The application of the durbin-watson test to the dynamic regression model under normal and non-normal errors

Until recently, a difficulty with applying the Durbin-Watson (DW) test to the dynamic linear regression model has been the lack of appropriate critical values. Inder (1986) used a modified small-disturbance distribution (SDD) to find approximate critical values. King and Wu (1991) showed that the exact SDD of the DW statistic is equivalent to the distribution of the DW statistic from the regression with the lagged dependent variables replaced by their means. Unfortunately, these means are unknown although they could be estimated by the actual variable values. This provides a justification for using the exact critical values of the DW statistic from the regression with the lagged dependent variables treated as non-stochastic regressors. Extensive Monte Carlo experiments are reported in this paper. They show that this approach leads to reasonably accurate critical values, particularly when two lags of the dependent variable are present. Robustness to non-normality is also investigated.     

A monte carlo comparison of tests based on the durbin-watson statistic with other autocorrelation tests in dynamic models

This paper examines the sampling properties of a number of serial correlation tests in dynamic linear models which include one or two lags of the dependent variable. Among the tests considered are the Durbin-Watson (DW) bounds test, modified versions of the DW proposed recently by King and Wu and Inder, Durbin's m test, Inder's point optimal test and a Hausman type test. Sampling designs include models with one or two lags of the dependent variable. The m, Hausman, and Inder's tests have the best performance, while Inder's modified DW test appears to be better than the other DW based tests. Results also suggest that tests are less powerful and more sensitive to design parameters in models with higher dynamics, with the DW-based tests being the most sensitive.     

The small sample properties of the reset test as applied to systems of equations

The RESET test for functional misspecification is generalised to cover systems of equations, and the properties of 7 versions are studied using Monte Carlo methods. The Rao F -test clearly exhibits the best performance as regards correct size, whilst the commonly used LRT (uncorrected for degrees-of-freedom), and LM and Wald tests (both corrected and uncorrected) behave badly even in single equations. The Rao test exhibits correct size even in ten equation systems, which is better than previous research concerning autocorrelation tests. The power of the test is low, however, when the number of equations grows and the correlation between the omitted variables and the RESET proxies is small.     

A cusum test in the linear regression model with serially correlated disturbances

This paper considers a modified CUSUM test, suggested by Dufour (1982) for parameter instability and structural change with an unknown change point in a linear model with serially correlated disturbances, in which a preliminary estimate of the autoregressive coefficient for the error process is obtained, and used to transform the data. Then the standard CUSUM statistic is calculated on the transformed data. This paper derives the asymptotic distribution of the modified CUSUM test. We show that the modified CUSUM test retains its asymptotic significance level, i.e., the modified CUSUM test has the same asymptotic distribution as the CUSUM test with serially uncorrelated errors.     

Consistent estimation of regression parameters under replicated ultrastructural model with non-normal errors

This article discusses the construction and efficiency properties of consistent estimators of regression parameters under replicated ultrastructural model with not necessarily normally distributed measurement errors. The variances of measurement errors associated with the study and explanatory variables are estimated from the replicated sample observations and are used for the consistent estimation of regression parameters. The asymptotic efficiency properties of the estimators are derived and analysed. The finite sample performance of the estimators is empirically studied through a Monte Carlo simulation.     

A test of the normality assumption in ordered probit model

This paper presents a Lagrange multiplier test of the normality assumption underlying the ordered probit model. The test is presented both for the standard ordered probit model and a version in which censoring is present in the dependent variable. The test is then compared to normality tests proposed here compares favorably to tests based on artificial regression techinques.     

Small sample properties of ridge estimators with normal and non-normal disturbances

In this paper we consider five well known and widely used ridge estimators when the convenient assumption of normality of the disturbances is abandoned and report on a Monte Carlo study of their small sample properties. The Monte Carlo experiment is applied to four different data sets with artificially varied degrees of multicollinearity, while the disturbances follow normal, lognormal, uniform and Laplace distributions with small and large variances. The results show that the best estimates are obtained for all ridge estimators when the disturbances follow the lognormal distribution. Also, none of the examined ridge estimators shows a consistent behavior under the different settings considered.     

Critical values improvement for the standard normal homogeneity test by combining Monte Carlo and regression approaches

The distribution of the test statistics of homogeneity tests is often unknown, requiring the estimation of the critical values through Monte Carlo (MC) simulations. The computation of the critical values at low α, especially when the distribution of the statistics changes with the series length (sample cardinality), requires a considerable number of simulations to achieve a reasonable precision of the estimates (i.e. 10⁶ simulations or more for each series length). If, in addition, the test requires a noteworthy computational effort, the estimation of the critical values may need unacceptably long runtimes.

To overcome the problem, the paper proposes a regression-based refinement of an initial MC estimate of the critical values, also allowing an approximation of the achieved improvement. Moreover, the paper presents an application of the method to two tests: SNHT (standard normal homogeneity test, widely used in climatology), and SNH2T (a version of SNHT showing a squared numerical complexity). For both, the paper reports the critical values for α ranging between 0.1 and 0.0001 (useful for the p-value estimation), and the series length ranging from 10 (widely adopted size in climatological change-point detection literature) to 70,000 elements (nearly the length of a daily data time series 200 years long), estimated with coefficients of variation within 0.22%. For SNHT, a comparison of our results with approximated, theoretically derived, critical values is also performed; we suggest adopting those values for the series exceeding 70,000 elements.     

Estimation and test of several multivariate normal means under an order restriction when the dimension is larger than two

Suppose that an order restriction is imposed among several p-variate normal mean vectors. We are interested in the problems of estimating these mean vectors and testing their homogeneity under this restriction. These problems are multivariate extensions of Bartholomew's (1959) ones. For the bivariate case, these problems have been studied by Sasabuchi et al. (1983) and (1998) and some others. In the present paper we examine the convergence of an iterative algorithm for computing the maximum likelihood estimator when p is larger than two. We also study some test procedures for testing homogeneity when p is larger than two.     

The robustness of the systemwise breusch-godfrey autocorrelation test for non-normal distributed error terms

Using Monte Carlo methods, the properties of systemwise generalisations of the Breusch-Godfrey test for autocorrelated errors are studied in situations when the error terms follow either normal or non-normal distributions, and when these errors follow either AR(1) or MA(1) processes. Edgerton and Shukur (1999) studied the properties of the test using normally distributed error terms and when these errors follow an AR(1) process. When the errors follow a non-normal distribution, the performances of the tests deteriorate especially when the tails are very heavy. The performances of the tests become better (as in the case when the errors are generated by the normal distribution) when the errors are less heavy tailed.     

Inferences on regression coefficients in a regression model under heteroscedasticity and robustness with respect to departure from normality

In this paper we consider a simple linear regression model under heteroscedasticity and nonnormality. A statistical test for testing the regression coefficient is then derived by assuming normality for the random disturbances and by applying Welch's method. Some Monte Carlo studies are generated for assessing robustness of this test. By combining Tiku's robust procedure with the new test, a robust but more powerful test is developed.     

Developing a Liu estimator for the negative binomial regression model: method and application

This paper introduces a new shrinkage estimator for the negative binomial regression model that is a generalization of the estimator proposed for the linear regression model by Liu [A new class of biased estimate in linear regression, Comm. Stat. Theor. Meth. 22 (1993), pp. 393–402]. This shrinkage estimator is proposed in order to solve the problem of an inflated mean squared error of the classical maximum likelihood (ML) method in the presence of multicollinearity. Furthermore, the paper presents some methods of estimating the shrinkage parameter. By means of Monte Carlo simulations, it is shown that if the Liu estimator is applied with these shrinkage parameters, it always outperforms ML. The benefit of the new estimation method is also illustrated in an empirical application. Finally, based on the results from the simulation study and the empirical application, a recommendation regarding which estimator of the shrinkage parameter that should be used is given.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.     

Comment on basic structure of the asymptotic theory in dynamic nonlinear econometric models

In this paper we compare the performance of the exogeneity tests of Revankar, Revankar and Hartley and Wu-Hausman for the cases of two and three included endogenous variables. The distribution and power functions are evaluated using the conditional distributions given in Kariya and Hodoshima. Our results indicate that the Revankar's test is the most powerful for large values of the concentration parameter and the Revankar and Hartley test is the most powerful for small values of the concentration parameter.     

Estimation of the coefficient of variation for non-normal model using progressive first-failure-censoring data

The coefficient of variation (CV) is extensively used in many areas of applied statistics including quality control and sampling. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on progressive first-failure-censored data, we study the behavior of the CV of a random variable that follows a Burr-XII distribution. Specifically, we compute the maximum likelihood estimations and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.     

Extended and updated tables for the Friedman rank test

The Friedman's test is used for assessing the independence of repeated experiments resulting in ranks, summarized as a table of integer entries ranging from 1 to k, with k columns and N rows. For its practical use, the hypothesis testing can be derived either from published tables with exact values for small k and N, or using an asymptotic analytical approximation valid for large N or large k. The quality of the approximation, measured as the relative difference of the true critical values with respect those arising from the asymptotic approximation is simply not known. The literature review shows cases where the wrong conclusion could have been drawn using it, although it may not be the only cause of opposite decisions. By Monte Carlo simulation we conclude that published tables do not cover a large enough set of (k, N) values to assure adequate accuracy. Our proposal is to systematically extend existing tables for k and N values, so that using the analytical approximation for values outside it will have less than a prescribed relative error. For illustration purposes some of the tables have been included in the paper, but the complete set is presented as a source code valid for Octave/Matlab/Scilab etc., and amenable to be ported to other programming languages.     

The application of stochastic approximation methods to the bio-assay problem

A new general model for the bio-assay problem is introduced. It is shown that when the slope of the dose-response curve and the median effective dose is known, the Robbins-Monro method yields an asymptotically optimal estimation procedure. Adaptive procedures are discussed for the case of unknown slope. Results of Monte Carlo studies are given.     

Hypothesis testing for the dispersion parameter of the hyper-Poisson regression model

The Poisson distribution is widely used to deal with count data, however, it is unable to capture the dispersion problems. The hyper-Poisson distribution is a particular case of the extended Conway–Maxwell distribution which takes into account the dispersion phenomena of the count data. The main motivation to consider this model is the possibility to link the mean to the regressor variables in very natural way to solve testing problems. So, this paper will be focalized in the gradient statistics to detect dispersions and to compare with the classical likelihood ratio statistic. Two illustrative applications are considered.     

Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model

The use of the Cox proportional hazards regression model is wide-spread. A key assumption of the model is that of proportional hazards. Analysts frequently test the validity of this assumption using statistical significance testing. However, the statistical power of such assessments is frequently unknown. We used Monte Carlo simulations to estimate the statistical power of two different methods for detecting violations of this assumption. When the covariate was binary, we found that a model-based method had greater power than a method based on cumulative sums of martingale residuals. Furthermore, the parametric nature of the distribution of event times had an impact on power when the covariate was binary. Statistical power to detect a strong violation of the proportional hazards assumption was low to moderate even when the number of observed events was high. In many data sets, power to detect a violation of this assumption is likely to be low to modest.     

A test for the parametric form of the variance function in a partial linear regression model

We consider the problem of testing for a parametric form of the variance function in a partial linear regression model. A new test is derived, which can detect local alternatives converging to the null hypothesis at a rate n^-1/2$n^{-1/2}$ and is based on a stochastic process of the integrated variance function. We establish weak convergence to a Gaussian process under the null hypothesis, fixed and local alternatives. In the special case of testing for homoscedasticity the limiting process is a scaled Brownian bridge. We also compare the finite sample properties with a test based on an L²$L^2$-distance, which was recently proposed by You and Chen [2005. Testing heteroscedasticity in partially linear regression models. Statist. Probab. Lett. 73, 61–70].