Changes of Variance in First-Order Autoregressive Time Series Models-With an Application

Wichern, Dean W.; Miller, Robert B.; Hsu, D. A.

doi:10.2307/2347232

Cited by 106 publications

(58 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• A division by is performed; is an intervention function acting on the standard deviation of the innovations in order to represent, for example, a troubled time interval during which the scatter is higher (Mélard, 1981a;Wichern et al, 1976).…”

Section: The Modelmentioning

confidence: 99%

Exact maximum likelihood estimation for extended ARIMA models

Azrak¹,

Mélard²

1993

Developments in Time Series Analysis

View full text Add to dashboard Cite

Several extensions to autoregressive integrated moving average (ARIMA) models have been considered in the recent years. Many of them are special cases of the extended ARIMA model treated in this paper. The main features are time-dependent coefficients in the autoregressive and moving average polynomials, various types of interventions (including the usual Box and Tiao form and the innovation interventions but also interventions acting on the scale), trends on the level or on the scale, built-in deterministic seasonal components and variable transformations. In the past, estimation procedures were limited to least squares although the evaluation of the likelihood function is available for special cases, including the ARMA model with time dependent coefficients. This paper deals with maximum likelihood estimation when several features of the extended ARIMA model are taken together. INTRODUCTIONSeveral extensions of ARIMA models have been considered in the recent years, including (a) the use of time-dependent coefficients in the autoregressive and moving average polynomials (Quenouille, 1957;Whittle, 1965;Abdrabbo and Priestley, 1967;Miller, 1968 and1969;Subba Rao, 1970; Mélard and Kiehm, 1981;Tyssedal and Tjøstheim, 1982;Grillenzoni, 1990), (b) various types of interventions, including the usual Box and Tiao (1975) formulation and the innovational interventions (Fox, 1972) but also interventions acting on the scale (Mélard, 1981a;Tsay, 1988), (c) additive (level) or multiplicative (scale) trend (Mélard, 1977), (d) built-in deterministic seasonal components on the variable (Abraham and Box, 1978) or on the innovation (Mélard, 1981b), (e) variable transformations (Box and Cox, 1964). 1 We are grateful to the referee for his/her comments, and more especially for the summary that we have used at the end of Section 1. 2The purpose of that model is to encompass several deterministic variations with respect to time in the framework of the usual stochastic ARIMA models. Other extensions not explicitly considered in this paper are ARMA models with GARCH errors (Bollerslev, 1986), threshold AR models (Tong, 1983), bilinear models (Subba Rao, 1981), fractional differencing ARIMA models (Granger and Joyeux, 1980). It should be noted, however, that some of these extensions can be handled using the same approach. For instance, threshold ARMA models (Mélard and Roy, 1988) can be seen as time-dependent ARMA models. Other approaches for time-dependent models include spectral density estimation (Priestley 1981(Priestley , 1988, recursive estimation (Ljung and Söderström, 1983;Young, 1984) and models with random coefficients (Nicholls and Quinn, 1982;Bougerol, 1992).Motivations for the extended ARIMA model which is used here have already been discussed elsewhere (Mélard, 1982a(Mélard, , 1985a). An illustration has already been provided (Mélard, 1985b). The estimation procedure was however limited to the conditional least squares approach, generalizing the approach of Box and Jenkins (1976). In this paper, an algorithm f...

show abstract

Section: The Modelmentioning

confidence: 99%

Exact maximum likelihood estimation for extended ARIMA models

Azrak¹,

Mélard²

1993

Developments in Time Series Analysis

View full text Add to dashboard Cite

show abstract

“…Typical examples include the discrimination between stages of high and low asset volatility or the detection of changes in the parameters of an AR(p) model in order to obtain superior forecasting procedures. Wichern et al (1976) studied the change point problem for the variance in a first order autoregressive model. These authors pointed out that -even if log-return data exhibits a stationary behavior in the mean -the variability is often not constant and as a consequence any conclusions based on the assumption of homoscedasticity could be misleading.…”

Section: Introductionmentioning

confidence: 99%

Change Point Analysis of Correlation in Non-stationary Time Series

Dette¹,

Wu²,

Zhou³

2019

STAT SINICA

View full text Add to dashboard Cite

A restrictive assumption in change point analysis is "stationarity under the null hypothesis of no change-point", which is crucial for asymptotic theory but not very realistic from a practical point of view. For example, if change point analysis for correlations is performed, it is not necessarily clear that the mean, marginal variance or higher order moments are constant, even if there is no change in the correlation. This paper develops change point analysis for the correlation structures under less restrictive assumptions. In contrast to previous work, our approach does not require that the mean, variance and fourth order joint cumulants are constant under the null hypothesis. Moreover, we also address the problem of detecting relevant change points.

show abstract

“…For instance, Wichern et al (1976) considered a detection procedure for a variance change at an unknown position in a first order autoregressive model. Abraham and Wei (1984) studied inference in the parameters of a time series under the presence of a variance change at an unknown point.…”

Section: Introductionmentioning

confidence: 99%

Covariance changes detection in multivariate time series

Galeano

Peña

2007

Journal of Statistical Planning and Inference

View full text Add to dashboard Cite

This paper studies the detection of step changes in the variances and in the correlation structure of the components of a vector of time series. Two procedures based on the likelihood ratio test (LRT) statistic and on a cumulative sums (cusum) statistic are considered and compared in a simulation study. We conclude that for a single covariance change the cusum procedure is more powerful in small and medium samples, whereas the likelihood ratio test is more powerful in large samples. However, for several covariance changes the cusum procedure works clearly better. The procedures are illustrated in two real data examples.

show abstract

Changes of Variance in First-Order Autoregressive Time Series Models-With an Application

Cited by 106 publications

References 7 publications

Exact maximum likelihood estimation for extended ARIMA models

Exact maximum likelihood estimation for extended ARIMA models

Change Point Analysis of Correlation in Non-stationary Time Series

Covariance changes detection in multivariate time series

Contact Info

Product

Resources

About