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The classical assumptions of linear regression are that the errors are uncorrelated and the variance of errors is constant (homoskedastic). So regress the returns against the indicators and test for autocorrelation and heteroskedasticity in the errors. If you don't observe any, then there's no issue with conventional hypothesis testing. If you do, use White ...

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In general I would answer your question in the following way: Alternatives to VaR which share most of its helpful properties but not its shortcomings are the so called coherent risk measures. They have the following properties: monotonicity sub-additivity homogeneity and translational invariance One example would be the conditional value-at-risk. You ...

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You can find a backtest for expected shortfall detailed in the paper below Kerkhof, F.L.J., & Melenberg, B. (2004). Backtesting for risk-based regulatory capital. Journal of Banking and Finance, 28, 1845-1865. Best, JK

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you should backtest in the future. Thus you calculate your VaR based on the last 250 business days and then look at the return tomorrow. You have to do this in a rolling/sliding fashion. Your approach is in-sample and what you should do is out-of-sample. The number of violations should be binomial. Furthermore you could do a runs test to test whether your ...

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Let's say your return realization for path $i$ is $r_i = \beta\cdot f_i$, where $f_i=(f_{1i}, f_{2i}, f_{3i})$ - factors realizations, and $\beta$ - factor coefficients. So, your VaR is $VaR=percentile(r_i,\alpha)$, where $\alpha$ - confidence. The simplest Monte Carlo stopping criterion is to keep adding paths $i$ and computing VaR on the growing sample ...

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Do $N$ MC simulations of $M$ samples, calculating your estimate of VaR for each one $\{\widehat{VaR}_i\}_{i=1}^N$ and you now have an IID sample! Take the sample (or unbiased) standard deviation for your estimate of VaR (this is probably what you mean by error) $SD(\widehat{VaR})=\sqrt{\frac{1}{N-1} \sum_{i=1}^N (\widehat{VaR}_i - \overline{VaR})^2}$ and of ...

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