Hodges–Lehmann Estimation of Static Panel Models with Spatially Correlated Disturbances

Abstract

Several studies point out a substantial downward bias of the Maximum Likelihood (ML) estimator of the spatial correlation parameter under strongly connected spatial structures. This paper proposes Hodges–Lehmann (HL) type interval and point estimators for the spatial parameter in static panel models with spatially autoregressive or moving average disturbances. HL estimators are implemented by means of ‘inverting’ common diagnostics for spatial correlation. Exact inference is implemented by means of Monte Carlo testing. A simulation study covering models with distinct degrees of spatial connectivity shows that HL confidence intervals are characterized by less size distortions and appear more robust against spatial connectivity in comparison with ML interval estimates. In addition, the bias of the HL point estimator based on the Moran’s I statistic is markedly smaller than its ML counterpart.

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Fig. 1

Notes

  1. 1.

    Further test statistics have also been considered: the Lagrange Multiplier (LM) test of Burridge (1980), the LM statistic proposed by Breusch and Pagan (1980), two non-parametric tests of Friedman (1937) and Frees (1995), the multivariate independence test of Tsay (2004), and the statistics proposed by Dufour and Khalaf (2002), i.e. an extension of the exact independence test of Harvey and Phillips (1982), Shiba and Tsurumi’s (1988) likelihood ratio test and Breusch and Pagan’s (1980) LM test. To economize on space the analysis is concentrated on a set of 3 alternative test approaches, which have shown most accurate performance in terms of power (MI) or robustness under diverse (misspecified) spatial structures (CD, \(CD^W\)).

  2. 2.

    Notably, the exact distribution of MI can also be derived by numerical integration (Bivand et al. 2009). In comparison, the MC approach is immediate to implement for all considered diagnostics, and, thus, facilitates the evaluation of simulation results over a set of rival diagnostics \({\mathcal {T}}\in \{MI, CD, CD^{W}\}\).

  3. 3.

    For this specification of \({\varvec{W}}\) the maximum number of neighbors for a given cross-sectional unit is not restricted as \(N\rightarrow \infty \), since J increases with N. As a consequence no additional information is gained from individual observations as N increases (Smith 2009). Thus, the bias of the ML estimator might not vanish asymptotically and asymptotic normality of MI might not be established. However, the main focus of this paper is rather on the estimation bias of \(\hat{\theta }\) than on asymptotic test and estimation properties. Moreover, even in this case the MC approach offers full control over the stochastic model features.

  4. 4.

    Rejection frequencies are adjusted by tuning the critical values of the tests such that empirical rejection frequencies are exactly 5% under \(H_0\).

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Correspondence to Christoph Strumann.

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Strumann, C. Hodges–Lehmann Estimation of Static Panel Models with Spatially Correlated Disturbances. Comput Econ 53, 141–168 (2019). https://doi.org/10.1007/s10614-017-9728-y

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Keywords

  • Panel data
  • Spatial correlation
  • Specification tests
  • Monte Carlo test
  • Exact confidence sets
  • Hodges–Lehmann estimators

JEL Classification

  • C12
  • C15
  • C21
  • C23