Bootstrapping the Grainger Causality Test With Integrated Data Richard Ti n University of Reading July 26, 2006 Abstract A Monte-carlo experiment is conducted to investigate the small sample performance of the Grainger causality test with cointegrated data with critical values obtained using the stationary bootstrap. It is shown that the empirical size of the test is close to the nominal size in samples as small as 25 observations but that this is achieved the cost of a reduction in power. Key words: Grainger Causality; Bootstrap. J.E.L. Classi cation: C12; C22. 1 Introduction Zapata & Rambaldi (1997) conduct a Monte Carlo exercise comparing the small sample performance of alternative procedures for testing Granger causality in the presence of integrated data. Their results show that all three tests have an empirical size which is close to the nominal size in sample sizes in excess of a hundred. In this paper we investigate the use of bootstrap critical values as a means of improving the performance of the test in smaller samples. 2 Testing for non-causality The Granger causality test is based on the following VAR model: z t = x t + u t (1) where z t and u t are (m 1) vectors, the latter being i.i.d multivariate normal. x t = (1; z t 1 ; : : : ; z t p ) and is a ((pm + 1) p) matrix of parameters. The test for non-causality entails testing the hypotheses that certain elements of are zero. The test can also be performed using the VAR model in VECM form: z t = ~x t + z t p + u t (2) Department of Agricultural and Food Economics, University of Reading, Earley Gate, P.O. Box 237, Reading, RG6 6AR, Reading, U.K. e-mail: j.r.ti n@reading.ac.uk. 1
where ~x t = (1; z t 1 ; : : : z t p+1 ), is (((p 1) m + 1) p) and is (m m) in which case the test is of zero restriction on both and. Where there is cointegration is of reduced rank (r) and can be decomposed as = 0 where and are (m r) matrices. The distribution of the test statistic in the presence of integrated data has been considered by Sims, Stock & Watson (1990), Lutkepohl & Reimers (1992), and Toda & Phillips (1993). In general the conclusion is that without information on the number of unit roots and the rank of certain submatrices in the cointegrating matrix, the distribution of the test statistic is unknown. In certain special cases the statistic can however be shown to have an asymptotic 2 distribution. For example, when there is cointegration and the model is bivariate (Sims et al. (1990), Lutkepohl & Reimers (1992)) or more generally when there is cointegration and the sub-matrix of the cointegrating matrix corresponding to the variables that are presumed causal under the null hypothesis is of full rank (Toda & Phillips (1993)). Toda & Yamamoto (1995) show that when the true model is a VAR(p) and a following VAR(p + d) is estimated, where d is the maximum order of integration in the variables of the model, the distribution of the Wald statistic based only on the coe cients of the rst p terms has an asymptotic 2 (p) distribution regardless of the properties of the cointegrating vectors. 3 Bootstrapping Interest in the use of the bootstrap as a method of improving the small sample performance of tests in a cointegrating framework stems from the work of Li & Maddala (1997). In particular they consider ways in which dependencies between the observed variables in the model may be preserved in the bootstrap sample. Li & Maddala (1997) also argue that the residuals used in generating the bootstrap sample should be generated using the parameter values assumed under the null hypothesis. In devising the methods we use here we follow Li & Maddala (1997) rst in using the stationary bootstrap as our method of resampling, and second, in using a sampling scheme in which the residuals generated under the null hypothesis are resampled and used to produce pseudo-data using the estimates obtained with the null hypothesis imposed. We obtain bootstrap critical values for causality test as follows. First the VECM is estimated under the joint restrictions required for Granger non-causality and cointegration. This in itself entails two steps, the rst follows Johansen & Juselius (1990, pp. 199-200) in estimating and under the restrictions that the rank of is one and = ( 1 ; 0) 0. The second step is to estimate following Mosconi & Gianini (1992, p. 407). 1 A bootstrap sample fu itjt = 1; : : : ; T g is obtained using the stationary bootstrap by drawing u i1 randomly from f^u it g, where f^u it g is the set of estimated residuals from the restricted VECM. u i2 is then 1 Zapata & Rambaldi (1997) note there are errors in the formula given by Mosconi & Gianini (1992). We note that Zapata & Rambaldi (1997) do not correct all of the errors and apply the following here: ^ = ~ ~s 1 s 0 ~ 1 1 s 1 s 0 1 ~V V 0 (xx 0 ) 1 V V 0 (xx 0 ) 1 (3) 2
25 30 40 50 75 100 200 400 p = 1 S A 0.146 - - 0.087-0.062 0.054 0.048 B 0.030 0.029 0.027 0.026 0.032 0.028 0.027 0.030 P A 0.477 - - 0.835-0.988 1.000 1.000 B 0.030 0.025 0.022 0.024 0.023 0.031 0.149 0.616 p = 2 S A 0.229 - - 0.108-0.070 0.052 0.051 B 0.030 0.032 0.037 0.043 0.043 0.054 0.050 0.050 P A 0.645 - - 0.937-0.998 1.000 1.000 B 0.141 0.258 0.481 0.693 0.937 0.987 1.000 0.999 p = 3 S A 0.344 - - 0.140-0.078 0.064 0.043 B 0.028 0.039 0.046 0.046 0.048 0.047 0.047 0.050 P A 0.727 - - 0.930-0.999 1.000 1.000 B 0.082 0.114 0.155 0.179 0.284 0.416 0.895 1.000 Table 1: Comparison of size (S) and power (P) of the modi ed Wald test with asymptotic (A) and bootstrap (B) critical values for Zapata and Rambaldi s model 3, nominal size 0.05. obtained according to the following: Pr [u i2 = u ij ] = 1 ' (5) Pr [u i2 = u ik ] = ' (6) where j = i + 1 if i < T, j = 1 otherwise and k 6= j. The bootstrap sample is used with the estimated parameters to produce pseudo-data satisfying the hypotheses of cointegration and Granger non-causality. By resampling repeatedly and testing the null hypothesis using the modi ed Wald statistic, critical values for the test are obtained. 4 Monte Carlo Experiment In order to assess the extent to which the bootstrap improves the performance of the non-causality test we repeat Zapata & Rambaldi s (1997) Monte Carlo experiment with their DGP(3) and DGP(4). We carry out 5000 Monte Carlo trials and for each trial we generate critical values using a bootstrap sample of 1000 observations. The probability used in the stationary bootstrap (') is 0:1. Tables 1 and 2 give the results of the trial along with the results reported by Zapata & Rambaldi (1997) for comparison. Results are reported with the order of the estimated VAR equal to the true order, p = 2; 3 for DGP 3 and 4 respectively, with cases where it is over- tted p = 3; 4 and under- tted p = 1; 2. In all cases the nominal size of the tests is 0.05. The results show that the use of the bootstrap gives where x = (~x 0 1; : : : ; ~x 0 T ), V = I p 0 1 s s = (1; 0) 0 and ~; ~ are as de ned in Mosconi & Gianini (1992). ; (4) 3
25 30 40 50 75 100 200 400 p = 2 S A 0.243 0.104 0.063 0.063 0.051 B 0.036 0.039 0.046 0.050 0.051 0.054 0.047 0.044 P A 0.450 0.783 0.987 1.000 1.000 B 0.087 0.116 0.194 0.246 0.352 0.473 0.799 0.990 p = 3 S A 0.369 0.106 0.071 0.057 0.048 B 0.035 0.038 0.041 0.043 0.051 0.049 0.050 0.047 P A 0.592 0.899 0.998 1.000 1.000 B 0.091 0.128 0.197 0.249 0.359 0.461 0.803 0.991 p = 4 S A 0.546 0.151 0.080 0.067 0.051 B 0.037 0.036 0.045 0.052 0.042 0.047 0.044 0.051 P A 0.723 0.907 0.998 1.000 1.000 B 0.076 0.139 0.196 0.246 0.356 0.473 0.810 0.992 Table 2: Comparison of size (S) and power (P) of the modi ed Wald test with asymptotic (A) and bootstrap (B) critical values for Zapata and Rambaldi s model 4, nominal size 0.05. an improvement in size over the use of asymptotic critical values at the expense of the power of the test. In the case of DGP(3) the power of the bootstrap test is reasonable with a sample size of 40 provided the correct number of lags are used in the estimation. The consequences of underestimating the number of lags is more severe than overestimation. Underestimation results in a severe reduction in the power of the test and under-rejection of the null hypothesis. Overestimation has only a small impact on the size of the test relative to the correctly parameterised model and there is also some reduction in power. The impacts of mis-specifying the lag length is less severe for DGP(4). The power of the test is uniformly lower for DGP(4) however. 5 Conclusion The paper has outlined a method of using the stationary bootstrap to generate critical values for the Granger causality test. A Monte Carlo experiment demonstrates the e cacy of this procedure in correcting the size distortion of the test in samples as small as 25 observations. It is noted that this improvement comes at a cost in terms of the power of the test and that this is worsened by incorrect speci cation of the lag length in the VAR. References Johansen, S. & Juselius, K. (1990). Maximum Likelihood Inference on Cointegration with Applications to the Demand for Money, Oxford Bulletin of Economics and Statistics 52: 169 210. Li, H. & Maddala, G. (1997). Bootstrapping Cointegrating Regressions, Journal of Econometrics 80: 297 318. 4
Lutkepohl, H. & Reimers, H. (1992). Granger-Causality in Cointegrated VAR Processes, Economics Letters 40: 263 68. Mosconi, R. & Gianini, C. (1992). Non-Causality in Cointegrated Systems: Representation Estimation and Testing, Oxford Bulletin of Economics and Statistics 54: 399 417. Sims, C., Stock, J. & Watson, M. (1990). Inference in Linear Time Series Models with some Unit Roots, Econometrica 58: 113 144. Toda, H. & Phillips, P. (1993). Vector Autoregressions and Causality, Econometrica 61: 1367 1393. Toda, H. & Yamamoto, T. (1995). Statistical Inference in Vector Autoregressions with Possibly Integrated Processes, Journal of Econometrics 66: 225 250. Zapata, H. & Rambaldi, A. (1997). Monte Carlo Evidence on Cointegration and Causation, Oxford Bulletin of Economics and Statistics 59: 285 298. 5