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Wild bootstrapping variance ratio tests
Economics Letters 92 (2006) 38 – 43 www.elsevier.com/locate/econbase
Wild bootstrapping variance ratio tests Jae H. Kim * Department of Econometrics and Business Statistics, Monash University, Caulfield East, VIC 3145, Australia Received 28 May 2005; received in revised form 5 December 2005; accepted 5 January 2006 Available online 2 May 2006
Abstract The wild bootstrap is proposed as a means of improving small sample properties of variance ratio tests. It is found that the wild bootstrap tests have desirable size properties and exhibit higher power than their alternatives in most cases. D 2006 Elsevier B.V. All rights reserved. Keywords: Conditional heteroskedasticity; Market efficiency; Monte Carlo experiment; Wild bootstrap JEL classification: C12; C15; G14
1. Introduction The variance ratio (VR) test has been widely used as a means of testing for the martingale property of financial time series, which is closely related to market efficiency in the weak form. The conventional VR tests include the Lo and MacKinlay (1988) test and the multiple VR test of Chow and Denning (1993). Notable applications of these VR tests include Liu and He (1991) and Yilmaz (2003). These VR tests are asymptotic tests, which can show small sample deficiencies. Recently, Wright (2000), based on ranks and signs, and Whang and Kim (2003), using the subsampling method of Politis et al. (1997), proposed the VR tests which do not rely on asymptotic approximations. They found that their VR tests exhibit small sample properties superior to those of the existing VR tests.
* Tel.: +61 3 99031596; fax: +61 3 99032007. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.01.007
J.H. Kim / Economics Letters 92 (2006) 38–43
39
In this paper, the wild bootstrap is proposed as an alternative. It is a resampling method that approximates the sampling distribution of a statistic, and is applicable to data with unknown forms of conditional and unconditional heteroskedasticity (see Mammen, 1993; Davidson and Flachaire, 2001). It has been found to be highly effective for econometric problems recently (see Goncalves and Kilian, 2004). The next section presents alternative VR tests including the wild bootstrap tests and Section 3 presents Monte Carlo simulation results.
2. Variance ratio test and wild bootstrapping Let X t be a martingale difference sequence that satisfies Assumption H* of Lo and MacKinlay (1988). The time series belonging to this class are serially uncorrelated with a fairly general form of conditional and unconditional heteroskedasticity that includes GARCH-type variances and variances with deterministic changes. Following Wright (2000), the VR statistic is written as ( ) ( ) T T 1 X 1 X 2 2 ðXt þ Xt1 þ ::: þ Xtkþ1 k lˆ Þ H ðXt lˆ Þ ; VRð X ; k Þ ¼ Tk t¼k T t¼1 P where lˆ ¼ T 1 Tt¼1 Xt . This is an estimator for the population VR, denoted as V(k), which is the ratio of 1/k times the variance of k-period return to the variance of one-period return. Lo and MacKinlay (1988) showed that, under Assumption H*, !1=2 k1 X 2ðk jÞ 2 dj ð1Þ M ð X ; k Þ ¼ ðVRð X ; k Þ 1Þ k j¼1 followsPthe standard normal distribution asymptotically under the null hypothesis that V(k) = 1, where 2 P dj ¼ f Tt¼jþ1 ðXt lˆ Þ2 Xtj lˆ gHf½ Tt¼1 ðXt lˆ Þ2 2 g. Chow and Denning (1993) proposed a joint test whose null hypothesis is V(k i ) = 1 for i = 1, . . . , l. The test statistic can be written as MVð X ; ki Þ ¼ max jM ð X ; ki Þj 1ViVl
ð2Þ
which asymptotically follows the studentized maximum modulus distribution with l and T degrees of freedom under Assumption H* of Lo and MacKinlay (1988). In this paper, the wild bootstrap is applied to M(X,k) and MV(X,k i ) tests given above, as they are asymptotically pivotal when the underlying time series is a martingale difference sequence. The advantage of bootstrapping asymptotically pivotal statistic is well known (see MacKinnon, 2002; p. 622). The wild bootstrap test based on MV(X,k i ) can be conducted in three stages as below: (i) Form a bootstrap sample of T observations X t* = g t X t (t = 1, . . . , T) where g t is a random sequence with E(g t ) = 0 and E(g2t ) = 1. (ii) Calculate MV* u MV(X*,k i ), the MV(X,k i ) statistic obtained from the bootstrap sample. (iii) Repeat (i) and many, say m, times to form a bootstrap distribution of the test (ii) sufficiently m statistic fMV X 4; kj ; j gj¼1 .
40
J.H. Kim / Economics Letters 92 (2006) 38–43
The p-value of the test can be obtained as the proportion of fMV X 4; kj ; j gm j¼1 greater than the in a similar sample value of MV(X,k i ). The wild bootstrap version of M(X,k) test can be implemented manner as a two-tailed test, where we obtain M*uM(X*,k) in Stage (ii) and fMV X 4; kj ; j gm j¼1 in Stage (iii). Conditionally on X t , X t* is a serially uncorrelated sequence with zero mean and variance X2t , which is a special case of Assumption H* of Lo and MacKinlay (1988). As such, M* and MV* respectively have the same asymptotic distributions as M(X,k) and MV(X,k i ). Since X t* is a serially uncorrelated sequence, wild bootstrapping approximates the sampling distributions under the null hypothesis, which is a desirable property for a bootstrap test. Note that the wild bootstrap is valid and the test statistics being bootstrapped are pivotal asymptotically, under the condition that X t follows a martingale difference sequence satisfying Assumption H* of Lo and MacKinlay (1988). In the next section, Monte Carlo simulation results are presented, comparing small sample properties of the wild bootstrap tests (M* and MV*) with those of the M test given in (1), the MV test given in (2), Wright’s (2000) sign test (S1) and subsampling test (WK) of Whang and Kim (2003). The latter two are respectively individual and multiple VR tests which are applicable to martingale difference time series. The simulation results associated with the VR tests assuming iid or conditionally homoskedastic time series are not presented for simplicity. They include iid versions of M and MV test and the Wald test of Richardson and Smith (1991). Simulations conducted in an earlier study revealed that they are severely over-sized and their size-adjusted power is lower than the wild bootstrap tests proposed here, under the models simulated in this study.
3. Monte Carlo results The simulation design loosely follows that of Wright (2000). The sample sizes considered are 100, 500 and 1000, with the value of holding period k set to 2, 5 and 10. The number of bootstrap iterations m is set to 1000, and so is the number of Monte Carlo trials. Table 1 presents the models simulated. Models 1 and 2 are GARCH(1,1) and stochastic volatility (SV) models respectively, which are used to evaluate the size properties. In generating Models 1 and 2, two different types of random errors are used; the standard normal distribution and Student-t distribution with three degrees of freedom. Models 1 and 2 are also used as error terms for Models 3 and 4, which are used to calculate the power. The level of significance a is set to 5%. For the wild bootstrap tests, only the results associated with the standard normal g t are reported. It is because simulation results are found to be largely insensitive to different choices of g t , including the two-point distribution of Mammen (1993), and the Rademacher distribution Table 1 Models used in Monte Carlo experiment Model 1 2 3 4
Xt = ut; pffiffiffiffi ut ¼ ht et ; ht ¼ 0:5 þ 0:75ht1 þ 0:1e2t1 X t = u t ; u t = exp(0.5h t )e t ; h t = 0.95h t1 + e t X t =0.1X t1 + u t (1 B)0.1X t = u t
e t ~ standard normal or Student-t distribution with three degrees of freedom.
GARCH(1,1) Stochastic volatility AR(1) Long memory
J.H. Kim / Economics Letters 92 (2006) 38–43
41
discussed in Davidson and Flachaire (2001). The Whang–Kim test requires the choice of the block length. They recommended taking a number of block lengths, but the results associated with one block length (the fourth in the grid suggested by the authors) are reported, as the results are not sensitive to different choices. The details of size properties are not reported due to page limitation. It is found that the M*, MV* and S1 tests show little sign of size distortion, while the M and MV test are slightly under-sized when the sample size is small. The WK test is seriously over-sized, except when the GARCH errors are used when the sample size is 1000. Table 2 reports the power of alternative individual VR tests for Models 3 and 4. Under GARCH errors, the M* test shows higher power than the S1 and M tests, while the S1 test is superior to the others under SV errors. Table 3 presents the case for the multiple VR tests. The power of the MV* test is always higher than the MV test. The WK test gives higher power than the MV* test, only when the sample size is 100. Additional models and holding periods are simulated in an earlier study, but the results, available from the author on request, are found to be qualitatively similar to those reported in this paper. In conclusion, Table 2 Power properties of alternative individual VR tests (level of significance = 5%) K
GARCH1 2
Model 3 T = 100 M S1 M* T = 500 M S1 M* T = 1000 M S1 M* Model 4 T = 100 M S1 M* T = 500 M S1 M* T = 1000 M S1 M*
GARCH2
5
10
2
SV1
5
10
2
SV2 5
10
2
5
10
8.3 10.7 8.3
8.1 7.5 8.2
4.9 5.9 5.9
12.2 9.1 12.7
7.1 6.9 7.7
4.4 6.2 5.8
8.5 9.4 9.3
8.5 7.3 10.2
5.9 6.4 7.7
9.7 10.4 10.5
9.8 9.1 9.1
5.9 7.6 5.7
50.0 30.3 49.0
32.3 18.5 32.1
18.7 10.1 19.8
48.4 30.3 48.7
30.6 20.4 31.1
19.4 11.3 20.6
32.6 29.3 33.5
21.3 17.7 22.2
15 12.5 16.4
35.8 45.1 38.8
22.3 28.4 23.5
13.9 17.7 15.0
79.0 50.1 79.2
55.7 31.7 55.7
34.8 18.5 35.1
79.6 47.1 79.7
54.1 31.9 54.3
33.8 16.6 34.4
55.7 52.6 57.0
38.1 36.3 38.1
23.9 19.8 24.5
58.3 73.9 62.1
40.2 51.5 41.4
25.9 29.0 26.4
12.0 12.1 11.9
13.2 14.1 13.1
10.4 14.1 12.1
11.5 11.5 11.7
13.8 13.1 14.5
12.5 14.1 14.9
9.9 12.4 10.7
13.3 16.1 14.3
9.7 16.6 12.1
12 16.3 13.8
16.6 21.7 17.2
12.7 22.6 13.9
53.3 34.6 53.3
63.6 43.5 63.6
60.5 42.4 61.6
53.3 35.7 53.8
64.8 44.0 65.2
63.4 43.9 64.1
38.4 37.2 40.5
47.2 49.4 47.9
45 48.5 47.0
43.5 60.4 47.4
53.7 72.3 56.5
51.3 71.0 53.3
84.6 56.8 84.9
91.9 70.1 92.1
90.5 68.8 90.6
84 57.3 83.9
91.5 71.0 91.7
89.6 67.8 90.2
61.3 63.2 62.8
71.7 77.7 72.8
70.3 75.9 71.4
65.9 86.9 68.4
76 93.9 76.5
73.9 92.3 74.9
Entries are in percentages. Bold entries indicate the highest value in each category.
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J.H. Kim / Economics Letters 92 (2006) 38–43
Table 3 Power properties of alternative multiple VR tests (level of significance = 5%) Model 3 T = 100 MV MV* WK T = 500 MV MV* WK T = 1000 MV MV* WK Model 4 T = 100 MV MV* WK T = 500 MV MV* WK T = 1000 MV MV* WK
GARCH1
GARCH2
SV1
SV2
5.8 9.5 15.2
7.4 10.7 15.9
6.9 10.9 23.3
6.4 10.9 21.3
36.6 41.9 21.8
37.9 44.9 21.5
23.3 29.8 25.4
26.1 33.6 26.6
68.5 72.8 33.8
66.9 73.2 34.5
44.9 51.0 31.3
48.9 55.3 36.2
10.8 14.5 23.5
11.1 15.2 21.9
10.3 14.3 29.5
13.3 17.0 26.4
61.5 65.4 53.8
62.4 67.4 56.1
43.4 49.7 52.2
50.9 57.6 50.2
91.2 93.2 80.1
90.3 92.0 79.1
69.8 74.4 70.1
75.8 78.7 71.8
Entries are in percentages. Bold entries indicate the highest value in each category.
it is found that the wild bootstrap tests are superior alternatives to the conventional VR tests over a wide range of sample sizes and error terms considered. This result suggests that the wild bootstrap tests be routinely used in practice, along with Wright’s (2000) sign test. Acknowledgements I would like to thank Brett Inder, Chris Orme, Param Silvapulle, Bill Zhang, and a referee for helpful comments and discussions. References Chow, K.V., Denning, K.C., 1993. A simple multiple variance ratio test. Journal of Econometrics 58, 385 – 401. Davidson, R., Flachaire, E., 2001. The wild bootstrap: tamed at last GREQAM. Document de Travail 99, A32. Goncalves, S., Kilian, L., 2004. Bootstrapping autoregressive with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 83 – 120.
J.H. Kim / Economics Letters 92 (2006) 38–43
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Liu, C.-Y., He, J., 1991. A variance-ratio test of random walks in foreign exchange rates. Journal of Finance 46, 773 – 785. Lo, A.W., MacKinlay, A.C., 1988. Stock market prices do not follow random walks: evidence from a simple specification test. The Review of Financial Studies 1, 41 – 66. MacKinnon, J.G., 2002. Bootstrap inference in econometrics. Canadian Journal of Economics 35, 615 – 645. Mammen, E., 1993. Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics 21, 255 – 285. Politis, D.N., Romano, J.P., Wolf, M., 1997. Subsampling for heteroskedastic time series. Journal of Econometrics 81, 281 – 317. Richardson, M., Smith, T., 1991. Tests of financial models in the presence of overlapping observations. The Review Financial Studies 4, 227 – 254. Whang, Y.-J., Kim, J., 2003. A multiple variance ratio test using subsampling. Economics Letters 79, 225 – 230. Wright, J.H., 2000. Alternative variance-ratio tests using ranks and signs. Journal of Business and Economic Statistics 18, 1 – 9. Yilmaz, K., 2003. Martingale property of exchange rates and central bank intervention. Journal of Business and Economic Statistics 21, 383 – 395.
Wild bootstrapping variance ratio tests Jae H. Kim * Department of Econometrics and Business Statistics, Monash University, Caulfield East, VIC 3145, Australia Received 28 May 2005; received in revised form 5 December 2005; accepted 5 January 2006 Available online 2 May 2006
Abstract The wild bootstrap is proposed as a means of improving small sample properties of variance ratio tests. It is found that the wild bootstrap tests have desirable size properties and exhibit higher power than their alternatives in most cases. D 2006 Elsevier B.V. All rights reserved. Keywords: Conditional heteroskedasticity; Market efficiency; Monte Carlo experiment; Wild bootstrap JEL classification: C12; C15; G14
1. Introduction The variance ratio (VR) test has been widely used as a means of testing for the martingale property of financial time series, which is closely related to market efficiency in the weak form. The conventional VR tests include the Lo and MacKinlay (1988) test and the multiple VR test of Chow and Denning (1993). Notable applications of these VR tests include Liu and He (1991) and Yilmaz (2003). These VR tests are asymptotic tests, which can show small sample deficiencies. Recently, Wright (2000), based on ranks and signs, and Whang and Kim (2003), using the subsampling method of Politis et al. (1997), proposed the VR tests which do not rely on asymptotic approximations. They found that their VR tests exhibit small sample properties superior to those of the existing VR tests.
* Tel.: +61 3 99031596; fax: +61 3 99032007. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.01.007
J.H. Kim / Economics Letters 92 (2006) 38–43
39
In this paper, the wild bootstrap is proposed as an alternative. It is a resampling method that approximates the sampling distribution of a statistic, and is applicable to data with unknown forms of conditional and unconditional heteroskedasticity (see Mammen, 1993; Davidson and Flachaire, 2001). It has been found to be highly effective for econometric problems recently (see Goncalves and Kilian, 2004). The next section presents alternative VR tests including the wild bootstrap tests and Section 3 presents Monte Carlo simulation results.
2. Variance ratio test and wild bootstrapping Let X t be a martingale difference sequence that satisfies Assumption H* of Lo and MacKinlay (1988). The time series belonging to this class are serially uncorrelated with a fairly general form of conditional and unconditional heteroskedasticity that includes GARCH-type variances and variances with deterministic changes. Following Wright (2000), the VR statistic is written as ( ) ( ) T T 1 X 1 X 2 2 ðXt þ Xt1 þ ::: þ Xtkþ1 k lˆ Þ H ðXt lˆ Þ ; VRð X ; k Þ ¼ Tk t¼k T t¼1 P where lˆ ¼ T 1 Tt¼1 Xt . This is an estimator for the population VR, denoted as V(k), which is the ratio of 1/k times the variance of k-period return to the variance of one-period return. Lo and MacKinlay (1988) showed that, under Assumption H*, !1=2 k1 X 2ðk jÞ 2 dj ð1Þ M ð X ; k Þ ¼ ðVRð X ; k Þ 1Þ k j¼1 followsPthe standard normal distribution asymptotically under the null hypothesis that V(k) = 1, where 2 P dj ¼ f Tt¼jþ1 ðXt lˆ Þ2 Xtj lˆ gHf½ Tt¼1 ðXt lˆ Þ2 2 g. Chow and Denning (1993) proposed a joint test whose null hypothesis is V(k i ) = 1 for i = 1, . . . , l. The test statistic can be written as MVð X ; ki Þ ¼ max jM ð X ; ki Þj 1ViVl
ð2Þ
which asymptotically follows the studentized maximum modulus distribution with l and T degrees of freedom under Assumption H* of Lo and MacKinlay (1988). In this paper, the wild bootstrap is applied to M(X,k) and MV(X,k i ) tests given above, as they are asymptotically pivotal when the underlying time series is a martingale difference sequence. The advantage of bootstrapping asymptotically pivotal statistic is well known (see MacKinnon, 2002; p. 622). The wild bootstrap test based on MV(X,k i ) can be conducted in three stages as below: (i) Form a bootstrap sample of T observations X t* = g t X t (t = 1, . . . , T) where g t is a random sequence with E(g t ) = 0 and E(g2t ) = 1. (ii) Calculate MV* u MV(X*,k i ), the MV(X,k i ) statistic obtained from the bootstrap sample. (iii) Repeat (i) and many, say m, times to form a bootstrap distribution of the test (ii) sufficiently m statistic fMV X 4; kj ; j gj¼1 .
40
J.H. Kim / Economics Letters 92 (2006) 38–43
The p-value of the test can be obtained as the proportion of fMV X 4; kj ; j gm j¼1 greater than the in a similar sample value of MV(X,k i ). The wild bootstrap version of M(X,k) test can be implemented manner as a two-tailed test, where we obtain M*uM(X*,k) in Stage (ii) and fMV X 4; kj ; j gm j¼1 in Stage (iii). Conditionally on X t , X t* is a serially uncorrelated sequence with zero mean and variance X2t , which is a special case of Assumption H* of Lo and MacKinlay (1988). As such, M* and MV* respectively have the same asymptotic distributions as M(X,k) and MV(X,k i ). Since X t* is a serially uncorrelated sequence, wild bootstrapping approximates the sampling distributions under the null hypothesis, which is a desirable property for a bootstrap test. Note that the wild bootstrap is valid and the test statistics being bootstrapped are pivotal asymptotically, under the condition that X t follows a martingale difference sequence satisfying Assumption H* of Lo and MacKinlay (1988). In the next section, Monte Carlo simulation results are presented, comparing small sample properties of the wild bootstrap tests (M* and MV*) with those of the M test given in (1), the MV test given in (2), Wright’s (2000) sign test (S1) and subsampling test (WK) of Whang and Kim (2003). The latter two are respectively individual and multiple VR tests which are applicable to martingale difference time series. The simulation results associated with the VR tests assuming iid or conditionally homoskedastic time series are not presented for simplicity. They include iid versions of M and MV test and the Wald test of Richardson and Smith (1991). Simulations conducted in an earlier study revealed that they are severely over-sized and their size-adjusted power is lower than the wild bootstrap tests proposed here, under the models simulated in this study.
3. Monte Carlo results The simulation design loosely follows that of Wright (2000). The sample sizes considered are 100, 500 and 1000, with the value of holding period k set to 2, 5 and 10. The number of bootstrap iterations m is set to 1000, and so is the number of Monte Carlo trials. Table 1 presents the models simulated. Models 1 and 2 are GARCH(1,1) and stochastic volatility (SV) models respectively, which are used to evaluate the size properties. In generating Models 1 and 2, two different types of random errors are used; the standard normal distribution and Student-t distribution with three degrees of freedom. Models 1 and 2 are also used as error terms for Models 3 and 4, which are used to calculate the power. The level of significance a is set to 5%. For the wild bootstrap tests, only the results associated with the standard normal g t are reported. It is because simulation results are found to be largely insensitive to different choices of g t , including the two-point distribution of Mammen (1993), and the Rademacher distribution Table 1 Models used in Monte Carlo experiment Model 1 2 3 4
Xt = ut; pffiffiffiffi ut ¼ ht et ; ht ¼ 0:5 þ 0:75ht1 þ 0:1e2t1 X t = u t ; u t = exp(0.5h t )e t ; h t = 0.95h t1 + e t X t =0.1X t1 + u t (1 B)0.1X t = u t
e t ~ standard normal or Student-t distribution with three degrees of freedom.
GARCH(1,1) Stochastic volatility AR(1) Long memory
J.H. Kim / Economics Letters 92 (2006) 38–43
41
discussed in Davidson and Flachaire (2001). The Whang–Kim test requires the choice of the block length. They recommended taking a number of block lengths, but the results associated with one block length (the fourth in the grid suggested by the authors) are reported, as the results are not sensitive to different choices. The details of size properties are not reported due to page limitation. It is found that the M*, MV* and S1 tests show little sign of size distortion, while the M and MV test are slightly under-sized when the sample size is small. The WK test is seriously over-sized, except when the GARCH errors are used when the sample size is 1000. Table 2 reports the power of alternative individual VR tests for Models 3 and 4. Under GARCH errors, the M* test shows higher power than the S1 and M tests, while the S1 test is superior to the others under SV errors. Table 3 presents the case for the multiple VR tests. The power of the MV* test is always higher than the MV test. The WK test gives higher power than the MV* test, only when the sample size is 100. Additional models and holding periods are simulated in an earlier study, but the results, available from the author on request, are found to be qualitatively similar to those reported in this paper. In conclusion, Table 2 Power properties of alternative individual VR tests (level of significance = 5%) K
GARCH1 2
Model 3 T = 100 M S1 M* T = 500 M S1 M* T = 1000 M S1 M* Model 4 T = 100 M S1 M* T = 500 M S1 M* T = 1000 M S1 M*
GARCH2
5
10
2
SV1
5
10
2
SV2 5
10
2
5
10
8.3 10.7 8.3
8.1 7.5 8.2
4.9 5.9 5.9
12.2 9.1 12.7
7.1 6.9 7.7
4.4 6.2 5.8
8.5 9.4 9.3
8.5 7.3 10.2
5.9 6.4 7.7
9.7 10.4 10.5
9.8 9.1 9.1
5.9 7.6 5.7
50.0 30.3 49.0
32.3 18.5 32.1
18.7 10.1 19.8
48.4 30.3 48.7
30.6 20.4 31.1
19.4 11.3 20.6
32.6 29.3 33.5
21.3 17.7 22.2
15 12.5 16.4
35.8 45.1 38.8
22.3 28.4 23.5
13.9 17.7 15.0
79.0 50.1 79.2
55.7 31.7 55.7
34.8 18.5 35.1
79.6 47.1 79.7
54.1 31.9 54.3
33.8 16.6 34.4
55.7 52.6 57.0
38.1 36.3 38.1
23.9 19.8 24.5
58.3 73.9 62.1
40.2 51.5 41.4
25.9 29.0 26.4
12.0 12.1 11.9
13.2 14.1 13.1
10.4 14.1 12.1
11.5 11.5 11.7
13.8 13.1 14.5
12.5 14.1 14.9
9.9 12.4 10.7
13.3 16.1 14.3
9.7 16.6 12.1
12 16.3 13.8
16.6 21.7 17.2
12.7 22.6 13.9
53.3 34.6 53.3
63.6 43.5 63.6
60.5 42.4 61.6
53.3 35.7 53.8
64.8 44.0 65.2
63.4 43.9 64.1
38.4 37.2 40.5
47.2 49.4 47.9
45 48.5 47.0
43.5 60.4 47.4
53.7 72.3 56.5
51.3 71.0 53.3
84.6 56.8 84.9
91.9 70.1 92.1
90.5 68.8 90.6
84 57.3 83.9
91.5 71.0 91.7
89.6 67.8 90.2
61.3 63.2 62.8
71.7 77.7 72.8
70.3 75.9 71.4
65.9 86.9 68.4
76 93.9 76.5
73.9 92.3 74.9
Entries are in percentages. Bold entries indicate the highest value in each category.
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J.H. Kim / Economics Letters 92 (2006) 38–43
Table 3 Power properties of alternative multiple VR tests (level of significance = 5%) Model 3 T = 100 MV MV* WK T = 500 MV MV* WK T = 1000 MV MV* WK Model 4 T = 100 MV MV* WK T = 500 MV MV* WK T = 1000 MV MV* WK
GARCH1
GARCH2
SV1
SV2
5.8 9.5 15.2
7.4 10.7 15.9
6.9 10.9 23.3
6.4 10.9 21.3
36.6 41.9 21.8
37.9 44.9 21.5
23.3 29.8 25.4
26.1 33.6 26.6
68.5 72.8 33.8
66.9 73.2 34.5
44.9 51.0 31.3
48.9 55.3 36.2
10.8 14.5 23.5
11.1 15.2 21.9
10.3 14.3 29.5
13.3 17.0 26.4
61.5 65.4 53.8
62.4 67.4 56.1
43.4 49.7 52.2
50.9 57.6 50.2
91.2 93.2 80.1
90.3 92.0 79.1
69.8 74.4 70.1
75.8 78.7 71.8
Entries are in percentages. Bold entries indicate the highest value in each category.
it is found that the wild bootstrap tests are superior alternatives to the conventional VR tests over a wide range of sample sizes and error terms considered. This result suggests that the wild bootstrap tests be routinely used in practice, along with Wright’s (2000) sign test. Acknowledgements I would like to thank Brett Inder, Chris Orme, Param Silvapulle, Bill Zhang, and a referee for helpful comments and discussions. References Chow, K.V., Denning, K.C., 1993. A simple multiple variance ratio test. Journal of Econometrics 58, 385 – 401. Davidson, R., Flachaire, E., 2001. The wild bootstrap: tamed at last GREQAM. Document de Travail 99, A32. Goncalves, S., Kilian, L., 2004. Bootstrapping autoregressive with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 83 – 120.
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