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 dynamics³², 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, ya, 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

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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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