What are some typical approaches to handling incomplete data in the Kalman Filter? I'm talking about the situation where some elements of the observed vector $y_t$ are missing, distinct from the case where an entire observed vector $y_t$ is missed. Another way of thinking about this would be that the dimension $p$ of the observed vector is different for each time point.
To explain my context a bit further, the observations are estimated parameters from a logistic regression performed at each time point. While each logistic regression includes the same covariates, sometimes the estimates are undefined due to collinearities in the data for that time point.
Best Answer
What is needed is simply to have a variable observation matrix, i.e. in the observation equation: $$ \boldsymbol{Y_t} = \boldsymbol{A_t}\boldsymbol{\theta_t} + \boldsymbol{R_t}\boldsymbol{e_t} $$ matrix $\boldsymbol{A_t}$ (and $\boldsymbol{R_t}$) should omit at time $t$ the rows corresponding to
NAentries in $\boldsymbol{Y_t}$. Most packages in R, for instance, will take care of that: you can have the observed multivariate time series withNAvalues without problems.