Semi†Parametric Estimation for Non†Gaussian Non†Minimum Phase ARMA Models

We consider inference for the parameters of general autoregressive moving average (ARMA) models which are possibly non†causal/non†invertible (also referred to as non†minimum phase) and driven by a non†Gaussian distribution. For non†minimum phase models, the observations can depend on both the past and future shocks in the system. The non†Gaussianity constraint is necessary to distinguish between causal†invertible and non†causal/non†invertible models. Many of the existing estimation procedures adopt quasi†likelihood methods by assuming a non†Gaussian density function for the noise distribution that is fully known up to a scalar parameter. To relax such distributional restrictions, we borrow ideas from non†parametric density estimation and propose a semi†parametric maximum likelihood estimation procedure, in which the noise distribution is projected onto the space of log†concave measures. We show the maximum likelihood estimators (MLEs) in this semi†parametric setting are consistent. In fact, the MLE is robust to the misspecification of log†concavity in cases where the true distribution of the noise is close to its log†concave projection (Cule and Samworth, 2010; Dümbgen et al., 2011). We derive a lower bound for the best asymptotic variance of regular estimators at rate n−12 for autoregressive (AR) models and construct a semi†parametric efficient one†step estimator. The estimation procedure is illustrated via a simulation study and an empirical example illustrating the methodology is also provided.