On the Distribution of the Sample Autocorrelation Coefficients

52 Pages Posted: 24 Nov 2008 Last revised: 10 Aug 2009

See all articles by Raymond Kan

Raymond Kan

University of Toronto - Rotman School of Management

Xiaolu Wang

Iowa State University

Date Written: November 21, 2008

Abstract

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation based tests are provided. Actual size properties of the Box-Pierce and Ljung-Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box-Pierce test statistic, we propose an adjusted Box-Pierce test that has a far superior size property than the traditional Box-Pierce and Ljung-Box tests.

Keywords: Sample autocorrelation coefficient, Finite sample distribution, Rank one update

JEL Classification: C13, C16

Suggested Citation

Kan, Raymond and Wang, Xiaolu, On the Distribution of the Sample Autocorrelation Coefficients (November 21, 2008). Available at SSRN: https://ssrn.com/abstract=1305312 or http://dx.doi.org/10.2139/ssrn.1305312

Raymond Kan (Contact Author)

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S3E6
Canada
416-978-4291 (Phone)
416-971-3048 (Fax)

Xiaolu Wang

Iowa State University ( email )

2167 Union Drive
Ames, IA 50011
United States

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