I'm writing a thesis about conditional VaR of Standard & Poor's 500 index. I have fitted my log-returns with GARCH(1,1)-proces and then made some conditional VaR-forecast (500 observations) with both standard normal noise and scaled t-distributed noise (with four degrees of freedom). However, my backtest is kinda weird.
My log-returns are nicely fitted with some t-distribution (surprise), and therefore I should be aware of the standard normal noise-assumption would overestimate VaR of both 5 % and 10 % quantile. These are my results
Can somebody explain to me, why I have so many violations for my t-distributed assumption?