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: Taylor & Francis, Ltd. on behalf of the American Statistical Association
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Distribution of the Estimators for
Autoregressive Time Series
With a Unit Root
DAVID A. DICKEY and WAYNE A. 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
autoregressive time series whose characteristic equations
variance a2. Properties of the regression estimator of p are obtained have a single root exceeding one and remaining roots less
under the assumption that p = 4 1. Representations for the limit than one in absolute value. Anderson (1959) obtained
distributions of the estimator of p and of the regression t test are
derived. The estimator of p and the regression t test furnish methods
the limiting distributions of estimators for higher-order
of testing the hypothesis that p = 1. processes with more than one root exceeding one in
KEY WORDS: Time series; Autoregressive; Nonstationary;
absolute value.
Random walk; Differencing. The hypothesis that p = 1 is of some interest in ap-
plications because it corresponds to the hypothesis that
1. INTRODUCTION it is appropriate to transform the time series by differ-
encing. Currently, practitioners may decide to difference
Consider the autoregressive model
a time series on the basis of visual inspection of the auto-
Yt = pYt-, + et , t = 1, 2, (I. 1) correlation function. For example, see Box and Jenkins
(1970, p. 174). The autocorrelation function of the devia-
where Yo = 0, p is a real number, and {et} is a sequence tions from the fitted model is then investigated as a test
of independent normal random variables with mean zero of the appropriateness of the model. Box and Jenkins
and variance 0-2 [i.e., et NID(O, -2)]. (1970, p. 291) suggested the Box and Pierce (1970) test
The time series Yt converges (as t -* oo) to a stationary
statistic
time series if IpI < 1. If IPI = 1, the time series is not K
stationary and the variance of Yt is to-2. The time series QK=nErk2, (1.3)
with p = 1 is sometimes called a random walk. If I p I > 1, where k=1 n n
the time series is not stationary and the variance of the
rk = Y
e~~ E
eet-k e2)-
time series grows exponentially as t increases. t=1 t=k+l
Given n observations Y1, Y2, ..., Yn, the maximum
and the et's are the residuals from the fitted model. Under
likelihood estimator of p is the least squares estimator
the null hypothesis, the statistic QK is approximately dis-
n n
tributed as a chi-squared random variable with K - p
A= (E Yt_12)l E YtYt . (1.2)
t=1 t=1
degrees of freedom, where p is the number of parameters
estimated. If I Yt} satisfies (1.1) then p = 0 under the
Rubin (1950) showed that A is a consistent estimator null hypothesis and et = Yt- Yt-1.
for all values of p. White (1958) obtained the limiting The likelihood ratio test of the hypothesis Ho: p = 1
joint-moment generating function for the properly nor- is a function of
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 function to obtain the limiting distribution of
where
p- p. For I p I < 1 the limiting distribution of nG2 p _ p) n
is normal. For IP I > 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 represent the limiting distribution of n (- 1) as
In this article we derive representatiolns for the limiting
that of the ratio of two integrals defined on the Wiener
distributions of p and of T, given that I p = 1. The
process.
representations permit construction of tables of the
percentage points for the statistics. The statistics A and T
* Wayne A. Fuller is Professor of Statistics at Iowa State Uni-
versity, Ames, IA 50011. David A. Dickey is Assistant Professor of
? Journal of the American Statistical Association
Statistics at North Carolina State University, Raleigh, NC 27650.
This research was partially supported by Joint Statistical Agreement June 1979, Volume 74, Number 366
No. 76-66 with the Bureau of the Census. Theory and Methods Section
427
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428 Journal of the American Statistical Association, June 1979
are also generalized to models containing intercept and i > 1, aj -l = aj?j+l = -1 for all j, and ai, = 0 other-
time terms. wise. By a result of Rutherford (1946), the roots of An are
In Section 4 the Monte Carlo method is used to com-
Xi, = ( ) sec2((n - i)r/(2n - 1))
pare the power of the statistics T and A with that of QK.
Examples are given in Section 5. i=l 21,2 ... , n-1
Let M be the t - 1 by n - 1 orthonormal matrix whose
2. MODELS AND ESTIMATORS
ith row is the eigenvector of An corresponding to Xin. The
The class of models we investigate consists of (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 express the normalized denominator sum of
and (c) the model squares appearing in A as
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 observations Yi, Y2, ..., Yn are available where Z = (Zn, Z2n ... , Zn-1,n) = Me,,.
Let
for analysis and define the (n - 1) dimensional vectors,
Hn-n
= (1- (n/2), 2- (n/2), 3 - (n/2), n2 n2 2 n2
. .. , n - 1- (n/2)),
n n(n -1) n (n-2) n (n-3) ... 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
define p as the last entry in the vector
= n-d (Yn_ n- 1 YE1, n-2 y E (n - j) (
(U2/U2)-lU2/yt, (2.3) t=2 j=1
and define PT as the last entry in the vector = Hnen = HnM-1Z . (3.3)
(U3'U3)-lU3'Yt . (2.4) Then
The statistics analogous to the regression t statistics for n - 1) = (2rn)-l(Tn2 -1) + Op(n-1) , (3.4)
the test of the hypothesis that p = 1 are
n(- 1) = (2r - 2Wn2)-l(Tn2- 1 - 2TnW
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 regression residual mean 3.2 Representations for the Limit Distributions
square
Having expressed n(,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-
and Ck is the lower-right element of (Uk'Uk)> tribution of the vector random variable. The following
lemma will be used in our derivation of the limit
3. LIMIT DISTRIBUTIONS distribution.
As the first step in obtaining the limit distributions we Lemma 1: Let {Zijjf l be a sequence of independent
investigate the quadratic forms appearing in the sta- random variables with zero means and common variance
tistics. Because the estimators are ratios of quadratic A Let {wi; i = 1, 2, . . } be a sequence of real numbers
forms we lose no generality by assuming o2 = 1 in the and let {lwin; i = 1, 2, ..., n - 1; n = 1, 2, ...} be a
sequel. triangular array of real numbers. If
3.1 Canonical Representation of the Statistics E Wi2 < 00
i=i
Given that p = 1, the quadratic form Et=2 Yt-12 can
n-i
be expressed as en'Anen, where en' _ (e1, e2, ..., e_)
lim EI wij2 = o Wi2
the elements at3 of An-l satisfy all = 1, a,, = 2 for n-*Oo i=l i=l
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Dickey and Fuller: Time Series With Unit Root 429
and For fixed i,
lim Win = wi
nx -000 lim in = =i (ai, bi, 9i)'
n -o
then Lil, wiZ, is well defined as a limit in 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 b 2 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 2X
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, for example, 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
Therefore, by (3.7) and Lemma 1, Tn* converges in
M
probability to T. It follows by analogous arguments that
aI (win - w,)2 < E/9
i=1 (Pn*, Tn*, Wn*, Vn*) converges in probability to
and (r, T, W, V). Because the distribution of (rn*, Tn*, Wn*,
n Vn*) is the same as that of nqn we obtain the conclusion.
a W172 < 3/9
i=M+l Corollary 1: Let Y, satisfy (1.1) with p 1. Then
Hence, for all n > No,
(p 1) I(P-'(T2-1),
n 00
var{ wi,-Zi - wiZiI < E n(- 1) A ( (r- 2W (T2 - 1)-2W]
i 1 i-1l
AU T 2TW]2-l)
and the result follows by Chebyshev's inequality.
and
Theorem 1: Let {Zi},ll be a sequence of NID(O, 1)
T A 2 T- 1) - 2TW]
random variables. Let qn' = (Pn, Tn0, Wn, V70), where the
elements of the vector are defined in (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
00 00
and *[(T - 2W)(T - 6V) - 1]
(F, T) - (, iy*2Z,2, , 2-yiZi)
TTA2(I7 - W2 - 3V2)U[(T - 2W) (T - 6V) - 1]
00 ~~~~00
Proof: The proof is an immediate consequence of
(W, V) = (E 2 yi2Zi, E 2F[2yi3- I2]Zi)
Theorem 1 because the denominator quadratic forms
'Yi = -1 'i\ol in p pA, PJ are continuous functions of q that have pro
ability 1 of being positive and the Sek2 of (2.8) converge
and
in probability to 02.
Y 2 lim n-2Xin= 4E (2i - l 7r]-2
70 --ic
The numerator and denominator of the limit repre-
sentation of n(A- 1) are consistent with White's (1958)
Then )n converges in distribution to -q, that is,
limit joint-moment generating function.
Note that the limiting distributions of A, and TA are
?In >7
obtained under the assumption that the constant term
,u is zero. Likewise, the limiting distributions of Pr, and
Proof: Note that q is a well-defined random variable
because FJt? iik < ?O for k = 2, 3, ..., 6. Let kin 7
beare
thederived under the assumption that the coefficient
ith column of HRM-1, where for time, R3, is zero. The distributions of P and TT are
unaffected by the value of ,u in (2.2). If ,u # 0 for (2.1)
in= (ai,e, bin0 gj0)' or A3 z 0 for (2.2), the limiting distributions of $, and tr
= [cov(T70, Zi0), cov(W70 Zin), cov(V70, Zi0)]' are normal. Trhus if (2.1) is the mainltainled model and
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430 Journal of the American Statistical Association, June 1979
the statistic Ty is used to test the hypothesis p = 1, the Monte Carlo Power of Two-Sided
hypothesis will be accepted with probability greater than Size .05 Tests of p = 1
the nominal level where ,u # 0.
By the results of Fuller (1976, p. 370), the limiting dis- p
tributions of p, p,J, and p', given that p = -1, are identi-
n Test .80 .90 .95 .99 1.00 1.02 1.05
cal and equal to the mirror image of the limiting distribu-
tion of A given that p = 1. Likewise, the limiting dis- 50 Q1 .09 .05 .05 .04 .04 .07 .47
Q5 .07 .04 .03 .03 .04 .08 .53
tributions of T, T, and Tr for p = -1 are identical and
QIO .05 .04 .03 .03 .03 .09 .54
equal to the mirror image of the limiting distribution of Q20 .03 .02 .02 .02 .02 .08 .52
T for p = 1. .57 .18 .08 .05 .05 .14 .71
T .57 .18 .08 .04 .05 .23 .70
In our derivations Y0 is fixed. The distributions of 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
distribution of A does not depend on Yo, but the small- 100 Q1 .15 .07 .05 .04 .05 .26 .94
sample distribution of A will be influenced by Yo. Q5 .13 .08 .05 .04 .04 .34 .95
In the derivations we assumed the et to be NID (0, a2). QIO .11 .06 .05 .03 .04 .37 .95
Q20 .08 .05 .04 .03 .03 .38 .95
The limiting distributions also hold for et that are inde- / .99 .55 .17 .05 .05 .54 .98
pendent and identically distributed nonnormal random T .99 .55 .17 .04 .05 .59 .97
.86 .30 .10 .05 .05 .49 .98
variables with mean zero and variance o-. White (1958)
.73 .18 .06 .04 .05 .51 .98
and Hasza (1977) have discussed this generalization.
The statistic T is a monotone function of the likelihood 250 Q1 .34 .12 .06 .05 .06 .94 1.00
Q5 .45 .13 .07 .04 .05 .95 1.00
ratio when Y0 is fixed under the null model of p = 1 and
QIo .34 .12 .06 .04 .05 .95 1.00
under the alternative model of p 5 1. Tests based on the Q20 .24 .10 .05 .04 .04 .95 1.00
p5 1.00 1.00 .74 .08 .05 .98 1.00
r statistics are not likelihood ratios and not necessarily
T 1.00 1.00 .74 .08 .05 .97 1.00
the most powerful that can be constructed if, for example, pg. 1.00 .96 .43 .06 .05 .98 1.00
the alternative model is that (Y0, Yi, . . ., Yn) is a portion 1.00 .89 .28 .04 .05 .98 1.00
of a realization from a stationary autoregressive process.
A set of tables of the percentiles of the distributions is
given in Fuller (1976, pp. 371,373) and a slightly more statistics. It is not surprising
accurate set in Dickey (1976). Dickey also presents to p, and T, because p and T
true value of the intercept in the regression is zero.
details of the table construction and gives estimates of
Third, for p < 1 the statistic A, yielded a more power-
the sampling error of the estimated percentiles.
ful test than the statistic TA. For p > 1 the ranking was
reversed and the Ty statistic was more powerful.
4. POWER COMPARISONS
For sample sizes of 50 and 100, and p < 1, Qi was the
The powers of the statistics studied in this article most powerful of the Q statistics studied. For sample size
were compared with that of the Box-Pierce Q statistic in 250, Q5 was the most powerful Q statistic. The size of
a Monte Carlo study using the model the Q tests for K > 5 was considerably less 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 hypothesis more than 95 percent of the
where the et - NID (0, a2) and Yo = 0. Four thousand
time for p close to, but less than, one. Because the tests
samples of size n = 50, 100, 250 were generated for
are consistent, the minimum point of the power function
p = .80, .90, .95, .99, 1.00, 1.02, 1.05. The random-
is moving toward one as the sample size increases.
number generator SUPER DUPER from McGill Uni-
versity was used to create the pseudonormal variables.
Eight two-sided size .05 tests of the hypothesis p = 1 5. EXAMPLES
were applied to each sample. The tests were p, T, pA, r,,Gould and Nelson (1974) investigated the stochastic
Q1, Q5, QIO, Q20, where QK is the Box-Pierce Q statistic
structure of the velocity of money using the yearly ob-
defined in (1.3) with et = Yt - Yt-i.
servations from 1869 through 1960 given in Friedman and
There are several conclusions to be drawn from the
Schwartz (1963). Gould and Nelson concluded that the
results presented in the table. First, the Q statistics are
logarithm of velocity is consistent with the model
less powerful than the statistics introduced in this article.
X, = Xt-- + et, where et - N(0, a2) and Xt is the
For example, when n = 250 and p = .8 the worst of the
velocity of money.
statistics introduced in this article rejected the null
Two models,
hypothesis 100 percent of the time, while the best of the
Q statistics rejected the null hypothesis in only 45 percent Xt - = p(Xt -X1) -- et (5.1)
of the samples.
and
Second, the performances of p$ and T were similar, and
they were uniformly more powerful than the other test Xt= ,U + pXt_.1+C e, (5.2)
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Dickey and Fuller: Time Series With Unit Root 431
were fit to the data. For (5.1) the estimates were n(p- 1). Also the "t statistic" constructed by dividing
the coefficient of Yt-, by the regression standard error is
t- Xi = 1.0044(Xt, - X1)i 2- .0052
approximately distributed as ,. For this example we 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 known that no intercept
enters the model if X1 is subtracted from all observations. Both statistics lead to rejection of the null hypothesis of
Model (5.2) permits an intercept in the model. The a unit root at the 5 percent level if the alternative hy-
numbers in parentheses are the "standard errors" output pothesis is that both roots are less than one' in absolute
by the regression program. For (5.1) we compute value. The Monte Carlo study of Section 4 indicated that
tests based on the estimated p were more powerful for
n(A- 1) = 91(.0044) = .4004 tests against stationarity than the T statistics. In this
and example the test based on A rejects the hypothesis at a
T = (. 0094) (.0044) = .4681 smaller size (.025) than that of the T statistic (.05).
Using either Table 8.5.1 or 8.5.2 of Fuller (1976), the
hypothesis that p = 1 is accepted at the .10 level. [Received November 1976. Revised November 1978.]
For (5.2) we obtain the statistics
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P(- - 1) = 92(.9702 - 1) = - 2.742 Anderson, Theodore W. (1959), "On Asymptotic Distributions of
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