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This paper considers the estimating problem of a panel data partially linear model with spatial autoregressive errors and fixed effects. In addition, we allow the idiosyncratic errors to be intertemporally correlated. By combining the polynomial spline series approximation, the semiparametric least squares method and the difference based technique, a new generalized moment estimator for the autoregressive parameter of the spatial model is constructed. Its consistency and asymptotic normality are established. In order to avoid the incidental parameter problem, a difference based intertemporal covariance matrix estimator is proposed. Based on the estimated spatially and time-wise correlated error structure, we further construct a weighted difference based semiparametric least squares estimator (WDSLSE) and a weighted difference based polynomial spline series estimator (WDPSSE) for the parametric and nonparametric components of the mean model, respectively. We develop an asymptotic theory for these two estimators, including the asymptotic normality, asymptotic efficiency and convergence rate. In particular, we show that the parametric component estimator has the same asymptotic distribution as that based on completely known spatial autoregressive parameter and intertemporal covariance matrix. Simulation studies demonstrate that our asymptotic theory is applicable for finite samples, and the analysis of a real data set illustrates the usefulness of our developed methodology.

In this paper, we propose a framework of outer product least squares for covariance (COPLS) to directly estimate covariance in the growth curve model based on an analogy, between the outer product of a data vector and covariance of a random vector, and the ordinary least squares technique. The COPLS estimator of covariance has an explicit expression and is shown to have the following properties: (1) following a linear transformation of two independent Wishart distribution for a normal error matrix; (2) having asymptotic normality for a nonnormal error matrix; and (3) having unbiasedness and invariance under a linear transformation group. And, a corresponding two-stage generalized least squares (GLS) estimator for the regression coefficient matrix in the model is obtained and its asymptotic normality is investigated. Simulation studies confirm that the COPLS estimator and the two-stage GLS estimator of the regression coefficient matrix are satisfying competitors with some evident merits to the existing maximum likelihood estimator in finite samples.