Inconsistent test-statistic formula between Jorion (Market Risk) and Bodie (Investments)

[kchristo]

Why is it that the test stat for VaR backtesting (Jorion) equals the sample estimate minus the population estimate divided by the standard error, while in Bodie, it equals the sample estimate times (n^.5) divided by the standard error?

In other words - Jorion: (actual exceptions - expected exceptions)/std. error
Bodie: (actual alpha * sqrt(n))/std. error

 

[David Harper CFA FRM]

Hi @kchristo

The t-statistic is given by: (X - X0)/SE(X) where X = is the observed statistic, X0 is the null hypothesized population parameter, and SE is the standard deviation of X's sampled distribution; aka, standard error. If it's a regression instance, we typically (but not necessarily) null hypothesize that the slope parameter is zero, such that the t-stat = (β - 0)/SE(β) = β/SE(β). Recall that the SE = S/sqrt(N) where the standard error is "shrinking" per CLT with the sample size by the 1/sqrt(N).

Jorion's matches the template. Bodie performed some simplification per: t-statistic = (X - X0)/SE(X) = (α - 0)/σ(α) = α/σ(α) but σ(α) = σ(e)/sqrt(N) such that α/σ(α) = α*sqrt(N)/σ(e). Hope that's helpful,

 

[kchristo]

very helpful. Thank you for the quick reply and explantion!