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.     

Bayesian monitoring of local residual autocorrelations taking into account the run-length

This paper proposes new two-sided monitoring algorithms for detecting the presence of first order residual autocorrelations in Dynamic Normal Models. The methodology uses a Bayesian decision approach with loss function which takes into account the run-length of the process. The power and mean run-length of the proposed algorithms are analysed by Monte Carlo methods. The results obtained improve those corresponding to the monitoring algorithm for residual autocorrelations proposed in Gargallo and Salvador [2003. Monitoring residual autocorrelations in dynamic linear models. Comm. Statist. Simulation Comput. 32(4), 1079–1104.] with respect to the run-length, and also exhibit more homogeneous behaviour.     

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.     

Partial and inverse autocorrelations in portmanteau-type tests for time series

We present a decomposition of the correlation coefficient between x_t and x_t?k into three terms that include the partial and inverse autocorrelations. The first term accounts for the portion of the autocorrelation that is explained by the inner variables {x_t?1 , x_t?2 , …, x _{t? k+1}}, the second one measures the portion explained by the outer variables {x _t+1, x _t+2, } ∪ {x _t?k?1, x _t?k?2,…} and the third term measures the correlation between x _t and x_t?k given all other variables. These terms, squared and summed, can form the basis of three portmanteau-type tests that are able to detect both deviation from white noise and lack of fit of an entertained model. Quantiles of their asymptotic sample distributions are complicated to derive at an adequate level of accuracy, so they are approximated using the Monte Carlo method. A simulation experiment is carried out to investigate significance levels and power of each test, and compare them to the portmanteau test.     

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.     

Minimum Mean Squared Error Estimation of the Noise in Unobserved Component Models

In model-based estimation of unobserved components, the minimum mean squared error estimator of the noise component is different from white noise. In this article, some of the differences are analyzed. It is seen how the variance of the component is always underestimated, and the smaller the noise variance, the larger the underestimation. Estimators of small-variance noise components will also have large autocorrelations. Finally, in the context of an application, the sample autocorrelation function of the estimated noise is seen to perform well as a diagnostic tool, even when the variance is small and the series is of relatively short length.     

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.     

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.     

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.     

Some weighted mixed portmanteau tests for diagnostic checking in linear time series models

The popular diagnostic checking methods in linear time series models are portmanteau tests based on either residual autocorrelation functions (acf) or partial autocorrelation functions (pacf). In this paper, we device some new weighted mixed portmanteau tests by appropriately combining individual tests based on both acf and pacf. We derive the asymptotic distribution of such weighted mixed portmanteau statistics and study their size and power. It is found that the weighted mixed tests outperform when higher order ARMA models are fitted and diagnostic checks are performed via testing lack of residual autocorrelations. Simulation results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of the mixed test.     

Sample autocorrelations of nonsationary fractionally integrated series

We derive the asymptotic distribution of the sample autocor-relations of nonstationary fractionally integrated processes of order d. If d≥1, the sample autocorrelations approach their probability limit one with a rate equal to the sample size. If d<1, the rate is slower and depends on d. These findings carry over to the case of detrended series. Monte Carlo evidence and an empirical example illustrate the theoretical results.     

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.     

Generating autocorrelated pseudo-random numbers with specific distributions

An easily implemented and computationally efficient procedure is presented for the generation of autocorrelated pseudo-random numbers with specific probability distributions. A plot illustrates the relationship among the autocorrelations of the uniform, Rayleigh, and exponential distributions corresponding to a given autocorrelation in the normal generating distribution.     

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.     

Durbin–Watson and Generalized Durbin–Watson Tests for Autocorrelations and Randomness

The Durbin–Watson (DW) test for lag 1 autocorrelation has been generalized (DWG) to test for autocorrelations at higher lags. This includes the Wallis test for lag 4 autocorrelation. These tests are also applicable to test for the important hypothesis of randomness. It is found that for small sample sizes a normal distribution or a scaled beta distribution by matching the first two moments approximates well the null distribution of the DW and DWG statistics. The approximations seem to be adequate even when the samples are from nonnormal distributions. These approximations require the first two moments of these statistics. The expressions of these moments are derived.     

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.     

A robust closed-form estimator for the GARCH(1,1) model

In this paper we extend the closed-form estimator for the generalized autoregressive conditional heteroscedastic (GARCH(1,1)) proposed by Kristensen and Linton [A closed-form estimator for the GARCH(1,1) model. Econom Theory. 2006;22:323–337] to deal with additive outliers. It has the advantage that is per se more robust that the maximum likelihood estimator (ML) often used to estimate this model, it is easy to implement and does not require the use of any numerical optimization procedure. The robustification of the closed-form estimator is done by replacing the sample autocorrelations by a robust estimator of these correlations and by estimating the volatility using robust filters. The performance of our proposal in estimating the parameters and the volatility of the GARCH(1,1) model is compared with the proposals existing in the literature via intensive Monte Carlo experiments and the results of these experiments show that our proposal outperforms the ML and quasi-maximum likelihood estimators-based procedures. Finally, we fit the robust closed-form estimator and the benchmarks to one series of financial returns and analyse their performances in estimating and forecasting the volatility and the value-at-risk.     

Estimation du comportement asymptotique des autocovariances et autocorrelations empiriques de processus multivariéeas

Statistics based on the sample autocovariances are widely used in time-series analysis. Estimators of the asymptotic covariance between the sample autocovariances are commonly derived from the so-called Bartlett's formula. However, this formula essentially holds for linear processes. This entails that for a wide range of nonlinear time series the above-mentioned estimators are not suitable. In this paper the behaviour of an alternative estimator is studied within the framework of centered or uncentered multivariate strongly mixing processes. Applications to differential functions of sample autocovariances, such as the sample autocorrelations, are considered.     

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.     

A New Test for Short Memory in Long Memory Time Series

This article considers short memory characteristics in a long memory process. We derive new asymptotic results for the sample autocorrelation difference ratios. We used these results to develop a new portmanteau test that determines if short memory parameters are statistically significant. In simulations, the new test can detect short memory components more often than the Ljung-Box test when these short memory components are in fact within a long memory process. Interestingly, our test finds short memory autocorrelations in U.S. inflation rate data, whereas the Ljung-Box test fails to find these autocorrelations. Modeling these short memory autocorrelations of the inflation rate data leads to improved model accuracy and more precise prediction.