Improved multivariate portmanteau test

A new portmanteau diagnostic test for vector autoregressive moving average (VARMA) models that is based on the determinant of the standardized multivariate residual autocorrelations is derived. The new test statistic may be considered an extension of the univariate portmanteau test statistic suggested by Peňa and Rodríguez (2002) . The asymptotic distribution of the test statistic is derived as well as a chi‐square approximation. However, the Monte–Carlo test is recommended unless the series is very long. Extensive simulation experiments demonstrate the usefulness of this test as well as its improved power performance compared to widely used previous multivariate portmanteau diagnostic check. Two illustrative applications are given.     

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.     

A JOINT PORTMANTEAU TEST FOR CONDITIONAL MEAN AND VARIANCE TIME‐SERIES MODELS

In this article, we propose a new joint portmanteau test for checking the specification of parametric conditional mean and variance functions of linear and nonlinear time‐series models. The use of a joint test is motivated for complete control of the asymptotic size since marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new test is based on an asymptotically distribution‐free transformation on the sample autocorrelations of both normalized residuals and squared normalized residuals. This makes it unnecessary to full detail the asymptotic properties of the estimates used to obtain residuals, which could be inefficient two‐step ones, avoiding also choices of maximum lag parameters increasing with sample length to control asymptotic size. The robust versions of the new test also properly account for higher‐order moment dependence at a reduced cost. The finite‐sample performance of the new test is compared with that of well‐known tests through simulations.     

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.     

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).     

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.     

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.     

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.     

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.     

Testing Parameter Change in General Integer‐Valued Time Series

We consider the structural change in a class of discrete valued time series, which the conditional distribution belongs to the one‐parameter exponential family. We propose a change point test based on the maximum likelihood estimator of the model's parameter. Under the null hypothesis (of no change), the test statistic converges to a well‐known distribution, allowing the calculation of the critical value of the test. The test statistic diverges to infinity under the alternative, meaning that the test has asymptotic power one. Some simulation results and real data applications are reported to show the effectiveness of the proposed procedure.     

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.     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

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.     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.     

Recursive adjustment,unit root tests and structural breaks

In this paper, we introduce unit root tests for time series with a potential structural break computed from test regressions in which the deterministic components have been recursively adjusted. We present finite sample critical values as well as Monte Carlo results on the size and power performance of the new procedures, and compare these with other available tests in the literature, such as OLS and quasi‐differenced based tests (see, for instance, Perron, (1997) Perron and Rodriguez, (2003) and Carrion‐i‐Silvestre et al. (2009) ). The small sample behaviour of the tests is evaluated in a known and an unknown break date context allowing for negligible and non‐negligible initial conditions. In the unknown break date case, two break date estimation procedures are considered, one based on the minimum unit root t‐statistic and the other based on the minimum sum of squared residuals obtained from a regression on a set of deterministic variables. The size and power performance of the recursive adjustment based procedure in the unknown break date case is encouraging. A further result of this paper relates to the aditional finite sample evidence on the performance of quasi‐differenced unit root tests, complementing the results in Perron and Rodriguez (2003) .     

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.     

High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

Abstract. In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change‐point problems and goodness‐of‐fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque–Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model.     

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.     

THE ESTIMATION AND PREDICTION OF SUBSET BILINEAR TIME SERIES MODELS WITH APPLICATIONS

Abstract. In this paper we define subset bilinear time series models, and then describe an algorithm for the estimation of these models. It is also pointed out that for this class of non-linear time series models, it is possible to obtain optimal several step predictors. The estimation technique of these models is illustrated with respect to three time series, and the optimal several steps ahead forecasts of these time series models are calculated. A comparison of these forecasts is made with the forecasts obtained by the best linear autoregressive and threshold autoregressive models. The residuals obtained from the models are tested for independence and Gaussianity using higher order moments.     

A test for second‐order stationarity of a time series based on the discrete Fourier transform

We consider a zero mean discrete time series, and define its discrete Fourier transform (DFT) at the canonical frequencies. It can be shown that the DFT is asymptotically uncorrelated at the canonical frequencies if and only if the time series is second‐order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic has approximately a chi‐square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a generalized non‐central chi‐square, where the non‐centrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power.