Abstract
Newton-step approximations to pseudo maximum likelihood estimates of spatial autoregressive models with a large number of parameters are examined, in the sense that the parameter space grows slowly as a function of sample size. These have the same asymptotic efficiency properties as maximum likelihood under Gaussianity but are of closed form. Hence they are computationally simple and free from compactness assumptions, thereby avoiding two notorious pitfalls of implicitly defined estimates of large spatial autoregressions. When commencing from an initial least squares estimate, the Newton step can also lead to weaker regularity conditions for a central limit theorem than some extant in the literature. A simulation study demonstrates excellent finite sample gains from Newton iterations, especially in large multiparameter models for which grid search is costly. A small empirical illustration shows improvements in estimation precision with real data.
Highlights
Spatial autoregressive (SAR) models, introduced by Cliff and Ord (1973), are popular tools for modelling cross-sectionally dependent economic data
Such dependence need not be geographic in nature, the spatial weight matrix is known by other terms such as ‘adjacency matrix’, ‘network link matrix’ and ‘sociomatrix’
We demonstrate that the Newton step can lead to much improved estimates in finite samples, both in terms of bias and efficiency
Summary
Spatial autoregressive (SAR) models, introduced by Cliff and Ord (1973), are popular tools for modelling cross-sectionally dependent economic data. Lee (2002) studied ordinary least squares (OLS) estimation of SAR models, stressing the need for lack of sparsity in the spatial weight matrix to establish desirable asymptotic properties such as consistency and efficiency. A theory that allows the model dimension to grow with sample size provides a more incisive analysis of large models in practice, much as typical asymptotic theory with a fixed parameter space itself can be thought as providing an approximation in finite samples The estimation of such increasing-order SAR models has been studied by Gupta and Robinson (2015, 2018) using IV, OLS and PMLE approaches. We denote true parameter values with 0 subscript and suppress the argument for a quantity evaluated at a true parameter value, i.e. f (τ0) ≡ f
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