+Extending the model
+
+Incorporating time dynamics
+
+The DFM is typically generalized to include autoregressive components
+
+
\[
+\begin{split}\begin{align}
+y_t & = \Lambda f_t + B x_t + u_t \\
+f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \qquad \eta_t \sim N(0, I)\\
+u_t & = C_1 u_{t-1} + \dots + C_q u_{t-q} + \varepsilon_t \qquad \varepsilon_t \sim N(0, \Sigma)
+\end{align}\end{split}
+\]
+
+
+
+
Where \(y_t\) is observed, \(f_t\) are unobserved latent factors, \(x_t\) are optional (unused for our case) exogenous variables, and the dynamic evolution of latent factors is expressed using the transition matrix \(A\) with \(\eta_t\) representing new information or random shocks. \(u_t\) is the error or “idiosyncratic” process
+
+
+
+
This model is then cast into state space form and the unobserved factors estimated via the Kalman filter. The likelihood can be evaluated as a byproduct of the filtering recursions with maximum likelihood estimation used to estimate the parameters.
+
+