Nonparametric conditional density specification testing and quantile estimation; with application to S&P500 returns

This paper develops a two stage procedure to test for correct dynamic conditional specification. It exploits nonparametric likelihood for an exponential series density estimator applied to the in-sample Probability Integral Transforms obtained from a fitted conditional model. The test is shown to be asymptotically pivotal, without modification. Numerical experiments illustrate both this and also that it can have significantly more power than equivalent tests based on the empirical distribution function, when applied to a number of simple time series specifications. In the event of rejection, the second stage nonparametric estimator can both consistently estimate quantiles of the data, under empirically relevant conditions, as well as correct the predictive log-scores of mis-specified models. Both test and estimator are applied to monthly S&P500 returns data. The estimator leads to narrower predictive confidence bands which also enjoy better coverage and contributes positively to the predictive log-score of Gaussian fitted models. Additional application involves risk evaluation,such as Value at Risk calculations or estimation of the probability of a negative return. The contribution of the nonparametric estimator is particularly clear during the financial crisis of 2007/8 and highlights the usefulness of a specification procedure which offers the possibility of partially correcting rejected specifications.