Mean-Centered State Equations in Kalman Filters for Dynamic Nelson-Siegel Models
Summary
The document asks whether two state-space formulations used with Kalman filters are equivalent when the latent state has a nonzero mean. One formulation expresses the transition in deviations from a mean; the other uses a direct autoregressive transition and specifies a distribution for the initial state. The observation equation is presented separately, and the author notes that common implementations appear to use the direct form.
The author reports good parameter estimates when fitting simulated data generated with a zero-mean state process, then asks whether empirical data should always be de-meaned. The post does not supply an algebraic derivation or a recommendation, so it is best read as a modeling question rather than a worked solution. Its practical focus is the treatment of the latent-state mean and initial conditions in dynamic Nelson-Siegel estimation; the appropriate specification depends on how the transition, intercept or centering, and initial-state distribution are defined together.
Key ideas
- A mean-centered transition models deviations of the latent state from a specified mean.
- A direct transition formulation may require different treatment of an intercept or initial-state distribution to represent a nonzero mean.
- The post contrasts formulations used in a dynamic Nelson-Siegel state-space setting.
- Good estimates on simulated zero-mean states do not settle how to specify the model for empirical data.
- The document poses the equivalence and de-meaning questions without resolving them.
Tags
Full text
# Correct form for State Space Equation for Kalman Filter for DNS
# Correct form for State Space Equation for Kalman Filter for DNS
In this paper:
http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf
in eqns 3,5 the state eqn has the mean removed.
$(z_t-\mu)=A(z_{t-1}-\mu) + \epsilon_t$
$y_t=C z_t + \delta_t$
However I have looked at several implementations of Kalman Filters for state space models I haven't seen this "de-meaned" version any where?
Moreover if you look at implementations such as this by Kevin Murphy he doesn't de-mean the state process either:
https://github.com/kmatzen/FullBNT-1.0.1/blob/master/Kalman/kalman_update.m
Thus Murphy (and most other authors) take the state space model to be i.e the state equation is NOT demeaned:
$z_t=Az_{t-1} + \epsilon_t$
$y_t=Cz_t + \delta_t$
However this model also includes an initial observation x0 which is distributed as $N(\xi, \Lambda)$. (Note that in the paper the choice over what x0 to use is not covered.)
Q1.) From a theoretical point of view are the two representations equivalent? Clearly they are if $\mu=0$ but more generally for non zero values of $\mu$?Does the mean of $x_0=\xi$ offset the process so that the rest of the observations cab be treated as mean zero? If so can someone show me how this makes the processes equivalent algebraically!
When I test the model I use data generated using the non demeaned version of the state equation, with the appropriate x_0, and use the toolbox to estimate it (which also does not demean the state equation) and I get good estimates. This is to be expected as $\mu$ in this case is zero by design. However when I work with empirical data I can't be sure that the empirical state process will be mean zero.
Q.2) Therefore should I always de-mean the state process when working with empirical data.
Kind Regards
BazShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.