Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte Carlo experiments illustrate the finite sample performance of the modified portmanteau test.     

Diagnostic Checking for GARCH-Type Models

The asymptotic distributions of squared and absolute residual autocorrelations for GARCH model estimated by M-estimators are derived. Two diagnostic tests are developed which can be used to check the adequacy of GARCH model fitted by using M-estimators. Simulation results show that the empirical sizes of both tests are close to the nominal size in most of the cases. The power of test based on absolute residual autocorrelation is found better than test based on squared residual autocorrelations. Our results reveal that there are estimators that can fit GARCH-type models better than the commonly used quasi-maximum likelihood estimator under non normal errors. An application to real data set is also presented.     

Testing for Uncorrelated Residuals in Dynamic Count Models With an Application to Corporate Bankruptcy

This article proposes new model checks for dynamic count models. Both portmanteau and omnibus-type tests for lack of residual autocorrelation are considered. The resulting test statistics are asymptotically pivotal when innovations are uncorrelated but possibly exhibit higher order serial dependence. Moreover, the tests are able to detect local alternatives converging to the null at the parametric rate T^{? 1/2}, with T the sample size. The finite sample performance of the test statistics are examined by means of Monte Carlo experiments. Using a dataset on U.S. corporate bankruptcies, the proposed tests are applied to check if different risk models are correctly specified. Supplementary materials for this article are available online.     

A Cauchy estimator test for autocorrelation

This article presents a new test for serial correlation in an observed stationary time series. Rather than using the traditional portmanteau tests based on the sample autocorrelation function, we propose a test based on the Cauchy estimator of correlation. A goodness-of-fit statistic for fitted autoregressive moving average models is also derived and the asymptotic distribution of this statistic is quantified. The test can be employed using either this asymptotic distribution or by using Monte-Carlo quantiles. The small sample behaviour is studied via simulation and the Monte-Carlo-based test seems to be more precise. The method is demonstrated on monthly asset returns for Facebook, Incorporated.     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.     

Portmanteau tests based on quadratic forms in the autocorrelations

Many white noise and goodness-of-fit tests are (asymptotically) written as quadratic forms in the ordinary autocorrelation estimates. The properties of such tests are studied by investigating the structure of the matrix of the quadratic form. We suggest to choose the matrix of the quadratic form in such a way that the power is maximized according to the information available about the alternative hypothesis. A simulation study sheds some light on the behavior of the test in finite samples. It is generally found more powerful than the most popular portmanteau tests, i.e., the Box and Pierce and the Ljung and Box tests.     

Simple correlated arma processes

The generalization of the simple correlated autoregressive processes introduced by RISAGEE ( 1980, 1981 ) is presented in the paper. The correlation structure of the model is investigated for the purpose of its identification. Various methods of estimation are discussed. Verification of the model is based on the portmanteau statistics whose behaviour is derived using the results for the simple correlated autoregressive processes and the prin¬ciple 0f Box and PIEBCE concerning the relation of estimated residual correlations in auto-regressive and ARMA models.     

Effects of a single outlier on arma identification

Fox (1972), Box and Tiao (1975), and Abraham and Box (1979) have proposed methods for detecting outliers in time series whose ARMA form is known (or identified). We show that the existence of a single aberrant observation, innovation, or intervention causes an ARMA model to be misidentified using unadjusted autocorrelation (acf) and partial autocorrelation estimates. The magnitude, location, type of outlier, and in some cases the ARMA's parameters, affect the identification outcome. We use variance inflation, signal-to-noise ratios, and acf critical values to determine an ARMA model's susceptibility to misidentifi-cation. Numerical and simulation examples suggest how to iteratively use the outlier detection methods in practice.     

Goodness of Fit Tests for Multivariate Counting Process Models with Applications

In this paper, we develop some distribution-free tests for checking the adequacy of the parametric forms of the intensity processes of a multivariate counting process model. The proposed tests, based in Khmaladze's transformations, are derived from the transforms of weighted aggregated martingale residual processes. The transformed processes converge weakly to independent Gaussian martingales under the assumed model. The distribution-free tests, such as Kolmogorov–Smirnov and Cramer–von Mises type tests, are appropriately defined to account for deviations in each of the transformed aggregated martingale residual processes. Consistency of the tests are discussed. The tests are applicable to multiplicative intensity models such as a competing risks model as well as to non-multiplicative intensity models such as a constant relative or excess mortality model. A small simulation study is conducted and an illustration to a real data example is provided.     

Power transformations to induce normality and their applications

Summary.  Random variables which are positive linear combinations of positive independent random variables can have heavily right-skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson–Hilferty cube root transformation for χ ² random variables as a special case. We also provide some important examples, including test statistics of goodness-of-fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box–Ljung–Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box–Ljung–Pierce portmanteau test, unlike the uncorrected versions.     

Testing for autocorrelation and random-effects in nonlinear mixed effects models based on M-estimation

M-estimation (robust estimation) for the parameters in nonlinear mixed effects models using Fisher scoring method is investigated in the article, which shares some of the features of the existing maximum likelihood estimation: consistency and asymptotic normality. Score tests for autocorrelation and random effects based on M-estimation, together with their asymptotic distribution are also studied. The performance of the test statistics are evaluated via simulations and a real data analysis of plasma concentrations data.     

Improved functional portmanteau tests

Functional time series is a popular method of forecasting in functional data analysis. The Box-Jenkins methodology for model building, with the aim of forecasting, includes three iterative steps of model identification, parameter estimation and diagnostic checking. Portmanteau tests are one of the most popular diagnostic checking tools. In particular, they are applied to find if the residuals of the fitted model are white noise. Gabrys and Kokoszka [Portmanteau test of independence for functional observations. J Am Stat Assoc. 2007;102(480):1338–1348.] proposed a portmanteau test of independence for functional observation based on Box and Pierce's statistic. Their statistic is too sensitive to the lag value, specially when the sample size is small. Here, two modifications of Gabrys and Kokoszka statistic are presented, which have superior properties in small samples. The efficiency of the modified statistics is demonstrated through a simulation study.     

A self-adjusted weighted likelihood ratio test for global clustering of disease

Compared to tests for localized clusters, the tests for global clustering only collect evidence for clustering throughout the study region without evaluating the statistical significance of the individual clusters. The weighted likelihood ratio (WLR) test based on the weighted sum of likelihood ratios represents an important class of tests for global clustering. Song and Kulldorff (Likelihood based tests for spatial randomness. Stat Med. 2006;25(5):825–839) developed a wide variety of weight functions with the WLR test for global clustering. However, these weight functions are often defined based on the cell population size or the geographic information such as area size and distance between cells. They do not make use of the information from the observed count, although the likelihood ratio of a potential cluster depends on both the observed count and its population size. In this paper, we develop a self-adjusted weight function to directly allocate weights onto the likelihood ratios according to their values. The power of the test was evaluated and compared with existing methods based on a benchmark data set. The comparison results favour the suggested test especially under global chain clustering models.     

Testing for Multivariate Autocorrelation

This paper concerns the problem of assessing autocorrelation of multivariate (i.e. systemwise) models. It is well known that systemwise diagnostic tests for autocorrelation often suffers from poor small sample properties in the sense that the true size overstates the nominal size. The failure of keeping control of the size usually stems from the fact that the critical values (used to decide the rejection area) originate from the slowly converging asymptotic null distribution. Another drawback of existing tests is that the power may be rather low if the deviation from the null is not symmetrical over the marginal models. In this paper we consider four quite different test techniques for autocorrelation. These are (i) Pillai's trace, (ii) Roy's largest root, (iii) the maximum F-statistic and (iv) the maximum t² test. We show how to obtain control of the size of the tests, and then examine the true (small sample) size and power properties by means of Monte Carlo simulations.     

A partial score test for difference among heterogeneous populations

In event time data analysis, comparisons between distributions are made by the logrank test. When the data appear to contain crossing hazards phenomena, nonparametric weighted logrank statistics are usually suggested to accommodate different-weighted functions to increase the power. However, the gain in power by imposing different weights has its limits since differences before and after the crossing point may balance each other out. In contrast to the weighted logrank tests, we propose a score-type statistic based on the semiparametric-, heteroscedastic-hazards regression model of Hsieh [2001. On heteroscedastic hazards regression models: theory and application. J. Roy. Statist. Soc. Ser. B 63, 63–79.], by which the nonproportionality is explicitly modeled. Our score test is based on estimating functions derived from partial likelihood under the heteroscedastic model considered herein. Simulation results show the benefit of modeling the heteroscedasticity and power of the proposed test to two classes of weighted logrank tests (including Fleming–Harrington's test and Moreau's locally most powerful test), a Renyi-type test, and the Breslow's test for acceleration. We also demonstrate the application of this test by analyzing actual data in clinical trials.     

Book Reviews

The diagnostic tools examined in this article are applicable to regressions estimated with panel data or cross-sectional data drawn from a population with grouped structure. The diagnostic tools considered include (a) tests for the existence of group effects under both fixed and random effects models, (b) checks for outlying groups, and (c) specification tests for comparing the fixed and random effects models. A group-specific counterpart to the studentized residual is introduced. The methods are illustrated using a hedonic housing price regression.     

Comparison of failure probabilities in the presence of competing risks

Typically, differences in the effect of treatment on competing risks are compared by a weighted log-rank test. This test compares the cause specific hazard rates between the groups. Often the test does not agree with the impressions gained from plots of the cumulative incidence functions. Here we discuss several K-sample tests allowing us to directly compare cumulative incidence functions. These include tests based on the weighted integrated difference between the subdistribution hazards or cumulative incidence functions, Kolmogorov-Smirnov type test, and Renyi type test. In addition to unadjusted comparison techniques, tests based on the regression modeling of the cumulative incidence functions are considered. A simulation study is used to compare the various tests and to assess their power against different alternatives. The methods are illustrated using real data examples.     

Sensitivity of the portmanteau statistic in time series modeling

The portmanteau statistic is commonly used for testing goodness-of-fit of time series models. However, this lack of fit test may depend on one or several atypical observations in the series. We investigate the sensitivity of the portmanteau statistic in the presence of additive outliers. Diagnostics are developed to assess both local and global influence. Three practical examples demonstrate the usefulness of the proposed diagnostics.     

On testing for independence between the innovations of several time series

Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions. In order to obtain simple asymptotic covariance structures, Möbius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramér–von Mises test statistics and tests based on non‐parametric measures. The ranks of the residuals are considered in the new methods, giving test statistics which are asymptotically margin‐free. Generalized cross‐correlations are introduced, extending the concept of cross‐correlation to an arbitrary number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log‐returns of Apple, Intel and Hewlett‐Packard traded on the Nasdaq financial market. The Canadian Journal of Statistics 40: 447–479; 2012 © 2012 Statistical Society of Canada     

Kernel‐based Generalized Cross‐validation in Non‐parametric Mixed‐effect Models

Abstract. Although generalized cross‐validation (GCV) has been frequently applied to select bandwidth when kernel methods are used to estimate non‐parametric mixed‐effect models in which non‐parametric mean functions are used to model covariate effects, and additive random effects are applied to account for overdispersion and correlation, the optimality of the GCV has not yet been explored. In this article, we construct a kernel estimator of the non‐parametric mean function. An equivalence between the kernel estimator and a weighted least square type estimator is provided, and the optimality of the GCV‐based bandwidth is investigated. The theoretical derivations also show that kernel‐based and spline‐based GCV give very similar asymptotic results. This provides us with a solid base to use kernel estimation for mixed‐effect models. Simulation studies are undertaken to investigate the empirical performance of the GCV. A real data example is analysed for illustration.