  A new mixture autoregressive model based on Student's $t$-distribution is
proposed. A key feature of our model is that the conditional $t$-distributions
of the component models are based on autoregressions that have multivariate
$t$-distributions as their (low-dimensional) stationary distributions. That
autoregressions with such stationary distributions exist is not immediate. Our
formulation implies that the conditional mean of each component model is a
linear function of past observations and the conditional variance is also time
varying. Compared to previous mixture autoregressive models our model may
therefore be useful in applications where the data exhibits rather strong
conditional heteroskedasticity. Our formulation also has the theoretical
advantage that conditions for stationarity and ergodicity are always met and
these properties are much more straightforward to establish than is common in
nonlinear autoregressive models. An empirical example employing a realized
kernel series based on S&P 500 high-frequency data shows that the proposed
model performs well in volatility forecasting.
