Estimating Measurement Noise in a Two-Factor Kalman Model
Summary
The document discusses maximum-likelihood estimation for a linear Gaussian two-factor state-space model of commodity prices, using the Schwartz-Smith framework. The observed log price is modeled as the sum of two latent state components plus measurement error, while the states evolve according to a transition matrix with process noise. The question is how to interpret the observation-error variance in an economic setting.
The response treats both state-process variance and measurement variance as quantities to estimate from data alongside the model parameters. Their relative size indicates how much observed movement is attributed to changing latent states versus noise: low process variance relative to observation variance implies noisy observations and slowly moving states, while the reverse implies observations closely reflect the states. The reported experiments illustrate that very small measurement variance can force the filter to track observations closely and yield extreme likelihood values, while large variance can permit implausible parameter regions. These are qualitative cautions; the document gives no calibration procedure, numerical diagnostics, or safeguards for maximum-likelihood optimization.
Key ideas
- A two-factor commodity-price model can represent observed prices as latent states plus measurement noise.
- The observation-error variance is estimated from data rather than assigned a universal economic meaning.
- The ratio of process noise to observation noise describes how much observed variation is attributed to state changes.
- Very small observation variance can make likelihood calculations numerically challenging, while large variance can weaken state recovery.
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# Maximum Likelihood using a Kalman filter for two factor model
# Maximum Likelihood using a Kalman filter for two factor model
I'm trying to implement a Kalman Filter for the parameter estimation of a linear gaussian two factor model in Matlab. (Schwartz Smith model for commodity prices) In other words: I try to compute the log-likelihood of the parameters.
My model:
$X_t = A X_{t-1} + \epsilon_X$ , with X beeing two-dimensional.
$Y_t = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^T X_t + \epsilon_Y$ , with Y beeing one-dimensional.
$A$ is time invariant and only depends on the parameters $\theta$, which I would like to determine.
My question:
Whats the variance of $\epsilon_Y$ ? I know, that it normally represents the noise of the measurement process, but I don't know what would be the equivalent in an economic context?
I managed to implement the filter and the results for experimenting with the variance are (mean = 0):
- For very (!) small variance: Reasonable results close to real values, but extremly small log-likelihoods around $-10^{12}$. I fear, that this will cause numerical issues in the upcoming maximum likelihood estimation. (I plan to use Metropolis Hastings)
Any help would be appreciated! Thanks!
## Answer by Denis (score 2)
https://quant.stackexchange.com/a/9579
Usually $var(e_x), var(e_y)$ variances are calibrated by maximum likelihood from data similar as you want to calibrate your parameters $\theta$.
Ratio $var(e_x)/var(e_y)$ tells you what are changes in your time-series
- $var(e_x)/var(e_y)$ is small: changes in time-series of observations are just noise and underlying state doesn't change much;
- $var(e_x)/var(e_y)$ is large: changes are due to change of state and observations data represent states almost exactly, without noise.
What you describe is reasonable:
- "For high variance: High log-likelihood in unreasonable areas far away from the real values" - if noise is high it is likely to have observations in a wide band around realized observations. In that case if you filter with KF it will smooth your data hardly to recover true states.
- "For very (!) small variance: Reasonable results close to real values, but extremly small log-likelihoods around" - if noise variance is small, then your observations are unlikely to deviate anywhere from expected and this means you assume that your realized observations are almost exactly reflect your states so if you filter with KF it will pass via all you observations without any smoothing.Shown 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.