Adjusted Empirical Likelihood for Time Series Models

Empirical likelihood method has been applied to dependent observations by Monti (Biometrika, 84, 395–405 1997) through the Whittle’s estimation method. Similar asymptotic distribution of the empirical likelihood ratio statistic for stationary time series has been derived to construct the confidence regions for the parameters. However, Monti’s approach is valid only when the error terms follow a Gaussian distribution. Nordman and Lahiri (Ann. Statist., 34, 3019–50 2006) derived estimating functions and empirical likelihood ratio statistic using frequency domain empirical likelihood approach for non-Gaussian error term distributions. Nonetheless, the required numerical problem of computing profile empirical likelihood function which involves constrained maximization has no solution sometimes, which leads to the drawbacks of using the original version of the empirical likelihood ratio. In this paper, we propose an adjusted empirical likelihood ratio statistic to modify the one proposed by Nordman and Lahiri so that it guarantees the existence of the solution of the required maximization problem, while maintaining the similar asymptotic properties as Nordman and Lahiri obtained. Simulations have been conducted to illustrate the coverage probabilities obtained by the adjusted version for different time series models which are competitive to the ones based on Nordman and Lahiri’s version, especially for small sample sizes. © 2017, Indian Statistical Institute.