Mean estimation bias in least squares estimation of autoregressive processes

Estimation of the parameters of an autoregressive process with a mean that is a function of time is considered. Approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived. For the case of a mean function that is a polynomial in time, a reparameterization that isolates the bias is given. Using the approximate expressions, a method of modifying the least squares estimator is proposed. A Monte Carlo study of the second-order autoregressive process is presented. The Monte Carlo results agree well with the approximate theory and, generally speaking, the modified least squares estimators performed better than the least squares estimator. For the second-order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one, two and three periods in the future.     

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.     

Functional Coefficient Cointegration Models Subject to Time–Varying Volatility with an Application to the Purchasing Power Parity

This paper analyses functional coefficient cointegration models with both stationary and non‐stationary covariates, allowing time‐varying (unconditional) volatility of a general form. The conventional kernel weighted least squares (KLS) estimator is subject to potential efficiency loss, and can be improved by an adaptive kernel weighted least squares (AKLS) estimator that adapts to heteroscedasticity of unknown form. The AKLS estimator is shown to be as efficient as the oracle generalized kernel weighted least squares estimator asymptotically, and can achieve significant efficiency gain relative to the KLS estimator in finite samples. An illustrative example is provided by investigating the Purchasing Power Parity hypothesis.     

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.     

Announcement

In this paper we obtain a small-disturbance approximation to the moment matrix of the limiting distribution of an operational generalized least squares (OGLS) estimator of the mean response vector in a random coefficient model.It is shown that for small samples the moment matrix of the limiting distribution underestimates the small-disturbance approximate moment matrix of the limiting distribution of the OGLS estimator. This suggests that for small samples the ‘standard errors’ of the OGLS estimates should be obtained from the small-disturbance approximate moment matrix of the limiting distribution rather than from the conventional asymptotic moment matrix.     

Partially generalized least squares and two-stage least squares estimators

A class of partially generalized least squares estimators and a class of partially generalized two-stage least squares estimators in regression models with heteroscedastic errors are proposed. By using these estimators a researcher can attain higher efficiency than that attained by the least squares or the two-stage least squares estimators without explicitly estimating each component of the heteroscedastic variances. However, the efficiency is not as high as that of the generalized least squares or the generalized two-stage least squares estimator calculated using the knowledge of the true variances. Hence the use of the term partial.     

Optimality of least squares in the seemingly unrelated regression equation model

In this article the authors have investigated the situations in which the single-equation least squares estimator is identical with the generalized least squares estimator in the seemingly unrelated regression model. The condition obtained turned out to be advantageous from an empirical point of view as it permits one to decide whether to go for a single-equation least squares method or Zellner's method with estimated disturbance variance covariance matrix for estimating the coefficients in the model.     

Improved forecasting of autoregressive series by weighted least squares approximate REML estimation

Restricted maximum likelihood (REML) estimation has recently been shown to provide less biased estimates in autoregressive series. A simple weighted least squares approximate REML procedure has been developed that is particularly useful for vector autoregressive processes. Here, we compare the forecasts of such processes using both the standard ordinary least squares (OLS) estimates and the new approximate REML estimates. Forecasts based on the approximate REML estimates are found to provide a significant improvement over those obtained using the standard OLS estimates.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

The asymptotic behaviour of the estimated generalized least squares method in the linear regression model

Abstract  In the linear regression model the generalized least squares (GLS) method is only applicable if the covariance matrix of the errors is known but for a scalar factor. Otherwise an estimator for this matrix has to be used. Then we speak of the estimated generalized least squares (EGLS) method. In this paper the asymptotic behaviour of both methods is compared. Results are applied to some standard models commonly used in econometrics     

Consistency and asymptotic normality of least squares estimators in generalized STAR models

Space–time autoregressive (STAR) models, introduced by Cliff and Ord [Spatial autocorrelation (1973) Pioneer, London] are successfully applied in many areas of science, particularly when there is prior information about spatial dependence. These models have significantly fewer parameters than vector autoregressive models, where all information about spatial and time dependence is deduced from the data. A more flexible class of models, generalized STAR models, has been introduced in Borovkova et al. [Proc. 17th Int. Workshop Stat. Model. (2002), Chania, Greece] where the model parameters are allowed to vary per location. This paper establishes strong consistency and asymptotic normality of the least squares estimator in generalized STAR models. These results are obtained under minimal conditions on the sequence of innovations, which are assumed to form a martingale difference array. We investigate the quality of the normal approximation for finite samples by means of a numerical simulation study, and apply a generalized STAR model to a multivariate time series of monthly tea production in west Java, Indonesia.     

Efficiency of the OLSE for regressions on two-dimensional grids with sinusoidal regressors and spatially correlated errors

For spatial regressions with sinusoidal surfaces, the ordinary least squares estimator (OLSE) is shown to be asymptotically as efficient as the generalized least squares estimator (GLSE) in that the covariance matrices of the two estimators have the same nontrivial limit under the same normalization.     

Simultaneous equations with error components

In this paper we derive a limited as well as a full information estimator for the structural parameters of a simultaneous equations model with error components. Under this model, the gain in efficiency by performing these estimators rather than the classical two-stage and three-stage least squares procedures is demonstrated. It is shown that the full information estimator will reduce to the limited information estimator when the disturbances of different structural equations are uncorrelated with each other but not necessarily when all structural equations are just identified. This is different from the analogous situation in the classical case.     

Relative efficiencies of some simple Bayes estimators of coefficients in a dynamic equation with serially correlated errors - II

Generalized least squares estimators, with estimated variance-covariance matrices, and maximum likelihood estimators have been proposed in the literature to deal with the problem of estimating autoregressive models with autocorrelated disturbances. In this paper we compare the small sample efficiencies of these estimators with those of some approximate Bayes estimators. The comparison is done with the help of a sampling experiment applied to a model specification. Though these Bayes estimators utilize very weak prior information, they out-perform the sampling theory estimators in every case we consider.     

A note on least squares estimation and the blue in a generalized linear regression model

In a generalized linear regression model, least squares and Gauss-Markov estimators differ, in general, if the variance-covariance matrix of the disturbances is singular. In the present note it is shown that, nevertheless, the conventional least squares procedure leads to a Gauss-Markov estimator if it is applied to a modified model which results from adding dummy constraints to the original model. These constraints reflect the effects of the singularity of the variance- convariance matrix. As a consequence, a Gauss-Markov estimate may always be obtained by standard least squares minimization, which offers considerable computational advantages.     

Fully modified semiparametric GLS estimation for regressions with nonstationary seasonal regressors

Regression models with seasonally integrated and possibly endogenous regressors and serially correlated regression errors are studied. Spectral decompositions of generalized sums of cross products of regressors and regression errors are used to develop a feasible generalized least squares estimator (FGLSE) which does not require parametric specifications for error processes. Using the FGLSE and following the spirit of “Fully Modified estimation” of Phillips and Hansen (Rev. Econ. Stud. 57 (1990) 99), a fully modified GLSE (FM-GLSE) and inference procedures are constructed. The distribution of the FM-GLSE is shown to be asymptotically a mixed normal distribution which validates standard inference based on the FM-GLSE with normal theory. A Monte-Carlo simulation shows that the FM-GLSE is more efficient than the ordinary least squares estimator (OLSE) in the cases of endogeneity or serial correlation and more efficient than the FM-estimator based on the OLSE in the case of serial correlation.     

Estimation and testing for a Poisson autoregressive model

This article considers statistical inference for a Poisson autoregressive model. A condition for ergodicity and a necessary and sufficient condition for the existence of moments are given. Asymptotics for maximum likelihood estimator and weighted least squares estimators with estimated weights or known weights of the parameters are established. Testing conditional heteroscedasticity and testing the parameters under a simple ordered restriction are noted. A simulation study is also given.     

A note on amemiya's partially generalized least squares

This note explores the relationship between the generalized least squares estimator and Amemiya's partially generalized least squares estimator and establishes the conditions under which the two estimators are equal.     

Variable selection via composite quantile regression with dependent errors

We propose composite quantile regression for dependent data, in which the errors are from short‐range dependent and strictly stationary linear processes. Under some regularity conditions, we show that composite quantile estimator enjoys root‐n consistency and asymptotic normality. We investigate the asymptotic relative efficiency of composite quantile estimator to both single‐level quantile regression and least‐squares regression. When the errors have finite variance, the relative efficiency of composite quantile estimator with respect to the least‐squares estimator has a universal lower bound. Under some regularity conditions, the adaptive least absolute shrinkage and selection operator penalty leads to consistent variable selection, and the asymptotic distribution of the non‐zero coefficient is the same as that of the counterparts obtained when the true model is known. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.