Weighted Maximum Likelihood for Mixed Causal Non-causal Autoregressive Models with Application to the Forecasting of Bubbles in Financial Time Series

The aim of this paper is to propose a weighted maximum likelihood approach in order to improve forecast precision in the context of locally explosive behaviours, also referred to as speculative bubbles, in financial time series. To model such behaviours we rely on the mixed causal non-causal (MAR) framework. MAR models are a class of autoregressive models that allow a process to depend on its leads in addition to its lags and are allowed to capture such non-linear features. We show how, when estimating a MAR model, weighting differently the information coming from our data allows us to target an estimator that improves the forecast with respect to the standard MLE estimator. We apply our methodology both in the context of a simulation study and to two different financial time series comparing the forecasting performance.