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.     

The Mean Squared Error of the Instrumental Variables Estimator When the Disturbance Has an Elliptical Distribution

This paper generalizes Nagar's (1959) approximation to the finite sample mean squared error (MSE) of the instrumental variables (IV) estimator to the case in which the errors possess an elliptical distribution whose moments exist up to infinite order. This allows for types of excess kurtosis exhibited by some financial data series. This approximation is compared numerically to Knight's (1985) formulae for the exact moments of the IV estimator under nonnormality. We use the results to explore two questions on instrument selection. First, we complement Buse's (1992) analysis by considering the impact of additional instruments on both bias and MSE. Second, we evaluate the properties of Andrews's (1999) selection method in terms of the bias and MSE of the resulting IV estimator.     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

Regression spline bivariate probit models: A practical approach to testing for exogeneity

Bivariate probit models can deal with a problem usually known as endogeneity. This issue is likely to arise in observational studies when confounders are unobserved. We are concerned with testing the hypothesis of exogeneity (or absence of endogeneity) when using regression spline recursive and sample selection bivariate probit models. Likelihood ratio and gradient tests are discussed in this context and their empirical properties investigated and compared with those of the Lagrange multiplier and Wald tests through a Monte Carlo study. The tests are illustrated using two datasets in which the hypothesis of exogeneity needs to be tested.     

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matr ix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.     

Preliminary test Liu-type estimators based on W,LR, and LM test statistics in a regression model

This article discusses the preliminary test approach for the regression parameter in multiple regression model. The preliminary test Liu-type estimators based on the Wald (W), Likelihood ratio (LR), and Lagrangian multiplier(LM) tests are presented, when it is supposed that the regression parameter may be restricted to a subspace. We also give the bias and mean squared error of the proposed estimators and the superior of the proposed estimators is also discussed.     

Estimator Choice and Fisher's Paradox: A Monte Carlo Study

This paper argues that Fisher's paradox can be explained away in terms of estimator choice. We analyse by means of Monte Carlo experiments the small sample properties of a large set of estimators (including virtually all available single-equation estimators), and compute the critical values based on the empirical distributions of the t-statistics, for a variety of Data Generation Processes (DGPs), allowing for structural breaks, ARCH effects etc. We show that precisely the estimators most commonly used in the literature, namely OLS, Dynamic OLS (DOLS) and non-prewhitened FMLS, have the worst performance in small samples, and produce rejections of the Fisher hypothesis. If one employs the estimators with the most desirable properties (i.e., the smallest downward bias and the minimum shift in the distribution of the associated t-statistics), or if one uses the empirical critical values, the evidence based on US data is strongly supportive of the Fisher relation, consistently with many theoretical models.     

Constrained design strategies for improving normal approximations in nonlinear regression problems

In nonlinear regression problems, the assumption is usually made that parameter estimates will be approximately normally distributed. The accuracy of the approximation depends on the sample size and also on the intrinsic and parameter-effects curvatures. Based on these curvatures, criteria are defined here that indicate whether or not an experiment will lead to estimates with distributions well approximated by a normal distribution. An approach is motivated of optimizing a primary design criterion subject to satisfying constraints based on these nonnormality measures. The approach can be used either to I) find designs for a fixed sample size or to II) choose the sample size for the optimal design based on the primary objective so that the constraints are satisfied. This later objective is useful as the nonnormality measures decrease with the sample size. As the constraints are typically not concave functions over a set of design measures, the usual equivalence theorems of optimal design theory do not hold for the first approach, and numerical implementation is required. Examples are given, and a new notation using tensor products is introduced to define tractable general notation for the nonnormality measures.     

The effect of fat-tailed error terms on the properties of systemwise RESET test

The small sample properties of the systemwise RESET (Regression Specification Error Test) test for functional misspecification are investigated using normal and non-normal error terms. When using normally distributed or less heavy tailed error terms, we find the Rao's multivariate F-test to be best among all other alternative test methods (i.e. Wald, Lagrange Multiplier and Likelihood Ratio). Using the bootstrap critical values, however, all test methods perform satisfactorily in almost all situations. However, the test methods perform extremely badly (even the RAO test) when the error terms are very heavy tailed.     

Preliminary test almost unbiased two-parameter estimators with student’s t errors and conflicting test statistics

Abstract

In this paper, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model under multicollinearity situation. The preliminary test almost unbiased two-parameter estimators based on the Wald, the Likelihood ratio, and the Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student’s t errors. The bias and quadratic risk of the proposed estimators are derived and compared. Furthermore, a Monte Carlo simulation is provided to illustrate some of the theoretical results.     

Evaluating bias correction in weighted proportional hazards regression

Often in observational studies of time to an event, the study population is a biased (i.e., unrepresentative) sample of the target population. In the presence of biased samples, it is common to weight subjects by the inverse of their respective selection probabilities. Pan and Schaubel (Can J Stat 36:111–127, 2008) recently proposed inference procedures for an inverse selection probability weighted (ISPW) Cox model, applicable when selection probabilities are not treated as fixed but estimated empirically. The proposed weighting procedure requires auxiliary data to estimate the weights and is computationally more intense than unweighted estimation. The ignorability of sample selection process in terms of parameter estimators and predictions is often of interest, from several perspectives: e.g., to determine if weighting makes a significant difference to the analysis at hand, which would in turn address whether the collection of auxiliary data is required in future studies; to evaluate previous studies which did not correct for selection bias. In this article, we propose methods to quantify the degree of bias corrected by the weighting procedure in the partial likelihood and Breslow-Aalen estimators. Asymptotic properties of the proposed test statistics are derived. The finite-sample significance level and power are evaluated through simulation. The proposed methods are then applied to data from a national organ failure registry to evaluate the bias in a post-kidney transplant survival model.     

Some comparative studies on testing parallelism of several straight lines under heteroscedastic variances

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as F_β test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.     

Robustness of Inference for One-sample Problem with Correlated Observations

The inference about the population mean based on the standard t-test involves the assumption of normal population as well as independence of the observations. In this paper we examine the robustness of the inference in the presence of correlations among the observations. We consider the simplest correlation structure AR(1) and its impact on the t-test. A modification of the t-test suitable for this structure is suggested, and its effect on the inference is investigated using Monte Carlo simulation.     

Effects of nonnormality on the spart for the correlation coefficient: bivariate edgeworth series distribution

The effects of sampling from the bivariate Edgeworth series distribution (BVESD) on the sequential probability ratio test (SPRT) for the correlation coefficient are assessed. The values of the average sample number (ASN) and the operating characteristic (OC) are determined by simulation from such populations. The robustness of the SPRT to this type of nonnormality is demonstrated.     

Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions

Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large‐sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo‐outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.     

Some robust anova procedures under heteroscedasticity and nonnormality

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber‘s M-estimators with the Brown-Forsythe test, several robust procedures were developed; these procedures were compared through computer simulation studies with the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test. The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions.     

The Size and Power of Bootstrap and Bartlett-Corrected Tests of Hypotheses on the Cointegrating Vectors

In this paper we compare Bartlett-corrected, bootstrap, and fast double bootstrap tests on maximum likelihood estimates of cointegration parameters. The key result is that both the bootstrap and the Bartlett-corrected tests must be based on the unrestricted estimates of the cointegrating vectors: procedures based on the restricted estimates have almost no power. The small sample size bias of the asymptotic test appears so severe as to advise strongly against its use with the sample sizes commonly available; the fast double bootstrap test minimizes size bias, while the Bartlett-corrected test is somehow more powerful.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.     

The finite-sample performance of robust unit root tests

This paper investigates the relative small sample performance of several robust unit root tests by means of a simulation study. It is confirmed that the traditional least-squares based Dickey-Fuller test has substantially lower power than several robust alternatives if the error distribution is fat-tailed while its power gain is small at the normal model. Particularly good results are achieved by a quasi-maximum likelihood test. However, all robust tests under consideration exhibit severe size distortions if the disturbances follow a skewed distribution. Moreover, under additive outliers, robust tests fail to produce stable sizes and good power properties. Consequently, the value of using robust unit root tests depends heavily of the type of nonnormality at hand.     

A robust procedure for comparing several means under heteroscedasticity and nonnormality

By applying Tiku's MML robust procedure to Brown and Forsythe's (1974) statistic, this paper derives a robust and more powerful procedure for comparing several means under hetero-scedasticity and nonnormality. Some Monte Carlo studies indicate clearly that among five nonnormal distributions, except for the uniform distribution, the new test is more powerful than the Brown and Forsythe test under nonnormal distributions in all cases investigated and has substantially the same power as the Brown and Forsythe test under normal distribution.