backtest_var: Backtest Value at Risk and Expected Shortfall

In stochvolTMB: Likelihood Estimation of Stochastic Volatility Models

Backtest Value at Risk and Expected Shortfall

Description

Tests if the exceedances of the Value at Risk (returns below -var) have the expected frequency and are independent over time.

Usage

backtest_var(x)

Arguments

x	A data.table returned from var_es. Rows without an observed return are ignored.

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.

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.

References

Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2), 73-84. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3905/jod.1995.407942")}

Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841-862. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/2527341")}

See Also

var_es

stochvolTMB documentation built on Nov. 7, 2026, 5:08 p.m.