How does the use of a logistic (or probit) regression help address the endogeneity concerns: explanatory variable is correlated with error term?
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:
- Whether someone wins or loses a sports game
- 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, 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.