<p>Logit and probit regressions do not help with endogeneity at all. What they do is bound the predicted value (aka conditional expectation) of your response variable between 0 and 1. As such they are appropriate for performing regression analysis for binary outcomes like:</p>

<ol>
<li>Whether someone wins or loses a sports game</li>
<li>Whether someone is going to be employed or unemployed</li>
</ol>

<p>To understand this, consider the univariate regression model</p>

<p><span class="math-container">$$y_i = \beta x_i + u_i$$</span></p>

<p>Say <span class="math-container">$y_i = 1$</span> if player <span class="math-container">$i$</span> wins a tennis match and <span class="math-container">$y_i$</span> = 0 if a player loses a tennis match, and you are trying to work out whether they will win based on the difference between their world ranking and that of their opponent, <span class="math-container">$x_i$</span>.</p>

<p>If you perform ordinary linear regression you might get an estimate of <span class="math-container">$\beta$</span> which is, say, <span class="math-container">$0.05$</span>. Perhaps most of your observations for <span class="math-container">$x_i$</span> (i.e. the difference in world rankings) are between <span class="math-container">$0$</span> and <span class="math-container">$20$</span>, which means that your predicted value of <span class="math-container">$y_i$</span> (i.e. <span class="math-container">$\beta x_i$</span>) is between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. You can then easily say whether you are predicting a win or a loss for your player depending on whether that predicted value is greater than or less than <span class="math-container">$0.5$</span>. However there may be certain situations where the difference in world rankings is <span class="math-container">$&gt;20$</span>, in which case you will get a prediction >1.</p>

<p>In one sense this is unproblematic because based on your model this player is clearly the forecast winner. However you may prefer not to have unbounded predicted values, in which case a probit or logit transformation or model is appropriate.</p>

<p>Taking probit as an example, you specify that the probability that a player wins, conditional on the difference in rankings is not <span class="math-container">$\beta x_i$</span> as before. Instead you state</p>

<p><span class="math-container">$$p(y_i = 1 | x_i) = \Phi(\beta x_i)$$</span></p>

<p>where <span class="math-container">$\Phi$</span> is the cumulative distribution function (CDF) of the normal distribution. As you can see in the normal CDF diagram <a href="https://en.wikipedia.org/wiki/Normal_distribution" rel="nofollow noreferrer">here</a>, no matter how high or low the value of <span class="math-container">$\beta x_i$</span> is, the variable on the y-axis never exceeds <span class="math-container">$1$</span>!</p>

<p>In addition, if the distribution of your error term <span class="math-container">$u_i$</span> is logistically distributed and you use logit, or normally distributed and you use probit, your estimation technique will be more efficient (i.e. you will have smaller standard errors) than if you performed the simple linear regression. This point is more complex though so I don't provide a proof.</p>

CC BY-SA 4.0
2020-03-21T20:41:37.373Z
2020-03-21T20:41:37.373Z
Slom
<p>How does the use of a logistic (or probit) regression help address the endogeneity concerns: explanatory variable is correlated with error term? </p>

CC BY-SA 4.0
2020-03-05T18:55:30.627Z
Do you have some reason to think that it does? It seems like it shouldn't generally help with endogeneity.
BKay
<econometrics>
Logistic (or probit) regression
AKS
# Logistic (or probit) regression

(Asked by: username_0 on 2020-03-05)

How does the use of a logistic (or probit) regression help address the endogeneity concerns: explanatory variable is correlated with error term?

**Tags:** econometrics
> **Comments:**
> **username_1**: Do you have some reason to think that it does? It seems like it shouldn't generally help with endogeneity.  

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## 1 answers

### Answer by username_2 on 2020-03-21. Score: 1

Logit and probit regressions do not help with endogeneity at all. What they do is bound the predicted value (aka conditional expectation) of your response variable between 0 and 1. As such they are appropriate for performing regression analysis for binary outcomes like:

1. Whether someone wins or loses a sports game
2. Whether someone is going to be employed or unemployed

To understand this, consider the univariate regression model

$$y\_i = \beta x\_i + u\_i$$

Say $y\_i = 1$ if player $i$ wins a tennis match and $y\_i$ = 0 if a player loses a tennis match, and you are trying to work out whether they will win based on the difference between their world ranking and that of their opponent, $x\_i$.

If you perform ordinary linear regression you might get an estimate of $\beta$ which is, say, $0.05$. Perhaps most of your observations for $x\_i$ (i.e. the difference in world rankings) are between $0$ and $20$, which means that your predicted value of $y\_i$ (i.e. $\beta x\_i$) is between $0$ and $1$. You can then easily say whether you are predicting a win or a loss for your player depending on whether that predicted value is greater than or less than $0.5$. However there may be certain situations where the difference in world rankings is $>20$, in which case you will get a prediction >1.

In one sense this is unproblematic because based on your model this player is clearly the forecast winner. However you may prefer not to have unbounded predicted values, in which case a probit or logit transformation or model is appropriate.

Taking probit as an example, you specify that the probability that a player wins, conditional on the difference in rankings is not $\beta x\_i$ as before. Instead you state

$$p(y\_i = 1 | x\_i) = \Phi(\beta x\_i)$$

where $\Phi$ is the cumulative distribution function (CDF) of the normal distribution. As you can see in the normal CDF diagram [here](https://en.wikipedia.org/wiki/Normal_distribution), no matter how high or low the value of $\beta x\_i$ is, the variable on the y-axis never exceeds $1$!

In addition, if the distribution of your error term $u\_i$ is logistically distributed and you use logit, or normally distributed and you use probit, your estimation technique will be more efficient (i.e. you will have smaller standard errors) than if you performed the simple linear regression. This point is more complex though so I don't provide a proof.


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