Forecasting VaR and ES using a joint quantile regression and its implications in portfolio allocation

Less disagreement, better forecasts: Adjusted risk measures in the energy futures market

“…where refers to the indicator function, which is 1 if r v ≤ , that is, a VaR exceedance occurs, and 0 otherwise. Of the broad FZ family, the aforementioned loss function FZ0 is the primary option in terms of risk estimation and backtesting; see related papers, Nolde and Ziegel (2017), Patton et al (2019), andMerlo et al (2021).…”

“…$${L}_{FZ0}$$ is the $$FZ0$$ loss function considered in this paper, formulated as follows: $$${L}_{FZ0}(r,v,e;\alpha )=-\frac{1}{\alpha e}{\mathbb{1}}\{r\le v\}(v-r)+\frac{v}{e}+\mathrm{log}(-e)-1,$$$ where $${\mathbb{1}}$$ refers to the indicator function, which is 1 if $$r\le v$$, that is, a VaR exceedance occurs, and 0 otherwise. Of the broad $$FZ$$ family, the aforementioned loss function $$FZ0$$ is the primary option in terms of risk estimation and backtesting; see related papers, Nolde and Ziegel (2017), Patton et al (2019), and Merlo et al (2021).…”

Expectile hidden Markov regression models for analyzing cryptocurrency returns

“…Among the different methods considered throughout the literature, quantile regression, introduced by Koenker & Bassett (1978), has represented a valid approach for modeling the entire distribution of returns while accounting for the well-known stylized facts, i.e., high kurtosis, skewness and serial correlation, that typically characterize financial assets. In the financial literature, the quantile regression framework has been positively applied to estimate and forecast Value at Risk (VaR) and quantile-based risk measures (Engle & Manganelli 2004, White et al 2015, Taylor 2019, Merlo et al 2021. Several generalizations of the concept of quantiles have also been introduced over the years.…”

Marginal M-quantile regression for multivariate dependent data

“…In particular, dependency of observations may be seen as a clustering effect (Bergsma et al 2009) which arises in a number of sampling designs, including clustered, multilevel, spatial, and repeated measures (Heagerty et al 2000, Bergsma et al 2009, Geraci & Bottai 2014. In this context, quantile methods for modeling dependent-type data have been considered in a wide range of different applications spanning from medicine (Smith et al 2015, Farcomeni 2012, Alfò et al 2017, Marino et al 2018, Merlo, Maruotti & Petrella 2021, social inequality (Heise & Kotsadam 2015), economics (Bassett & Chen 2002, Kozumi & Kobayashi 2011, Bernardi et al 2015, Giovannetti et al 2018, Merlo, Petrella & Raponi 2021, environmental modeling (Hendricks & Koenker 1992, Pandey & Nguyen 1999, Reich et al 2011) and education (Kelcey et al 2019). When the interest of the research is on the entire conditional distribution, in addition to the classical quantile regression, a possible alternative approach is to consider the M-quantile regression proposed by Breckling & Chambers (1988).…”