**State Space form**

**Measurement Equation**

**Transition Equation**

**Future 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.

then

where

**Predictive distribution of **

**Updating equations**

and

**Contemporaneous filter:**

**Predictive filter: **

In the latter case

or

where the gain matrix is given by

and

**Initialization**

Start Kalman at with diffuse prior

**Prediction**

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 .

—————————————————————————————————————–

Further Reading:

Time Series Analysis

Time Series: Theory and Methods (Springer Series in Statistics)

An Introduction to State Space Time Series Analysis (PRACTICAL ECONOMETRICS SERIES)

Forecasting, Structural Time Series Models and the Kalman Filter

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