What are the limitations of the linear probability model (LPM)? Explain how these limitations are taken care of by the probit model.

The Linear Probability Model (LPM) is a commonly used method for estimating binary outcome models in econometrics. Despite its simplicity and ease of interpretation, LPM has several limitations that can lead to unreliable results. Understanding these limitations and how the probit model addresses them is crucial for econometric analysis.

1. Predicted Probabilities Outside the [0, 1] Range

The most fundamental limitation of the LPM is that it can predict probabilities outside the [0, 1] range. Since the LPM assumes a linear relationship between the independent variables and the probability of the dependent event, there are no inherent restrictions on the predicted values. This means that for some values of the explanatory variables, the model can predict probabilities less than 0 or greater than 1, which is nonsensical because probabilities must lie within this range. 

For instance, if the LPM predicts a probability of 1.2 or -0.3, this would be an impossible outcome. The LPM fails to capture the natural boundaries that probabilities should respect.

The probit model addresses this limitation by using a cumulative normal distribution function to model the probability. This ensures that the predicted probabilities are always constrained within the [0, 1] range. The probit model applies a non-linear transformation to the linear prediction, mapping the predicted values into a _______ ____ ________ ______ ___ ________ ______ ____ _____ _____ __________ __________.
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