<p>Let's take an example. Assume there are 5 periods, then the dummies take the following values:
<span class="math-container">$$
\begin{align*}
&amp;d_1 = \begin{bmatrix} 1 &amp;0 &amp;0 &amp;0 &amp;0 \end{bmatrix}\\
&amp;d_2 = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 \end{bmatrix}\\
&amp;d_3 = \begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\end{bmatrix}\\
&amp;d_4 = \begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\end{bmatrix}\\
&amp;d_5 = \begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}
\end{align*}
$$</span>
Now for <span class="math-container">$t = 3,\ldots, 5$</span> we define the vector:
<span class="math-container">$$
d^\ast_t = d_t + (t-2) d_1 - (t-1)d_2.
$$</span>
This gives:
<span class="math-container">$$
\begin{align*}
d^\ast_3 = \begin{bmatrix} 1 &amp; -2 &amp; 1 &amp; 0 &amp; 0\end{bmatrix}\\
d^\ast_4 = \begin{bmatrix} 2 &amp; -3 &amp; 0 &amp; 1 &amp; 0\end{bmatrix}\\
d^\ast_5 = \begin{bmatrix} 3 &amp; -4 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}
\end{align*}
$$</span>
For every vector <span class="math-container">$d^\ast_t$</span> the coefficients sum to zero.</p>
<p>In general, the sum of the coefficients in <span class="math-container">$d^\ast_t$</span> will be equal to:
<span class="math-container">$$
-(t-1) + (t-2) + 1 = 0.
$$</span></p>

CC BY-SA 4.0
2024-01-20T12:27:21.327Z
2024-01-20T12:27:21.327Z
tdm
<p>paper :<a href="https://www.journals.uchicago.edu/doi/epdf/10.1086/696225" rel="nofollow noreferrer">https://www.journals.uchicago.edu/doi/epdf/10.1086/696225</a></p>
<p>deaton imposes linear constraints in order to deal with the collinearity of age-period-cohort. How to implement it through stata?</p>
<p>Why  can the coefficients of <span class="math-container">$d_{t}^{*}=d_{t}-[(t-1)d_{2}-(t-2)d_{1}] $</span> sum to 0?</p>
<p><a href="https://i.sstatic.net/JLHJY.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JLHJY.png" alt="The analysis of household surveys: A microeconometric approach to development policy (p126)" /></a></p>
<p>Image Source:The analysis of household surveys: A microeconometric approach to development policy (p126)</p>

<mathematical-economics><labor-economics><stata>
Ask about the HLT method of "Life cycle wage growth across countries"
Barry Alen
# Ask about the HLT method of "Life cycle wage growth across countries"

(Asked by: username_0 on 2024-01-19)

paper :<https://www.journals.uchicago.edu/doi/epdf/10.1086/696225>

deaton imposes linear constraints in order to deal with the collinearity of age-period-cohort. How to implement it through stata?

Why can the coefficients of $d\_{t}^{\*}=d\_{t}-[(t-1)d\_{2}-(t-2)d\_{1}] $ sum to 0?

[![The analysis of household surveys: A microeconometric approach to development policy (p126)](https://i.sstatic.net/JLHJY.png)](https://i.sstatic.net/JLHJY.png)

Image Source:The analysis of household surveys: A microeconometric approach to development policy (p126)

**Tags:** mathematical-economics, labor-economics, stata


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## 1 answers

### Answer by username_1 on 2024-01-20. Score: 3

Let's take an example. Assume there are 5 periods, then the dummies take the following values:
$$
\begin{align\*}
&d\_1 = \begin{bmatrix} 1 &0 &0 &0 &0 \end{bmatrix}\\
&d\_2 = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \end{bmatrix}\\
&d\_3 = \begin{bmatrix} 0 & 0 & 1 & 0 & 0\end{bmatrix}\\
&d\_4 = \begin{bmatrix} 0 & 0 & 0 & 1 & 0\end{bmatrix}\\
&d\_5 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \end{bmatrix}
\end{align\*}
$$
Now for $t = 3,\ldots, 5$ we define the vector:
$$
d^\ast\_t = d\_t + (t-2) d\_1 - (t-1)d\_2.
$$
This gives:
$$
\begin{align\*}
d^\ast\_3 = \begin{bmatrix} 1 & -2 & 1 & 0 & 0\end{bmatrix}\\
d^\ast\_4 = \begin{bmatrix} 2 & -3 & 0 & 1 & 0\end{bmatrix}\\
d^\ast\_5 = \begin{bmatrix} 3 & -4 & 0 & 0 & 1\end{bmatrix}
\end{align\*}
$$
For every vector $d^\ast\_t$ the coefficients sum to zero.

In general, the sum of the coefficients in $d^\ast\_t$ will be equal to:
$$
-(t-1) + (t-2) + 1 = 0.
$$


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