A general decomposition formula with interaction effects

Abstract

This note presents a general way to decompose differences over time or between objects into the ceteris paribus effects and the interaction effects of an arbitrary number of factors. The decomposition addresses the issue of interaction effects between factors which have been neglected in the decomposition literature. It has the additional advantage of being path-independent and aggregation consistent. An empirical application studying distributional change demonstrates that interaction effects may be a relevant feature of reality that is overlooked if other decomposition methods are used.

Keywords:

• ceteris paribus effects
• sequential decomposition
• distributional change

JEL Classification:

• J31
• J71
• C02
• D31

Acknowledgement

I am grateful to Andos Juhasz for helpful comments and research assistance. The data used in this paper (SOEP v27, 1998–2010) were made available by the German Socio-Economic Panel Study (SOEP) at the German Institute for Economic Research (DIW), Berlin.

Notes

¹ It seems that a recent paper by Rothe (2012) introduced (independently from this article) a similar definition of interaction effects in the context of decomposing the composition effect in between-group decompositions. Note, however, that the factors are defined somewhat differently in Rothe’s paper. An important further difference is that Rothe specifically considers the composition effect in a between-group decomposition, while the formula given here is formulated in terms of general and unspecified counterfactual outcomes, independently of how they are generated (e.g. they may be generated by simulation, see empirical example below).

² See more detailed working paper version Biewen (2012).

³ For more examples involving interaction effects, see working paper Biewen (2012).

⁴ The results are shown for the Theil coefficient. Results for other inequality indices are very similar.