For the purposes of the CVA calculation, someone might need to proxy the CDS spreads (and their associated implied hazard rates) for counterparty cases with illiquid CDSs.
A common approach (leaving outside the whole bootstrapping procedure) followed by the banking industry is a cross sectional regression of the following form, trained or estimated on a sample derived from liquid CDSs:
$$\boldsymbol{y_{\tau}}= \boldsymbol{X_{\tau}} \boldsymbol{\beta_{\tau}} + \boldsymbol{\varepsilon_{\tau}}, \quad \varepsilon_{\tau,i}\sim N(0,1)$$
where
- $\boldsymbol{y}$ is an $n \times 1$ vector containing the natural logarithms of the hazard rates of each liquid CDS $i$ for a specific tenor $\tau$.
- $\boldsymbol{X}$ is an $n \times k$ matrix containing instrumental/dummy variables like geographical location, rating or industry that correspond to the liquid CDSs. First columns of the matrix consists of ones (intercept).
- $\boldsymbol{\beta}$ is the $k \times 1$ vector of the coefficients, including the intercept, which should be estimated via OLS.
- Finally, the vector $\boldsymbol{\varepsilon_{\tau}}$ contains the error terms which follow the standard normal distribution.
My (open) question is, apart from the "usual" checks about the statistical significance of the parameters, the normality of the residuals, the multicollinearity, the goodness of fit, etc. are there any other aspects of such a regression model that can be assessed or checked (and how), especially about the performance of its out of sample "predictions" $\boldsymbol{\hat{y_{\tau}}}$?
