Statistics & Probability Letters 19 (1994) 221-231 North-Holland 22 February 1994 A simple form of Bartlett’s formula for autoregressive processes Roland0 Cavazos-Cadena zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Departamento de Estadistica y Ca’lculo, Universidad Aut&oma Agraria Antonio Narro, Buenavista, Saltillo, Coah, Mexico Received July 1992 Revised May 1993 Abstract: An autoregressive process of finite order is considered. In this context it is shown that Bartlett’s formula for the asymptotic covariance matrix B of a vector of sample autocorrelations reduces to a matrix product, and a recursive method for computing B is given. Keywords: Autoregressive processes; sample autocorrelations; asymptotic distribution; asymptotic covariance matriu; Bartlett’s formula; recursive calculation of a Bartlett matrix. 1. Introduction This work concerns the following problem: Given an autoregressive process of finite order, find an algebraic procedure to determine the asymptotic covariance matrix of a vector of sample autocorrela- tions. For the sake of motivation, consider first an arbitrary (second order) stationary process {X,). From a second order point of view, the association structure of IX,} is characterized by the autocorrelations p(h), h = 0, 1, 2,. . . ; usually, however, p(. ) is unknown and should be estimated. Thus, after observing X r, . . . , X,, the sample autocorrelations b(h) are constructed (see Section 3 for details) and large-sample statistical inference about p(.> is based on the following result, valid under conditions satisfied by the models considered below: For each positive integer r, the asymptotic distribution (as n * m) of the vector n1/2(p^(1) -p(l), . . . , b(r) - p(r))’ is normal with mean zero and covariunce matrix B, = [ b(i, zyxwvutsrqponmlkjihgfedcbaZYXWV j)Ii,j= 1,2,. _, r, where b(i, j) = ? (p(k+i) +p(k-i) -%(k)p(i)}{p(k+j) +p(k-j) -2p(k)p(j)}, (1) k=l This equation can be traced back to Bartlett (1946) and is presently known as Bartlett’s formula; see, for instance, Brockwell and Davis (19871, pp. 214-215, or Anderson (1971) pp. 494-495. Bartlett’s formula is analytical (since it involves an infinite series) and, in general, difficult to evaluate exactly. On the other hand, when {X,} obeys a model determined by a finite number of parameters it might be Correspondence to: Dr. C. Cavazos-Cadena, Departamento de Estadistica y Calculo, Universidad Autonoma Agraria Antonio Narro, Buenavista, Saltillo, Coah, 25315, Mexico. This work was kindly supported by PSFO under Grant 02-06-300/02 and by MAXTOR Foundation for Applied Probability and Statistics (MAXFAPSI under Grant No. Ol-Ol-56/93-3. 0167-7152/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0167-7152(93)E0107-5 221