Distribution of the Estimators for Autoregressive Time Series With a Unit Root
Author(s): David A. Dickey and Wayne A. Fuller
Reviewed work(s):
Source: Journal of the American Statistical Association, Vol. 74, No. 366 (Jun., 1979), pp. 427431
Published by: American Statistical Association
Stable URL: http://www.jstor.org/stable/2286348 .
Accessed: 04/03/2013 13:31
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 forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal
of the American Statistical Association.
http://www.jstor.org
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions
Distribution
of the Estimators
for
Autoregressive
TimeSeries
Witha UnitRoot
DAVIDA. DICKEYand WAYNEA. FULLER*
Let n observations Yi, Y2, ...,
Yn be generated by the model
Yt = pYt-1 + et, where Y0 is a fixed constant and {et t_ln is a sequence of independent normal random variables with mean 0 and
variance a2. Properties of the regressionestimator of p are obtained
under the assumption that p = 4 1. Representations for the limit
distributionsof the estimator of p and of the regressiont test are
derived. The estimatorof p and the regressiont test furnishmethods
of testingthe hypothesisthat p = 1.
Rao (1961) extended White's results to higher-order
autoregressivetime serieswhose characteristicequations
have a singleroot exceedingone and remainingrootsless
than one in absolute value. Anderson (1959) obtained
the limitingdistributionsof estimatorsfor higher-order
processes with more than one root exceeding one in
value.
absolute
WORDS: Time series; Autoregressive; Nonstationary;
KEY
The hypothesisthat p = 1 is of some interestin apRandom walk; Differencing.
plicationsbecause it correspondsto the hypothesisthat
it is appropriateto transformthe time series by differ1. INTRODUCTION
encing.Currently,practitionersmay decide to difference
Considerthe autoregressivemodel
a timeserieson the basis of visual inspectionofthe auto(I. 1) correlationfunction.For example, see Box and Jenkins
Yt = pYt-, + et , t = 1,2,
(1970, p. 174). The autocorrelationfunctionofthe deviawhere 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
K
time series if IpI < 1. If IPI = 1, the time series is not
QK=nErk2,
(1.3)
stationaryand the variance of Yt is to-2. The time series
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 =
eet-k
YE e~~
E
eIt'
e2)-1
time series growsexponentiallyas t increases.
t=1
t=k+l
Given n observationsY1, Y2, ..., Yn, the maximum
and the et'sare the residualsfromthe fittedmodel. Under
likelihoodestimatorof p is the least squares estimator
the null hypothesis,the statisticQK is approximatelydisn
n
tributedas a chi-squaredrandom variable with K - p
A=
(1.2) degreesof freedom,wherep is the numberof parameters
(E Yt_12)l E YtYt .
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.
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
For
malized numerator and denominator of A-p.
n
invert
the
1
to
able
he
was
#
genjoint-moinent
IpI
A =
_
Se
1)
(,E Yt-2)
(Ap
erating functionto obtain the limitingdistributionof
_
where
p)
ofnG2p
p- p. For Ip I < 1 the limitingdistribution
n
is normal. For IPI > 1 the limiting distribution of
Se2 = (n - 2) E (Yt - pYt i)2
t=2
is Cauchy. For p = 1, White was
-p)
(
|pln(p2-I)-1
1) as In this article we derive representatiolns
able to representthe limitingdistributionof n (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
percentagepointsforthe statistics.The statisticsAand T
A
* Wayne A. Fuller is Professor of Statistics at Iowa State University,Ames, IA 50011. David A. Dickey is Assistant Professorof
Statistics at North Carolina State University,Raleigh, NC 27650.
This researchwas partiallysupported by JointStatistical Agreement
No. 76-66 with the Bureau of the Census.
? Journalof the AmericanStatistical Association
June 1979,Volume74, Number366
Theoryand Methods Section
427
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions
428
Journalof the AmericanStatisticalAssociation,June 1979
are also generalizedto models containinginterceptand i > 1, aj -l = aj?j+l = -1 forall j, and ai, = 0 otherwise. By a resultofRutherford(1946), the rootsofAnare
timeterms.
In Section 4 the Monte Carlo methodis used to comXi, = ( ) sec2((n - i)r/(2n - 1))
pare the powerof the statisticsT and Awith that of QK.
i=l 21,2... , n-1
Examples are given in Section 5.
matrixwhose
Let M be the t - 1 by n - 1 orthonormal
to Xin.The
ith rowis the eigenvectorofAncorresponding
M
is
of
itth
element
The class of models we investigateconsistsof (a) the
model (1.1), (b) the model
2(2n -1)Mit=
2. MODELS AND ESTIMATORS
2
Yt =,u + pYt-, + et,
t = 1, 2,...
(2.1)
0
Yo=
1A+
Yo =
0 .
2)-1(2t - 1)(2i -)r]
-
(3.1)
,
and we can expressthe normalizeddenominatorsum of
squares appearingin as
A
and (c) the model
Yt =
cos [(4n
t = 1, 2, ...
ft + pYt- + et,
(2.2)
=
rn
n
n-I
E Yt_12
n-2
n-2
-
E
XinZin2
X
(3.2)
t=2
Assume n observationsYi, Y2, ..., Yn are available whereZ
Let
foranalysis and definethe (n - 1) dimensionalvectors,
=
(Zn,
...
Z2n
Zn-1,n)
,
Me,,.
=
Hn-n
=
(1-
yt = (Y2, Y3, Y4,
Yt-11 =
(n/2), 3 - (n/2),
(n/2), 2-
(Y1, Y2, Y3,
. .. , n - 1-
. ..
I Y.)
.
I
Yn_1),
(n/2)),
n -1)
n(n
I
Let U= Yt-1, U2 = (1, Yt-1), and U3 = (1, t,Yt-1). We
definep as the last entryin the vector
(Tny Wn,Vn)
=
=
and definePT as the last entryin the vector
(U3'U3)-lU3'Yt
(2.4)
.
The statisticsanalogousto the regressiont statisticsfor
the test of the hypothesisthat p = 1 are
A
=
(A-
A
=
(pAA
Tr=(r-
1)
-
(Sel2Cl)
1)(Se22C2)
1) (Se32C3)4
(2.5)
,
,
2,
(2.6)
(2.7)
where Sek2 is the appropriate regressionresidual mean
square
Sek2= (n - k - 1) '[Y1'(I
-
Uk(Uk'Uk)'lUk')Yt]
n (n-2)
n- 2
n2
n (n-3)
...
2(n-3)
...
n
n-2
and
(2.3)
(U2/U2)-lU2/yt,
2
n2
n2
n-d (Yn_
n-i
n
y
YE1,
1n-
E
n-2
t=2
Hnen = HnM-1Z
(n
-
j=1
j) (j - )ej)'
(3.3)
.
Then
n
-
1) = (2rn)-l(Tn2 -1) +
1) =
n(-
Op(n-1)
(2r - 2Wn2)-l(Tn2-
1 - 2TnWn)
+
n(Pr
1) =
*[(Tn-
[2(Pr
-
-Wn2
2Wn)(Tn -
3
6Vn) -
(3.4)
,
(3.5)
Op(n-6),
2)]-1
1] + Op(n-)
.
(3.6)
forthe LimitDistributions
3.2 Representations
Having expressedn(,6 - 1), n(AM-1),
and
nC(A'-
1)
(2.8) in terms of (rn, Tn, Wn, Vn) we obtain the limiting dis-
elementof (Uk'Uk)>
and Ck is the lower-right
3. LIMIT DISTRIBUTIONS
As the firststep in obtainingthe limitdistributionswe
investigatethe quadratic formsappearing in the statistics. Because the estimatorsare ratios of quadratic
formswe lose no generalityby assumingo2 = 1 in the
sequel.
tributionof the vector random variable. The following
lemma will be used in our derivation of the limit
distribution.
Lemma 1: Let {Zijjf l be a sequence of independent
randomvariables withzero means and commonvariance
A Let {wi; i = 1, 2, . . } be a sequence of real numbers
and let
{lwin; i =
1, 2, ...,
-
1; n
=
triangulararray of real numbers.If
of the Statistics
3.1 Canonical Representation
Given that p = 1, the quadratic formEt=2 Yt-12 can
be expressedas en'Anen,where en' _ (e1, e2, ..., e_)
the elements at3 of An-l satisfy all = 1, a,, = 2 for
n
E
i=i
Wi2
<
00
n-i
limEI wij2= o Wi2
n-*Ooi=l
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions
i=l
1, 2, ...}
be a
Dickeyand Fuller:TimeSeriesWithUnitRoot
and
lim Win
429
For fixedi,
= wi
lim in=
nx-000
n
then Lil, wiZ, is well definedas a limitin mean square
and
00
n
plim{LwZ}=W
i=1
Proof: Let
such that
e
i_l1
wiZ
> 0 be given. Then we can choose an M
21(yi,7y2, 2yi3-
E (a
2 b2
Let
(
E
Wi2 <
E/9
n-I
n-1
n 2XinZZ,
and
(,
=
binZi,
i=i
00
n
WZ
a21
2-
1j
i=1
W,21 <
for all n > M. Furthermore,given Ml, we can choose
No > M such that n > No implies
aI
(win - w,)2 < E/9
i=1
and
n
a
i=M+l
W172 <
lim
n-I
var{
i 1
i-1l
3/9
E
(r, T, W, V)
-
iy*2Z,2,,
(W, V)
=
'Yi =
and
Y
2
(E
-1
2-yiZi)
2 yi2Zi, E 2F[2yi3-
I2]Zi)
'i\ol
lim n-2Xin= 4E (2i -l
7r]-2
(T2
2W
-
1)-2W]
2TW]2-l)T
and
T
A
-
1)
(
-
W2-
1) -
T-
2
Let Yt satisfy(2.1) with p
TTA2(I7
~~~~00
00
A ((r-
=
1. Then
*[(T
-
W2
-
2TW]
3V2)-1
and
00
(,
1)
1. Then
I(P-'(T2-1),
AU
n
where
(F, T)
1)
n(-
Theorem1: Let {Zi},ll be a sequence of NID(O, 1)
randomvariables. Let qn' = (Pn, Tn0,Wn, V70),wherethe
elementsof the vectorare definedin (3.2) and (3.3). Let
00
n-o-
Corollary 1: Let Y, satisfy (1.1) with p
(p
wiZiI <
1
Tn}
-limvarI
(Pn*, Tn*, Wn*, Vn*) converges in probability to
(r, T, W, V). Because the distributionof (rn*, Tn*, Wn*,
Vn*) is the same as that of nqnwe obtain the conclusion.
and the resultfollowsby Chebyshev'sinequality.
=
ai2 _ limvar{Tn*}
n-ow
Therefore,by (3.7) and Lemma 1, Tn* converges in
probabilityto T. It followsby analogous argumentsthat
00
wi,-Zi -
E
n--*= i=1
Hence, forall n > No,
n
.
ginZi)
E
i=1
Now, forexample,by (3.3)
f/9
i=1
M
ainZi)
n-I
n-1
Vn*)
(Wn*,
1/30)
(1, 1/3,
=
g 2)
(Pn*, Tn*) =
i=M+l
(3.7)
y2)*
By Jolley (1-961,p. 56, #307,308) we have
00
2
(ai, bi,9i)'
=i
-o
3V2)U[(T
-
2W)(T
-
6V)
-
1]
- 2W) (T - 6V) - 1]
Proof: The proof is an immediate consequence of
Theorem 1 because the denominatorquadratic forms
in p pA, PJ are continuousfunctionsof q that have prob
ability 1 of being positiveand the Sek2 of (2.8) converge
in probabilityto 02.
The numeratorand denominatorof the limit representation
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
for time, R3,is zero. The distributionsof P and TT are
ith columnof HRM-1,where
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
model and
are normal. Trhusif (2.1) is the mainltainled
=
[cov(T70,
Zi0),cov(W70 Zin),cov(V70, Zi0)]'
70 --ic
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions
Journal
of the American
StatisticalAssociation,
June1979
430
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.
p
By the resultsof Fuller (1976, p. 370), the limitingdistributionsof p, p,J,and p', giventhat p = -1, are identi- n
.95
Test
.80
.90
.99
1.00
1.02
1.05
cal and equal to the mirrorimage ofthe limitingdistribu.05
.04
Q1
.09
.05
.04
.07
.47
tion of given that p = 1. Likewise, the limitingdis- 50
.07
.04
.03
.04
.03
.53
.08
Q5
tributionsof T, T, and Tr forp = -1 are identical and
.05
.04
.03
.03
.03
.09
.54
QIO
equal to the mirrorimage of the limitingdistributionof
.03
.02
.02
.02
.02
.52
.08
Q20
.57
.18
.08
.05
.05
.14
.71
T forp = 1.
.57
.18
.04
.05
T
.08
.23
.70
In our derivationsY0 is fixed.The distributionsof pA
.11
.28
.10
.06
.05
.06
.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
.15
Q1
.07
.05
.04
.05
.94
.26
.13
.08
.05
.04
sample distributionof will be influencedby Yo.
.04
.34
.95
Q5
.11
.06
.05
.03
.04
.37
.95
QIO
In the derivationswe assumed the etto be NID (0, a2).
.08
.05
.04
.03
.03
.38
.95
Q20
The limitingdistributionsalso hold foret that are inde/
.99
.55
.17
.05
.05
.54
.98
.17
.04
.05
pendent and identicallydistributednonnormalrandom
.99
.55
.97
T
.59
.05
.86
.30
.10
.05
.49
.98
variables with mean zero and variance o-. White (1958)
.73
.18
.06
.04
.05
.51
.98
and Hasza (1977) have discussedthis generalization.
.34
.12
250
.06
.05
.06
.94
1.00
Q1
The statisticT is a monotonefunctionofthe likelihood
.45
.07
1.00
.13
.04
.05
.95
Q5
ratio when Y0 is fixedunderthe null model of p = 1 and
.34
.12
.06
.04
.05
.95
1.00
QIo
.24
.10
.05
.04
.04
.95
1.00
underthe alternativemodel ofp 5 1. Tests based on the
Q20
.74
.05
p5
1.00
1.00
.98
1.00
.08
r statisticsare not likelihoodratios and not necessarily
1.00
1.00
1.00
.74
.05
.97
T
.08
the mostpowerfulthat can be constructedif,forexample,
1.00
.06
.05
pg.
1.00
.96
.43
.98
1.00
.89
.28
.04
.05
.98
1.00
thealternativemodelis that (Y0, Yi, . . ., Yn) is a portion
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.
Third,forp < 1 the statistic A, yieldeda more powerthe samplingerrorof the estimatedpercentiles.
ful test than the statistic For p > 1 the rankingwas
reversedand the Ty statisticwas more powerful.
4. POWERCOMPARISONS
For sample sizes of 50 and 100, and p < 1, Qi was the
most
powerfulofthe Q statisticsstudied.For sample size
The powers of the statistics studied in this article
250,
was the most powerfulQ statistic. The size of
Q5
were comparedwiththat of the Box-Pierce Q statisticin
the
tests
Q
forK > 5 was considerablyless than .05 for
a Monte Carlo studyusingthe model
A
A
A
TA.
1,2, ..., n
n = 50.
There is evidence that T and y, are biased tests, acthe null hypothesismore than 95 percentof the
cepting
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 Universitywas used to create the pseudonormalvariables.
5. EXAMPLES
Eight two-sidedsize .05 tests of the hypothesisp = 1
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 yearlyobin (1.3) withet = Yt - Yt-i.
defined
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
Xt = p(Xt -X1) -- et
(5.1)
Qstatisticsrejectedthe nullhypothesisin only45 percent
of the samples.
and
Second, the performancesof p$and T were similar,and
(5.2)
Xt= ,U+ pXt_.1+C
e,
they were uniformlymore powerfulthan the other test
Yt=
pYt-+
et
,
t=
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions
Dickeyand Fuller:TimeSeriesWithUnitRoot
431
were fitto the data. For (5.1) the estimateswere
Xi = 1.0044(Xt,
(.0094)
and for (5.2),
t-
=
-
X1)i
2-
n(p- 1). Also the "t statistic" constructedby dividing
the coefficient
of Yt-, by the regressionstandarderroris
approximatelydistributedas ,. For thisexamplewe have
.0052
(n-p
.0141+ .9702Xt_-,
.0050
2=
(.0176) (.0199)
n(A-
1)
T =
(. 0094)
=
91(.0044) = .4004
(.0044) = .4681
Using either Table 8.5.1 or 8.5.2 of Fuller (1976), the
hypothesisthat p = 1 is accepted at the .10 level.
For (5.2) we obtain the statistics
P(-- 1)
and
TA
=
=
92(.9702- 1) = - 2.742
(.0199)l'(.9702 - 1.0)
=
- 1.50
Again the hypothesisis accepted at the .10 level.
As a second example we study the logarithmof the
quarterlyFederal Reserve Board Production Index for
the period 1950-1 through1977-4. We assume that the
time seriesis adequately representedby the model
Yt =
fo
+ ,#it
+ aiYt-i +
a2Yt-2 +
et
whereet are NID(0, -2) randomvariables.
On the basis of the resultsof Fuller (1976, p. 379) the
coefficient
of Yt-, in the regressionequation
Yt-
Yt-i = ,Bo+ fit + (ca,+
ca2--)Yt-1
- a2(Yt-l.-
Yt-2) +
et
can be used to test the hypothesisthat p = a, + ca2 = 1.
This hypothesisis equivalentto the hypothesisthat one
of the roots of the characteristicequation of the process
is one. The least squares estimateof the equation is
Pt'- Yt-, = .52 + .00120t- .119Yt_
(.15) (.00034) (.033)
+ .498(Yt-,
(.081)
-Yt_2),
-2
= .033
Thereare 110 observationsin theregression.The numbers
in parenthesesare the quantities output as "standard
errors"by the regressionprogram.On the basis of the
resultsof Fuller,the statistic(n - p) (A - 1) (1 + 62)',
of Yt-, and p is the numberof
where Ais the coefficient
parametersestimated, is approximatelydistributedas
+
a2)-'
=
and
Model (5.1) assumes that it is knownthat no intercept
entersthe modelifX1 is subtractedfromall observations.
Model (5.2) permits an interceptin the model. The
numbersin parenthesesare the "standard errors"output
by the regressionprogram.For (5.1) we compute
and
-1)(1
T, =
106(-.119)(.502)-l
= -25.1
(.033) '(-.119) = -3.61
Both statisticslead to rejectionof the null hypothesisof
a unit root at the 5 percentlevel if the alternativehypothesisis that both roots are less than one' in absolute
value. The Monte Carlo studyofSection 4 indicatedthat
tests based on the estimated p were more powerfulfor
tests against stationaritythan the T statistics. In this
example the test based on Arejects the hypothesisat a
smallersize (.025) than that of the T statistic (.05).
[ReceivedNovember1976. RevisedNovember1978.]
REFERENCES
Anderson,TheodoreW. (1959), "On Asymptotic
Distributions
of
Estimatesof Parametersof StochasticDifferenceEquations,"
AnnalsofMathematical
Statistics,
30, 676-687.
Box, GeorgeE.P., and Jenkins,GwilymM. (1970), Time Series
AnalysisForecasting
and Control,
San Francisco:Holden-Day.
Box, GeorgeE.P., and Pierce,David A. (1970), "Distributionof
Residual Autocorrelations
in Autoregressive-Integrated
Moving
AverageTime SeriesModels,"JournaloftheAmericanStatistical
Association,
65, 1509-1526.
David, HerbertA. (1970),OrderStatistics,
New York: JohnWiley
& Sons.
Dickey,David A. (1976), "Estimationand HypothesisTestingin
NonstationaryTime Series," Ph.D. dissertation,Iowa State
University.
Friedman,Milton,and Schwartz,A.J. (1963),A Monetary
History
of theUnitedStates1867-1960,Princeton,N.J.: PrincetonUniversityPress.
Fuller,WayneA. (1976),Introduction
toStatistical
TimeSeries,New
York: JohnWiley& Sons.
Gould,JohnP., and Nelson,CharlesR. (1974), "The Stochastic
ofthe Velocityof Money,"AmericanEconomicReview,
Structure
64, 405-417.
Hasza, David P. (1977),"Estimationin Nonstationary
TimeSeries,"
Ph.D. dissertation,
Iowa State University.
Jolley,L.B.W. (1961), Summation
ofSeries (2nd ed.), New York:
Dover Press.
ofEstimaRao, M.M. (1961),"Consistency
and LimitDistributions
torsofParametersin ExplosiveStochasticDifference
Equations,"
AnnalsofMathematical
Statistics,
32, 195-218.
ofan Estimatorofthe
Rao, M.M. (1978),"Asymptotic
Distribution
BoundaryParameterofan UnstableProcess,"AnnalsofStatistics,
6, 185-190.
Rubin, Herman (1950), "Consistencyof Maximum-Likelihood
Estimatesin the Explosive Case," in StatisticalInferencein
DynamicEconomicModels,ed. T.C. Koopmans,New York: John
Wiley& Sons.
D.E. (1946), "Some ContinuantDeterminants
Rutherford,
Arising
in Physicsand Chemistry,"
of theRoyal Societyof
Proceedings
Edinburgh,
Sect. A, 62, 229-236.
White,JohnS. (1958), "The LimitingDistributionof the Serial
in the ExplosiveCase," AnnalsofMatheCoefficient
Correlation
maticalStatistics,
29, 1188-1197.
This content downloaded on Mon, 4 Mar 2013 13:31:08 PM
All use subject to JSTOR Terms and Conditions