# Prediction covariance matrix with Kalman filter

1. Jun 28, 2011

### mtal

Hello all.

I have set up a model using the Kalman filter to estimate automobile prices. I'm having difficulty in figuring out how to formulate a prediction covariance matrix based on the model, i.e. given a set $y_{new} = y_1, \ldots, y_N$ of $N$ cars, finding the covariance matrix based on the predictions for the given set of cars.

The model is set up in the following way:
$$y_t = X_t \alpha_t + \epsilon_t$$
is the observation process, and
$$\alpha_t = T\alpha_{t-1} + \nu_t$$
is the state process. Here $T$ is the identity matrix and the error terms, $\epsilon, \nu$ are uncorrelated mean zero processes, constant over time.

From what I know, prediction variance for a single $y_{new}$ is calculated as
$$Var(y_{\textrm{pred}}) = \sigma^2(X_t P_t X_t^T + 1)$$
where $\sigma^2$ is the model variance and $P_t$ is the covariance matrix for $X_t$, which is the characteristic coefficient matrix.

But if I have multiple cars, $y_{new} = y_1, \ldots, y_N$, is it correct that the corresponding covariance matrix would be calculated as
$$Var(y_{\textrm{new}}) = \sigma^2(X_t P_t X_t^T + I)$$
where $I$ is a $N\times N$ identity matrix, or am I missing something?

Help much appreciated!