Osman Dogan

GMM inference in spatial autoregressive models

ABSTRACT In this study, we investigate the finite sample properties of the optimal generalized method of moments estimator (OGMME) for a spatial econometric model with a first-order spatial autoregressive process in the dependent variable and the disturbance term (for short SARAR(1, 1)). We show that the estimated asymptotic standard errors for spatial autoregressive parameters can be substantially smaller than their empirical counterparts. Hence, we extend the finite sample variance correction methodology of Windmeijer (2005) to the OGMME for the SARAR(1, 1) model. Results from simulation studies indicate that the correction method improves the variance estimates in small samples and leads to more accurate inference for the spatial autoregressive parameters. For the same model, we compare the finite sample properties of various test statistics for linear restrictions on autoregressive parameters. These tests include the standard asymptotic Wald test based on various GMMEs, a bootstrapped version of the Wald test, two versions of the C(α) test, the standard Lagrange multiplier (LM) test, the minimum chi-square test (MC), and two versions of the generalized method of moments (GMM) criterion test. Finally, we study the finite sample properties of effects estimators that show how changes in explanatory variables impact the dependent variable.

Simple Tests for Social Interaction Models with Network Structures

ABSTRACT We consider an extended spatial autoregressive model that can incorporate possible endogenous interactions, exogenous interactions, unobserved group fixed effects and the correlation of unobservables. In the generalized method of moments (GMM) and the maximum likelihood (ML) frameworks, we introduce simple gradient-based robust test statistics that can be used to test for the presence of the endogenous effects, the correlation of unobservables and the contextual effects. These test statistics are robust to local parametric misspecifications and only require consistent estimates from a transformed linear regression model to compute. We carry out an extensive Monte Carlo study to investigate the size and power properties of the proposed tests. The results show that the proposed tests have good finite sample properties, and are useful for testing the presence of the various effects in a social interaction model.

Adjustments of Rao's Score Test for Distributional and Local Parametric Misspecifications

Raos (1948) seminal paper introduced a fundamental principle of testing based on the score function and the score test has local optimal properties. When the assumed model is misspecified, it is well known that Raos score (RS) test loses its optimality. A model could be misspecified in a variety of ways. In this paper, we consider two kinds: distributional and parametric. In the former case, the assumed probability density function differs from the data generating process. Kent (1982) and White (1982) analyzed this case and suggested a modified version of the RS test that involves adjustment of the variance. In the latter case, the dimension of the parameter space of the assumed model does not match with that of the true one. Using the distribution of the RS test under this situation, Bera and Yoon (1993) developed a modified RS test that is valid under the local parametric misspecification. This involves adjusting both the mean and variance of the standard RS test. This paper considers the joint presence of the distributional and parametric misspecifications and develops a modified RS test that is valid under both types of misspecification. Earlier modified tests under either type of misspecification can be obtained as the special cases of the proposed test. We provide three examples to illustrate the usefulness of the suggested test procedure. In a Monte Carlo study, we demonstrate that the modified test statistics have good finite sample properties.

Teaching Size and Power Properties of Hypothesis Tests Through Simulations

Abstract In this study, we review the graphical methods suggested in Davidson and MacKinnon (Davidson, Russell, and James G. MacKinnon. 1998. “Graphical Methods for Investigating the Size and Power of Hypothesis Tests.” The Manchester School 66 (1): 1–26.) that can be used to investigate size and power properties of hypothesis tests for undergraduate and graduate econometrics courses. These methods can be used to assess finite sample properties of various hypothesis tests through simulation studies. In addition, these methods can be effectively used in classrooms to reinforce students’ understanding of basic hypothesis testing concepts such as Type I error, Type II error, size, power, p-values and under-or-over-sized tests. We illustrate the procedural aspects of these graphical methods through Monte Carlo experiments, and provide the implementation codes written in Matlab and R for the classroom applications.

Asymptotic Distribution of Test Statistics: The Estimating Equation Approach and the Delta Method

The delta method that consists of a Taylor approximation can be used to determine the asymptotic variance and distribution of test statistics. In an alternative approach, the test statistic can be combined with some estimating equations in the M-estimation framework for the purpose of deriving its asymptotic variance and distribution. Pierce (1982) uses this alternative approach and shows how the nuisance parameters can be replaced by their efficient estimators in deriving the asymptotic variance and distribution of certain test statistics. In this article, we show how these approaches are related for test statistics that can be written as the sample average of data.

Simple Tests for Endogeneity of Spatial Weights Matrices

In this study, we propose a Raos score (RS) statistic (Lagrange multiplier (LM) statistic) to test for endogeneity of the spatial weights matrix in a spatial autoregressive model. To achieve this, we start with a spatial autoregressive model with an acceptable form for the generating process for the elements of the endogenous spatial weights matrix as in Qu and Lee (2015). Our test statistic is simple to calculate because it requires computationally simple estimations. By construction, the test statistic is robust in the sense that its asymptotic null distribution is a centered chi-square distribution regardless of the (local) presence of a spatial autoregressive parameter in the alternative model. We summarize the asymptotic properties of our test statistic under the null and the alternative hypotheses. To investigate its finite sample size and power properties, we conduct a Monte Carlo study. The results are in line with our theoretical findings and indicate that the robust test has good size and power properties.

Robust LM Tests for Spatial Dynamic Panel Data Models

In this study, we introduce adjusted Raos score test statistics (Lagrange multiplier (LM) tests) for a spatial dynamic panel data (SDPD) model that includes a contemporaneous spatial lag, a time lag and a spatial-time lag. The maximum likelihood estimator for the estimation of SDPD models can have asymptotic bias because of individual and time fixed effects. Bias arises since the limiting distribution of the score functions derived from the corresponding concentrated log-likelihood functions are not centered on zero. First, we show how the score functions should be adjusted to avoid the effect of asymptotic bias on the standard LM test statistics. Second, we further adjust score functions such that the resulting LM test statistics are valid when there is local parametric misspecification in the alternative model. Our adjusted LM test statistics can be used to test the presence of the contemporaneous spatial lag, time lag and spatial-time lag in an SDPD model. In a Monte Carlo study, we demonstrate that our suggested test statistics have good finite sample size and power properties. Finally, we illustrate implementation of these tests in an application on public capital productivity in contiguous 48 US states.

Lagrange Multiplier Tests for Non-Linear Hypotheses Under Distributional and Local Parametric Misspecification

In this study, we give a general account to the asymptotic properties of Raos score statistic for testing a non-linear hypothesis under both distributional and parametric misspecifications. The distributional misspecification arises if the parametric family of distribution functions that is used to formulate a quasi log-likelihood function does not include the data generating process (DGP). In the case of parametric misspecification, some of nuisance parameters, that are relevant for describing the DGP, are excluded from the model under the alternative hypothesis. We suggest a modified version of score statistic that is robust to both types of misspecifications for a general non-linear null hypothesis. We establish the asymptotic properties of the modified test statistic under both null and local alternative hypotheses. We illustrate the derivation of modified score statistics within the context of a spatially augmented Solow growth model. Finally, through a simulation study, we demonstrate that the modified score statistic has good finite sample properties both in terms of size and power.

Heteroskedasticity Consistent Covariance Matrix Estimators for Spatial Autoregressive Models

In the presence of heteroskedasticity, conventional test statistics based on the ordinary least square estimator lead to incorrect inference results for the linear regression model. Given that heteroskedasticity is common in cross-sectional data, the test statistics based on various forms of heteroskedasticity consistent covariance matrices (HCCMs) have been developed in the literature. In contrast to the standard linear regression model, heteroskedasticity is a more serious problem for spatial econometric models, generally causing inconsistent extremum estimators of model coefficients. In this paper, we investigate the finite sample properties of the heteroskedasticity-robust generalized method of moments estimator (RGMME) for a spatial econometric model with an unknown form of hetereoskedasticity. In particular, we develop various HCCM-type corrections to improve the finite sample properties of the RGMME and the conventional Wald test. Our Monte Carlo results indicate that the HCCM-type corrections can produce more accurate results for inference on model parameters and the impact effects estimates in small samples.