Download UNIT ROOT TESTING USING COVARIATES: SOME THEORY AND

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 61, 4 (1999)
0305-9049
UNIT ROOT TESTING USING COVARIATES: SOME
THEORY AND EVIDENCE{
Guglielmo Maria Caporale and Nikitas Pittis
I. INTRODUCTION
In their seminal study, Nelson and Plosser (1982) argued that most US
macroeconomic series could be characterised as difference-stationary (DS)
as opposed to trend-stationary (TS) processes, which implied that shocks
were persistent, and hence that the data were consistent with Real Business
Cycle (RBC) models, where ¯uctuations are driven by technology shocks.
However, the power of unit tests was soon called into question, and it
became clear that in any ®nite sample it is impossible to discriminate
between a unit root and one which is very close to unity (see, e.g., Stock
(1991), Miron (1991), Campbell and Perron (1991), Rudebusch (1993)), or
even as low as 0.8 (see West (1988)).
Stock (1994) evaluates alternative testing methods, and shows that the
augmented Dickey-Fuller (ADF) t-test has lower power than most other unit
root tests, both asymptotically and in ®nite samples, but exhibits the lowest
size distortion of all univariate tests. This would suggest devising a test
which combines the desirable size properties of the ADF test with higher
power. Such a test has been proposed by Hansen (1995), who argues that
univariate tests ignore potentially useful information from related time
series, and that the exclusion of related stationary covariates from the
regression equation may lead to a power loss, which results in the overacceptance of the unit root null.1 He carries out a Monte Carlo study which
shows that his covariate-ADF (CADF) test is more powerful than the ADF
test and at same time does not suffer from size distortion for most
correlation structures. Therefore, the CADF test has the appealing property
that it delivers sizeable power gains but not at the expense of large size
distortions, which are typical instead of other unit root tests (see Stock
(1994)).
{Financial support from ESRC grant number L116 25 10103, Macroeconomic Modelling and
Policy Analysis in a Changing World, is gratefully acknowledged. We also wish to thank Anindya
Banerjee, Stephen Hall, Ron Smith and Aris Spanos for useful comments and suggestions. The
usual disclaimer applies.
1
A similar point had been made by Spanos (1990) and Caporale and Pittis (1996), who showed
that including an extra Granger-causing variable in the conditioning information set results in a
reduction in the size of the autoregressive parameter of an AR(1) process.
583
# Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford
OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
584
BULLETIN
The purpose of this paper is twofold. Firstly, we analyse the correlation
structure that must characterise the series involved if power gains are to be
obtained. We show that the inclusion of covariates affects unit root testing
by: (a) reducing the standard error of the estimate of the autoregressive
parameter without affecting the estimate itself, and/or (b) reducing both the
standard error and the absolute value of the estimate itself. Conditions in
terms of Granger causality and contemporaneous correlation are then
derived for case (a) or (b) to arise. Secondly, we investigate whether the
®nding of a unit root in a number of US macroeconomic series is reversed
when the more powerful CADF test, rather than the standard ADF method,
is used. It is shown that employing the former results in the rejection of the
unit root null in most cases, although high persistence is still found.
The layout of the paper is the following. Section II reviews Hansen's
(1995) Covariate Augmented Dickey Fuller (CADF) testing procedure for
unit roots. Section III analyzes the conditions under which using covariates
yields power gains, and also discusses the merits of this approach relative to
univariate unit root tests. As an illustration, Section IV applies the CADF
test to a number of US macroeconomic series. Section V offers some
concluding remarks.
II.
THE CADF TEST
Consider the following AR(1) model:
Äyt ˆ äytÿ1 ‡ ut where ut is i:i:d: (0, ó 2u ):
(1)
The null hypothesis of a unit root in the characteristic polynomial of fyt g
takes the form H0 : ä ˆ 0. Next, assume that (Äxt , ut ), where xt is an I(0)
process, is i.i.d and zero mean. By introducing the covariate Äxt into the
systematic part of (1) we have:
Äyt ˆ äytÿ1 ‡ bÄxt ‡ ùt
(2)
It can be shown (see Hansen (1995) that the error variance will be lower
than in the univariate regression equation (1). This suggests that ä can be
estimated more precisely in the context of (2), and that the t-test for the
hypothesis H0 : ä ˆ 0 will have more power.
Hansen (1995) extends the analysis to the general case in which the
univariate series contains deterministic components (dt ) and lagged dependent variables:
yt ˆ dt ‡ St
(3)
dt ˆ ì, or dt ˆ ì ‡ Wt
(4)
where
and
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UNIT ROOT TESTING USING COVARIATES
585
a(L)ÄSt ˆ äStÿ1 ‡ ít
(5)
a(L) being a pth order polynomial in the lag operator. The error term in (5)
is a white noise process which, however, covariates with Äxt , both contemporaneously and temporally, according to:
ít ˆ b(L)9(Äxt ÿ ìx ) ‡ ùt
(6)
where Äxt is an m-vector, and b(L) is a lag polynomial which allows (but
does not require) for both q1 lags and q2 leads.
Again the t-test for the hypothesis ä ˆ 0 is expected to have more power
when (5) is taken into account in the estimation of (3). The power gains are
a function of the extent to which the regressors Äxt explain the zerofrequency movement of ít . A measure of the explanatory power of Äxt is
given by the following squared correlation coef®cient between ít and ùt :
r2 ˆ
ó 2íù
ó 2ù ó 2í
(7)
When the regressors Äxt explain nearly all the zero-frequency movement in
ít , then r2 ˆ 0, whereas if they have no explanatory power, r2 ˆ 1.
Hansen (1995) shows that the asymptotic distribution of the t-statistic
^ ˆ ä=s(
^ ä),
^ under the null hypothesis ä ˆ 0, is a convex mixture of the
t(ä)
standard normal and the Dickey-Fuller (DF) distribution, with the weights
determined by the nuisance parameter r2 :
^ ˆ r(DF) ‡ (1 ÿ r2 )1=2 N(0, 1)
(8)
t(ä)
^ ) N(0, 1), whereas as r2 ) 1,
It can be seen from (8) that as r2 ) 0, t(ä)
^
t(ä) ) DF. Hansen (1995) refers to this as the CADF (p, q1 , q2 ) statistic,
where CADF stands for `covariate augmented Dickey-Fuller', and
(p, q1 , q2 ) are the orders of the polynomials a(L) and b(L) as given by (5)
and (6). He also computes appropriate critical values for alternative values
of r2 in steps of 0.1, the upper and lower bounds being the DF and N(0, 1)
ones respectively. These critical values are reported in Hansen's Table 1,
and show that the conventional Dickey-Fuller critical values are conservative when a regression has stochastic covariates.
It is important to note that, in the context of the general model (4)±(7),
the inclusion of covariates may affect not only the standard error but also
the estimates of ä. The extent to which power gains stem from a reduction
in the standard error of the estimate alone depends on the contemporaneous
and temporal correlation structure between yt and its covariates xtÿô ,
ô ˆ 0, 1, 2, . . . , as we show in the next section.2
2
Hansen's (1995) test is related to the cointegration test provided by Kremers, Ericsson and Dolado
(1992), who show that for large (small) values of the signal-t-noise ratio the asymptotic distribution
of the t-statistic in an ECM converges to the standard normal (Dickey-Fuller) distribution. The
method developed by Pesaran, Shin and Smith (1996) to test for the existence of a long-run
relationship under the null of no cointegration is also closely related to the approach described above.
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BULLETIN
III.
SOME THEORY
Whilst Hansen (1995) examines only experimentally how the power of unit
root tests is affected by the inclusion of covariates, we investigate this issue
analytically. Below we distinguish between a number of cases which can
arise, and also explain how our typology is related to Hansen's (1995)
Monte Carlo analysis.
(i) Unit Root Tests and Granger Causality
Following Hansen's (1995) experimental design, we assume that the data
are generated from the following VAR model:
Äyt ˆ ä1 ytÿ1 ‡ åt with ä1 ˆ ÿc=T , 0
åt
a11 a12
åtÿ1
e
ˆ
‡ 1t
Äxt
a21 a22 Äxtÿ1
e2t
where
e1t
e2t
0
1
0 ó 21
ó 21
1
(9)
(9a)
(9b)
and also the eigenvalues of the matrix [aij ], i, j ˆ 1, 2, are restricted to be
less than one in absolute value, which ensures that the VAR is stable by
making Äxt an I(0) variable.
The above model implies that
åt ˆ (a11 ÿ ó 21 a21 )(Äytÿ1 ÿ ä1 ytÿ2 ) ‡ (a12 ÿ ó 21 a22 )Äxtÿ1 ‡ ó 21 Äxt ‡ í1t
(10)
where:
í1t ˆ e1t ÿ ó 21 e2t
(10a)
Substituting (10) into (9) we obtain:
Äyt ˆ [(1 ÿ a11 )ä1 ‡ ó 21 a21 ä1 ]ytÿ1 ‡ [a11 ÿ ó 21 a21 ](1 ‡ ä1 )Äytÿ1
‡ [a12 ÿ ó 21 a22 ]Äxtÿ1 ‡ ó 21 Äxt ‡ í1t
(11)
Now we examine the following cases concerning the correlation structure of
the variables involved:
Case 1: ó 21 ˆ a21 ˆ a12 ˆ 0 (no contemporaneous correlation and no
Granger causality). Under these restrictions, equation (11) becomes:
Äyt ˆ (1 ÿ a11 )ä1 ytÿ1 ‡ a11 (1 ‡ ä1 )Äytÿ1 ‡ í1t
(11a)
Obviously, in this case the CADF test is equivalent to the ADF(1) test. The
inclusion of covariates does not affect either the estimate of the coef®cient
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587
on ytÿ1 or its standard error. Therefore, unit root inference is not affected by
the inclusion of such covariates. This is also shown empirically by Hansen
(1995) in his experiment 1.
Case 2: ó 21 ˆ 0, a12 , a21 6ˆ 0 (no contemporaneous correlation but Granger causality in both directions). Under these restrictions, equation (11)
becomes:
Äyt ˆ (1 ÿ a11 )ä1 ytÿ1 ‡ a11 (1 ‡ ä1 )Äytÿ1 ‡ a12 Äxtÿ1 ‡ í1t
(11b)
In this case a CADF(1, 1, 0) test is more powerful than the ADF(1) test.
This is because, although the coef®cient of ytÿ1 is the same as in the
ADF(1) case, its standard error will be smaller, and therefore it will be
estimated with a greater degree of precision. This case is not examined by
Hansen (1995) in his experiments.
Case 3: ó 21 6ˆ 0, a12 ˆ a21 ˆ 0 (contemporaneous correlation but no Granger causality in any direction). Equation (11) then becomes:
Äyt ˆ (1 ÿ a11 )ä1 ytÿ1 ‡ a11 (1 ‡ ä1 )Äytÿ1 ÿ ó 21 a22 Äxtÿ1 ‡ ó 21 Äxt ‡ í1t
(11c)
Here a CADF(1, 1, 0) test is again more powerful than an ADF(1) test. The
coef®cient on ytÿ1 is the same as in the ADF(1) equation, but the power
gains stem from a decrease in the variance of its estimate. This case
corresponds to Hansen's (1995) design 7, where the power of the ADF(2)
test is found to be 19% as opposed to 27% for a CADF(2, 1, 1) test.
Case 4: ó 21 6ˆ 0, a21 6ˆ 0, a12 ˆ 0 (contemporaneous correlation and oneway Granger causality running from åt to Äxt ). Equation (11) therefore
becomes:
Äyt ˆ [(1 ÿ a11 ) ‡ ó 21 a21 ]ä1 ytÿ1 ‡ (a11 ÿ ó 21 a21 )(1 ‡ ä1 )Äytÿ1
ÿ ó 12 a22 Äxtÿ1 ‡ ó 21 Äxt ‡ í1t
(11d)
In this case a CADF(1, 1, 0) test produces different inference from an
ADF(1) test ± both the standard error and the estimate of the coef®cient on
ytÿ1 will be different. This is due to the presence of the additional term
ó 21 a21 ä1 in the coef®cient on ytÿ1 . The extent to which the inclusion of
covariates results in a reduction in the value of this coef®cient depends on
whether the extra term ó 21 a21 ä1 is negative, i.e. on whether ó 21 and a21
have the same sign. If they do, the coef®cient on ytÿ1 increases in absolute
value. As the unit root test is one-sided, this leads to power gains. The
opposite is true, however, if ó 21 and a21 have opposite signs. Then, the extra
term tends to make the coef®cient positive. In fact, Hansen's (1995)
simulations show that a positive ó 21 and negative a21 (or a12 ) (experiments
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BULLETIN
2, 3, 4, 5, pp. 1161±1164) are associated with an over-rejection of the null
in the univariate framework.
Case 5: ó 21 6ˆ 0, a12 6ˆ 0, a21 ˆ 0 (contemporaneous correlation and oneway Granger causality running from Äxt to åt ). Under these restrictions,
equation (11) becomes:
Äyt ˆ (1 ÿ a11 )ä1 ytÿ1 ‡ a11 (1 ‡ ä1 )Äytÿ1 ‡ (a12 ÿ ó 21 a22 )Äxtÿ1
‡ ó 21 Äxt ‡ í1t
(11e)
It can be seen that the coef®cient on ytÿ1 in the CADF(1, 1, 0) equation is
now the same as the one in the ADF(1) equation. Power gains result from a
higher degree of precision in the estimation of this coef®cient when
covariates are included.
To sum up, we have shown analytically (unlike Hansen (1995), who does
so experimentally) that the coef®cient on ytÿ1 in a CADF framework will be
the same as the one in an ADF framework if there is no contemporaneous
correlation between åt and Äxt and/or no Granger causality running from åt
to Äxt . However, even in these cases, the variance of the estimate of this
coef®cient will be lower in a CADF framework than in an ADF framework,
thus resulting in power gains. The CADF test will be equivalent to the ADF
test only in the case where there is neither contemporaneous nor temporal
dependence between åt and Äxt .
(ii) Power Comparisons
Stock (1994) compares the local asymptotic power of a number of classes
of univariate unit root tests, the members of which have the same local
asymptotic power functions, but can perform quite differently in ®nite
samples. He carries out Monte Carlo simulations and ®nds that, although
the Dickey-Fuller ^ô-statistic exhibits the greatest ability to control size for
alternative designs, it also has the lowest power ± in fact, it has the worst
size-adjusted power of all tests considered. However, the higher power of
other tests, except perhaps the DF-GLS devised by Elliott et al. (1992), is
achieved at the expense of large size distortions.
These results suggest that the inclusion of covariates in the ADF-model
could be a simple way to increase power without incurring large size
distortions. Consider, for instance, our case 3, where ó 21 6ˆ 0, a12 ˆ
a21 ˆ 0. Hansen's experimental design 7 shows that CADF(2, 0, 0),
CADF(2, 1, 0), CADF(2, 0, 1), and CADF(2, 1, 1) all have the same ®nite
sample size as an ADF(2) test with nominal size of 5 percent. However, the
power of these tests is 28, 27, 28 and 27 percent respectively, whereas the
power of the ADF(2) test is only 19 percent. These ®ndings are even more
striking if we take into account that for this particular design the value of r
is equal to 0.84, which means that the relative contribution of the covariate
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UNIT ROOT TESTING USING COVARIATES
589
to the innovations in the DF model is very small. As an another example,
consider our case 5, where ó 21 6ˆ 0, a12 6ˆ 0 and a21 ˆ 0, which corresponds
to Hansen's design 15. Here the ®nite sample size for the CADF(2, 1, 0)
and CADF(2, 1, 1) is still 5 percent, whereas the size for CADF(2, 0, 0)
and CADF(2, 0, 1) is only 1 percent. Nevertheless, despite these size
distortions, the power gains from including covariates are huge. For
example, the power of CADF(2, 1, 0) is 64 percent, whereas the power of
the ADF(2) is only 20 percent.
In a system-of-equations framework, multivariate tests have lower power
compared to the univariate CADF test if the covariates are I(1), because
fewer parameters have to be estimated and hence there are extra degrees of
freedom (see Horvath and Watson (1995)). The reason is that the multivariate procedure tests the joint null hypothesis that both the series of
interest and the covariates are I(1), whilst the CADF tests the unit root
hypothesis only for the series of interest. Also, in a single equation framework, provided the cointegrating vector is correctly prespeci®ed, conditional
ECM-based t-tests for no-cointegration have higher power than tests for
cointegration based on estimating the cointegrating vector, and the power
gain is of the same order of magnitude as that achieved by using a CADF
(rather than an ADF) test in the case of univariate unit roots (see Zivot
(1996)).3
IV. EMPIRICAL RESULTS
As an illustration, in this section we carry out both ADF and CADF tests in
order to analyze the stationarity properties of a number of U.S. macroeconomic time series including output, unemployment, government spending,
money, prices and interest rates, as in the Nelson and Plosser (1982) paper.
The univariate results presented above lead us to believe that even in a
multivariate setup, with more than one covariate, covariate augmentation
might be advantageous, and in fact we ®nd that in most cases it does reverse
the ®nding of a unit root. As there is no evidence of misspeci®cation of the
selected univariate ADF models, this can only be attributed to the inclusion
of appropriately selected covariates in the ADF regressions, which results in
the correct de®nition of the largest root and of its standard error, as shown
in Section III, and hence in power gains. However, the multivariate nature
of the empirical analysis does mean that there is no one-to-one mapping
between the theory presented above and the empirical ®ndings. In other
words, our typology cannot be directly relied upon for the interpretation of
the results.
The selected sample consists of quarterly data, obtained from the
3
For another t-test of cointegration in a conditional ECM framework, based on the estimation
rather than imposition of the cointegrating vector, and for its power properties compared to
dimensional invariant tests for cointegration, see Banerjee, Dolado and Mestre (1998).
# Blackwell Publishers 1999
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BULLETIN
Business Conditions Digest of the U.S. Department of Commerce, and cover
the period 1948.1 to 1994.4. All series except the bond yield are in logs.
Table 1 reports standard univariate Augmented Dickey-Fuller (ADF) tests,
with the optimal lag-length of the ADF regression equation being determined by the Schwarz Information Criterion (SIC). It can be noted that the
null hypothesis of a unit root is rejected in the case of the unemployment
TABLE 1
Univariate ADF(1) Tests
DF
ADF (1)
Y‡
ADF (2)
ADF (3)
ADF (4)
Optimal R2 for the
selected l
l
ÿ1.77
ÿ2.38
ÿ2.67
ÿ2.48
ÿ2.72 1
(ÿ9.15) (ÿ9.28) (ÿ9.26) (ÿ9.24) (ÿ9.23)
IP‡
ÿ1.75
ÿ3.03
ÿ2.49
ÿ2.81
ÿ2.17 4
(7.55) (ÿ7.80) (ÿ7.81) (ÿ7.78) (ÿ7.88)
E‡
ÿ2.43
ÿ3.08
ÿ3.18
ÿ2.93
ÿ2.60 1
(10.30) (ÿ10.58) (ÿ10.55) (ÿ10.52) (ÿ10.49)
UR‡ ÿ2.13
ÿ4.72 ÿ4.08 ÿ2.66 ÿ3.02 1
(ÿ5.04) (ÿ5.55) (ÿ5.52) (ÿ5.49) (ÿ5.44)
P‡
ÿ2.47
ÿ2.10
ÿ2.48
ÿ1.96
ÿ1.61 4
(ÿ9.49) (ÿ10.13) (ÿ10.14) (ÿ10.38) (ÿ10.39)
RW
ÿ6.51 ÿ5.43 ÿ5.25 ÿ2.08 ÿ3.98 1
(ÿ10.13)
(10.14) (ÿ10.11) (ÿ10.10) (ÿ10.08)
M‡
ÿ0.21
ÿ1.54
ÿ1.76
ÿ2.21
ÿ1.84 1
(ÿ8.74) (ÿ9.18) (ÿ9.16) (ÿ9.16) (ÿ9.14)
R
ÿ1.25
ÿ1.42
ÿ1.40
ÿ1.51
ÿ1.50 1
(ÿ11.03) (ÿ11.08) (ÿ11.03) (ÿ11.01) (ÿ10.98)
SP‡ ÿ1.93
ÿ2.27
ÿ2.25
ÿ2.18
ÿ2.33 1
(ÿ5.67) (ÿ5.76) (ÿ5.74) (ÿ5.71) (ÿ5.68)
GR
ÿ4.53 ÿ3.87 ÿ3.51 ÿ2.88 ÿ2.88 1
(ÿ5.61) (ÿ6.18) (ÿ6.15) (ÿ6.15) (ÿ6.17)
XE‡ ÿ3.72 ÿ3.82 ÿ4.46 ÿ4.80 ÿ4.56 2
(ÿ5.41) (ÿ5.40) (ÿ5.42) (ÿ5.39) (ÿ5.40)
MR
ÿ1.10
ÿ1.33
ÿ1.23
ÿ1.08
ÿ1.29 1
(ÿ5.86) (ÿ5.87) (ÿ5.86) (ÿ5.82) (ÿ5.82)
0.16
0.35
0.28
0.42
0.65
0.41
0.37
0.05
0.10
0.49
0.09
0.03
Notes 1. The series codes are Y ˆ Real GDP; IP ˆ Industrial Production; E ˆ Employment;
UR ˆ Unemployment Rate; P ˆ Consumer Price Index; RW ˆ Real Wages; M ˆ Money
Supply (M1); R ˆ Bond Yield; SP ˆ Common Stock Prices; GR ˆ Real Government Expenditure; XR ˆ Real Exports; MR ˆ Real Imports
2. A `‡' indicates cases in which a linear trend (found to be signi®cant) is included in the
ADF equations.
3. The values of the Schwarz Information Criterion (SIC) for selecting the optimum lag-length
in the ADF equations are reported in parentheses.
4. 5% critical values: a) constant: ÿ2.87
b) constant plus a linear trend: ÿ3.43.
A ` ' indicates rejection of the null, based on the optimum ADF regression, at the 5%
signi®cance level.
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UNIT ROOT TESTING USING COVARIATES
rate, real wages, real government expenditure, and real exports for any laglength in the ADF regression equations. For the remaining eight series,
namely, real output, industrial production, employment, consumer price
index, money supply, long-term interest rate, a composite stock price index,
and real imports, the null of a unit root cannot be rejected on the basis of
the ADF test.
Next we examine whether the parameter of interest, ä, and its standard
error can be estimated more precisely by including covariates in the ADF
regression equations. We estimate the cross-correlations between the white
residuals from the optimal ADF regression and each of the ®rst differenced
series (for lags 0, 1, 2, 3, and 4) as a guide to selecting the appropriate
covariates in the Covariate Augmented Dickey Fuller (CADF) regression
equations ± the covariates with the highest degree of (contemporaneous and
temporal) correlation are then included. First differences are required
because, under the null, all series are I(1). For example, the residuals from
the univariate ADF regression for real (GNP) (^ít ) appear to covariate with
®rst differenced industrial production and unemployment rate at lag zero,
with correlation coef®cients equal to 0.65 and ÿ0.53 respectively. No
signi®cant correlation is observed between ^
ít and any of the variables for
higher lags. In the case of industrial production, ^ít is correlated with output,
employment, unemployment rate, interest rate, and real government spending at zero lag, and with money supply, and government spending at lag
one.4
The analysis based on the sample cross-correlations suggests alternative
combinations of variables to be included as covariates in the CADF
regression equations. We select the one which minimises the SIC, and the
results are reported in Table 2. In order to use the correct critical values
from Hansen's (1995) Table 1, we need a consistent estimate of r2 . Hansen
(1995) suggests the non-parametric estimator:
^2 ˆ
r
where
^ ˆ
Ù
ó^ 2í
ó^ íù
ó^ íù
ó^ 2ù
ˆ
ó^ 2íù
ó^ 2í ó^ 2ù
M
X
kˆÿM
w(k=M)(T ÿ1 )
(12)
X
t
^tÿk ç
^9t
ç
(12a)
^ t )9 are least squares estimates of the error terms í1 and ùt
^t ˆ (^
ít ù
and ç
from the regression equations (5) and (6) respectively. We employ both
Bartlett and Parzen kernel weights. It must be noted, however, that the
choice of the kernel is not as important as the selection of the bandwidth
parameter M. In consistency proofs, it is usually assumed that M ) 1 as
4
Detailed results are not reported for reasons of space, but can be found in Caporale and Pittis
(1997).
# Blackwell Publishers 1999
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# Blackwell Publishers 1999
TABLE 2
Covariate ADF Tests
I. Estimation Results
^t
1. Real GNP (Yt ): ÄYt ˆ 0:031 ÿ 0:0031 Y tÿ1 ˆ 0:058ÄY tÿ1 ‡ 0:254Ä(I t ) t ÿ 0:038ÄURt ‡ ù
(0:009)
(0:0011)
(0:055)
(0:032)
(0:010)
2. Ind. Production (IPt ): Ä(IP) t ˆ 0:025 ÿ 0:0052(IP) tÿ1 ‡ 0:046Ä(IP) tÿ1 ÿ 0:148Ä(IP) tÿ2 ‡ 0:095Ä(IP) tÿ3 ÿ 0:136Ä(IP) tÿ4
(0:007)
(0:0017)
(0:054)
(0:046)
(0:047)
(0:042)
^t
‡ 0:891ÄY t ÿ 0:119Ä(UR) t ‡ 0:451ÄRt ÿ 0:054ÄSPt ÿ 0:026ÄSPtÿ1 ‡ ù
(0:110)
(0:017)
(0:202)
(0:018)
(0:018)
^t
3. Employment (Et ): ÄE t ˆ 0:264 ÿ 0:0244 E tÿ1 ‡ 0:196ÄE tÿ1 ‡ 0:220ÄRt ‡ 0:204ÄY t ‡ 0:173ÄY tÿ2 ‡ 0:00013 t ‡ ù
(0:107)
(0:0099)
(0:062)
(0:075)
(0:031)
(0:036)
(0:0004)
4. Consumer Price Index (Pt ): ÄPt ˆ 0:0019 ÿ 0:0049 Ptÿ1 ‡ 0:425ÄPtÿ1 ÿ 0:044ÄPtÿ2 ‡ 0:269ÄPtÿ3 ÿ 0:038ÄPtÿ4
(0:006)
(0:0020)
(0:065)
(0:064)
(0:061)
(0:054)
^t
ÿ 0:405Ä(RW ) t ÿ 0:129ÄM t ‡ 0:000057Ät ‡ ù
(0:034)
(0:000024)
(0:040)
(0:006)
(0:045)
(0:070)
(0:140)
(0:000016)
^t
6. Long-Term Interest Rate (Rt ): ÄRt ˆ 0:0003 ÿ 0:0077 Rtÿ1 ‡ 0:216ÄRtÿ1 ‡ 0:046Ä(IP) t ‡ ù
(0:0006)
(0:0093)
(0:069)
(0:012)
^t
7. Stock-Price Index (SPt ): Ä(SP) t ˆ 0:158 ÿ 0:0467 ‡ 0:229Ä(SP) t ‡ 1:540ÄM t ‡ 0:00065 t ‡ ù
(0:048)
(0:0156)
(0:068)
(0:323)
(0:00025)
^t
8. Real Imports (MRt ): Ä(MR) t ˆ 0:045 ÿ 0:0089(MR) tÿ1 ÿ 0:162Ä(MR) tÿ1 ‡ 0:396Ä(XR) t ‡ 1:021ÄPt ‡ ù
II.
Diagnostics and Tests
^
t(ä)
ÿ2.73
Yt
ÿ2.95
IPt
ÿ2.47
E‡t
ÿ2.39
P‡t
ÿ3.35
M ‡t
ÿ0.83
Rt
ÿ2.99
SP‡t
ÿ2.78
MRt
(0:017)
(0:0032)
^ uw
r
0.40
0.27
0.64
0.61
0.49
0.96
0.90
0.61
(0:053)
SIC
ÿ10.08
ÿ8.91
ÿ10.96
ÿ10.80
ÿ9.84
ÿ11.13
ÿ5.84
ÿ6.22
Notes: A `‡' indicates cases in which a linear trend is included in the CADF equation.
(0:048)
(0:382)
R2
0.64
0.78
0.54
0.78
0.70
0.12
0.19
0.34
5% c.v.
ÿ2.51
ÿ2.40
ÿ3.10
ÿ3.10
ÿ2.99
ÿ2.84
ÿ3.33
ÿ2.64
BULLETIN
(0:059)
^2
5. Money Supply (Mt ): ÄM t ˆ 0:139 ÿ 0:022 M tÿ1 ‡ 0:376ÄM tÿ1 ÿ 0:626ÄPt ÿ 0:955ÄRtÿ1 ‡ 0:000095 t ‡ ù
UNIT ROOT TESTING USING COVARIATES
593
T ) 1, such that M=T 1=2 ) 0, although such an assumption should not be
treated as a guide to the optimal selection of the lag truncation parameter
(for a discussion of this issue, see Andrews (1991)).
To make the strongest possible case against the null we report, in the
second part of Table 2, the highest estimate of r (which is accompanied by
the highest critical value) for alternative kernel weights and values of the
bandwidth parameter. As an illustration, we discuss in detail the results
from the estimation of the CADF regression equations for real GNP, and
then only summarise those for the other variables.
In the case of real GNP the SIC is minimised when the ®rst differences of
industrial production and unemployment rate at lag zero are included as
covariates in the CADF regression, i.e. the relevant test is a CADF(1, 1, 0).
The SIC reaches the value of ÿ10.08, which is much lower than the
corresponding value of ÿ9.28 in the univariate ADF regression. Moreover,
the adjusted R2 jumps from 0.16 in the univariate ADF to 0.64 in the CADF
regression. Both industrial production and the unemployment rate appear to
be highly signi®cant. The inclusion of trending covariates results in the time
trend becoming insigni®cant, and the latter is therefore excluded from the
CADF regression equation. The explanatory power of these particular
covariates is also re¯ected in the relatively small estimate of r (0.4), which
suggests that the included covariates explain a substantial amount of the
movement of ít at the zero frequency. This leads to the rejection of the null
hypothesis of a unit root for real GNP, since the t-statistic is ÿ2.73, the
critical value corresponding to r ˆ 0:4 being ÿ2.51. However, it must be
noted that the point estimate of ä is very close to zero, which implies that,
although real GNP is not I(1), it is still highly persistent.
As for the other series, Table 2 shows which variables should be included
as covariates in order to minimise the SIC. Of the eight series for which the
unit root null was not rejected in the univariate framework, three more,
namely industrial production, money supply and real imports, were found to
be I(0) when the unit root test was carried out in a covariate framework.
However, even for these series the point estimate of ä is very small, namely
0.0052, 0.022, and 0.0089 for industrial production, money supply and real
imports respectively. Consequently, the largest root is signi®cantly smaller
than but very close to one, and the series exhibit a high degree of
persistence.
V.
CONCLUSIONS
Most of the recent literature agrees that univariate unit root tests have low
power. Hansen (1995) suggests that adopting a multivariate framework
might result in large power gains, and presents some Monte Carlo evidence
indicating that his recommended CADF test does produce more precise
estimates of the autoregressive coef®cient than a conventional ADF test.
This paper focuses on the theoretical conditions under which power can
# Blackwell Publishers 1999
594
BULLETIN
be increased by using covariates, and hence the CADF test should be used
in preference to the ADF test. More speci®cally, we show that power gains
are a function of the correlation structure of the VAR, and that they will be
achieved as long as either contemporaneous or temporal dependence are
present. We also stress that a major advantage of the CADF test compared
to univariate unit root tests is the fact that power can be increased without
incurring large size distortions. Contrary to what standard ADF tests
suggest, a number of US macroeconomic time series, such as real GNP,
industrial production, money supply, and real imports, can be characterised
as stationary when a CADF test using appropriate covariates is carried out,
although they still appear to be highly persistent.
These ®ndings can be seen as a challenge to the prevailing wisdom of the
1980s, namely that non-stationarity is a feature of most macroeconomic
series (which is interpreted as supportive of RBC models). Furthermore,
they suggest that the statistical properties of a time series should not be
considered in isolation, but in the context of multivariate models, and that
economic theory should also be relied upon as a guide to model speci®cation (see McCallum (1993)). In the context of a more sensible structural
economic modelling the question of whether I(0) or I(1) is the appropriate
univariate description of a series then becomes less interesting (see Pesaran
and Shin (1994)).
University of East London.
University of Cyprus
Date of Receipt of Final Manuscript: October 1998.
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