I'm currently working on replicating a scientific paper for practice in which they estimate a multivariate random walk plus noise model, apparently using OLS. I have no clue however how they would do this using OLS, so I would love some help!
The details:
Model: $\mathbf y_t=\mathbf \mu_t + \epsilon_t$, $\mu_t=\mu_{t-1}+\beta+\eta_t$, with both $\epsilon$ and $\eta$ being normally distributed and with $\mathbf y_t$ being a row vector with dimensions 12x1 and T=240.
Now, normally when using OLS in the multivariate case I'd use the famous equation $\beta=(X'X)^{-1}X'y$ to obtain the parameters, BUT I don't see how this would work when we have two different innovations, namely $\epsilon$ and $\eta$.
Any help would be lovely, since I'm trying to get the same results as in the research paper in order to try and grasp the methods :-). In case it is useful, the paper is "Seasonal Changes in Central England Temperatures - Proietti & Hillebrand (2015)" and the relevant pages for this question are from 9-10.
Okay, so I figured it out myself after a few hours of headaches. Incase anyone wondered: I just ran a KF and I set $\eta_t$ to 0 for all I time series, which in turn gave the correct table results.