White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case

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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 .

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THE LIMITINGDISTRIBUTION OF THE SERIAL CORRELATIONCOEFFICIENT IN THE EXPLOSIVE CASE

BY JOHN . WHITE

Aero ivision, inneapolis oneywellegulatorompany, inneapolisMinnesota

1. Introductionnd summary. everalauthorshave studied he discretestochasticrocessxt) nwhich hex's arerelated ythe stochasticifferenceequation

(1.1) Xt= axt,- + u, tX 1, 2, * Ty

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



LIMITING DISTRIBUTIONS 1189

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

x =(X1, X2X*** X)

is easily found o be

(2.1) f(x') = exp [(- 1/20o2)? (Xt-aXt_1)2](2.1) AX')

~~~~~~~~~(2ircr2)/2

The maximumikelihood stimator or is then the least-squares stimator '.

Sincewe shall be considering nlythedistributionf

, E Xt Xt-ia 2 '

Xt-1

we may,without oss ofgenerality,ake 2 = 1. For the timebeingwe shallalso setxo= 0.

We maynow write 2.1) in matrix orm s follows:

(2.2) f(x') - ~~exp-I x'Px)(2.2) AX -

(2r) T/2

whereP is the T X T matrix

-1 a 2 -a - 0

-a 1 + at2 _cot 0

(2.3) P [ O -+a 1 + a2 -a .IC + ay2 xIa -+a 1

o -a Ii

Sincea is a consistent stimator or , we shallconsider he distributionf - arather han thatofa alone. We have

A E XtXt.ia-a -C% a

(2.4) = E XtXt_i_ a EXt_

EXt-l

x'Axx'Bx'



1190 JOHN S. WHITE

whereA andB are theT X T matrices2 a - 1 0-1 22a -10 - 1 2a

-1 2ca -1

(2.5)0 -1 0

-1 0 010 1 01

B O 0 1

o 1 ?1

Let m(u,v) be the jointmomentgenerating unction f x'Ax and x'Bx. Wehave

m(u,v) = E (exp {x'Axu + x'Bxv})

(2.6) = (27r)-T12

fxp (x'Axu+ x'Bxv x'Px/2)dx

= (27r)fT12f xp (-x'Dx/2) dx,

whereD is the T X T matrix

p q Oq P q

(2.7) D=P-2Au-2Bv= 0 q P p

q P q

0 q 1p= 1+ a2- 2v+ 2au, q= -(a + u).

By a well-knownntegrationormulaCramer[1],Eq. (11.12.2.), p. 120) we

have

(2.8) m(u,v) = (27r)T12 exp ( x2Dx) dx = (detD)-1.

If wenowwrite etD D(T), we note thatexpanding2.7) bytheelements f

thefirst olumn ivesthedifferencequation

(2.9) D(T) = pD(T - 1) - q2D(T - 2).

From the nitialvaluesD(1) = 1 andD(2) = p - q2,we obtain

1 _ s T 1 - r T(2.10) D r)=-s s-r~~




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 .



1192 JOHN S. WHITE

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

- 4(1 - a2)U2/g2 4(1 + a2)V/l2 8aUV/g3+ 4V2/g}"1/2]

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

r = 1 U2+ 2V+ 0(9-3)

(4.5)~ r= -(1 - al)g2+Og)

(4.5) ~ ~ ~ 2U2 + 2a2Vs = a + 2aU/g+ (1 - 2)g2 + O(g )

Substitutingheappropriate alues ofg(T) from3.2), we have

U2 +2Vr = 1 T +0 (T-') for a I < 1,

(4.6) ___ TS- a2 + 2a - a2 U+ U + 2a2V+ 0(T-').

+ - +0(I ) forlal > 1,

(4.7)2_ U 2V a2 2aU(a - 1) (U2 2a2V) 1) +2 a -3T).= i + Ia |I a2T

II




If lal=

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

(4.11) m(w)= E(exp w} , m*(u, ) = E(exp Xu + Yv} ,

then

m(w) E(exp{X - zY}w) = E(exp{Xw Yzw}) - m*(w, zw).

To apply his echniqueo theproblemt hand,we set W = x'Ax/gzx'Bx/g2.rom4.1), 4.2) and 4.9)wehave

m(w)= M(w,-zw),

limm(w)= exp -zw + w2/2) for a j < 1,

(4.12) = (1 + 2zw- w2)-' for a > 1,= {exp (V2caw) cos 2V-zw - ___ sin2-\/-Zw)}

forlal =a



1194 JOHN S. WHITE

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)).




Hence the limiting istributionf g(T)(ci -a), for a I > 1, is the Cauchydistribution.We have beenunable to invert imm(w) whenIa = 1. In thenext ection

certain esults oncerninghis imit nd moregeneralproblems f this typewill

be discussed.Ifwenow etxo= c,a non-zeroonstant, heanalysisproceedsmuch s before.

Let A, B, P, and D be theT X T matrices efinedn (2.3), (2.5) and (2.7). We

thenhave, analogousto (2.1) and (2.4),

f(x') = (2)-T2 exp (cx,a - 2c2/2 -x'Px/2),

(4.19) A x'Ax+ cx, - aea a x'Bx+c2

The jointmoment eneratingunctionfx'Ax + cx1 - W22andx'Bx+ C2 is

m(u,v) = E (exp {(xAx + cx - aeC2)u+ (x'Bx + c2)v})

= {exp (c2v - cau - a c2)} (2X)-T2

2 ~ ~ ~ ~(4.20) xfexp([uD+]cxi x2D

- aC ~~~~~ +x [u .JIx7)(r-dxI

=exp (C2V- ae)U-!) p + 2 D (T)JD 1

limm(U/g,V/g2)= limM(U, V)

(4.21) --f(T(a 2C2 Y~-D(T - 1)1,= limD(T)-4 exp (2 1 D(T)]JJ

whereD(T) is as definedn (4.2) whileD'(T - 1) is definedn a similar ashion

butwithg = g(T).

For ja j < 1,itfollows rom4.6) and (4.8) that,sinceg(T) andg(T - 1) areofthesameorder,

limD(T) = limD'(T - 1)and hence

(4.22) lim m(U/g, V/g2) = limM(U, V) = limD(T)-112.

We seethatthis imit s the sameas thatfor o= 0 as given n (4.9) andhence

the limiting istributionfg(T)(& - a) does not dependon the initialvalue

xofor at < 1.For

ja > 1 wehave,from4.7),

lim D(T) = 1- (U + 2V),

(4.23) lim 'T- 1 = (U + 2V)



1196 JOHN S. WHITE

and nplaceof 4.22)wehave

limM(U, V) = Jim (T) "12 exp 2c [ _ D' (T )])

= (1 - U2- 2Vf112 exp{( -l)c ( Y2+ V)}

Thismomenteneratingunction aybe invertedythemethodsfSectiontogive

(4.25) f(x) = vr(1+X2)

1 ( ) ( q =

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-

stated) heoremhatthe imitingistributionfa functionfa sequence findependentandom ariables, ith uitable estrictionsnthese andom ari-ables, ependsnly nthe ormf he unctionnd sthe ame s thedistributionof relatedunctionalna stochasticrocess.

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

fx t) dx It)

G[x(.)J =2 x1f(t)dt f (t)dt

ontheWienerrocess,ndependentf hedistributionf heu's. Thisdistribu-tionwillbe consideredna future aper.

REFERENCES

[11H. CRAMER, Mathematical ethods fStatistics, rincetonUniversity ress,1946.[21W. J.DIXON, "Further ontributionso theproblem f erial correlation," nn.Math.

Stat.,Vol. 15 1944),pp. 119-144.

White John S. the Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case

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