RDP 2019-08: The Well-meaning Economist 4. Existing Preferences for Targets Conflict

How did Santos Silva and Tenreyro (2006) justify their assumption that the arithmetic mean is the appropriate focus in the trade application?

The problem, of course, is that economic relations do not hold with the accuracy of physical laws. All that can be expected is that they hold on average. Indeed, here we interpret economic models like the gravity equation as yielding the expected value of the variable of interest, y ≥ 0, for a given value of the explanatory variables … (Santos Silva and Tenreyro 2006, p 643)

They then reference material from a textbook by Arthur Goldberger:

When the theorist speaks of Y being a function of X, let us say that she means that the average value of Y is a function of X. If so, when she says that g(X) increases with X, she means that on average, the value of Y increases with X. (Goldberger 1991, p 5)

Santos Silva and Tenreyro later dismiss medians as an option, on account of the high incidence of zeros in comprehensive trade samples. No consideration is given to other quasilinear means, which I establish in this paper as being feasible targets.

There have, however, been several papers that have already identified the feasibility of econometrically targeting the special case of the geometric mean (when there are no zeros), particularly since the Santos Silva and Tenreyro paper was published. Two are in health applications (Basu, Manning and Mullahy 2004; Manning et al 2005). Another two are in intergenerational mobility applications (Jäntti and Jenkins 2015; Mitnik and Grusky 2017). And another two again are in labour applications (Petersen (2017); Hansen (2019), the latter is a draft manuscript). Judging by sentiment in Olivier, Johnson and Marshall (2008), geometric mean targeting seems more common in the medical sciences.^[10]

So what are the views of these different authors on the merits of targeting geometric means? Petersen (2017) is most dismissive:

In terms of best practice, the coefficients for the conditional geometric mean of the dependent variable are rarely of substantive interest, whereas those for the conditional arithmetic mean are. (p 150)

Another group is less dismissive but still favours arithmetic means. Inertia seems to play a role:

We cannot rule out that this elasticity, were it estimated robustly and without bias (a point to which we will return), might be of interest under some circumstances. But a case for estimating it has not, to our knowledge, been made. (Mitnik and Grusky (2017, p 8); the point to which they return is about zeros.)

In what follows, we adopt the perspective that the purpose of the analysis is to say something about how the expected outcome, E(y|x), responds to shifts in a set of covariates x. Whether E(y|x) will always be the most interesting feature of the joint distribution $\varphi \left(y,x\right)$ to analyze is, of course, a situation-specific issue. However, the prominence of conditional-mean modelling in health econometrics renders what we suggest below of central practical importance. (Basu et al 2004, p 751)

In contrast, Olivier et al (2008), Jäntti and Jenkins (2015) and Hansen (2019) seem to approve of targeting geometric means. Hansen writes that the arithmetic mean ‘arises naturally in many economic models’ (p 13), but later notes that in labour research it helps to model the log wage because its conditional distributions are less skewed. Estimated relationships become more robust, i.e. less sensitive to small changes to tails of the conditional wage distribution. In a subsequent footnote he makes a connection to the geometric mean, so the approval is implicit. The Jäntti and Jenkins (2015) approval is also implicit. The Olivier et al paper recommends targeting the geometric mean when a log transformation helps to normalise the conditional distribution of the outcome variable, facilitating inference in small samples. They also cite several other medical science papers that already take this approach.

What to make of these implicit disagreements? A resolution is important for policymakers, but the papers with conflicting views do not discuss each other. Moreover, they do not consider other quasilinear mean options, or other relevant decision criteria. This is an odd state of affairs. In the gravity case, Baldwin and Taglioni (2007) write that omitting controls for multilateral resistance is a ‘gold medal mistake’. Yet Tables 1 and 2 show that the choice of mean is at least as influential.

Footnote

Gorajek (2018) also shows that index functions can be understood as coming from econometric estimators of different quasilinear means. [10]