Modelling bid-ask spread conditional distributions using hierarchical correlation reconstruction

Gramatyka 10, 30-067 Krak ow, Poland Abstract --While we would like to predict exact values, available incomplete information is rarely sufficient - usually allowing only to predict conditional probability distributions. This article discusses hierarchical correlation reconstruction (HCR) methodology for such prediction on example of usually unavailable bid-ask spreads, predicted from more accessible data like closing price, volume, high/low price, returns. In HCR methodology we first normalize marginal distributions to nearly uniform like in copula theory. Then here we model each moment (separately) of predicted variable as a linear combination of mixed moments of known variables using least squares linear regression - getting accurate description with interpretable coefficients describing linear relations between moments. Combining such predicted moments we get predicted density as a polynomial, for which we can e.g. There were performed 10-fold cross-validation log-likelihood tests for 22 DAX companies, leading to very accurate predictions, especially when using individual models for each company as there were found large differences between their behaviors. Additional advantage of the discussed methodology is being computationally inexpensive, finding and evaluation a model with hundreds of parameters and thousands of data points takes a second on a laptop. I NTRODUCTION While it is more convenient to work on exact values, real life predictions usually have some uncertainty, controlling of which could allow e.g. to distinguish nearly certain predictions from the practically worthless ones. Generally, wanting to predict Y variable from X ( X 1,...,X d) variables, if there is no a strict relation, they often come from some complicated joint probability distribution - knowing X x, we can only predict Pr ( Y X x) conditional probability distribution.