A consistent bootstrap test for conditional density functions with time-series data

This paper presents a new test for evaluating conditional density functions for time-series data, thereby being applicable to forecasting problems. We show that the test statistic is asymptotically distributed standard normal under the null hypothesis, and diverges to infinity when the null hypothesis is false. We use a bootstrap algorithm to approximate the distribution of the test statistic, and show that the bootstrap distribution converges to the asymptotic distribution of the test statistic in probability. An application to inflation forecasting is also presented to demonstrate the usefulness of the test.