regression model $m(t) = \beta(l)z(t-l) $, the lag regression coefficient $\beta$ is, $$\renewcommand{\theequation}{6}
\beta(l) = \frac{\gamma_{zm}(l)}{\gamma_{z}(0)}$$

With Equation 3, the right hand side of Equation 6 can be decomposed into two parts: $$\renewcommand{\theequation}{7}
\beta(l) = \frac{\gamma_{z\tilde{m}}(l)}{\gamma_{z}(0)} +b\frac{\gamma_{z}(l)}{\gamma_{z}(0)},$$ in which the first term on the right hand side [is negligible]{} if $z$ is decorrelated with $\tilde{m}$ beyond lag $l$ days, and therefore the feedback strength can be estimated as,

$$\renewcommand{\theequation}{8}
b_{S} = \beta(l) \frac{\gamma_{z}(0)} {\gamma_{z}(l)}$$

Figure 8b shows the strength of eddy-jet feedback calculated using Equation 8, with 95$\%$ confidence intervals estimated with bootstrapping as in Section 4b. While the margin of error grows with lag time, the strength of eddy-jet feedback is largely underestimated, and the bias results from the quasi-oscillatory nature of the eddy forcing. Using lag regression, we are also able to estimate the pattern of anomalous eddy fluxes associated with the annular mode. The pressure-latitude distribution of eddy flux anomaly generally agrees with the results from LRF, with a pattern correlation over 0.9 through a wide range of lag days (figures not shown).

Low-pass filtering
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The bias with lag regression suggests that the correlation between $\tilde{m}$ and $z$ is not negligible [relative to the correlation between 