# Estimating Lifetime Earnings Distributions Using Copulas Lorraine Dearden Emla Fitzsimons Alissa Goodman Greg Kaplan ESRC Methods Festival 19 July 2006.

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Estimating Lifetime Earnings Distributions Using Copulas Lorraine Dearden Emla Fitzsimons Alissa Goodman Greg Kaplan ESRC Methods Festival 19 July 2006

Motivation 1.To illustrate the use of copulas as a tool for modeling earnings dynamics 2.Example of an application: –estimating the distribution of lifetime earnings for graduates and non-graduates in the UK

Modelling Earnings Dynamics Approach 1 1.Specify a dynamic model for earnings e.g. 2.Estimate paramers – usually requires T>=3 for identification 3.Simulate earnings paths from estimated model Approach 2 1.Estimate conditional cross- sectional distributions directly: 2.Estimate a mobility process, e.g. with transition matrices 3.Simulate earnings paths using cross-sectional distributions and assumed mobility process Copula methodology is a formalization of approach 2

Background on Copulas Consider 2 RVs (X,Y) with continuous joint CDF F(x,y) Sklars Theorem (1959) allows us to decompose joint distribution into a unique function of marginal distributions: F(x,y) = C(F X (x), F y (y)) where C:[0,1] 2 [0,1] A simple example: –Independence Copula:C(u,v) = uv Think of joint distribution as being decomposed into: –Elements relating only to marginal distributions –Elements relating to dependence structure (copula)

More on Copulas In terms of densities: Decomposition generalizes to any n-dimensional distribution Likelihood function for data from {(x i, y i ) } i=1..N

Why copulas? 1.Technique for generating rich dependence structures with few parameters and short panels 2.Allows for marginals to be estimated nonparametrically so that estimation of dependence structure is robust to misspecification of marginals 3.Allows for combination of panel and cross-sectional data, which each type of data employed where it is best suited 4.Allows us to model non-linearities in mobility, such as tail dependence. The degree of linear correlation in wages can vary across the cross-sectional age- specific distribution

A Copula Model For Earnings Aim is to obtain reliable estimate of X-sectional distribution of vector of residuals w={w a }, a=1…T. Denote conditional density f(w|X) wherea is lab mkt experience; T is total # years in labour force; gender, graduate status X 1

Copula Decomposition of Earnings By copula decomposition, we can write density of vector of earnings residuals as Assume a first-order Markov process for :

Parametric Assumptions 1.Marginal Distributions –Estimate nonparametrically 2.Copula Function 1.Gaussian copula: dependence structure implicit in a bivariate normal dist 1 parameter: rho = rank correlation Tail dependence (correlation in extremes of dist) approaches 0 Linear correlation only 2.T-copula: depedence structure implicit in a a bivariate t- distiribution 2 parameters: rho + df (degrees of freedom) df governs additional dependence in tails of the distribution

t-copula vs Gaussian copula Rank correlation is equal in both samples (0.75)

Estimation Procedure 3 stages: 1.First stage regression to estimate g(a,X) and generate the vectors of residuals, 2.Use re-scaled ECDFS to estimate 3.Maximize likelihood with respect to copula parameters Estimate separately for males, females, graduates and non- graduates g(.) controls for race, region and time effects All copula parameters allowed to vary by labor market experience Consistent but not efficient.

Data Sources (1) Labour Force Survey (LFS): q uarterly survey; each hh interviewed for 5 successive qrtrs since spring 1992 Before spring 1997: earnings data in 5 th qrtr only Afterwards: earnings data in 1 st & 5 th qrtrs (2) British Household Panel Survey (BHPS): annual survey; ongoing panel since 1991 Up to 11 years of earnings obs (to 2001)

Sample Selection Sample of 19-60 yr old employees Trim earnings values above (below) 99 th (1 st ) percentile for each group Use residuals from regression of log real weekly wages on dummies for year, race, region of residence Convert residuals to annual units

Choosing a copula 1.Pseudo-Likelihood Based Test - Chen and Fan (2005) Accounts for estimation error in marginal distribution, non-nested models and fact that none of models under consideration is true DGP 2.Aikaike Information Criterion

Parameter Estimates

Simulations & Employment Transitions Dist of lifetime earnings is specified by piecing together sequences of bivariate dists 10,000 draws of earnings paths are made from this lifetime dist (separately by group) Moreover, we incorporate labour market mobility into these trajectories Mobility parameters obtained from 2 employment transition models (1) Prob of becoming employed at age a given ue at a-1 (2) Prob of becoming unemployed at age a given emp at a-1 each estimated using up to 11 years of BHPS data Also estimate re-entry rank upon re-employment

Copula vs ARMA models Copula framework v flexible and general –a/c for tail dependence –different dependence in different parts of dist; allow this to vary by age and other Xs –non-parametric marginal distributions …but at expense of not explicitly allowing for unobserved heterogeneity Natural to ask whether in moving to copula model, we are neglecting important features of dynamic process And what features we capture that a linear model can not

Comparison of Models Strategy 1 1.Estimate rich ARMA model on BHPS 2.Estimate same ARMA model on data simulated from copula model Strategy 2 1.Estimate copula model on BHPS 2.Estimate copula model on data simulated from ARMA model Use BHPS because 3 periods required for identification of ARMA model

Linear Model of Earning Dynamics

Copula Estimates on Linear Simulations t-copula and gauss give same likelihood – df maximsed at +Inf Gaussian ARMA model can not capture the non- linear dependence structure in wages

Conclusions and Summary Illustrated the use of copulas in modelling earnings dynamics Highlighted departures of copula based models from traditional linear ARMA models Provided and example of an application for constructing lifetime earnings distributions

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