Adam Gorajek of RBA in this paper writes why choosing “mean” matters:

*Economists usually inform policymakers with conclusions that come from studying the conditional expectation, i.e. arithmetic mean, of some potential outcome. But there are other means to study, from the same ‘quasilinear’ family. And they can support very different conclusions. In trade research, for instance, studying other means can transform the perceived roles of colonial history, geography, and trade wars. In wages research, studying other means can reverse perceived earnings differentials between groups. Similar scenarios will be common in other tasks of policy evaluation and forecasting. To choose means well I propose selection criteria, which also consider options that are outside of the quasilinear family, such as quantiles. Optimal choices are application-specific and ideally accommodate the preferences of the relevant policymaker. In the wages case, policymaker aversion to inequality makes it sensible to reject the arithmetic mean for another quasilinear one.*

Example:

*Suppose we discover today that in 1990 a random subset of Australian schoolchildren were given a badly misprinted version of the standard mathematical textbook. Its answers were wrong and the explanations were nonsense. Suppose also that today we can survey these and the unaffected schoolchildren (all now **adults) about their incomes. Besides objecting to the injustice of the misprint, an economic researcher might see this as a unique opportunity to assess the value of effective educational materials for career outcomes.*

*Conducting the survey, our hypothetical researcher records that half of the affected students now have annual incomes of $40k and half have $100k. For the unaffected students, half have annual incomes of $60k and half have $80k. For reasons that I leave to the paper, the standard strategy in this simplified situation would be to summarise the salaries of each group with their so-called ‘arithmetic mean’, which is a basic type of average. Since both groups have arithmetic mean incomes of $70k, the headline conclusion for the policymaker is that the misprint was unimportant. Even in complex research situations, summarising outcomes with numbers akin to these arithmetic means is a standard strategy.*

*But what if we choose a different summary measure, like a ‘geometric mean’, or any other mean in the ‘quasilinear’ family? Leaving an explanation of these concepts aside, the point is that often the conclusions will change. For instance, in the textbook misprint case, the geometric mean income for the affected students is $63k and for the unaffected students is $69k. Hence the headline conclusion for the policymaker is that the misprint was detrimental. The reason for the change is that the geometric mean penalises inequality, which is higher among the affected students. The penalty is an attractive feature here, because in western democracies it is evident from tax and social security systems that policymakers view income inequality as undesirable. The question is only what amount of penalty is appropriate.*

Hmm..

September 12, 2019 at 3:57 pm |

Brilliant! It gets more interesting too if we also consider conditional variance. Good luck explaining the matter to policymakers