Estimation for spatial dynamic panel data with fixed effects: The case of spatial cointegration

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Abstract

Yu et al. (2008) establish asymptotic properties of quasi-maximum likelihood estimators for a stable spatial dynamic panel model with fixed effects when both the number of individuals n and the number of time periods T are large. This paper investigates unstable cases where there are unit roots generated by temporal and spatial correlations. We focus on the spatial cointegration model where some eigenvalues of the data generating process are equal to 1 and the outcomes of spatial units are cointegrated as in a vector autoregressive system. The asymptotics of the QML estimators are developed by reparameterization, and bias correction for the estimators is proposed. We also consider the 2SLS and GMM estimations when T could be small.

Introduction

Spatial econometrics deals with (spatial) interactions of economic units in cross section data, and it can be extended to panel data models. Baltagi et al., 2003, Baltagi et al., 2007 and  Kapoor et al. (2007) investigate the estimation and testing of different spatial panel data models under random effects specifications. An alternative specification for a panel data model assumes fixed effects, which has the advantage of robustness in that the individual effects are allowed to correlate with included regressors in the model (Hausman, 1978). For static models, Lee and Yu (2010a) study the spatial panel data model with fixed effects, spatial lags and spatial autoregressive (SAR) disturbances. For dynamic models, we have the following spatial dynamic panel data (SDPD) model:Ynt=λ10WnYnt+γ0Yn,t1+ρ0WnYn,t1+Xntβ0+cn0+αt0ln+Untwith Unt=λ20MnUnt+Vnt for t=1,2,,T. Here, Ynt=(y1t,y2t,,ynt) and Vnt=(v1t,v2t,,vnt) are n×1 column vectors, and vit’s are i.i.d. across i and t with zero mean and variance σ02. The Wn and Mn are n×n nonstochastic spatial weights matrices, Xnt is an n×k matrix of nonstochastic regressors,1 and cn0 is an n×1 column vector of individual fixed effects. αt0 is a scalar representing a time effect, and ln is an n×1 column vector of ones. The disturbances Unt follow an SAR process with the spatial weights matrix Mn, which may or may not be the same as Wn.2 The Wn and Mn are row-normalized with zero diagonals. A row-normalized Wn (resp. Mn) has the property Wnln=ln (resp. Mnln=ln), which ensures that all the weights are between 0 and 1, and a weighting operation can be interpreted as an average of the neighboring values. The practical specification of Ord (1975) is constructed by the row-normalization of a symmetric matrix. Ord (1975) showed that such a row-normalized matrix is diagonalizable and all of its eigenvalues are real. We consider such a case that Wn is diagonalizable with all real eigenvalues.

Define Sn(λ1)=Inλ1Wn for an arbitrary value λ1 and SnSn(λ10) at the true parameter value. Presuming Sn is invertible,3 by denoting An=Sn1(γ0In+ρ0Wn), (1) can be rewritten as Ynt=AnYn,t1+Sn1(Xntβ0+cn0+αt0ln+Unt).

Assumption 1

(i) Wn and Mn are row-normalized nonstochastic spatial weights matrices with zero diagonals; (ii) (Inλ10Wn) and (Inλ20Mn) are invertible with |λj0|<1 for j=1 and 2; (iii) Wn is diagonalizable, i.e., Wn=ΓnϖnΓn1 where Γn is the eigenvector matrix, ϖn is the diagonal eigenvalue matrix, and all the eigenvalues are real.

With ϖn in Assumption 1, the eigenvalue matrix of An is Dn=(Inλ10ϖn)1(γ0In+ρ0ϖn). Depending on the eigenvalues of An, we have three cases of SDPD models. When all the roots are less than 1 in absolute value, we call it a stable case. When all the roots are equal to 1, we term it a pure unit root case, which generalizes the unit root dynamic panel data model in the time series literature to include spatial elements. When some of the roots (but not all) are equal to 1, we define it as a spatial cointegration case, where the unit roots in the process are generated with mixed time and spatial dimensions. This terminology of spatial cointegration can be justified by regarding the spatial panels as a vector autorgressive (VAR) system and we can show the existence of cointegration relationships among spatial units (see Section 2.2).

For the SDPD model in this paper, the cointegrating space is completely known (determined by the spatial weights matrix) when cointegration occurs, while in the conventional cointegration time series literature it is the main object of inference (see Johansen, 1991 and Phillips, 1995, etc.). In the conventional cointegration, the dimension of a VAR process is fixed and relatively small; in the present paper, it can be large and asymptotically it tends to infinity. Such a system is of particular interest for the study of market integration across regions. Due to different eigenvalue matrices of the models, the asymptotics of the spatial cointegration case differs from those of the stable case and the pure unit root case. In Yu et al. (2008), the consistency and asymptotic distribution of the maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators are established for the stable SDPD model. These estimators are nT-consistent, and have biases of the order O(1/T). Bias correction for the estimator is possible; when T grows faster than n1/3, the bias corrected estimator yields a centered confidence interval.

When there are unit roots in the process, we find that the appropriate reparameterization motivated by Sims et al. (1990) provides a good device for our investigation.4 The current paper’s main finding is that, for the spatial cointegration case, all the estimates are nT-consistent; but the variance matrix of the estimates is singular in the limit, and the sum of the spatial and dynamic effects estimates is superconsistent. For the pure unit root case, the dynamic effect estimate is superconsistent and other estimates are nT-consistent, while the sum of spatial lag effect and spatial time lag effect estimates is also superconsistent. Bias corrections for spatial cointegration or pure unit root cases can also be constructed, but with different procedures as compared to the stable case.5

There is an interest in nonstationary panels for both independent panels and cross-sectionally correlated panels with common time factors. That literature is now large. The present paper differs from that literature in that it covers an unstable panel data model with the cross sectional dependence specified by local spatial interactions as well as (fixed) time effects. There are already extensive empirical applications for nonstationary panel data.6 We expect that our model contributes to existing nonstationary panel data models of empirical interest. For example, Keller and Shiue (2007) investigate how the spatial feature influences the expansion of interregional trade by studying historical data on Chinese rice markets. The spatial effects are significant and the sum of the estimates of the spatial and dynamic effects is approximately 1. By applying the SDPD model to the rice price, we find that the regional prices follow a spatial cointegration process (Lee and Yu, 2010b).

The present paper is organized as follows. Section 2 further discusses the spatial cointegration model. Section 3 considers the QML estimation of the spatial cointegration model. We derive the consistency and asymptotic distribution of the parameter estimates via a reparameterization approach. The asymptotics of the estimator relies on both n and T tending to infinity. Also, a bias correction procedure is proposed. The asymptotics for the pure unit root case is summarized in Appendix D. Section 4 presents the GMM estimation for comparison. For the asymptotics of the GMM estimator, T is allowed to be finite and fixed. Monte Carlo results on the finite sample properties of these estimators are presented in Section 5. Conclusions are in Section 6. Some useful lemmas and proofs are collected in Appendix A Notations, Appendix B Some lemmas, Appendix C Some algebra and proof for, Appendix D Unit root SDPD.

Section snippets

The spatial cointegration model

The following subsections provide some different views on implied structures of the model (1).

QML estimation

In this section, we consider the quasi-maximum likelihood approach where individual effects are jointly estimated with common parameters in the model. In order for the QML estimates to be consistent, T is assumed to be large. Denote Xnt=h=0t1Xnh and t̃=tT+12.

Assumption 8

T goes to infinity and n is a strictly increasing function of T, and limT1nT3t=1T(cn0t̃+X̃ntβ0)WnuRnJnRnWnu(cn0t̃+X̃ntβ0)0.

Assumption 8 specifies that n as T. We say that n,T simultaneously.16

GMM estimation

Instead of the QML, we may use GMM for estimation. For the SDPD models, the GMM uses lagged values and exogenous variables to construct linear moments. Contrary to the QML estimation where consistency requires T tending to infinity, the GMM is applicable even when T is small, because GMM estimation can be applied to an equation after the elimination of the fixed effects so that the GMME does not have the O(1/T) bias as in MLE.18

Monte Carlo

We conduct a small Monte Carlo experiment to evaluate the performance of our QML estimators, their bias corrected estimators, 2SLSE, G2SLSE and GMME for the spatial cointegration model. Samples are generated from (1) using θ0=(0.4,0.2,1,0.4,0.2,1) where θ0=(γ0,ρ0,β0,λ10,λ20,σ02), and Xnt,cn0,αt0 and Vnt are generated from independent normal distributions.21

Conclusion

This paper investigates unstable SDPD models where there are unit roots generated by temporal and spatial correlations in the DGP. The spatial cointegration model refers to the case where some (but not all) eigenvalues of the DGP are equal to 1; the pure unit root model refers to the case where all the eigenvalues are equal to 1. We develop the consistency, superconsistency, and asymptotics of the QML and GMM estimators by reparameterization and data transformation, and also propose bias

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We would like to thank participants of seminars at Ohio State University, University of Michgan, Yale University, University of Washington at Seattle, and the Third Symposium on Econometric Theory and Applications at HKUST; we would also like to thank Professor Peter Phillips and the co-editor, Professor Peter M. Robinson, of this journal for helpful comments as well as two anonymous referees. Yu acknowledges funding from National Science Foundation of China (Grant No. 71171005) and support from Center for Statistical Science of Peking University.

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