- Email: [email protected]
A chi-square test for a unit root
37
Economics Letters 34 (1990) 37-42 North-Holland
A hi-square
test for a unit root
James A. Kahn and Masao Ogaki * Universityof Rochester, Rochester, NY 14627, USA Received 14 February 1990 Accepted 21 March 1990
The present paper develops a cl&square test for a unit root that has higher power than Dickey-Fuller size is small and the autoregressive root is close to one.
tests when the sample
1. Introduction
Much work has been done to modify Dickey and Fuller (1979, 1981) tests for unit roots to allow for general distributions and serial correlation of the disturbance term and various order of deterministic trends [e.g., Said and Dickey (1984), Phillips (1987) and Phillips and Perron (1988)]. Dejong et al. (1988) study small sample properties of Said-Dickey and Phillips-Perron tests and conclude that these tests are practically useless against trend stationary alternatives which are plausible for annual, quarterly, and monthly macroeconomic time series. The problem of these tests is that they are not powerful when the autoregressive root is close to one and the sample size is small. Rather than modifying Dickey-Fuller tests, we develop a x2 test for a unit root that has greater power than Dickey-Fuller tests when the autoregressive root is close to one and the sample size is small. Though it is beyond the scope of the present paper to develop tests that are practically useful in the sense of DeJong et al. (1988), it is important to investigate alternative tests given that modified versions of Dickey-Fuller tests given that modified versions of Dickey-Fuller tests do not have high power in small samples. Our tests does not rely on a functional central limit theorem as most of the other existing test do, but on a general central limit theorem. Though functional central limit theorems apply under general conditions [e.g., McLeish (1975)], central limit theorems apply under even more general conditions. For example, we can allow for deterministic seasonal fluctuations in the time series to be tested while Dickey-Fuller tests do not. This is useful because seasonally unadjusted data are of interest in most applications.
* The authors are grateful to Mahmoud El-Gamal, John Heaton, Charles Plosser, Danny Quah, Mark Watson, and especially to Lars Hansen and Adrian Pagan for helpful discussions. 016%1765/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
38
J.A. Kahn, M. Ogaki /A
chi-square test
2. Regression properties Consider a stochastic process {x I : t 2 l} generated in discrete time according to x, = (Y&-l + u,,
t=l,2,...,
(1)
where { uI} is a sequence of random variables with mean zero. We are interested in the following two hypotheses: HU: cw= 1, and HS: 1CY1c 1. The initial value, x0, is allowed to be any random variable. To motivate the test we develop, let us consider the regression ~~=bAx,+e,.
(2)
First, let us consider the case where hypothesis HS is true. For now, assume that x, is covariance stationary. In this case the best linear predictor of x, given Ax, is 0.5Ax,. To see this, define n,=Ax,[x,-0.5Ax,]
=0.5(x:-x;_,).
(3)
Then E(nI) = 0 as long as xt is covariance stationary so that E(x:) = E(x:_,). Thus PAX, satisfies the orthogonality condition E{ Ax,[x, - PAX,]} = 0 at /3= 0.5. This result is convenient to treat a composite null hypothesis of ( a 1 -c 1 because the true value of p does not depend on the value of (Y. Second, let us consider the case where hypothesis HU is true. For now, assume that u, is serially uncorrelated and that x0 has a finite second moment, so that the least square prediction theory is applicable. In this case E(T),) f 0 because E(x:) # E(xf_i). Hence j3 + 0.5. In fact, E(~h)/Wk)~l= E(x,u,)/EKu,)21= 1 if Wx,+,) = 0 for t 2 1. In this sense, /3= 1 when (Y= 1. This discontinuity of j3 as (Y approaches one motivates our tests.
3. Asymptotic properties of OLS estimators In this section, properties of the OLS estimator of /3 in eq. (5) are studied. Let b, be the OLS estimator:
Since CT=tq, = 0.5(x$ - xi),
(5) The key requirement for our test, when the null is that of a unit root (hypothesis HU), is that a central limit theorem applies to the partial sum of u,. To be precise, let
P, = c j=l
uj
(6)
J.A. Kahn, M. Ogaki /A
be the partial sum. We require that (l/T)‘/*Pz
a*=T+CC lim
E(T-‘PG)
39
chi-square test
converges in distribution to N(0,
a*), where
>O
as the sample size T approaches cc (all the limits in this paper is taken as T + 00). This requirement is satisfied under very general conditions. In the following, we assume that u, is stationary and ergodic with finite second moments, and.apply Gordin’s theorem [see, e.g., Hansen (1985)] so that
Alternatively, we can use a central limit theorem for weakly dependent random variables [see, e.g., Serfling (1968) and White and Domowitz (1984)]. Let (tin, F, P) be the underlying probability space. Let S denote a measurable transformation mapping s2 into itself. Then an arbitrary random variable, say h,,, and S together generates a time series via the relation h,(w) = h,( S’( w)). We assume that S is constructed so that u,(w) = u,,( S’( w)). Let &, be a subsigma algebra of F and B, = { b, in F: b, = { w: S’(w) is in b,} for some b,, in B,,}. Define the set of G = {g: g is a random variable that is measurable with respect to B, has a finite second moment, and E( g 1B _,) = 0 for some non-negative 7 }, and the notation P,( h,) = CT=,h,. Assumption 1. The transformation u0 has a finite second moment, inf
lim
geG
Tern
S is stationary and ergodic, u0 is measurable with respect to B,,,
supE[T-‘{P,(g-u,)}*]
=O,
and
Assumption 1 allows us to apply Gordin’s theorem. The next theorem shows that b, does not converge in probability, but converges in distribution to a non-degenerate random variable under hypothesis HU. Let u,” = E(uF) and PO = 0. Then xt = x0 + P,, and T-‘P, converges in distribution to zero and (l/T)“*a-‘P, converges in distribution to a normally distributed random variable with mean zero and variance one. Since x+- xi = P; + x,T-‘P,, by (5): -1
O.~(~J*(T-~~*~‘P,)*-X~(T-~P~.
Thus b, - 0.5 converges in distribution to 0.5( u’/u,‘)x~.
(8)
We have shown the next result.
* Noting that the seasonal dummy can be artificially viewed as stationary and ergodic stochastic process [see, e.g., Ogaki (1988, p. 26, fn. l)], analysis in frequency domain by Hansen (1985) shows Gordin’s theorem is applicable to stochastic processes with detemum ’ ‘stic seasonal fluctuations.
40
J.A. Kahn, M. Ogaki / A chi-square test
Theorem 1. Suppose that Assumption 1 is satisfied and hypothesis HU is true. Then (b, - 0.5 : T 2 1) converges in distribution to 0.5( u 2/o~)~~, where xf is a random variable with the Chi-square distribution with one degree of freedom.
It should be noted that the term which involves x0 in (9) is negligible asymptotically but may have a strong impact on the small sample properties of b,. In eq. (5), T-‘CT=I(Ax,)2 converges almost surely to E[(Ax,)~], and T’-‘(xg - x,‘) converges in probability to zero. The next theorem follows immediately from eq. (5) and shows that b, is a consistent estimator of /3 under hypothesis HS. Theorem 2. Suppose that Assumption 1 is satisfied and that hypothesis 0.5): T 2 l} converges to zero in probability for any E c 1.
HS is true. Then (T’(b,
-
There are two types of discontinuity when a approaches one from below which we exploit in this paper. First, b, converges in probability to 0.5 when 1a 1 < 1, while b, has mean 0.5(1 + a2/u$-) when a = 1. Second, b,- 0.5 is of order T-’ in probability if 1a 1 -e 1, while b, - 0.5 is of order 1 in probability if a = 1.
4. A consistent test for a unit root Theorem 1 and 2 suggest a simple x2 test based on the OLS estimator b, with the null of a unit root (hypothesis HU). Under the null of a unit root, 2(u:/u2)(br - 0.5) converges in distribution to xz. Since the ratio z/u’ is an unknown parameter in general, we need to estimate this parameter to construct a test statistic. Suppose that &. and s$ are consistent estimators for u,’ and u2 under the null hypothesis HU. We also require that s&‘s: converges in probability to a positive real number under the alternative hypothesis HS. These requirements are met by the following estimators. Let aT be the OLS estimator of a in eq. (1) and T
2 = T-’ c (xt - aTx,_1)2,
S UT
00)
1=1 l(T)
c
s2 T = T-l T==
[j(T) -
-l(T)
,T I/I(T)]
il [h
- az-XI-d-L
where l(T) is the lag truncation number that satisfies Z(T) = 0(T’j4). is J,=
2(s;,/s$)(b,-
0.5).
- aTxt-l-d
3
(11)
The test statistic we propose
(12)
Under the null of a unit root, JT converges in distribution to xy in the light of Theorem 1. This test rejects the null of a unit root when JT is smaller than some critical value. Now let us consider the asymptotic property of JT under the alternative hypothesis HS. Theorem 2 implies that JT is O,(T-‘) in this case. Hence this test is consistent. ’
We used the RNDN function of GAUSS to create normal (pseudo) random variables.
J.A. Kahn, M. Ogaki /A
41
chi-square test
5. Finite sample properties We compare finite sample properties of the JT tests with those of conventional Dickey-Fuller tests, using simulations based on 40000 replications. Data were generated by the model (1) with the u, independent and identically distributed N(0, 1). 2 We consider a situation in which an econometrician knows that uI is serially uncorrelated. Since u 2 = u,’ in this case, we define JT = 2( b, - 0.5). Two versions of Dickey-Fuller tests used are the test statistic T( ur - l), which we denote by a(l), and the t-test for the hypothesis (Y= 1 in the regression (l), which we denote by t(1). In the following, x0 is an N(0, l/(1 - a2)) random variable that is independent of the U, (t = 1,. . . , 7”) when 1a 1 < 1, so that X, is stationary. Table 1 reports finite sample powers when the five percent critical values implied by asymptotic theories are used. The results are relevant when the econometrician does not know the true distributions and does not correct for small sample size distortions. In this case, our K, test has much greater power than the a(l) and t(1) tests when (Y is close to one and the sample size is small. On the other hand, Dickey-Fuller tests have higher power than the K, test when the sample size is 200 and (Y is 0.95. Table 2 reports finite sample size calculations when the five percent critical values implied by asymptotic theories are used. The initial value, x0, was either fixed to zero or a N(0, l/(1 - p’)) random variable with p = 0.95 or p = 0.99. When the initial value is fixed to zero, there are little size
Table 1 Empirical powers using five percent nominal critical values. a OL
KT
T(a,-1)
11
50 50
0.95 0.99
0.518 0.505
0.096 0.021
0.184 0.085
100 100
0.95 0.99
0.533 0.507
0.282 0.041
0.373 0.103
200 200
0.95 0.99
0.552 0.514
0.745 0.083
0.791 0.146
T
’ Estimates obtained from 40000 replications.
Table 2 Empirical sizes using five percent nominal values. a T
KT
T(cYT-~~)
11
0.051 0.051 0.051
50 50 50
x,=0 p = 0.95 p = 0.99
0.049 0.241 0.356
0.042 0.028 0.016
100 100 100
xg = 0 p = 0.95 p = 0.99
0.051 0.193 0.308
0.045 0.038 0.024
0.050 0.051 0.051
200 200 200
xg = 0 p = 0.95 p = 0.99
0.051 0.148 0.261
0.048 0.043 0.032
0.050 0.050 0.050
a Estimates obtained from 40000 replications.
42 Table 3 Size adjusted
J.A. Kahn, h4. Ogaki / A chi-square test
powers
of five percent
tests. a
a=p
KT
va,
50 50
0.95 0.99
0.116 0.098
0.166 0.068
0.180 0.083
100 100
0.95 0.99
0.141 0.109
0.354 0.085
0.365 0.101
200 200
0.95 0.99
0.174 0.117
0.789 0.131
0.790 0.148
T
a Estimates
obtained
-1)
'1
from 40000 replications.
distortions for the K, and t(1) tests while the a(l) test is conservative when the sample size is small. When the initial value is a N(0, l/(1 - p’)) random variable, the K, test has significant size distortions and is very liberal. The a(l) test is very conservative while the t(1) test has little size distortions. Table 3 reports size adjusted powers. The K, test has higher power than Dickey-Fuller tests when (Y= 0.99 and the sample size is 50 or 100. The a(1) and t(1) tests have higher power then the K, then when a = 0.95 for any sample size and when (Y= 0.99 and the sample size is 200. The present paper developed a &i-square test for a unit root that has higher power than Dickey-Fuller tests when the sample size is small and the autoregressive root is close to one. Since this is exactly where unit root tests have most difficulties, our test may have some merit.
References Dejong, David N., John C. Nankervis, N.E. Savin, and Charles H. Whiteman, 1988, Unit root tests or coin tosses for time series with autoregressive errors?, Working paper series no. 89-14 (University of Iowa, Iowa City, IA). Dickey, David A. and Wayne A. Fuller, 1979, Distributions of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association 74, 427-431. Dickey, David A. and Wayne A. Fuller, 1981, Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica 49,1057-1072. Hansen, Lam P., 1985, A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators, Journal of Econometrics 30, 203-208. McLeish, D.L., 1975, Invariance principles for dependent variables, Z. Wahrscheilichkeitstheorie und Verw. Gebiete 32, 165-178. Phillips, Peter, C.B., 1987, Time series regression with a unit root, Econometrica 55, 277-301. Phihips, Peter C.B. and Pierre Perron, 1988, Testing for a unit root in time series regression, Biometrica 75, 335-346. Said, Said E. and David A. Dickey, 1984, Testing for unit roots in autoregressive-moving average models of unknown order, Biometrica 71, 599-607. Serfling, R.J., 1968, Contributions to central Iimit theory for dependent variables, Annals of Mathematical Statistics 39, 1158-1175. White, Halbert and Ian Domowitz, 1984, Nonlinear regression with dependent observations, Econometrica 52,143-162.
Economics Letters 34 (1990) 37-42 North-Holland
A hi-square
test for a unit root
James A. Kahn and Masao Ogaki * Universityof Rochester, Rochester, NY 14627, USA Received 14 February 1990 Accepted 21 March 1990
The present paper develops a cl&square test for a unit root that has higher power than Dickey-Fuller size is small and the autoregressive root is close to one.
tests when the sample
1. Introduction
Much work has been done to modify Dickey and Fuller (1979, 1981) tests for unit roots to allow for general distributions and serial correlation of the disturbance term and various order of deterministic trends [e.g., Said and Dickey (1984), Phillips (1987) and Phillips and Perron (1988)]. Dejong et al. (1988) study small sample properties of Said-Dickey and Phillips-Perron tests and conclude that these tests are practically useless against trend stationary alternatives which are plausible for annual, quarterly, and monthly macroeconomic time series. The problem of these tests is that they are not powerful when the autoregressive root is close to one and the sample size is small. Rather than modifying Dickey-Fuller tests, we develop a x2 test for a unit root that has greater power than Dickey-Fuller tests when the autoregressive root is close to one and the sample size is small. Though it is beyond the scope of the present paper to develop tests that are practically useful in the sense of DeJong et al. (1988), it is important to investigate alternative tests given that modified versions of Dickey-Fuller tests given that modified versions of Dickey-Fuller tests do not have high power in small samples. Our tests does not rely on a functional central limit theorem as most of the other existing test do, but on a general central limit theorem. Though functional central limit theorems apply under general conditions [e.g., McLeish (1975)], central limit theorems apply under even more general conditions. For example, we can allow for deterministic seasonal fluctuations in the time series to be tested while Dickey-Fuller tests do not. This is useful because seasonally unadjusted data are of interest in most applications.
* The authors are grateful to Mahmoud El-Gamal, John Heaton, Charles Plosser, Danny Quah, Mark Watson, and especially to Lars Hansen and Adrian Pagan for helpful discussions. 016%1765/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
38
J.A. Kahn, M. Ogaki /A
chi-square test
2. Regression properties Consider a stochastic process {x I : t 2 l} generated in discrete time according to x, = (Y&-l + u,,
t=l,2,...,
(1)
where { uI} is a sequence of random variables with mean zero. We are interested in the following two hypotheses: HU: cw= 1, and HS: 1CY1c 1. The initial value, x0, is allowed to be any random variable. To motivate the test we develop, let us consider the regression ~~=bAx,+e,.
(2)
First, let us consider the case where hypothesis HS is true. For now, assume that x, is covariance stationary. In this case the best linear predictor of x, given Ax, is 0.5Ax,. To see this, define n,=Ax,[x,-0.5Ax,]
=0.5(x:-x;_,).
(3)
Then E(nI) = 0 as long as xt is covariance stationary so that E(x:) = E(x:_,). Thus PAX, satisfies the orthogonality condition E{ Ax,[x, - PAX,]} = 0 at /3= 0.5. This result is convenient to treat a composite null hypothesis of ( a 1 -c 1 because the true value of p does not depend on the value of (Y. Second, let us consider the case where hypothesis HU is true. For now, assume that u, is serially uncorrelated and that x0 has a finite second moment, so that the least square prediction theory is applicable. In this case E(T),) f 0 because E(x:) # E(xf_i). Hence j3 + 0.5. In fact, E(~h)/Wk)~l= E(x,u,)/EKu,)21= 1 if Wx,+,) = 0 for t 2 1. In this sense, /3= 1 when (Y= 1. This discontinuity of j3 as (Y approaches one motivates our tests.
3. Asymptotic properties of OLS estimators In this section, properties of the OLS estimator of /3 in eq. (5) are studied. Let b, be the OLS estimator:
Since CT=tq, = 0.5(x$ - xi),
(5) The key requirement for our test, when the null is that of a unit root (hypothesis HU), is that a central limit theorem applies to the partial sum of u,. To be precise, let
P, = c j=l
uj
(6)
J.A. Kahn, M. Ogaki /A
be the partial sum. We require that (l/T)‘/*Pz
a*=T+CC lim
E(T-‘PG)
39
chi-square test
converges in distribution to N(0,
a*), where
>O
as the sample size T approaches cc (all the limits in this paper is taken as T + 00). This requirement is satisfied under very general conditions. In the following, we assume that u, is stationary and ergodic with finite second moments, and.apply Gordin’s theorem [see, e.g., Hansen (1985)] so that
Alternatively, we can use a central limit theorem for weakly dependent random variables [see, e.g., Serfling (1968) and White and Domowitz (1984)]. Let (tin, F, P) be the underlying probability space. Let S denote a measurable transformation mapping s2 into itself. Then an arbitrary random variable, say h,,, and S together generates a time series via the relation h,(w) = h,( S’( w)). We assume that S is constructed so that u,(w) = u,,( S’( w)). Let &, be a subsigma algebra of F and B, = { b, in F: b, = { w: S’(w) is in b,} for some b,, in B,,}. Define the set of G = {g: g is a random variable that is measurable with respect to B, has a finite second moment, and E( g 1B _,) = 0 for some non-negative 7 }, and the notation P,( h,) = CT=,h,. Assumption 1. The transformation u0 has a finite second moment, inf
lim
geG
Tern
S is stationary and ergodic, u0 is measurable with respect to B,,,
supE[T-‘{P,(g-u,)}*]
=O,
and
Assumption 1 allows us to apply Gordin’s theorem. The next theorem shows that b, does not converge in probability, but converges in distribution to a non-degenerate random variable under hypothesis HU. Let u,” = E(uF) and PO = 0. Then xt = x0 + P,, and T-‘P, converges in distribution to zero and (l/T)“*a-‘P, converges in distribution to a normally distributed random variable with mean zero and variance one. Since x+- xi = P; + x,T-‘P,, by (5): -1
O.~(~J*(T-~~*~‘P,)*-X~(T-~P~.
Thus b, - 0.5 converges in distribution to 0.5( u’/u,‘)x~.
(8)
We have shown the next result.
* Noting that the seasonal dummy can be artificially viewed as stationary and ergodic stochastic process [see, e.g., Ogaki (1988, p. 26, fn. l)], analysis in frequency domain by Hansen (1985) shows Gordin’s theorem is applicable to stochastic processes with detemum ’ ‘stic seasonal fluctuations.
40
J.A. Kahn, M. Ogaki / A chi-square test
Theorem 1. Suppose that Assumption 1 is satisfied and hypothesis HU is true. Then (b, - 0.5 : T 2 1) converges in distribution to 0.5( u 2/o~)~~, where xf is a random variable with the Chi-square distribution with one degree of freedom.
It should be noted that the term which involves x0 in (9) is negligible asymptotically but may have a strong impact on the small sample properties of b,. In eq. (5), T-‘CT=I(Ax,)2 converges almost surely to E[(Ax,)~], and T’-‘(xg - x,‘) converges in probability to zero. The next theorem follows immediately from eq. (5) and shows that b, is a consistent estimator of /3 under hypothesis HS. Theorem 2. Suppose that Assumption 1 is satisfied and that hypothesis 0.5): T 2 l} converges to zero in probability for any E c 1.
HS is true. Then (T’(b,
-
There are two types of discontinuity when a approaches one from below which we exploit in this paper. First, b, converges in probability to 0.5 when 1a 1 < 1, while b, has mean 0.5(1 + a2/u$-) when a = 1. Second, b,- 0.5 is of order T-’ in probability if 1a 1 -e 1, while b, - 0.5 is of order 1 in probability if a = 1.
4. A consistent test for a unit root Theorem 1 and 2 suggest a simple x2 test based on the OLS estimator b, with the null of a unit root (hypothesis HU). Under the null of a unit root, 2(u:/u2)(br - 0.5) converges in distribution to xz. Since the ratio z/u’ is an unknown parameter in general, we need to estimate this parameter to construct a test statistic. Suppose that &. and s$ are consistent estimators for u,’ and u2 under the null hypothesis HU. We also require that s&‘s: converges in probability to a positive real number under the alternative hypothesis HS. These requirements are met by the following estimators. Let aT be the OLS estimator of a in eq. (1) and T
2 = T-’ c (xt - aTx,_1)2,
S UT
00)
1=1 l(T)
c
s2 T = T-l T==
[j(T) -
-l(T)
,T I/I(T)]
il [h
- az-XI-d-L
where l(T) is the lag truncation number that satisfies Z(T) = 0(T’j4). is J,=
2(s;,/s$)(b,-
0.5).
- aTxt-l-d
3
(11)
The test statistic we propose
(12)
Under the null of a unit root, JT converges in distribution to xy in the light of Theorem 1. This test rejects the null of a unit root when JT is smaller than some critical value. Now let us consider the asymptotic property of JT under the alternative hypothesis HS. Theorem 2 implies that JT is O,(T-‘) in this case. Hence this test is consistent. ’
We used the RNDN function of GAUSS to create normal (pseudo) random variables.
J.A. Kahn, M. Ogaki /A
41
chi-square test
5. Finite sample properties We compare finite sample properties of the JT tests with those of conventional Dickey-Fuller tests, using simulations based on 40000 replications. Data were generated by the model (1) with the u, independent and identically distributed N(0, 1). 2 We consider a situation in which an econometrician knows that uI is serially uncorrelated. Since u 2 = u,’ in this case, we define JT = 2( b, - 0.5). Two versions of Dickey-Fuller tests used are the test statistic T( ur - l), which we denote by a(l), and the t-test for the hypothesis (Y= 1 in the regression (l), which we denote by t(1). In the following, x0 is an N(0, l/(1 - a2)) random variable that is independent of the U, (t = 1,. . . , 7”) when 1a 1 < 1, so that X, is stationary. Table 1 reports finite sample powers when the five percent critical values implied by asymptotic theories are used. The results are relevant when the econometrician does not know the true distributions and does not correct for small sample size distortions. In this case, our K, test has much greater power than the a(l) and t(1) tests when (Y is close to one and the sample size is small. On the other hand, Dickey-Fuller tests have higher power than the K, test when the sample size is 200 and (Y is 0.95. Table 2 reports finite sample size calculations when the five percent critical values implied by asymptotic theories are used. The initial value, x0, was either fixed to zero or a N(0, l/(1 - p’)) random variable with p = 0.95 or p = 0.99. When the initial value is fixed to zero, there are little size
Table 1 Empirical powers using five percent nominal critical values. a OL
KT
T(a,-1)
11
50 50
0.95 0.99
0.518 0.505
0.096 0.021
0.184 0.085
100 100
0.95 0.99
0.533 0.507
0.282 0.041
0.373 0.103
200 200
0.95 0.99
0.552 0.514
0.745 0.083
0.791 0.146
T
’ Estimates obtained from 40000 replications.
Table 2 Empirical sizes using five percent nominal values. a T
KT
T(cYT-~~)
11
0.051 0.051 0.051
50 50 50
x,=0 p = 0.95 p = 0.99
0.049 0.241 0.356
0.042 0.028 0.016
100 100 100
xg = 0 p = 0.95 p = 0.99
0.051 0.193 0.308
0.045 0.038 0.024
0.050 0.051 0.051
200 200 200
xg = 0 p = 0.95 p = 0.99
0.051 0.148 0.261
0.048 0.043 0.032
0.050 0.050 0.050
a Estimates obtained from 40000 replications.
42 Table 3 Size adjusted
J.A. Kahn, h4. Ogaki / A chi-square test
powers
of five percent
tests. a
a=p
KT
va,
50 50
0.95 0.99
0.116 0.098
0.166 0.068
0.180 0.083
100 100
0.95 0.99
0.141 0.109
0.354 0.085
0.365 0.101
200 200
0.95 0.99
0.174 0.117
0.789 0.131
0.790 0.148
T
a Estimates
obtained
-1)
'1
from 40000 replications.
distortions for the K, and t(1) tests while the a(l) test is conservative when the sample size is small. When the initial value is a N(0, l/(1 - p’)) random variable, the K, test has significant size distortions and is very liberal. The a(l) test is very conservative while the t(1) test has little size distortions. Table 3 reports size adjusted powers. The K, test has higher power than Dickey-Fuller tests when (Y= 0.99 and the sample size is 50 or 100. The a(1) and t(1) tests have higher power then the K, then when a = 0.95 for any sample size and when (Y= 0.99 and the sample size is 200. The present paper developed a &i-square test for a unit root that has higher power than Dickey-Fuller tests when the sample size is small and the autoregressive root is close to one. Since this is exactly where unit root tests have most difficulties, our test may have some merit.
References Dejong, David N., John C. Nankervis, N.E. Savin, and Charles H. Whiteman, 1988, Unit root tests or coin tosses for time series with autoregressive errors?, Working paper series no. 89-14 (University of Iowa, Iowa City, IA). Dickey, David A. and Wayne A. Fuller, 1979, Distributions of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association 74, 427-431. Dickey, David A. and Wayne A. Fuller, 1981, Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica 49,1057-1072. Hansen, Lam P., 1985, A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators, Journal of Econometrics 30, 203-208. McLeish, D.L., 1975, Invariance principles for dependent variables, Z. Wahrscheilichkeitstheorie und Verw. Gebiete 32, 165-178. Phillips, Peter, C.B., 1987, Time series regression with a unit root, Econometrica 55, 277-301. Phihips, Peter C.B. and Pierre Perron, 1988, Testing for a unit root in time series regression, Biometrica 75, 335-346. Said, Said E. and David A. Dickey, 1984, Testing for unit roots in autoregressive-moving average models of unknown order, Biometrica 71, 599-607. Serfling, R.J., 1968, Contributions to central Iimit theory for dependent variables, Annals of Mathematical Statistics 39, 1158-1175. White, Halbert and Ian Domowitz, 1984, Nonlinear regression with dependent observations, Econometrica 52,143-162.