Random effects probit and logit: understanding predictions and marginal effects

ABSTRACT

Random effects probit and logit are nonlinear models, so we need predicted probabilities and marginal effects to communicate the economic significance of results. In these calculations, how one treats the individual-specific error term matters. Should one (i) set them equal to zero or (ii) integrate them out? We argue that (ii) is the quantity that most readers would expect to see. We discuss these in the context of the statistical package Stata, which changed its default predictions from (i) to (ii) in version 14. In Appendix 5, we illustrate how to calculate predictions and marginal effects using method (ii) in Stata 13 and earlier.

KEYWORDS:

• Random effects
• probit
• logit
• marginal effect
• prediction
• Stata

JEL CLASSIFICATION:

• C23
• C24
• C87

Acknowledgements

The authors acknowledge helpful comments and advice from Tim Cason, Justin Tobias, Ronald Oaxaca, Robert Slonim, Mohitosh Kejriwal, and three anonymous referees.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Here we follow the notation of Oaxaca and Dickinson (2016).

² For further information on estimating and extending RE probit, see Moffatt (2015, section 10.4) and Train (2009, chapter 7). Arulampalam (1999) discusses alternative normalizations for estimating random effects probit and subsequent marginal effects calculations.

³ For context, we estimate in Bland and Nikiforakis (2015), and in Ivanov, Levin, and Peck (2013; Table 6), we estimate in 6 of 16 cases.

⁴ We derive these results explicitly in Appendix 1.

⁵ For example in Appendix 3, we replicate Table 3 of Cason, Masters, and Sheremeta (2010). We estimate for all columns of this table. Unsurprisingly, the mean and median predictions are not particularly different.

⁶ See Appendix 2 for more information.

⁷ In their Table 6, Ivanov, Levin, and Peck (2013) report marginal effects estimated from 16 subsets of their sample. In Appendix 4, we estimate the ratio of these two marginal effects for these specifications. It varies from 0.77 to 1.22.

⁸ We thank an anonymous referee for helping us make this distinction.

⁹ In contrast, we estimate for the data from Bland and Nikiforakis (2015). While Cason, Masters, and Sheremeta (2010) has very little between-subject heterogeneity, this cannot be seen in their Table 3, with omits lnsig2u.

¹⁰ In replicating their table, it became evident that Ivanov, Levin, and Peck (2013) used Stata’s now superseded mfx command. We compute both mean and median marginal effects using margins.