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P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 384 Handbook of Empirical Economics and Finance Conditional on h, the MLE of ␦ = ( , ,  ) is identical to the GLS, ˆ ␦ GLS = N i=1 ˜ X i ∗−1 ˜ X i −1 N i=1 ˜ X i ∗−1 y i , (13.43) where ˜ X i = x i 00 0 y i−1 X i . (13.44) When h is unknown, one can use a two-step procedure. In the first step, we regress y i1 on x i to obtain ˆ 2 v ∗ and apply GMM to obtain ˆ 2 ⑀ . In the second step, we substitute estimated ˆ h for h in Equation 13.43. However, the feasible GLS is not as efficient as GLS (for detail, see Hsiao, Pesaran, and Tahmiscoglu 2002). 13.7 Models with Both Individual- and Time-Specific Additive Effects When time-specific effects also appear in v it as in Equation 13.2, the estimators ignoring the presence of t like those discussed in Sections 13.13 to 13.6 are no longer consistent when T is finite. For notational ease and without loss of generality, we illustrate the fundamental issues of dynamic model with both individual- and time-specific additive effects model by restricting  = 0 in Equation 13.1, thus the model becomes y it = y i,t−1 + v it , (13.45) v it = ␣ i + t + ⑀ it ,i= 1, ,N,t= 1, ,T,y i0 observable. (13.46) The panel data estimators discussed in Sections 13.5 and 13.6 assume no presence of t (i.e., t = 0∀t). When t are indeed present, those estimators are not consistent if T is finite when N →∞. For instance, the consistency of GMM (Equation 13.33) is based on the assumption that 1 N N i=1 y i,t−j v it converges to the population moments (Equation 13.32). However, if t are also present as in Equation 13.46, this condition is likely to be violated. To see this, taking first difference of Equation 13.45 yields y it = y i,t−1 + v it = y i,t−1 + t + ⑀ it , (13.47) i = 1, ,N, t = 2, ,T. P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 Dynamic Panel Data Models 385 Although E(y i,t−j v it ) = 0 for j = 2, ,t, (13.48) the sample moment, as N −→ ∞ , 1 N N i=1 y i,t−j v it = 1 N N i=1 y i,t−j t + 1 N N i=1 y i,t−j ⑀ it (13.49) converges to ¯y t−j t , which in general is not equal to zero, in particular, if y it has mean different from zero, 5 where ¯y t = 1 N N i=1 y it . To obtain consistent estimators of , we need to take explicit account of the presence of t in addition to ␣ i .If␣ i and t are random and satisfy Equation 13.4, because Ey i0 v it = 0, we either have to write Equation 13.45 conditional on y i0 or to complete the system (Equation 13.45) by deriving the marginal distribution of y i0 . By continuous substitutions, we have y i0 = 1 − m 1 − ␣ i + m−1 j=0 −j j + m−1 j=0 ⑀ i,−j j = v i0 , (13.50) assuming the process started at period −m. Under Equation 13.4, Ey i0 = Ev i0 = 0, Var (y i0 ) = 2 0 , E(v i0 v it ) = 1− m 1− 2 ␣ = c, Ev it v jt = d. Stacking the T + 1 time series observations for the ith indi- vidual into a vector, y i = (y i0 , ,y iT ) and y i,−1 = (0,y i0 , ,y i,T−1 ) ,v i = (v i0 , ,v iT ) .Let y = (y 1 , ,y N ) ,y −1 = (y 1,−1 , ,y N,−1 ),v = (v 1 , ,v N ) , then y = y −1 + v , (13.51) Ev = 0 , Ev v = 2 ⑀ I N ⊗ 0 0 I T + 2 ␣ I N ⊗ 0 c ∗ e T c ∗ e T e T e T + 2 e N e N ⊗ d ∗ 0 0 I T , (13.52) = 2 0 − d 2 ⑀ ,d ∗ = d 2 ,c ∗ = c 2 ␣ , (13.53) where ⊗ denotes the kronecker product. The system (Equation 13.51) has a fixed number of unknowns (, 2 ⑀ , 2 ␣ , 2 , 2 0 ,c,d)asN and T increase. There- fore, the MLE (or quasi-MLE or GLS) of Equation 13.51 is consistent and asymptotically normally distributed. 5 For instance, if y it is also a function of exogenous variables as Equation 13.1. P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 386 Handbook of Empirical Economics and Finance When ␣ i and t are fixed constants, we note that first differencing only eliminates ␣ i from the specification. The time-specific effects, t , remain at Equation 13.47. To further eliminate t , we note that the cross-sectional mean y t = 1 N N i=1 y it is equal to y t = y t−1 + t + ⑀ t , (13.54) where ⑀ t = 1 N N i=1 ⑀ it . Taking deviation of Equation 13.47 from Equa- tion 13.54 yields y ∗ it = y ∗ i,t−1 + ⑀ ∗ it , i = 1, ,N, t = 2, ,T, (13.55) wherey ∗ it = (y it −y t ) and ⑀ ∗ it = (⑀ it −⑀ t ).Thesystem (Equation 13.55) no longer involves ␣ i and t . Since E[y i,t−j ⑀ ∗ it ] = 0 for j = 2, ,t, t = 2, ,T, (13.56) the 1 2 T(T −1) orthogonality conditions can be represented as E(W i ˜⑀ ∗ i ) = 0 , (13.57) where ˜⑀ ∗ i = (⑀ ∗ i2 , , ⑀ ∗ iT ) , W i = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q i2 0 ··· 0 0 q i3 ·· . . . . . . . . . 0 0 q iT ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,i= 1, ,N, and q it = (y i0 ,y i1 , ,y i,t−2 ) ,t= 2, 3, ,T. Following Arellano and Bond (1991), we can propose a generalized method of moments (GMM) estimator, 6 ˜ GMM = 1 N N i=1 ˜y ∗ i,−1 W i ˆ −1 1 N N i=1 W i ˜y ∗ i,−1 −1 1 N N i=1 ˜y ∗ i,−1 W i ˆ −1 1 N N i=1 W i ˜y ∗ i , (13.58) 6 For ease of exposition, we have only considered the GMM that makes use of orthogonality conditions. Foradditionalmoments conditions suchas homoscedasticity orinitial observations see, e.g., Ahn and Schmidt (1995), Blundell and Bond (1998). P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 Dynamic Panel Data Models 387 where ˜y ∗ i = (y ∗ i2 , , y ∗ iT ) , ˜y ∗ i−1 = (y ∗ i1 , , y ∗ i,T−1 ) , and ˆ = 1 N 2 N i=1 W i ˆ ˜⑀ ∗ i N i=1 W i ˆ ˜⑀ ∗ i (13.59) and ˆ ˜⑀ ∗ i = ˜y ∗ i − ˜y ∗ i,−1 ˜, and ˜ denotes some initial consistent estimator of , say a simple instrumental variable estimator. The asymptotic covariance matrix of ˜ GMM can be approximated by asy. cov (˜ GMM ) = N i=1 ˜y ∗ i,−1 W i ˆ −1 N i=1 W i ˜y ∗ i,−1 −1 . (13.60) To implement the likelihood approach, we need to complete the system (Equation 13.55) by deriving the marginal distribution of y ∗ i1 through con- tinuous substitution, y ∗ i1 = m−1 j=0 ⑀ ∗ i,1−j j = ˜⑀ ∗ i1 ,i= 1, ,N. (13.61) Let y ∗ i = (y ∗ i1 , , y ∗ iT ), y ∗ i = (0, , y ∗ i,T−1 ), ˜⑀ ∗ i = (˜⑀ ∗ i1 , , ⑀ ∗ iT ), the system y ∗ i = y ∗ i,−1 + ˜⑀ ∗ i , (13.62) does not involve ␣ i and t . The MLE conditional on = Var (y ∗ i1 ) 2 ⑀ is identical to the GLS ˆ GLS = N i=1 y ∗ i,−1 ˜ A −1 y ∗ i,−1 −1 N i=1 y ∗ i,−1 ˜ A −1 y ∗ i . (13.63) where ˜ A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −10 0··· 00 −12−10··· · · 0 −12−1 ··· · · ·····2 −1 0 ··· −12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (13.64) The GLS isconsistent and asymptotically normally distributed with covari- ance matrix equal to Var(ˆ GLS ) = 2 ⑀ N i=1 y ∗ i,−1 ˜ A −1 y ∗ i,−1 −1 . (13.65) P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 388 Handbook of Empirical Economics and Finance Remark 13.7 The GLS with present is basically of the same form as the GLS without the time-specific effects (i.e., = 0 ) (Hsiao, Pesaran, and Tahmiscioglu 2002), (Equation 13.25). However, there is an important dif- ference between the two. The estimator (Equation 13.63) uses y ∗ i,t−1 as the regressor for the equation y ∗ it (Equation 13.62), not uses y i,t−1 as the regres- sor for the equation y it (Equation 13.47). If there are indeed common shocks that affect all the cross-sectional units, then the estimator Equation 13.25 is inconsistent while Equation 13.63 is consistent (for detail, see Hsiao and Tahmiscioglu 2008). Note also that even though when there are no time- specific effects, Equation 13.63 remains consistent, although it will not be as efficient as Equation 13.25. Remark13.8 Theestimator(Equation13.63)andtheestimatorEquation13.58 remain consistent and asymptotically normally distributed when the effects are random because the transformation (Equation 13.54) effectively removes the individual- and time-specific effects from the specification. However, if the effects are indeed random,then the MLE or GLS of Equation 13.51 is more efficient. Remark 13.9 The GLS (Equation 13.63) assumes known .If is unknown, one may substitute it by a consistent estimator ˆ, then apply the feasible GLS. However, there is an important difference between the GLS and the feasible GLS in a dynamic setting. The feasible GLS is not asymptotically equivalent to the GLS when T is finite. However, if both N and T →∞and lim ( N T ) = c > 0, then the FGLS will be asymptotically equivalent to the GLS. (Hsiao and Tahmiscioglu 2008). Remark 13.10 The MLE or GLS of Equation 13.63 can also be derived by treating t as fixed parameters in the system (Equation 13.47). Through continuous substitution, we have y i1 = ∗ 1 + ˜⑀ i1 , (13.66) where ∗ 1 = m j=0 j 1−j and ˜⑀ i1 = m j=0 j ⑀ i,1−j . Let y i = (y i1 , , y iT ), y i,−1 = (0, y i1 , , y i,T−1 ), ⑀ i = (˜⑀ i1 , , ⑀ iT ), and = ( ∗ 1 , 2 , , T ), we may write y = NT × 1 ⎛ ⎜ ⎜ ⎝ y 1 . . . y N ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ y 1,−1 . . . y N,−1 ⎞ ⎟ ⎟ ⎠ + (e N ⊗ I T ) + ⎛ ⎜ ⎜ ⎝ ⑀ 1 . . . ⑀ N ⎞ ⎟ ⎟ ⎠ = y −1 + (e N ⊗ I T ) + ⑀ , (13.67) If ⑀ it is i.i.d. normal with mean 0 and variance 2 ⑀ , then ⑀ i is independently normally distributed across i with mean 0 and covariance matrix 2 ⑀ ˜ A, and = Var (˜⑀ i1 ) 2 ⑀ . P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 Dynamic Panel Data Models 389 The log-likelihood function of y takes the form log L =− NT 2 log 2 ⑀ − N 2 log | ˜ A|− 1 2 2 ⑀ [y − y −1 − (e N ⊗ I T ) ] (I N ⊗ ˜ A −1 )[y − y −1 − (e N ⊗ I T ) ]. (13.68) Taking partial derivative of Equation 13.68 with respect to and solving for yields ˆ = (N −1 e N ⊗ I T )(y − y −1 ). (13.69) Substituting Equation 13.69 into Equation 13.68 yields the concentrated log- likelihood function. log L c =− NT 2 log 2 ⑀ − N 2 log | ˜ A| − 1 2 2 ⑀ (y ∗ − y ∗ −1 ) (I N ⊗ ˜ A −1 )(y ∗ − y ∗ −1 ). (13.70) Maximizing Equation 13.69 conditional on yields Equation 13.63. Remark 13.11 When approaches to 1 and 2 ␣ is large relative to 2 ⑀ , the GMM estimator of the form (Equation 13.68) suffers from the weak instru- mental variables issues and performs poorly (e.g., Binder, Hsiao, and Pesaran 2005). On the other hand, the performance of the likelihood or GLS estimator (Equation 13.63) is not affected by these problems. Remark 13.12 Hahn and Moon (2006) propose a bias corrected estimator as ˜ b = ˜ ∗ cv + 1 T (1 + ˜ ∗ cv ). (13.71) They show that when N/T → c, as both N and T tend to infinity where 0 < c < ∞, √ NT(˜ b − ) ⇒ N(0, 1 − 2 ). (13.72) The limited Monte Carlo studies conducted by Hsiao and Tahmiscioglu (2008) to investigate the finite sample properties of the feasible GLS (FGLS), GMM, bias corrected (BC) estimator of Hahn and Moon (2006) have shown that in terms of bias and root mean square errors, FGLS dominates. However, the BC rapidly improves as T increase. In terms of the closeness of actual size to the nominal size, again FGLS dominates and rapidly approaches the nominal size when N or T increases. The GMM also has actual sizes close to nominal sizes except for the cases when is close to unity (here = 0.8). The BC has significant size distortion, presumably because of the correction of bias being based on ˆ ∗ cv and the use of asymptotic covariance matrix which is significantly downward biased in finite sample. P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 390 Handbook of Empirical Economics and Finance Remark 13.13 Hsiao and Tahmiscioglu (2008) also compared the FGLS and GMM with and without the correction of time-specific effects in the presence of both individual- and time-specific effects or in the presence of individual- specific effects only. It is interesting to note that when both individual- and time-specific effects are present, the biases and root mean squares errors are largeforestimatorsassumingno time-specific effects. On the other hand, even in the case of no time-specific effects in the true data generating process, there is hardly any efficiency loss for the FGLS or GMM that makes the correction of presumed presence of time-specific effects. Therefore, if an investigator is not sure if the assumption of cross-sectional independence is valid or not, it might be advisable to use estimators that take account both individual- and time-specific effects. 13.8 Estimation of Multiplicative Models In this section we consider the estimation of Equation 13.1, where v it is as- sumed to be of the form v it = ␣ i t + ⑀ it . (13.73) When ␣ i is independently distributed across i with mean 0 and variance 2 ␣ and t isindependentlydistributed overt withmean0 andvariance 2 ,Ev it = 0,Ev 2 it = 2 ⑀ + 2 ␣ 2 = 2 v , and Ev it v is = 0 for t = s, Ev it v js = 0 for i = j.In other words, Equation 13.1 has error terms that are uncorrelated over time and across individuals, with constant variance 2 v . Hence the least squares estimator is consistent and asymptotically normally distributed either N or T or both tend to infinity. When ␣ i and t are treated as fixed constants, the MLE are inconsistent if T is finite for the same basic reason as the additive model (Equation 13.2). Ahn, Lee, and Schmidt (2001), Bai (2007), Kiefer (1980), etc., have proposed a nonlinear GMM and iterative LS estimators for the static model with multi- plicative effects. Their nonlinear GMM approach can be similarly generalized to obtain a consistent estimator of (e.g., Hsiao 2008). Let t = t / t−1 , then (y it − t y i,t−1 ) = (y i,t−1 − t y i,t−2 ) + (⑀ it − t ⑀ i,t−1 ),t= 2, ,T. (13.74) It follows that E[y i,t−j (⑀ it − t ⑀ i,t−1 )] = 0, for j = 2, ,t. (13.75) P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 Dynamic Panel Data Models 391 Let W i = T(T −1) 2 × (T −1) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ q i2 0 ·· 0 0 q i3 ·· 0 0 0 ·· · ····· 0 ···q iT ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , = (T −1) × (T −1) ⎡ ⎢ ⎢ ⎢ ⎣ 2 0 ·· 0 0 3 ·· · ····· ···· T ⎤ ⎥ ⎥ ⎥ ⎦ , q it = (y i0 , ,y i,t−2 ),t= 2, ,T, ⑀ i = (⑀ i2 , , ⑀ iT ) , ⑀ i,−1 = (⑀ i1 , , ⑀ i,T−1 ) . Then a GMM estimator of and can be obtained from the moment condi- tions E[W i (⑀ i − ⑀ i,−1 )] = 0 . (13.76) The nonlinear GMM estimators of and amount to applying nonlinear three-stage least squares to the system y i = [I T−1 + ]y i,−1 − y i,−2 + ⑀ i − ⑀ i,−1 ,i= 1, ,N, (13.77) using W i as instruments, where y i = (y i2 , ,y iT ) ,y i,−1 = (y i1 , ,y i,T−1 ) , and y i,−2 = (y i0 , ,y i,T−2 ) . ThenonlinearGMMestimatorsof and t areconsistentandasymptotically normally distributed as N →∞. From the t , we can solve for t through the normalization rule 1 = 1or T t=1 2 t = 1. From and t , we obtain ˆ␣ i = 1 T t=1 ˆ 2 t T t=1 ˆ t y it − ˆ T t=1 ˆ t y i,t−1 ,i= 1, ,N. (13.78) The estimator (Equation 13.78) is consistent if T →∞. The implementationofnonlinear GMM is quitecomplicated,Pesaran (2006, 2007) notes that ¯y t = ¯y t−1 + ¯␣ t + ¯⑀ t , (13.79) where ¯y t = 1 N N i=1 y it , ¯␣ = 1 N N i=1 ␣ i , ¯⑀ t = 1 N N i=1 ⑀ it . P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 392 Handbook of Empirical Economics and Finance When N →∞, ¯⑀ t −→ 0. Assuming ¯␣ = 0, substituting t = ¯␣ −1 (¯y t − ¯y t−1 ) into Equation 13.45 yields, y it = y i,t−1 + ␥ 1i ¯y t + ␥ 2i ¯y t−1 + ⑀ it (13.80) Therefore, Pesaran (2006, 2007) suggests estimating the cross-sectional mean augment regression (Equation 13.80) and shows that as both N and T →∞, the least squares estimator of Equation 13.80 yields consistent and asymptot- ically normally distributed ˆ. 13.9 Test of Additive versus Multiplicative Model Multiplicative model implies departure from additivity in their effects on outcomes. It is shown by Bai (2007) that the additive model is embedded into the model of multiple common factors with heterogeneous response by letting ␣ i = ␣ i 1 , t = 1 t , then Equation 13.2 becomes v it = ␣ i t + ⑀ it . (13.81) When N −→ ∞ , one may solve t from Equation 13.79 that yields ˆ t = (¯␣ ¯␣ ) − ¯␣ (υ t − ¯y t−1 ), (13.82) where ( ¯␣ ¯␣ ) − denotes the generalized inverse of ( ¯␣ ¯␣ ). Substituting Equa- tion 13.82 into Equation 13.45 again yields Equation 13.80. Therefore, the Pesaran cross-sectional mean augmented regression of Equation 13.80 is con- sistent whether the unobserved heterogeneity is additive or multiplicative, but Equation 13.80is inefficient if the unobserved heterogeneities are additive compared to Equation 13.58 or Equation 13.63. However, if the underlying model is multiplicative, Equation 13.80 is consistent, butnotEquation13.58or Equation 13.63. Therefore, a Hausman type specification test can be proposed to test the null: H 0 : Equation 13.2 holds versus H 1 : Equation 13.2 does not hold by considering the test statistic ˆ A − ˆ m √ Var ( ˆ m ) − Var(ˆ A ) ∼ N(0, 1), (13.83) P1: BINAYA KUMAR DASH November 3, 2010 16:25 C7035 C7035˙C013 Dynamic Panel Data Models 393 where ˆ A denotes the efficient estimator of Equation 13.1 under the additive assumption (Equation 13.2) and ˆ m is the estimator (Equation 13.1) under the multiplicative assumption (Equation 13.73). 13.10 Concluding Remarks In this chapter we review three fundamental issues of modeling dynamic paneldatain thepresenceof unobservedheterogeneityacrossindividualsand over time—the fixed effects of modeling unobserved individual- and time- specific heterogeneity versus random effects; additive versus multiplicative effects and the likelihood versus methods of moments approach. Wehavenotdiscussedissues ofmodeling multivariatedynamicpanel mod- els (e.g., Binder, Hsiao, and Pesaran (2005), panel unit root tests (e.g., Breitung and Pesaran 2008; Moon and Perron 2004; Phillips and Sul 2003); parameter heterogeneity (e.g., Hsiao and Pesaran 2008), etc. However, inprinciple, those issues can also be put in these perspectives. The advantage of the fixed effects specification is that there is no need to specify the relations between the unobserved effects and observed condi- tional (or explanatory) variables. The disadvantages are that (1) unless both cross-sectionaldimensionandtime dimension of panels arelarge, the fixed ef- fects specification introduces incidental parameters issues on the individual- specific effects, ␣ i , if the time dimension is fixed and on the time-specific ef- fects, t if the cross-sectional dimension is small; (2) the impact of time-invariant but individual-specific variables such as gender or socio- demographic background variables with the presence of additive individual- specific effects and the impact of time-specific but individual invariant such as price and some macro-variables with the presence of additive time-specific effects are unidentified; and (3) the fixed effects inference only makes use of within-group variation. The between group information is ignored. The advantages of random-effects specification are (1) there are no inciden- tal parameter issues; (2) the impacts of observed individual-specific but time- invariant and individual-invariant but time-varying variables can be iden- tified; (3) both the within-group and between group information are used for inference. Since the between group variation in general is much larger than the within group variation, the RE specification can lead to much more efficient use of sample information. The disadvantage is that the relationship between the unobserved effects and observed conditional variables need to be specified. In short, the advantages of random effects specification are the disadvantage of fixed effects specification and the advantages of fixed effects specification are the disadvantages of random effects specification. Statistical inference procedures for additive effects models are simpler than themultiplicativeeffects models.However,ifthe data generatingprocesscalls for a multiplicative effects specification, statistical inference proceduresbased on additive effects specification will be misleading. On the other hand, if the [...]... variance of ( In − Wn )Vnt is 2 n , where n = ( In − Wn )( In − Wn ) This 0 transformed equation has less degrees of freedom than n Denote the degree of freedom of Equation 14.4 as n∗ Then, n∗ is the rank of the variance matrix of ( In − Wn )Vnt , which is the number of nonzero eigenvalues of n Hence, P1: NARESH CHANDRA November 12, 2010 18:3 C7035 402 C7035˙C014 Handbook of Empirical Economics and Finance. .. increasing the number of blocks, the number of unit eigenvalues of the block diagonal matrix will also increase, but the percentage remains a constant In our simulation, when n = 18, n∗ = 16; when n = 54, n∗ = 48 P1: NARESH CHANDRA November 12, 2010 412 18:3 C7035 C7035˙C014 Handbook of Empirical Economics and Finance times We also compare the empirical standard deviation (SD) and the empirical mean square... number of non-unit eigenvalues6 of Wn Thus, the transformed variables do not have time effects and are all stable even when 0 + ␥0 + 0 is equal to or greater than 1 Let [Fn , Hn ] be the orthonormal matrix of eigenvectors and n be the diagonal matrix of nonzero eigenvalues of n such that n Fn = Fn n and n Hn = 0 That is, the columns of Fn consist of eigenvectors of nonzero eigenvalues and those of Hn... 16:25 C7035 C7035˙C013 Handbook of Empirical Economics and Finance Phillips, P.C.B., and D Sul 2003 Dynamic Panel Estimation and Homogeneity Testing Under Cross-Section Dependence Econometrics Journal 6: 217–259 Ziliak, J P 1997 Efficient Estimation with Panel Data When Instruments Are Predetermined: An Empirical Comparison of Moment-Condition Estimators, Journal of Business and Economic Statistics... extended model, where serial correlation on each spatial unit over time and spatial dependence across spatial units are allowed in the disturbances Su and Yang (2007) study the dynamic panel data with spatial error and random 397 P1: NARESH CHANDRA November 12, 2010 18:3 C7035 398 C7035˙C014 Handbook of Empirical Economics and Finance effects These panel models specify the spatial correlation by including... Econometrica 54, 869–880 Anderson, T W and C Hsiao 1981 Estimation of Dynamic Models with Error Components Journal of American Statistical Association 76:598–606 Anderson, T W., and C Hsiao 1982 Formulation and Estimation of Dynamic Models Using Panel Data Journal of Econometrics 18:47–82 Arellano, M., and S R Bond 1991 Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to... }, i = 1, 2, , n and t = 1, 2, , T, are i.i.d across i and t with zero mean, variance 2 and E|vit |4+ < ∞ for some 0 > 0 Assumption 3 Sn () is invertible for all ∈ and the true parameter 0 is in the interior of Furthermore, is compact Assumption 4 The elements of Xnt are nonstochastic and bounded, uniformly T 1 ˜ ∗˜ in n and t, and the limit of nT t=1 Xnt J n Xnt exists and is nonsingular... the number of eigenvalues of Wn different from 1 With the VECM representation, one may 6 This is so, because (1) the set K n of eigenvectors corresponding to the zero eigenvalues of ( In − Wn )( In − Wn ) is the same as that of ( In − Wn ) ; (2) the dimension of K n is the number of −1 −1 unit eigenvalues of Wn ; (3) Wn = Rn ϖn Rn if and only if Wn = Rn ϖn Rn , i.e., the eigenvalues of Wn and Wn are... a sum of a possible stable part, a possible unstable or explosive part, and a time effect part (see Appendix A.2 for proof) u s ␣ Ynt = Ynt + Ynt + Ynt , (14.3) 3 When ␥0 + 0 + 0 = 1 and ␥0 = 1, the asymptotic properties of estimators are considered in √ Yu and Lee (2010) The QML estimate of the dynamic coefficient is nT 3 consistent and the √ estimates of other parameters are nT consistent, and they... (see Phillips and Moon 1999; Hahn and Kuersteiner 2002; Alvarez and Arellano 2003; Hahn and Newey 2004, etc.) For the panel data with spatial interactions, Kapoor, Kelejian, and Prucha (2007) extend the asymptotic analysis of the method of moments estimators to a spatial panel model with error components, where T is finite Baltagi, Song, Jung, and Koh (2007) consider the testing of spatial and serial dependence . Handbook of Empirical Economics and Finance Remark 13.13 Hsiao and Tahmiscioglu (2008) also compared the FGLS and GMM with and without the correction of time-specific effects in the presence of. disturbances. Su and Yang (2007) study the dynamic panel data with spatial error and random 397 P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C 014 398 Handbook of Empirical Economics and Finance effects C7035˙C 014 402 Handbook of Empirical Economics and Finance n ∗ = n − m n is also the number of non-unit eigenvalues 6 of W n . Thus, the transformed variables do not have time effects and are
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