State Space form
1.2. Kalman Filter
Recursive procedure for computing the optimal estimator of the state vector at time t. When the model is Gaussian Kalman filter can be interpreted as updating the mean and covariance matrix of the conditional distribution of the state vector as new observations become available.
Predictive distribution of
In the latter case
where the gain matrix is given by
Start Kalman at with diffuse prior
Taking conditional expectations in the measurement equation for
with MSE matrix
MLE and prediction error decomposition
Prediction errors or innovations
Prediction error decomposition
Diagnostic tests can be based on the standardized innovations
which are serially independent if is known
* is maximized w.r.t. numerically. Diffuse prior exact likelihood.
* and run independently on .