Bayesian inference of time varying parameters in autoregressive processes

In the autoregressive process of first order AR(1), a homogeneous correlated time series $u_t$ is recursively constructed as $u_t = q\; u_{t-1} + \sigma \;\epsilon_t$, using random Gaussian deviates $\epsilon_t$ and fixed values for the correlation coefficient $q$ and for the noise amplitude $\sigma$. To model temporally heterogeneous time series, the coefficients $q_t$ and $\sigma_t$ can be regarded as time-dependend variables by themselves, leading to the time-varying autoregressive processes TVAR(1). We assume here that the time series $u_t$ is known and attempt to infer the temporal evolution of the 'superstatistical' parameters $q_t$ and $\sigma_t$. We present a sequential Bayesian method of inference, which is conceptually related to the Hidden Markov model, but takes into account the direct statistical dependence of successively measured variables $u_t$. The method requires almost no prior knowledge about the temporal dynamics of $q_t$ and $\sigma_t$ and can handle gradual and abrupt changes of these superparameters simultaneously. We compare our method with a Maximum Likelihood estimate based on a sliding window and show that it is superior for a wide range of window sizes.