1.7.4.md

Solution to Review Question

by Qiang Gao, updated at May 20, 2017

---

Chapter 1 Finite-Sample Properties of OLS

Section 7 Application: Returns to Scale in Electricity Supply

...

Review Question 1.7.4 (Recovering left-out coefficients from restricted OLS)

Calculate the restricted OLS estimate of $$\beta_5$$ from

$$ \log \left( \frac{ TC_i }{ p_{i3} } \right) =

• \underset{ (0.88) }{ 4.7 } + \underset{ (0.017) }{ 0.72 } \log ( Q_i ) + \underset{ (0.20) }{ 0.59 } \log \left( \frac{ p_{i1} }{ p_{i3} } \right) - \underset{ (0.19) }{ 0.007 } \log \left( \frac{ p_{i2} }{ p_{i3} } \right). \tag{1.7.8} $$

How do you calculate the standard error of $$ b_5 $$ from the printout of the restricted OLS?

Solution

Because of the restriction

$$ \beta_3 + \beta_4 + \beta_5 = 1, $$

the restricted OLS estimate of $$ \beta_5 $$ is

$$ b_5 = 1 - b_3 - b_4 = 1 - 0.59 - (- 0.007) = 0.417. $$

We can write

$$ b_5 = 1 + \mathbf{c}' \mathbf{b}, $$

where

$$ \mathbf{c} \equiv \begin{bmatrix} 0 \ 0 \ -1 \ -1 \end{bmatrix}, \qquad \mathbf{b} \equiv \begin{bmatrix} b_1 \ b_2 \ b_3 \ b_4 \end{bmatrix}, $$

then

$$ \mathrm{Var} ( b_5 \mid \mathbf{X} ) = \mathrm{Var} ( 1 + \mathbf{c}' \mathbf{b} | \mathbf{X}) = \mathbf{c}' \mathrm{Var} ( \mathbf{b} \mid \mathbf{X} ) \mathbf{c}. $$

From the printout of the restricted OLS regression, we have the estimate $$ \widehat{ \mathrm{Var} ( \mathbf{b} \mid \mathbf{X} ) } $$, then we can calculate the standard error of $$b_5$$ as

$$ \mathrm{SE} ( b_5 ) = \sqrt{ \mathbf{c}' \widehat{ \mathrm{Var} ( \mathbf{b} \mid \mathbf{X} ) } \mathbf{c} }. $$

---

Copyright ©2017 by Qiang Gao