Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

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.     

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n ^?3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n ^?1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.     

Heteroscedasticity and/or autocorrelation diagnostics in nonlinear models with AR(1) and symmetrical errors

In this paper, we discuss tests of heteroscedasticity and/or autocorrelation in nonlinear models with AR(1) and symmetrical errors. The symmetrical errors distribution class includes all symmetrical continuous distributions, such as normal, Student-t, power exponential, logistic I and II, contaminated normal, so on. First, score test statistics and their adjustment forms of heteroscedasticity are derived. Then, the asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. The properties of test statistics are investigated through Monte Carlo simulations. Finally, a real data set is used to illustrate our test methods.     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

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.     

Asymptotic properties of the score test for varying dispersion in exponential family nonlinear models

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].     

Tests for the order of integration against higher order integration

Locally best invariant tests for the null hypothesis of I(p) against the alternative hypothesis of I(q), < q, are developed for models with independent normal errors. The tests are semiparametrically extended for models with autocorrelated errors. The method is illustrated by two real data sets in terms of double unit roots. The proposed tests can be used for determining integration orders of nonstationary time series.     

Higher-order asymptotics under model misspecification

Most of the higher-order asymptotic results in statistical inference available in the literature assume model correctness. The aim of this paper is to develop higher-order results under model misspecification. The density functions to O(n^?3/2) of the robust score test statistic and the robust Wald test statistic are derived under the null hypothesis, for the scalar as well as the multiparameter case. Alternate statistics which are robust to O(n^?3/2) are also proposed.     

Non Null Asymptotic Distributions of Some Criteria in Censored Exponential Regression

In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n^{? 1/2} of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.     

Local power of some tests in exponential family nonlinear models

In this paper we obtain asymptotic expansions up to order n^−1/2 for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in exponential family nonlinear models (Cordeiro and Paula, 1989), under a sequence of Pitman alternatives. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the dispersion parameter, thus generalising the results given in Cordeiro et al. (1994) and Ferrari et al. (1997). We also present Monte Carlo simulations in order to compare the finite-sample performance of these tests.     

A Joint Score Test for Heteroscedasticity in the Two Way Error Components Model

This article extends the work by Holly and Gardiol (2000) (A score test for individual heteroscedasticity in a one-way error component model. In: Krishnakumar, J., Ronchetti, E., Eds. Panel Data Econometrics: Future Directions. Elsevier, North-Holland, Amsterdam, pp. 199–211, Ch. 10) to the two-way error components model. It deals exclusively with a joint heteroscedasticity test by first deriving Rao's efficient score statistics. Then, based on appropriate set of assumptions, we deduce the asymptotic distribution of the score under contiguous alternatives. Finally, we provide the expression for the score test statistic in the presence of heteroscedasticity and discuss its asymptotic local power.     

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σ _u ² . The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σ _u ² , both estimators have identical asymptotic covariance matrices up to the order of σ _u ² . We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σ _u ² ).     

Small-sample likelihood inference in extreme-value regression models

We deal with a general class of extreme-value regression models introduced by Barreto-Souza and Vasconcellos [Bias and skewness in a general extreme-value regression model, Comput. Statist. Data Anal. 55 (2011), pp. 1379–1393]. Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as χ² with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite-sample performance of the three classical tests (likelihood ratio, Wald, and score), the gradient test that has been recently proposed by Terrell [The gradient statistic, Comput. Sci. Stat. 34 (2002), pp. 206–215], and the adjusted likelihood ratio test obtained in this article. Our simulations favour the latter. Applications of our results are presented.     

On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qL?L test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald (SW) test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the SW test does not show this non-monotonicity and its power is far superior to the qL?L test when the break is not very small. We claim that the optimality of the qL?L test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the SW test indeed dominates the qL?L test and, in many cases, the latter has zero relative asymptotic efficiency.     

Conditional Score Tests for Heteroscedasticity in the Two-Way Error Components Model

This paper extends the one-way heteroskedasticity score test of Holly and Gardiol (2000, In: Krishnakumar, J, Ronchetti, E (Eds.), Panel Data Econometrics: Future Directions, North-Holland, Amsterdam, pp. 199–211) to two conditional Lagrange Multiplier (LM) tests of heteroskedasticity under contiguous alternatives within the two-way error components model framework. In each case, the derivation of Rao's efficient score statistics for testing heteroskedasticity is first obtained. Then, based on a specific set of assumptions, the asymptotic distribution of the score under contiguous alternatives is established. Finally, the expression for the score test statistic in the presence of heteroskedasticity and related asymptotic local powers of these score test statistics are derived and discussed.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. 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.     

An observation on regression-based specification tests

The use of regression-based specification tests, such as the nR² form of the Lagrange Multiplier test, has become quite widespread over the last 20 years. The popularization of the nR² form of the Lagrange Multiplier (LM) test, perhaps the most widely used class of regression-based tests, has come about in large part from the ease of its application to many tests of nonlinear restrictions and its asymptotic equivalence to Likelihood Ratio and Wald tests. Properly performed, these regression-based tests invariably include regressors which are orthogonal by construction to the dependent variable of the regression. The purpose of this paper is to motivate the inclusion of such variables by investigating implications for the test size and power if these regressors are erroneously omitted. It is straightforward to show that both the size and power of the test are adversely affected by omitting these regressors.     

Maximum studentized score tests for the detection of outliers in time series regression models

Efficient score tests exist among others, for testing the presence of additive and/or innovative outliers that are the result of the shifted mean of the error process under the regression model. A sample influence function of autocorrelation-based diagnostic technique also exists for the detection of outliers that are the result of the shifted autocorrelations. The later diagnostic technique is however not useful if the outlying observation does not affect the autocorrelation structure but is generated due to an inflation in the variance of the error process under the regression model. In this paper, we develop a unified maximum studentized type test which is applicable for testing the additive and innovative outliers as well as variance shifted outliers that may or may not affect the autocorrelation structure of the outlier free time series observations. Since the computation of the p-values for the maximum studentized type test is not easy in general, we propose a Satterthwaite type approximation based on suitable doubly non-central F-distributions for finding such p-values [F.E. Satterthwaite, An approximate distribution of estimates of variance components, Biometrics 2 (1946), pp. 110–114]. The approximations are evaluated through a simulation study, for example, for the detection of additive and innovative outliers as well as variance shifted outliers that do not affect the autocorrelation structure of the outlier free time series observations. Some simulation results on model misspecification effects on outlier detection are also provided.