Higher education funding reforms in England: the distributional effects and the shifting balance of costs



12. Appendix: Copula Model for Earnings Dynamics

The approach that uses copula functions is relatively new to the literature on
modelling earnings dynamic
s32, and an outline of the methodology is provided here.

Let a be the labor market experience of an individual and let X be a vector of
observed characteristics of the individual. We can write the logarithm of the observed
wage, y
a, of an individual as

\* MERGEFORMAT (.)

O103

where is the conditional expectation. The aim is to estimate a statistical model for the
distribution of the vector of residuals , where T is the total number of years spent in
the labor force. We denote this conditional density , for a subset . In our application,
the variables contained in include gender and whether or not the individual is a
graduate.

Obj1087

The curse of dimensionality renders a fully nonparametric estimator of infeasible for
large
T. Rather than assume a multivariate normal distribution, our approach is to use
Sklar's (1959) theorem to decompose into a sequence of marginal densities and a
copula density which completely describes the inter-temporal dependence structure in
the vector w. We refer the reader to Nelsen (1999) and Joe (1997) for details on the
definition of a copula and examples of parametric forms.

\* MERGEFORMAT (.)

If wages follow an n-order Markov process, the copula density may be written as

Obj110

32 The exception to this is Bonhomme and Robin (2005). The use of copula functions is much more
common in financial econometrics. See for example Patton (2006a,b).

36



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