Toshihisa Toyoda

Estimation of variance after a preliminary test of homogeneity and optimal levels of significance for the pre-test

The question of whether to pool two samples in variance estimation is often decided via a preliminary F test. In this paper we show that the optimal pre-test F value is unity for a onesided alternative, where the objective function is to minimize average relative risk. The outcome is independent of numbers of degrees of freedom in each sample. Optimal significance levels vary somewhat but are close to + for most d.f. and equal to : when numerator and denominator d.f. are equal. The results also apply to regression variance estimation across two data regimes.

Testing equality between sets of coefficients after a preliminary test for equality of disturbance variances in two linear regressions

Abstract In this paper we propose a two-stage test which consists of testing equality of regression coefficients after a preliminary test for equality of disturbance variances in two linear regressions. The optimal procedure of the two-stage test is presented. The sampling performance of the optimal two-stage test is compared with that of the Jayatissa test and with that of the size-corrected Wald test.

Estimation of regression coefficients after a preliminary test for homoscedasticity

Abstract When heteroscedasticity of the variances of disturbances in a regression model is suspected, we perform a preliminary test for homoscedasticity prior to estimation of regression coefficients. According to the result of the pre-test, we use either the ordinary least squares estimator or the two-stage Aitken estimator ( 2SAE ). In this paper, using orthonormal regressors, we derive the mean square error ( MSE ) of the pre-test estimator and show that the 2SAE is inadmissible when the MSE is used as a criterion. Further, we seek the optimal critical value of the pre-test in the sense of minimizing the average relative risk which is based on the MSE .

Testing linear hypothesis on regression coefficients after a pre-test for disturbance variance

Abstract In this note, we analyze the sampling performance of the two-stage test that tests the linear hypothesis on regression coefficients after a pre-test for disturbance variance. It is shown that the two-stage test has a serious upward bias in size when the degrees of freedom is small, but it gets smaller as the degrees of freedom gets larger and the power of the two-stage test is higher than that of the conventional F test for the linear hypothesis which does not conduct the pre-test.

A monte carlo study of the wald, lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.

Pre-testing on part of the data

Reserving part of a data set for final estimation, using the other part for pre-testing in a regression model is not good practice from a risk function point of view, even though such procedure has been advocated and practised by some econometricians and statisticians. Comparing quadratic risk functions for estimators based on pre-tests on part of the data with estimators based on pre-tests on all the data shows that neither dominate. However, the part sample estimator is decidedly inferior in average relative risk.