Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints.

In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.     

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.     

Finite-sample distributions of the Wald,likelihood ratio,and Lagrange multiplier test statistics in the classical linear model

In this note, it is shown that the finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier statistics in the classical linear regression model are members of the generalized beta model introduced by McDonald and Xu (1995a McDonald, J.B., Xu, Y.J. (1995a). A generalization of the beta distribution with applications. J. Econom. 66:133–152.[Crossref], [Web of Science ®] , [Google Scholar]). This is useful for examining the properties of these test statistics. For example, this characterization makes it easy to find distribution, quantile, and density functions for each test statistic, makes it clear why Wald tests may overreject the null hypothesis using asymptotic critical values, and formalizes the fact that the Lagrange multiplier statistic follows a distribution with bounded support.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

On Small Sample Properties of the Wald,LR and LM Tests in a Linear Model with AR(1) Errors

When the error terms are autocorrelated, the conventional t-tests for individual regression coefficients mislead us to over-rejection of the null hypothesis. We examine, by Monte Carlo experiments, the small sample properties of the unrestricted estimator of ρ and of the estimator of ρ restricted by the null hypothesis. We compare the small sample properties of the Wald, likelihood ratio and Lagrange multiplier test statistics for individual regression coefficients. It is shown that when the null hypothesis is true, the unrestricted estimator of ρ is biased. It is also shown that the Lagrange multiplier test using the maximum likelihood estimator of ρ performs better than the Wald and likelihood ratio tests.     

Pooling multivariate data under W, LR and LM tests

Two independent random samples are drawn from two multivariate normal populations with mean vectors μ1 and μ2 and a common variance-covariance matrix Σ. Ahmed and Saleh (1990) considered preliminary test maximum likelihood estimator (PMLTE) for estimating μ1 based on the Hotelling's T _N ², when it is suspected that μ1=μ2. In this paper, the PTMLE based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. Using the quadratic risk function, the conditions of superiority of the proposed estimator for departure parameter are derived. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the PTMLE based on W test produces the highest minimum guaranteed efficiencies compared to UMLE among the three test procedures.     

Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W,LR and LM tests

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on the Wald (W), the likelihood ratio (LR) and the Lagrangian multiplier (LM) tests are considered in this paper. The bias and the risk functions of the proposed estimators are derived. The regions of optimality of the estimators are determined under the quadratic risk function. Under the null hypothesis, the SPTRRE based on LM test has the smallest risk, followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge and departure parameters are discussed. The optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameters.     

Local power and size properties of the LR,Wald, score and gradient tests in dispersion models

We derive asymptotic expansions for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Alternate test statistics for a functional errors-in-variables model

Some test statistics for the structural coefficients of simultaneous equations model often referred to as the multivariate linear functional relationship model are proposed in this article. The following cases are considered: the covariance matrix of errors is either unknown, known up to a proportionality factor, or completely known. The exact and approximate distributions of the proposed test statistics, as well as those of some that are known, are also given.     

Asymptotic distributions of some test statistics for a linear functional relationship

We consider the testing problems of the structural parameters for the multivariate linear functional relationship model. We treat the likelihood ratio test statistics and the test statistics based on the asymptotic distributions of the maximum likelihood estimators. We derive their asymptotic distributions under each null hypothesis respectively. A simulation study is made to evaluate how we can trust our asymptotic results when the sample size is rather small.     

因子模型门槛效应的LM和Wald检验及其统计性质研究

门槛因子模型可以有效地刻画高维度时间序列的共变特征和区制转换行为,具有良好的可解释性和预测能力。针对因子载荷矩阵存在的门槛效应,本文提出了拉格朗日乘子和沃尔德检验方法,并给出了渐近分布,相关结果表明以上检验统计量具有良好的大样本性质和有限样本表现。在实证部分,以我国股市的行业指数作为研究对象,通过构建门槛因子模型来刻画我国股票市场波动的共变性特征和非对称效应。实证结果表明基于门槛因子模型可以很好地刻画中国股市行业收益率波动的共变特征和区制转换行为。     

On the distribution of some test statistics connected with the multivariate linear functional relationship model

An alternate representation of the densities of some test statistics for the structural coefficients of the multivariate linear functional relationship model is proposed in this article. These statistics are distributed as the ratio of a linear combination of chi-square variÂtes over the root of a product of chi-square variÂtes. A computable representation of their densities has already been derived by Provost (1984) with the help of the technique of the inverse Mellin transform. The connection of the alternate representation to the densities of products of independent beta type-2 and of independent F-random variables is also discussed.     

An alternative Wald type test for two linear restrictions with applications to non-linear models

This paper proposes a new test procedure called the rel test to resolve the problem of small-sample local biasedness and non-monotonic power behavior of the Wald test for two linear restrictions caused by inaccuracy of the estimated covariance matrix of the estimator. This new test procedure, which does not need the covariance matrix of the estimator, involves finding the critical region based on contour points of the percentile confidence limit of a rel utilizing the bootstrap in order to obtain a test with the desired size and good power properties. Simulation results indicate that this new test procedure, the rel test, performs rather well both with respect to controlling size and having monotonic increasing power.     

A new interval mapping approach based on a general sib-pair regression model with a modified Wald test

Various regression models based on sib-pair data have been developed for mapping quantitative trait loci (QTL) in humans since the seminal paper published in 1972 by Haseman and Elston. Fulker and Cardon [D.W. Fulker, L.R. Cardon, A sib-pair approach to interval mapping of quantitative trait loci, Am. J. Hum. Genet. 54 (1994) 1092–1103] adapted the idea of interval mapping [E.S. Lander, D. Botstein, Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps, Genetics 121 (1989) 185–199] to the Haseman–Elston regression model in order to increase the power of QTL mapping. However, in the interval mapping approach of Fulker and Cardon, the statistic for testing QTL effects does not obey the classical statistical theory and hence critical values of the test can not be appropriately determined. In this article, we consider a new interval mapping approach based on a general sib-pair regression model. A modified Wald test is proposed for the testing of QTL effects. The asymptotic distribution of the modified Wald test statistic is provided and hence the critical values or the $p$-values of the test can be well determined. Simulation studies are carried out to verify the validity of the modified Wald test and to demonstrate its desirable power.     

Sensitivity of Rao's score test,the Wald test and the likelihood ratio test to nuisance parameters

The purpose of this paper is to compare the sensitivity of the likelihood ratio test, Rao's score test, and the Wald test to the change of the nuisance parameters. The main result is that, with an error of magnitude O(n⁻¹), the null distributions and the local alternative distributions of these tests are equally sensitive to nuisance parameter. We will also give accurate factorizations of these test statistics as quadratic forms, which are themselves useful for asymptotic analyses.     

A non-iterative approach to estimating parameters in a linear structural equation model

The research described herein was motivated by a study of the relationship between the performance of students in senior high schools and at universities in China. A special linear structural equation model is established, in which some parameters are known and both the responses and the covariables are measured with errors. To explore the relationship between the true responses and latent covariables and to estimate the parameters, we suggest a non-iterative estimation approach that can account for the external dependence between the true responses and latent covariables. This approach can also deal with the collinearity problem because the use of dimension-reduction techniques can remove redundant variables. Combining further with the information that some of parameters are given, we can perform estimation for the other unknown parameters. An easily implemented algorithm is provided. A simulation is carried out to provide evidence of the performance of the approach and to compare it with existing methods. The approach is applied to the education example for illustration, and it can be readily extended to more general models.     

Local influence on a test of linear hypothesis in multiple regression model

SUMMARY The local influence method is adapted to investigate the influence of observations on testing the linear hypothesis. The method provides information about individually or jointly influential observations in performing the test, which the usual diagnostic methods do not yield. An example is presented for illustration of the method.     

Specification test for a linear regression model with ARCH process

ARCH models are used widely in analyzing economic and financial time series data. Many tests are available to detect the presence of ARCH; however, there is no acceptable procedure available for testing an estimated ARCH model.. In this paper we develop a test for a linear regression model with ARCH disturbances using the framework of the information matrix (IM) test. For the ARCH specification, the covariance matrix of the indicator vector is not block diagonal, and the IM test is turned out to be a test for variation in the fourth moment, i.e., a test for heterokurtosis. An illustrative example is provided to demonstrate the usefulness of the proposed test.     

Dimension test approach of heteroscedasticity in the linear model

Heteroscedasticity testing has a long history and is still an important matter in the linear model. There exist many types of tests, but they are limited in use to their own specific cases and sensitive to normality. Here, we propose a dimension test approach to heteroscedasticity. The proposed test overcomes the shortcomings of the existing methods, so that it is robust to normality and is unified in sense that it is applicable in the linear model with multi-dimensional response. Numerical studies confirm that the proposed test is favorable over the existing tests with moderate sample sizes, and real data analysis is presented.     

Consistent and robust variable selection in regression based on Wald test

Selection of relevant predictor variables for building a model is an important problem in the multiple linear regression. Variable selection method based on ordinary least squares estimator fails to select the set of relevant variables for building a model in the presence of outliers and leverage points. In this article, we propose a new robust variable selection criterion for selection of relevant variables in the model and establish its consistency property. Performance of the proposed method is evaluated through simulation study and real data.