Swanson and Williams (2014): restricting based period response to one I would like to replicate the results as in Swanson and Williams (AER 2014): &quot;Measuring the Effect of the Zero Lower Bound on Medium- and Longer-Term Interest Rates&quot;
https://www.aeaweb.org/articles?id=10.1257/aer.104.10.3154.
However, I don't understand their specification for equation (9) (p. 11)
$\Delta y_{t} = \gamma^{\tau_{i}}+ \delta^{\tau_{i}}\boldsymbol \beta \boldsymbol X_{t}+\varepsilon_{t}$ in the paper. &quot;Where $\gamma^{\tau_{i}}$ and $ \delta^{\tau_{i}}$ are scalars that are allowed to take on different
values in each calendar year $i$&quot;. $t$ indexex days and $\Delta y_{t}$ is the chagne in yield, $\boldsymbol X_{t}$ is a set of macroeconomin data surprise. They wrote

&quot;We normalize the $\delta^{\tau_{i}}$ so that they have an average value of unity from 1990–2000, ...&quot;

What chould they mean that the coefficients have an average value of one? Could it mean simply set $\delta^{\tau_{1990}} = 1$, $\delta^{\tau_{1991}} = 1$ ... $\delta^{\tau_{2000}} = 1$?
They have also posted their MATLAB code on the (bottom of the) website. But I still don't fully understand the specification. Would be nice if someone can help me out here.
I'm no empiricist (surprise), but skimming through the paper, I'm pretty sure that just means that the $\delta^{\tau_i}$'s are chosen so that

$$ \frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1 $$

as I stated in a comment above. This assumption is necessary in order to separately identify (estimate) $\delta^{\tau_i}$ and $\mathbf{\beta}$.

Without this assumption, all you are able to estimate is $\delta^{\tau_i}\mathbf{\beta}$, so you need additional assumptions to solve for the two factors separately.