Even while using historical simulation VaR, 1 day VaR is converted into 10 day VaR by multiplying 1 day VaR by Sqrt(10) for regulatory reporting purposes.
What are the underlying assumptions for doing this and how can those assumptions be tested statistically?
What are the underlying assumptions for doing this
Assumption: Historical returns are lognormally distributed with no autocorrelation.
can those assumptions be tested statistically
Testing: $\sqrt{xy} = \sqrt{x} \sqrt{y}$
Substitute time $t$ and variance $\sigma^2$ for $x$ and $y$ respectively
$\sqrt{t\sigma^2} = \sqrt{t} \sqrt{\sigma^2} = \sigma\sqrt{t}$
Some links for you to check out if you would like to investigate further:
Practically, I can tell you the sqare root assumption doesn't actually hold in practice--vol is not actually homoskedastic as a result of underlying returns not being iid (the scale tends to fall just short of the square of 12 in equities as a result of heterskedasticity).
A quick google turned up this, which seems to walk through precisely what you're asking about. Would probably be as good as any place to start.
Do we actually need lognormal returns as amdopt states? As long as returns are i.i.d., we have $\textrm{E}(r_tr_{t+1})=0$ and as a result $\textrm{Variance}(\sum_1^{10}r_t)=10\textrm{Variance}(r_t)$, so the VaR which is the threshold for a left tail weight of (say) $\alpha$ is scaled by $\sqrt{10}$.
one-day VaR cannot be converted into ten-day VaR, as Z ~ N(r, sigma) should provide a different distribution over one-day, limited time horizon.
