tailieunhanh - foundations of econometrics phần 9

Chúng tôi sẽ không cố gắng để chứng minh kết quả này, Davidson và MacKinnon (năm 1993, mục 8,8). Tuy nhiên, chúng tôi sẽ thảo luận về nó một thời gian ngắn. Xem xét bất kỳ khác root-n phù hợp và tiệm ước tính không thiên vị, ~ nói θ. Nó có thể được hiển thị | 550 Methods for Stationary Time-Series Data The result makes it clear that P1 and P2 are not the autocorrelations of an AR 2 process. Recall that for an AR 1 process the same P that appears in the defining equation ut put-1 t is also the correlation of ut and ut-1. This simple result does not generalize to higher-order processes. Similarly the autocovariances and autocorrelations of ut and ut-i for i 2 have a more complicated form for AR processes of order greater than 1. They can however be determined readily enough by using the Yule -Walker equations. Thus if we multiply both sides of equation by ut-i for any i 2 and take expectations we obtain the equation Vi P1 Vi-1 P2 Vi-2. Since Vo v1 and v2 are given by equations this equation allows us to solve recursively for any vi with i 2. Necessary conditions for the stationarity of the AR 2 process follow directly from equations . The 3 X 3 covariance matrix Vo V1 V2 V1 Vo V1 V2 V1 Vo of any three consecutive elements of an AR 2 process must be a positive definite matrix. Otherwise the solution to the first three Yule-Walker equations based on the hypothesis of stationarity would make no sense. The denominator D evidently must not vanish if this solution is to be finite. In Exercise readers are asked to show that the lines along which it vanishes in the plane of P1 and P2 define the edges of a stationarity triangle such that the matrix is positive definite only in the interior of this triangle. The stationarity triangle is shown in Figure . Copyright 1999 Russell Davidson and James G. MacKinnon Autoregressive and Moving Average Processes 551 Moving Average Processes A qth order moving average or MA q process with a constant term can be written as yt A O0 t O1 t 1 Oq t q where the t are white noise and the coefficient O0 is generally normalized to 1 for purposes of identification. The expectation of the yt is readily seen to be A and so we can write X Ut yt