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Abstract
This paper investigates the quasi-maximum likelihood estimation of spatial dynamic panel data models where spatial weights matrices can be time varying. We find that QML estimate is consistent and asymptotically normal. We investigate marginal impacts of explanatory variables in this system via space–time multipliers. Monte Carlo results are reported to investigate the finite sample properties of QML estimates and marginal effects. When spatial weights matrices are substantially varying over time, a model misspecification of a time invariant spatial weights matrix may cause substantial bias in estimation. Slowly time varying spatial weights matrices would be of less concern.
RÉSUMÉ la présente communication se penche sur l'estimation du quasi maximum de vrai semblance de modèles de données du groupe des dynamiques spatiales, où les matrices de poids spatiales peuvent varier en fonction du temps. Nous relevons que l'estimation de QML est homogène et normale sur un plan asymptotique. Nous nous penchons sur des impacts marginaux de variables causales dans ce système, par le biais de multiplicateurs spatio-temporels. Des résultats Monte Carlo sontfournis pour l'examen d’échantillons finis d'estimations QML et d'effets marginaux. Lorsque les matrices de poids spatiales varient de façon substantielle avec le temps, une erreur de spécification de modèle d'une matrice de poids spatiale ne variant pas avec le temps risquerait de fausser sensiblement les estimations. Les matrice de poids spatiale variant avec le temps auraientune importance moindre.
RESUMEN Este estudio investiga la estimación casi-máxima de probabilidad de semejanza de modelos dinámicos de datos de panel en donde las matrices ponderadas espaciales pueden variar con el tiempo. Indicamos que la estimación QML es constante y asimptóticamente normal. Investigamos impactos marginales de variables explicativas en este sistema mediante multiplicadores espacio-temporales. Se informan los resultados de Monte Carlo para investigar las propiedades de muestra finitas de las estimaciones QML y los efectos marginales. Cuando las matrices ponderadas espaciales varían considerablemente en el tiempo, los errores de especificación del modelo para una matriz ponderada espacial invariable en el tiempopodrían causar una considerable parcialidad en la estimación. Las matrices de pesos espaciales variables lentos serían menos preocupantes.
Acknowledgements
We would like to thank two anonymous referees for helpful comments. Yu ackowledges funding from National Science Foundation of China (Grant No.71171005) and support from Center for Statistical Science of Peking University.
Notes
1. Matlab code is available on request.
2. We would like to emphasize that the time varying W nt is due to changes of economic environments, but not due to missing observations. Missing observations would create an observable unbalanced panel, which is beyond the scope of this paper.
3. As W
n
is row-normalized, whether elements of D
n
are larger than 1 or not depends on the value of . See Yu et al. (Citation2012).
4. When W
nt
is row-normalized for all t, unit roots will present when . This is so, because
is the eigenvalue of A
nt
with corresponding eigenvector of ones l
n
for all t when W
nt
is row-normalized (so that
), where we have
for all t. Hence, when
, we have at least one unit root in A
nt
for all t.
5. A formal condition which guarantees the infinite sums being well defined is Assumption 6 below.
6. This follows from row-normalized W
nt
, which implies that for all h and
.
7. For asymptotic analysis, we assume that both n and T are large. If n is finite, the model with spatial interaction with finite n would be of less interest. If T is finite, the MLE would have an incidental parameter problem and alternative estimates such as the GMM shall be designed.
8. We say a (sequence of n×n) matrix B
n
is uniformly bounded in row and column sums if and
, where
is the row sum norm and
is the column sum norm.
9. When W
nt
is time invariant, becomes
and a
θ,n
in (14) can be written as
where
Thus, for the time invariant W
n
, as is shown in Lee & Yu (Citation2010c), we have
where the
term comes from small terms in (14).
10. In LeSage & Pace (Citation2009), for a cross-sectional SAR model Y
n
=λ
0
W
n
Y
n
+X
n
β
0+V
n
with S
n
=I
n
−λ0
W
n
, the average total impact is , which can be decomposed into the average direct impact
and the average indirect impact
.
11. If we have changes of x for all time from the infinite past, such a total impact will be .
12. We generate the data with 20+T periods and take the last T periods as our sample. And the initial value is generated as N(0, I n ) in the simulation.
13. For the estimation of the information matrix, we use to approximate
14. The rook matrix represents a square tessellation with a connectivity of four for the inner fields on the chessboard and two and three for the corner and border fields, respectively.
15. For the misspecification with the rook matrix only, the CP is very small when λ0 is large.
16. The following statistics reported are all after bias correction.