom) in their rst step of -step GMMs D GIV IT = O = I O I T ; (33) T which yields the simple GIV estimator This, as we already indicated, is certainly not optimal either, but is easy and could perhaps suit well as rst step in a -step procedure Blundell and Bond (998) mention that, in most of the cases they examined, -step and -step GMMs gave similar results, suggesting that the weighting matrix to be used in combination with Zi BB seems of minor concern under homoskedasticity However in Kiviet (007), in less restrained simulation experiments, it is demonstrated that di erent initial weighting matrices can lead to huge di erences in the performance of -step and -step GMMs There it is argued that a better (though yet suboptimal and unfeasible) weighting matrix would be W = " BB P N _ (Zi BB i= ) 0 D = " Z BB i ; (34) 9

11 with D = " = H C C 0 I T + " T 0 T! : (35) This can be made operational by choosing some value for = ": From the simulations it follows that this value should not be chosen too low, and reasonably satisfying results were obtained by choosing = " = 0: 3 The optimal weighting matrix for GMMs The obtain the optimal weighting matrix W opt of (6) for the generic model (0) one has to consider the matrices Q i Zi 0 " i " 0 i Zi and, ideally, evaluate E(Q i ) = E(Z 0 i " i " 0 i Z i ) = Var(Z 0 i " i ): Denoting the typical element of the L L matrix Q i by q i;lm ; where l; m = ; :::; L ; and de ning q s i;lm E s (q i;lm ); where E s denotes expectation conditional on information available at time-period s; we have, by the LIE (law of iterated expectations), and by the LLN (law of large numbers) plim N! N NP i= E(q s i;lm) = E(q i;lm ) = [Var(Z 0 i " i )] l;m (36) q i;lm s = lim N! N NP i= E(q i;lm) s = lim N! N NP [Var(Zi 0 " i )] lm : (37) Hence, N P N i= qs i;lm has probability limit equal to the corresponding element of the covariance matrix of the limiting distribution of N P = N i= Z0 i " i : So, we do not necessarily have to evaluate E(q i;lm ); nding q i;lm s for a convenient s and calculating N P N i= qs i;lm for all l; m = ; :::; L su ces for the present purpose Of course, other types of conditioning sets are allowed too, and di erent conditioning sets may be used for the individual elements q i;lm of Q i ; and even for separate components of these individual elements Since a GMM estimator is invariant to the scale of the weighting matrix we may multiply all elements by N= P "; and thus W opt can be obtained by inverting the matrix that has elements " N i= qs i;lm : 3 Ordinary GMM We rst derive formally (most published earlier derivations are rather sketchy) the wellknown optimal weighting matrix W opt AB, given in (3), to be used when the instruments Zi AB ; given in (3), are employed in model (5) In this case Zi 0 " i " 0 i Zi consists of components which are of the following four forms: (y t i ) 0 " it " is y s i ; (y t i ) 0 " it " is X s i ; (X t i ) 0 " it " is y s i ; or (X t i ) 0 " it " is X s i ; where t; s = ; :::; T: Because of the symmetry of Zi 0 " i " 0 i Zi we only have to consider components for t s: 0 i=

12 We nd for t > s + E t [(y t i ) 0 " it " is y s i ] = (y t i ) 0 y s E t [(y t i ) 0 " it " is X s i ] = (y t i ) 0 X s i " is E t (" it ) = O; i " is E t (" it ) = O; i = O; i = O: E s [(X t i ) 0 " it " is y s i ] = " is E s [(X t i ) 0 " it ]y s E s [(X t i ) 0 " it " is X s i ] = " is E s [(X t i ) 0 " it ]X s For s = t we obtain E t [(y t i ) 0 " it " i;t y t 3 i ] = (y t i ) 0 y t 3 i [E t (" it " i;t ) " i;t E t (" it )] = "(y t i ) 0 y t 3 i ; E t [(y t i ) 0 " it " i;t X t i ] = (y t i ) 0 X t i E t (" it " i;t ) = "(y t i ) 0 X t i ; E t 3 [(X t i ) 0 " it " i;t y t 3 i j X t i ] = (X t i ) 0 y t 3 i E[" it " i;t j X t i ] = "(X t i ) 0 y t 3 i ; E[(X t i ) 0 " it " i;t X t i j X t i ] = (X t i ) 0 X t i E(" it " i;t j X t i ) = "(X t i ) 0 X t i ; because E(" it " i;t j X t i ) = E(" it " i;t " i;t " it " i;t + " i;t " i;t j X t i ) = "; since for j = ; we nd E(" it " i;t j j X t i ) = E[E t (" it " i;t j j X t i )] = E[" i;t j E t (" it j X t i )] = 0 and E(" i;t " i;t j X t i ) = E t [E(" i;t " i;t j X t i )] = E t " i;t [E(" i;t j X t i )] = 0; whereas E(" i;t j X t i ) = ": Finally, for s = t = ; :::; T; we get E t [(y t i ) 0 (" it ) y t i ] = (y t i ) 0 y t i E t [(" it ) ] = "(y t i ) 0 y t i ; E t [(y t i ) 0 (" it ) X t i ] = "(y t i ) 0 X t i ; E t [(X t i ) 0 (" it ) y t i j X t i ] = (X t i ) 0 y t i E t [(" it ) j X t i ] = "(X t i ) 0 y t i ; E[(X t i ) 0 (" it ) X t i j X t i ] = "(X t i ) 0 X t i : P Collecting from the above all elements " N i= qs i;lm ; we obtain P N i= ZAB0 i HZi AB : Hence, its inverse (4) is proportional to the optimal weighting matrix W opt AB when " i iid(0; "I T ) for all i: 3 System GMM For GMMs, where the instruments Zi BB given in (7) are applied to the system (0) with hybrid disturbances as in (6), the derivation of an optimal weighting matrix W opt BB is much more involved, because the various components of Zi 0 " i " 0 i Zi are of a very di erent nature The elements associated with " it " is yield results equivalent to those just obtained Of the additional components we rst examine those that involve " it ( i +" is ); viz (y t i ) 0 " it ( i +" is )y i;s ; (X t i ) 0 " it ( i +" is )y i;s ; (y t i ) 0 " it ( i +" is )x 0 is and (X t i ) 0 " it ( i +" is )x 0 is: Next we will examine components that involve ( i +" it )( i +" is ); viz y i;t ( i + " it )( i + " is )y i;s ; y i;t ( i + " it )( i + " is )x 0 is; the transpose of the latter, and nally x it ( i +" it )( i +" is )x 0 is: We examine these eight types of components successively, for t; s = ; :::; T, upon assuming e ect stationarity, which implies (3) The rst component can be decomposed into two contributions, viz (y t i ) 0 " it ( i + " is )y i;s = i (y t i ) 0 y i;s " it + (y t i ) 0 y i;s (" it )" is : For t = s we nd for these E[ i (y t i ) 0 y i;t " it ] = E[ i (y t i ) 0 y i;t " i;t ] = " y t E[(y t i ) 0 y i;t (" it )" it ] = E t [(y t i ) 0 y i;t (" it )" it ] = "(y t i ) 0 y i;t [E t (" t ) " t E t (" t )] = "(y t i ) 0 y i;t ;

13 for s = t they give for s < t we nd and for s = t + we obtain E t [ i (y t i ) 0 y i;t " it ] = 0 E t [(y t i ) 0 y i;t (" it )" i;t ] = "(y t i ) 0 y i;t ; Substituting (5) and using the notation where x";j = 0 for j 0; we nd thus E s [(y t i ) 0 " it ( i + " is )y i;s ] = 0; E[ i (y t i ) 0 y it " it ] = E(y it " it ) y t E t [(y t i ) 0 y it (" it )" i;t+ ] = 0: x";j E(x it " t j ); for j = :::; ; 0; ; ; :::; (38) E(y it " it ) = E(y i;t " it ) + E[" it (x it ) 0 ] + E[(" it ) ] Now for s t + we nd where = E(y i;t " i;t ) 0 x"; + " = ( ) " 0 x";; E[ i (y t i ) 0 y it " it ] = y [( ) " 0 x";] t : E[(y t i ) 0 " it ( i + " is )y i;s ] = E[ i (y t i ) 0 " it y i;s ] = y t E(y i;s " it ) = y s t t ; j E(y i;t+j " it ); for j 0: (39) We already found 0 = ( ) " 0 x";; and further obtain whereas for j > = E(y i;t+ " it ) = E(y i;t " it ) + E[" it (x i;t+ ) 0 ] + E[(" i;t+ )(" it )] = x"; 0 x"; " = [( ) 0 x"; 0 x";] ( ) "; j = E(y i;t+j " it ) = E(y i;t+j " it ) + E[" it (x i;t+j ) 0 ] + E[(" i;t+j )(" it )] = j + ( 0 x";j 0 x";j+ 0 x";j ): For the second component we obtain (X t i ) 0 " it ( i + " is )y i;s = i (X t i ) 0 " it y i;s + (X t i ) 0 (" it )" is y i;s :

14 For s = t we nd E[ i (X t i ) 0 " it y i;t ] = " t x E t [(X t i ) 0 (" it )" it y i;t ] = "(X t i ) 0 y i;t ; for s = t E[ i (X t i ) 0 " it y i;t ] = 0 E t [(X t i ) 0 (" it )" i;t y i;t j X t i ] = "(X t i ) 0 y i;t ; for s < t for s = t + E[(X t i ) 0 " it ( i + " is )y i;s ] = 0; and for s > t + E[ i (X t i ) 0 " it y i;t ] = [( ) " 0 x";] t x E t [(X t i ) 0 (" it )" i;t+ y i;t ] = 0; E[(X t i ) 0 " it ( i + " is )y i;s ] = E[ i (X t i ) 0 " it y i;s ] = E(" it y i;s ) t x = s t t x : The third component is (y t i ) 0 " it ( i + " is )x 0 is = i (y t i ) 0 " it x 0 is + (y t i ) 0 (" it )" is x 0 is: For s = t we nd for the rst term i (y t i ) 0 " it x 0 it = i (y t i ) 0 " it x 0 it i (y t i ) 0 " it x 0 i;t ; where the second sub-term has expectation zero and the rst yields and for the second term we nd E[ i (y t i ) 0 " it x 0 it] = y t 0 x";; (y t i ) 0 (" it )" it x 0 it = (y t i ) 0 " itx 0 it (y t i ) 0 " it " i;t x 0 i;t ; where the second sub-term has expectation zero and the rst yields E t [(y t i ) 0 " itx 0 it j x it ] = "(y t i ) 0 x 0 it: For s = t we nd E[ i (y t i ) 0 " it x 0 i;t ] = 0 E t [(y t i ) 0 (" it )" i;t x 0 i;t j x i;t ] = "(y t i ) 0 x 0 i;t ; for s < t E[(y t i ) 0 " it ( i + " is )x 0 is] = 0; 3

15 for s = t + E[ i (y t i ) 0 " it x 0 i;t+] = y t E(" it x 0 i;t+) = y t E(" it x 0 i;t+ " i;t x 0 i;t+ " it x 0 it + " i;t x 0 it) E t [(y t i ) 0 (" it )" i;t+ x 0 i;t+] = 0; and for s > t + = y t ( 0 x"; 0 x";) E[ i (y t i ) 0 " it x 0 is] = y t E(" it x 0 is " i;t x 0 is " it x 0 i;s + " i;t x 0 i;s ) = y t ( 0 x";s t 0 x";s t+ 0 x";s t ) E[(y t i ) 0 " it (" is )x 0 is] = 0: The fourth component is (X t i ) 0 " it ( i + " is )x 0 is = i (X t i ) 0 " it x 0 is + (X t i ) 0 (" it )" is x 0 is: For t = s we nd and since we also have For s = t for s < t for s = t + and for s > t + E[ i (X t i ) 0 " it x 0 it] = ( t x ) 0 x";; (X t i ) 0 (" it )" it x 0 it = (X t i ) 0 " itx 0 it (X t i ) 0 " it " i;t x 0 it; E[(X t i ) 0 " itx 0 it j Xi t ] = "(X t i ) 0 x 0 it E t [(X t i ) 0 " it " i;t x 0 it] = O: E[ i (X t i ) 0 " it x 0 i;t ] = O E[(X t i ) 0 (" it )" i;t x 0 i;t j X t i ] = "(X t i ) 0 x 0 i;t ; E[ i (X t i ) 0 " it x 0 is] = O E[(X t i ) 0 (" it )" is x 0 is] = O; E[ i (X t i ) 0 " it x 0 i;t+] = ( t x )E(" it x 0 i;t+) = ( 0 x"; 0 x";) t x E t [(X t i ) 0 (" it )" i;t+ x 0 i;t+] = O; E[ i (X t i ) 0 " it x 0 is] = ( 0 x";s t 0 x";s t+ 0 x";s t ) t x E[(X t i ) 0 " it (" is )x 0 is] = O: Next we commence with the second group of four components The fth, y i;t ( i + " it )( i + " is )y i;s = y i;t y i;s [ i + i (" it + " is ) + " it " is ]; is symmetric in s and t: Just examining t s we nd E[y i;t y i;s i j y i;t ; y i;s ] = y i;t y i;s E[y i;t y i;s i (" it + " is )] = 0: 4

16 For y i;t y i;s " it " is we nd when s = t and for s < t where E t [y i;t y i;t " it] = "(y i;t ) E t [y i;t " it " is y i;s ] = 0: The sixth and seventh component follow from and regarding " it " is y i;t for t > s and for s > t y i;t x 0 is[ i + i (" it + " is ) + " it " is ]; E[ i y i;t x 0 is j y i;t ; x 0 is] = y i;t x 0 is E[ i (" it + " is )y i;t x 0 is] = 0 0 ; x 0 is we nd, for t = s E t [" ity i;t x 0 it j x it ] = "y i;t x 0 it; E t [" it " is y i;t x 0 is] = 0 0 ; E s [" it " is y i;t x 0 is j x is ] = 0 0 : The eighth and nal component is again symmetric in t and s: From we nd and for t = s x it (" it + i )(" is + i )x 0 is = x it x 0 is[ i + i (" it + " is ) + " it " is ]; E[x it x 0 is i j x it ; x 0 is] = x it x 0 is E[x it x 0 is i (" it + " is )] = O; E[x it x 0 it" it j x it ] = "x it x 0 it; while for s < t E t [x it x 0 is" it " is ] = O: P Collecting all the above results and taking for all elements " N i= qs i;lm, we nd for the GMMs estimator that exploits N(T ) L instrument matrix Z BB and has the disturbances given in (6) that the optimal weighting matrix is given by W opt BB _ [ZBB0 (I N D opt )Z BB + ND opt ] ; (40) where D opt is the (T ) (T ) matrix D opt = H C C 0 I T + " T 0 T! ; (4) with C the (T ) (T ) matrix given in (30), and hence D opt = D = " given in (35), and D opt is the L L matrix D opt = O C 0 ; (4) C " O 5

17 with C the (K + )T (T )= (K + )(T ) matrix C = ( C C ); (43) with C the (K + )T (T )= (T ) matrix 3 " y 0 y y T 3 y 0 " y 0 y T 4 y 0 0 " 3 y 0 0 T y C = " T y ; (44) " x 0 x x T 3 x 0 " x 0 x T 4 x 0 0 " 3 x T x " T x and C the (K + )T (T )= K(T ) matrix which is 3 y 0 x"; 0 y 0 x"; y 0 x"; T 3 y 0 x";t O y 0 x"; 0 y 0 x"; T 4 y 0 x";t O O 3 y 0 x"; O 0 T y 0 x"; O O O O T y 0 x"; ( x ) 0 x"; ( x ) 0 x"; ( x ) 0 x"; ( x ) 0 ; x";t O ( x ) 0 x"; ( x ) 0 x"; ( x ) 0 x";t O O ( 3 x ) 0 x"; O (T x ) 0 5 x"; O O O O ( T x ) 0 x"; (45) where we used the short-hand notation 0 x"; 0 x"; 0 x"; 0 x";t 0 x";t 0 x";t+ 0 x";t ; for t = ; :::; T : Note that the weighting matrix (34) explored in Kiviet (007) is suboptimal because it sets C = O; which would only be true if = 0: When the model is pure autoregressive, ie K = 0; the optimal weighting matrix is much simpler Then it is obvious that C is void It leads to the following simpli cations C = C 0 = ( ) " = ( ) " j = j ; for j y = =( ): 6 (46)

18 Substituting these we nd that " C simpli es to the T (T )= (T ) matrix C = " C 3(); with C 3 () given by: 3 ( ) ( ) : : : : : : T 5 ( ) T 4 ( ) 0 ( ) ( ) : : : : : : T 6 ( ) T 5 ( ) ( ) 3 : : : : : : T 7 ( ) 3 T 6 ( ) : : : : : : T 8 ( ) 4 T 7 ( ) : T ( ) T : : : : : : 0 T (47) A more promising operational alternative to D DP D or D GIV in the pure rst-order autoregressive model could be the following Instead of capitalizing on the assumption = 0; as in D W ; one could proceed as follows: capitalize on particular chosen values = ; " = " and = that seem relevant for the situation at hand and then use the operational point-optimal weighting matrix W popt BB (; ; ") _ Z BB0 [I N D = " ]Z BB + N O C3 () C 3 () 0 : (48) O This is optimal for the chosen values of ; and ": Note that the optimal weighting matrix derived above allows a di erent approach in obtaining a -step GMMs estimator (under e ects stationarity and homoskedasticity) than the usually employed ^ cw of (0) based on rst step residuals, viz by simply substituting in D opt and D opt the rst-stage estimators of ; and ": Below, by GMMs opt we indicate the unfeasible GMMs estimator using in the optimal weighting matrix the true values of ; and "; whereas GMMs popt is the feasible point-optimal -step GMMs estimator using particular values of ; and ": 4 Monte Carlo study In the simulation experiments below we have severely restricted ourselves regarding the generality of the Monte Carlo design At this stage we only focussed on the case K = 0, ie like Blundell and Bond (998) we just examine the fully stationary pure rst-order autoregressive case This implies that the panel data have to be generated such that for all i = ; :::; N rst start-up values are obtained according to y i0 = i + and next in a recursion the T observations r " i0; y it = y i;t + i + " it ; 7

19 from the N white-noise series i iid(0; ) and the N distinct (T + ) whitenoise series " it iid(0; "); t = 0; :::; T: All these white-noise series have to be mutually independent from each other Note that this yields Var(y it ) = =( ) + "=( ); so there is both e ect stationarity and stationarity with respect to the idiosyncratic disturbances We will investigate just a few particular combinations of values for the design parameters Without loss of generality we may scale with respect to " and choose " = : We will examine only positive stable values of ; viz = 0:(+0:)0:9; and just a few di erent values of the ratio = ": This variance ratio we vary, for reasons explained in Kiviet (995, 007), by examining f0:5; ; ; 4g; where = : (49) " Hence, we x = ( ) For the sample sizes N and T we choose N = 00 and T = 4(+)0 For all cells we ran 000 replications and all the results for all separate cells are obtained by precisely the same series of random numbers All results are presented graphically In 6 Figures we give results for 6 di erent weighting matrices, viz D GIV ; D DP D ; D = " for = " = 0; D = " for = " = 0; D = " for the true value of = " (which is not feasible in practice) and nally the nonoperational optimal W opt BB matrix Each gure contains 6 diagrams The top 8 contain -step GMMs results and the bottom 8 the related -step GMMs results, where residuals have been used obtained from the -step GMMs procedure in that same gure The gures have four columns of diagrams, which correspond from left to right to the four examined values of = f0:5; ; ; 4g: Each group of 8 diagrams consists of two rows of 4 diagrams: the rst row presents bias, ie the Monte Carlo estimate of E(^ ); the second row depicts relative precision, which is expressed as the Monte Carlo estimate of RMSE(^)= in % In Figure we investigate the operational but sub-optimal simple weighting matrix based on D GIV = I T : We nd negative bias for moderate ; but positive bias for larger and small The bias does not change much between -step and -step estimation, as already noticed by Blundell and Bond (998) However, the precision (measured by relative root mean squared error) improves slightly by -step estimation (which is in agreement with asymptotic theory) Whether these are general phenomena or is special for the GIV weighting matrix will follow when examining other weighting devices We note that the precision gets poor when is small (which is natural when one divides by ); but is especially poor when is small while large Figure shows that D DP D yields bias gures shifted in upward direction (especially for large values), when compared with those in Figure Therefore the bias is smaller and thus the precision higher for moderate Now -step is hardly better than -step Figure 3 is based on the weighting matrix presented in Windmeijer (000), which assumes = 0 For the results are slightly better than those of Figures and But for larger actual values of very poor results are obtained when is small or moderate From Figure 4 we learn that overstating the true value of = " seems less harmful than understating it Aiming at point-optimality at = " = 0; while neglecting the D opt contribution given in (4) to the weighting matrix, is better than the methods in Figures through 4 when is small The method examined in Figure 5 di ers from 8

20 that of Figure 4 in just one respect, viz that the true value of = " has been used in weighting matrix (35) This non-operational method removes almost all bias and clearly shows the best precision results obtained at this stage Finally we examine the qualities of the optimal weighting matrix W opt BB given in (40) In performing the calculations we experienced a serious problem, viz that this matrix is not always positive de nite Although we found W opt BB to be non-singular in all our experiments, it was not always positive de nite due to the contribution of D opt in (4) For = 0; when D opt = O; the optimal weighting matrix was always positive de nite, but not for > 0: We found that, although the matrix is asymptotically proportional to a covariance matrix which is certainly positive de nite, in nite sample it may not be, apparently due to the hybrid nature of W opt BB ; which is partly determined by data moments and partly by parameters Whenever W opt BB was not positive de nite in our simulations, we removed the D opt contribution from the weighting matrix, which then proved to be positive de nite For = we could retain D opt for the larger and smaller T values, but we had to remove it in 60% of the replications for = 0: and T = 0: However, at = 4 this problem emerged much more frequently: in fact in 99% of the replications when T large and small, though hardly ever for = 0:9 Hence, strictly speaking we were only able for the case = 0 to fully examine GMM with optimal weight matrix Figure 6 presents results for the adapted procedure The results do not di er much from those in Figure 5 In a next version of the paper we will also simulate panel data models with further predetermined regressors, cf Bun and Kiviet (006) 5 Conclusions The GMM system estimator for e ect stationary dynamic panel data models exploits two hybrid sets of orthogonality conditions The covariance matrix of all individual orthogonality condition expressions determines the optimal weighting matrix, and due to the hybrid nature of the orthogonality conditions deriving those covariances is not straightforward We show that obtaining a conditional covariance su ces, and that different conditioning sets may be chosen for all separate elements of this variance matrix By choosing for all individual covariances conditioning sets that simplify the analytical derivation of them we nd an expression for the optimal weighting matrix for the model that contains the rst lag of the dependent variable as an explanatory variable and an arbitrary number of further predetermined regressors The resulting weighting matrix depends on the in practice unknown parameters of the model and on further characteristics of the data series, viz the covariance between the regressors and the unobserved individual e ects and the autocovariance between predetermined regressors and idiosyncratic disturbances Hence, it cannot be used to obtain optimal -step GMM estimators, but it can serve as a guide in nding close to optimal or point-optimal operational weighting matrices for GMMs We performed a number of Monte Carlo experiments in the context of the very speci c but simple stationary zero-mean panel data AR() model and examined and compared the results of a few implementations of and -step GMMs, which di er in the weighting matrix employed in -step estimation We found that the results of -step estimation hardly improve upon the corresponding -step GMMs results Hence, the quality of the weighting matrix used in -step estimation is found to be very important Apparently, 9

21 the theoretical result of asymptotic optimality of -step estimation, irrespective of the weighting matrix used in -step estimation, has limited relevance in this model for the sample sizes examined For reasons that still require further study we were not able yet to exploit in a really satisfactory way the derived optimal weighting matrix for GMMs However, it did lead to an operational though suboptimal weighting matrix which seems superior in many cases to those presently in use (the one that yields the GIV estimator and the weighting matrix used in the DPD program) For this new weighting matrix one has to make a reasonable prior guess of the error-component variance ratio = ": References Ahn, SC, Schmidt, P, 995 E cient estimation of models for dynamic panel data Journal of Econometrics 68, 5-7 Ahn, SC, Schmidt, P, 997 E cient estimation of dynamic panel data models: Alternative assumptions and simpli ed estimation Journal of Econometrics 76, Arellano, M, 003 Panel Data Econometrics Oxford University Press Arellano, M, Bond, S, 99 Some tests of speci cation for panel data: Monte Carlo evidence and an application to employment equations Review of Economic Studies 58, Arellano, M, Bover, O, 995 Another look at the instrumental variable estimation of error-components models Journal of Econometrics 68, 9-5 Baltagi, BH, 005 Econometric Analysis of Panel Data, third edition John Wiley & Sons Blundell, R, Bond, S, 998 Initial conditions and moment restrictions in dynamic panel data models Journal of Econometrics 87, 5-43 Blundell, R, Bond, S, Windmeijer, F, 000 Estimation in dynamic panel data models: improving on the performance of the standard GMM estimators In: Baltagi, BH (Eds), Nonstationary Panels, Panel Cointegration, and Dynamic Panels Advances in Econometrics 5, Amsterdam: JAI Press, Elsevier Science Bun, MJG, Kiviet, JF, 006 The e ects of dynamic feedbacks on LS and MM estimator accuracy in panel data models Journal of Econometrics 3, Doornik, JA, Arellano, M, Bond, S, 00 Panel data estimation using DPD for Ox mimeo, University of Oxford Doran, HE, Schmidt, P, 005 GMM estimators with improved nite sample properties using principal components of the weighting matrix, with an application to the dynamic panel data model Forthcoming in Journal of Econometrics Hansen, LP, 98 Large sample properties of generalized method of moment estimators Econometrica 50, Kiviet, JF, 995 On bias, inconsistency, and e ciency of various estimators in dynamic panel data models Journal of Econometrics 68, Kiviet, JF, 007 Judging contending estimators by simulation: Tournaments in dynamic panel data models Chapter (pp8-38) in Phillips, GDA, Tzavalis, E (Eds) The Re nement of Econometric Estimation and Test Procedures; Finite Sample and Asymptotic Analysis Cambridge University Press Lancaster, T, 00 Orthogonal parameters and panel data Review of Economic Studies 69,

22 Windmeijer, F, 000 E ciency comparisons for a system GMM estimator in dynamic panel data models In: RDH Heijmans, DSG Pollock and A Satorra (eds), Innovations in Multivariate Statistical Analysis, A Festschrift for Heinz Neudecker Advanced Studies in Theoretical and Applied Econometrics, vol 36, Dordrecht: Kluwer Academic Publishers (IFS working paper W98/)

23 Figure GMMs and GMMs employing D GIV ; for = 0:5; ; ; 4:

24 Figure GMMs and GMMs employing D DP D ; = 0:5; ; ; 4: 3

25 Figure 3 GMMs and GMMs employing D = " with = " = 0; for = 0:5; ; ; 4: 4

26 Figure 4 GMMs and GMMs employing D = " with = " = 0; for = 0:5; ; ; 4: 5

27 Figure 5 GMMs and GMMs employing D = " (non-operational); for = 0:5; ; ; 4: 6

28 Figure 6 GMMs and GMMs employing W opt BB-adapted (non-operational); = 0:5; ; ; 4: 7