  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 we model (joint)
densities as linear combinations of orthonormal polynomials, getting its
decomposition into (mixed) moments. 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. calculate expected value, but also variance to evaluate
uncertainty of such prediction, or we can use the entire distribution e.g. for
more accurate further calculations or generating random values. 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.
