Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.     

Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix

This paper considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimator introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The finite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation.     

Finite sample properties of maximum likelihood estimator in spatial models

We investigate the finite sample properties of the maximum likelihood estimator for the spatial autoregressive model. A stochastic expansion of the score function is used to develop the second-order bias and mean squared error of the maximum likelihood estimator. We show that the results can be expressed in terms of the expectations of cross products of quadratic forms, or ratios of quadratic forms in a normal vector which can be evaluated using the top order invariant polynomial. Our numerical calculations demonstrate that the second-order behaviors of the maximum likelihood estimator depend on the degree of sparseness of the weights matrix.     

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.     

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.     

Regression Models with Data‐based Indicator Variables*

Ordinary least squares estimation of an impulse‐indicator coefficient is inconsistent, but its variance can be consistently estimated. Although the ratio of the inconsistent estimator to its standard error has a t‐distribution, that test is inconsistent: one solution is to form an index of indicators. We provide Monte Carlo evidence that including a plethora of indicators need not distort model selection, permitting the use of many dummies in a general‐to‐specific framework. Although White's (1980) heteroskedasticity test is incorrectly sized in that context, we suggest an easy alteration. Finally, a possible modification to impulse ‘intercept corrections’ is considered.     

Inference in semiparametric binary response models with interval data

This paper studies the semiparametric binary response model with interval data investigated by Manski and Tamer (2002). In this partially identified model, we propose a new estimator based on MT’s modified maximum score (MMS) method by introducing density weights to the objective function, which allows us to develop asymptotic properties of the proposed set estimator for inference. We show that the density-weighted MMS estimator converges at a nearly cube-root-n rate. We propose an asymptotically valid inference procedure for the identified region based on subsampling. Monte Carlo experiments provide supports to our inference procedure.     

The method of elimination and substitution in the GMM estimation of mixed regressive,spatial autoregressive models

This paper proposes a computationally simple GMM for the estimation of mixed regressive spatial autoregressive models. The proposed method explores the advantage of the method of elimination and substitution in linear algebra. The modified GMM approach reduces the joint (nonlinear) estimation of a complete vector of parameters into estimation of separate components. For the mixed regressive spatial autoregressive model, the nonlinear estimation is reduced to the estimation of the (single) spatial effect parameter. We identify situations under which the resulting estimator can be efficient relative to the joint GMM estimator where all the parameters are jointly estimated.     

GMM estimators with improved finite sample properties using principal components of the weighting matrix,with an application to the dynamic panel data model

GMM estimators have poor finite sample properties in highly overidentified models. With many moment conditions the optimal weighting matrix is poorly estimated. We suggest using principal components of the weighting matrix. This effectively drops some of the moment conditions. Our simulations, done in the context of the dynamic panel data model, show that the resulting GMM estimator has better finite sample properties than the usual two-step GMM estimator, in the sense of smaller bias and more reliable standard errors.     

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.     

Robust estimation for structural spurious regressions and a Hausman-type cointegration test

This paper analyzes an approach to correcting spurious regressions involving unit-root nonstationary variables by generalized least squares (GLS) using asymptotic theory. This analysis leads to a new robust estimator and a new test for dynamic regressions. The robust estimator is consistent for structural parameters not just when the regression error is stationary but also when it is unit-root nonstationary under certain conditions. We also develop a Hausman-type test for the null hypothesis of cointegration for dynamic ordinary least squares (OLS) estimation. We demonstrate our estimation and testing methods in three applications: (i) long-run money demand in the U.S., (ii) output convergence among industrial and developing countries, and (iii) purchasing power parity (PPP) for traded and non-traded goods.     

Prewhitening Bias in HAC Estimation*

Heteroskedasticity and autocorrelation consistent (HAC) estimation commonly involves the use of prewhitening filters based on simple autoregressive models. In such applications, small sample bias in the estimation of autoregressive coefficients is transmitted to the recolouring filter, leading to HAC variance estimates that can be badly biased. The present paper provides an analysis of these issues using asymptotic expansions and simulations. The approach we recommend involves the use of recursive demeaning procedures that mitigate the effects of small‐sample autoregressive bias. Moreover, a commonly used restriction rule on the prewhitening estimates (that first‐order autoregressive coefficient estimates, or largest eigenvalues, >0.97 be replaced by 0.97) adversely interferes with the power of unit‐root and [ Kwiatkowski, Phillips, Schmidt and Shin (1992) Journal of Econometrics, Vol. 54, pp. 159–178] (KPSS) tests. We provide a new boundary condition rule that improves the size and power properties of these tests. Some illustrations of the effects of these adjustments on the size and power of KPSS testing are given. Using prewhitened HAC estimates and the new boundary condition rule, the KPSS test is consistent, in contrast to KPSS testing that uses conventional prewhitened HAC estimates [ Lee, J. S. (1996) Economics Letters, Vol. 51, pp. 131–137].     

Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

ARCH/GARCH with persistent covariate: Asymptotic theory of MLE

The paper considers a volatility model which introduces a persistent, integrated or near-integrated, covariate to the standard GARCH(1, 1) model. For such a model, we derive the asymptotic theory of the quasi-maximum likelihood estimator. In particular, we establish consistency and obtain limit distribution. The limit distribution is generally non-Gaussian and represented as a functional of Brownian motions. However, it becomes Gaussian if the covariate has innovation uncorrelated with the squared innovation of the model or the volatility function is linear in parameter. We provide a simulation study to demonstrate the relevance and usefulness of our asymptotic theory.     

Asymptotics of the principal components estimator of large factor models with weakly influential factors

This paper introduces a drifting-parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components (PC) estimator in situations when the factors’ explanatory power does not strongly dominate the explanatory power of the cross-sectionally and temporally correlated idiosyncratic terms. Under our asymptotics, the PC estimator is inconsistent. We find explicit formulae for the amount of the inconsistency, and propose an estimator of the number of factors for which the PC estimator works reasonably well. For the special case when the idiosyncratic terms are cross-sectionally but not temporally correlated (or vice versa), we show that the coefficients in the OLS regressions of the PC estimates of factors (loadings) on the true factors (true loadings) are asymptotically normal, and find explicit formulae for the corresponding asymptotic covariance matrix. We explain how to estimate the parameters of the derived asymptotic distributions. Our Monte Carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n$n$ and T$T$. We apply our theoretical results to test a hypothesis about the factor content of the US stock return data.     

Fixed effects instrumental variables estimation in correlated random coefficient panel data models

We provide a set of conditions sufficient for consistency of a general class of fixed effects instrumental variables (FE-IV) estimators in the context of a correlated random coefficient panel data model, where one ignores the presence of individual-specific slopes. We discuss cases where the assumptions are met and violated. Monte Carlo simulations verify that the FE-IV estimator of the population averaged effect performs notably better than other standard estimators, provided a full set of period dummies is included. We also propose a simple test of selection bias in unbalanced panels when we suspect the slopes may vary by individual.     

Testing for serial correlation,spatial autocorrelation and random effects using panel data

This paper considers a spatial panel data regression model with serial correlation on each spatial unit over time as well as spatial dependence between the spatial units at each point in time. In addition, the model allows for heterogeneity across the spatial units using random effects. The paper then derives several Lagrange multiplier tests for this panel data regression model including a joint test for serial correlation, spatial autocorrelation and random effects. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin and Bera [1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York] and in the panel data context by Baltagi et al. [2003. Testing panel data regression models with spatial error correlation. Journal of Econometrics 117, 123–150]. The second is the LM tests for the error component panel data model with serial correlation derived by Baltagi and Li [1995. Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics 68, 133–151]. Hence, the joint LM test derived in this paper encompasses those derived in both strands of earlier works. In fact, in the context of our general model, the earlier LM tests become marginal LM tests that ignore either serial correlation over time or spatial error correlation. The paper then derives conditional LM and LR tests that do not ignore these correlations and contrast them with their marginal LM and LR counterparts. The small sample performance of these tests is investigated using Monte Carlo experiments. As expected, ignoring any correlation when it is significant can lead to misleading inference.