C Reducing estimation error

C.1 Quantile factor modelling example

Consider the QFM (from equation (??)) with two factors, the market risk factor \(R_m\) and the squares of that factor as an additional factor \(R_m^2\). The factor loadings can be estimated by a procedure outlined by Koenker and Bassett Jr (1978). Once the quantile errors have been deduced, the quantile factor sensitivities have been estimated, and the error quantiles set to the median quantile as in the main text the following quantile prediction model can be used: \[\begin{align} \hat{\mathcal{Q}}(\tau) = \hat{\alpha} (\tau) + \hat{\beta}_1^\intercal(\tau)R_m + \hat{\beta}_2^\intercal(\tau)R_m^2 + \epsilon_t(0.5), \end{align}\] which yields \(|\tau|\) sample covariance matrices \(\hat{\Sigma}^{(\tau)}\).

C.2 Estimating the ERC portfolio Lagrangian multiplier

We want to choose a value of \(\eta\) that minimises all of the distances between the total risk contributions so that they are minimised, but this should be done across all \(K\) folds using cross-validation. The general optimisation is the same as before: \[\begin{align} \eta^* = \underset{\eta}{\text{argmin}} \Big \{ \frac{1}{K} \sum_{k = 1}^K \hat{h}_k^{erc}(\eta, \mathcal{I}_k, \mathcal{I}_{-k})\Big \}. \end{align}\] The procedure of estimating \(h\) still needs to be specified. The function \(f(\cdot|\textbf{X})\) used to find the ERC portfolio can be used as a distance to minimise. Hence: \[\begin{align} h_k^{erc}(\eta, \mathcal{I}_k, \mathcal{I}_{-k}) = \sum_{i = 1}^N \sum_{j \geq i}^N(w_{i, \mathcal{I}_{-k}}(\Sigma_{\mathcal{I}_k} w_{\mathcal{I}_{-k}})_i - w_{j, \mathcal{I}_{-k}}(\Sigma_{\mathcal{I}_k} w_{\mathcal{I}_{-k}})_j )^2, \end{align}\] where \(w\) and \(\Sigma\) can be replaced by their sample estimations to find \(\hat{h}\) given in the main text.

References

Koenker, Roger, and Gilbert Bassett Jr. 1978. “Regression Quantiles.” Econometrica: Journal of the Econometric Society, 33–50.