# Kalman Filter Estimate vs ACF Least Squares Estimate

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?