Improving Moving Average Trading Rules with Boosting and Statistical Learning Methods, страница 4

Finally, observe that there is a simple and classical solution to our model-averaging problem, looking for weights which minimize a squared error loss of predictions in the training set Z. Thus, given predictions hˆ₁(x), hˆ₂(x),  .  .  .  , hˆ_T(x), we can seek the weights w = (w₁, w₂,  .  .  .  , w_T) so that

                                                                               M                T                        ²

                                    wˆ = argmin^∑= ^ y_i ^{− ∑}t₌1 w h_t ^ˆ_t ( )x_i ^                                 (5)

                                                                      w         i 1

If we denote by X the M × T matrix with each row (hˆ₁(x_i), hˆ₂(x_i),  .  .  .  , hˆ_T(x_i)), and similarly let Y = (y₁,  .  .  .  , y_M) be the M-vector of outputs in the training set, the solution of minimization problem (5) is the linear regression of Y on hˆ₁(x), hˆ₂(x),  .  .  .  , hˆ_T(x), that is:

                                                         wˆ = (X^TX)⁻¹X^TY                                                     (6)

Nevertheless, this classical combining predictions procedure is hardly useful in our problem because the matrix X^TX is singular and in practice it is impossible to obtain the weights in (6).

FITNESS MEASURES AND SIGNAL FILTERING

With the purpose of evaluating the forecasting performance of technical trading rules and all learning methods considered above, we have used statistical and economic criteria. As Satchell and Timmermann (1995) point out, the use of statistical or economic criteria can lead to very different outcomes because standard forecasting criteria are not necessarily particularly well suited for assessing the economic value of predictions of a nonlinear process.

With the aim of assessing the economic signifi cance of the market directional predictions obtained by the different methods, we have transformed it on simple market trading strategies, consisting of investing total funds in either the stock market or a risk-free security. Following Allen and Karjalainen (1999), the forecast from each predictor is used to classify each trading day into periods ‘in’ (earning the market return) or ‘out’ of the market (earning the risk-free rate of return security). The trading strategy specifi es the position to be taken the following day, given the current position and the ‘buy’ or ‘sell’ signals generated by the different predictors. On the one hand, if the current state is ‘in’ (i.e., holding the market) and the share prices are expected to fall on the basis of a sell signal generated by one particular predictor, then shares are sold and the proceeds from the sale invested in the risk-free security (earning the risk-free rate of return r_ft). Alternatively, if the current state is ‘out’ and the predictor indicates that share market prices will increase in the near future, the rule returns a ‘buy’ signal and then the risk-free security is sold and shares are bought (earning the market rate of return r_t). Finally, in the other two cases, the current state is preserved.

The trading rule net return over the predicted period of 1 to N can be calculated as follows:

                                       r = ∑ ∑tN1 r It ⋅ bt + tN1 rft ⋅Ist + n⋅log11+−cc                             (7)

                                                                  =                     =

where r_t is the market rate of return constructed over the closing price (or level of the NYSE Composite Index, P_t) on day t; I_bt and I_st are indicator variables equal to one when the predictor signals are to buy and sell respectively, and zero otherwise, satisfying the relation I_bt · I_st = 0, t ∈ [1, N]; n is the number of transactions; and c denotes the one-way transaction costs (expressed as a fraction of the price). Regarding the transaction costs, results by Sweeney (1988) suggest that large institutional investors could achieve in the mid 1970s one-way transaction costs in the range of [0.1–0.2%]. Even though there had been substantial reductions in costs in recent decades, we used one-way transaction costs of 0.2%. As for the risk-free rate of return, following the literature we use the 3-month Treasury Bill Rate (see e.g. Bodie et al., 2002).^[1]