Logit and probit regressions

$${\eta }_{Yx}=\frac{x}{Y}\frac{\partial Y}{\partial x}={\beta }_1\frac{x}{Y}.$$

Clearly, researchers need to choose the levels of Y and x at which to report this elasticity; it is traditional to calculate the elasticity at the means. Thus, economists typically report

$${\eta }_{Yx}={\beta }_1\frac{\overline{x}}{\overline{Y}}.$$

Constant elasticities

Consider the following demand equation:

where q is the quantity demanded, p is the price the good is sold at, $\alpha ,\beta >0,$ and $\epsilon$ is an error term. The price elasticity of demand is given by

$${\eta }_{qp}=\frac{p}{q}\frac{\partial q}{\partial p}=\frac{p}{\alpha p^{-\beta }{e}^{\epsilon }}\left(-\beta \alpha p^{-\beta -1}{e}^{\epsilon }\right)=-\beta .$$

In other words, this demand curve has a constant price elasticity of demand equal to $-\beta .$ Moreover, we can convert the estimation of this equation into a linear regression by taking the natural logarithm of both sides of (5) to get $\ln q=\ln \alpha -\beta \ln p+\epsilon .$

The logit equation and the quasi-elasticity

It is not appropriate to use the normal formula for an elasticity with (3) because the dependent variable is itself a number without units between 0 and 1. As an alternative it makes more sense to calculate the quasi-elasticity , which is defined as:

Since

$$\ln \left(\frac{\mathrm{Pr}\left(x_i\right)}{1-\mathrm{Pr}\left(x_i\right)}\right)={\beta }_0+{\beta }_1{x}_i+{\epsilon }_i,$$

we can calculate this elasticity as follows:

$$\frac{\partial \left(\ln \left(\frac{\mathrm{Pr}\left(x_i\right)}{1-\mathrm{Pr}\left(x_i\right)}\right)\right)}{\partial x}={\beta }_1.$$

Focusing on the left-hand-side, we get:

$$\frac{1-\mathrm{Pr}\left(x_i\right)}{\mathrm{Pr}\left(x_i\right)}\frac{\left(1-\mathrm{Pr}\left(x_i\right)\right)\frac{\partial \mathrm{Pr}\left(x_i\right)}{\partial x}+\mathrm{Pr}\left(x_i\right)\frac{\partial \mathrm{Pr}\left(x_i\right)}{\partial x}}{{\left(1-\mathrm{Pr}\left(x_i\right)\right)}^2}={\beta }_1$$

or

$$\frac{1}{\mathrm{Pr}\left(x_i\right)\left(1-\mathrm{Pr}\left(x_i\right)\right)}\frac{\partial \mathrm{Pr}\left(x_i\right)}{\partial x}={\beta }_1$$

or

Thus, we see from (6) that the quasi-elasticity is given by:

The quasi-elasticity measures the percentage point change in the probability due to a 1 percent increase of x . Notice that it is dependent on what value of x it is evaluated at. It is usual to evaluate (8) at the mean of x . Thus, the quasi-elasticity at the mean of x is:

$$\eta \left(\overline{x}\right)={\beta }_1\overline{x}\mathrm{Pr}\left(\overline{x}\right)\left(1-\mathrm{Pr}\left(\overline{x}\right)\right),$$

where

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

Hypothesis testing

The researcher using the logit model (and any regression estimated by ML) has three choices when constructing tests of hypotheses about the unknown parameter estimates—(1) the Wald test statistic, (2) the likelihood ratio test, or (3) the Lagrange Multiplier test. We consider them in turn.

The wald test

The Wald test is the most commonly used test in econometric models. Indeed, it is the one that most statistics students learn in their introductory courses. Consider the following hypothesis test:

Quite often in these test researchers are interested in the case when $\beta =0$ —i.e., in testing if the independent variable’s estimated parameter is statistically different from zero. However, $\beta$ can be any value. Moreover, this test can be used to test multiple restrictions on the slope parameters for multiple independent variables. In the case of a hypothesis test on a single parameter, the t-ratio is the appropriate test statistic. The t-statistic is given by

$$t=\frac{{\stackrel{⌢}{\beta }}_i-\beta }{\text{s}\text{.e}\text{.}\left({\stackrel{⌢}{\beta }}_i\right)}~t_{n-k-1},$$

where k is the number of parameters in the mode that are estimated. The F-statistic is the appropriate test statistic when the null hypothesis has restrictions on multiple parameters. See Cameron and Trivedi (2005: 224-231) for more detail on this test. According to Hauck and Donner (1977) the Wald test may exhibit perverse behavior when the sample size is small. For this reason this test must be used with some care.