When Kalman Filters May Not Improve on Linear Regression
Summary
The document addresses whether replacing a statistically significant linear regression with a Kalman filter will necessarily improve predictions. Its central answer is conditional: performance depends on the data and on how the filter is calibrated. A Kalman filter requires choices for its state transition and observation matrices and its process and measurement noise parameters, which can make tuning more involved than fitting a basic regression.
The replies caution that added model complexity does not guarantee greater forecast accuracy. One explains that a regression with coefficients modeled as a random walk can be represented as a Kalman filter, linking the methods and showing why the filter is not automatically a fundamentally better model. Another notes that linear regression can be formulated as a Kalman estimate. The discussion cites general forecasting competition evidence in support of simpler methods sometimes performing well, but supplies no direct test on the questioner’s data. Model comparison and calibration therefore remain necessary.
Key ideas
- A Kalman filter is not guaranteed to improve predictions over linear regression.
- Performance depends on the data and on calibration of the filter’s state and noise parameters.
- A regression with random-walk coefficients can be expressed as a Kalman filtering problem.
- Greater model complexity does not ensure greater forecast accuracy.
- The discussion gives no empirical comparison on the specific dataset, so evaluation must be data dependent.
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Full text
# Does Kalman filter always improve over linear regression? # Does Kalman filter always improve over linear regression? If I have a simple linear regression that has statistical signification but I would like to improve the overall prediction results. Will a Kalman filter be always an improvement or as least achieve similar results to linear regression? Edit: Relevant Threads: How to tune Kalman filter's parameter? ## Answer by Robert Szóstakowski (score 6) https://quant.stackexchange.com/a/20828 There is no a "yes/no answer" to that question. Generally Kalman Filter tends to be better than linear regression, but everything depends on - the data which you have, - how you calibrate your model. I expect that you have used some library for estimating linear regression parameters. Now you need to think how will you "tune" Kalman filter - the constants F, H, R, Q. See Wiki Page of Kalman Filter. I have asked a related question and Kalman Filter parameters tuning is not as easy as in the linear regression example. General rule is - simple models tends to be better than complicated ones. Take a look at the quote from Makridakis Competitions. > "The most interesting test of how academic methods fare in the real world was provided by Spyros Makridakis, who spent part of his career managing competitions between forecasters who practice a "scientific method" called econometrics -- an approach that combines economic theory with statistical measurements. Simply put, he made people forecast in real life and then he judged their accuracy. This led to a series of "M-Competitions" he ran, with assistance from Michele Hibon, of which M3 was the third and most recent one, completed in 1999. Makridakis and Hibon reached the sad conclusion that "statistically sophisticated and complex methods do not necessarily provide more accurate forecasts than simpler ones."" ## Answer by Craig (score 5) https://quant.stackexchange.com/a/20839 There is no magic in the Kalman Filter. The linear regression model usually assumes the coefficients follow a random walk and as such it essentially boils down to an estimation followed by exponential smoothing of the coefficients. ## Answer by LazyCat (score 1) https://quant.stackexchange.com/a/20831 Yes, linear regression can be cast as a Kalman filter estimate. I believe, D. Simons book "Optimal State Estimation: .. " has all the details.
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