Furthermore, we show that if we assume that the marginal cost of information is fixed, capacity will be elastic with respect to a change in policy; consequently, the Kalman gain in this case will also adjust in response to the policy change.
Specifically, in the presence of the correlation, a change in the variance of the exogenous shock does not change the dynamic behavior of the model in the fixed capacity case, whereas it changes the model's dynamics in the fixed information-processing cost case in which both the variance of noise and the Kalman gain are affected by the interactions between the correlation, the variance of the fundamental shock, and the conditional variance.
After computing the optimal steady-state conditional variance-covariance matrix, we can recover the variance-covariance matrix of the noise vector and then determine the Kalman gain.
It is worth noting that in the multivariate RI problem, the agent's preference, budget constraint, and information-processing constraints jointly determine the values of the conditional variance of the state, the variance of the noise, and the Kalman gain, whereas in the multivariate SE problem given the variance of the noise, the propagation equation updating the conditional variance based on the budget constraint is used to determine the conditional variance and then the Kalman gain.
In addition, the innovation form of Kalman gain is employed for state estimation with no prior knowledge of noise properties.
Thus, the Kalman gain matrix for optimal state estimation under stochastic disturbance can be obtained without acquiring knowledge on noise properties or computing the discrete Riccati equation.
o](kT) in Equation (13) are the Kalman gain matrices and the optimally estimated state.
The difference between the predicted observable variable and the realized observable variable, known as the prediction error, is then distributed among the predicted state variables by a Kalman gain
vector to form the updated state variables.