A simple approximation to the average effect of the treatment on the treated in panel settings with selective enrolment

ABSTRACT

This article shows that, in panel settings when the probability of receiving a treatment is low, simple comparisons of outcomes between those who are treated in a given period and those who are untreated in that period, but receive the treatment in some other period, approximately identify the average effect of the treatment on the treated. Monte Carlo simulations reveal that this approximate selection-correction substantially reduces selection bias relative to naive comparisons of outcomes between treated and untreated units.

KEYWORDS:

• Causal effect
• treatment effect
• selection bias

JEL CLASSIFICATION:

• C1
• C2
• C13
• C14

Acknowledgment

The author thanks an anonymous referee for helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ The time-invariance assumption is for simplicity and can be relaxed.

² In the terminology of Angrist, Imbens, and Rubin (1996), this approximation separates the never-takers from the compliers, defiers and always-takers.

³ This approximate selection-correction approach can also be viewed as an approximation to a formal latent-class analysis (see Leisch 2004, for an introduction).

⁴ E.g. consider a two-type model where a fraction of the population enrols with probability and enrols with probability . In broad terms, a small population probability implies that either and or . In the latter case, unobserved heterogeneity would not introduce much selection bias, however, so only the former case is of interest here.

⁵ In the plots in Figure 1, the bold lines represent the medians, the boxes represent the interquartile range and the ‘whiskers’ represent the minimum and maximum estimates (with outliers excluded according to R’s default algorithm).

⁶ I chose this threshold because it was a whole number that generated a reasonably low treatment probability.

⁷ I exclude observations for which the first-generation member of the family was born in a Northern state. Throughout my analysis, I use the Census Bureau’s definition of the North and South. To account for the possibility that Southern-born members of the third generation (who are the youngest in the sample) might have migrated North after they were interviewed, I exclude third-generation respondents who were 18 or younger at the time of the survey.

⁸ The data do not identify the birthplaces of first-generation respondents’ parents, and the lack of return migration means that third-generation members of families who ever migrate to the North, but whose second-generation parents were Southern-born, must themselves be Northern-born.