GMM and 2SLS estimation of mixed regressive,spatial autoregressive models

The GMM method and the classical 2SLS method are considered for the estimation of mixed regressive, spatial autoregressive models. These methods have computational advantage over the conventional maximum likelihood method. The proposed GMM estimators are shown to be consistent and asymptotically normal. Within certain classes of GMM estimators, best ones are derived. The proposed GMM estimators improve upon the 2SLS estimators and are applicable even if all regressors are irrelevant. A best GMM estimator may have the same limiting distribution as the ML estimator (with normal disturbances).     

Estimating and Forecasting with a Dynamic Spatial Panel Data Model*

This study focuses on the estimation and predictive performance of several estimators for the dynamic and autoregressive spatial lag panel data model with spatially correlated disturbances. In the spirit of Arellano and Bond (1991) and Mutl (2006) , a dynamic spatial generalized method of moments (GMM) estimator is proposed based on Kapoor, Kelejian and Prucha (2007) for the spatial autoregressive (SAR) error model. The main idea is to mix non‐spatial and spatial instruments to obtain consistent estimates of the parameters. Then, a linear predictor of this spatial dynamic model is derived. Using Monte Carlo simulations, we compare the performance of the GMM spatial estimator to that of spatial and non‐spatial estimators and illustrate our approach with an application to new economic geography.     

Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances

This study develops a methodology of inference for a widely used Cliff–Ord type spatial model containing spatial lags in the dependent variable, exogenous variables, and the disturbance terms, while allowing for unknown heteroskedasticity in the innovations. We first generalize the GMM estimator suggested in  and  for the spatial autoregressive parameter in the disturbance process. We also define IV estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GMM estimator. Much of the theory is kept general to cover a wide range of settings.     

GMM Estimation with Non‐causal Instruments under Rational Expectations*

Lanne and Saikkonen [Oxford Bulletin of Economics and Statistics (2011a) Vol. 73, pp. 581–592], show that the generalized method of moments (GMM) estimator is inconsistent, when the instruments are lags of variables that admit a non‐causal autoregressive representation. This article argues that this inconsistency depends on distributional assumptions, that do not always hold. In particular under rational expectations, the GMM estimator is found to be consistent. This result is derived in a linear context and illustrated by simulation of a nonlinear asset pricing model.     

A Generalized Method of Moments Estimator for a Spatial Panel Model with an Endogenous Spatial Lag and Spatial Moving Average Errors

Abstract

This paper proposes a new generalized method of moments (GMM) estimator for spatial panel models with spatial moving average errors combined with a spatially autoregressive dependent variable. Monte Carlo results are given suggesting that the GMM estimator is consistent. The estimator is applied to English real estate price data.     

Asymptotic distributions of impulse response functions in short panel vector autoregressions

This paper establishes the asymptotic distributions of the impulse response functions in panel vector autoregressions with a fixed time dimension. It also proves the asymptotic validity of a bootstrap approximation to their sampling distributions. The autoregressive parameters are estimated using the GMM estimators based on the first differenced equations and the error variance is estimated using an extended analysis-of-variance type estimator. Contrary to the time series setting, we find that the GMM estimator of the autoregressive coefficients is not asymptotically independent of the error variance estimator. The asymptotic dependence calls for variance correction for the orthogonalized impulse response functions. Simulation results show that the variance correction improves the coverage accuracy of both the asymptotic confidence band and the studentized bootstrap confidence band for the orthogonalized impulse response functions.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Indirect inference for dynamic panel models

Maximum likelihood (ML) estimation of the autoregressive parameter of a dynamic panel data model with fixed effects is inconsistent under fixed time series sample size and large cross section sample size asymptotics. This paper proposes a general, computationally inexpensive method of bias reduction that is based on indirect inference, shows unbiasedness and analyzes efficiency. Monte Carlo studies show that our procedure achieves substantial bias reductions with only mild increases in variance, thereby substantially reducing root mean square errors. The method is compared with certain consistent estimators and is shown to have superior finite sample properties to the generalized method of moment (GMM) and the bias-corrected ML estimator.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

GMM estimation of spatial autoregressive models with moving average disturbances

In this paper, we introduce the one-step generalized method of moments (GMM) estimation methods considered in Lee (2007a) and Liu, Lee, and Bollinger (2010) to spatial models that impose a spatial moving average process for the disturbance term. First, we determine the set of best linear and quadratic moment functions for GMM estimation. Second, we show that the optimal GMM estimator (GMME) formulated from this set is the most efficient estimator within the class of GMMEs formulated from the set of linear and quadratic moment functions. Our analytical results show that the one-step GMME can be more efficient than the quasi maximum likelihood (QMLE), when the disturbance term is simply i.i.d. With an extensive Monte Carlo study, we compare its finite sample properties against the MLE, the QMLE and the estimators suggested in Fingleton (2008a).     

GMM Estimation with Non‐causal Instruments*

This note provides a warning against careless use of the generalized method of moments (GMM) with time series data. We show that if time series follow non‐causal autoregressive processes, their lags are not valid instruments, and the GMM estimator is inconsistent. Moreover, endogeneity of the instruments may not be revealed by the J‐test of overidentifying restrictions that may be inconsistent and has, in general, low finite‐sample power. Our explicit results pertain to a simple linear regression, but they can easily be generalized. Our empirical results indicate that non‐causality is quite common among economic variables, making these problems highly relevant.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

Frequency domain estimation of temporally aggregated Gaussian cointegrated systems

This paper considers joint estimation of long run equilibrium coefficients and parameters governing the short run dynamics of a fully parametric Gaussian cointegrated system formulated in continuous time. The model allows the stationary disturbances to be generated by a stochastic differential equation system and for the variables to be a mixture of stocks and flows. We derive a precise form for the exact discrete analogue of the continuous time model in triangular error correction form, which acts as the basis for frequency domain estimation of the unknown parameters using discrete time data. We formally establish the order of consistency and the asymptotic sampling properties of such an estimator. The estimator of the cointegrating parameters is shown to converge at the rate of the sample size to a mixed normal distribution, while that of the short run parameters converges at the rate of the square root of the sample size to a limiting normal distribution.     

Nonparametric spatial regression under near-epoch dependence

This paper establishes asymptotic normality and uniform consistency with convergence rates of the local linear estimator for spatial near-epoch dependent (NED) processes. The class of the NED spatial processes covers important spatial processes, including nonlinear autoregressive and infinite moving average random fields, which generally do not satisfy mixing conditions. Apart from accommodating a larger class of dependent processes, the proposed asymptotic theory allows for triangular arrays of heterogeneous random fields located on unevenly spaced lattices and sampled over regions of arbitrary configuration. All these features make the results applicable in a wide range of empirical settings.     

Choosing instrumental variables in conditional moment restriction models

Properties of GMM estimators are sensitive to the choice of instrument. Using many instruments leads to high asymptotic asymptotic efficiency but can cause high bias and/or variance in small samples. In this paper we develop and implement asymptotic mean square error (MSE) based criteria for instrument selection in estimation of conditional moment restriction models. The models we consider include various nonlinear simultaneous equations models with unknown heteroskedasticity. We develop moment selection criteria for the familiar two-step optimal GMM estimator (GMM), a bias corrected version, and generalized empirical likelihood estimators (GEL), that include the continuous updating estimator (CUE) as a special case. We also find that the CUE has lower higher-order variance than the bias-corrected GMM estimator, and that the higher-order efficiency of other GEL estimators depends on conditional kurtosis of the moments.     

Panel data models with spatially correlated error components

In this paper we consider a panel data model with error components that are both spatially and time-wise correlated. The model blends specifications typically considered in the spatial literature with those considered in the error components literature. We introduce generalizations of the generalized moments estimators suggested in Kelejian and Prucha (1999. A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509–533) for estimating the spatial autoregressive parameter and the variance components of the disturbance process. We then use those estimators to define a feasible generalized least squares procedure for the regression parameters. We give formal large sample results for the proposed estimators. We emphasize that our estimators remain computationally feasible even in large samples.     

Testing Convergence in Income Distribution*

The generalized method of moments (GMM) estimator is often used to test for convergence in income distribution in a dynamic panel set‐up. We argue that though consistent, the GMM estimator utilizes the sample observations inefficiently. We propose a simple ordinary least squares (OLS) estimator with more efficient use of sample information. Our Monte Carlo study shows that the GMM estimator can be very imprecise and severely biased in finite samples. In contrast, the OLS estimator overcomes these shortcomings.     

Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients

Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients. Spatial Economic Analysis. The spatial model with space-varying coefficients proposed by Sun et al. in 2014 has proved to be useful in detecting the location effects of the impacts of covariates as well as spatial interaction in empirical analysis. However, Sun et al.’s estimator is inconsistent when heteroskedasticity is present – a circumstance that is more realistic in certain applications. In this study, we propose a kind of semi-parametric generalized method of moments (GMM) estimator that is not only heteroskedasticity robust but also takes a closed form written explicitly in terms of observed data. We derive the asymptotic distributions of our estimators. Moreover, the results of Monte Carlo experiments show that the proposed estimators perform well in finite samples.