RDP 2019-08: The Well-meaning Economist 7. Conclusion

The arithmetic mean is a well-established measure of central tendency and economists now use it for many purposes. I show that an alternative option is to target other means in the quasilinear family. The family is infinitely large, containing the arithmetic and geometric means as special cases, and to target them we can use standard tools. The researcher monotonically transforms the outcome variable of interest, uses standard tools to estimate the conditional arithmetic mean function, and transforms the estimated function back again. The same applies to confidence intervals.

The choice to depart from the arithmetic mean can matter a lot for the conclusions that researchers offer to policymakers. For instance, across different models and samples of trade, targeting alternatives to the arithmetic mean increases the estimated effects of physical distance and colonial ties a lot. The estimated effects of FTAs move a lot too, in directions that are more model-dependent. In a study about the relationship between self-employment status and wages, key parameter estimates remain statistically and economically significant, but change sign. These differences invite an important question: how can we researchers worry so much about the confounding effects of omitted variables and not choose our means carefully? While sometimes we do target alternative types of quasilinear means, we tend to do so unconsciously.

An ideal way to choose targets is on the basis of policymaker utility. Each quasilinear mean can be justified as a certainty equivalent of an outcome distribution under a particular specification of policymaker preferences, or risk aversion, over outcomes. For example, in western democracies, governments have revealed in their tax and social security systems an aversion to income inequality (income ‘risk’). It is hence ideal to focus most income research on quasilinear means that are certainty equivalents under risk aversion. Examples are the geometric and IHS means.

We can also choose targets using the perspective of loss functions, i.e. considering the relative costs for the policymaker of different over- and under-predictions. If the costs of over- and under-predictions are symmetric, the arithmetic mean is usually sensible. The median is also an option. Costs will not always be symmetric though. Indeed, ‘an assumption of symmetry is probably a poor one’ (Granger 1999, p 166). If the policy objective relates to long-term growth rates, and the economist is modelling shorter-term outcomes, the asymmetric loss function of the geometric mean will be optimal. Prime examples are models of inflation, for central bankers, and models of financial returns, for pension fund managers.

But it is not always practical to choose targets solely on the basis of policymaker utility, loss functions, or other mathematical criteria that I introduce. And with so many options to choose from, it is often hard to do objectively. A pragmatic approach will also consider the simplicity of statistical inference. What comes of these considerations will be application-specific, just as optimal transformations in the work of Box and Cox (1964) are application-specific. Whatever the final choice, it helps to state it clearly. If there are several options on the table, as with the gravity model, targeting each can be a useful form of sensitivity analysis.

Readers might still be sceptical about the value of targeting alternative quasilinear means. Quantile regression provides a lot of flexibility already and my proposed selection criteria often do not point forcefully to a particular type of quasilinear mean. However, to outright reject alternative quasilinear means for all applications is to take some other uncomfortable positions. One is to dismiss the ubiquitous estimation method of OLS after log transformation, or in fact any power transformation. Another is to dismiss a growing literature that uses IHS transformations. Both sets of techniques already do effectively target different quasilinear means.

If we do choose to work with other quasilinear mean types, logical consistency will dictate changes to several aspects of our analysis. For instance, instead of choosing estimators partly on the basis of their unbiasedness, the appropriate criteria will be quasi-unbiasedness. The same logic concludes that an existing bias correction, argued as appropriate for log-linear models, is a counterproductive complication to research.