The Roy model is one of the earliest works in economics on
self-selection
In statistics, self-selection bias arises in any situation in which individuals select themselves into a group, causing a biased sample with nonprobability sampling. It is commonly used to describe situations where the characteristics of the peop ...
due to
A. D. Roy. The basic model considers two types of workers that choose occupation in one of two sectors.
Original model
Roy's original paper deals with workers selecting into fishing and hunting professions, where there is no uncertainty about the amount of goods (fish or rabbits) that will be caught in a given period, but fishing is more costly as it requires more skill. The central question that Roy tries to answer in the original paper is whether the best hunters will hunt and the best fishermen will fish. While the discussion is non-mathematical, it is observed that choices will depend on the distribution of skills, the correlation between these skills in the population, and the technology available to use these skills.
Further developments
George Borjas was the first to formalize the model of Roy in a mathematical sense and apply it to self-selection in
immigration
Immigration is the international movement of people to a destination country of which they are not natives or where they do not possess citizenship in order to settle as permanent residents or naturalized citizens. Commuters, tourists, a ...
. Specifically, assume source country 0 and destination country 1, with log earnings in a country ''i'' given by ''w
i= a
i + e
i'', where ''e
i∼N(0,
)''. Additionally, assume there is a cost ''C'' associated with migrating from country 0 to country 1 and workers know all parameters and their own realization of e
0 and e
1. Borjas then uses the implications of the Roy model to infer something about what wages for immigrants in country 1 would have been had they stayed in country 0 and what wages for non-immigrants in country 0 would have been had they migrated. The third, and final, element needed for this is the correlation between the wages in the two countries, ''ρ''. A worker will choose to immigrate if
which will happen with probability ''1-Φ(v)'' where ''v'' is
, ''s
v'' is the standard deviation of ''e
1 – e
0'', and ''Φ'' is the standard normal cdf. This leads to the famous central result that the expected wage for immigrants depends on the selection mechanism, as shown in equation (1), where ''ϕ'' is the standard normal pdf and, like before, ''Φ'' is the standard normal cdf.
:
(1)
While Borjas was the first to mathematically formalize the Roy model, it has guided thinking in other fields of research as well. A famous example by
James Heckman
James Joseph Heckman (born April 19, 1944) is a Nobel Prize-winning American economist at the University of Chicago, where he is The Henry Schultz Distinguished Service Professor in Economics and the College; Professor at the Harris School of Pu ...
and
Bo Honoré who study
labor market participation using the Roy model, where the choice equation leads to the
Heckman correction
The Heckman correction is a statistical technique to correct bias from non-randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data. Concep ...
procedure. More generally, Heckman and Vytlacil propose the Roy model as an alternative to the LATE framework proposed by
Joshua Angrist
Joshua David Angrist (born September 18, 1960) is an Israeli-American economist and Ford Professor of Economics at the Massachusetts Institute of Technology. Angrist, together with Guido Imbens, was awarded the Nobel Memorial Prize in Economics ...
and
Guido Imbens.
References
{{Reflist
Economics models