These questions arose when I was reading Online Appendix D for the paper Missing Events in Event Studies: Identifying the Effects of Partially-Measured News Surprises by R.S. Gurkaynak, B. Kisacikoglu and J.H. Wright.
From what I have learnt, the usual exercise of Expectation-maximization with kalman filter is an iterative repetition of
E-step: based on the estimates of parameters from the last M-step, use kalman filter to update the estimates of the latent variable;
M-step: use the estimates of latent variable from the E-step to get new estimates of parameters.
In the paper cited above, however, the authors seem to use EMKF algorithm to estimate the parameters with one single M-step. Specifically, the authors aim to uncover the latent variable $f_t$ in the equation $$y_t=\beta's_t+\gamma d_tf_t+\varepsilon_t,$$ where $y_t$ is an $n\times1$ vector, $s_t$ is a $K\times1$ vector, which are observed without error, $d_t$ is a dummy that is $1$ if there is an event at time $t$ or $0$ otherwise, $f_t$ is an iid $N(0,\sigma_f^2)$ latent variable, $\beta$ is a $K\times n$ matrix, $\gamma$ is an $n\times1$ vector, and $\varepsilon_t$ is iid with mean zero and diagonal variance-covariance matrix of $\Sigma_\varepsilon.$
The log-likelihood function is as follows: $$\begin{aligned}l(\theta)=-\frac{1}{2}\Sigma_{t=1}^{T}\mathbf{1}(d_{t}=1)\{(y_{t}-\beta^{\prime}s_{t})^{\prime}(\Sigma_{\varepsilon}+\gamma\gamma^{\prime})^{-1}(y_{t}-\beta^{\prime}s_{t})\\&+\log(|\Sigma_{\varepsilon}+\gamma\gamma^{\prime}|)\}\\&+\mathbf{1}(d_{t}=0)\{y_{t}^{\prime}\Sigma_{\varepsilon}^{-1}y_{t}+\log(|\Sigma_{\varepsilon}|)\}\end{aligned}.$$ To identify the parameters, the authors solve the following system of equations, which is obtained by maximize the log-likelihood function: $$\beta=E(\mathbf{1}(d_t=1)s_ts_t^{\prime})^{-1}E(\mathbf{1}(d_t=1)s_ty_t^{\prime}),$$ $$\Sigma_{\varepsilon}=E(y_{t}y_{t}^{\prime}\mathbf{1}(d_{t}=0)),$$ $$\gamma\gamma^{\prime}=E((y_{t}-\beta^{\prime}s_{t})(y_{t}-\beta^{\prime}s_{t})^{\prime}\mathbf{1}(d_{t}=1))-\Sigma_{\varepsilon}.$$
But $f_t$ does not appear in the equations above. Doesn't this mean to solve for $\beta$, $\gamma$ and $\Sigma_{\epsilon}$ one does not actually need the kalman filter? Can anyone explain this to me?
