Logit and probit regressions

Interpretation of the logit model parameters

The interpretation of the economic meaning of the parameter values in a logit model is not very obvious. See Stata Library, Categorical and Count Data Analysis Utilities for useful utilities and an excellent discussion of how to interpret categorical and count regression results at http://www.ats.ucla.edu/stat/stata/library/longutil.htm/ (accessed July 19, 2009). One simple, but not often used, interpretation comes from taking the first-derivative of (3) with respect to x :

Thus, in the labor force participation model one interpretation is that ${\beta }_1$ is equal to the change in the natural logarithm of the odds that the wife is in the labor force due to a one unit change in the independent variable x. This interpretation is both awkward and not really economically informative.

Stata offers two command for estimating a logit regression—logit and logistic. The logit command returns the parameter estimates as shown in (3). The logistic command returns the odds ratio rather than the parameter estimates. The odds ratio is equal to $e^{{\beta }_1}$ . Thus, one can go from the odds ratio reported by the logistic command to the parameter estimates merely by taking the natural logarithm of the odds ratio. The interpretation of the odds ratio is straightforward. For example, assume that $y=1$ means that the birth weight of an individual is less than 2,500 grams and $y=0$ means that the birth weight is greater than 2,500 grams. A logit parameter estimate of -0.27 is equivalent to an odds ratio of 0.97 (i.e., $e^{-0.27}=0.97$ ). An odds ratio of 0.97 means that odds of a baby being underweight are 0.97 times those of the odds of a baby being of normal weight. To see what is being said re-write (2.3) as:

$$\frac{\mathrm{Pr}\left(x\right)}{1-\mathrm{Pr}\left(x\right)}=e^{{\beta }_0+{\beta }_{1}x+\epsilon }.$$

A one unit change in x implies that:

$$\frac{\mathrm{Pr}\left(x+1\right)}{1-\mathrm{Pr}\left(x+1\right)}=e^{{\beta }_0+{\beta }_1\left(x+1\right)+\epsilon }$$

or

$$\frac{\mathrm{Pr}\left(x+1\right)}{1-\mathrm{Pr}\left(x+1\right)}=e^{{\beta }_0+{\beta }_{1}x+\epsilon }{e}^{{\beta }_1}$$

or

$$\frac{\mathrm{Pr}\left(x+1\right)}{1-\mathrm{Pr}\left(x+1\right)}=e^{{\beta }_1}\left(\frac{\mathrm{Pr}\left(x\right)}{1-\mathrm{Pr}\left(x\right)}\right).$$

Thus, $e^{{\stackrel{⌢}{\beta }}_1}$ is equal to the percent change in the odds that y equals 1 (a baby is born underweight) due to a one unit change in x . The logistic command reports $e^{{\stackrel{⌢}{\beta }}_1}$ while the logit command reports ${\stackrel{⌢}{\beta }}_1.$ Because of the ease of interpretation of the odds ratio, Stata argues that the logistic command is the proper one to use.

Elasticities

Another route to follow is to try to find something that can be interpreted as an elasticity. Elasticities are important enough topic in economics for us to discuss them here in some detail. The reason they are so attractive to economists is that they have no units and, thus, can be compared across different commodities. For instance, it is quite reasonable to compare the demand elasticity for apples with the demand elasticity for pearl necklaces in spite of the fact that the units of measuring apples and necklaces are different. There are a few important ways that elasticities appear in regressions.

Linear regression elasticities

In a linear regression of the form (ignoring the subscripts and the error term)

$$Y={\beta }_0+{\beta }_{1}x,$$

we would calculate the elasticity of Y with respect to x to be