This paper shows, analytically and numerically, the effects of a misspecification in the degree of integration on testing for cointegration. Johansen LR tests tend to find too much spurious cointegration while the Engle-Granger test shows a more robust performance than the LR tests

Gonzalo, Jesús

Lee, Tae-Hwy

English

Name not available

On the robustness of cointegration tests when series are fractionally integrated

Universidad Carlos III de Madrid e-Archivo

Working Paper 96-07 Departamento de Estadística y Econometría Statistics and Econometrics Series 03 Universidad Carlos III de Madrid January 1996 Calle Madrid, 126 28903 Getafe (Spain) Fax (341) 624-9849 ON THE ROBUSTNESS OF COINTEGRATION TESTS WHEN SERIES ARE FRACTlONALLy INTEGRATED Jesús Gonzalo and Tae-Hwy Lee· Abstract _ This paper shows, analytically and numerically, the effects of a misspecification in the degree of integration on testing for cointegration. Johansen LR tests tend to find too much spurious cointegration while the Engle-Granger test shows a more robust performance than the LR tests. Key Words Cointegration; fractional unit roots; size of test; Johansen LR test; EG test ·Gonzalo, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid; Lee, Department of Economics, University of California. JEL c1assification: C32; C22. 1. Introduction It is well established that many economic series contain dominant, smooth components, even after removal of simple deterministic trends. A stochastic process with no deterministic components is defmed to be integrated of order d, denoted I(d), if it has a stationary and invertible ARMA representation after applying the differencing operator (1 - B/. The components of the vector Xl are said to be cointegrated of order (d, b), if all of components of Xl are I(d) and there exists a vector a (;!: O) such that a' Xl is I(d - b), b > O. Usually the case with d = b = 1 is considered. This paper is concerned with the robustness of testing for cointegration when series are not exactly 1(1) but fractionally integrated with d ;!: 1. When d is not an integer, the series is said to be fractionally integrated and discussed by Granger and Joyeux (1980) and Hosking (l981). The model has been used in applied econometric work where flexible characterization of low frequency dynamics is important, and has yielded new empirical regularities and insights into understanding behavior of many economic time series. There is considerable evidence on long memory properties in macroeconomic and fmancial time series data such as GDP, interest rate spreads, inflation rates, forward premiums, (power transformations of) stock retums, and exchange rates. See Baillie (1995) for a survey. We investigate two methods to test for cointegration when the series are fractionally integrated. One method is the one suggested by Engle and Granger (1987, EG hereafter), which looks for a linear combination of level series that minimizes the variance of the linear combination using OLS. Second method is Johansen's (1988, 1991) procedure, which maximizes the canonical correlation between the f11'st differenced series and the level series. The main assumption of the both tests is that series are exactly 1(1). When variables are I(d) with d ;!: 1 but if they are wrongly assumed as 1(1), it is found that the Johansen tests tend to fmd cointegration too often. 1 Consider a 2 x 1 1(1) vector..XI · = (YI xJ'., Since the .. ,variance. oLan .. 1(1), series,,· goes to infinity as t goes to infmity while variance of an 1(0) series is fmite, if an 1(1) vector XI is cointegrated, there exists a vector a (;1: O) such that the variance of a' XI is fmite. EG suggest to test for a unit root in the residual ZI from the OLS cointegration regression YI = ao + alXI + ZI' The EG test is based on the Dickey-Fuller (1979, DF) statistic augmented with k lagged differences, the t-value for ~, from the OLS regression, 'ó'ZI = PZI_l + Pl.ó.z'l_l + ... + P~I.k + error. (1) An 1(1) series and an 1(0) series are not correlated asymptotically. The Johansen method exploits the fact that a' XI may be correlated with MI' and look for a that maximizes the canonical correlation between MI and XI' If the maximum canonical correlation between MI and XI is not zero, a' XI is 1(0) and XI is cointegrated. A cointegrated 2 x 1 vector XI can be generated from a vector error correction model (VECM) (2) where El = (e lt e2J' is a 2 x 1 Gaussian vector white noise. If XI is cointegrated, it can be shown that TI = ya' where a and y are 2 x r matrices with r being the rank of TI. Testing for cointegraLion is equivalent to testing for the hypothesis on the rank of TI. r¡'s can be eliminated by regressing MI and XI_l on M I_l, ... , MI_k' and 1. This can -1 T ,give residuals ROl and respectively. Let Sij = T LI=lR¡rRjl (i, j = O, 1). TheRlt, two squared canonical correlations (1 ~ ~l ~ ~ ~ O) between ROl and Rlt can be estimated [rom the eigenvalues of M== s~ ~ S10 S~ SOl' The Johansen method is designed to test if the maximum squared canonical correlation Al can be positive, which is zero if XI is not cointegrated. 2 The likelihood ratio statistics for testing for the null ,hypothesis Ho: r = °are Ql =- T ln(1 - ~1)(1 - ~) (3a) and Q2 = - T ln( 1 - ~l). (3b) Ql is to test the null hypothesis against the altemative hypothesis H1: r > 0, and Q2 is to test the null hypothesis against H1: r = 1. Note that Q2 ~ 'li1 and Ql ~ T(~l + ~. 2. Fractionally integrated processes Suppose XI = 0'L XJ' are generated from (1 - B)dYI = el! (4a) (1 - B)dXI = e21· (4b) An I(el) process )'1 generated from (4a) has the following properties. (a) YI is covariance stationary if d < 0.5. (b) YI has an invertible moving average representation if d > - 0.5. (e) YI is mean-reverting when d < 1. (d) If d > 0, YI has long-memory, the autocovariances of YI are not absolutely surnmable, and the power spectrum of YI is unbounded for frequencies approaching zero. (e) YI has an infinite variance when d > 0.5. (f) The DF-t statistic diverges to -co if d < 1 and diverges to if d > 1 as T ~+00 00 (Sowell, 1990, Theorem 4). Thus if d > 1, the DF tests has no power asymptotically to test for a unit root against a stationary reoL The fractional dilTerence operator (1 - B)d is defmed by its Maclaurin series 3 ,.._------------------,------------------------00 00 (1'- B)d.._:::, Lr(~(~~u+:\) ~= L'dj~; ~:: j -~' - d'dj~l' do = 1 (5) j=O j=O where r(·) is the gamma function, and dj are squared surnmable if d > - 0.5. Since (1 - B)d-l.1YI = el!' itcan be written (6) where aj = - ru - d + l)/[rU + 1)r(- d + 1)]. We may then write a VECM of the form (7) For simplicity, in the following proposition, we consider only the case with k = O in the VECM to compute the LR statistics. Any fmite k that is not sufficiently large enough to make the error El a vector white noise will lead to the same results. Proposition: Suppose (Yl Xt)' are l(d) processes generated Irom (4), and we estimate a VECM with k = O. a. JI d ~ 1.5, then ~l -~~ O as T -7 oo. (3-2d) l'  b. JI 1 < d < 1.5, then T Al = Op(l). 1\ c. JI d = 1, then 71..1 = Op(l). (2d-l) l'  d. JI 0.5 < d < 1, then T Al = Op(l). Proof: (a) Following Gourieroux et al (1989), for d ~ 1.5, Su = OpC¡..2d-l), Soo = Op(¡..2d-J), and Sl!> = Op(¡..2d-2). Therefore ti = Si~SlOS~Ol = Op(l) , and the result 4 follows. (b) For 1 < d < 1.5, Su = 0lrd-I), Soo = 0ll), and SlO = Op(~-2). 1\ '11.3Therefore M = Op(T'" ). (e) For d = 1, Su = Op(n, SIO = Op(1), Soo = Op(1), and 1\ -1 ......2d-1 therefore M = Op(T). (d) For 0.5 < d < 1, Su = Op(l ), Soo = Op(1), and SIO = 1\ 12dOp(l). Therefore M = Op(T - ), and the result follows. If d = 1, then IA.I = Op(1). If d * 1, ni ~ 00 as T ~ 00, and the size of the LR tests inereases to one as T ~ 00 beeause Q2 i!: ni and QI i!: T(~I +~. Note that if 1 < d < 1.5 then O < 3 - 2d < 1, and if 0.5 < d < 1 then O < 2d - 1 < 1. Thus in these two cases, ~I ~ O but at the rate slower than T so that ni diverges, and therefore the size of the LR tests goes lO one. 3. A monte carlo We generate XI = (YI XJ' from (4) assuming that ell and e21 are U.d. N(O, 1), where t = 1, ... , T + q. The frrst q = 2000 observations are disearded. We approximated (1 - B)J = Lj:O dj ¡¡i by assuming ~ = O for j > 1000. In Tables 1 and 2, we report simulated size of the eointegration tests for various values of d. When d < 1, the size is large for both EG and the Johansen tests. Although Sowell (1990) shows that thc DF t statistic diverges to - 00 if d < 1 as T ~ 00, Diebold and Rudebuseh (1991a), and Hassler and Wolters (1994) provide evidenee that the power of the DF tests are poor against fraetional altematives in fmite samples. Table 3 also shows that it is hard to tell I(d) with d < 1 from the behavior of 1(1) using the augmented DF (ADF) tests in a small sample. Thus if the variables are fraetionally integrated with d < 1, then it is likely that we proeeed assuming the series are 1(1) and may get ineorreet eonelusion that the system has a long-ron relationship. When d > 1, the Johansen tests tend to fmd too mueh spurious eointegration while the EG test does noL Sowell (1990) shows that if d > 1 then the DF test has zero power 5 asymptotically to tesl for a unit root against a stationary rool. Thus if the variables are fractionally integrated with d > 1, then it is likely that we proceed assuming they are 1(1) (see Tab1e 3), and the results may suggest cointegration incorrectly. The VECM (7) shows that the lrue number of lagged differences is infmity. If not enough k is used in computing the Johansen LR tests, aH the omitted dynamics will be included in the residual El and the model suffers dynamic misspecification. We thus report the results computed with k = 3 and 9, but the problem remains even with k = 9. What are the rclevant values of d for economic variables? For example, the fractional differencing parameter d estimated in the literature are: d = 1.17 for annual disposable income (Diebold and Rudebusch (1991b», d = 1.29 for quarterly real GNP (Sowell (1992», and d is ranged from 1.04 to 1.36 for various nominal spot exchange rates (Cheung (1993». Also d is estimated about 0.6 for money growth rates (Tieslau (1991», and d is ranged from 0.40 to 0.57 for inflation rates in several developed countries (Hassler and Wolters(1995». This indicates money stock and price series may have d greater than olle. 4. Conclusions This paper shows that when variables are fractionally integrated with d > 1, the Johansen tests tend 10 fmd too much spurious cointegration while a standard Dickey-Fuller test on the residual of an Engle-Granger regression does noto If the both tests (EG and LR) produce the same results in testing for cointegration, it is likely that d :S 1. If they yield different answers, then it may indicate that d > 1. 6 ----------------------...---------------------------'i I 1 I ! I References Baillie, RT. (1995), "Long Memory Processes and Fractional Integration in Econometrics," Joumal 01 Econometrics, forthcoming. Cheung, Y.W. (1993), "Long Memory in Foreign Exchange Rates," Joumal 01 Business and Economic Statistics, 11, 93-101. Cheung, Y.W. and K.S. Lai (1993), "A Fractional Cointegration Analysis of Purchasing Power Parity," Joumal 01 Business and Economic Statistics, 11, 103-112. Dickey, D.A. and W.A. Fuller (1979), "Distribution of the Estimators for Autoregressive Time Series with a Unit Root," Joumal 01 American Statistical Association, 74, 427-31. Diebold, EX. and G.D. Rudebusch (1991a), "On the Power of Dickey-Fuller Tests against Fractional Alternatives," Economics Letters, 35, 155-160. Diebold, EX. and G.D. Rudebusch (1991b), "1s Consumption Too Smooth ? Long Memory and the Deaton Paradox," Review 01 Economics and Statistics, 71, 1-9. Engle, R.F. and C.W.J. Granger (1987), "Cointegration and Error Correction: Representation, Estimation, and Testing," Econometrica, SS, 251-276. Gourieroux, C., F. Maurel, and A. Monfort (1989), "Least Squares and Fractionally 1ntegrated Regressors," 1NSEE, Working Paper No. 8913. Granger, C.W.J. and R Joyeux (1980), "An Introduction to Long Memory Time Series Models and Fractional Differencing," Joumal 01 Time Series Analysis, 1, 15-29. Hassler, U. and L Wolters (1994), "On the Power bf Unit Root Tests Against Fractional Alternatives," Economics Letters, 45, 1-5. Hassler, U. and J. Wolters (1995), "Long Memory in Inflation Rates: International Evidence," Joumal 01 Business and Economic Statistics, 13, 37-45. Hosking, J.RM. (1981), "Fractional Differencing," Biometrika, 68, 165-176. Johansen, S. (1988), "Statistical Analysis of Cointegration Vectors", Joumal 01 Economic Dynamics and Control, 12, 231-254. Johansen, S. (1991), "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models", Econometrica, 59, 1551-1580. Sowell, F.B. (1990), uThe Fractional Unit Root Distribution," Econometrica, 58, 495-505. Sowell, F.B. (1992), "Modeling Long Run Behavior with the Fractiona1 ARIMA Model," Joumal 01 Monetary Economics, 29, 277-302. Ties1au, M.A. (1991), "Long Memory Models and Macroeconomic Time Series," Michigan State University, Working Paper, No. 9005. 7 Table l. Fractionally integrated processes (T = 100) k=O k = 3 k=9 EG QI Q2 EG QI Q2 EG QI Q2 d=0.5 .986 1.000 .997 .361 .651 .432 .083 .233 .171 d=0.6 .900 .991 .947 .223 .406 .262 .060 .180 .141 d=0.7 .636 .832 .652 .146 .224 .167 .043 .152 .132 d=0.8_ .341 .447 .340 .094 .142 .105 .042 .129 .129 d=0.9 .143 .147 .132 .065 .087 .076 .038 .132 .122 d=1.0 .047 .048 .052 .048 .068 .072 .032 .133 .130 d=1.1 .016 .058 .069 .037 .077 .078 .034 .153 .149 d=1.2 .010 .152 .166 .028 .090 .085 .036 .163 .165 d=1.3 .031 .337 .351 .021 .105 .113 .030 .171 .171 d=l.4 .053 .563 .592 .024 .110 .128 .027 .188 .184 d=1.5 .076 .753 .774 .019 .135 .144 .032 .221 .206 d=1.6 .100 .860 .873 .016 .165 .185 .037 .236 .230 d=1.7 .111 .921 .931 .012 .185 .196 .048 .274 .267 d=1.8 .135 .952 .957 .016 .191 .213 .045 .312 .294 d=1.9 .150 .969 .973 .029 .231 ·.243 .053 .357 .336 d=2.0 .176 .982 .981 .039 .271 .254 .054 .400 .385 The frequency of rejecting the null hypothesis in 1000 replications is reported at the 5% leve!. The critical values for T = 100 are simulated from 90000 replications using the DGP with d = 1. Table 2. Fractionally integrated processes (T = 1000) k=O k = 3 k = 9 EG Q¡ Q2 EG Q¡ Q2 EG Q¡ Q2 d=O.5 1.000 1.000 1.000 1.000 1.000 1.000 .984 1.000 1.000 d=0.6 1.000 1.000 1.000 .994 1.000 1.000 .851 .992 .936 d=0.7 .998 1.000 1.000 .879 .984 .941 .572 .815 .648 d=0.8 .872 .978 .946 .535 .708 .565 .269 .370 ..284 d=0.9 .384 .455 .376 .193 .213 .186 .134 .140 .116 d=1.0 .063 .056 .056 .059 .055 .055 .059 .053 .062 d=l.l .020 .118 .124 .026 .057 .066 .037 .051 .058 d=1.2 .047 .439 .464 .026 .160 .175 .023 .078 .104 d=1.3 .102 .768 .805 .029 .304 .332 .023 .141 .154 d=1.4 .166 .925 .925 .040 .441 .483 .021 .210 .248 d=1.5 .211 .974 .977 .030 .522 .546 .018 .248 .283 d=1.6 .253 .991 .990 .017 .543 .573 .014 .257 .281 d=1.7 .302 .994 .995 .010 .483 .517 .008 .245 .258 d=1.8 .331 .999 1.000 .004 .380 .390 .009 .216 .215 d=1.9 .345 .999 .999 .007 .245 .256 .015 .196 .199 d=2.0 .349 .999 .999 .029 .204 .192 .034 .208 .196 The frequency of rejecting the null hypothesis in 1000 replications is reponed at the 5% level. The critical values for T = 1000 are simulated from 90000 replications using the DGP with d = 1. Table 3. Power of ADF tests when a series is I(d) , , T = 100 T = 1000 ADF(O) ADF(3) ADF(Paid ADF(O) ADF(3) ADF(Paid [mean(Paid' sd(Paie)] [mean(Paid, sd(Paid] d=O.5 .999 .553 .696 [1.875, 2.805] 1.000 1.000 .963 [7.937, 3.997] d=0.6 .941 .355 .556 [1.926,2.874]. 1.000 .998 .867 [7.647, 3.860] d=0.7 .691 .223 .399 [1.811, 2.925] .999 .887 .674 [6.602, 3.606] d=0.8 .354 .129 .258 [1.500, 2.807] .825 .521 .412 [4.981, 3.148] d=0.9 .141 .080 .117 [1.258, 2.805] .331 .177 .187 [2.959, 2.705] d=1.0 .047 .055 .069 [1.229, 2.842] .055 .055 .050 [1.021, 2.400] d=l.l .032 .049 .058 [1.513, 2.846] .032 .026 .030 [3.368, 2.993] d=1.2 .047 .038 .047 [2.150, 3.013] .104 .036 .042 [5.701, 3.394] d=1.3 .087 .041 .056 [2.605, 2.983] .205 .071 .050 [7.322, 3.598] d=l.4 .148 .047 .067 [2.832, 2.916] .285 .112 .064 [8.124, 3.651] d=1.5 .214 .053 .067 [2.842, 2.884] .330 .133 .068 [8.452, 3.688] d=1.6 .257 .056 .080 [2.706, 2.759] .368 .138 .066 [8.045, 3.624] d=1.7 .317 .052 .079 [2.563, 2.674] .383 .126 .069 [7.081, 3.372] d=1.8 .342 .049 .077 [2.339, 2.646] .388 .095 .062 [5.698, 3.068] d=1.9 .366 .052 .062 [2.207, 2.706] .405 .065 .052 [3.813, 2.688] d=2.0 .387 .050 .061 [2.301, 2.817] .398 .055 .061 [2.017, 2.359] 5% leve!. 1000 replications. ADF(p) denotes the DF tests augmented with P 1agged frrst differences. p = O, 3, or Paie' Paie is chosen using the Ale among P = O, 1, , 19.oo' When P = Paie is used, the mean and the standard deviation of Paie in 1000 replications are reported in brackets, [mean(Paie), sd(Paie)]' ,".,-,-,----------,-------------¡-----

1s Consumption Too Smooth ? Long Memory and the Deaton Paradox,&quot;

A Fractional Cointegration Analysis of Purchasing Power Parity,&quot;

An Introduction to Long Memory Time Series Models and Fractional Differencing,&quot;

Cointegration and Error Correction: Representation, Estimation, and Testing,&quot;

Distribution of the Estimators for Autoregressive Time Series with a Unit Root,&quot;

Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models&quot;,

On the robustness of cointegration tests when series are fractionally integrated

Abstract

Similar works

Full text

Available Versions

Name not available

Name not available

Universidad Carlos III de Madrid e-Archivo