Chi‐squared portmanteau tests for structural VARMA models with uncorrelated errors

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving average models with uncorrelated errors. Under the assumption that errors are uncorrelated but non‐independent, it is known that the Ljung–Box (or Box–Pierce) portmanteau test statistic is asymptotically distributed as a weighted sum of chi‐squared random variables which can be far from the chi‐square distribution usually employed. We therefore propose a new portmanteau statistic that is asymptotically chi‐squared even in the presence of uncorrelated but non‐independent errors. Monte Carlo experiments illustrate the finite sample performance for the proposed portmanteau test.     

Mixed Portmanteau Tests for Time‐Series Models

Abstract. This paper obtains the joint limiting distribution of residuals and squared residuals of a general time‐series model. Based on this, we propose a mixed portmanteau statistic for testing the adequacy of fitted time‐series models. In some cases, it is shown that this statistic can be simply approximated by the sum of well‐known portmanteau statistics. The finite‐sample performance of the new test is compared with those of well‐known tests through simulations.     

An Improvement of the Portmanteau Statistic

Abstract. The portmanteau statistic is based on the first m‐residual autocorrelations, and is used for diagnostic checks on the adequacy of fit of a model. In this article, we propose a modified portmanteau statistic with a correction term that allows for the use of small values of m for the chi‐squared test. For this modification, we take a different approach to that suggested by Ljung [Biometrika (1986), Vol. 73, pp. 725–30]. Their empirical behaviour is clarified using asymptotic theory.     

A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series

Abstract. In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite‐order vector autoregressive (VAR(∞)) series, which is an extension of Hong's [Biometrika (1996c) vol. 83, 615–625] univariate results. We first filter each series by a finite‐order autoregression and the test statistic is a standardized version of a weighted sum of quadratic forms in the residual cross‐correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393–411], the asymptotic distribution of the test statistic is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross‐correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy [Statistics and Probability Letters (2006) vol. 76, pp. 58–68] that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented.     

Multivariate Portmanteau Test For Autoregressive Models with Uncorrelated but Nonindependent Errors

Abstract. We study the asymptotic behaviour of the least squares estimator, of the residual autocorrelations and of the Ljung–Box (or Box–Pierce) portmanteau test statistic for multiple autoregressive time series models with nonindependent innovations. Under mild assumptions, it is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi‐squared random variables. When the innovations exhibit conditional heteroscedasticity or other forms of dependence, this asymptotic distribution can be quite different from that of models with independent and identically distributed innovations. Consequently, the usual chi‐squared distribution does not provide an adequate approximation to the distribution of the Box–Pierce goodness‐of‐fit portmanteau test in the presence of nonindependent innovations. 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.     

On multiple portmanteau tests*

Abstract. The portmanteau statistic based on the first m residual autocorrelations is used for diagnostic checks on the adequacy of fitting a model with varying m. In this article, we propose an approximation of the joint probability of multiple portmanteau tests with different degrees of freedom (DF). This distribution is easy to compute when all DF are even integers; its empirical behaviour is clarified in terms of asymptotic theory.     

Nonparametric methods for analysing the accuracy of diagnostic tests with multiple readers

The evaluation of diagnostic agents or imaging procedures is governed by the same scientific and regulatory rules as that of other medical products. Receiver operating characteristic (ROC) curves, and especially the area under these ROC curves, are indices for the accuracy of a diagnostic test for continuous as well as ordinal data. The methodology of multivariate rank statistics for the nonparametric Behrens-Fisher problem is used to evaluate the accuracy of a diagnostic test in a complex factorial design with repeated measurements. Hypotheses are formulated by means of relative treatment effects and are tested by a multivariate extension of the Mann-Whitney statistic in a heteroscedastic model. The application of this method is demonstrated by the analysis of a data set from a diagnostic clinical trial.     

Portmanteau tests for ARMA models with infinite variance

Abstract. Autoregressive and moving‐average (ARMA) models with stable Paretian errors are some of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking of fitted models has not been addressed. In this article, we develop portmanteau tests for checking the randomness of a time series with infinite variance and for ARMA diagnostic checking when the innovations have infinite variance. It is assumed that least squares or an asymptotically equivalent estimation method, such as Gaussian maximum likelihood, is used. It is also assumed that the distribution of the innovations is identically and independently distributed (i.i.d.) stable Paretian. It is seen via simulation that the proposed portmanteau tests do not converge well to the corresponding limiting distributions for practical series length so a Monte Carlo test is suggested. Simulation experiments show that the proposed Monte Carlo test procedure works effectively. Two illustrative applications to actual data are provided to demonstrate that an incorrect conclusion may result if the usual portmanteau test based on the finite variance assumption is used.     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).     

DISTRIBUTION OF RESIDUAL AUTOCORRELATIONS IN NONSTATIONARY AUTOREGRESSIVE PROCESSES

Abstract. The residual autocorrelations in nonstationary autoregressive processes with autoregressive characteristic roots on the unit circle are considered. Limiting distributions of the residual autocovariances and the residual autocorrelations are shown to be the same as the limiting distributions when parameters are estimated with all roots on the unit circle known. The portmanteau statistic is shown to have a x² limiting distribution. The Canadian lynx data set is analysed to illustrate our theory. The portmanteau test seems also useful when the characteristic roots are close to the unit circle.     

Diagnostic checking of nonlinear multivariate time series with multivariate arch errors

Multivariate time series with multivariate ARCH errors have been found useful in many applications. In order to check the adequacy of these models, we define the sum of squared (standardized) residual autocorrelations and derive their asymptotic distribution. The results are used to derive several new multivariate portmanteau tests. Simulation results show that the asymptotic standard errors are quite satisfactory compared with empirical standard errors and that the tests have reasonable empirical size and power. The distribution of the standardized residual autocorrelations is also derived.     

Distributions for residual autocovariances in parsimonious periodic vector autoregressive models with applications

The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking‐Li‐McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi‐square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi‐square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.     

A mixed portmanteau test for ARMA‐GARCH models by the quasi‐maximum exponential likelihood estimation approach

This paper investigates the joint limiting distribution of the residual autocorrelation functions and the absolute residual autocorrelation functions of ARMA‐GARCH models. This leads a mixed portmanteau test for diagnostic checking of the ARMA‐GARCH model fitted by using the quasi‐maximum exponential likelihood estimation approach in Zhu and Ling (2011) . Simulation studies are carried out to examine our asymptotic theory, and assess the performance of this mixed test and other two portmanteau tests in Li and Li (2008) . A real example is given.     

A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.     

LINEAR AND QUADRATIC SERIAL RANK TESTS FOR RANDOMNESS AGAINST SERIAL DEPENDENCE

Abstract. The problem of testing randomness against specified and unspecified contiguous ARMA alternatives is considered. In the case of specified alternative, the linear serial rank tests proposed by the authors (Hallin, Ingenbleek and Puri 1985) are shown to be asymptotically most powerful within the class of all possible tests (at the required level). In the case of unspecified alternative, however, any of the above optimal tests is unfortunately completely insensitive against a whole subclass of the alternative. Quadratic serial rank statistics , providing tests of the χ²-type are therefore introduced. Their asymptotic distributions are derived, under the null hypothesis as well as under contiguous ARMA alternatives. The asymptotically maximin most powerful quadratic serial rank tests (for a given density) are then obtained. Because of their close similarity with the Box-Pierce parametric test, we call them rank portmanteau tests . The asymptotic relative efficiencies (AREs) of the rank portmanteau tests with respect to one another and their AREs with respect to the corresponding Box-Pierce and quadratic Spearman tests are derived.     

ON WEIGHTED PORTMANTEAU TESTS FOR TIME‐SERIES GOODNESS‐OF‐FIT

Recent work in the literature has shown weighted variants of the classic portmanteau test for time series can be more powerful in many situations. In this article, we study the asymptotic distribution of weighted sums of the squared residual autocorrelations where both the sample size n and maximum lag of the statistic m grow large. Several weighting schemes are introduced, including a data‐adaptive statistic in which the weights are determined by a function of the sample partial autocorrelations. These statistics can provide more power than other portmanteau tests found in the literature and are much less sensitive to the choice of the maximum correlation lag. The efficacy of the proposed methods is further demonstrated through an analysis of Australian red wine sales.     

Moment tests for window length selection in singular spectrum analysis of short– and long–memory processes

In this paper, we propose a new methodology for selecting the window length in Singular Spectral Analysis in which the window length is determined from the data prior to the commencement of modelling. The selection procedure is based on statistical tests designed to test the convergence of the autocovariance function. A classical time series portmanteau type statistic and two test statistics derived using a conditional moment principle are considered. The first two are applicable to short–memory processes, and the third is applicable to both short– and long–memory processes. We derive the asymptotic null and alternative distributions of the statistics under fairly general regularity conditions. Consistency of the tests implies that the selection criteria will identify true convergence with a finite window length with probability arbitrarily close to one as the sample size increases. Results obtained using Monte Carlo simulation point to the relevance of the asymptotic theory and show that the conditional moment tests will choose a window length consistent with the Whitney embedding theorem. Application to observations on the Southern Oscillation Index shows how observed experimental behaviour can be reflected in features seen with real world data sets.     

Self-Weighted Lad-Based Inference for Heavy-Tailed Continuous Threshold Autoregressive Models

This note investigates the self-weighted least absolute deviation estimation (SLADE) of a heavy-tailed continuous threshold autoregressive (TAR) model. It is shown that the SLADE is strongly consistent and asymptotically normal. The SLADE is global in the sense that the convergence rate is first obtained before deriving its limiting distribution. Moreover, a test for the continuity of TAR model is considered. A sign-based portmanteau test is developed for diagnostic checking. An empirical example is given to illustrate the usefulness of our method. Combined with the results (Yang and Ling, 2017), a complete asymptotic theory on the SLADE of a heavy-tailed TAR model is established. This enriches asymptotic theory of statistical inference in threshold models.     

Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.     

Tests for conditional heteroscedasticity of functional data

Functional data objects derived from high-frequency financial data often exhibit volatility clustering. Versions of functional generalized autoregressive conditionally heteroscedastic (FGARCH) models have recently been proposed to describe such data, however so far basic diagnostic tests for these models are not available. We propose two portmanteau type tests to measure conditional heteroscedasticity in the squares of asset return curves. A complete asymptotic theory is provided for each test. We also show how such tests can be adapted and applied to model residuals to evaluate adequacy, and inform order selection, of FGARCH models. Simulation results show that both tests have good size and power to detect conditional heteroscedasticity and model mis-specification in finite samples. In an application, the tests show that intra-day asset return curves exhibit conditional heteroscedasticity. This conditional heteroscedasticity cannot be explained by the magnitude of inter-daily returns alone, but it can be adequately modeled by an FGARCH(1,1) model.