Joint Asymptotic Distributions of Sample Autocorrelations for Time Series of Martingale Differences

Abstract

It is well known fact for theiid data that the limiting standard errors ofsample autocorrelations are all unity for all time lags and they are asymptot-ically independent for dierent lags (Brockwell and Davis, 1991). It is alsousual practice in time series modeling that this fact continues to be valid forwhite noise series which is a sequence of uncorrelated random variables. Thispaper contradicts this usual practice for white noise. We consider a sequenceof martingale dierences which belongs to white noise time series and deriveexactjoint asymptotic distributions of sample autocorrelations. Some im-plications of the result are illustrated for conditionally heteroscedastic timeseries.AMS 2000 subject classications.Primary 62M10; Secondary 62E20.Keywords. Limiting standard errors, sample autocorrelations, exact joint asymp-totic distribution, conditionally heteroscedastic time series.1. IntroductionThe sample autocorrelation function (ACF) for a given time series plays acrucial role in identifying an appropriate model for the data. To evaluate theadequacy of the model, the \residual" is recommended to be carefully examined.Here the term residual is dened as the dierence of observations and the ttedvalues obtained after tting an appropriate model. As discussed by Box andPierce (1970), the ACF based on residuals can be used as a useful diagnostic tool.See also Hwanget al.(1994). For the case ofiid data, it is a well known fact thatReceived June 2006; accepted November 2006.yThis work was supported by the SRC program of KOSEF (R11-2000-073-0000).1Corresponding author. Department of Statistics, Sookmyung Women's University, Seoul140-742, Korea (e-mail: shwang@sookmyung.ac.kr)