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Considering following linear model \begin{equation} y_t = X_t f_t + \varepsilon_t, \qquad t=1,\cdots, T \end{equation} where $y_t\in\Re^{300\times 1}$ and $X_t\in\Re^{300\times 60}$ are two given sequences for $t=1,\cdots, T$. Error term $\varepsilon_t\sim \mathcal{N}(0,Q)$ and $Q$ is a $300\times 300$ unknown diagnal matrix.

I was wondering, is it possible to efficiently estimate the weights $W = Q^{-1}$ or the noise covariance of residuals? so that I could minimise the variance of the $\hat{f}_t$ by solving following weighted linear least squares: \begin{equation} \min_{f_t}\|y_t - X_t f_t\|^2_W \end{equation}

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  • $\begingroup$ I use Feasible generalized least squares to estimate the diagnal noise covariance iteratively: Feasible generalized least squares. It quickly converged within 50 iterations $\endgroup$
    – Stephen Ge
    Jan 6 at 2:38
  • $\begingroup$ Is there any structure on Q? If completely unrestricted, I don’t see how you’d estimate it. A typical strategy is to model Q as a parametric function of the regressors. $\endgroup$
    – David
    Jan 10 at 5:15

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