In the financial econometrics literature, the use of quantile regression to estimate risk measures, such as the Value-at-Risk (VaR), has enormously increased in the last years. The goal of this research proposal is twofold. First, we aim at including in the quantile regression model variables observed at different frequencies than the dependent one by introducing a MI(xed)-DA(ta) Sampling terms in the AutoRegressive Conditional Heteroskedasticity (ARCH) model, estimated using quantile regressions. The MIDAS term allows in fact for the inclusion of variables usually observed at lower frequencies than the dependent variable. This mixing frequency approach can accommodate contexts where the dependent variable is observed daily and some covariates are observed at lower frequencies which is quite common in financial framework where macroeconomic variables are usually observed at monthly or quarterly frequencies and considered as driving forces of the assets' volatility. Therefore, the inclusion of such information may largely improve the estimation of the risk measures of interest in particular the VaR calculated using quantile regression approach. The estimation of the VaR measures through the quantile regression tools may benefit of many advantages. The proposed model, named Q-ARCH-MIDAS, will be extensively evaluated from the statistical point of view and in a large empirical application. The second aim of the present research proposal deals with the choice of the best model to forecast the VaR. The problem of ranking and evaluating several models is largely recognized in the financial econometrics literature. In this context, we aim at introducing the Partially Ordered SETs (POSETs) to analyze different risk models (among which there is also the proposed Q-ARCH-MIDAS). The POSET-based approach uses different loss functions characterized by peculiar aspects to rank the models controlling their degree of incomparability.
Notwithstanding the increased interest from the literature on the MIDAS and GARCH-MIDAS models, to the best of our knowledge, this is the first time that the MIDAS terms are incorporated within the quantile regression framework to obtain accurate VaR measures. The proposed model will be deeply investigated. From a statistical point of view, we will investigate the conditions which assure the strict stationarity of the process. Moreover, we will deal with the problem of the weight parameter of the MIDAS term in line with the literature (see Section 2.4 in Ghysels et al. (2007)) is defined as a nuisance parameter. We aim at solving this problem by profiling out such a nuisance parameter. Another crucial statistical aspect we will take into consideration is the sequential procedure to find the optimal number of daily return lags to be included into the model. To solve this problem we will build a likelihood ratio type test configured for a quantile regression model with the inclusion of additional MIDAS variables. From an empirical viewpoint, the finite sample properties of the proposed Q-ARCH-MIDAS model will be investigated through a comprehensive Monte Carlo simulation. In this exercise, we will simulate the data generating process as a Q-ARCH-MIDAS model in order to evaluate the bias of the resulting estimates, according to different quantile levels and number of observations. The empirical application will be focused on the VaR forecasts for different financial stock indexes. The set of competing models that we will compare will be sufficiently large: we aim at including not only the standard ARCH model, always estimated in the quantile regression framework but also several well-known models always used to analyze financial data. The evaluation of a large number of models, under different evaluation criteria, will be carried out introducing, for the first time, the POSET-based approach to compare the performances of the models, ranking them and monitory their degree of incomparability.
The research could be further expanded in many directions. First of all, it would be of interest to include an additional exogenous variable, observed at the same frequency of the variable of interest, in the equation of the Q-ARCH-MIDAS model typically represented by a realized measure useful for modeling and forecasting future volatility. Realized measures of volatility may play a key role in calculating accurate VaR forecasts and several realized measures employing high frequency intra-day data are already developed in the financial literature (Gerlach and Wang (2020)). Another extension would be the inclusion of the GARCH term in the proposed Q-ARCH-MIDAS model.
References
Gerlach, R. and C. Wang (2020). Semi-parametric dynamic asymmetric Laplace models for tail risk forecasting, incorporating realized measures. International Journal of Forecasting 36, 489-506