THE STÄTIONARITY OF AN ESTIMATED AUTOREGRESSIVE PROCESS BY T. W. ANDERSON TECHNICAL REPORT NO. 7 NOVEMBER 15, 1971

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1 H. V. JOHNS JR. THE STÄTIONARITY OF AN ESTIMATED AUTOREGRESSIVE PROCESS BY T. W. ANDERSON TECHNICAL REPORT NO. 7 NOVEMBER 15, 1971 PREPARED UNDER CONTRACT NQ A (NR ) FOR THE OFFICE OF NAVAL RESEARCH THEODORE W. ANDERSON, PROJECT DIRECTOR DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA

2 THE STATIONARITY OF AN ESTIMATED AUTOREGRESSIVE PROCESS by T. W. Anderson Stanford University TECHNICAL REPORT NO. 7 NOVEMBER 15, 1971 PREPARED UNDER THE AUSPICES OF OFFICE OF NAVAL RESEARCH CONTRACT #N A Q T. W. ANDERSON, PROJECT DIRECTOR DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA

3 A stationary stochastic process {y } with mean %y t "= 0 satisfies a stochastic difference equation if there exist constants ß =1, 3,,..., 3 such that {u } defined by k (1) I 3 r y t _ r - u fc, t -...,-1, 0, 1,..., r=0 consists of independently identically distributed random variables. The process {y.} is stationary and y is independent of u,, t t t+x u +2,... if and only if the roots of the associated polynomial equation (2) f 3 r xp- r = 0 r=0 r are less than 1 in absolute value. The process is autoregressive of order p. We assume u - 0 and, u - O with 0 < a <. Let y..,..., y be T successive observations on the process. To estimate the coefficients 3, s..., 3 one can solve the linear equations j=l i_:j J X where 1 T ~ i < 4 > c i " K±* T "?' y t+i y t' i=0, lf '? See, for example, Section 5.4 of T. W. Anderson (1971). We assume that there are at least p different nonzero values of t observed. The purpose of this note is to show that the solution of (3) yields coefficients- corresponding to a stationary process; that is, the roots of

4 (5) b x p " r = 0 r=0 are less than 1 in absolute value. Pagano (1971) has shown this result by a different method. Let y-p+i = y-p+2 : = y o = and y T+i = y T+2 = = y_ = 0. Define the vectors (6) 7«. " l'< \ 't-1 C~" V j). f «e * y J. "I P «\ y t-p+l The equations (3) can be written (7) V T+p I t=l y t y t s T+p I t=l y t y t-r The equation (7) is the first row of (8) T+p 5. y t zl = fc t-i * T+p I t=l It lt-1 and b' is the first row of B. The other p-d rows of B constitute the matrix (-1 0). Theorem 1. The matrix B defined by (8) has characteristic roots less than 1 in absolute value. Proof. If u is a characteristic vector of B corresponding to a characteristic root X

5 T+p T+p w i?' I it zi = " xu c c ' I y t yj r Normalize u so that t-i - t=i ~ fc ~ fc x '. T+p _ T+p (10) 1 = u' I y. y' u = I (u«y ) (u«y ) " t=l ~ C ~.. t=l - -c - -c T +p _ T +p. u ' z Li yl.i u - z (-' y f _i) (-' y^.-.)» t=l t=l where u is the complex conjugate of u. Then multiplication of (9) on the right by u gives T+p _ T+p (11) x = u ' I hk-i*~- I (u ' V (u ' W- t=i ~ ~ ~ t=i ~ ~ ~ ~ By the Cauchy-Schwarz Inequality X j_ 1. We can have equality only if u' y. = u 1 y -, t=l,..., T+p } which is impossible. Q.E.D. ~ ~t ~ ~t x Since the characteristic roots of B are the roots of (5), the desired result has been proved. [See Section 5.4 of T. W. Anderson (1971).] Theorem 2. The roots of (5), where b.,,... b are the solu- _. p.. tion to (3), are less than 1 in absolute value. The result can be extended to the vector-valued autoregressive process {y } satisfying (12) _. B y = u, t =... -1, 0, 1,..., r_0 ~ r ~ c r ~ t. where y and u are q-component vectors and B 0 - I, R.,... s B are q x q matrices, ^u = 0, and # u u' - Z, positive definite ~t ~ f ~t t ~ and finite. The analogue of (2) is

6 (13) I I P r xp" r = 0. r=0 We observe y.,..., y and define < 14 > i- li-f J x ZtKJi- i=0 ' x > ' Then the estimates B.,..., B are the solution to ~1 ~p (15) ) B. c.. = -c, i=l,..., p. The roots of (13) with B replaced by B, r=l,..., p, have roots *-r ~r less than 1 in absolute value.

7 REFERENCES Anderson, T. W. (1971), The Statistical Analysis of Time Series, John Wiley & Sons, Inc. Pagano, Marcello (1971), "When is an Autoregressive Scheme Stationary?", Research Report No.53, Department of Statistics, State University of New York at Buffalo.

8 UNCLASSIFIED - SecwHv Classification gjgw^.wj^acsbgbg^^ DOCUMSNT OM»iL Ä7A.' RAP jscaalt/'rt Ut-ttai ol Ml*. he<& *mtfitm* ««*j»tf?gg^^.:^.?ff? f _ 1^T? T?.? 1 '?/!*^<* * lm,tf< *, 0 3. OniCSK'ATJH 6 ACTIVITY (Corporal«öitiket} a*. :««$» «? oaeju'ritv CLASSIFICATION DEPARTMENT OF STATISTICS STANFORD UNIVERSITY, CALIF. 3*..-««U5» 3. REPORT TITLE.»WH»'"»"'««' wm«sw»»e THE STATIONARITY OF AN ESTIMATED AUTOREGRESSIVE PROCESS * j 4. DESCRIPTIVE MOTES (Type öl'evwfws'jäs^siw^^' TECHNICAL REPORT 5. AUTHORS) (Last nemo, tint reine, tnltiel) ANDERSON, T.W. WM W «M»» WBJWWWKWWWW^^W S- REPORT DATE November 15, Ba. CONTRACT OB GRANT NO. N A-0U h. PROJECT NO. NR gs. ORMMNATeR'O S8KTOST NUMSBRftl #7 76. NO. P Ksr»» Opts 8 SSKPSJRT NOCSJ f4nyefiser numtore s?ss may 6s assigned 10. AVAILABILITY/LIMITATION NOTICES Reproduction in whole or in part is permitted for any purpose of the UNITED STATES GOVERNMENT : H.SUPPLEMENTARY NOTES %%. SPONSORING MILITARY ACTIVITY OFFICE OF NAVAL RESEARCH ARLINGTON, VIRGINIA 13. ABSTRACT, The estimated coefficients of an autoregressive process define a stationary process if they are computed from the (Toeplitz) matrix of sample moments computed from all available observations using the same divisor. im, F Ä 1473 UNCLASSIFIED

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