Tests if the exceedances of the Value at Risk (returns below -var) have the expected frequency and are
independent over time.
Arguments
- x
A
data.tablereturned fromvar_es. Rows without an observed return are ignored.
Value
A data.table with one row for each level and the columns level, n,
expected and observed (number of exceedances), rate (observed rate), lr_uc,
p_uc, lr_ind, p_ind, lr_cc, p_cc and es_ratio.
Details
Let \(I_t = 1\) if \(y_t < -VaR_t\) and \(p = 1 - \alpha\) the expected exceedance rate.
Unconditional coverage (Kupiec 1995): a likelihood ratio test of \(P(I_t = 1) = p\). A small p-value means too many or too few exceedances.
Independence (Christoffersen 1998): a likelihood ratio test that \(I_t\) is a Markov chain with \(P(I_t = 1 \mid I_{t-1} = 0) = P(I_t = 1 \mid I_{t-1} = 1)\). A small p-value means that exceedances come in clusters, which happens when the model reacts too slowly to changes in volatility.
Conditional coverage: the sum of the two statistics, with two degrees of freedom.
es_ratio is the mean of \(-y_t / ES_t\) over the exceedances. It is close to one if the Expected
Shortfall is correct, and larger than one if the model underestimates the size of the losses in the tail.
References
Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2), 73-84. doi:10.3905/jod.1995.407942
Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841-862. doi:10.2307/2527341