Autoregressive Models with Non-Causal ARCH Volatility

This paper introduces a novel non-causal (forward-looking) ARCH specification in which conditional heteroskedasticity depends on leads of the process. When observed in calendar time, this time inversion allows large past shocks to affect the entire conditional distribution rather than only its scale, as in standard ARCH models. The resulting forward-looking dynamics can generate symmetric bimodal predictive densities, providing a new interpretation of high-volatility regimes. We establish the stochastic properties of autoregressive processes with errors following the proposed non-causal ARCH specification and derive sufficient conditions for consistency and asymptotic normality of an Approximate Quasi-Maximum Likelihood Estimator (AQMLE). A kernel-based estimator of the marginal error distribution and the predictive density is also developed. Simulation results demonstrate good finite-sample performance. An empirical application to monthly CPI data shows that the proposed model captures distributional dynamics more effectively than traditional approaches, particularly during periods of elevated uncertainty and volatility.