A procedure foe improving upon the performance of the traditional heteroskedast1c1ty pretest estimator

In heteroskedasticity pretesting, if the null hypothesis of homoskedasticity is accepted, the OLS estimator rather than the 2SAE is used. However, if the degree of severity of heteroskedasticity were so mild that the OLS estimator would still outperform the 2SAE, then this methodology would produce results that are inferior to the OLS estimator.

This paper suggests that instead of pretesting for the presence of heteroskedasticity alone, researchers should in addition use a relative efficiency criterion to compare the performance of both estimators for improved results.     

On choosing the level of significance for the goldfeld and quandt heteroskedasticity pretesting

When heteroskedasticity is suspected pretests are performed to investigate its presence in the data. If the null hypothesis of homoskedasticity is accepted through pretests based on an arbitrarily selected level of significance, the OLS estimator rather than the two stage Aitken Estimator (BSAE) is used. This methodology defines the traditional heteroskedasticity pretest estimator. Until now nothing is known about a generally accepted optimal level of significance for heteroskedasticity pretesting. This paper through exact numerical analysis suggests what the level of significance should be for improved heteroskedasticity pretesting.     

Measuring the degree of severity of heteroskedasticity and the choice between the ols estimator and the 2sae

This paper dwells on the choice between the ordinary least squares and the estimated generalized least squares estimators when the presence of heteroskedasticity is suspected. Since the estimated generalized least squares estimator does not dominate the ordinary least squares estimator completely over the whole parameter space, it is of interest to the researcher to know in advance whether the degree of severity of heteroskedasticity is such that OLS estimator outperforms the estimated generalized least squares (or 2SAE). Casting the problem in the non-spherical error mold and exploiting the principle underlying the Bayesian pretest estimator, an intuitive non-mathematical procedure is proposed to serve as an aid to the researcher in deciding when to use either the ordinary least squares (OLS) or the estimated generalized least squares (2SAE) estimators.     

Towards Uniformly Efficient Trend Estimation Under Weak/Strong Correlation and Non‐stationary Volatility

In this paper, we consider the deterministic trend model where the error process is allowed to be weakly or strongly correlated and subject to non‐stationary volatility. Extant estimators of the trend coefficient are analysed. We find that under heteroskedasticity, the Cochrane–Orcutt‐type estimator (with some initial condition) could be less efficient than Ordinary Least Squares (OLS) when the process is highly persistent, whereas it is asymptotically equivalent to OLS when the process is less persistent. An efficient non‐parametrically weighted Cochrane–Orcutt‐type estimator is then proposed. The efficiency is uniform over weak or strong serial correlation and non‐stationary volatility of unknown form. The feasible estimator relies on non‐parametric estimation of the volatility function, and the asymptotic theory is provided. We use the data‐dependent smoothing bandwidth that can automatically adjust for the strength of non‐stationarity in volatilities. The implementation does not require pretesting persistence of the process or specification of non‐stationary volatility. Finite‐sample evaluation via simulations and an empirical application demonstrates the good performance of proposed estimators.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

Comparison of estimators for heteroskedastic models

Ordinary least squares (OLS) yield inefficient parameter estimates and inconsistent estimates of the covariance matrix in case of heteroskedastic errors. Robinson's adaptive estimator and the Cragg estimator avoid any explicit parameterization of heteroskedasticity, and reduce the danger of misspecification. A small Monte Carlo experiment is performed to compare the behavior of the adaptive estimator with the performance of the Cragg estimator. The Monte Carlo experiment includes simulations of the Generalized Least Squares (GLS) estimator. Indeed, an interesting question is how more sophisticated techniques, like the adaptive estimator, compare with GLS when the latter relies on an incorrect specification of the heteroskedastic process. It turns out that the regression parameters, when estimated adaptively, display small mean squared errors and great efficiency in case of medium or high heteroskedasticity. The covariance matrix, instead, is better estimated by the Cragg estimator or by GLS based on a misspecified error term, since the adaptive estimator overpredicts the standard errors of the regression parameters.     

Alternative tests for heteroscedasticity of disturbances:a comparative study

This paper presents three small sample tests for testing the heteroscedasticity among regression disturbances. The power of these tests are compared with two of the leading tests for this hypothesis, one by Goldfeld and Quandt [5] and the other by Theil [17]. We also provide a heuristic method of selecting the number of middle observations to be deleted for the Goldfeld-Quandt type of tests.     

Consequences of effect size heterogeneity for meta-analysis: a Monte Carlo study

In this article we use Monte Carlo analysis to assess the small sample behaviour of the OLS, the weighted least squares (WLS) and the mixed effects meta-estimators under several types of effect size heterogeneity, using the bias, the mean squared error and the size and power of the statistical tests as performance indicators. Specifically, we analyse the consequences of heterogeneity in effect size precision (heteroskedasticity) and of two types of random effect size variation, one where the variation holds for the entire sample, and one where only a subset of the sample of studies is affected. Our results show that the mixed effects estimator is to be preferred to the other two estimators in the first two situations, but that WLS outperforms OLS and mixed effects in the third situation. Our findings therefore show that, under circumstances that are quite common in practice, using the mixed effects estimator may be suboptimal and that the use of WLS is preferable.     

A suggestion for measuring heteroskedasticity

The lack of a generally-accepted measure of the degree of severity of heteroskedasticity is shown to have caused some Monte Carlo studies to draw misleading conclusions. An attractive measure of heteroskedasticity is suggested.     

Asymptotic properties of the ols and grls estimator for the replicated functional relationship model

For the simple linear functional relationship model with replication, the asymptotic properties of the ordinary (OLS) and grouping least squares (GRLS) estimator of the slope are investi- gated under the assumption of normally distributed errors with unknown covariance matrix. The relative performance of the OLS and GRLS estimator is compared in terms of the asymptotic mean square error, and a set of critical parameters are identified for determining the dominance of one estimator over the other. It is also shown that the GRLS estimator is asymptoticallyequivalent to the maximum likelihood (ML) estimator under the given assumptions.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.     

More Efficient Tests Robust to Heteroskedasticity of Unknown Form

In the presence of heteroskedasticity of unknown form, the Ordinary Least Squares parameter estimator becomes inefficient, and its covariance matrix estimator inconsistent. Eicker (1963) and White (1980) were the first to propose a robust consistent covariance matrix estimator, that permits asymptotically correct inference. This estimator is widely used in practice. Cragg (1983) proposed a more efficient estimator, but concluded that tests basd on it are unreliable. Thus, this last estimator has not been used in practice. This article is concerned with finite sample properties of tests robust to heteroskedasticity of unknown form. Our results suggest that reliable and more efficient tests can be obtained with the Cragg estimators in small samples.     

On the sensitivity of pre-test estimators to covariance misspecification

In statistical and econometric practice it is not uncommon to find that regression parameter estimates obtained using estimated generalized least squares (EGLS) do not differ much from those obtained through ordinary least squares (OLS), even when the assumption of spherical errors is violated. To investigate if one could ignore non-spherical errors, and legitimately continue with OLS estimation under the non-spherical disturbance setting, Banerjee and Magnus (1999) developed statistics to measure the sensitivity of the OLS estimator to covariance misspecification. Wan et al. (2007) generalized this work by allowing for linear restrictions on the regression parameters. This paper extends the aforementioned studies by exploring the sensitivity of the equality restrictions pre-test estimator to covariance misspecification. We find that the pre-test estimators can be very sensitive to covariance misspecification, and the degree of sensitivity of the pre-test estimator often lies between that of its unrestricted and restricted components. In addition, robustness to non-normality is investigated. It is found that existing results remain valid if elliptically symmetric, instead of normal, errors are assumed.     

Comparisons of the <Emphasis Type="Italic">r</Emphasis> − <Emphasis Type="Italic">k</Emphasis> class estimator to the ordinary least squares estimator under the Pitman’s closeness criterion

In the presence of multicollinearity, the r − k class estimator is proposed as an alternative to the ordinary least squares (OLS) estimator which is a general estimator including the ordinary ridge regression (ORR), the principal components regression (PCR) and the OLS estimators. Comparison of competing estimators of a parameter in the sense of mean square error (MSE) criterion is of central interest. An alternative criterion to the MSE criterion is the Pitman’s (1937) closeness (PC) criterion. In this paper, we compare the r − k class estimator to the OLS estimator in terms of PC criterion so that we can get the comparison of the ORR estimator to the OLS estimator under the PC criterion which was done by Mason et al. (1990) and also the comparison of the PCR estimator to the OLS estimator by means of the PC criterion which was done by Lin and Wei (2002).     

Peculiar bias properties of the ols estimator when applied to a dynamic model with autocorrelated disturbances

This study reveals that contrary to the conventional wisdom among econometricians, the bias of the OLS estimator can be quite small when the estimator is applied to a geometrically distributed lag model, y_t<ce:glyph name="dbnd6"/> α + βx _t+ λy _t-₁. + u_t, with autocorrelated disturbances, be they AR(1), MA(1), MA(2), AR(2), and ARMA(1,1). This happens when λ is large and x_tis smoothly trended (e.g., a real GNP series). In fact, the bias of the OLS estimator becomes zero at one parameter combination, and the OLS estimator performs well over a wide range around this parameter combination. By decomposing the disturbance term into two parts, the paper also explains why OLS shows such an unexpected property. These findings have both pedagogical and practical significance.     

THE EFFICIENCY OF THE COCHRANE-ORCUTT PROCEDURE

It is well known that the ordinary least squares (OLS) estimator, though unbiased, is inefficient in the presence of autocorrelated disturbances. Further, it is also widely accepted that the Cochrane-Orcutt (C-O) estimator is more efficient than the OLS estimator. However, Kadiyala (1968) and Maeshiro (1976, 1978) have argued that OLS is more efficient than C-O when the independent variable is trended and the autocorrelation coefficient is positive. We re-examine this issue and show that C-O is more efficient than OLS for the model without an intercept term.     

Confidence regions based on inefficient estimators

We consider an unbiased estimator of a function of mean value parameters, which is not efficient. This inefficient estimator is correlated with a residual vector. Thus, if a unit dispersion is unknown, it is impossible to determine the correct confidence region for a function of mean value parameters via a standard estimator of an unknown dispersion with the exception of the case when the ordinary least squares (OLS) estimator is considered in a model with a special covariance structure such that the OLS and the generalized least squares (GLS) estimator are the same, that is the OLS estimator is efficient. Two different estimators of a unit dispersion independent of an inefficient estimator are derived in a singular linear statistical model. Their quality was verified by simulations for several types of experimental designs. Two new estimators of the unit dispersion were compared with the standard estimators based on the GLS and the OLS estimators of the function of the mean value parameters. The OLS estimator was considered in the incorrect model with a different covariance matrix such that the originally inefficient estimator became efficient. The numerical examples led to a slightly surprising result which seems to be due to data behaviour. An example from geodetic practice is presented in the paper.     

OLS and IV estimation of regression models including endogenous interaction terms

We analyze a class of linear regression models including interactions of endogenous regressors and exogenous covariates. We show how to generate instrumental variables using the nonlinear functional form of the structural equation when traditional excluded instruments are unknown. We propose to use these instruments with identification robust IV inference. We furthermore show that, whenever functional form identification is not valid, the ordinary least squares (OLS) estimator of the coefficient of the interaction term is consistent and standard OLS inference applies. Using our alternative empirical methods we confirm recent empirical findings on the nonlinear causal relation between financial development and economic growth.     

Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate<Emphasis Type="Italic">t</Emphasis> errors under the Pitman nearness criterion

In this paper we consider a linear regression model with omitted relevant regressors and multivariatet error terms. The explicit formula for the Pitman nearness criterion of the Stein-rule (SR) estimator relative to the ordinary least squares (OLS) estimator is derived. It is shown numerically that the dominance of the SR estimator over the OLS estimator under the Pitman nearness criterion can be extended to the case of the multivariatet error distribution when the specification error is not severe. It is also shown that the dominance of the SR estimator over the OLS estimator cannot be extended to the case of the multivariatet error distribution when the specification error is severe. This research is partially supported by the Grants-in-Aid for 21st Century COE program.     

The effects of the proxy information on the iterative Stein-rule estimator of the disturbance variance

In a regression model with proxy variables, we consider the iterative estimator of the disturbance variance to obtain more precise estimates. In the formula of the estimator of the disturbance variance, the estimator is obtained by using Stein-rule (SR) estimator instead of OLS (ordinary least squares) estimator is called Iterative estimator of the disturbance variance. It is shown that, in a regression model with proxy variables the mean square error (MSE) of the iterative estimator of the disturbance variance is greater than the MSE of the disturbance variance related to the OLS estimator under certain conditions.