Methodology

The 2252 prices w_t^(k) for each of the risk factor series k are then log differenced to produce 2251 consecutive log differenced price changes

$$x_{t+1}^{_{(k)}} = \ln(w_{t+1}^{_{(k)}})-\ln(w_t^{_{(k)}}).$$ (3)

In the univariate context, the primary focus of each candidate model is to predict the log difference series x_t+1^(k) in terms of a conditional probability density function denoted by p_t^(k). In the multivariate context, however, the primary focus of each candidate model is to predict x_t^(P) defined by  

$$x_t^{_{(P)}} = {1\over10}\sum_{k=1}^{10}x_t^{_{(k)}}$$ (4)

in terms of a conditional probability density function denoted by p_t^(P).

It is convenient at this point to establish the following nomenclature and notational guidelines.

• Price changes are numbered from -1249 to 1001. Specific, price change events are described as a subset of $t\in[-1249,1001]$.
• The notations (k) and (P) are used as a superscript to reference the univariate portfolios k and the multivariate portfolio P respectively.
• The notation (k,P) is used as a superscript in an expression valid in both contexts.
• The superscripts - and + are used to reference the long and short positions of a portfolio.
• The superscript $\pm$ is used in an expression valid for both positions.
• The nomenclature prediction-realization pair refers to the ordered mathematical object (p_t^(k,P),x_t+1^(k,P)). It should be kept in mind that by definition - p_t^(k,P) are the forecast distributions for realizations x_t+1^(k,P) and depend only on data until t.

The first 1250 of the price changes x_t^(k,P), $(t\in[-1249,0])$,constitute the in-sample period of those models requiring buildup and/or optimization. The final 1001 data points, $t\in[1,1001]$ are used to construct 1000 out-of-sample prediction-realization pairs. This latter set of 1000 pairs are then analyzed in different ways to evaluate the performance of the models.