A review of empirical likelihood methods for time series

• Several versions of empirical likelihood (EL) for time series are summarized.
• Methods are illustrated and compared with data and numerical examples.
• EL for high-dimensional data and for long-range time dependence are reviewed.
• Several open research problems for EL with dependent data are presented.

We summarize advances in empirical likelihood (EL) for time series data. The EL formulation for independent data is briefly presented, which can apply for inference in special time series problems, reproducing the Wilks phenomenon of chi-square limits for log-ratio statistics. For more general inference with time series, versions of time domain block-based EL, and its generalizations based on divergence measures, are described along with their distributional properties; some approaches are intended for mixing time processes and others are tailored to time series with a Markovian structure. We also present frequency domain EL methods based on the periodogram. Finally, EL for long-range dependent processes is reviewed as well as recent advantages in EL for high dimensional problems. Some illustrative numerical examples are given along with a summary of open research issues for EL with dependent data.