The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case Author(s): John S. White Reviewed work(s): Source: The Annals of Mathematic al Statistics, Vol. 29, No. 4 (Dec., 1958), pp. 1188-1197 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2236955 . Accessed: 10/01/2012 00:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematical Statistics. http://www.jstor.org
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8/3/2019 White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
Author(s): John S. WhiteReviewed work(s):Source: The Annals of Mathematical Statistics, Vol. 29, No. 4 (Dec., 1958), pp. 1188-1197Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2236955 .
Accessed: 10/01/2012 00:44
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The
whereheu's are unobservableisturbances,ndependentndidenticallyis-tributed ithmean ero nd variance2, nda is anunknownarameter.
The statisticalroblems tofind ome ppropriateunctionfthex's as anestimatoror andexaminetsproperties.
Wemay ewrite1.1)as
(1.2) Xt=Ut + aUt_ + * +a+Ct lU + &XO.
From 1.2) we see that the distributionf the successive 's is notuniquelydeterminedythat f heu'salone.The distributionfxomust lsobespecified.Three istributionshich avebeenproposedor o re thefollowing:
(A) xo= a constantwith robabilityne),(B) xo s normally istributed ithmean zeroand varianceo2/(l - a2),
(C) Xo= XT.
DistributionB) is perhapshemost ppealing rom physical oint fview,sincefxohas thisdistributionnd ftheu's arenormallyistributed,hen heprocesssstationarye.g., eeKoopmans4]).However,here re everalnalyticdifficultieshich rise nthe tatisticalreatmentfthisprocess. istribution(C), the o-calledircularistribution,asbeenproposeds an approximationto (B) and s much asier oanalyze e.g., ee Dixon 2]).DistributionA) has
beenstudied xtensivelyy Mann and Wald [5].An interestingeature fdistributionA) is that may ssume nyfinitealue,while or istributions(B) and (C) a mustbe between 1 and 1. From 1.2) weseethat processsatisfying1.1)and(A) has
(1.3) var xt) = 2(1 + a2 + * + a2"'1).
If a I > 1, imt-.var xt) = ooand theprocesss said to be "explosive."Mann and Wald [5] considered nlythe case Ia I < 1. They showedthat the
least quares stimatoror isthe erial orrelationoefficient'
(1.4)a
22xsx_iE X9_1
Received ecember0,1955; evisedMay 27,1958.1 Inthis aper, he ummationignE will lways efero ummationrom= 1tot= T.
1188
8/3/2019 White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
and that for aI< 1) thisestimators asymptotically ormally istributed ith
mean a and variance (1 - a2)/T. Rubin [6] showed that the estimatora isconsistent i.e., plim a = a) for all a.
In this paper the asymptotic istribution f& will be studied under the as-sumption hat the u's are normally istributed. or Ia I > 1, t is shown hat heasymptotic istributionfa is the Cauchy distribution.or Ia I = 1, a momentgenerating unction s found, he inversion f whichwillyield the asymptoticdistribution.
2. The distributionf & - a. From equation (1.1) and condition A) thejoint distributionf
where ands areroots f he quation - px + q2 = 0, that s(2.11) r,s = (p 4 p / 4q2)/2.
The nversionfm(u,v) =(T)-- seems utof he uestion or inite . Theinversionf certainimitingormfm(u,v)willbe discussednSection .
3. The standardizingunction(T). Since&i s consistenthe imiting istri-butionf& - a is theunitaryistribution.he firstroblemhenstofind omefunctionfT, sayg(T), such hat he imitingistributionfg(T) (& - a) isnon-degenerate.enote hat heresultsfMannandWald Eq. (1.4) above)
giveg(T) = (T/I{1 - a2})l for ja j < 1, since (T/{1 -a2})* (c> ) has alimitingormal istribution.he function2(T) correspondsoughlyo thereciprocalf he symptoticariancef & - a), or nFisher'serminologyhe"information"n supplied ythe ample.
The "information"n a maybe obtainedxplicitlys follows. etf be thedensityunction2.1)with O= 0 and o,2 = 1. The"information,"ay (a), isthen efineds
I(a) = (E d( 2)
= E ( xt_j)(3.1) (T 1a2r) if
T(T-1) if a I=1.2
If thex'shadbeen ndependentandomariables,hen(a) (a& a) would easymptotically (0, 1) (Cramer1],Eq.(33.3.4), p. 503). This,ofcourse,snot he ase.This pproach oes,however,ive nheuristic ethodor inding
function(T) such hat (T) (& - a) has a non-degenerateimitingistribution.Wemightow akeg(T) = [1(a)]*;however,twill implifyhe omputationsto useslightmodificationshich reasymptoticallyquivalento [1(a)]1. Wechoose
g T) = <for a I < 1,
T(3.2) fora I = 1,
- aTa a-1 forjal > 1.
In the next ectiontwillbe shown hatg(T) (ca a) has a non-degeneratedistributionor ll values f .
8/3/2019 White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
4. The limiting istributionf g(T) (& - a). We shallfirst onsider he ointdistribution fx'Ax/g(T) and x'Bx/g2(T).et M(U, V) be the jointmomentgenerating unctionf thesetwo statistics.We thenhave
M(U, V) = E[expx'AxU/g(T) + x'BxV/g2(T)]
= m[U/g(T),V/g'(T)j,
wherem(u,v) is the ointmoment eneratingunction2.6).From 2.10) and (2.11) withg = g(T), u = U/g and v = Vlg2,we have
M(U,V)
=D(T)-1(4.2) 1-srT + 1-rTr-s s-r
r S 2[1 + a2 + 2aU/g - 2V/g2? {(1 - a2)2 - 4a(1 -a2)U/g
ForsufficientlyargeT and Ia I $ 1,wemayfactor1 - a2) out oftheradical n(4.3) and expandtheremainingadicalbythebinomial heorem.We thenhave,up to terms f orderO(g-3)
r, = 1[ + a2 + 2aU/g - 2V/92(4.4)
2- 2 - 2aU/g 2(1 + a2)V _ 2U2 + g,fJ]
(1 - oil)g2 (1- a.2) lTakingrwiththeplus sign and s withthe minus ignwe have
1, theexpansionn (4.4) is notvalid; however, rom4.3), wehave1 4 TaU + 2iV + O(T2) for I =1,
(4.8) V VS = 1 + Val_2i-\V O(T-2)=1+ T T-
Substitutinghese esultsn 4.2),wehave
limM(U, V) =exp (V + U2/2) for al < 1,
(1- u2 - 2V)-1/2 for(aj > 11
(4.9)= exp VaU (cos 2V-VV-2U
2\in
for ax = 1.
The nextproblems toobtain he imitingistributionifg(T)(a& a) fromlimM(U, V). Sinceg(T)(&$* a) = g(T)x'Ax/x'bX,heproblems one offindinghedistributionfthe ratiooftwo random ariables. nemethod fsolutionasbeen roposedyGurland3].LetX andYbetwo andomariables,Prob Y > 0) = 1. Wewish odeterminehedistributionf Z = X/Y. Let
W = = X - zY. ThenwehaveProb Z < z) = Prob X/Y < z)
(4.10) = Prob X - zY < 0)
= Prob W. < 0).
If the distributionfW can be found,hedistributionfZ will mmediatelyfollow. requentlyhedistributionfWcanbefoundrom hat fX and Y bymeans fmomenteneratingunctions.et
The inversion f imm(w) s trivial or a I < 1. The moment eneratingunctionexp (-zw+w2/2) is immediately ecognized s that of a randomvariablewhich s normally istributedwithmean -z and variance 1. Hence we have
lim rob (W < 0) - (27r)-L xp (-{t + z}2/2)dt
00(4.13) = (21rr-112fLxp -t2/2)dt
=limProb {g(T)(8' - a) <z},
i.e., g(T) (' - a) is asymptoticallyormalwithmean 0 and variance1.For Ia I > 1, the nverse f im m(w) mightbe obtaineddirectlyn termsof
Besselfunctions; owever,t is more ppealingfrom statistical ointofviewtoproceed s follows. et X and Y be independent hi-squared ariableswithonedegreeoffreedom. hen E(exp {Xw ) = E(exp { Yw ) = (1 - 2w)112 is theircommonmomentgenerating unction.Now set R = aX - bY, the momentgeneratingunction f R willbe
mR(w) E(exp{Rw}) = E(exp{aX - bYiw)
= ({1 - 2aw} 11+ 2bw)-1/2
In particularfwe set
(4.15) 2a = V/1 z2-z, 2b=V/1 + Z2+ Z,
wehave
(4.16) mR(w) = (1 + 2zw W2)112 = lim m(w).
Hence,the imiting istributionf W, for a j > 1, s the ame s thedistributionof R = aX - bY.We thenhave
rimrob (W < 0) = Prob (aX - bY < 0)
= Prob (X < bY/a)
(4.17) 1 fo a exp - x/2- y/2)=- Z I / ~~~~dxy
- limProb {g(T)(a' - a) < z} = say F(z).
The density unction orrespondingo F(z) is
f(z) dz 2)_Vlo/5 xp (-by/2a - y/2)4d(b/a dy
(4.18) 1~V7la (ba)d(/a2ir 1 + (bla) dz
1r1 + Z2 (by (4.15)).
8/3/2019 White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
as the imitingistributionfg(T)(& - a). Wenote hatfor = 0, f(x) is theCauchy istributions obtainedn 4.18).
6. Finalremarks. heresultsfMannandWald[5]show hat he imitingdistributionfg(T)(a - a), forca < 1, s alsoN(O,1) if, atherhan ssumingthat he"errors"tare normallyistributed,e merelyssume hat ll ofthemomentsf heu's arefinite.his sanotherxamplef ninvariancerinciplewhicheems oholdquitegenerallyor he imitingistributionsffunctionfrandomariables. oughlypeaking,here eems obe an unprovedandun-
Ageneralesultf his ormsDonsker's heorem7]which ives he imitingdistributionfany functionf sumsof independentdenticallyistributedrandom ariableswithfinite ariancess thedistributionfa correspondingfunctionalntheWiener rocess.t is conjecturedhat his ype freasoningwill how hat heresultsfMann ndWaldwill tillhold ftheu's aremerelyassumedohavefiniteariances.
Fora = 1, pplicationfDonsker's heoremhowshat he imitingistribu-tion fg(T)(a& a) is the ame s thedistributionf hefunctional