pantula sastry a comparison of unit roor test criteria

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A Comparison of Unit-Root Test Criteria

Author(s): Sastry G. Pantula, Graclela Gonzalez-Farias and Wayne A. Fuller

Source: Journal of Business & Economic Statistics, Vol. 12, No. 4 (Oct., 1994), pp. 449-459

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? 1994 American Statistical Association Journal of Business & Economic Statistics, October 1994, Vol. 12, No. 4

A Comparison of Unit-Root Test Criteria

Sastry G. PANTULA

Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203

Graciela GONZALEZ-FARIAS

Departmento de Mathematicas, ITESM, Monterrey, NL 64849, Mexico

Wayne A. FULLER

Department of Statistics, Iowa State University, Ames, IA 50011

During the past 15 years, the ordinary least squares estimator and the corresponding pivotal

statistic have been widely used for testing the unit-root hypothesis in autoregressive processes.

Recently, several new criteria, based on maximum likelihood estimators and weighted symmetric

estimators, have been proposed. In this article, we describe several different test criteria. Results

from a Monte Carlo study that compares the power of the different criteria indicate that the

new tests are more powerful against the stationary alternative. Of the procedures studied, the

weighted symmetric estimator and the unconditional maximum likelihood estimator provide the

most powerful tests against the stationary alternative. As an illustration, the weekly series of

one-month treasury-bill rates is analyzed.

KEY WORDS: Autoregressive processes; Maximum likelihood estimators; Weighted symmetric

estimators.

Testing for unit roots in autoregressive (AR) processes

has received considerable attention since the work by Fuller

(1976) and Dickey and Fuller (1979), who considered tests

based on the ordinary least squares (OLS) estimator and the

corresponding pivotal statistic. Several extensions of the

procedures that they suggested exist in the literature. See

Diebold and Nerlove (1989) for a survey of the unit-root

literature. Recently, Gonzalez-Farias (1992) and Dickey

and Gonzalez-Farias (1992) considered maximum likelihood

(ML) estimation of the parameters of the AR process and

suggested tests for unit roots based on these estimators. El-

liott and Stock (1992), Elliott, Rothenberg, and Stock (1992),

and Elliott (1993) developed most powerful invariant tests for

testing the unit-root hypothesis against a particular alternative

and used these tests to obtain an asymptotic power envelope.

Both approaches produced tests against the alternative of a

root less than 1 with much higher power than the test criteria

based on the OLS estimators. See also Hwang and Schmidt

1993).

We summarize the new approaches, introduce a new test,

and use Monte Carlo methods to compare the power of the

test criteria in finite samples. In Section 1, we introduce the

model and present different unit-root test criteria. In Section

2, we present a Monte Carlo study that compares the power

of the new approaches to that of existing methods. In Section

3, we give extensions. In Section 4, we analyze a data set to

illustrate the different test criteria. In Section 5, we present

our conclusions.

1. TEST CRITERIA

Consider the model

Yt = M(1 - p) + pYt-1 + et, t > 2, (1.1)

where the et's are independent random variables with mean 0

and variance o2. Assume that YI is independent of et for

t > 2. We are interested in testing the null hypothesis

that p = 1. Different estimators and test criteria are ob-

tained depending on what is assumed about Y1. Test criteria

are typically constructed using likelihood procedures under

the assumption that the et's are normally distributed. The

asymptotic distributions of the test statistics are, however,

valid under much weaker assumptions on the distribution of

et. Moreover, the limiting distributions do not depend on

Y1. We present some different test statistics and summarize

their asymptotic distributions. We refer the reader to Dickey

and Fuller (1979, 1981), Elliott et al. (1992), Elliott (1993),

Fuller (1992), and Gonzalez-Farias (1992) for the proofs of

the asymptotic results.

1.1 Y1 Fixed

When YI is considered fixed and et ,, NI(0, a2), maximiz-

ing the log-likelihood function is equivalent to minimizing

Qc(P, p, a2) = (n - 1) log a2

n

+- [,-(1 - p) -pt_1]1

t= 2

=(n - 1) log a2 1-2 Qs(#, p). (1.2)

In this case, the conditional ML estimator of p is the same

as the OLS estimator p,,OLS, obtained by regressing Yt on 1

and Yt_1 for t = 2, 3,..., n. The OLS estimator of p is

nin

P,-OLS - Y(l)) - Z - Y(1))Yt.

There are three common approaches for constructing a test2

There are three common approaches for constructing a test

449



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450 Journal of Business & Economic Statistics, October 1994

Table 1. Limiting Null Distributions of Statistics

Procedure, statistic Limiting distribution

OLS

n(1A,OLS - 1) [G - H2]-[5 TH]

A,,o [G - H2]-1/2[_ TH]

4IOLS 72 + [G - H2]- [ - TH]2

MLE for Y1 N(IA, #2)

nI,M- 1) G-'1

I,ML G-1/2t

MGI2

Elliott and Stock

nP7,GLS - 1 G-

77,GLS G-1/2

4rES 49G + 14t + 7

Weighted symmetric

n(Aws - 1) [G - H2]-1[C - TH - G + 2H2]

fws [G - H2]-1/2[t - TH - G + 2H2]

of the hypothesis that p = 1. For the first-order process, one

can construct a test based on the distribution of the estimator

of p. A second test is obtained by constructing a pivotal statis-

tic for p by analogy to the usual t test of regression analysis.

This is the most used test in practice because the pivotal ap-

proach extends immediately to higher-order processes. The

pivotal statistic associated with the OLS estimator is

OLS = [O (S,_ltZ - )] (12 P,OLs - 1), (1.3)

L -t=2j

where [Y(o), (-1)] = (n - 1)- En2 Yt, Yt-11] and

n

Ls= (n - 3) [Y, - Yo) - ,OLS (t-1 -

t=2

A third test can be constructed on the basis of the likelihood

ratio. The null model with p = 1 reduces (1.1) to the random

walk. The sum of squares associated with the null model

is _t2(Yt - Y1_1)2, and a likelihood-ratio-type statistic for

testing p = 1 is

n

COLS = (2LS2s)- Yt -- Yt-1) -- (n - 3)2LS (1.4)

The limiting distributions of the statistics, derived by Dickey

and Fuller (1979, 1981), are given in Table 1. For example,

4OLS converges in distribution to T2 + [G - H2]-1 T - 2,

where = .5[T2 - 1], T = W(1),

G = W2(t)dt, H = W(t)dt,

and W(t) is a standard Brownian motion on [0, 1]. Empirical

percentiles for n(p,,OLS - 1) and ,OLs may be found in

tables 8.5.1 and 8.5.2 of Fuller (1976). The percentiles for

OLS were given by Dickey and Fuller (1981).

1.2 Y1 -a N(p, a2)

If Y1 is assumed to be a normal random variable with mean

yi and variance a2, maximizing the log of the likelihood func-

tion is equivalent to minimizing

Q0(7l, p, a2) = log a2 + -2(y1 - )2+ Qc(L , pa2), (1.5)

where Qc(pp, p, a2) is defined in (1.2). The first observation

enters the quantity (1.5) but not (1.2) because Y1 is random

with variance a2 under the model that leads to (1.5). The ML

estimators minimize Ql (i/, p, a2) and satisfy

1,ML = ln )- (p,ML) 2(1.6)

1+(n- 1) P1 -),

En ,-;....) .0_

P1,ML t (YtI-,-,L) (1.7)

and

,ML = n-1 ( ML)

n

S [Y - Pl,ML - P,ML)

t= 2

- 1,MLYt-1] 2 (1.8)

Substituting (1.6) in (1.7) and simplifying, we get that

n(p1l,ML - 1) is a solution to a fifth-degree polynomial. Us-

ing the arguments of Gonzalez-Farias (1992), it is possible

to show that the limiting distribution of n(p'f,ML - 1) is that

of G-'I. Recall that G-' is also the limiting representation

of n(FOLS - 1), where PoLS is the OLS estimator obtained by

regressing Y, on Yt_1 without an intercept. The percentiles of

G-lI may be found in table 8.5.1 of Fuller (1976). The piv-

otal and the likelihood-ratio-type statistics for testing p = 1

are

n n 1/2

71,ML [Y-rY t1 - ,I,ML)2 ( 1,ML - 1) (1.9)

t=2

and

4)i1,ML = n(s2 - ( )1, (1.10)

where s2 = (n 1)-1 2(Yt - Yt_1)2. The limiting distri-

butions of these statistics are given in Table 1. We make a

few remarks before considering the next case.

Remark 1.1. Assume that p = 1. Let pbe any estimator

of p such that = 1 +O,(n-'). Then, 2 in (1.6) evaluated at

is Y1 + Op(n-1/2). Likewise, if / is fixed at j, then the value

of p that maximizes the likelihood is obtained by regressing

Y - j on Yt-1 - j. If j is such that j = Yi + O,(n-1/2), then

n[p^(j) - 1] - G-'1 and the estimator has the same large-

sample behavior as the estimator obtained by regressing Yt

-Y1 on Yt-1 - Y1, which was suggested by Dickey and Fuller

(1979). See also Chan and Wei (1987) and Chan (1988).

Remark 1.2. Based on Remark 1.1, several approxima-

tions to the ML estimator are possible. We consider the

following estimators in our study.



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Pantula, Gonzalez-Farias, and Fuller: A Comparison of Unit-Root Test Criteria 451

1. Let P = Pp,OLS. Compute iteratively 4' M and

p- using (1.6) and (1.7) with pl,ML and iI,ML replaced by

pi- and A,.. This iterative procedure, if it converges,

converges to the ML estimator. In our study, we use the

statistics obtained at the tenth iteration (i = 10). From Re-

mark 1.1, it follows that the statistics have the asymptotic

representations given in Table 1. In our simulations, we call

these estimators the ML estimators and omit the superscripts.

2. Elliott and Stock (1992) suggested using the statistic

(ES = ns-2 [6S - s2] + 7, (1.11)

where A7 = i(PT) is the A of (1.6) evaluated with p7 in place

of ,ML, P7 = 1 - 7n- s is the estimator (1.8) with p1,ML

replaced with 17, and Pl,ML is replaced with p7. They argued

that IES is the most powerful invariant test for testing p = 1

against the alternative Ha: p = P7. The alternative p7 was

selected by finding the point that is approximately tangent to

the asymptotic power envelope at a power of 50%.

3. Elliott et al. (1992) considered

P7,GLS =P(k7), (1.12)

obtained by regressing Y, - '7 on Y,_1 - I 7, without an

intercept. They called the estimator P7,GLS the Dickey-Fuller

generalized least squares (GLS) estimator of p. Let '7,GLS

denote the corresponding pivotal statistic for testing p = 1.

The limiting distribution of r7,GLS is given in Table 1.

1.3 YV 1 N[/A, (1 - p2)-1o2]

Suppose that Y1 is a normal random variable with mean IL

and variance (1 - p2)-1a2, for Ipl < 1. Then maximizing

the log of the likelihood function is equivalent to minimizing

Qu (/l, p, 2) = log a2 + log (1 - p2) + (1 - p2)a-2

x (Y1 - t)2 + QC(t,p, a2). (1.13)

Gonzalez-Farias (1992) showed that the unconditional ML

estimator (UMLE) u,ML of p is a solution to a fifth-degree

polynomial. She showed that the asymptotic distribution of

n[5jU,ML - 1] is that of the unique negative root of

Ebix =0, (1.14)

i=O

where bo = -4, b4 = 2(G - H2), bi = -8(Q - TH + H2)

+2(T - 2H)2 + 4, b2 = 8(G - H2) + 8( - TH + H2) - 1, and

b3 = -2( - TH + H2) - 8(G - H2). For a given p, the value

of l that minimizes (1.13) is

Y1 + (1 - p) _21Yt + Yn

2 + (n - 2)(1 - p)

Moreover, for a given p, (1.13) is maximized at the p that

is the solution to a cubic equation (see Hasza 1980). The

equation (1.15) and the cubic equation can be solved itera-

tively. If the iterations converge, the estimators converge to

the UMLE'S of p and p. Gonzalez-Farias (1992) also de-

rived the asymptotic distribution of = tu(i t ), obtained

at the ith iteration, starting with an initial estimator of p. The

distribution of U(O) depends on the initial estimator and

on the iteration number i. Even though the asymptotic dis-

tribution of n[f) - 1] obtained for a finite i is not the same

as that of n[fu,ML - 1], the observed empirical distribution

and the power are similar for the two procedures. In this

article, we consider n[p - 1] obtained in the 6th iteration,

starting the iterations with the simple symmetric estimator

given in Section 1.4. The choice of the initial estimator and

the number of iterations are the same as the ones considered

by Gonzalez-Farias (1992). We shall call (6) the UMLE and

omit the (6) exponent. The corresponding pivotal statistic is

S= [V{}-]-1/2n[U- 1], (1.16)

where V{(iU} is the variance of iju computed from the es-

timated information matrix. The limiting distribution of Tj

was given by Gonzalez-Farias (1992).

1.4 Symmetric Estimators

If a normal stationary AR process satisfies (1.1), it also

satisfies the equation Yt = p(1 - p) + p Yt+ + ut, where ut

, NI(O, a2). This symmetry leads one to consider estimators

of p that minimize

nn1

= Z (yt - Pyt-2+ (1- Wt+1)(Yt- PYt+1)

=2 t=

where w,, t = 2,3,..., n, are weights yt = Y, - y, and

S= n-' 1 -, Yt. Observe that the OLS estimator is a member

of this class with all wt equal to 1. Dickey, Hasza, and Fuller

(1984) discussed the properties of the simple symmetric es-

timator obtained by setting wt = .5. Another member of the

class of symmetric estimators was studied by Park and Fuller

(1993). The estimator is obtained by setting wt = n-'(t - 1)

and is called the weighted symmetric (WS) estimator. The

WS estimator for the first-order model is

Pws = Pws() = 1=2 t-tn , (1.17)

and

n-1 n -1/2

,rws() = [Pws - 1] y2 - n ]- 1 2 (1.18)

Lt=-2 =-I

is the pivotal statistic, where aw = (n - 2)-'1Qw(ws). The

limiting distributions of the statistics are given in Table 1.

The WS estimator and the ML estimator yield similar values

for samples that produce ML estimates not too close to 1,

say less than 1 - .iln. For samples that produce estimates

close to 1, the ML estimator is always less than 1, But the

WS estimator can exceed 1 for some models. The WS es-

timator is easier to compute than the ML estimator. Both

the simple symmetric and the weighted symmetric estima-

tors treat observations at the beginning of the sample period

in the same way as observations at the end of the sample

period.

The limiting distribution of the two-step WS estimator

of p that replaces y with the estimator of p, denoted by

Zws, obtained by substituting pws(Y) into (1.15) has been ob-

tained. Our simulations indicate that pws(,ws) and ,ws(jws)



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have about the same power as p(ys0) and '(y), respec-

tively. Moreover, iterating the procedure did not produce

any significant change in empirical power, and hence the it-

erated estimators are not included in the reported simulation

results.

The simple symmetric estimator was included in our orig-

inal Monte Carlo simulations. Because the power of the WS

estimator always exceeded that of the simple symmetric esti-

mator, we do not report the results for the simple symmetric

estimator.

1.5 Other Criteria

We considered two other estimation criteria. The test

criteria associated with the /P obtained by regressing Yt -

Uo(Pu,oLS) on Yt-1 - iU(P,,OLS), where 2u() is defined in

(1.15), had powers much smaller than those based on the

ML and WS estimators, and hence the powers are not re-

ported here. Moreover, we investigated an estimator P7,ULS

similar to P7,GLS that is obtained by regressing Yt - 'u(P7)

on Yt-1 - iu(P7), where p7 = 1 - n-17. The powers of

criteria based on p7,ULS are small compared to the powers

of the WS estimator and hence are not reported. Elliott

(1993) presented extensions of the procedures of Elliott et al.

(1992) to the model in which the alternative is a stationary

process.

2. POWER STUDY, THE FIRST-ORDER PROCESS

In this section, we compare the empirical power of the

different test criteria described in Section 1. For tests based

on the estimators of p, the test is the test of p = 1 against the

alternative that p < 1. We first present the percentiles of the

test criteria and then study their empirical powers.

2.1 Empirical Percentiles

Our model is

Yt = A(1 - p) + pYt-1 + et, t > 2, (2.1)

where et , NI(O, a2). To construct the percentiles, we let

Y1 = el, I = 0, p = 1, and generate the et as independent

standard normal random variables. The RANNOR function

in SAS (1985) is used to generate the et's. For a given sample

size n, we generated 20,000 samples of size n and computed

the different test statistics. The empirical critical value for

a 5%-level test is computed for the set of 20,000 samples.

The procedure was replicated three times and the average

of the three critical values is reported in Table 2. Empir-

ical percentiles for the infinite case are obtained from the

asymptotic representation, as described by Dickey and Fuller

1979).

We consider the test criteria based on the following.

1. OLS-n(p,,OLS - 1), TOLS, 4OLS

2. MLE for Y1 , N(t, 2 2)-n(j,ML - 1), ',ML, I,ML

3. Elliott and Stock-n(7,GLS - 1), T7,GLS, 4ES

4. MLE for Y1 NI/1, (1 - p2) - 2]__ n(U 1), ?u

5. WS -- n('ws - 1), ws

Table 2. Empirical 5% Critical Values for Different

Unit-Root Test Criteria

Test

statistic 25 50 100 250 oo

nlP,OLS- 1] -12.50 -13.30 -13.70 -14.00 -14.10

n[p,ML - 1] -12.4 -11.93 -10.41 -8.95 -8.10

n[P7,GS - 1] -10.95 -10.16 -9.35 -8.64 -8.10

n[p.- 1] -12.02 -12.49 -12.76 -12.95 -13.13

n[pws - 1] -12.03 -12.48 -12.69 -12.88 -13.07

rA,,oLS -3.00 -2.93 -2.89 -2.88 -2.86

,M -2.75 -2.53 -2.28 -2.07 -1.95

7tGLS -2.56 -2.30 -2.14 -2.03 -1.95

7 u -2.75 -2.69 -2.66 -2.65 -2.64

ws -2.66 -2.61 -2.56 -2.54 -2.50

4OLS 5.25 4.86 4.72 4.61 4.59

1,ML 6.46 5.51 4.86 4.39 4.39

(DES 2.87 2.98 3.09 3.19 3.96

NOTE: Except for (DOLS and 4N1.ML, reject Ho for values of the statistic less than the

critical values.

Except for (OLS and P,ML, we reject Ho: p = 1 for values of

the statistic less than the critical values given in Table 2.

2.2 Empirical Power

In this section, we study the empirical power of the statis-

tics described in Section 2.1. Critical values for 5 -level

tests are given in Table 2, Section 2. The percentiles for finite

samples are estimated by generating et's from a standard nor-

mal distribution. We considered two cases, (1) Y1 , N(0, 1)

and (2) Y1 N[0O, (1 - p2)-1 and generated samples of size

n = 25, 50, 100, and 250, with p = .98, .95, .93, .90, .85, .80,

and .70. The powers are computed based on 5,000 Monte

Carlo replications. Empirical powers are summarized in

Tables 3-6. In generating Tables 3-6, the same et's were

used in both cases, except that Yi = el in case (1), whereas

Yi = el(1 - p2)-1/2 in case (2). The power is higher for

case (1) than for case (2) for all test statistics. The tables

contain only the pivotal statistics and 4ES. Except for OLS,

the pivotal statistics generally have power that is comparable

to or greater than that based on n(p'- 1). For OLS, the power

of n(p',oLS - 1) is much greater than that of T,OLS but is less

than that of n(&ws - 1).

(1) Y, s N(0, 1)

The powers of the pivotal statistics ,OLS, T7,GLS, and ?ws

are plotted in Figure 1, page 455. From the tables, we observe

the following:

1. The pivotal based on OLS has low power compared

to the powers of the remaining criteria.

2. The test criterion based on the unconditional likeli-

hood, u, has powers very similar to those of ws, with 7u

being somewhat superior in small samples (n < 50).

3. For sample sizes n = 25 and 50, tests based on the

WS and the unconditional likelihood estimators have higher



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Table 3. Empirical Powers for 5%-Level Test Criteria (n = 25, 5,000 replications)

p

Statstc 98 95 93 90 85 80 70

Y, - N 0, 1)

A,, OLS 6.54 6.22 6.34 6.94 8.46 12.14 20.64

, ,ML 6.14 8.50 8.58 10.26 14.64 20.52 34.42

7,GLS 6.22 8.68 8.94 11.24 16.56 23.34 37.86

,nE 6.24 8.68 9.28 11.32 17.42 25.24 39.70

Ts 612 864 926 1130 1728 2516 3966

IES 6.30 8.84 9.96 12.28 18.20 26.64 41.54

Y, N[O, (1 -2)-'

OL,oLS 6.40 6.08 6.98 7.42 942 13.18 21.68

ri,Mr 5.74 6.66 7.70 8.70 12.84 19.20 32.90

77,GLS 5.82 7.12 7.46 8.88 13.86 20.34 34.78

nEi 5.78 7.08 7.84 914 14.10 20.98 36.28

Ir s 578 708 778 910 1386 2080 3610

I ES 5.82 6.98 7.96 9.16 14.08 20.94 35.52

power than those based on the ML estimator (?ji,ML) derived

under the assumption that Y1 has variance 2.

4. For sample sizes n = 100 and 250, the tests based on

the true model, fP,ML and ir,ML, have higher power than those

based on the WS and the unconditional likelihood estimators.

5. For small sample sizes and p close to 1, 'EPs has

slightly higher power than T7,GLS, and these two test crite-

ria have higher power than the remaining criteria. For values

of p smaller than .9 and n > 100, these criteria tend to have

lower power than the criteria based on the unconditional like-

lihood and WS estimators.

(2) Y,, vN[0, (1- )-'1]

Empirical powers for this case are summarized in Tables

3-6. In Figure 2, page 456, we plot the powers of the pivotal

statistics TsoLS, ,GLS, and ws. From the tables, we observe

that comments 1-3 for Case (1) are also applicable for Case

2).

4. For all sample sizes, the criteria based on the uncondi-

tional likelihood and WS estimators have higher power than

those based on P1,,ML and l,,ML.

5. The criteria based on the WS and the unconditional

likelihood have higher power than the criteria suggested by

Elliott et al. (1992).

6. The WS criteria have slightly greater power than

the criteria based on the unconditional stationary likeli-

hood for n = 100 and 250. The likelihood procedure has

slightly greater power than the ws procedure for n = 25 and

5 0.

3. EXTENSIONS

The test statistics presented here can be extended to higher-

order autoregressive and moving average (ARMA) processes.

They can also be extended to the case in which a time trend

is included in the model.


p

Statstc 98 95 93 90 85 80 70

Y,r N(0, 1)

TA,,OLS 6.24 6.84 8.04 11.76 19.50 32.30 64.70

.1,ML 7.32 12.34 15.56 23.64 37.98 56.06 86.12

T,oLS 7.40 13.06 17.74 26.62 43.12 61.50 88.60

7ru .7.46 12.90 17.00 25.50 40.84 59.32 88.02

rws 7.30 12.36 16.54 24.90 40.00 58.38 87.60

QES 7.52 13.42 1922 27.40 44.58 61.82 85.02

Y, , N[O, (1-p2)-'

?,A,OLS 6.08 7.28 8.64 12.88 21.04 34.12 66.18

I,ML 6.38 952 13.18 1944 33.26 51.64 84.72

7,GLS 6.08 996 13.50 1998 33.48 51.66 82.42

ru 6.20 10.06 13.46 20.18 33.78 52.66 84.44

Irws 6.06 9.78 13.16 19.68 33.10 51.84 84.80

4DEs 6.04 10.32 13.88 20.24 32.38 48.60 74.90



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P

Statstc 98 95 93 90 85 80 70

Y N (, 1)

TAOLS 6.68 11.70 17.80 31.06 62.10 87.70 99.70

1,ML 10.92 27.60 42.02 64.40 90.54 98.84 100.00

,GoLS 10.94 29.62 44.08 67.28 90.80 97.90 99.72

ri 10.30 25.86 38.82 59.98 88.38 98.34 100.00

Ts 10.48 26.08 39.16 60.22 88.72 98.36 100.00

IDEs 11.02 29.76 44.66 67.44 88.56 95.28 98.76

YV 1 N[O, (1-p2)-']

T,,OLS 7.08 11.80 19.06 30.14 56.20 78.92 97.64

71,ML 8.12 1948 31.18 51.92 82.08 95.88 9996

1t,GLS 8.10 1950 30.20 4980 75.32 88.36 97.60

,r 8.00 19.48 30.68 52.10 83.70 97.16 100.00

Tws 8.10 19.62 31.08 52.18 83.86 97.08 100.00

DES 8.24 19.44 28.42 47.22 69.78 82.84 93.76

3.1 Extensions to Higher-Order Processes

Dickey and Fuller (1979) gave extensions to higher-order

AR processes. Said and Dickey (1984) extended the results

to ARMA models of unknown order. They approximated an

ARMA process by a long AR process and used the test criteria

suggested by Dickey and Fuller (1979). This procedure is

known as the augmented Dickey-Fuller approach. Elliott

et al. (1992) and Elliott (1993) extended their procedures to

include AR and MA processes. They derived the limiting dis-

tribution of n(I,GLS - 1), '7,GLs, and IES under the assump-

tion that et satisfy some stationary and mixing conditions.

The limiting distributions depend on a nuisance parameter

w2, the limiting variance of n-1/2 C1 et . They use two

approaches to estimate w2. One approach approximates et

by a higher-order AR process, whereas the second approach

uses a weighted sum of covariance-type spectral estimator

suggested by Phillips and Perron (1988). Elliott (1993) also

presented a similar correction for our WS estimator.

Gonzalez-Farias (1992) gave an extension of the uncondi-

tional ML estimation for higher-order AR processes. Under

some regularity conditions on the roots, the ML estimation

of the largest root has the same asymptotic distribution as

that of pU,ML in the first-order AR model. This procedure,

however, requires a search for the largest root on the real line

that maximizes the likelihood. Typically, this is not a severe

problem in moderately large samples. Such an estimator can

be obtained, for example, in SAS (1985) using ESTIMATE

P = (1)(p - 1) METHOD = ML. Moreover, the largest root

can be obtained by solving a pth degree polynomial in which

the coefficients of the characteristic polynomial are the ML

estimates.

For a pth-order AR process, a pivotal statistic ,ws ,pfor

testing 01 = 1 in the WS regression is obtained by minimizing

Table 6. Empirical Pbwers for 5%-Level Test Criteria (n = 250, 5,000 replications)

p

Statstc 98 95 93 90 85 80 70

Y N(O, 1)

r/,OLS 11.72 44.08 73.62 96.74 100.00 100.00 100.00

1,ML 31.42 85.84 97.88 99.96 100.00 100.00 100.00

77,GLS 31.60 85.66 97.52 9988 9998 9998 100.00

Sru 26.18 76.88 95.10 99.90 100.00 100.00 100.00

rws 26.70 77.88 95.34 99.94 100.00 100.00 100.00

(DES 31.06 85.02 96.78 9982 9994 9994 9998

Y, , N[O, (1 -p2)-J

OI,OLS 12.26 46.30 75.58 97.02 100.00 100.00 100.00

7,ML 1954 60.82 80.64 94.00 9958 9996 100.00

7,OGLS 19.16 58.80 76.44 89.88 96.68 98.96 99.90

ru 1928 67.92 91.38 9972 100.00 100.00 100.00

rws 1988 68.76 91.68 9976 100.00 100.00 100.00

4?ES 18.78 56.26 73.32 87.06 95.28 98.26 9980



8/12


n=25, Y1 - N0,1) n=50, Y1 -N0,1)

o

OCCO

(0

SOLS(0

00G

0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.70 0.75 0.80 0.85 0.90 0.95 1.00

rr

n=Figure 1. Pbwers of Pivotal Statistics When Y, s Generated From N(, 1): OLS.n=250,Y WLS; --(--, GLS.)

o \ \ o

S OLS

.. .....WLS

o-- GLS

070 075 080 085 090 095 100 070 075 080 085 090 095 100

rr

Figure 1. Powers of Pivotal Statistics When YI Is Generated From N(O, 1): , OLS;......., WLS; -.-.-., GLS.

nP12

(t-p) y, - ly-, 1- OiZt-i

t=p+1 L i=2

nP2

+ ](n - t +p) yp - Olyt-p+1 +LOiZt p2 (3.1)

t=p+1=21

where y, = Y, - y and Z, = Y, - Yt-1. The test statistic has

the same asymptotic distribution as 's(y) for the first-order

process. One could use a GLS estimator of / instead of y.

In our simulations, however, we saw little improvement in

power from the use of an alternative estimator of the mean.

Hence we only present results for is(y). The pivotal statistic

can be obtained with any standard regression package using

the weights, dependent variable, and independent variables

given in Table 7.

In Section 3.3, we present simulation results that compare

the WS pivotal statistic with the OLS estimator. For the case

of ARMA processes, we consider the augmented Dickey-

Fuller approach for WS estimation. When the order of the

process is unknown, we approximate the model by an AR

process in which the orderp is taken to be the order p selected

by the Akaike information criterion (AIC) plus 2. That is,

our choice ofp is PAc + 2, where pAIc is the order selected by

the AIC. This choice of p seems to maintain the level of the

test criteria better than the use of a smaller order. Of course,

the use of a high order for the pure AR processes reduces the

power relative to the AIC for such processes. Because we

wish to compare the powers when the size is maintained, we

present results for the order PAIC + 2.

3.2 Extensions to the Time Trend Case

Dickey and Fuller (1979), Said and Dickey (1984), Elliott

et al. (1992), and Elliott (1993) presented extensions of their

procedures for the case in which linear trend is included.

Elliott et al. (1992) considered testing against the particular

alternative p = (1 - 13.5/n) for the trend case.

The WS estimator can be easily extended to the case in

which a linear trend is included in the fitted model. Let

y, = Y, - ' - bt, where a and b are the OLS estimators

obtained by regressing Y, on 1 and t. If yt is used in (3.1),

then the limiting distribution of the pivotal statistic ws,p,t, for



9/12


n=25, Y1 -Stationary n=50, Y1 -Stationary

oo

T\

OoO

070 075 080 085 090 095 100 070 075 080 085 090 095 100

rr

n= 00,Yof Piv1 otalStationary n=250,Ys Generated From N[1 OLS; .........WLStationary; ------

o

o 0

1 OLS

S--- GLS

0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.70 0.75 0.80 0.85 0.90 0.95 1.00

rh rh

Figure 2. Powers of Pivotal Statistics When Y1 Is Generated From N[O,(1 -p2)-'J: OLS;.......... WLS; .-...., GLS.

testing 01 = 1, is

[G- H2 - 3K2] -1/2

x [( - 3TK - HT - 12HK + 2H2 - G + 12K2], (3.2)

Table 7. Data Array for Construction of Weighted

Symmetric Regression

Dependent

Weight variable O1 02 ... OP

Wp+I Yp+i Yp Z ... 22

Wp+2 Yp+2 Yp+l Zp+l ... Z3

Wn-p Yn Yn-1 Zn-1 ... Zn-p+I

1- w2 Y Y2 -Z3 . Zp+I

1 - w3 Y2 Y3 -Z4 ... -Zp+2

1 - Wn-p+l Yn-p Yn-p+l -Zn-p+2 ... -Zn

where

K = 2 tw(t)dt - w(t)dt.

See Fuller (1992). We have considered test statistics based

on alternative estimators of (a, b) and found little diference in

the powers. For example, using the estimated GLS estimators

of (a, b), or including the intercept and trend directly in the

WS squares regression, did not lead to an improvement in

the power relative to the use of the OLS estimator of trend.

Hence only results based on OLS estimators of (a, b) are

presented.

3.3 Power Study for Extensions

In this power study, we compare oLS,p and ws, when the

order of the underlying process is unknown. We consider

both the cases in which an intercept is included and those in

which an intercept and trend are included in the alternative

model. The subscripts t and r are used to identify the mean-

adjusted and trend-adjusted cases, respectively. For each

method, the order p between 1 and 10 with the smallest AIC

is determined. Then' = min(AIC +2, 10) is used as the order



10/12


Table 8. Empirical 5% Percentiles for the Pivotal Statistics When

the Order is Estimated

Statistic n = 100 n = oo

OLS,pA -289 -286

Tw,pIA-260 -250

IOLS,pr -346 -341

rw,pr -336 -319

of the process. Note thatpoLs may be different fromws. We

report results for sample size n = 100.

We first constructed empirical percentiles for the test statis-

tics by generating 20,000 samples of size n = 100 from the

model (2.1) and then computing the empirical percentiles

of the different test statistics for the set of 20,000 samples.

The empirical percentiles for the mean and the trend cases are

summarized in Table 8. These empirical percentiles take into

account the fact that p is estimated and hence are different

from those given in Table 2. The last column, for the infinite

case, contains entries from Table 2.

To study the power, we consider both AR and mixed mod-

els. The models are:

1. AR(1)-- Yt = PYt-1 + et, t > 2,

2. AR(2)- Yt = PYt-1 + qYt-1 - Yr-2 + et, t > 3,

3. ARMA(1, I)-- Y, = pYt-,_1 + et - Oet-l, t > 2,

where et ' NID(O, 1); p = 1, .95, .90, .80, .70; = -.5,.5; 0

= -.5, .5. For the cases IpI < 1, the initial observations are

generated from the stationary distributions so that the process

is stationary. For the cases in which p = 1, a corresponding

stationary process Xt(= Yt - Yt-1) is generated for t > 1 and

Yt then is computed as i=, Xi. Empirical powers, based on

1,000 replications, are summarized in Table 9. We plot the

empirical powers in Figure 3.

From Table 9, we observe the following:

1. The WS procedure has power uniformly higher than

that of OLS. The gain in power is greater in the mean case

than in the trend case.

2. Comparing the power for the AR(1) case with those

given in Table 5, we see that there is a significant loss in power

due to estimating a long order process. We have observed a

similar loss in power in the AR(2) case whenp is estimated as

opposed to using p = 2. See also Park and Fuller (1993). The

power for the AR processes increases if we use PAIC instead

of PAMC + 2. With PAIC, however, the size of the test for the

ARMA models was greater than the nominal level.

3. Even with PAIC + 2, the level of both test criteria are

much higher than the nominal level for the trend case when

the MA parameter is .5. Similar results were observed by

Elliott et. al. (1992) and Elliott (1993).

4. For the mean case, the average of fOLS is 3.8 for

the AR(1) model, 4.7 for both AR(2) cases, 5.6 for the

Table 9. Empirical Power of the OLS and ws Pivotal Statistics (n = 100)

Man case Trend case

Model P TOLS,p,AA Tws3,p,/ TOLS,p,r Tws,p,r

R1 100 58 53 5446

95 108 207 92 88

90 239 474147 166

80 682 847 431 516

70 875 941 696734

AR 2 )

= -5 100 55 46 48 45

95 89 195 65 66

90 214410 129 162

80 639 816400 452

70 875 936686755

5 100 58 55 50 45

95 79 184 57 61

90 179 383 115 133

80 447 704259 364

70 658 861 430 551

AR MA 1 , 1)

=-5 100 45 5458 58

95 89 195 71 78

90 204415 138 168

80 487 699 295 364

70 714858 482 572

=5 100 60 68 107 99

95 160 248 147 150

90 392 514268 305

80 832 858 691 673

70 943 946894888



11/12


AR1) AR2) Ph=-05

o

070 075 080 085 090 095 100 070 075 080 085 090 095 100

hrh

AR2) Ph=05 ARMAThta=-05

81

0oo? OSm

. I . WSmto

- OLSt

- WLSt

070 075 080 085 090 095 100 070 075 080 085 090 095 100

hrh

Figure 3. Powers of Pivotal Statistics in Higher Order Processes for Both the Mean and the Trend Cases. Sample size is 100. OLSmn

= TOLS,p,, (- ); WLSmn = ws,p, (........); OLSt = OLS,pA, (-. ); WLSt = ws,p,(- -).

ARMA(1,1) model with 0 = .5, and 4.8 for the ARMA(1, 1)

model with q = -.5. The average ofws, is 3.9 for the AR(1)

model, 4.8 for the two AR(2) cases, 5.8 for the ARMA(1, 1)

model with 0 = .5, and 4.9 for the ARMA(1,1) model with

0 = -.5.

4. EXAMPLE

To illustrate the test criteria, we analyze the one-month

treasury-bill rate (Yt) measured weekly on Fridays. The

sample period is Friday, January 3, 1975, to Friday,

December 28, 1990. The data were analyzed by Bansal,

Gallant, Hussey, and Tauchen (1992). Those authors stated,

The one month treasury bill rate was obtained from the Fed-

eral Reserve Board. This rate, which is calculated on a daily

basis, is computed from the average price in the secondary

market for the treasury bill that matures four weeks from the

next Thursday. The price is the average from five dealers of

the bid prices at approximately 4 p.m. Eastern time. Because

there is a two business-day settlement period in the secondary

market, this rate is actually a forward rate on an investment

whose term is between 27 days and 31 days, depending on

the day of the week. In particular, for our Friday series, the

buyer of the bill on Friday would normally take delivery the

subsequent Tuesday. Hence our Friday one-month treasury

bill rate is most precisely characterized as a four-day forward

rate for a 30-day investment (p. 16).

Because large variability in Y, seems to be associated with

large values of Yt, we analyze Yt = log Y,. The results for

Yt are similar in nature and are not reported. Using an up-

Table 10. Test Statistics or Treasury-Bill Rate With Different

Orders of Autoregressive Fit

7TOLS, 7w

1 249 267

5 256 275

10 244 264

15 202 223

20 176 199

30 160 184

39 161 189



12/12


per bound of 40 for the order of an AR process, the AIC +2

criterion selected POLS = Pw, = 39. Because we have 834

observations, we use the critical values from the limiting dis-

tribution of oLS,, and 'ws,,. The 5% critical values for os,,

and ws,, are -2.86 and -2.50, respectively. Moreover, the

10% critical values are -2.57 and -2.25, respectively. The

results are summarized in Table 10. If we fit an AR process

of order < 10, the WS test statistic rejects the unit-root hy-

pothesis at the 5% level. For orders greater than or equal to

15, the WS test fails to reject at the 10% level. The OLS

test fails to reject the unit root at the 10% level for all of the

orders we considered. As the number of lags increases, the

test criteria decline in absolute value for p > 5.

5. SUMMARY

We have discussed criteria for testing the null hypothesis

of a unit root in a first-order AR process. Based on our simu-

lation study, it is clear that the OLS criteria are the least pow-

erful of the statistics studied. The criteria suggested by Elliott

et al. (1992) are very powerful under the model in which the

initial observations is N(0, a2) but are not the most powerful

against the alternative in which the process is a stationary AR

process with parameter less than 1 in absolute value.

We feel that there are a modest number of situations in

which one is willing to assume that the first observation has

variance equal to the residual variance. For the model in

which the alternative is a stationary process, the criteria based

on the unconditional likelihood and those based on the WS

estimator are the most powerful. These tests also have good

power when the first observation is N(O, cr2). Hence they are

the recommended test statistics for general use.

ACKNOWLEDGMENTS

We thank David Dickey, an associate editor, and the editor

for their comments. We are grateful to Cheng Su for helping

us with the figures. This research is partly supported by

the cooperative agreement 43-3AEU-3-80088 with National

Agricultural Statistics Service and the Bureau of the Census.

[Received December 1992. Revised January 1994. ]

REFERENCES

Bansal, R., Gallant, A. R., Hussey, R., and Tauchen, G. (1992), Nonpara-

metric Estimation of Structural Models for High-Frequency Market Data,

technical report, Duke University. Dept. of Economics.

Chan, N. H. (1988), The Parametric Inference for Nearly Nonstationary

Time Series' Journal of the American Statistical Association, 83, 857-

862.

Chan, N. H., and Wei, C. Z. (1987), Asymptotic Inference for Nearly

Nonstationary AR(1) Process', The Annals of Statistics, 15, 1050-

1063.

Dickey, D. A., and Fuller, W. A. (1979), Distribution of Estimators for

Autoregressive Time Series With a Unit Root', Journal of the American

Statistical Association, 74, 427-431.

(1981), Likelihood Ratio Tests for Autoregressive Time Series With a Unit

Root, Econometrica, 49, 1057-1072.

Dickey, D. A., and Gonzalez-Farias, G. (1992), A New Maximum Likeli-

hood Approach to Testing for Unit Roots, unpublished paper presented

at the 1992 Joint Statistical Meetings, August 9-13, Boston.

Dickey, D. A., Hasza, D. P., and Fuller, W. A. (1984), Testing for Unit Roots

in Seasonal Time Series, Journal of the American Statistical Association,

79, 355-367.

Diebold, F. X., and Nerlove, M. (1989), Unit Roots in Economic Time

Series: A Selective Survey, inAdvances in Econometrics: Cointegration,

Spurious Regression and Unit Roots, eds. T. B. Fomby and G. F. Rhodes,

Greenwich, CT: JAI Press, pp. 3-70.

Elliott, G. (1993), Efficient Tests for a Unit Root When the Initial Obser-

vation Is Drawn From its Unconditional Distribution, technical report,

Harvard University, Dept. of Economics.

Elliott, G., Rothenberg, T. J., and Stock, J. H. (1992), Efficient Tests for an

Autoregressive Unit Root:' unpublished paper presented at the NBER-

NSF Time Series Seminar, October 31, Chicago.

Elliott, G., and Stock, J. H. (1992), Efficient Tests for an Autoregressive

Unit Root, unpublished paper, presented at the 1992 Joint Statistical

Meetings, August, 9-13, Boston.

Fuller, W. A. (1976), Introduction to Statistical Time Series, New York: John

Wiley.

- (1992), Weighted Symmetric Estimators, unpublished notes, Iowa

State University, Dept. of Statistics.

Gonzalez-Farias, G. M. (1992), A New Unit Root Test for Autoregressive

Time Series, unpublished Ph.D. thesis, North Carolina State University,

Dept. of Statistics.

Hasza, D. P. (1980), A Note on Maximum Likelihood Estimation for the

First Order Autoregressive Process, Communications in Statistics, Part

A-Theory and Methods, 9, 1411-1415.

Hwang, J., and Schmidt, P. (1993), Alternative Methods of Detrending and

the Power of Unit Root Tests, unpublished manuscript, Michigan State

University, Dept. of Economics.

Park, H. J., and Fuller, W. A. (1993), Alternative Estimators for the Pa-

rameters of the Autoregressive Process, unpublished manuscript, Iowa

State University, Dept. of Statistics.

Phillips, P. C. B., and Perron, P. (1988), 'Testing for a Unit Root in a Time

Series Regression, Biometrika, 75, 335-346.

Said, S. E., and Dickey, D. A. (1984), Testing for Unit Roots in

Autoregressive-Moving Average Models of Unknown Order, Biometrika,

71,599-608.

SAS (1985), SAS Basics User Guide (Version 5), Cary, NC: SAS Institute.

pantula sastry a comparison of unit roor test criteria

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