Time domain estimation of non-fundamental ARMA models in the presence of heteroskedasticity of unknown form

This paper considers time domain estimation of possibly non-fundamental (that is, non-causal and/or non-invertible) non-Gaussian linear ARMA models with martingale difference innovations that may display conditional heteroskedasticity of unknown form. Instead of explicitly parametrizing the underlying volatility process (the higher order dependence) and employing maximum likelihood procedures, we propose a time domain minimum distance objective function based on innovations predictability using second and third powers of past innovations. Using the proposed efficient GMM estimator, which is consistent and asymptotically normal, we estimate possibly non-fundamental ARMA models to inflation data from 29 OECD countries and find widespread evidence on non-fundamentalness.