I am having trouble relating the estimate of the standard deviation to the precision of the estimate of the $y$-variable. For example:

$$\tilde{y}_t=a_{y,1}\,\tilde{y}_{t-1}+a_{y,2}\,\tilde{y}_{t-2}+\frac{a_r}{2} \sum_{j=1}^{2}r_{t-j}- r^*_{t-j}+ \varepsilon_{\tilde{y},t}$$

The error is assumed to be normally distributed with expectation zero and an estimated standard deviation of 0.931 (via maximum likelihood). Note that $\tilde{y}$ is estimated via the Kalman filter. What does this standard deviation tell me in this setting?


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