Goodness-of-fit test of copula functions for semi-parametric univariate time series models

In this paper, we propose a goodness-of-fit test, named pseudo “in-and-out-of-likelihood” (PIOL) ratio test, to check for misspecification in semi-parametric copula models for univariate time series. The proposed test extends the idea of the IOS test by Presnell and Boos (J Am Stat Assoc 99:216–227, 2004) and PIOS test by Zhang et al. (J Econom, 193:215–233, 2016), which are problematic for direct application to univariate time series. The PIOL test provides an integrated framework for both independent data and time series data. In addition, an approximation method is implemented to alleviate the computational burden of calculating the test statistics. Asymptotic properties of the proposed test statistics are discussed. The finite-sample performance is examined through simulation studies. We also demonstrate the proposed method through the analysis of a time series of daily transactions of Apple trade.