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.     

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.     

ON THE SQUARED RESIDUAL AUTOCORRELATIONS IN NON-LINEAR TIME SERIES WITH CONDITIONAL HETEROSKEDASTICITY

Abstract. Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.     

On modelling and diagnostic checking of vector periodic autoregressive time series models

Abstract.  Vector periodic autoregressive time series models (PVAR) form an important class of time series for modelling data derived from climatology, hydrology, economics and electrical engineering, among others. In this article, we derive the asymptotic distributions of the least squares estimators of the model parameters in PVAR models, allowing the parameters in a given season to satisfy linear constraints. Residual autocorrelations from classical vector autoregressive and moving-average models have been found useful for checking the adequacy of a particular model. In view of this, we obtain the asymptotic distribution of the residual autocovariance matrices in the class of PVAR models, and the asymptotic distribution of the residual autocorrelation matrices is given as a corollary. Portmanteau test statistics designed for diagnosing the adequacy of PVAR models are introduced and we study their asymptotic distributions. The proposed test statistics are illustrated in a small simulation study, and an application with bivariate quarterly West German data is presented.     

ADEQUACY OF ASYMPTOTIC THEORY FOR GOODNESS-OF-FIT CRITERIA FOR SPECTRAL DISTRIBUTIONS

Abstract. Any of the Cramér-von Mises, Anderson-Darling, and Kolmogorov-Smirnov statistics can be used to test the null hypothesis that the standardized spectral distribution of a stationary stochastic process is a specified one. The asymptotic distributions of the criteria have been characterized (Anderson, 1993). They are the same as for probability distributions if the observations are independent (all autocorrelations zero), but are different when there is dependence. In this paper simulation with 10000 replications has been used to determine the distributions of the criteria for samples of size 6, 10, 30 and 100 when the observations are independent. These empirical distributions have been compared with the asymptotic distributions in order to ascertain the sample sizes necessary for using the asymptotic tables. For practical purposes they are 30 for the Cramér-von Mises and Kolmogorov statistics and over 100 for Anderson-Darling.     

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Bartlett's formula for a general class of nonlinear processes

Abstract.  A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.     

ASYMPTOTIC PROPERTIES OF THE SAMPLE AUTOCORRELATIONS AND PARTIAL AUTOCORRELATIONS OF A MULTIPLICATIVE ARIMA PROCESS

Abstract. We shall investigate the asymptotic behaviour of the sample autocorrelations and partial autocorrelations of a multiplicative ARIMA process and derive their limiting distributions. Some simulations are presented to illustrate the results obtained.     

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.     

Goodness-of-fit tests for autoregressive processes

Goodness-of-fit tests for autoregressive processes can be based on the difference betwe en the empirical standardized spectral distribution of an observed time series and the standardized spectral distribution of the autoregressive process with parameters estimated from the series. The asymptotic covariance function of this difference, considered as a stochastic process on [0, π], is found. Methods to compute the asymptotic distribution of the Cramer--von Mises statistic are given.     

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.     

ARMA MODELLING WITH NON-GAUSSIAN INNOVATIONS

Abstract. The problem of modelling time series driven by non-Gaussian innovations is considered. The asymptotic normality of the maximum likelihood estimator is established under some general conditions. The distribution of the residual autocorrelations is also obtained. This gives rise to a potentially useful goodness-of-fit statistic. Applications of the results to two important cases are discussed. Two real examples are considered.     

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.     

DIAGNOSTIC CHECKING ARMA TIME SERIES MODELS USING SQUARED-RESIDUAL AUTOCORRELATIONS

Abstract. Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.     

Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

We derive tests of stationarity for univariate time series by combining change‐point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank‐based tests of stationarity by combining cumulative sum change‐point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second‐order characteristics are proposed next. The finite‐sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large‐scale Monte Carlo experiments, and illustrations on two real datasets are provided. Extensions to multi‐variate time series are briefly discussed as well.     

Estimation of stationary autoregressive models with the Bayesian LASSO

This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log‐likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate‐wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper‐parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method.     

Residual Autocorrelation Distribution in the Validation Data Set

Testing model performance on a data set other than the data set used for estimation is common practice in econometrics, technological stochastic modelling and environmetrics. In this paper, using an ARMAX model, the asymptotic distribution of the residual autocorrelations in the validation data set is given and a χ² test for overall residual incorrelation is considered.     

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.     

Efficient use of higher-lag autocorrelations for estimating autoregressive processes

The Yule–Walker estimator is commonly used in time-series analysis, as a simple way to estimate the coefficients of an autoregressive process. Under strong assumptions on the noise process, this estimator possesses the same asymptotic properties as the Gaussian maximum likelihood estimator. However, when the noise is a weak one, other estimators based on higher-order empirical autocorrelations can provide substantial efficiency gains. This is illustrated by means of a first-order autoregressive process with a Markov-switching white noise. We show how to optimally choose a linear combination of a set of estimators based on empirical autocorrelations. The asymptotic variance of the optimal estimator is derived. Empirical experiments based on simulations show that the new estimator performs well on the illustrative model.     

Threshold quantile autoregressive models

This article studies estimation and asymptotic properties of Threshold Quantile Autoregressive processes. In particular, we show the consistency of the threshold and slope parameter estimators for each quantile and regime, and derive the asymptotic normality of the slope parameter estimators. A Monte Carlo experiment shows that the standard ordinary least squares estimation method is not able to detect important nonlinearities produced in the quantile process. The empirical study concentrates on modelling the dynamics of the conditional distribution of unemployment growth after the second world war. The results show evidence of important heterogeneity associated with unemployment and strong asymmetric persistence of unemployment growth in the higher quantiles.