Bayesian analysis of restricted penalized empirical likelihood

Restricted Empirical Likelihood Estimation for Time Series Autoregressive Models

“…Second, by assuming sparsity of vectors and w (to reduce the number of parameters and the number of the estimating equations respectively) and − log REL being convex, we can use the REL method for high-dimensional data. For more details on the appealing properties of the REL L R (w, ), readers are referred to Bayati et al [24]. With these appealing properties, we propose the REL estimator of as given by…”

“…Nordman and Lahiri [22] has argued that Monti's results were based on the assumption of normal distribution for the error term, and thus has extended the classical empirical likelihood to the one based on spectral distribution via Fourier transforms. Their results apply to both short and long memory dependencies; see Nordman and Lahiri [24] for details.…”

“…To the best of our knowledge, all the studies on empirical likelihood for time series rely on the unbiasedness property of estimating equations. But it seems that the unbiasedness condition alone, when data contains outliers, will produce misleading estimates for the parameters of interest (Bayati et al [24]). In other words, in the usual empirical likelihood approach, features such as the dispersion of estimator functional, which depends on the dispersion of data itself, are not included in the statistical inference process.…”

“…In this paper, for the first time we introduce the restricted empirical likelihood (REL) for the AR(r) time series models and present its Bayesian analysis. To our best knowledge, Bayati et al [24] is the only one that has also used the idea of REL. Nevertheless, Bayati et al [24] and this paper are very different in the sense that Bayati et al [24] has studied the REL for independent observations while this paper investigates the REL for the more sophisticated case of dependent observations specifically from autoregressive models.…”

“…To our best knowledge, Bayati et al [24] is the only one that has also used the idea of REL. Nevertheless, Bayati et al [24] and this paper are very different in the sense that Bayati et al [24] has studied the REL for independent observations while this paper investigates the REL for the more sophisticated case of dependent observations specifically from autoregressive models. In this paper, we not only prove the asymptotic normality of REL estimator for general models, but also give the detailed Bayesian analysis of AR(r) time series models using the REL and as well develop efficient algorithm for computing posterior and estimating hyper-parameters.…”