A reality check on technical trading rule profits in the u,s, futures markets, страница 7

                                                mˆ ck  Yk15N12Yksˆk 22loglogN6, k1, . . . , m                              (10)

where 1_{} is an indicator function. Hansen argues that the threshold rate,

22loglogN, ensures _Mˆ^c is a consistent estimator that effectively captures all alternatives with m_k  0, and thus leads to a consistent estimate of the null distribution, which in turn improves the power of the test. Because different threshold rates may result in different p-values in finite samples, two additional estimators, mˆ ^l_k  min(Y_k, 0) and mˆ ^u_k  0 are considered. These estimators provide a lower bound and an upper bound for the distribution, respectively.

The distribution of the test statistic under the null hypothesis can be approximated by the empirical distribution derived from the following bootstrap resamples based on the stationary bootstrap:

                          Z*_k,b,t  Y*_k,b,t  g_i(Y_k),    i  l, c, u,            b  1, . . . , B,         t  1, . . . , N          (11)

where g_l(g)  max(0, g), g_c(g)g¹₅g 2_(sˆk2_N)2loglogN6, and g_u(g) g. Note that the expected values of Z*_k,b,t, i l,c,u, are given by Mˆ ^l, Mˆ ^c, and Mˆ ^u, respectively. The three estimators for M typically produce three different p-values and therefore result in three SPA tests: SPA_l, SPA_c, and SPA_u. The p-values of the three tests for

SPA can be obtained by calculating T_b^SPA* max5(max_k1,p,_m N¹²Z*_k,bsˆ _k), 06 for each i and b  1, . . . , B and then comparing T^SPA to the quantiles of T_b^SPA*:

B ¹5TbSPA* TSPA6

p_SPA  a    .       (12) b1    B

The p-value based on Mˆ ^c is consistent for the true p-value (SPA_c test), whereas p-values based on Mˆ ^l and Mˆ ^u provide a lower and upper bound for the true p-value (SPA_l test and SPA_u test), respectively.

Given N prediction observations and the set of 9,385 technical trading rules, implementation of both White’s and Hansen’s bootstrap tests begins with determining the smoothing parameter, q  q_N and the number of bootstrap samples, B. In practice, it is inevitable that the smoothing parameter q is chosen arbitrarily based on the data used. The parameter q is inversely related to the block length re-sampled by the stationary bootstrap. A larger value of q is relevant for data with little dependence and a smaller value of q for data with more dependence. Sullivan et al. (1999), however, found that p-values for their several combinations of different samples and performance criteria were insensitive to the choice of the smoothing parameter q.

On the other hand, the number of bootstrap samples B should be a sufficiently large number, because it may influence the accuracy of p-values estimated. Brock et al. (1992) and Kho (1996), however, demonstrated that their estimated bootstrap p-values were insensitive to the replication size B, once it was extended beyond 500. Sullivan et al. (1999) also used 500 replications. This study, therefore, sets q  0.1 and B  500, which are the same values of the smoothing parameter and the number of bootstrap samples applied in Sullivan et al. (1999). The value of the smoothing parameter delivers a mean block length of 10.

EMPIRICAL RESULTS

Tables I–III report the performance results of the best technical trading rule under the mean net return criterion for each of the sample periods. Each table includes White’s (2000) Reality Check bootstrap p-value (p_RC), Hansen’s (2005) Lower, Consistent, and Upper SPA bootstrap p-values (p_l, p_c, and p_u), and White’s nominal p-value (p_N,W)and Hansen’s nominal p-value (p_N,H). White’s and Hansen’s nominal p-values are obtained by applying each bootstrap procedure to the best trading rule only, thereby ignoring the effect of data snooping.