Sampled autocovariance and autocorrelation results for linear time processes. (English) Zbl 0695.62212

Anderson, Oliver D.; de Gooijer, Jan G.

Commun. Stat., Simulation Comput. 17, No. 2, 489-513 (1988).

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the covariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length \(n\to \infty\), our results for the ARMA class retrieve those of M. S. Bartlett [J. R. Stat. Soc. B 8, 27-41 (1946)]. Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett’s general asymptotic formula to finite series realisations.

MSC:

62M10

Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

ARMA; ARIMA; ARUMA models; Bartlett approximation; moments of serial correlations; Taylor expansions

References:

[1]	Anderson O.D., Cahiers du CERO 19 pp 223– (1977)
[2]	Anderson O.D., Cahiers du CERO 21 pp 221– (1979)
[3]	Anderson O.D., Sankhyā B 41 pp 177– (1979)
[4]	Anderson O.D., Statistica 39 pp 287– (1979)
[5]	Anderson O.D., Cahiers du CERO 22 pp 309– (1980)
[6]	Anderson O.D., Sankhyā B 45 pp 249– (1983)
[7]	DOI: 10.2307/2983611 · doi:10.2307/2983611
[8]	Davies N., Model Misspecification in Time Series Analsis (1977)
[9]	DOI: 10.1080/03610928008827975 · Zbl 0443.62071 · doi:10.1080/03610928008827975
[10]	DOI: 10.1016/0304-4076(80)90033-0 · Zbl 0457.62075 · doi:10.1016/0304-4076(80)90033-0
[11]	DOI: 10.1016/0304-4076(85)90155-1 · Zbl 0589.62080 · doi:10.1016/0304-4076(85)90155-1
[12]	Fuller W.A., Introduction to Statistical Time Series (1976) · Zbl 0353.62050
[13]	Kendall M.G., Biometrika 41 (1954)
[14]	Kendall M.G., The Advanced Theory of Statistics (1983)
[15]	Lancaster H.O., J. Roy. Statist. Soc. B 16 pp 247– (1954)
[16]	Moran P.A.P., Biometrika 35 pp 255– (1948)