Abstract
Nonlinear probability models, such as logits and probits for binary dependent variables, the ordered logit and ordered probit for ordinal dependent variables and the multinomial logit, together with log-linear models for contingency tables, have become widely used by social scientists in the past 30 years. In this chapter, we show that the identification and estimation of causal effects using these models present severe challenges, over and above those usually encountered in identifying causal effects in a linear setting. These challenges are derived from the lack of separate identification of the mean and variance in these models. We show their impact in experimental and observational studies, and we investigate the problems that arise in the use of standard approaches to the causal analysis of nonexperimental data, such as propensity scores, instrumental variables, and control functions. Naive use of these approaches with nonlinear probability models will yield biased estimates of causal effects, though the estimates will be a lower bound of the true causal effect and will have the correct sign. We show that the technique of Y-standardization brings the parameters of nonlinear probability models on a scale that we can meaningfully interpret but cannot measure. Other techniques, such as average partial effects, can yield causal effects on the probability scale, but, in this case, the linear probability model provides a simple and effective alternative.
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Notes
- 1.
In general, we do not use a subscript to indicate individual observations except where its omission might lead to confusion.
- 2.
Indeed, when we apply these models, we also assume that the latent error has a given distribution (e.g., logistic), and we cannot know whether this is an accurate assumption either. But, in general, it seems that these models are more robust (at least when we are concerned about comparisons of parameter values across models or samples) to violations of the assumption about the distributional form of the error than they are to violations of the assumptions about the standard deviation of that distribution (Cramer 2007).
- 3.
We can write the standard deviation of the error in this way even though, given that we assumed e in Eq. (10.1) had a logistic distribution, \( \nu \) will almost certainly not have a logistic distribution.
- 4.
But Robinson and Jewell (1991: 239) point out that “to test the null hypothesis of no treatment effect in a randomized study, it is always as or more efficient to adjust for the covariate [Z in our example] … when logistic models are used” (parentheses added by authors).
- 5.
Or, equally, (YX), if we collapse the three-way table over the Z margin.
- 6.
A third approach we do not discuss here is the use of average effects on the predicted probability. Wooldridge (2002) and Cramer (2007) show that average partial effects (APEs) are unaffected by the attenuation bias created by omitted covariates orthogonal to the treatment variable. See also the concluding section where we discuss the use of the linear probability model.
- 7.
Had we used the probit model for estimating c 1, then \( h=\sqrt{{c_1^2\operatorname{var}(X)+1}} \), reflecting the assumption placed on the latent error term which, for the probit, differs from that of the logit.
- 8.
As noted by Karlson et al. (2012), this can only be hold under the assumption that the latent error distribution of both models, (10.6b) and (10.13), is logistic and we know this cannot be true. However, as noted in footnote 2, violating this assumption appears not to be very consequential for the model’s ability to recover the parameters of interest (see also Cramer 2007).
- 9.
The probit is used here because the error terms, e 3 through e 8, and all the variables are normally distributed.
- 10.
To simplify exposition, in the following, we assume that the causal effect is constant across individuals in the population. Under this assumption, the IV identifies the average causal effect. Whenever that assumption does not hold, an additional assumption—monotonicity—is required in order for the IV to recover the average causal effect for a subset of the population that is affected or moved by the instrument (see Imbens and Angrist 1994; Blundell et al. 2005). However, the problem we sketch in the following also pertains to the situation in which we recover a local average treatment effect.
- 11.
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Breen, R., Karlson, K.B. (2013). Counterfactual Causal Analysis and Nonlinear Probability Models. In: Morgan, S. (eds) Handbook of Causal Analysis for Social Research. Handbooks of Sociology and Social Research. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6094-3_10
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