Asymptotic distribution of Durbin-Watson statistic

In this note the asymptotic distribution of Durbin–Watson statistic is established without any condition on the design matrix.     

The bias of the least squares estimator over interval constraints

We show that the interval constrained least squares estimator for a regression model is in general bias. The bias and some of its properties are given when the regression residuals are normally distributed.     

The CUSUM of squares test for the stability of regression models with non-stationary regressors

In this paper, we propose a modified CUSUM of squares test in time series regression models with a non-stationary regressor and show that the limiting distribution of this test is the sup of the absolute value of a Brownian bridge.     

Asymptotic distribution of the sup-Wald statistic under specification errors

Consider a simple structural break model where y_t=α₁+β₁f(x_t)+u_t for t≤k₀ and y_t=α₂+β₂f(x_t)+u_t for t>k₀. The timing of break and the structural parameters are unknown. Suppose the true functional form of the regressor f(·) is misspecified as g(·). We do not place too many restrictions on the functional forms of f(·) and g(·). A frequently encountered example in economics is that the true model is measured in level, but we estimate a log-linear model, i.e. when f(x_t)=x_t and g(x_t)=log(x_t) For any f(·) and g(·), we derive a nonstandard limiting null distribution of the sup-Wald test statistic under some very general regularity conditions. Monte Carlo simulations support our findings.     

Asymptotic bias reduction for a conditional marginal effects estimator in sample selection models

In this article we discuss the differences between the average marginal effect and the marginal effect of the average individual in sample selection models, estimated by the Heckman procedure. We show that the bias that emerges as a consequence of interchanging the measures, could be very significant, even in the limit. We suggest a computationally cheap approximation method, which corrects the bias to a large extent. We illustrate the implications of our method with an empirical application of earnings assimilation and a small Monte Carlo simulation.     

On least squares estimation with a particular linear function of the dependent variable

We consider the standard linear regression model where the endogenous variable y is substituted by Ty, T being a symmetric, idempotent matrix. Comparing the mean square error (MSE) matrices we show that a ‘naive’ LS-procedure may work better than a competing estimator usually proposed in the literature and may even perform better than the LS-estimator based on untransformed data. We derive necessary and sufficient conditions for MSE-dominance and outline some ideas for testing.     

The convergence of least squares learning in stochastic temporary equilibrium models

Summary. This paper provides conditions for the almost sure convergence of the least squares learning rule in a stochastic temporary equilibrium model, where regressions are performed on the past values of the endogenous state variable. In contrast to earlier studies, (Evans and Honkapohja, 1998; Marcent and Sargent, 1989), which were local analyses, the dynamics are studied from a global viewpoint, which allows one to obtain an almost sure convergence result without employing projection facilities. Received: April 7, 2001; revised version: September 5, 2001     

Inadmissibility of the iterative Stein-rule estimator of the disturbance variance in a linear regression

In this paper, we consider the estimator of the disturbance variance in a linear regression when the Stein-rule estimator is used in place of the OLS estimator (the iterative Stein-rule estimator of the disturbance variance). It is shown that the iterative Stein-rule estimator of the disturbance variance is dominated by the usual estimator of the disturbance variance based on the OLS estimator under the squared error loss criterion, if the number of regressors is greater than or equal to five.     

The exact estimation of logit models by non-linear least squares

The purpose of this paper is to examine the relationship between least squares and maximum likelihood estimation, where the likelihood function is the product of two explicit functions. We illustrate the correspondence for the particular case of the logit model, and show that this can be estimated by commonly accessible non-linear least squares estimation packages. Unlike the conventional non-linear least squares approach the estimates obtained following the proposed method are maximum likelihood for all sample sizes.     

A comparison of bias approximations for the two-stage least squares (2SLS) estimator

We consider the bias of the two-stage least squares (2SLS) estimator in linear instrumental variable regression with only one endogenous regressor. By using asymptotic expansion techniques, we approximate the 2SLS coefficient estimation bias under various scenarios regarding the number and strength of instruments.     

The relative efficiency of a robust generalized Bayes estimator in a linear regression model with multicollinearity

This paper uses Monte Carlo techniques to examine the performance of a robust generalized Bayes estimator for a linear regression model when multicollinearity is present. Unlike many improved estimators, this near-minimax estimator performs very well under squared error loss even when the data are ill-conditioned.     

Testing for normality in linear regression models using regression and scale equivariant estimators

In this paper we provide a general solution to the problem of controlling the probability of a type I error in normality tests for the disturbances in linear regressions when using robust-regression residuals. We show that many classes of well-known robust regression estimators belong to the class of regression and scale equivariant estimators. It is these equivariance properties that are used to reduce the nuisance parameter space under the null, from which we develop Monte Carlo and Maximized Monte Carlo tests for the null of disturbance normality. Finally, we illustrate in a simulation experiment the potential power gains from using robust-regression residuals in testing this null hypothesis.     

Results on the bias and inconsistency of ordinary least squares for the linear probability model

This note formalizes bias and inconsistency results for ordinary least squares (OLS) on the linear probability model and provides sufficient conditions for unbiasedness and consistency to hold. The conditions suggest that a “trimming estimator” may reduce OLS bias.     

On the asymptotic bias and mean squared error of an improved estimator for coefficients in linear regression

An estimator for regression coefficients of Kadiyala (1984) is considered. It is proved that the estimator is asymptotically unbiased. The asymptotic weak mean squared error of the estimator is also derived and it is proved that, under certain conditions, the estimator dominates a general class of estimators given by Vinod and Ullah (1981).     

Finite sample bias of the least squares estimator in an AR(p) model: estimation,inference, simulation and examples

This paper shows that the first order bias of least squares estimators of the coefficients of an AR(p) model is important for ‘typical’ macroeconomic time series and proposes a simple to apply method of bias reduction. Biases in individual coefficients often cumulate in the sum with far-reaching consequences for the cumulative impulse response function. This function, being nonlinear in the underlying coefficients, is particularly sensitive to biases when, as is often the case, the shocks are long-lived. Simulations and examples demonstrate some of the magnitudes involved.     

On Lindley-like mean correction in the improved estimation of linear regression models

The risks of estimators incorporating the correction to the sample mean in the spirit of Stein and Lindley are approximated in the case of small disturbances.     

The use of discounted least squares in technological forecasting

The use of growth curves in technological forecasting usually employs an equal weighting of all data points in the time series. This paper considers the benefits of weighting recent information more heavily through the utilization of discounted least squares. The method is used to model the growth of the percentage of households with CATV; discounting gives better results for short-term forecasting.     

Asymptotic inference in regression models with autoregressive errors having roots on the unit circle

An inference procedure is proposed for regression models with stationary regressors and non-stationary autoregressive errors. It is shown that the usual GLS or Cochrane–Orcutt procedure should be done in reverse order by starting the estimation from the error structure.     

The asymptotic efficiency of semiparametric estimators for censored linear regression models

This paper presents numerical comparisons of the asymptotic mean square estimation errors of semiparametric generalized least squares (SGLS), quantite, symmetrically censored least squares (SCLS), and tobit maximum likelihood estimators of the slope parameters of censored linear regression models with one explanatory variable. The results indicate that the SCLS estimator is less efficient than the other two semiparametric estimators. The SGLS estimator is more efficient than quantile estimators when the tails of the distribution of the random component of the model are not too thick and the probability of censoring is not too large. The most efficient semiparametric estimators usually have smaller mean square estimation errors than does the tobit estimator when the random component of the model is not normally distributed and the sample size is 500–1,000 or more.