# State space models: Advantage of Stationary State Vector?

Consider a State Space Model, where the observed process is $Y_t$ $$Y_t = B F_t + \epsilon_t \\ F_t = \Phi F_{t-1} + \nu_t$$ where the error terms are white noise. Later on, I want to compute the state vector $F_t$ with the Kalman Filter.

I often find the assumption that $(F_t)_t$ is stationary in the literature but it's not clear to me why this assumption is meaningful. So, what are the properties or advantages of the model if I chose $(F_t)_t$ to be a stationary process?

• I don't know about "often". The basic structural time series models have state variables that are explicitly nonstationary. For example, the local level model has $y_t = \mu_t + \epsilon_t$ with $\mu_t = \mu_{t-1} + \eta_t$. Here the state variable $\mu_t$ is a random walk. – The Laconic Apr 1 '18 at 16:42
• When the model is stationary, the Kalman Filter (KF) converges, which means that the prediction and filtering covariances P(t|t−1) and P(t|t) converge for t→∞. So if you find the stationary distribution, the KF update can be avoided in some cases. – Yves Apr 2 '18 at 15:06
• Thanks you @Yves, do you have a good reference for this finding? – ReinerZufall Apr 2 '18 at 17:51
• @Larry This relates to the so-called Riccati equation. See e.g. the famous book Harvey, A. Forecasting Structural Time Series Models and the Kalman Filter. I would encourage you to consider the stationary $\text{AR}(p)$ where the convergence is reached (exactly) in $p$ steps. – Yves Apr 2 '18 at 18:22

Kalman filter is a tool that can be still optimal even with non-stationary state or noise. The assumption of being stationary often aims to compute some specific equation. For example, in the state equation, $F_t = \Phi F_{t-1} + v_t$, if we want to compute the autocorrelation matrix $\mathbf{E}[F_tF_t^T]$, it can come from $\mathbf{E}[F_tF_t^T] = \Phi \mathbf{E}[F_{t-1}F_{t-1}^T] \Phi^T+ Q_t$ by setting $\mathbf{E}[F_tF_t^T] = \mathbf{E}[F_{t-1}F_{t-1}^T]$ (can only be adopted if $F_t$ is stationary).