In a stream of previous posts, see for example here and here, we have examined the ability of some simple models to forecast the signs of financial returns for the purpose of trading. In this post we explore another fundamental aspect of the above approach, namely the impact of data digitization.

When forecasting the sign of financial returns we essentially transform the forecasting problem into a digital one, i.e., into a mapping of the continuous forecast into the set of [math] \left\{-1, 1\right\}[/math] or [math] \left\{0, 1\right\}[/math]. So, why not use the data in digitized form instead of their continuous form when generating the forecast itself? We shall show that this digitized approach can lead to potential performance enhancements, so you should consider it in practical applications.

Let [math] y_{t}[/math] denote the monthly returns of the NASDAQ ETF, QQQ, or the monthly returns of the gold ETF, GLD. Then consider the digitized returns [math] z_{t} = \left[I(y_{t} > 0) - I(y_{t} \leq 0)\right][/math] (this formulation has the natural advantage that the threshold for deciding on positive or negative returns is zero). We shall apply two models in the [math] y_{t}[/math] and [math] z_{t}[/math] series as follows:

First, the "speculative constant" (M1) model as in [math] y_{t} = \mu + \epsilon_{t}[/math] or [math] z_{t} = \delta + \eta_{t}[/math]; second the "speculative constant plus negative drift" model (M2) as in [math] y_{t} = \mu_{0} + \mu_{1}I(y_{t-1} \leq 0) + \epsilon_{t}[/math] or $latex z_{t} = \delta_{0} + \delta_{1}I(z_{t-1} \leq 0) + \eta_{t}[/math]. The trading rule is the natural one based on the sign of the corresponding forecast. We use a rolling window of $latex R=12$ months and present some illustrative results in Tables 1 and 2 that follows (returns in percent). Note that we use the notation Mi-C or Mi-D for the model using continuous (C) or digitized (D) returns.