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# Understanding the Kalman Filter Application in Economic Time Series Data

The Kalman filter has been extensively used in Science for various applications, from detecting missile targets to just any changing scenario that can be learned.

I'm trying to understand how Kalman Filter can be applied on Time Series data with Exogenous variables - in a nutshell, trying to replicate PROC UCM in excel.

State-space equation :

To those familiar with the Kalman filter, it essentially consists of the following two steps,

Predict:

Update:

Most of the text on Kalman only introduce univariate analysis, with no exogenous variables. And most applications in control engg seem to suit that as well.

What I'm stuck figuring is -

1. How can I update the H matrix with every observation? Pretty much, MMSE or ML can help me do this, but I'm just unable to do this with just one observation! The problem of recursive estimation with just one observation if I could say...

2. How can I bring in the estimation of betas of other exogenous variables that also affect the Y variable, so, I'm going to be understating the latent state variable to be just a constant base or linear trend.

Any help would be greatly appreciated, and if you have some good docs/sites that explain this better for the econometrician, please do pass it on.

Thanks,

Arun

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### Replies to This Discussion

Arun,

As an Econometrician, why aren't you using a "Transfer Function" model?

Isn't Kalman Filtering a part of the Transfer Function models? Atleast, that's what I thought from all the research in the background..

Arun,

A Kalman Filter model is a State Space Transform of a Transfer Function model. It has been found to be useful when you have chunks of missing data and can incorporate transience in parameters.  Strictly speaking, the models are similar but in practice a TF model properly formed can lead to the idenitification of unspcecified deterministic structure which is a generalization of the UCM approach. For example, pulse outliers, changes in trend/seasonality/trend/parameters/variance can exist and need to be modeled.  The inclusion of lead and lags on the causals are also of importance.

The existence of multiple trends in a time series is as likely as multiple level shifts.  Addressing these issues our software, Autobox , addresses these automatically.  Send over your data and we will run it and send you back results.  My email is [email protected]