Modelling bid-ask spread conditional distributions using hierarchical correlation reconstruction

Abstract

While we would like to predict exact values, the information available, being incomplete, is rarely sufficient - usually allowing only conditional probability distributions to be predicted. This article discusses hierarchical correlation reconstruction (HCR) methodology for such a prediction using the example of bid-ask spreads (usually unavailable), but here predicted from more accessible data like closing price, volume, high/low price and returns. Using HCR methodology, as in copula theory, we first normalized marginal distributions so that they were nearly uniform. Then we modelled joint densities as linear combinations of orthonormal polynomials, obtaining their decomposition into mixed moments. Then we modelled each moment of the predicted variable separately as a linear combination of mixed moments of known variables using least squares linear regression. By combining these predicted moments, we obtained the predicted density as a polynomial, for which we can e.g. calculate the expected value, but also the variance to determine the uncertainty of the prediction, or we can use the entire distribution for, e.g. more accurate further calculations or generating random values. 10-fold cross-validation log-likelihood tests were conducted for 22 DAX companies, leading to very accurate predictions, especially when individual models were used for each company, as significant differences were found between their behaviours. An additional advantage of using this methodology is that it is computationally inexpensive; estimating and evaluating a model with hundreds of parameters and thousands of data points by means of this methodology takes only a second on a computer.

Multi-feature evaluation of financial contagion

“…HCR methodology, discussed e.g. in Duda ( 2018 ), Duda et al ( 2020 ), Duda and Bhatta ( 2020 ), is a practical approach to apply it to multivariate case thanks to first normalizing variables to nearly uniform (marginal) distributions, as in copula theory.…”

“… density model instead for some and normalization to integrate to 1, discussed e.g. in Duda et al ( 2020 ).…”

“…Thus for variables, to consider up to -th mixed moments of all subsets of variables, we would need coefficients—which in practice requires a basis of mixed moments to be optimized, as discussed e.g. in Duda et al ( 2020 ).…”

Low cost prediction of probability distributions of molecular properties for early virtual screening

“…Here, as in [3], [4], [5], we will directly predict these a i coefficients with linear regression from feature vector x:…”

“…The above-mentioned methodology was previously used for various applications e.g. demographic [3], financial [4], astronomical [5]. Here we often have more features than datapoints, requiring new regularization techniques proposed in this article.…”