RDP 2019-08: The Well-meaning Economist 1. Introduction

The statistical expectation, i.e. arithmetic mean, is one of the foundations of economics: policy evaluations usually focus on learning about how a change in some policy affects the arithmetic mean of an outcome variable; forecasts usually target the conditional arithmetic mean of an outcome variable; and the arithmetic mean is used to define statistical concepts like bias, dispersion, and skew.

But for none of these applications is it the only option. Policy evaluations sometimes target quantiles (following Koenker and Bassett (1978)), while the forecasting literature ventures further still (Varian (1975) is a classic example). In a similar spirit, this paper makes a case for targeting options from the so-called quasilinear family of means. The family is infinitely large and contains the arithmetic and geometric means as special cases. I show that all of the options can offer important advantages for policymakers and are feasible targets for researchers. Indeed, some approaches to estimation already target different quasilinear means, just not deliberately.

More deliberate approaches are important because switching between types can change the recommendations offered to policymakers. The effects are too large to leave to chance. In models of trade, for instance, switching to different quasilinear means can dramatically change the estimated effects of physical distance, colonial ties, and free trade agreements (FTAs). The estimates matter because distance is a basis for international development assistance (World Bank Group 2018).^[1] Recent US trade negotiations have also triggered widespread interest in the effects of trade policies.

The effects of changing targets are not always large though. I show, for example, the results of a study about the determinants of CEO earnings, in which switching makes little difference. Likewise for a study about the effect of a hospital intervention on the cost of patient care. But in a study about the wage premium for self-employment over contract employment, switching matters again; the key result changes sign and remains statistically and economically significant. Wage comparisons like these are important if we wish to have informed community dialogues about, say, industrial relations and gender or racial equity.

Similar observations have been made elsewhere in the literature. In particular, others point out that some existing estimation methods target geometric means and that switching to geometric targets from arithmetic ones can matter a lot. Some of my examples are theirs. However: the views expressed in those papers about the merits of geometric mean targeting are mixed; the papers with conflicting views do not discuss each other; the infinite number of other possible targets in the quasilinear mean family are not recognised; and decision criteria that I argue are important are not considered. So far the discussion has not done justice to the importance of the decision.

To judge the appeal of the different candidates and provide a coherent basis for choosing among them, I propose several decision criteria, one of which uses the expected utility framework of von Neumann and Morgenstern (1944). The idea is that each quasilinear mean is the certainty equivalent of an outcome distribution under a particular specification of policymaker preferences over potential outcomes. Equivalently, each quasilinear mean is the certainty equivalent of an outcome distribution under a particular specification of policymaker risk aversion. So a good choice of mean is one that reflects the preferences of the relevant policymaker. For example, governments in western democracies use their tax and social security systems to reduce income inequality, which reveals a form of risk aversion in income. Hence it is natural to focus most wages research on quasilinear means that reflect this risk aversion. In that case the arithmetic mean is a misleading standalone summary of potential policy outcomes. So are quantiles.

An alternative way to motivate choices is to use a loss function criteria, i.e. to consider the relative costs for the policymaker of different over- and under-predictions. If the policy objective relates to long-term growth rates, and the economist is modelling short-term outcomes, the geometric mean is better than the arithmetic one; the short-term outcomes are compounding and this feature is accommodated by the loss function of the geometric mean. Prime examples are models of inflation, for central bankers, and models of financial returns, for pension fund managers. When model fit is high, like it is for the inflation case, the decision tends to matter less.

A third set of criteria relates to useful mathematical behaviours of different means. For instance, many means produce conclusions that are invariant to arbitrary changes in the units of measurement. Some do not though.

Unfortunately, it is sometimes hard to choose means on the basis of any of these three sets of criteria, not least because there will often be many similarly attractive options. In these circumstances it is sensible to focus on the simplicity of statistical inference as the relevant decision criterion. A literature on power transformations, stemming from Box and Cox (1964), shows that variable transformation can simplify the task of statistical inference, partly by making residuals more normally distributed. I show that the same transformations implement switches between different quasilinear mean targets, hence some quasilinear means are easier targets than others. The easiest targets are application-specific.

When we do choose to depart from learning about arithmetic means, logical consistency will dictate changes to several aspects of our analysis. A surprising example is a change to the convention of choosing estimators partly on the basis of their unbiasedness. This result challenges a literature on bias corrections, most notably papers by Goldberger (1968), Kennedy (1981), and van Garderen and Shah (2002). That literature has influenced several areas of economics, including the measurement of key macroeconomic variables like inflation (International Labour Office et al 2004, p 118).

To sum up, it is a classic task of the economist to judge whether some model is a valid (or sufficiently close to valid) description of the data-generating process being studied. The conclusions in this paper rest on the premise that, even if the model is valid, there will be other descriptions that are equally so. The different options are distinguished by the characteristics of the data-generating process they describe. A characteristic described by most models is the conditional arithmetic mean, which is a decision that can have important implications for policymakers and is often poorly justified. I propose several ways to think more carefully about the decision and show that alternatives are easy to implement with existing tools.

Footnote

Currently 20 countries receive special assistance from the International Development Association of the World Bank, in recognition of development challenges that include remoteness. Small island states in the Pacific are extreme cases, as documented in Becker (2012). [1]