Spatial dependence in panel data models
When observations are available across space as well as over time, the additional dimension allows the estimation of the full covariance of one type of association, using the other dimension to provide the asymptotics (e. g. in SUR models with N << T). However, as in the pure cross-sectional case, there is insufficient information in the NT observations to estimate the complete (NT)2 covariance matrix cov[y;„ yj Ф 0, (with i Ф j and t Ф s) without imposing some structure. For small N and large T, the asymptotics in the time domain can be exploited to obtain a nonparametric estimate of cross-sectional dependence, while time dependence must be parameterized. Similarly, for large N and small T, the asymptotics in the spatial domain can be exploited to yield a nonparametric estimate of serial (time) dependence, while spatial dependence must be parameterized. As in the pure cross-sectional case, the latter requires the use of a spatial weights matrix. In each of these situations, asymptotics are only needed in one of the dimensions while the other can be treated as fixed.
When both spatial as well as serial dependence are parameterized, a range of specifications can be considered, allowing different combinations of the two. For ease of exposition, assume that the observations are stacked by time period,
i. e. they can be considered as T time slices of N cross-sectional units. Restricting attention to "lag" dependence, and with f(z) as a generic designation for the regressors (which may be lagged in time and/or space), four types of models can be distinguished.
1. pure space-recursive, in which the dependence pertains to neighboring locations in a different period, or,
Vu = Y [Wyt-i]i + f(z) + Cv (14.12)
where, using the same notational convention as before, [Wyt-1]i is the ith element of the spatial lag vector applied to the observations on the dependent variable in the previous time period (using an N x N spatial weights matrix for the cross-sectional units).
2. time-space recursive, in which the dependence relates to the same location as well as the neighboring locations in another period, or,
Vit = tyn-1 + Y [Wy t-1]i + f(z) + Ct (14.13)
3. time-space simultaneous, with both a time-wise and a spatially lagged dependent variable, or,
Vu = tyn-1 + P[Wyt]i + f(z) + Ct (14.14)
where [Wyt]; is the ith element of the spatial lag vector in the same time period.
4. time-space dynamic, with all forms of dependence, or,
Vtt = ^ Уи-і + P[Wyt]i + у [WytJi + f(z) + ги. (1415)
In order to estimate the parameters of the time-space simultaneous model, asymptotics are needed in the cross-sectional dimension, while for the time – space dynamic model, asymptotics are needed in both dimensions. For the other models, the type of asymptotics required are determined by the dependence structure in the error terms. For example, the pure space-recursive model with iid errors satisfies the assumptions of the classical linear model and can be estimated by means of OLS.
Spatial lag and spatial error dependence can be introduced into the crosssectional dimension of traditional panel data models in a straightforward way. For example, in a spatial SUR model, both autoregressive as well as regression parameters are allowed to vary by time period, in combination with a nonparametric serial covariance. The spatial lag formulation of such a model would be (in the same notation as before):
У и = Pt[Wyt] i + v ‘it Pt + e-it (14.16)
with var[eft] = о2 and Е[є„ ej = Ots.18
An important issue to consider when incorporating spatial dependence in panel data models is the extent to which fixed effects may be allowed. Since the estimation of the spatial process models requires asymptotics in the cross-sectional domain (N ^ «>), fixed effects (i. e. a dummy variable for each location) would suffer from the incidental parameter problem and no consistent estimator exists. Hence, fixed cross-sectional effects are incompatible with spatial processes and instead a random effects specification must be considered.