A study of estimator densities and performance under misspecification

The exact density functions of the OLS and 2SLS estimators when exogenous variables are wrongly excluded from the equation being estimated, where two endogenous variables appear in the equation of interest, are presented and tabulated. Misspecification does not appear to alter the relative concentrations of OLS and 2SLS — OLS is always more concentrated — but does change the relative biases with the result that under misspecification OLS may indeed be the superior estimation technique. In addition, misspecification can substantially increase the concentration of both estimators, as well as reducing their biases, therebyyielding an apparent improvement in estimator performance.     

Dynamic Structure of Disequilibrium Models

This paper investigates the dynamic structure of a standard disequilibrium model. By assuming that the model variables are non-stationary time series with respect to ample empirical evidence, we find the following: 1) It is the exogenous variables rather than the price adjustment process that form the real adjustment force of the model; 2) Quantity disequilibrium and price disequilibrium are isomeric in the model, and follow a weakly stationary process when all the variables areI (1) nonstationary; 3) The disequilibrium process has a none-zero mean when the weakly exogenous variables of the demand equation do not cointegrate with those of the supply equation, corresponding to certain 'chronic disequilibrium' phenomena; 4) The isomerism between quantity disequilibrium and price changes makes it unnecessary to lean on the 'min condition' to characterise disequilibrium.     

Edgeworth approximations in first-order stochastic difference equations with exogenous variables

This article obtains the Edgeworth approximate distribution of the OLS estimator of the autoregressive parameter of a first-order stochastic difference equation with exogenous variables. The approximate distribution is compared with the exact distribution computed using the Imhof algorithm. Phillips' results on the pure autoregressive process are also revised.     

Autocorrelated disturbances in the light of specification analysis

The paper reconsiders the problem of autocorrelation among disturbances caused by the omission of some regressors from a single-equation regression model. Regression coefficients and disturbances of the misspecified model have been redefined appropriately. Performance of the OLS method of estimation of these regression coefficients and that of the Durbin-Watson test of randomness of disturbances have been studied. Some of the alternative methods of estimating the regression coefficients in situations where the disturbances are autocorrelated heve been examined. It appears that these methods can no longer be used if the autocorrelation is due to omission of regressors.     

Bootstrap HAC Tests for Ordinary Least Squares Regression*

There is a need for tests that are derived from the ordinary least squares (OLS) estimators of regression coefficients and are useful in the presence of unspecified forms of heteroskedasticity and autocorrelation. A method that uses the moving block bootstrap and quasi‐estimators in order to derive a consistent estimator of the asymptotic covariance matrix for the OLS estimators and robust significance tests is proposed. The method is shown to be asymptotically valid and Monte Carlo evidence indicates that it is capable of providing good control of significance levels in finite samples and good power compared with two other bootstrap tests.     

Prediction in the Panel Data Model with Spatial Correlation: the Case of Liquor

Abstract

This paper considers the problem of prediction in a panel data regression model with spatial autocorrelation in the context of a simple demand equation for liquor. This is based on a panel of 43 states over the period 1965–1994. The spatial autocorrelation due to neighbouring states and the individual heterogeneity across states is taken explicitly into account. We compare the performance of several predictors of the states’ demand for liquor for 1 year and 5 years ahead. The estimators whose predictions are compared include OLS, fixed effects ignoring spatial correlation, fixed effects with spatial correlation, random-effects GLS estimator ignoring spatial correlation and random-effects estimator accounting for the spatial correlation. Based on RMSE forecast performance, estimators that take into account spatial correlation and heterogeneity across the states perform the best for forecasts 1 year ahead. However, for forecasts 2–5 years ahead, estimators that take into account the heterogeneity across the states yield the best forecasts.     

Model averaging based on James–Stein estimators

Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

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.     

Estimating a class of triangular simultaneous equations models without exclusion restrictions

This paper provides a control function estimator to adjust for endogeneity in the triangular simultaneous equations model where there are no available exclusion restrictions to generate suitable instruments. Our approach is to exploit the dependence of the errors on exogenous variables (e.g. heteroscedasticity) to adjust the conventional control function estimator. The form of the error dependence on the exogenous variables is subject to restrictions, but is not parametrically specified. In addition to providing the estimator and deriving its large-sample properties, we present simulation evidence which indicates the estimator works well.     

The exact distribution of exogenous variable coefficient estimators

This paper derives the exact probability density function of the instrumental variable (IV) estimator of the exogenous variable coefficient vector in a structural equation containing n + 1 endogenous variables and N degrees of overidentification. The derivations make use of an operator calculus which simplifies the algebra of invariant polynomials with multiple matrix arguments. A leading case of the general distribution that is more amenable to analysis and computation is also presented. Conventional classical assumptions of normally distributed errors and non-random exogenous variables are employed.     

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.     

Raising the bar (5)

Raising the bar (5). Spatial Economic Analysis. This editorial summarizes and comments on the papers published in this issue 12(1) so as to raise the bar in applied spatial economic research and highlight new trends. The first paper examines the impact of the level of education on the decision to migrate and finds that it is approximately twice as large if both variables are modelled simultaneously. The second paper is one of the first papers to introduce a spatial component to models of international environmental agreements and to develop an exciting overlap with New Economic Geography. The third paper provides a tool, applied to Beijing, with which urban economic planners can investigate the role of variation and selection mechanisms in cluster development and identify possible paths of growth. The fourth paper contributes to the existing literature on retail geography by examining the role of consumption possibilities as an urban amenity. The fifth paper develops a Bayesian estimator of a linear regression model with spatial lags among the dependent variable, the explanatory variables and the disturbances. Finally, the sixth paper develops a semi-parametric generalized method of moments (GMM) estimator for a spatial autoregressive model with space-varying coefficients of the explanatory variables and a spatial autoregressive coefficient common to all units.     

Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

Small sample considerations in estimation from panel data

Recent interest in statistical inference for panel data has focused on the problem of unobservable, individual-specific, random effects and the inconsistencies they introduce in estimation when they are correlated with other exogenous variables. Analysis of this problem has always assumed the variance components to be known. In this paper, we re-examine some of these questions in finite samples when the variance components must be estimated. In particular, when the effects are uncorrelated with other explanatory variables, we show that (i) the feasible Gauss-Markov estimator is more efficient than the within groups estimator for all but the fewest degrees of freedom and its variance is never more than 17% above the Cramer-Rao bound, (ii) the asymptotic approximation to the variance of the feasible Gauss-Markov estimator is similarly within 17% of the true variance but remains significantly smaller for moderately large samples sizes, and (iii) more efficient estimators for the variance components do not necessarily yield more efficient feasible Gauss-Markov estimators.     

Estimation of a dynamic system of seemingly unrelated regressions with autoregressive disturbances

This paper examines the small sample properties of the asymptotically efficient estimator due to Hatanaka (1976) and Dhrymes and Taylor (1976) applied to a system of seemingly unrelated regressions characterized by both autoregressive disturbances and lagged endogenous variables. The results of several Monte Carlo experiments suggest that, in general, this estimation procedure performs well in samples of modest size. Two important situations in which the results are mixed are also reported.     

Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu ( 2009a ) and is based on a Feasible Generalized Least Squares procedure that uses a super‐efficient estimator of the sum of the autoregressive coefficients α when α = 1. The resulting Wald test statistic asymptotically follows a chi‐square distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias‐corrected version of the OLS estimator of α proposed by Roy and Fuller ( 2001 ). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modelling the trend of global and hemispheric temperatures.     

Omitted variables,variability of estimated parameters and the appearance of autocorrelated disturbances

This paper shows that in a standard regression model with omitted variables, the OLS formula for the estimated variance matrix of the regression coefficients is more likely to underestimate the appropriate criterion of estimator reliability which is the Mean Square Errors matrix. Using examples of two and three regressor models, we show that overestimation, though possible, occurs in rather special cases. Throughout, our analysis is contrasted with that of Chaudhuri (1977) and clarifies some ambiguities of that paper. Finally, we disagree with Chaudhuri who distinguishes between the corresponding coefficients in the correct and the misspecified models. This distinction is inappropriate and leads to a misplaced criticism of some GLS variants when errors are serially correlated. A first-order Markov process is an inexact representation of serial correlation which is due to omitted regressors.     

The power of four tests of autocorrelation in the linear regression model

The power of each of four tests of first-order autocorrelation in the linear regression model is determined for a simple and multiple regression model whose parameters are presumed to be known. The tests are: Durbin-Watson bounds test, a test based on Theil's best linear unbiased scalar estimator, a test devised by Abrahamse, Koerts and Louter, and an exact test devised by Durbin.For positive values of the coefficient of autocorrelation the Durbin-Watson bounds test is generally better than the tests based on the estimator proposed by Abrahamse, Koerts and Louter, the best linear unbiased scalar estimator, and the Durbin exact test. For negative values of the coefficient of autocorrelation, the pattern of results is mixed for all four test procedures. A byproduct of these experiments is the demonstrated feasibility of enumerating the distribution of the Durbin-Watson test statistic for any regression matrix and thus eliminating the region of indeterminacy from the Durbin-Watson test procedure.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.