Kalman Filter Estimate vs ACF Least Squares Estimate

Question

I am currently reading Chapter 5, Applications to the Gas Markets, in Stochastic Modelling of Electricity and Related Markets by Benth, Benth and Koekebakker, World Scientific, 2008.

In the subchapter 5.3.1 Kalman filtering they assume that the detrended and deseasonalized logarithmic spot price dynamics for gas follow the process $X_1(t) + X_2(t)$ with $$dX_i(t) = -\alpha_i X_i(t) + \sigma_i dB_i(t)$$ for $i=1,2$ with two independent Brownian motions $B_1, B_2$.

They cite Barlow, Gusev and Lai, 2004, which have shown that the confidence bands on the mean reversion estimates are quite wide. For this reason, they are proposing to use the empirical ACF to estimate the mean reversion parameters, assuming stationarity.

The stationary ACF of $X_1(t) + X_2(t)$ is given by $$ \rho(\tau) = \hat{\omega}_1 e^{-\alpha_1 \tau} + \hat{\omega}_2 e^{-\alpha_2 \tau}$$ with the lag $\tau$ and the coefficients $$\hat{\omega}_i = \frac{\frac{\sigma_i^2}{2\alpha_i}}{\frac{\sigma_1^2}{2\alpha_1}+\frac{\sigma_2^2}{2\alpha_2}}$$ for $i=1,2$. Using least squares they arrive at the values $\hat{\omega}_1 = 0.73$, $\hat{\omega}_2 = 0.27$, $\alpha_1 = 0.02$, $\alpha_2 = 0.28$. Using $$\sigma_i^2 = 2\alpha_i \times \text{Var}(X_1(t)+X_2(t))$$ for $i = 1,2$ and estimating $\text{Var}(X_1(t)+X_2(t))$ to be $0.11$ they get as a starting point for the Kalman filter $\sigma_1 = 0.057$ and $\sigma_2 = 0.129$, which then yields $\hat{\sigma}_1 = 0.065$ and $\hat{\sigma}_2 = 0.573$.

How can the difference between the Kalman filter estimates $\hat{\sigma}_1$ and $\hat{\sigma}_2$ and the implicit least squares ACF values for $\sigma_1$ and $\sigma_2$ be explained? Where does it come from and why would the Kalman filter estimate be the right one in contrast to the ACF least squares estimate? In general, is the difference large?

Answer 1

As noted on p. 11 of this site: http://www.pages.drexel.edu/~pyo22/mem640/lecture04-Estimation/mem640Lecture-Estimation.pdf least squares is a "special" case of Kalman Filtering. Additional details can also be found on other web sites, such as: http://www.insidegnss.com/node/3445

Also, as noted in both links above - although least squares is a special case of Kalman Filtering - the two algorithms differ from each other. Note that the 'weighting matrix' W reduces to the identity matrix in the least squares case. So you might expect that the results would also differ when you process the data.

As to which one will be more accurate - it all depends on how closely the 'signal processing model' of your algorithm matches the process that generates your input data. I've programmed Kalman Filters whose results were completely wrong because: 1) it started with a bad first guess, or 2) the state model being used was wrong, or 3) the input data was far too noisy. I've also programmed 'hypothesis testing' arrays of Kalman Filters, where all filters operated on the same data, but each filter was set up to match a particular target motion (ie: zig right, zig left, go straight, etc.) - in that case, only the filter that closely predicted the actual input data was the correct one (ie: residuals were very small), and the residuals of all the others were way out of range.