  Value-at-risk (VaR) and expected shortfall (ES) are two commonly utilized
metrics for quantifying financial risk. In this study, we review the widely
employed Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
models. These models are explored with diverse distributional assumptions on
innovation, including parametric, non-parametric, and `semi-parametric' that
incorporates a parametric tail distribution based on extreme value theory.
Additionally, we introduce a non-parametric local linear quantile
autoregression (LLQAR) with kernel weights depending on the distance between
the current loss and past losses, and decreasing in the time lag.
  To evaluate the performance of different methods for VaR and ES estimation,
we employ a multi-criteria approach. This involves mean squared error
assessment using simulated data, backtesting on both simulated data and US
stocks, and application of the ESBootstrap test. The LLQAR method, which does
not necessarily require stationarity assumptions, seems to perform better for
simulated non-stationary data as well as real-world data, for estimating VaR
and ES.
