Portmanteau tests for linearity of stationary time series

This article considers the problem of testing for linearity of stationary time series. Portmanteau tests are discussed which are based on generalized correlations of residuals from a linear model (that is, autocorrelations and cross-correlations of different powers of the residuals). The finite-sample properties of the tests are assessed by means of Monte Carlo experiments. The tests are applied to 100 time series of stock returns.     

A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

Bootstrap estimates of the sample bivariate autocorrelation and partial autocorrelation distributions

This paper shows how the bootstrap method can be used to estimate the joint distribution of sample autocorrelations and partial autocorrelations. The exact joint distribution of sample autocorrelations is mathematically intractable and attempts at workable approximations are difficult and rely on special assumptions. The bootstrap offers an accurate solution to this problem without requiring special assumptions and in a way that avoids theoretical difficulties. The bootstrap-estimated joint distributions of the autocorrelations and partial autocorrelations of time series are shown to lead to better ARMA model identification. This is demonstrated using simulated series.     

A comparative study on a permutation statistic

We studied asymptotic distribution and finite sample properties of a randomly weighted permutation statistic. The asymptotic normality and the finite sample simulations derived from our studies provided theoretical and numerical justifications for distributional assumption of many useful test statistics used in identifying spatial autocorrelations of mapped data. We compared a new method in computing the mean and the approximated variance of the randomly weighted D statistic, a special permutation statistic, with the Walter’s conditional method. In the numerical illustration of the method, we calculated the standardized values of the D statistic by subtracting the mean from the D statistic and dividing the difference by the standard deviation for the standardized mortality ratios (SMRs) and the life expectancies among the 48 states of the continental USA. Spatial autocorrelations of the SMRs and the life expectancies were found to be statistically significant.     

A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process

In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.     

Generalized portmanteau statistics and tests of randomness

This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented in the paper were derived by usig a symbolic manipulation program.     

Asymptotic variance–covariance matrix of sample autocorrelations for threshold-asymmetric GARCH processes

In the field of financial time series, threshold-asymmetric conditional variance models can be used to explain asymmetric volatilities [C.W. Li and W.K. Li, On a double-threshold autoregressive heteroscedastic time series model, J. Appl. Econometrics 11 (1996), pp. 253–274]. In this paper, we consider a broad class of threshold-asymmetric GARCH processes (TAGARCH, hereafter) including standard ARCH and GARCH models as special cases. Since sample autocorrelation function provides a useful information to identify an appropriate time-series model for the data, we derive asymptotic distributions of sample autocorrelations both for original process and for squared process. It is verified that standard errors of sample autocorrelations for TAGARCH models are significantly different from unity for lower lags and they are exponentially converging to unity for higher lags. Furthermore they are shown to be asymptotically dependent while being independent of standard GARCH models. These results will be interesting in the light of the fact that TAGARCH processes are serially uncorrelated. A simulation study is reported to illustrate our results.     

A note on the distribution of mean square error of forecasting

Simulation has been a very important and widely used method in the study of misspecification or order determination in time series analysis. Mean square error of forecasting (MSEF) has been a major criterion for comparing the performance of different models. In simulation studies, standard deviations of MSEF's are calculated from the computed values of the MSEF's, In this note, the distribution of MSEF from simulation studies is established. Exact variance of the MSEF can be obtained from the prespecified values of the model selected for simulation. This variance should be a more appropriate criterion for evaluating the performance between models.     

Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data

In this paper, we propose a hidden Markov model for the analysis of the time series of bivariate circular observations, by assuming that the data are sampled from bivariate circular densities, whose parameters are driven by the evolution of a latent Markov chain. The model segments the data by accounting for redundancies due to correlations along time and across variables. A computationally feasible expectation maximization (EM) algorithm is provided for the maximum likelihood estimation of the model from incomplete data, by treating the missing values and the states of the latent chain as two different sources of incomplete information. Importance-sampling methods facilitate the computation of bootstrap standard errors of the estimates. The methodology is illustrated on a bivariate time series of wind and wave directions and compared with popular segmentation models for bivariate circular data, which ignore correlations across variables and/or along time.     

A note on diagnostic checking of the double autoregressive model

The double autoregressive model finds its use in the modelling of conditional heteroscedasticity of time series data. In view of its growing popularity, the goodness-of-fit of the model is examined. The asymptotic distributions of the residual and squared residual autocorrelations are derived. Two test statistics are then constructed which can be used to measure the adequacy of the conditional mean and conditional variance components of a fitted model. Our goodness-of-fit tests out-perform other benchmark tests such as the Ljung–Box test in simulation studies. To illustrate the testing procedure, the model is fitted to the weekly log-return series of the Hang Seng Index.     

Imputation of missing variance data using non-linear mixed effects modelling to enable an inverse variance weighted meta-analysis of summary-level longitudinal data: a case study

Missing variances, on the basis of the summary-level data, can be a problem when an inverse variance weighted meta-analysis is undertaken. A wide range of approaches in dealing with this issue exist, such as excluding data without a variance measure, using a function of sample size as a weight and imputing the missing standard errors/deviations. A non-linear mixed effects modelling approach was taken to describe the time-course of standard deviations across 14 studies. The model was then used to make predictions of the missing standard deviations, thus, enabling a precision weighted model-based meta-analysis of a mean pain endpoint over time. Maximum likelihood and Bayesian approaches were implemented with example code to illustrate how this imputation can be carried out and to compare the output from each method. The resultant imputations were nearly identical for the two approaches. This modelling approach acknowledges the fact that standard deviations are not necessarily constant over time and can differ between treatments and across studies in a predictable way.     

Portmanteau test for conditional heteroscedasticity, using ranks of squared residuals

Squared residual autocorrelations have been found useful in detecting departures from linearity in time series models. This is especially the case with data exhibiting heterogeneity in variances. A rank test is proposed which is much more robust than its parametric counterpart.     

On the partial autocorrelations of once integrated autoregressive-moving average processes

An observation, from practical experience with analysing univariate time series, suggests a simple relationship between the partial autocorrelations of a process realisation which requires first differencing, and those for that same sequence of differences. The asymptotic result is proved for a general once integrated autoregressive process, but an extension to twice integrated processes is shown not to be relevant for finite samples. The results are illustrated with examples from the literature.     

Sample moments of the autocorrelations of moving average processes and a modification to bartlett'sasymptotic variance formula

In this paper we express the sample autocorrelations for a moving average process of order q as a function of its own theoretical autocorrelations and the sample autocorrelations for the generating white noise series. Approximate analytic expressions are then obtained forthe moments of the sample autocorrelations of the moving average process.

Using these expressions, together with numerical evidence, we show that Bartlett's asymptotic formula for the variance of the sample autocorrelations of moving average processes, which is used widely in identifying these processes, is a large overestimate when considering finitesample sizes.

Our approach is for motivational purposes and so is purely formal, the amount of mathematics presented being kept to a minimum.     

A note on time series model specification in the presence of outliers

We investigate the usefulness of sample autocorrelations and partial autocorrelations as model specification tools when the observed time series is contaminated by an outlier. The results indicate that the specification power of these statistics could be significantly jeopardized by an additive outlier. On the other hand, an innovational outlier seems to cause no harm to them.     

Testing for autocorrelation among common stock returns

We consider the null distribution of autocorrelation coefficients for stock returns when the variance of the returns is infinite. We show that the empirical autocorrelations then tend to zero faster than in the standard case and that they tend, after suitable normalisation, in distribution to a rather complicated nonnormal law. An empirical application to the 14 most busy German stocks reveals that the significance of observed correlations is thereby in general reduced.     

On aggregation of strongly dependent time series

We consider cross-sectional aggregation of time series with long-range dependence. This question arises for instance from the statistical analysis of networks where aggregation is defined via routing matrices. Asymptotically, aggregation turns out to increase dependence substantially, transforming a hyperbolic decay of autocorrelations to a slowly varying rate. This effect has direct consequences for statistical inference. For instance, unusually slow rates of convergence for nonparametric trend estimators and nonstandard formulas for optimal bandwidths are obtained. The situation changes, when time-dependent aggregation is applied. Suitably chosen time-dependent aggregation schemes can preserve a hyperbolic rate or even eliminate autocorrelations completely.     

Simulating dependent discrete data

This article describes a method for simulating n-dimensional multivariate non-normal data, with emphasis on count-valued data. Dependence is characterized by either Pearson correlations or Spearman correlations. The simulation is accomplished by simulating a vector of correlated standard normal variates. The elements of this vector are then transformed to achieve the target marginal distributions. We prove that the method corresponds to simulating data from a multivariate Gaussian copula. The simulation method does not restrict pairwise dependence beyond the limits imposed by the marginal distributions and can achieve any Pearson or Spearman correlation within those limits. Two examples are included. In the first example, marginal means, variances, Pearson correlations, and Spearman correlations are estimated from the epileptic seizure data set of Diggle et al. [P. Diggle, P. Heagerty, K.Y. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press, Oxford, 2002]. Data with these means and variances are simulated to first achieve the estimated Pearson correlations and then achieve the estimated Spearman correlations. The second example is of a hypothetical time series of Poisson counts with seasonal mean ranging between 1 and 9 and an autoregressive(1) dependence structure.     

Distribution asymptotique des autocorrélations ďun processus saisonnier non stationnaire

We examine the behaviour of the sample autocorrelations of a seasonal time series for which the first difference of order s (s ≥ 1) is stationary. The asymptotic distribution of the autocorrelations r'(k) based on uncentered data and of the autocorrelations r(k) based on centered data are derived. In each case, the asymptotic distribution is characterized as a function of the lag k and the parameters of the process. A simulation study was conducted in order to investigate the rate of convergence of the finite sample distributions of r(k) and r'(k) to their asymptotic counterparts and to evaluate the effect of centering or not centering the data on the distribution of autocorrelations.     

One-step M-estimators in the linear model,with dependent errors

We consider the problem of robust M-estimation of a vector of regression parameters, when the errors are dependent. We assume a weakly stationary, but otherwise quite general dependence structure. Our model allows for the representation of the correlations of any time series of finite length. We first construct initial estimates of the regression, scale, and autocorrelation parameters. The initial autocorrelation estimates are used to transform the model to one of approximate independence. In this transformed model, final one-step M-estimates are calculated. Under appropriate assumptions, the regression estimates so obtained are asymptotically normal, with a variance-covariance structure identical to that in the case in which the autocorrelations are known a priori. The results of a simulation study are given. Two versions of our estimator are compared with the L₁ -estimator and several Huber-type M-estimators. In terms of bias and mean squared error, the estimators are generally very close. In terms of the coverage probabilities of confidence intervals, our estimators appear to be quite superior to both the L₁-estimator and the other estimators. The simulations also indicate that the approach to normality is quite fast.