Distribution of the Estimators for Autoregressive Time Series With a Unit Root

Author(s): David A. Dickey and Wayne A. Fuller

Source: Journal of the American Statistical Association, Vol. 74, No. 366 (Jun., 1979), pp. 427-
431
Published by: American Statistical Association
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 Distribution
of the Estimators
for
Autoregressive TimeSeries
Witha UnitRoot
DAVIDA. DICKEYand WAYNEA. FULLER*

Let n observations Yi, Y2, ..., Yn be generated by the model Rao (1961) extended White's results to higher-order
Yt = pYt-1 + et, where Y0 is a fixed constant and {et t_ln is a se-
quence of independent normal random variables with mean 0 and autoregressivetime serieswhose characteristicequations
have a singleroot exceedingone and remainingrootsless
variance a2. Properties of the regressionestimator of p are obtained
under the assumption that p = 4 1. Representations for the limit than one in absolute value. Anderson (1959) obtained
distributionsof the estimator of p and of the regressiont test are
derived. The estimatorof p and the regressiont test furnishmethodsthe limitingdistributionsof estimatorsfor higher-order
of testingthe hypothesisthat p = 1. processes with more than one root exceeding one in
KEY WORDS: Time series; Autoregressive; Nonstationary; absolute value.
Random walk; Differencing. The hypothesisthat p = 1 is of some interestin ap-
plicationsbecause it correspondsto the hypothesisthat
1. INTRODUCTION it is appropriateto transformthe time series by differ-
encing.Currently,practitionersmay decide to difference
Considerthe autoregressivemodel a timeserieson the basis of visual inspectionofthe auto-
Yt = pYt-, + et , t = 1,2, (I. 1) correlationfunction.For example, see Box and Jenkins
(1970, p. 174). The autocorrelationfunctionofthe devia-
where Yo = 0, p is a real number,and {et} is a sequence tions fromthe fittedmodel is then investigatedas a test
of independentnormalrandomvariables withmean zero of the appropriatenessof the model. Box and Jenkins
and variance 0-2 [i.e., et NID(O, -2)]. (1970, p. 291) suggestedthe Box and Pierce (1970) test
The timeseries Yt converges(as t -* oo) to a stationary statistic
time series if IpI < 1. If IPI = 1, the time series is not K

stationaryand the variance of Yt is to-2. The time series QK=nErk2, (1.3)

k=1
withp = 1 is sometimescalleda randomwalk. If Ip I > 1, where
n n
the time series is not stationaryand the variance of the rk = YE e~~
e2)-1 E eet-k
eIt' -
time series growsexponentiallyas t increases. t=1 t=k+l
Given n observationsY1, Y2, ..., Yn, the maximum
likelihoodestimatorof p is the least squares estimator and the et'sare the residualsfromthe fittedmodel. Under
the null hypothesis,the statisticQK is approximatelydis-
n
n
tributedas a chi-squaredrandom variable with K - p
A= (E Yt_12)l E YtYt . (1.2) degreesof freedom,wherep is the numberof parameters
t=1 t=1
estimated. If I Yt} satisfies(1.1) then p = 0 under the
Rubin (1950) showed that is a consistentestimator null hypothesisand et = Yt- Yt-1.
A

for all values of p. White (1958) obtained the limiting The likelihoodratio test of the hypothesisHo: p = 1
joint-momentgeneratingfunctionfor the properlynor- is a functionof
malized numerator and denominator of A-p. For
n
IpI # 1 he was able to invert the joint-moinent gen- _
A =
(Ap 1) Se (,E Yt-2)
erating functionto obtain the limitingdistributionof
p- p. For Ip I < 1 the limitingdistribution ofnG2p _ p) where
n
is normal. For IPI > 1 the limiting distribution of Se2 = (n - 2) E (Yt - pYt i)2
|pln(p2-I)-1 ( -p) is Cauchy. For p = 1, White was t=2

able to representthe limitingdistributionof n (- 1) as In this article we derive representatiolns forthe limiting

that of the ratio of two integralsdefinedon the Wiener distributionsof p and of T, given that p = 1. The
I
process. representationspermit constructionof tables of the
* Wayne A. Fuller is Professor of Statistics at Iowa State Uni- percentagepointsforthe statistics.The statisticsAand T
versity,Ames, IA 50011. David A. Dickey is Assistant Professorof
Statistics at North Carolina State University,Raleigh, NC 27650. ? Journalof the AmericanStatistical Association
This researchwas partiallysupported by JointStatistical Agreement June 1979,Volume74, Number366
No. 76-66 with the Bureau of the Census. Theoryand Methods Section
427
428 Journalof the AmericanStatisticalAssociation,June 1979

are also generalizedto models containinginterceptand i > 1, aj -l = aj?j+l = -1 forall j, and ai, = 0 other-
timeterms. wise. By a resultofRutherford(1946), the rootsofAnare
In Section 4 the Monte Carlo methodis used to com-
Xi, = ( ) sec2((n - i)r/(2n - 1))
pare the powerof the statisticsT and Awith that of QK.
Examples are given in Section 5. i=l 21,2... , n-1
Let M be the t - 1 by n - 1 orthonormalmatrixwhose
2. MODELS AND ESTIMATORS ith rowis the eigenvectorofAncorresponding
to Xin.The
The class of models we investigateconsistsof (a) the itth element of M is
model (1.1), (b) the model Mit= 2(2n -1)- 2

Yt =,u + pYt-, + et, t = 1, 2,... (2.1) cos [(4n - 2)-1(2t - 1)(2i -)r] , (3.1)
Yo= 0 and we can expressthe normalizeddenominatorsum of
and (c) the model squares appearingin as A

n n-I
Yt = 1A+ ft + pYt- + et, t = 1, 2, ... (2.2) rn = n-2 E Yt_12 - n-2 E XinZin2 X (3.2)
Yo = 0 . t=2

Assume n observationsYi, Y2, ..., Yn are available whereZ = (Zn, Z2n ... , Zn-1,n) = Me,,.
foranalysis and definethe (n - 1) dimensionalvectors, Let
Hn-n

= (1- (n/2), 2- (n/2), 3 - (n/2), n2 n2 2 n2

. .. , n - 1- (n/2)), n -1) n (n-2) n (n-3) ... n

n(n
yt = (Y2, Y3, Y4, . .. I Y.) I
n- 2 2(n-3) ... n-2
Yt-11 = (Y1, Y2, Y3, . I
Yn_1),
and
Let U= Yt-1, U2 = (1, Yt-1), and U3 = (1, t,Yt-1). We
(Tny Wn,Vn) n n-i
definep as the last entryin the vector
= n-d (Yn_ 1n- YE1,
y n-2 E (n - j) (j - )ej)'
(U2/U2)-lU2/yt, (2.3) t=2 j=1

and definePT as the last entryin the vector = Hnen = HnM-1Z . (3.3)

(U3'U3)-lU3'Yt . (2.4) Then

The statisticsanalogousto the regressiont statisticsfor n - 1) = (2rn)-l(Tn2 -1) + Op(n-1) , (3.4)

the test of the hypothesisthat p = 1 are (2r - 2Wn2)-l(Tn2- 1 - 2TnWn)
n(- 1) =
A = (A- 1) (Sel2Cl) , (2.5) + Op(n-6), (3.5)

A = (pAA
- 1)(Se22C2) , (2.6) n(Pr 1) = [2(Pr -Wn2 - 3 2)]-1
Tr=(r- 1) (Se32C3)4 2, (2.7) *[(Tn- 2Wn)(Tn - 6Vn) - 1] + Op(n-) . (3.6)

where Sek2 is the appropriate regressionresidual mean forthe LimitDistributions

3.2 Representations
square
Having expressedn(,6 - 1), n(AM-1), and nC(A'- 1)
Sek2= (n - k - 1) '[Y1'(I - Uk(Uk'Uk)'lUk')Yt] (2.8) in terms of (rn, Tn, Wn, Vn) we obtain the limiting dis-
elementof (Uk'Uk)>
and Ck is the lower-right tributionof the vector random variable. The following
lemma will be used in our derivation of the limit
3. LIMIT DISTRIBUTIONS distribution.

As the firststep in obtainingthe limitdistributionswe Lemma 1: Let {Zijjf l be a sequence of independent

investigatethe quadratic formsappearing in the sta- randomvariables withzero means and commonvariance
tistics. Because the estimatorsare ratios of quadratic A Let {wi; i = 1, 2, . . } be a sequence of real numbers
formswe lose no generalityby assumingo2 = 1 in the and let {lwin; i = 1, 2, ..., n - 1; n = 1, 2, ...} be a
sequel. triangulararray of real numbers.If

of the Statistics
3.1 Canonical Representation E Wi2 < 00
i=i
Given that p = 1, the quadratic formEt=2 Yt-12 can
n-i
be expressedas en'Anen,where en' _ (e1, e2, ..., e_) limEI wij2= o Wi2
the elements at3 of An-l satisfy all = 1, a,, = 2 for n-*Ooi=l i=l
Dickeyand Fuller:TimeSeriesWithUnitRoot 429
and For fixedi,
lim Win = wi
nx-000
lim in= =i (ai, bi,9i)'
n -o

then Lil, wiZ, is well definedas a limitin mean square 21(yi,7y2, 2yi3- y2)* (3.7)
and
n 00 By Jolley (1-961,p. 56, #307,308) we have
plim{LwZ}=W wiZ
i=1 i_l1
E (a 2 b2 g 2) = (1, 1/3, 1/30)
Proof: Let e > 0 be given. Then we can choose an M Let
such that n-I n-1
(Pn*, Tn*) = ( n 2XinZZ, ainZi)
00
2 E Wi2 < E/9
n-1 n-I
i=M+l = (, .
(Wn*, Vn*) binZi, E ginZi)
and i=i i=1
n 00

a21 WZ 2- 1j W,21 < f/9 Now, forexample,by (3.3)

i=1 i=1 n-I
for all n > M. Furthermore,given Ml, we can choose lim E ai2 _ limvar{Tn*} -limvarI Tn} 1
n--*= i=1 n-ow n-o-
No > M such that n > No implies
M
Therefore,by (3.7) and Lemma 1, Tn* converges in
aI (win - w,)2 < E/9
probabilityto T. It followsby analogous argumentsthat
i=1 (Pn*, Tn*, Wn*, Vn*) converges in probability to
and (r, T, W, V). Because the distributionof (rn*, Tn*, Wn*,
n Vn*) is the same as that of nqnwe obtain the conclusion.
a W172 < 3/9
i=M+l Corollary 1: Let Y, satisfy (1.1) with p 1. Then

Hence, forall n > No,

(p 1) I(P-'(T2-1),
n 00

var{ wi,-Zi - wiZiI < E

1) A ((r- 2W (T2
i 1 i-1l
n(- - 1)-2W]
AU 2TW]2-l)T
and the resultfollowsby Chebyshev'sinequality.
and
Theorem1: Let {Zi},ll be a sequence of NID(O, 1)
randomvariables. Let qn' = (Pn, Tn0,Wn, V70),wherethe T A 2 T- 1) - 2TW]
elementsof the vectorare definedin (3.2) and (3.3). Let Let Yt satisfy(2.1) with p = 1. Then
= (r, T, W, V)
n - 1) (
- W2- 3V2)-1
where
*[(T - 2W)(T - 6V) - 1]
00 00
and
(F, T) - (, iy*2Z,2,, 2-yiZi)
TTA2(I7 - W2 - 3V2)U[(T - 2W) (T - 6V) - 1]
00 ~~~~00
(W, V) = (E 2 yi2Zi, E 2F[2yi3- I2]Zi) Proof: The proof is an immediate consequence of
Theorem 1 because the denominatorquadratic forms
'Yi = -1 'i\ol
in p pA, PJ are continuousfunctionsof q that have prob
and ability 1 of being positiveand the Sek2 of (2.8) converge
in probabilityto 02.
Y
2
lim n-2Xin= 4E (2i -l 7r]-2
70 --ic
The numeratorand denominatorof the limit repre-
sentation of n(A- 1) are consistentwith White's (1958)
Then )n convergesin distributionto -q,that is,
limitjoint-momentgeneratingfunction.
Note that the limitingdistributionsof A, and TAare
?In >7
obtained under the assumptionthat the constant term
Proof: Note that q is a well-definedrandom variable ,u is zero. Likewise, the limitingdistributionsof Pr, and
because FJt? iik < ?O fork = 2, 3, ..., 6. Let kinbe the 7 are derivedunder the assumptionthat the coefficient
ith columnof HRM-1,where for time, R3,is zero. The distributionsof P and TT are
unaffectedby the value of ,uin (2.2). If ,u # 0 for (2.1)
in= (ai,e, bin0gj0)' or A3 z 0 for (2.2), the limitingdistributionsof $, and tr
= [cov(T70, Zi0),cov(W70 Zin),cov(V70, Zi0)]' are normal. Trhusif (2.1) is the mainltainled model and
430 Journal
of the American
StatisticalAssociation,
June1979

the statisticTy is used to test the hypothesisp = 1, the Monte Carlo Power of Two-Sided
hypothesiswill be accepted withprobabilitygreaterthan Size .05 Tests of p = 1
the nominallevel where,u# 0.
By the resultsof Fuller (1976, p. 370), the limitingdis- p
tributionsof p, p,J,and p', giventhat p = -1, are identi- n Test .80 .90 .95 .99 1.00 1.02 1.05
cal and equal to the mirrorimage ofthe limitingdistribu-
tion of given that p = 1. Likewise, the limitingdis- 50
A
Q1 .09 .05 .05 .04 .04 .07 .47
Q5 .07 .04 .03 .03 .04 .08 .53
tributionsof T, T, and Tr forp = -1 are identical and QIO .05 .04 .03 .03 .03 .09 .54
equal to the mirrorimage of the limitingdistributionof Q20 .03 .02 .02 .02 .02 .08 .52
T forp = 1. .57 .18 .08 .05 .05 .14 .71
T .57 .18 .08 .04 .05 .23 .70
In our derivationsY0 is fixed.The distributionsof pA .28 .10 .06 .05 .06 .11 .67
and T do not depend on the value of Y0. The limiting .18 .06 .04 .04 .05 .13 .68
distributionof does not depend on Yo, but the small- 100
A

Q1 .15 .07 .05 .04 .05 .26 .94

sample distributionof will be influencedby Yo.
A
Q5 .13 .08 .05 .04 .04 .34 .95
In the derivationswe assumed the etto be NID (0, a2). QIO .11 .06 .05 .03 .04 .37 .95
Q20 .08 .05 .04 .03 .03 .38 .95
The limitingdistributionsalso hold foret that are inde- / .99 .55 .17 .05 .05 .54 .98
pendent and identicallydistributednonnormalrandom T .99 .55 .17 .04 .05 .59 .97
variables with mean zero and variance o-. White (1958) .86 .30 .10 .05 .05 .49 .98
.73 .18 .06 .04 .05 .51 .98
and Hasza (1977) have discussedthis generalization.
The statisticT is a monotonefunctionofthe likelihood 250 Q1 .34 .12 .06 .05 .06 .94 1.00
Q5 .45 .13 .07 .04 .05 .95 1.00
ratio when Y0 is fixedunderthe null model of p = 1 and QIo .34 .12 .06 .04 .05 .95 1.00
underthe alternativemodel ofp 5 1. Tests based on the Q20 .24 .10 .05 .04 .04 .95 1.00
r statisticsare not likelihoodratios and not necessarily p5 1.00 1.00 .74 .08 .05 .98 1.00
T 1.00 1.00 .74 .08 .05 .97 1.00
the mostpowerfulthat can be constructedif,forexample, pg. 1.00 .96 .43 .06 .05 .98 1.00
thealternativemodelis that (Y0, Yi, . . ., Yn) is a portion 1.00 .89 .28 .04 .05 .98 1.00
of a realizationfroma stationaryautoregressiveprocess.
A set of tables of the percentilesof the distributionsis
given in Fuller (1976, pp. 371,373) and a slightlymore statistics.It is not surprisingthat p and T are superior
accurate set in Dickey (1976). Dickey also presents to p, and T, because p and T use the knowledgethat the
details of the table constructionand gives estimates of true value of the interceptin the regressionis zero.
the samplingerrorof the estimatedpercentiles. Third,forp < 1 the statistic A, yieldeda more power-
ful test than the statistic For p > 1 the rankingwas
TA.

reversedand the Ty statisticwas more powerful.

4. POWERCOMPARISONS For sample sizes of 50 and 100, and p < 1, Qi was the
The powers of the statistics studied in this article most powerfulofthe Q statisticsstudied.For sample size
were comparedwiththat of the Box-Pierce Q statisticin 250, Q5 was the most powerfulQ statistic. The size of
a Monte Carlo studyusingthe model the Q tests forK > 5 was considerablyless than .05 for
n = 50.
Yt= pYt-+ et , t= 1,2, ..., n There is evidence that T and y, are biased tests, ac-
cepting the null hypothesismore than 95 percentof the
where the et - NID (0, a2) and Yo = 0. Four thousand
time forp close to, but less than, one. Because the tests
samples of size n = 50, 100, 250 were generated for are
consistent,the minimumpoint of the powerfunction
p = .80, .90, .95, .99, 1.00, 1.02, 1.05. The random- is moving
toward one as the sample size increases.
number generatorSUPER DUPER fromMcGill Uni-
versitywas used to create the pseudonormalvariables.
Eight two-sidedsize .05 tests of the hypothesisp = 1 5. EXAMPLES
were applied to each sample. The tests were p, T, pA, r,,
Gould and Nelson (1974) investigatedthe stochastic
Q1, Q5, QIO, Q20, whereQK is the Box-PierceQ statistic structureof the velocityof money using the yearlyob-
defined in (1.3) withet = Yt - Yt-i.
servationsfrom1869through1960 givenin Friedmanand
There are several conclusionsto be drawn fromthe
Schwartz (1963). Gould and Nelson concluded that the
resultspresentedin the table. First,the Q statisticsare
logarithm of velocity is consistent with the model
less powerfulthan the statisticsintroducedin thisarticle.
X, = Xt-- + et, where et - N(0, a2) and Xt is the
For example, when n = 250 and p = .8 the worstof the
velocityof money.
statistics introduced in this article rejected the null
Two models,
hypothesis100 percentof the time,whilethe best of the
Qstatisticsrejectedthe nullhypothesisin only45 percent Xt - = p(Xt -X1) -- et (5.1)
of the samples.
and
Second, the performancesof p$and T were similar,and
they were uniformlymore powerfulthan the other test Xt= ,U+ pXt_.1+C e, (5.2)
Dickeyand Fuller:TimeSeriesWithUnitRoot 431
were fitto the data. For (5.1) the estimateswere n(p- 1). Also the "t statistic" constructedby dividing
the coefficient
of Yt-, by the regressionstandarderroris
t- Xi = 1.0044(Xt, - X1)i 2- .0052
approximatelydistributedas ,. For thisexamplewe have
(.0094)
and for (5.2), (n-p -1)(1 + a2)-'

= .0141+ .9702Xt_-, 2= .0050 = 106(-.119)(.502)-l = -25.1

(.0176) (.0199) and
T, = (.033) '(-.119) = -3.61
Model (5.1) assumes that it is knownthat no intercept
entersthe modelifX1 is subtractedfromall observations. Both statisticslead to rejectionof the null hypothesisof
Model (5.2) permits an interceptin the model. The a unit root at the 5 percentlevel if the alternativehy-
numbersin parenthesesare the "standard errors"output pothesisis that both roots are less than one' in absolute
by the regressionprogram.For (5.1) we compute value. The Monte Carlo studyofSection 4 indicatedthat
tests based on the estimated p were more powerfulfor
n(A- 1) = 91(.0044) = .4004 tests against stationaritythan the T statistics. In this
and example the test based on Arejects the hypothesisat a
T = (. 0094) (.0044) = .4681 smallersize (.025) than that of the T statistic (.05).
Using either Table 8.5.1 or 8.5.2 of Fuller (1976), the
hypothesisthat p = 1 is accepted at the .10 level. [ReceivedNovember1976. RevisedNovember1978.]
For (5.2) we obtain the statistics
REFERENCES
P(-- 1) = 92(.9702- 1) = - 2.742 Anderson,TheodoreW. (1959), "On Asymptotic Distributions
of
and Estimatesof Parametersof StochasticDifferenceEquations,"
TA = (.0199)l'(.9702 - 1.0) = - 1.50 AnnalsofMathematical Statistics,
30, 676-687.
Box, GeorgeE.P., and Jenkins,GwilymM. (1970), Time Series
Again the hypothesisis accepted at the .10 level. AnalysisForecasting and Control,San Francisco:Holden-Day.
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fo
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whereet are NID(0, -2) randomvariables. University.
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coefficient versityPress.
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