Exact and higher-order properties of the MLE in spatial autoregressive models, with applications to inference

Abstract

The quasi-maximum likelihood estimator for the autoregressive parameter in a spatial autoregression usually cannot be written explicitly in terms of the data. A rigorous analysis of the first-order asymptotic properties of the estimator, under some assumptions on the evolution of the spatial design matrix, is available in Lee (2004), but very little is known about its exact or higher-order properties. In this paper we first show that the exact cumulative distribution function of the estimator can, under mild assumptions, be written in terms of that of a particular quadratic form. Simple examples are used to illustrate important exact properties of the estimator that follow from this representation. In general models a complete exact analysis is not possible, but a higher-order (saddlepoint) approximation is made available by the main result. We use this approximation to construct confidence intervals for the autoregressive parameter. Coverage properties of the proposed confidence intervals are studied by Monte Carlo simulation, and are found to be excellent in a variety of circumstances.

Adjusted QMLE for the spatial autoregressive parameter

“…Consider an arbitrary nonzero real eigenvalue ω of W . If y / ∈ null(M X S(ω −1 )), the function λ → (n−k)(y W M X S(λ)y)/(y S(λ) M X S(λ)y) is continuous at λ = ω −1 because it is well defined at λ = ω −1 , and is well defined in a neighborhood of λ = ω −1 (the last claim follows by Lemma S.1.1 in the online supplement to Hillier and Martellosio, 2018a). The proof is completed on using Lemma A.3 to establish the limiting behavior of the term −tr(M X G(λ)) as λ → ω −1 .…”

“…Also, it is easy to see that, under Assumption 1, l(λ) a.s. goes to −∞ at each real zero of det(S(λ)) (cf. Hillier and Martellosio, 2018a). Thus, l(λ) has a.s. at least one critical point corresponding to a maximum in any interval between two consecutive real zeros of det(S(λ)).…”

“…The approximate cdf ofλ aML can then be inverted to obtain confidence intervals for λ. This approach to constructing confidence intervals has been applied by Hillier and Martellosio (2018a) to the (unadjusted) QMLE of λ. Whilst revising the present paper we discovered that essentially the same approach had previously been suggested by Paige et al (2009) for general quadratic estimating equations, and then by Jeganathan et al (2015) specifically for some spatial models. 2009) use the device Pr(θ ≤ z) = Pr(q(z) ≤ 0) under the assumption that q(θ) is monotonically decreasing in θ, whereas we have used it under the more general assumption that the function with derivative q(θ) is single-peaked.…”

“…It is interesting to note that, for the unadjusted profile likelihood l(λ), single-peakedness over Λ holds if ntr(G 2 (λ)) > [tr(G(λ))] 2 for all λ ∈ Λ, a condition that does not depend on X (seeHillier and Martellosio, 2018a).…”

SPECIAL ISSUE OF ECONOMETRIC THEORY IN HONOR OF PROFESSOR RICHARD J. SMITH: GUEST EDITORS’ INTRODUCTION

“…Grant Hillier and Federico Martellosio in their paper "Exact likelihood inference in group interaction spatial autoregressive models" explore the application of a general result on the quasimaximum likelihood estimator [QMLE] of the autoregressive parameter in a spatial autoregressive model provided in Hillier and Martellosio (2016) to a class of models based on spatial weights matrices that embody group-interaction. These models are important in various areas of application to the study of networks, and to panels with a spatial autoregressive component.…”