A reality check on technical trading rule profits in the u,s, futures markets, страница 6

⁵ In general, technical trading returns on futures contracts are either uncorrelated or have a small negative correlation with returns on stocks and bonds (Lukac & Brorsen, 1990). Then, the expected returns of technical trading rules are equal to the risk-free rate in the Capital Asset Pricing Model (CAPM). Since margin requirements can be posted in treasury-bills in the futures market, zero profit is considered a reasonable benchmark.

lim_NS                  var (N¹² Y). Note that the null hypothesis above is a composite hypothesis whose asymptotic distribution typically depends on nuisance parameters and thus is not uniquely known. White (2000), therefore, imposes the least favorable configuration (LFC), the points of the null least favorable to the alternative, on the null. In our application, the points under the LFC are m_k  0 for all k. Then, asymptotic behavior of the test statistic

                                                                             TRC  max N12Yk                                                                                       (6)

k1,p,m

is known, and an asymptotic p-value for the test of the null hypothesis can be obtained based on observations of Y_k,t, where t  1, . . . , N.

To find the asymptotic distribution of the test statistic, bootstrap resamples are generated using Politis and Romano’s (1994) stationary bootstrap. The resampling algorithm is directly related to selecting random indexes h(t) for t  1, . . . , N. Once the random indexes have been selected, the resampled performance statistic of each trading rule can be computed as

Y*_k  N¹ a^N_t1Y*_k,t, where Y*_k,t  Y_k,h(t). Under the appropriate conditions,

the distribution of N¹²(Y*  Y) converges to the distribution of N¹²(Y  M) as N increases (Politis & Romano, 1994). An estimate of the desired distribution of N(0, ) can be constructed by repeatedly drawing realizations of

N¹²(Y*  Y). White’s Reality Check p-value (p_RC) for testing H₀ then can be obtained by comparing T^RC to the quantiles of:

                                                                                      b1              B

where 1_{} denotes an indicator function that takes the value one if the expression in {} is true and the value zero otherwise. The null hypothesis is rejected at the a% significance level if p_RC  (a/100). For the Sharpe ratio criterion, White’s Reality Check p-value can be obtained using a similar manner as above (see Sullivan et al., 1999).

Hansen (2005) proposes another testing procedure based on the same framework as in White (2000), named the Superior Predictive Ability (SPA) test. According to Hansen (2003, 2005), LFC-based tests such as White’s test may reduce rejection probabilities of the test under the null by the inclusion of poor and irrelevant alternative models because it does not satisfy a relevant similarity condition that is necessary for a test to be unbiased. In research on technical trading rules, poor-performing trading rules are inevitably included because there is no theoretical guidance regarding the proper selection of parameters (i.e., trading rules). Hansen’s test is designed to reduce the influence of the poor alternatives by adopting a studentized test statistic and a data-dependent null distribution, and thus may improve the power of the test. The studentized test statistic is given by:

^N12Y

                                                        T^SPA  max ea max                 ^kb, 0f                                               (9)

k1,p,m s^ˆ _k

where sˆ ²_k is a consistent estimator of s²_k  var(N¹²Y_k).

Next, to reduce the influence of poor-performing models while preserving the influence of the alternatives with m_k  0, Hansen proposes the following estimator for M: