<p>I have a question related to IV methods. Suppose I have a regression </p>

<p><span class="math-container">$Y = a + b_1 X_1 + b_2 X_2 + \epsilon$</span>, </p>

<p>where <span class="math-container">$Y$</span> is a dependent variable and <span class="math-container">$X_1$</span> is an endogenous independent variable and <span class="math-container">$X_2$</span> is an exogenous one. I have found a variable <span class="math-container">$Z$</span> as the instrumental variable for <span class="math-container">$X$</span>. The question is as follows. Suppose I now set up new variables such that <span class="math-container">$X_1^l(X_1^h) = 1(0)$</span> if <span class="math-container">$X_1 \leq(&gt;) a_1$</span> and <span class="math-container">$X_2^l(X_2^h) = 1(0)$</span> if <span class="math-container">$X_2 \leq(&gt;) a_2$</span>. The my regression becomes</p>

<p><span class="math-container">$Y = a + c_1 X_1^l X_2^l + c_2 X_1^l X_2^h + c_3 X_1^h X_2^l + c_4 X_1^h X_2^h + \epsilon$</span>.</p>

<p>I want to know how to set up IV in this case. Thanks. </p>

CC BY-SA 4.0
2019-04-10T13:10:12.347Z
Two ideas: (i) Use $Z_1^j Z_2^k$ as IV's. (ii) Use the control function approach.
chan1142
CC BY-SA 4.0
2019-04-10T17:56:41.773Z
Do you mean $X_1^l Z^l$, $X_1^h Z^l$, $X_1^l Z^h$, $X_1^h Z^h$?
Zhu Chen
CC BY-SA 4.0
2019-04-11T13:52:46.397Z
Oh, $X_1$ is endogenous and $X_2$ is exogenous in your question. Then $Z^j X_2^k$. But I think the control function approach is perhaps better. Please also note that your are in the dummy variable trap. You will need to remove one variable (e.g., $X_1^l X_2^l$).
chan1142
<econometrics><instrumental-variable>
instrumental variables
Zhu Chen
# instrumental variables

(Asked by: username_0 on 2019-04-10)

I have a question related to IV methods. Suppose I have a regression

$Y = a + b\_1 X\_1 + b\_2 X\_2 + \epsilon$,

where $Y$ is a dependent variable and $X\_1$ is an endogenous independent variable and $X\_2$ is an exogenous one. I have found a variable $Z$ as the instrumental variable for $X$. The question is as follows. Suppose I now set up new variables such that $X\_1^l(X\_1^h) = 1(0)$ if $X\_1 \leq(>) a\_1$ and $X\_2^l(X\_2^h) = 1(0)$ if $X\_2 \leq(>) a\_2$. The my regression becomes

$Y = a + c\_1 X\_1^l X\_2^l + c\_2 X\_1^l X\_2^h + c\_3 X\_1^h X\_2^l + c\_4 X\_1^h X\_2^h + \epsilon$.

I want to know how to set up IV in this case. Thanks.

**Tags:** econometrics, instrumental-variable

> **Comments:**
> **username_1**: Two ideas: (i) Use $Z_1^j Z_2^k$ as IV's. (ii) Use the control function approach.  
> **username_0**: Do you mean $X_1^l Z^l$, $X_1^h Z^l$, $X_1^l Z^h$, $X_1^h Z^h$?  
> **username_1**: Oh, $X_1$ is endogenous and $X_2$ is exogenous in your question. Then $Z^j X_2^k$. But I think the control function approach is perhaps better. Please also note that your are in the dummy variable trap. You will need to remove one variable (e.g., $X_1^l X_2^l$).  

---

## 0 answers

