<p>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</p>

<p>$$ \frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1 $$</p>

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

<p>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.</p>

CC BY-SA 3.0
2016-12-17T13:37:39.667Z
2016-12-17T13:37:39.667Z
Theoretical Economist
<p>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;
<a href="https://www.aeaweb.org/articles?id=10.1257/aer.104.10.3154" rel="nofollow noreferrer">https://www.aeaweb.org/articles?id=10.1257/aer.104.10.3154</a>.</p>
<p>However, I don't understand their specification for equation (9) (p. 11)
<span class="math-container">$\Delta y_{t} = \gamma^{\tau_{i}}+ \delta^{\tau_{i}}\boldsymbol \beta \boldsymbol X_{t}+\varepsilon_{t}$</span> in the paper. &quot;Where <span class="math-container">$\gamma^{\tau_{i}}$</span> and <span class="math-container">$ \delta^{\tau_{i}}$</span> are scalars that are allowed to take on different
values in each calendar year <span class="math-container">$i$</span>&quot;. <span class="math-container">$t$</span> indexex days and <span class="math-container">$\Delta y_{t}$</span> is the chagne in yield, <span class="math-container">$\boldsymbol X_{t}$</span> is a set of macroeconomin data surprise. They wrote</p>
<blockquote>
<p>&quot;We normalize the <span class="math-container">$\delta^{\tau_{i}}$</span> so that they have an average value of unity from 1990–2000, ...&quot;</p>
</blockquote>
<p>What chould they mean that the coefficients have an average value of one? Could it mean simply set <span class="math-container">$\delta^{\tau_{1990}} = 1$</span>, <span class="math-container">$\delta^{\tau_{1991}} = 1$</span> ... <span class="math-container">$\delta^{\tau_{2000}} = 1$</span>?</p>
<p>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.</p>

CC BY-SA 3.0
2016-12-17T04:59:03.680Z
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 $\frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1$?
Theoretical Economist
CC BY-SA 3.0
2016-12-17T05:05:09.280Z
Here we go. Alternative link to the [paper](http://www.ericswanson.us/papers/hfzb.pdf) 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.
Theoretical Economist
<macroeconomics><econometrics><statistics>
Swanson and Williams (2014): restricting based period response to one
user100525
# Swanson and Williams (2014): restricting based period response to one

(Asked by: username_0 on 2016-12-16)

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.

**Tags:** macroeconomics, econometrics, statistics

> **Comments:**
> **username_1**: 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 $\frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1$?  
> **username_1**: Here we go. Alternative link to the [paper](http://www.ericswanson.us/papers/hfzb.pdf) 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.  

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## 1 answers

### Answer by username_1 on 2016-12-17. Score: 2 | Accepted answer

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.


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