Goodness-of-Fit Tests Focus on Value-at-Risk Estimation

A common statistical problem in finance is measuring the goodness-of-fit of a given distribution to real world data. This can be done using distances to measure how close an empirical distribution is from a theoretical distribution. The tails of the distribution should receive special importance if the focus is on Value-at-Risk (VaR) calculations. This paper analyzes the use of distances to test the goodness-of-fit of estimated distributions for VaR calculation purposes. The Crnkovic and Drachman (1996) distance and a new distance are used to perform goodness-of-fit tests. The critical values of the tests are obtained using Monte Carlo simulation, and goodness-of-fit tests are performed based on the distances. The power of the tests is assessed through Monte Carlo experiments, showing good results for sample sizes greater than 250. The US Dollar/Brazilian Real exchange rate and the Ibovespa index are used as examples of practical applications of how to test the hypothesis that an empirical distribution is equal to an estimated one. The estimated distributions considered are the Generalized Hyperbolic (GH), the NIG (Normal Inverse Gaussian) and Normal. The test results rejected the null hypothesis for the Normal distribution, but did not reject it for the Generalized Hyperbolic and NIG, both at a 1% significance level