The stationarity conditions for an autoregressive (AR) process in general are reduced to a remarkably simple inequality if the lag coefficients are restricted to be identical. The condition is not only analytically elegant but also applicable in checking the validity of the stationarity conditions for such a restricted AR process of any order

Hammes, David L

Im, Eric Iksoon

Wills, Doug

English

University of Washington: UW Tacoma Digital Commons

University of Washington TacomaUW Tacoma Digital CommonsBusiness Publications Milgard School of Business2-2006Stationarity Condition for AR Index ProcessEric Iksoon ImUniversity of Hawaii at HiloDavid L. HammesUniversity of Hawaii at HiloDoug WillsUniversity of Washington - Tacoma Campus, dtwills@uw.eduFollow this and additional works at: https://digitalcommons.tacoma.uw.edu/business_pubPart of the Economic Theory CommonsThis Article is brought to you for free and open access by the Milgard School of Business at UW Tacoma Digital Commons. It has been accepted forinclusion in Business Publications by an authorized administrator of UW Tacoma Digital Commons.Recommended CitationIm, Eric Iksoon; Hammes, David L.; and Wills, Doug, "Stationarity Condition for AR Index Process" (2006). Business Publications. 1.https://digitalcommons.tacoma.uw.edu/business_pub/1STATIONARITY CONDITIONFOR AR INDEX PROCESSERIC IKSOON IM AND DAVID L. HAMMESUniversity of Hawaii at HiloDOUGLAS T. WILLSUniversity of Washington at TacomaThe stationarity conditions for an autoregressive ~AR! process in general arereduced to a remarkably simple inequality if the lag coefficients are restricted tobe identical+ The condition is not only analytically elegant but also applicable inchecking the validity of the stationarity conditions for such a restricted AR pro-cess of any order+1. PROBLEM AND MOTIVATIONThe stationarity conditions for an unrestricted autoregressive ~AR! process interms of lag coefficients increase in number with the lag order+ This makes itprogressively more complex to identify exact bounds for lag coefficients ensur-ing the stationarity+ To reduce the level of complexity, we consider a restrictedAR process where the lag coefficients are all equal ~referred to in this note asan AR index process!+ Then, the AR index process of order n ~ 1! can bewritten asyt a (i1nyti  «t , (1)where «t ; iid~0, s 2! with s 2  `+Because «t is stationary, the stationarity of yt in ~1! is contingent upon theconvergence of yt in the corresponding homogeneous difference equation:yt a (i1nyti  0+ (2)In light of the Schur theorem ~Chiang and Wainwright, 2005, pp+ 589–590!,the necessary and sufficient conditions for convergence of yt in ~2! can beexpressed asWe are deeply indebted to Professor Paolo Paruolo, NP co-editor of Econometric Theory, and anonymous refer-ees for constructive comments and suggestions that led to significant improvements+ Errors, if any, are solelyours+ Address correspondence to Eric Iksoon Im, Department of Economics, University of Hawaii at Hilo,200 W+ Kawili Street, Hilo, Hawaii 96720-4091, USA; e-mail: eim@hawaii+edu+Econometric Theory, 22, 2006, 164–168+ Printed in the United States of America+DOI: 10+10170S0266466606060075164 © 2006 Cambridge University Press 0266-4666006 $12+006S2k~a!6  0 for all k ~1 k n!, (3)whereS2k~a!  Sk~a! Tk'~a!Tk~a! Sk'~a! , (4)in which Sk~a! defines a lower triangular matrix of order k whose ~i, j ! thelement is equal to a if i  j, 1 if i  j, and 0 if i  j, whereas Tk~a! is alsoa lower triangular matrix of order k whose ~i, j !th element is equal to a ifi  j and 0 if i  j+ That is,S2k~a! 1 0 J 0 a J J aa L L I 0 L I II L L 0 I L L Ia J a 1 0 J 0 aa 0 J 0 1 a J aI L L I 0 L L II L L 0 I L L aa J J a 0 J 0 1+ (5)In this simplified framework, the set of convergence conditions in ~3! reducesto a single inequality in terms of a and n only ~established formally as a theo-rem in the next section!:10n  a  1+ (6)If we let a  f in ~3! as is usual in the econometrics literature, the conver-gence condition ~6! is expressed in terms of f as1  f  10n+ (7)In addition to its analytical elegance, either ~6! or ~7!, depending on howthe AR index process is written, is applicable in practice+ The stationarity con-dition for an AR~1! process, yt  fyt1  «t with «t ; iid~0,s 2!, known as1  f  1, is immediately obtained from ~7! for n  1+ The stationarityconditions for an AR~2! process, yt  f1 yt1 f2 yt2 «t , are also known asf1  f2  1; f2  f1  1; 1  f2  1 ~e+g+, see Judge, Griffiths, Hill,Lütkepohl, and Lee, 1985, p+ 293!+ The solution of these three inequalities underthe restriction f1  f2  f can be worked out as 1  f  12_ , but the sameSTATIONARITY CONDITION FOR AR INDEX PROCESS 165result is immediately derived from ~7! for n  2+ Similarly, the condition canbe used to check the validity of the stationarity conditions for an AR indexprocess of any higher order+2. PROOF AND DISCUSSIONThe convergence condition in ~6! depends on analytical eigenvalues of S2k~a!that may well be useful elsewhere+ Hence, we deal with them separately in apair of lemmas in sequence prior to establishing our main result as a theorem+DEFINITION+ li~{! denote eigenvalues of any real square matrix of order rin descending order: l1~{!  l2~{!  {{{  lr~{! .LEMMA 1+ For any real square matrix M of order r, li~j1 M  j2 Ir ! j1li~M !  j2 where j1 and j2 are nonnull constants.Proof of Lemma 1+ Because 6lIr  ~j1 M  j2 Ir !6  6~l  j2!Ir  j1 M 6,we can deduce l  li~j1 M  j2 Ir ! and l  j2  li~j1 M !  j1li~M !+ Sub-stituting out l from either equation, li~j1 M  j2 Ir !  j1li~M !  j2+1 LEMMA 2+ For S2k~a! defined in (5), li~S2k~a!!  ka for i  1; 1 fori  2,3, + + + , k  1; 1 a for i  k  1, k  2, + + + , 2k  1, 2k.Proof of Lemma 2+ Define a square matrix of order 2k:A2k  S2k~1!Lk Lk'Lk Lk' , (8)where Lk is a lower triangular matrix of order k whose ~i, j !th element is equalto 1 if i  j and 0 if i  j+Then we can rewrite S2k~a! in terms of A2k asS2k~a!  aA2k ~1 a!I2k + (9)Hence, in view of Lemma 1li ~S2k~a!!  li ~aA2k ~1 a!I2k ! ali ~A2k ! ~1 a!+ (10)It is clear from ~10! that once analytical solutions for li~A2k! are obtained,their counterparts for li~S2k~a!! immediately follow+The upper k rows in A2k in ~8! are identical with the lower k rows, whichimplies rank~A2k!  k+ Hence, we can reduce the lower k rows to null vectors166 ERIC IKSOON IM ET AL.through pre- and postmultiplications of an appropriate nonsingular matrix andits inverse:P2k1 A2k P2k  B2kLk Lk' Lk'0kk 0kk , (11)whereP2k  Ik 0kkIk Ik ; P2k1 Ik 0kkIk Ik  + (12)Because P2k in ~12! is nonsingular, A2k and B2k in ~11! have the same set ofeigenvalues with the same multiplicities ~Magnus and Neudecker, 1990, p+ 14!+Hence, noting that B2k is block-diagonal,li ~A2k !  li ~B2k ! li ~Lk Lk' ! for i1,2, + + + , k;0 for i k1, k 2, + + + ,2k+ (13)Because Lk Lk'  ii'  Ik where i~k 1! ~1, 1, + + + ,1!' , in view of Lemma 1,li ~Lk Lk' !  li ~ii'  Ik ! li ~ii' !1, (14)in whichli ~ii' !  k for i1;0 for i 2,3, + + + , k+ (15)Recursive substitutions of ~15! into ~14!, ~14! into ~13!, and then ~13! into ~10!lead to the eigenvalues sought after:li ~S2k~a!!  ka1 for i1;1 for i 2,3, + + + , k;1 a for i k1, k 2, + + + ,2k1,2k+THEOREM+ For yt  a(i1n yti  0, the necessary and sufficient condi-tion for convergence of yt is 10n  a  1.Proof of Theorem+ yt in the theorem converges if and only if 6S2k~a!6  0for all k ~1 k  n! by virtue of the Schur theorem as stated in ~3!+ Reflectingthe eigenvalues from Lemma 2 on ~3!,STATIONARITY CONDITION FOR AR INDEX PROCESS 1676S2k~a!6  )i12kli ~S2k~a!!  ~ka1!1~1!k1~1 a!k ~ka1!~1 a!k  0 for all k ~1 k n!,(16)which holds if and only if 10n  a  1+ NOTE1+ We owe conciseness of the proof to an anonymous referee+REFERENCESChiang, A+C+ & K+ Wainwright ~2005! Fundamental Methods of Mathematical Economics+ 4th ed+McGraw-Hill+Judge, G+G+, W+E+ Griffiths, R+C+ Hill, H+ Lütkepohl, & T+ Lee ~1985! The Theory and Practiceof Econometrics+ Wiley+Magnus, J+R+ & H+ Neudecker ~1990! Matrix Differential Calculus: With Applications in Statisticsand Econometrics+ Rev+ ed+ Wiley+168 ERIC IKSOON IM ET AL.

Stationarity Condition for AR Index Process

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Stationarity Condition for AR Index Process

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