Probability space of regression models and its applications to financial time series

Abstract

We introduce a notion of a probability space of regression models and discuss its applications to financial time series. The probability space of regression models $ℒ=(ℳ,\mathrm{\wp })$ consists of a set of regression models $ℳ$ and a probability measure $\mathrm{\wp }$ , which is based on the model “quality”, i.e. its ability to “fit” into historical data and to forecast the future values of the target variable. The set of regression models $ℳ$ is assembled by selecting various combinations of input variables with different lags, transformations, etc., and varying historical data sets that are used for model building and validation. It is assumed that the model set $ℳ$ is “complete” in the sense that it exhausts all the “meaningful” regression models that are possible to be built given available historical data and independent variables. Each model $M$ from the set $ℳ$ yields a scenario $y(t;m)$ for the target variable $y$ , and thus the probability space of regression models $ℒ=(ℳ,\mathrm{\wp })$ allows one to build a probability distribution for $Y(t)$ for each projection time $t$ . We demonstrate how those distributions can be used to estimate risk capital reserves required by the regulators for large U.S. banks for credit and operational risks under the macroeconomic scenarios provided by the Federal Reserve Bank (FRB) for the Comprehensive Capital Analysis and Review (CCAR) stress testing.