UNIT ROOT TESTING USING COVARIATES: SOME THEORY AND EVIDENCE{

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: Äy t ˆ äy t 1 u t where ut is i:i:d: (0, ó 2 u ): (1) The null hypothesis of a unit root in the characteristic polynomial of fy t g takes the form H 0 : ä ˆ 0. Next, assume that (Äx t,u t ), where x t is an I(0) process, is i.i.d and zero mean. By introducing the covariate Äx t into the systematic part of (1) we have: Äy t ˆ äy t 1 bäx t ù t (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 H 0 : ä ˆ 0 will have more power. Hansen (1995) extends the analysis to the general case in which the univariate series contains deterministic components (d t ) and lagged dependent variables: y t ˆ d t S t (3) where d t ˆ ì, or d t ˆ ì Wt (4) and

UNIT ROOT TESTING USING COVARIATES 585 a(l)äs t ˆ äs t 1 í t (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 Äx t, both contemporaneously and temporally, according to: í t ˆ b(l)9(äx t ì x ) ù t (6) where Äx t is an m-vector, and b(l) is a lag polynomial which allows (but does not require) for both q 1 lags and q 2 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 Äx t explain the zerofrequency movement of í t. A measure of the explanatory power of Äx t is given by the following squared correlation coef cient between í t and ù t : r 2 ˆ ó 2 íù ó 2 ù ó 2 (7) í When the regressors Äx t explain nearly all the zero-frequency movement in í t, then r 2 ˆ 0, whereas if they have no explanatory power, r 2 ˆ 1. Hansen (1995) shows that the asymptotic distribution of the t-statistic t(^ä) ˆ ^ä=s(^ä), under the null hypothesis ä ˆ 0, is a convex mixture of the standard normal and the Dickey-Fuller (DF) distribution, with the weights determined by the nuisance parameter r 2 : t(^ä) ˆ r(df) (1 r 2 ) 1=2 N(0, 1) (8) It can be seen from (8) that as r 2 ) 0, t(^ä) ) N(0, 1), whereas as r 2 ) 1, t(^ä) ) DF. Hansen (1995) refers to this as the CADF (p, q 1,q 2 ) statistic, where CADF stands for `covariate augmented Dickey-Fuller', and (p, q 1,q 2 ) 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 r 2 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 y t and its covariates x t ô, 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.

586 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: Äy t ˆ ä 1 y t 1 å t with ä 1 ˆ c=t, 0 (9) å t ˆ a11 a 12 å t 1 e 1t (9a) Äx t a 21 a 22 Äx t 1 e 2t where e 1t 0 e 2t 0 1 ó 21 ó 21 1 (9b) and also the eigenvalues of the matrix [a ij ], i, j ˆ 1, 2, are restricted to be less than one in absolute value, which ensures that the VAR is stable by making Äx t an I(0) variable. The above model implies that å t ˆ (a 11 ó 21 a 21 )(Äy t 1 ä 1 y t 2 ) (a 12 ó 21 a 22 )Äx t 1 ó 21 Äx t í 1t (10) where: í 1t ˆ e 1t ó 21 e 2t (10a) Substituting (10) into (9) we obtain: Äy t ˆ [(1 a 11 )ä 1 ó 21 a 21 ä 1 ]y t 1 [a 11 ó 21 a 21 ](1 ä 1 )Äy t 1 [a 12 ó 21 a 22 ]Äx t 1 ó 21 Äx t í 1t (11) Now we examine the following cases concerning the correlation structure of the variables involved: Case 1: ó 21 ˆ a 21 ˆ a 12 ˆ 0 (no contemporaneous correlation and no Granger causality). Under these restrictions, equation (11) becomes: Äy t ˆ (1 a 11 )ä 1 y t 1 a 11 (1 ä 1 )Äy t 1 í 1t (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

UNIT ROOT TESTING USING COVARIATES 587 on y t 1 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, a 12,a 21 6ˆ 0 (no contemporaneous correlation but Granger causality in both directions). Under these restrictions, equation (11) becomes: Äy t ˆ (1 a 11 )ä 1 y t 1 a 11 (1 ä 1 )Äy t 1 a 12 Äx t 1 í 1t (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 y t 1 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, a 12 ˆ a 21 ˆ 0 (contemporaneous correlation but no Granger causality in any direction). Equation (11) then becomes: Äy t ˆ (1 a 11 )ä 1 y t 1 a 11 (1 ä 1 )Äy t 1 ó 21 a 22 Äx t 1 ó 21 Äx t í 1t (11c) Here a CADF(1, 1, 0) test is again more powerful than an ADF(1) test. The coef cient on y t 1 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, a 21 6ˆ 0, a 12 ˆ 0 (contemporaneous correlation and oneway Granger causality running from å t to Äx t ). Equation (11) therefore becomes: Äy t ˆ [(1 a 11 ) ó 21 a 21 ]ä 1 y t 1 (a 11 ó 21 a 21 )(1 ä 1 )Äy t 1 ó 12 a 22 Äx t 1 ó 21 Äx t í 1t (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 y t 1 will be different. This is due to the presence of the additional term ó 21 a 21 ä 1 in the coef cient on y t 1. 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 a 21 ä 1 is negative, i.e. on whether ó 21 and a 21 have the same sign. If they do, the coef cient on y t 1 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 a 21 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 a 21 (or a 12 ) (experiments

588 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, a 12 6ˆ 0, a 21 ˆ 0 (contemporaneous correlation and oneway Granger causality running from Äx t to å t ). Under these restrictions, equation (11) becomes: Äy t ˆ (1 a 11 )ä 1 y t 1 a 11 (1 ä 1 )Äy t 1 (a 12 ó 21 a 22 )Äx t 1 ó 21 Äx t í 1t (11e) It can be seen that the coef cient on y t 1 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 y t 1 in a CADF framework will be the same as the one in an ADF framework if there is no contemporaneous correlation between å t and Äx t and/or no Granger causality running from å t to Äx t. 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 å t and Äx t. (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, a 12 ˆ a 21 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

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, a 12 6ˆ 0 and a 21 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).

590 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) ADF (2) ADF (3) ADF (4) Optimal l R 2 for the selected l Y 1.77 2.38 2.67 2.48 2.72 1 0.16 ( 9.15) ( 9.28) ( 9.26) ( 9.24) ( 9.23) IP 1.75 3.03 2.49 2.81 2.17 4 0.35 (7.55) ( 7.80) ( 7.81) ( 7.78) ( 7.88) E 2.43 3.08 3.18 2.93 2.60 1 0.28 (10.30) ( 10.58) ( 10.55) ( 10.52) ( 10.49) UR 2.13 4.72 4.08 2.66 3.02 1 0.42 ( 5.04) ( 5.55) ( 5.52) ( 5.49) ( 5.44) P 2.47 2.10 2.48 1.96 1.61 4 0.65 ( 9.49) ( 10.13) ( 10.14) ( 10.38) ( 10.39) RW 6.51 5.43 5.25 2.08 3.98 1 0.41 ( 10.13) (10.14) ( 10.11) ( 10.10) ( 10.08) M 0.21 1.54 1.76 2.21 1.84 1 0.37 ( 8.74) ( 9.18) ( 9.16) ( 9.16) ( 9.14) R 1.25 1.42 1.40 1.51 1.50 1 0.05 ( 11.03) ( 11.08) ( 11.03) ( 11.01) ( 10.98) SP 1.93 2.27 2.25 2.18 2.33 1 0.10 ( 5.67) ( 5.76) ( 5.74) ( 5.71) ( 5.68) GR 4.53 3.87 3.51 2.88 2.88 1 0.49 ( 5.61) ( 6.18) ( 6.15) ( 6.15) ( 6.17) XE 3.72 3.82 4.46 4.80 4.56 2 0.09 ( 5.41) ( 5.40) ( 5.42) ( 5.39) ( 5.40) MR 1.10 1.33 1.23 1.08 1.29 1 0.03 ( 5.86) ( 5.87) ( 5.86) ( 5.82) ( 5.82) 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: 2.87 b) constant plus a linear trend: 3.43. A`' indicates rejection of the null, based on the optimum ADF regression, at the 5% signi cance level.

UNIT ROOT TESTING USING COVARIATES 591 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) (^í t ) appear to covariate with rst differenced industrial production and unemployment rate at lag zero, with correlation coef cients equal to 0.65 and 0.53 respectively. No signi cant correlation is observed between ^í t and any of the variables for higher lags. In the case of industrial production, ^í t 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 r 2. Hansen (1995) suggests the non-parametric estimator: where ^r 2 ˆ ^ó 2 íù ^ó 2 í ^ó 2 ù (12) ^Ù ˆ ^ó 2 í ^ó íù ^ó íù ^ó 2 ˆ XM w(k=m)(t 1 ) X ù kˆ M t ^ç t k^ç9 t (12a) and ^ç t ˆ (^í t ^ù t )9 are least squares estimates of the error terms í 1 and ù t 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 )1as 4 Detailed results are not reported for reasons of space, but can be found in Caporale and Pittis (1997).

TABLE 2 Covariate ADF Tests I. Estimation Results 1. Real GNP (Y t ): ÄY t ˆ 0:031 0:0031 Y t 1 ˆ 0:058 ÄY t 1 0:254 Ä(I t) t 0:038 ÄUR t ^ù t (0:009) (0:0011) (0:055) (0:032) (0:010) 2. Ind. Production (IP t ): Ä(IP) t ˆ 0:025 0:0052 (IP) t 1 0:046 Ä(IP) t 1 0:148 Ä(IP) t 2 0:095 Ä(IP) t 3 0:136 Ä(IP) t 4 (0:007) (0:0017) (0:054) (0:046) (0:047) (0:042) 0:891 ÄY t 0:119 Ä(UR) t 0:451 ÄR t 0:054 ÄSP t 0:026 ÄSP t 1 ^ù t (0:110) (0:017) (0:202) (0:018) (0:018) 3. Employment (E t ): ÄE t ˆ 0:264 0:0244 E t 1 0:196 ÄE t 1 0:220 ÄR t 0:204 ÄY t 0:173 ÄY t 2 0:00013t ^ù t (0:107) (0:0099) (0:062) (0:075) (0:031) (0:036) (0:0004) 4. Consumer Price Index (P t ): ÄP t ˆ 0:0019 (0:006) 0:0049 (0:0020) P t 1 0:425 (0:065) ÄP t 1 0:044 (0:064) ÄP t 2 0:269 (0:061) ÄP t 3 0:038 (0:054) ÄP t 4 0:405 Ä(RW) t 0:129 ÄM t 0:000057 Ät ^ù t (0:059) (0:034) (0:000024) 5. Money Supply (M t ): ÄM t ˆ 0:139 0:022 M t 1 0:376 ÄM t 1 0:626 ÄP t 0:955 ÄR t 1 0:000095 t ^ù 2 (0:040) (0:006) (0:045) (0:070) (0:140) (0:000016) 6. Long-Term Interest Rate (R t ): ÄR t ˆ 0:0003 0:0077 R t 1 0:216 ÄR t 1 0:046 Ä(IP) t ^ù t (0:0006) (0:0093) (0:069) (0:012) 7. Stock-Price Index (SP t ): Ä(SP) t ˆ 0:158 0:0467 0:229Ä(SP) t 1:540 ÄM t 0:00065 t ^ù t (0:048) (0:0156) (0:068) (0:323) (0:00025) 8. Real Imports (MR t ): Ä(MR) t ˆ 0:045 0:0089 (MR) t 1 0:162 Ä(MR) t 1 0:396 Ä(XR) t 1:021 ÄP t ^ù t (0:017) (0:0032) (0:053) (0:048) (0:382) II. Diagnostics and Tests t(^ä) ^r uw SIC R 2 5% c.v. Y t 2.73 0.40 10.08 0.64 2.51 IP t 2.95 0.27 8.91 0.78 2.40 E t 2.47 0.64 10.96 0.54 3.10 P t 2.39 0.61 10.80 0.78 3.10 M t 3.35 0.49 9.84 0.70 2.99 R t 0.83 0.96 11.13 0.12 2.84 SP t 2.99 0.90 5.84 0.19 3.33 MR t 2.78 0.61 6.22 0.34 2.64 Notes: A` ' indicates cases in which a linear trend is included in the CADF equation. 592 BULLETIN

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 10.08, which is much lower than the corresponding value of 9.28 in the univariate ADF regression. Moreover, the adjusted R 2 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 í t 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 2.73, the critical value corresponding to r ˆ 0:4 being 2.51. 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

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. REFERENCES Andrews, D. W. K. (1991). `Heteroskedasticity and autocorrelation consistent covariance matrix estimation', Econometrica, Vol. 59, pp. 817±58. Banerjee, A., Dolado, J. J. and Mestre, R. (1998). `Error. Correction mechanism tests for cointegration in single-equation framework', Journal of Time Series Analysis, Vol. 19, pp. 267±84. Campbell, J. Y. and Mankiw, N. G. (1987a). `Are output uctuations transitory?', Quarterly Journal of Economics, Vol. 102, pp. 857±80. Campbell, J. Y. and Mankiw, N. G. (1987b). `Permanent and transitory components in macroeconomic uctuations', American Economic Review Papers and Proceedings, Vol. 77, pp. 111±17. Campbell, J. Y. and Perron, P. (1991). `Pitfalls and opportunities: what macroeconomists should know about unit roots', NBER Macroeconomics Annual, pp. 141±201. Caporale, G. M. and Pittis, N. (1996). `Persistence in macroeconomic time series: is it a model invariant property?', D.P. no. 02-96, Centre for Economic Forecasting, London Business School. Caporale, G. M. and Pittis, N. (1997). `Unit root testing using covariates: some

UNIT ROOT TESTING USING COVARIATES 595 theory and evidence', D.P. no. 20-97, Centre for Economic Forecasting, London Business School. Christiano, L. J. and Eichenbaum, M. (1989). `Unit roots in GNP: do we know and do we care?', Carnegie-Rochester Conference Series on Public Policy, Vol. 32, pp. 7±62. De Long, J. B. and Summers, L. H. (1988). `On the existence and interpretation of a ``unit root'' in US GNP', NBER W.P. no. 2716. Elliott, G., Rothenberg, T. J. and Stock, J. H. (1992). `Ef cient tests for an autoregressive unit root', NBER Technical W.P. no. 130. Hansen, B. E. (1995). `Rethinking the univariate approach to unit root testing ± Using covariates to increase power', Econometric Theory, Vol. 11, pp. 1148±71. Horvath, M. T. K. and Watson, M. W. (1995). `Testing for cointegration when some of the cointegrating vectors are prespeci ed', Econometric Theory, Vol. 11, pp. 984±1014. Kremers, J. J. M., Ericsson, N. R. and Dolado, J. J. (1992). `The power of cointegration tests', bulletin, Vol. 54, pp. 325±48. McCallum, B. T. (1993). `Unit roots in macroeconomic time series: some critical issues', NBER W.P. no. 4368. Miron, J. A. (1991). `Comment on ``Pitfalls and opportunites: what macroeconomists should know about unit roots''', NBER Macroeconomics Annual, pp. 211± 19. Nelson, C. and Plosser, C. (1982). `Trends and random walks in macroeconomic time series: some evidence and implications', Journal of Monetary Economics, Vol. 10, pp. 139±62. Pesaran, M. H. and Shin, Y. (1994). `Long-run structural modelling', D.P. no. 94-19, Department of Applied Economics, University of Cambridge. Pesaran, M. H., Shin, Y. and Smith, R. J. (1996). `Testing for the existence of a long-run relationship', mimeo, Department of Applied Economics, University of Cambridge. Rudebusch, G. D. (1993). `The uncertain unit root in real GNP', American Economic Review, Vol. 83, pp. 264±72. Spanos, A. (1990). `Unit roots and their dependence on the conditioning information set', Advances in Econometrics, Vol. 8, pp. 271±92. Stock, J. H. (1991). `Con dence intervals for the largest autoregressive root in US macroeconomic time series', Journal of Monetary Economics, Vol. 28, pp. 435± 59. Stock, J. H. (1994). `Unit roots, structural breaks and trends', in Engle, R. F. and McFadden, D. L. (eds.), Handbook of Econometrics, Vol. 4, chapter 46, pp. 2379±841, Elsevier, Amsterdam. West, K. D. (1988). `On the interpretation of near random walk behavior in GNP', American Economic Review, Vol. 778, pp. 202±08. Zivot, E. (1996). `The power of single equation tests for cointegration when the cointegrating vector is prespeci ed', mimeo, University of Washington.