# Swanson and Williams (2014): restricting based period response to one

I would like to replicate the results as in Swanson and Williams (AER 2014): "Measuring the Effect of the Zero Lower Bound on Medium- and Longer-Term Interest Rates" 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. "Where $$\gamma^{\tau_{i}}$$ and $$\delta^{\tau_{i}}$$ are scalars that are allowed to take on different values in each calendar year $$i$$". $$t$$ indexex days and $$\Delta y_{t}$$ is the chagne in yield, $$\boldsymbol X_{t}$$ is a set of macroeconomin data surprise. They wrote

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

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.

• Could you post more details about what $\Delta y_t$ and $\mathbf{X}$ are? I'm having trouble accessing the article at the moment. I don't think they've restricted all the $\delta^{\tau_i}$'s to $1$, though. That would defeat the purpose of including them in the specification. Surely it just means that $\frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1$? Commented Dec 17, 2016 at 4:59
• Here we go. Alternative link to the paper on Swanson's website. I'll try to have a closer look at this, but I'm putting the link up here in case someone else comes across this question and is able to help. Commented Dec 17, 2016 at 5:05

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.