NON‐NEGATIVE AUTOREGRESSIVE PROCESSES

Jiří Anděl

Journal of Time Series Analysis, 1989, vol. 10, issue 1, 1-11

Abstract: Abstract. Consider a stationary autoregressive process given by Xt=b1Xt‐1+…+bpXt‐p+Yt, where the Yt are independent identically distributed positive variables and b1,…,bp are non‐negative parameters. Let the variables X1,…,Xn be given. If p= 1 then it is known that b1*= min(Xt/Xt‐1) is a strongly consistent estimator for b1 under very general conditions. In this paper the case p= 2 is analysed in detail. It is proved that min(Xt/Xt‐1)→b1 almost surely (a.s.) and min(Xt/Xt‐2)→b2+b12 a.s. as n→ 8. The convergence is very slow. Denote by b1* and b2* values of b1 and b2 respectively which maximize b2+b2 under the conditions Xt‐b1Xt‐1‐b2Xt‐2≥ 0 for t= 3,…, n. We prove that b1*b1 and b2*b2 a.s. Simulations show that b1* and b2* are better than the least‐squares estimators of the autoregressive coefficients when the distribution of Yt is exponential.

Date: 1989
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