Empirical likelihood for martingale differences

In this article, we consider an empirical likelihood with vector observations that are martingale differences and prove a Wilks' type theorem under a conditional Lindeberg condition. We then generalize this result to approximate martingale differences. As applications of the first result we discuss the construction of confidence regions for various time series models, including linear and nonlinear autoregressive models, an ARCH(1) model, and nonlinear autoregressive models with heteroskedasticity. The second result is used to derive a Wilks' type theorem for a blockwise empirical likelihood for Markov chains.