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Nonlinear regression models

Sometimes the dependent variable is not a continuous variable. For example, we could be interested in whether persons work at a large law firm (value 1) or a small law firm (value 0) after graduation (Sander and Bambauer, 2012). This is an example of a binary or binomial dependent variable, i.e. taking two values.

Binary dependent variable

When the independent variables are binary variables, we can still use linear regression models, but if the dependent variable is a 1/0 variable, other models are usually taken into account. This is because linear regression models will also predict values other than 0 and 1, including negative values.

In case of a binary (1/0) dependent variable, the binary logit model is often used. In this model, the probability of belonging to category 1 versus category 0 – the odds of working at a large versus a small law firm – plays a central role rather than the values 1 and 0 themselves.

Interpretation

The estimated coefficients resulting from binary logit models cannot be interpret directly due to the nonlinear nature of the model. If a coefficient is significantly larger than 0, we know that there is a significant positive relationship between the independent variable and the probability of belonging to category 1 versus 0. Odds ratios play an important role in logit models. If the odds ratio for an age variable equals 1, it means that the odds of working at a large versus a small firm are similar for older and younger individuals. If the odds ratio is larger than 1, it means that older people are more likely than younger people to work for a large versus a small firm.

Marginal effects may provide more information about the effects as implied by the estimated coefficients. Marginal effect capture how the probability of belonging to category 1 changes given a one-unit increase in an independent variable.

Multinomial dependent variable

If the dependent variable consists of more than 2 categories, the multinomial logit model can be used. For example, the decision to own a noncorporate business, a corporate business, or no business at all, is a multinomial dependent variable with three categories (Fan and White, 2003).

Odds ratios are always interpreted relative to a base category. Marginal effects are interpreted in the same way as for binary logit models.

Other models

Ordered dependent variables can be modelled with ordered logit models. An example relates to risk perceptions (no risk, low risk, medium risk, high risk) or the degree to which a person agrees with a certain statement (totally disagree, disagree, agree, totally agree).

Although these are frequently used models there are many alternative models depending on the specific properties of the dependent variable, such as variables indicating the survival time of a firm.

Resources

    • Sander, R., & Bambauer, J. (2012). The secret of my success: How status, eliteness, and school performance shape legal careers. Journal of Empirical Legal Studies, 9(4), 893-930.

    • Fan, W., & White, M. J. (2003). Personal bankruptcy and the level of entrepreneurial activity. The Journal of Law and Economics, 46(2), 543-567.

    • Sloan, F. A., Stout, E. M., Liang, L., & Whetten-Goldstein, K. (2000). Liability, risk perceptions, and precautions at bars. The Journal of Law and Economics, 43(2), 473-502.

    • Finkelstein, M. O., & Levin, B. (2001). Statistics for Lawyers. 2nd edition. Springer. Chapter 14.7.