Consider a $T \times n$ matrix of data $Y_{t}$ and its lagged values $Y_{t-1}$. The matrix $Y_{t}$ can be split into $$Y_{t}\equiv\left[\begin{array}{cc} Y_{t,1} & Y_{t,2}\end{array}\right] $$ where $Y_{t,1}$ has no missing data, but $Y_{t,2}$ has missing data such that only the most recent $T_{2}$ data is available (for both $Y_{t}$ and $Y_{t-1}$).
I want to estimate the VAR model $$Y_{t}=\alpha+Y_{t-1}\beta+\varepsilon_{t}$$ where $\varepsilon_{t}\sim N\left(0,\Sigma\right) $ with full-information maximum likelihood. When I estimated this model, I replaced the missing data with zeros in order to ensure that the matrix math works. As a result, I keep finding the FIML estimate of $\alpha$ and $\beta$ equal to the OLS estimate using the dataset adjusted with the zeros (I still need to use an FIML estimator for the covariance matrix of the error).
I'm not sure this is the correct way to do it. Can anyone confirm?
In particular, there is a term in the log-likelihood function where one takes the sum of $$\left(Y_{t}-\alpha-Y_{t-1}\beta\right)\Sigma^{-1}\left(Y_{t}-\alpha-Y_{t-1}\beta\right)'$$ for each period. Instead of replacing with zeros, I could only calculate it using the data that's available in each period. For instance, in the first part of the data it would only use the variables from $Y_{t,1}$ that are available and the part of the covariance matrix that is available to take the inverse. I do not think that is the same as replacing with zeros.
