There’s a genuinely interesting paper on the use of Kalman filters in econometrics here. Applications considered including estimating the demand for international reserves, the persistence of shocks to stock returns, decomposing the trend and cyclic components of GDP, and measuring volatility in financial markets.
Math is definitely required so it’s not for the casual reader. This post is basically a place-marker.
As a simply technical note, I assume that the “kinks” reference in the conclusion as a modification to the symmetrical loss function means that the loss function is temporarily disturbed.
In any event, this is what quants do all the time, but it sure seems like other variables could dominate the improved estimate of unknown/unmeasured variables. But it’s not my field.
My criticism of the use of Kalman filters in econometrics is a bit different. I doubt that the “known” parameters are actually known in many cases. Inaccuracies in the knowns could well overwhelm the improvements in the estimating of unknowns or unmeasured variables.
Dave –
That’s actually my point, although perhaps I stated it less clearly.
Process engineers are trained to take all the rigorous science and transfer it into a production environment, so they can laugh – OK, not really laugh – so they can make the science and the practicalities of a production environment converge.
As a purely intellectual puzzle, I thought the article was interesting. And perhaps it’s applicable to certain financial analytical exercises. But I’m a cynic. When you don’t even know if the manganese bin is really all manganese then these second and third order estimating techniques tend to leave you non-plussed. but a steel mill isn’t an option trading desk…..
Oh, and I’ve never heard this “kink” thing. I didn’t know if was a reference to a temporary disruption to the Gaussian distribution assumption, or an acknowledgement that a “kink” has occurred in the dominating variables and set the data stream on a new path, despite continuation of Gaussian distribution of unknowns or white noise.
Actually what you are talking about is measurement error and that problem is something to worry about whether you are using a Kalman filter or any sort of algorithm or estimation technique.
As for the kinks they are referring to the loss function, basically a loss function is defined as:
C(x-hat,x) and C(x,x) =0–i.e. there are no losses when your estimate is exactly on. You also want the losses to non-decreasing of a norm of the sampling error. For example,
C(x-hat,x) = (x-hat – x)^2.
A quadratic loss function. The further your estimate from x-hat is the larger the losses. You treat both over and under estimates the same (i.e. this is a symmetrical loss function). It is also a polynomial of order 2 hence not only is it continuous it is also differentiable (i.e. it is smooth–no kinks).
An asymmetric continuous loss function would be along the lines of:
C(x-hat,x) = c|x-hat – x| for x-hat x.
If b > c then it is the case that you consider over-estimation as being more costly than an underestimation.
In estimation procedures the goal is always to find an estimator, x-hat, which is a function of the observed data that minimizes your cost function.
Cost/loss functions can even be like a utility function used in economics–i.e. a subjective function, that is two people might have different cost/loss functions even when it comes to the same issue. However, this need not be the case. Mean square error is one candidate for a loss function–i.e. we want an estimator that minimizes the mean square error (this is the quadratic loss function noted above).
HTH….
Just realized there is an error in the above post, the second loss function did not come through completely
C(x-hat,x) = c|x-hat – x| for x-hat greater than or equal to x.
C(x-hat,x) = b|x-hat – x| for x-hat less than x.
There we go.