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Time series of returns, $r_t$, in finance are often modeled with some type of conditional heteroskedasticity model, e.g. ARCH(1):

$$r_t = \sigma_t z_t$$ $$\sigma_t^2 = a_0 +a_1 r_{t-1}^2$$

where, say, $z_t \tilde{} N(0,1)$, which implies that

$$r_t = z_t \sqrt{a_0 +a_1 r_{t-1}^2}$$

and hence the returns are not independent. However, Value-at-risk (VaR) seems to be estimated by constructing a single empirical distribution of the observed returns and taking some quantile. But since the returns are not independent, they cannot be represented by one univariate distribution, so why this procedure to calculate VaR is considered valid? It seems to me that, at the very least, VaR estimated in this way would be biased, but I am not sure I have seen corrections being applied / discussed for this.

Add 1 It seems to me that given a time series with a persistent volatility (e.g. one with a high order of the ARCH term above, say 100), any finite period of observations, say 250, is likely to have explored a smaller part of the total space compared to IID returns, so I would have thought that one has to correct somehow for this when estimating VaR from historical observations in this case.

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    $\begingroup$ There will typically exist a long-run (or unconditional or ergodic) distribution associated with this model, and that may be what the empirical example you cite attempts to estimate and work with. Can you give a concrete example of the practice or claim you are questioning? $\endgroup$ – Drew Saunders Aug 4 '17 at 12:21
  • $\begingroup$ @Drew Historical Simulation approach to VaR estimation for market risk is done on about 250 observations of the most recent riskfactor returns. (Now the BCBS has added a requirement for estimation of SVaR, which is VaR estimated on a "period of stress", which might be trying to address precisely the point of smaller space of outcome explored in persistent volatility time series.) $\endgroup$ – Confounded Aug 4 '17 at 12:29
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The fact that you have a univariate distribution does not imply that the returns are independent. If fact it is plenty of univariate models that take into account the dependence of the time series. In volatility models the innovations (${{Z}_{t}}$ in your question) are independent, not the returns.

The procedure to caluclate VaR with volatiltiy models is consistent. When you compute quantiles, you may think that the time dependence is broken apart by ordering the returns, but this is not the case, because the returns are modeled using time dependence variance.

I try to explain with an example. There are closed formulas to complute VaR using simple volatility models. Most of the time, for advanced models you have to use simulation. For example you estimate a GARCH model with your data. Then you simulate N (N>1000) paths from this model and for each path you compute VaR. Finally you can compute the mean of tha VaRs or better you compute a confidence interval.

In multivariate models the procedure is the same. You have just more parameters to model the time-varying correlation between assets (see GARCH-DCC model)

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