A reality check on technical trading rule profits in the u,s, futures markets, страница 5

Among Lukac, Brorsen, and Irwin’s 12 technical trading systems, 7 trading systems (L-S-O Price Channel, Directional Indicator, Range Quotient, Reference Deviation, Directional Movement, Parabolic Time/Price, and Directional Parabolic) remain the same, but 5 trading systems (Simple Moving Average with a Percentage Price Band, Dual Moving Average Crossover, Outside Price Channel, M-II Price Channel, and Alexander’s Filter Rule) are slightly modified to include a confirmation device as an additional parameter. According to Schwager (1996), confirmation devices make a simple yet important modification that prevents a basic trend-following system from making potential whipsaw losses and will not bring about selection bias because they have long been used for many technical trading systems. Three additional trading systems encompass the Exponential Moving Average Crossover (EMC), the Moving Average Convergence-Divergence (MACD), and the Relative Strength Index (RSI) systems. These three systems are selected because they have been prominently featured in well-known books on technical analysis, such as Murphy (1986), Schwager (1996), and Kaufman (1998). A total of 9,385 trading rules are drawn from these trading systems, all of which were available to investors before the beginning of the full sample period. Details of the trading mechanism and parameter values for each of the 14 trading systems are described in Appendix Table AI.

Data-Snooping-Adjusted Statistical Tests

White’s (2000) Bootstrap Reality Check test and Hansen’s (2005) Superior Predictive Ability (SPA) test provide comprehensive tests across all trading rules considered and directly quantify the effect of data snooping by testing the null hypothesis that the performance of the best trading rule is no better than the performance of the benchmark. The best rule is identified by applying a performance measure to the full universe of trading rules, and then a desired p-value is obtained from comparing the performance of the best rule to approximations of the asymptotic distribution of the performance statistic. Previous studies that used the standard bootstrap or recursive bootstrap methods construct bootstrap samples by resampling raw price or return series and then applying a trading rule to the resampled series, whereas White’s and Hansen’s bootstrap tests applied in this study allow researchers to directly resample observations of a performance statistic.

To evaluate the performance of each of k technical trading rules, where k  1, . . . , m and m  9,385 in the present case, the following form of a performance measure can be utilized:

                                                                           Yk,t1  rk,t1  r0,t1                                                                                          (4)

where r_k,t1 and r_0,t1 denote daily net returns of each trading rule and the benchmark rule, respectively, at time t  1. Following the literature on the futures market, we assume “zero profit” as the benchmark, and therefore the second term on the right hand side in Equation (4) always has zero value.⁵ The null hypothesis is then defined as follows: provided that for each trading rule k, where k  1, . . . , m, m_k  E(Y_k) is well-defined,

                                                                      H₀:kmax1,p,mm_k  0                                                   (5)

which states that the best technical trading rule does not generate a mean net return greater than zero.

White’s (2000) test is based on the m  1 performance statistic

Y  N¹ a^Nt₁Y_k,t.It is assumed that N¹²(Y  M) S^d N(0, ),where S^d indicates convergence in distribution as N Sq, and a variance–covariance matrix