Logit and probit regressions

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This module reviews the basic concepts needed to estimate and understand logit and probit regressions using Stata. It is intended for advanced undergraduates.

Introduction

Consider a model that “explains” whether a wife enters the work force. It is straight forward to think of potential explanatory variables—her potential wage rate, the income of her partner, the number of children under the age of 6 in the household, and the number of children in the household between the ages of 6 and 18 are candidates to be independent variables used to explain the wife’s decision to enter the labor force. The dependent variable, Y , however, is a dummy variable because the wife chooses either to enter the labor force $\left(Y=1\right)$ or not to enter the labor force $\left(Y=0\right).$ An OLS model of the form:

${Y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i}+{\epsilon }_{i}$

does not make sense. Figure 1 shows what the data of this model might look like when graphed against one of the explanatory variables. Figure 1 also includes the regression line that an OLS estimation of (1) will yield. It is easy to see one problem with this approach—the predicted values of Y that can be greater than 1 and less than 0. In addition, special properties must be attributed to the error term and it is the simple properties ascribed to the error term that make the OLS model so attractive. J. S. Cramer (2003) Logit Models from Economics and Other Fields (Cambridge: Cambridge University Press): 10. The linear regression line can be a poor representation of a discrete dependent variable.

The logit model

There does exist another approach to the modeling problem—assume that the dependent variable is the probability that the wife is in the labor force . For instance we might assume that we have a linear probability model of the form $\mathrm{Pr}\left({x}_{i}\right)={\beta }_{0}+{\beta }_{1}{x}_{i}+{\epsilon }_{i}.$ This model can be estimated reasonably successfully if the observed frequencies are well away from their bounds of 0 and 1. For a full discussion of this model see Ladd, G. W. (1966) “Linear Probability Functions and Discriminant Functions,” Econometrica 34 : 873-888. However, is more appealing to assume that the probability varies monotonically with x and remains within the bounds of [0,1], as shown in Figure 2. This S-shaped curve is known as the sigmoid curve and can be represented algebraically for some variable z by: $\mathrm{Pr}\left(z\right)=\frac{{e}^{z}}{1+{e}^{z}}.$ The signoid function forces the dependent variable to be between 0 and 1.

We can simplify our analysis by using a bit of algebra. First, the inverse probability is $1-\mathrm{Pr}\left(z\right)=1-\frac{{e}^{z}}{1+{e}^{z}}=\frac{1}{1+{e}^{z}}.$ Thus,

$\frac{\mathrm{Pr}\left(z\right)}{1-\mathrm{Pr}\left(z\right)}=\frac{\frac{{e}^{z}}{1+{e}^{z}}}{\frac{1}{1+{e}^{z}}}={e}^{z}.$

Taking the natural logarithm of (2) gives $\mathrm{ln}\left(\frac{\mathrm{Pr}\left(z\right)}{1-\mathrm{Pr}\left(z\right)}\right)=z.$ Assuming that z is a linear function of x (and, more generally, of other variables) gives the logit model:

$\mathrm{ln}\left(\frac{\mathrm{Pr}\left({x}_{i}\right)}{1-\mathrm{Pr}\left({x}_{i}\right)}\right)={\beta }_{0}+{\beta }_{1}{x}_{i}+{\epsilon }_{i}.$

We can estimate the parameters of this model using maximum likelihood methods . In the probit model the error term is assumed to be normally distributed with a mean of zero and a unit variance. The assumption that the variance is equal to 1 is due to technical considerations. See [Cramer, 22]. In the logit model the error term is assumed to have a standardized logistic distribution . This distribution has a mean of 0 and a variance of 1 and is very similar to a normal distribution with the same mean and variance. The pdf of a logistic distribution is $f\left(x\right)=\frac{\lambda {e}^{-\lambda x}}{{\left(1+{e}^{-\lambda x}\right)}^{2}}$ , where $\lambda =\frac{\pi }{\sqrt{3}}\approx 1.814$ . See Cramer, 24-26 for a fuller discussion of the logistic distribution. While the choice of which model to use generally is personal, it should be noted that the ratio of the parameter of a logit model to the parameter of a probit model (using the same data set) usually varies between 1.6 and 2.0. We focus on the logit model in the balance of this discussion.

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