Noncausal <scp>AR</scp> ‐ <scp>ARCH</scp> Model and Its Applications to Financial Time Series

ABSTRACT We extend the noncausal autoregressive models by introducing noncausality into the variance component, allowing the volatility to depend on future prices as well. We refer to this model as the noncausal AR‐ARCH model, and it enables us to account for shocks arising from market agents who possess more information and engage in forward‐looking trading behaviours, leading to a better fit for financial time series. In terms of parameter estimation, we develop a quasi‐maximum likelihood estimation method and establish its asymptotic properties. Building on this, we propose three hypothesis testing statistics to determine whether the data exhibits a noncausal AR structure and whether the innovation term follows a noncausal ARCH model. The simulation results demonstrate the consistency of the parameter estimation as well as the good size control and high power of the hypothesis tests in detecting noncausal structures. In our empirical applications, we employ the proposed model in both stock markets and crude oil futures markets. Our empirical findings indicate that the variance is causal in the US stock market but noncausal in the Chinese stock market. Furthermore, we observe a noticeable distinction between Brent and WTI crude oil futures, as Brent exhibits noncausality in both its mean and variance, whereas WTI follows a purely causal process.