Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root

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

Reviewed work(s):
Source: Econometrica, Vol. 49, No. 4 (Jul., 1981), pp. 1057-1072
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1912517 .
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 Econometrica,Vol. 49, No. 4 (July, 1981)

LIKELIHOOD RATIO STATISTICS FOR AUTOREGRESSIVE

TIME SERIES WITH A UNIT ROOT

BY DAVID A. DICKEY AND WAYNE A. FULLER

Let the time series Y, satisfy Y, = a + pY,_- + e,, where Y1 is fixed and the e, are
normal independent (0, a2) random variables. The likelihood ratio test of the hypothesis
that (a, p) = (0, 1) is investigated and a limit representation for the test statistic is pre-
sented. Percentage points for the limiting distribution and for finite sample distributions
are estimated. The distribution of the least squares estimator of a is also discussed. A
similar investigation is conducted for the model containing a time trend.

1. INTRODUCTION

LET Y, SATISFY THE MODEL

(1.1) Yt=a+pYt-I+et (t n),

where Y, is fixed and {et} is a sequence of normal independent random variables
with mean 0 and variance a2, [et,NI(0, a2)]. The maximum likelihood estimators
of p and a, conditional on Y1, are the least squares estimators
n -In
(1.2) p, = (Y,_l I9(1))2 E(,Y0)Y,I-(1)

= Y(o) - - )9
% pj(_

where (i) = (n - lEnt2Y+ for i= -1,0.

The statistic constructed by analogy to the regression "t statistic" for the
estimated a is
A
IA

a, a,i /L

where

Sa e14\ - 1)1 + '' (2 t-1 Ye,)} ) j,

n

(1.) t ot +;(n--
e n) + (- t- (t = , ,. , )
An alternative model for Y is 2

where Y1 is fixed and e, NI(0, 2). Let X denote the (n-1) x 3 matrix whose
ith row is (1,i - In, Y,) and let Y' = (Y2, Y3, . .. , Yn).Then the least squares
1057
1058 D. A. DICKEY AND W. A. FULLER

estimator of 0 = (a, ,B,p)' is

(1.4) = (XX)-
as= (&x,I3s,,$s)' x Y.
Let C, denote the ijth element of (X'X)'. Then the "regression t statistics" are

(1.5) Ta( = ClSe) a,

(1.6) TI7(= C22SeT) /39

where

(1.7) = (n-4)-1Y'[I-X(X'X)-IX']Y.
Se2T
We shall study the likelihood ratio test of the hypothesis that (a, p) = (0, 1) for
model (1.2), the likelihood ratio test of the hypothesis that (a, 3,p) = (0,0, 1) for
model (1.3), and the likelihood ratio test of the hypothesis that (a, A, p) = (a, 0, 1)
for model (1.3). We also investigate the distributions of a,_,,Talk &a, 8,-,Tar, and T,,T
under the null model.
The likelihood ratio statistics are derived in Section 2 and the limiting
distributions presented in Section 3. Percentage points for the distributions
obtained by Monte Carlo methods are given in Section 4. In Section 5, it is
shown that the limit distributions of the test statistics are unchanged when {et} in
(1.1) and (1.3) is replaced by a stationary pth order autoregressive process whose
coefficients must be estimated. In Section 6 the powers of the likelihood ratio
tests (DI,02 and 03 are compared to the powers of other test statistics. Section 7
contains an illustration of the use of the test statistics.

2. LIKELIHOOD RATIO STATISTICS

We construct the likelihood ratio statistics for the null hypothesis that the true
model is a random walk with zero drift. We consider first the test that (a, 3,p)
= (0,0, 1) in model (1.3) against the alternative that the null is not true. The
logarithm of the likelihood function for a sample of n observations from model
(1.3), conditional on Y1, is
log L = (n - l)log(27) - (n - l)log a

n
t
-(2a) E Y- ,Bt -2 n)_ _-1]2.
t=2

Under the null hypothesis, Ho: (a, 3,p) = (0,0, 1), the likelihood is maximized
with respect to a2 to obtain
n
aO2= (n -l1)- Y-t_ 1)2.
t=2

Under the alternative hypothesis the maximum of the likelihood occurs at

 LIKELIHOOD RATIO STATISTICS 1059

( 0, '), where O, was defined in (1.4) and 62 = (n - 4)(n - 1)- 'S2. Therefore the
likelihood ratio is

(TO a] =I[ + 3(n -4) _'2]21

where

(2 = (3Se2 )'[(n - l)a -(n -4)Se,].

Thus, the likelihood ratio test rejects the null hypothesis for large values of
02' where '2 iS the usual regression "F test" of the hypothesis HO:(a, /, p) =
(0,0,1).
In a similar manner, it can be shown that the likelihood ratio statistic for
testing HO:(a, p) = (0, 1) against HA: Not HO, for the model (1.1) is

[1 + 2(n 3)-'(DI 2

where
= (2S
4?1 )'[(n - l)0- (n - 3)SJ.j

The likelihood ratio test of the hypothesis HO:(a, /3,p) = (a, 0, 1) in model
(1.3) is a monotone function of

2 _ _ y( _ 1))2)}(-)S2
4> Ser)[(-1 ) {2 (y(o)

The statistics 02 and 0I3 are the common regression "F tests" one would
construct for the hypotheses. The null hypothesis for test D3 iS that the time series
is a random walk with drift a. It is easily demonstrated that the distribution of
the test statistic 03 does not depend upon a.

3. LIMITING DISTRIBUTIONS

Under the null hypotheses, the statistics introduced in Sections 1 and 2 can be
expressed as functions of a few sample statistics. Let
n t-1 2
(3.1) Fn = (n-1)-2 ej

n
Tn = (n- 1) 2 et= (n- 1)f(0),
t=2

t =2

n-1
n-i
(n - 51)/2 -
Vn= E (n -t)(t I)e,.
t= 1
1060 D. A. DICKEY AND W. A. FULLER

Then, for example,

(n -T)nTA = - (n-l)(5 -l)Wn

and

(n -
l)(A, -1) = (In-W 2) 4 [(Tn + (n - 1)2e,)

-(n-1 1 et2] -TnWn}

-(rn- Wn2) {2(T2a2) TnWn} + OP(nA).

Because Tn and Wn are odd functions of (el, ~2., en) = en and Fn is an

even function of en, the distributions of a and TTa,are symmetric. Given that
a2= 1, Dickey and Fuller [7] have shown that [F, Tn,Wn, V,n(p, - 1)] con-
verges in distribution to (F, T, W, V, 8), where
00 00

r = yi2Zi2 T= 2
i=l i=l
00 00

W= 22-yZ,Z, V= E (2?'7 - 2?y')Zi,

i=l i=l

(2i 2), -
and {Z,}j is a sequence of normal independent (0, 1) random variables.
Therefore, given that (a, p) = (0, 1),

n 2a1- a,-- T- SW,

and

(3.3) Ta- (T- SW)(P - W2)2r-2,

because Sell converges in probability to a .

For model (1.3) with the assumption that 0' = (0,0, 1), we have

1 0 (n- 1) Wn
X'X = (n-1) 0 12-'n(n-2) 2(n-1)2Vn

(n-1)2Wn I(n-1)3Vn (n-1)F

 LIKELIHOOD RATIO STATISTICS 1061

Letting D,, = diag[(n - 1)7, (n - 1), (n - 1)a], we obtain

l3~~~~~

(3.4) Dn 'X'XDn' - A,

(3.5) a-Dn1'(X'YX/XO ) f
where

1 0 W
0 1 !2 v
A= ? 12
w Iv r
2
and f' [T, IT- W, (T2 -1)]. The matrix A is invertible with probability 1
and it is readily verified that

Q + W2 6VW -w
(3.6) A-'=Q-[ 6VW 12Q + 36V2 -6Vj,
-W -6V I
where Q = -W2 -3 V2. Thus

aDn(A-_ 0) -A f
The third element of A -!f is the limit random variable for n ( - 1) given in
Dickey and Fuller [7]. Using the fact that S2 converges in probability to a2, we
obtain
A 2
,< Q2- ( Q + W2)2 (1, 0, O)A-f,

2 1,O)A- ,
--r Q 1Q+3 V2) (O,

F1-* T2 + a2( W2)},

P2
9 tAI=-l[2+ 1(2TW)2 + 2

?39 -2- 1(f'A -!f- T2 = 2-'[ 12(l T

T-W) ]

and
A W2-3V2) 2[(2T-W)(T-6V)-1].

The limiting distributions hold for any fixed Y1 and for e, a sequence of
independent identically distributed random variables.
1062 D. A. DICKEY AND W. A. FULLER

TABLE I
EMPIRICAL DISTRIBUTION OF 7;A FOR (a, p) = (0, 1) IN Yt = a + pYt-i + e,.
(Symmetric Distribution)

Sample Probability of a smaller value

size
n 0.90 0.95 0.975 0.99

25 2.20 2.61 2.97 3.41

50 2.18 2.56 2.89 3.28
100 2.17 2.54 2.86 3.22
250 2.16 2.53 2.84 3.19
500 2.16 2.52 2.83 3.18
00 2.16 2.52 2.83 3.18
s.e. 0.003 0.004 0.006 0.008

TABLE II
EMPIRICAL DISTRIBUTION OF TaT FOR (a, fi, p) = (0, 0, 1) IN Yt = a + 8it + pYt-l + e,.
(Symmetric Distribution)

Sample Probability of a smaller value

size
n 0.90 0.95 0.975 0.99

25 2.77 3.20 3.59 4.05

50 2.75 3.14 3.47 3.87
100 2.73 3.11 3.42 3.78
250 2.73 3.09 3.39 3.74
500 2.72 3.08 3.38 3.72
00 2.72 3.08 3.38 3.71
s.e. 0.004 0.005 0.007 0.008

TABLE III
EMPIRICAL DISTRIBUTION OF 7;6, FOR (a, /3, p) = (0, 0, 1) IN Yt = a + 8it + pYt-l + e,
(Symmetric Distribution)

Sample Probability of a smaller value

size
n 0.90 0.95 0.975 0.99

25 2.39 2.85 3.25 3.74

50 2.38 2.81 3.18 3.60
100 2.38 2.79 3.14 3.53
250 2.38 2.79 3.12 3.49
500 2.38 2.78 3.11 3.48
00 2.38 2.78 3.11 3.46
s.e. 0.004 0.005 0.006 0.009
 LIKELIHOOD RATIO STATISTICS 1063

TABLE IV
EMPIRICALDISTRIBUTIONOF (DI FOR (a, p) = (0, 1) IN Yt = a + pY,_ 1 + et

Sample Probability of a smaller value

size
n 0.01 0.025 0.05 0.10 0.90 0.95 0.975 0.99

25 0.29 0.38 0.49 0.65 4.12 5.18 6.30 7.88

50 0.29 0.39 0.50 0.66 3.94 4.86 5.80 7.06
100 0.29 0.39 0.50 0.67 3.86 4.71 5.57 6.70
250 0.30 0.39 0.51 0.67 3.81 4.63 5.45 6.52
500 0.30 0.39 0.51 0.67 3.79 4.61 5.41 6.47
00 0.30 0.40 0.51 0.67 3.78 4.59 5.38 6.43
s.e. 0.002 0.002 0.002 0.002 0.01 0.02 0.03 0.05

TABLE V
EMPIRICALDISTRIBUTIONOF 02 FOR (a, /3, p) = (0, 0, 1) IN Yt = a + 8t + pY_ I+ et

Sample Probability of a smaller value

size
n 0.01 0.025 0.05 0.10 0.90 0.95 0.975 0.99

25 0.61 0.75 0.89 1.10 4.67 5.68 6.75 8.21

50 0.62 0.77 0.91 1.12 4.31 5.13 5.94 7.02
100 0.63 0.77 0.92 1.12 4.16 4.88 5.59 6.50
250 0.63 0.77 0.92 1.13 4.07 4.75 5.40 6.22
500 0.63 0.77 0.92 1.13 4.05 4.71 5.35 6.15
oo 0.63 0.77 0.92 1.13 4.03 4.68 5.31 6.09
s.e. 0.003 0.003 0.003 0.003 0.01 0.02 0.03 0.05

TABLE VI
EMPIRICALDISTRIBUTIONOF (D3FOR (a, /3, p) = (a, 0, 1) IN Yt = a + 8t + pY,_ I + et

Sample Probability of a smaller value

size --
n 0.01 0.025 0.05 0.10 0.90 0.95 0.975 0.99

25 0.74 0.90 1.08 1.33 5.91 7.24 8.65 10.61

50 0.76 0.93 1.11 1.37 5.61 6.73 7.81 9.31
100 0.76 0.94 1.12 1.38 5.47 6.49 7.44 8.73
250 0.76 0.94 1.13 1.39 5.39 6.34 7.25 8.43
500 0.76 0.94 1.13 1.39 5.36 6.30 7.20 8.34
00 0.77 0.94 1.13 1.39 5.34 6.25 7.16 8.27
s.e. 0.004 0.004 0.003 0.004 0.015 0.020 0.032 0.058
1064 D. A. DICKEY AND W. A. FULLER

4. SIMULATION

Tables I-VI contain percentiles for the null distributions of the "regression t
A A A

statistics" (TaL, Ta,r,, iT,), and "regression F tests" ((D,, (21)03).The null model is
given in each table.
The empirical distributions of the statistics for finite samples were created
from statistics for samples generated by the model with Y, = 0 and Y, = Y- l +
e, t = 2,3, ... , n, for n = 25, 50, 100, 250, and 500. Three replicates of 50,000
samples were generated for n = 25, two for n = 50, 100, and 250, and one for
n = 500. The simulation of the limit case was conducted using the procedure
given in Dickey [6]. Three replicates of 50,000 were generated for the limit case.
For symmetric distributions, cells equidistant from zero were pooled to create a
symmetric histogram.
For each of the six estimators and for each sample size, the 0.01, 0.025, 0.05,
0.10, 0.90, 0.95, 0.975, and 0.99 percentage points of the distributions were
calculated. These empirical percentiles were then plotted against n. Based on the
plots, regression functions of the form P = a + /8ny were fitted to the percentiles
of the empirical distributions. Because several observations on each percentile
were available for n = 25, 50, 100, 250, and for the limit case, regression F tests
for lack of fit for the smoothing regressions were computed. Of the 36 lack of fit
statistics computed, 7 were significant at the 0.25 level, 2 at the 0.05 level, and
none at the 0.01 level. The regression smoothed percentiles are given in Tables I
through VI.
David [5, Section 2.5] gives a method for constructing distribution free
confidence intervals for the percentiles of a distribution based on empirical
percentiles. In Tables I through VI the number in the row labeled "s.e" is the
largest of the two half lengths of the 68.26% confidence intervals constructed for
n = 25 and for the limit case. These entries provide an upper bound for the
estimated standard errors of the regression smoothed percentiles.
The histogram1TaI, for 50,000sampleswith n = 25 is shownin Figure 1. Figure
2 contains the histogram for i-, constructed from 50,000 samples of size n = 25

F---- I I _ _
-6 -3 0 3 6
FIGURE 1.-Histogram for 50,000 values of T. constructed with n = 25.
 LIKELIHOOD RATIO STATISTICS 1065

-6 ~~-3 0 6
FIGURE 2.-Histogram for 50,000 values of q, constructed with n = 25.

generated with 0' = (0,0, 1). The distributions are symmetric and the histograms
were constructed to be symmetric. The distributions of the T statistics are
distinctive in two respects; the distribution is bimodal and the "spread" of the
distribution is much larger than that of Student's t distribution.

5. DISTRIBUTIONSFOR HIGHER ORDER PROCESSES

In this section we demonstrate that the test statistics investigated in the

previous sections can be applied in higher order autoregressive processes. Con-
sider data generated by the model

(5.1) Y=0,
Yt=Yt - I + Zt (t = 2, 39. . . ),

where

Zt =lZt- I + 02-2 +*** + Z + et

is a stationary autoregressive process and the et are NID(O, a2) random variables.
The model can also be written
p
Yt = pYt-1 + E 0(Y-i-Y, I) + et,

where p = I and Zt = Y--Y>_-. To simplify the presentation we assume,

without loss of generality, a 2= 1.
Consider the regression equation
p
Yt+p= a + /3[t - I(n -p + 1)] + pYt+p- + 9iZZ+p-i+ et+p

t = 1, 2, ... , n-p. Let Hn denote the (p + 3) X (p + 3) sums of squares and

products matrix needed to compute the regression, let Mn denote the square roots
1066 D. A. DICKEY AND W. A. FULLER

= (a,
of the diagonal elements of H1, let Y'n /8, P, 01, 2 ... 9 9)9 and let Yndenote
the least squares estimator of 'yn.Then

Mn(Yn-Y,)
=[MYn
HTOMMnnMMn 7- 1,
where
n-p
( Is[ t -
-
gn (n -p + 1)] Y+_l Zt+-lS*** Zt )'et+P.
t=1

Fuller [8, p. 374] has demonstrated that n -2 Yt is converging to n- 2

(l- X_l= )0 _ ej as t increases. By the results of Fuller, we have

n
n-1 YZt= Op(n-2),
t=2
n
E yt2 I1= Op(n 2)
t=2

n
n - I
tt-(n -
+p )] Zt+p-j= OP(n-22)
t=-

n
1: Yt_lZt-1= Op(n).
t= 2

Therefore

plim Mn- 'HM-' = block diag(HI I, H22),

[1 0 -2
HI, 0 l rF-232 V 9

,r-2wW -232 v I

H22 is the p x p correlation matrix of the process Zt, and r, W, and V are as
defined in (3.2). It follows that the limiting distribution of the vector composed of
the first three elements of M A(, -yn) is the same as the limiting distribution of

n 2an2 -t ( (
1)]

discussed in Section 3. Similar results are easily obtained for the regression that
does not contain the time trend.

6. EMPIRICAL POWER OF TESTS

Tables VII-IX were constructed to give information on the power of the tests.
In Table VII the power was computed for samples of size n = 100 generated for
model (1.1) with p = 0.8, 0.9, 0.95, 1.00, 1.02, and 1.05 and a = 0.0, 0.5, and 1.0.
LIKELIHOOD RATIO STATISTICS 1067

ForFor Ti T
a a n(T- n(-
4D3 42 41
Statistic
=0 0 1) 1)

0.710.86
power 0.460.57
power 0.410.78 0.00
0.57
is is p
a =
0.500.550.770.87 0.470.83 0.50
0.57 0.8
computed
computed
0.620.530.890.88 0.650.93 1.00
0.72
from
from

3000 0.100.140.180.29 0.090.22 0.00

0.15
10,000
a EMPIRICAL
0.120.100.270.25 0.130.36 0.50
0.19 p=0.9
samples.
samples.

0.220.050.560.13
0.43
0.390.73 1.00 POWER
OF

0.040.060.060.11
0.07
0.040.08 0.00 Two
p
=
a
0.11
0.050.040.120.02 0.090.21 0.50 0.95 SIDED

0.090.010.530.00 0.630.91 1.00

0.37 SIZE
0.05
0.050.050.040.05
0.05
0.030.04 0.00 TABLE
p
= VII
TESTS
a
0.060.070.070.03 0.670.76 0.50
0.06 0.99
FOR
0.110.120.140.00
0.13
1.001.00 1.00

SAMPLES
0.050.050.060.05 0.050.05 0.00
0.05
OF
a p=
0.050.050.260.14
0.05
0.970.98 0.50 1.0 SIZE

0.050.050.350.07 100(
1.001.00 1.00
0.04 YI

0.170.180.490.47 0.450.43 0.00

0.08
FIXED)
p=
a
0.970.991.001.00 1.001.00 0.50
1.00 1.02

1.001.001.001.00 1.001.00 1.00

1.00

0.960.960.970.97 0.950.95 0.00

0.94
a
p=
1.001.001.001.00 1.001.00 0.50
1.00 1.05

1.001.001.001.00 1.001.00 1.00

1.00
 1068 D. A. DICKEY AND W. A. FULLER

TABLE VIII
EMPIRICAL POWER OF Two SIDED SIZE 0.05 TESTS AGAINST THE STATIONARY ALTERNATIVE
FOR SAMPLESOF n = 50, 100, AND 250 (20,000 SAMPLES)

p = 0.8 p = 0.9 p=0.95 p=0.99 p= 1.0

n n n n n

Statistic 50 100 250 50 100 250 50 100 250 50 100 250 50 100 250

>1 0.25 0.80 1.0 0.09 0.24 0.94 0.05 0.09 0.37 0.05 0.05 0.05 0.05 0.05 0.05
>2 0.09 0.44 1.0 0.04 0.10 0.64 0.04 0.04 0.15 0.04 0.04 0.04 0.05 0.05 0.05
4>3 0.17 0.59 1.0 0.08 0.16 0.78 0.06 0.07 0.23 0.06 0.05 0.06 0.05 0.05 0.05
n(p - 1) 0.30 0.87 1.0 0.10 0.29 0.97 0.05 0.09 0.43 0.05 0.04 0.05 0.05 0.05 0.05
T9 0.20 0.74 1.0 0.07 0.19 0.91 0.04 0.06 0.31 0.04 0.04 0.04 0.05 0.05 0.05
n(pA- 1) 0.14 0.57 1.0 0.06 0.14 0.78 0.05 0.05 0.21 0.05 0.04 0.04 0.05 0.05 0.05
iT 0.11 0.48 1.0 0.05 0.11 0.69 0.04 0.05 0.16 0.05 0.04 0.04 0.05 0.05 0.05

TABLE IX
EMPIRICAL POWER OF ONE SIDED SIZE 0.05 TESTS FOR SAMPLES OF SIZE 100( YI FIXED)

p=0.8 p=0.9 p = 0.95 p=0.99 p= 1.00

a a a a a

Statistic 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00

d 0.97 0.95 0.89 0.53 0.31 0.03 0.23 0.01 0.00 0.08 0.00 0.00 0.05 0.00 0.00
n(pA- 1) 0.95 0.95 0.96 0.46 0.42 0.29 0.19 0.06 0.00 0.07 0.00 0.00 0.05 0.00 0.00
i 0.86 0.90 0.95 0.30 0.43 0.73 0.12 0.21 0.66 0.06 0.06 0.20 0.05 0.00 0.00
n(pA- 1) 0.73 0.72 0.72 0.24 0.20 0.12 0.10 0.06 0.01 0.05 0.04 0.02 0.05 0.05 0.06
iT 0.64 0.67 0.78 0.18 0.22 0.34 0.08 0.09 0.15 0.05 0.04 0.03 0.05 0.05 0.05

For a = 0 power is computed from 10,000 samples.

For a #0 power is computed from 3000 samples.

The statistic (DIis the likelihood ratio test of (a, p) = (0, 1) against the alterna-
tive (a, p) 7P(0, 1) for model (1.1). The statistic )2 iS the likelihood ratio test of
(a, 3,p) = (0,0, 1) against the general alternative of model (1.3). Note that in
models (1.1) and (1.3) the initial value Y1 is fixed. Because the alternative is
broader for 02, 02 displays smaller power than J, in Table VII where the
parameter /3 = 0 for all examples of the table. Both 01l and 02 display bias,
having power less than the size for p = 0.99. The power of both tests increases as
a increases.
The statistic I3 is the likelihood ratio test of (a, /, p) = (a, 0, 1) against the
general alternative of model (1.3). In Table VII the power of I3 is between those
of (D, and (2 for p <.99. At p = 1.02 and a = 0, the power of I3 is considerably
less than the powers of (D and (2. No bias is evident inA3.
We have included in Table VII the statistics A,,, A-n, and T discussed by
A
Fuller [8, Section 8.5]. The null distributions of the statistics and T are
computed under model (1.1) with the assumption that (a, p) = (0, 1). The distri-
butions of the statistics for (a, 1), a 7 0 differ from those with a = 0. The null
 LIKELIHOOD RATIO STATISTICS 1069

distributions of the statistics 3 pT, and T are independent of a and therefore

they maintain their size for p = I and a #0 O. The tests used for Table VII were
constructed from p,,, p,,T, and T b'y removing equal areas from the two tails of
the distribution. The tests T,, and T, are generally less powerful than the corre-
sponding tests P1and (D2 when p < 1.
Table VIII contains the estimated power for the test statistics against the
stationary first order autoregressive time series. The tests are the two sided tests
of Table VII. The observations for p 1 for the samples of Table VIII were
generated using the model

Y, = pYt-I + et (t = 23, ...

* ,n),

Y =
-p2)-2e (t=1),
where the et are NID(0, 1) random variables. The power was computed for
20,000 samples of each of the three sample sizes. Generally speaking pL is the
most powerful of the tests considered. The test I3 is the most powerful of the
tests that permit the null model to contain drift.
Table IX contains the power for one sided tests when the true model is (1.1)
with p < 1. Included in this table is the von Neumann ratio
n -1n
d = n[E Y-n] ( Yt-t l)29
t=l ~~t=2
where
n
Yn=n-l .
t= 1

Sargan [15] gives percentiles for d when Yt is generated by model (1.1) with
(a, p) = (0, 1). For sample sizes 50 and 100 and significance level 0.05 we use the
percentiles from Sargan's paper. The jth percentile of the limit distribution for d
is the reciprocal of the (100-j)th percentile in the table of Anderson and
Darling [2, p. 203]. For finite sample sizes not considered by Sargan, we use the
asymptotic percentiles as critical values for the power calculations of Table IX.
Fuller [9] has constructed modifications of the statistic d that are applicable to
higher order autoregressive processes and to model (1.3). The methods used to
generate the samples of Table IX are those used to generate the samples of Table
VII with p < 1. The statistic d is an appropriate test when the alternative is that
Y, is a stationary first order autoregressive time series. It displays good power for
this alternative (that is, when a = 0 and p < 1). The statistic n(p - 1) is only
slightly less powerful than d for a = 0 and maintains somewhat better power for
a &0. For p < 1 and a # 0 the estimator P is closer to one on the average than
the corresponding estimator associated with a = 0. Therefore, the tests A andT,
display poor power for values of p close to one and a #0 O.Because the estimator
p, converges to p for p < 1, there is some sample size for any p < 1 for which the
1070 D. A. DICKEY AND W. A. FULLER

TABLE X
MODELS AND TEST STATISTICS

Test
Null Model Alternative Model Statistic

Yt= YA_I+e, Yt =i+pYtAI+e, (1

Yt = Yt- I + et Yt = ai + At + pYt- I + et (D2
Yt=a+ Yt-5+et Y,=a+8t+pYt-I+et 4'3
Y= +e
+pYt- I

= c + Yt- I +Y
Yt= a +,_t + pY,I+e, A A

Yt e,t PY1 t PTas

"

Yt = Yt- I + et Yt =a +pYt- I + et d
P < I

aPoor power for Y, not stationary, a =, 0, small n, and p less than, but close to one.

statistics will have power greater than the size. Because the null distributions are
derived under the assumption that (a, p) = (0, 1), there is no sample size for
which the tests d, p, and are appropriate if the alternative includes a :# 0 and
p= 1.
The test statistics discussed in this section and the hypotheses for which they
are appropriate are summarized in Table X.

7. EXAMPLE
To illustrate the use of the tables we study the logarithm of the quarterly
Federal Reserve Board Production Index 1950-1 through 1977-4. We assume
that the time series is adequately represented by the model

(7.1) Yt =,Bo + Pi1t + aI Y>-1 + a2(YA I- Yt2) + ev,

where e, are independent identically distributed (0, a2) random variables. The
ordinary least squares estimates are

Yt- Yt- = 0.52 + 0.00120t - 0.119 Yt -+ 0.498 Yt_I- Yt-2)

(0.15) (0.00034) (0.033) (0.081)
R.S.S. = 0.056448,

Y-Y,-I= 0.0054 + 0.447 (Yt -I Yt -2), R.S.S. = 0.063211,

(0.0025) (0.083)

Yt -Yt-i= 0.511 (Yt_, -Yt2), R.S.S.=0.065966,

(0.079)
where R.S.S. denotes the residual sum of squares. The numbers in parentheses
are the quantities output as "standard errors" by the regression program.
To test the hypothesis that 80 = /31= 0 and al = 1 against the general alterna-
 LIKELIHOOD RATIO STATISTICS 1071

tive (7.1) we compute

-
= 0.065966 - 0.056448 5-95
=
3(0.000533)

where 0.000533 = 0.056448/106 is the residual mean square for the full model
regression. As there are 110 observations in the regression the 97.5 per cent point
of the distribution of (12, as given in Table V, is 5.59. Therefore the hypothesis
Po= ,B = 0 and a1 = 1 is rejected at the 2.5 per cent level.
To test the hypothesis that f,3 = 0 and a1 = 1 against the general alternative
(7.1) we compute

(, = 0.063211 - 0.056448 = 6.34.

3 2(0.000533)

The 95 per cent point of the distribution is given in Table VI as 6.49 and the
90 per cent point as 5.47. Therefore at the 5 per cent level one could accept the
hypothesis that the second order autoregressive process has a unit root with
possible drift under the maintained hypothesis that the process is second order.
The null hypothesis would be rejected at the 10 per cent level. We note that on
the basis of Table 8.5.2 of Fuller [8] the statistic
;.-0.119.
0? 9- 6
3.61
TT=
0.033
would lead to rejection of the hypothesis of a unit root at the 10 per cent level if
a two sided test is performed. If the alternative is that both roots are less than
one in absolute value the hypothesis of a unit root is rejected at the 5 per cent
level.

North Carolina State University

and
Iowa State University

ManuscriptreceivedJune, 1978; final revisionreceivedApril, 1980.

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1072 D. A. DICKEY AND W. A. FULLER

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