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 11. 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 21. 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

, , ,

, , 1 ,

log i a i i a i a

i a i a i a

y z

z z u

1{ (log | )}Ta aF y X

Background on Copulas

• Consider 2 RV’s (X,Y) with continuous joint CDF F(x,y)• Sklar’s Theorem (1959) allows us to decompose joint

distribution into a unique function of marginal distributions:

F(x,y) = C(FX(x), Fy(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 {(xi, yi) }i=1..N

1 1 1

log ( ) log ( ) log ( ( ), ( ))N N N

i i i ii i i

f x f y c F x F y

( , ) ( ) ( ) ( ( ), ( ))X Y X Yf x y f x f y c F x F y

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={wa}, a=1…T.

• Denote conditional density f(w|X)

where a is lab mkt experience;

T is total # years in labour force;

gender, graduate status X1

, ,log ,i a i i ay g a X w

Copula Decomposition of Earnings

• By copula decomposition, we can write density of vector of earnings residuals as

• Assume a first-order Markov process for :

1

( | ) | , ,..., | , | | ,T

a T aa

f w X c F w a X F w T X X f w a X

1

1 1 1 1 1 11 1

( | ) | , , | 1, | , | ,T T

a a aa a

f w X c F w a X F w a X a X f w a X

1{ }Ta aw

Parametric Assumptions

1. Marginal Distributions– Estimate nonparametrically

2. Copula Function1. 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.

, , , 1 , 1 ,1 1

ˆ ˆˆ ˆlog ( ( | ), ( | ) | )T N

i a i a i a i a i aa i

c F w X F w X X

1ˆ ˆ{ ( | )}Ta aF w X

1ˆ{ }Ta aw

Data Sources

(1) Labour Force Survey (LFS): quarterly survey; each hh interviewed for 5 successive qrtrs since spring 1992

Before spring 1997: earnings data in 5th qrtr only Afterwards: earnings data in 1st & 5th 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) 99th (1st) 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

t stat p-value

Male Graduates 21.66 0.00Male Non-graduates 30.13 0.00Female Graduates 22.22 0.00Female Non-graduates 35.46 0.00

T-Copula Gauss

Male Graduates 1.77 1.28Male Non-graduates 1.29 1.01Female Graduates 1.78 1.37Female Non-graduates 1.77 1.39

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

Lifetime Earnings Distributions0

.01

.02

.03

.04

.05

Pro

po

rtio

n

0 1 2 3Total lifetime earnings, £millions

Males

0.0

1.0

2.0

3.0

4.0

5

Pro

po

rtio

n

0 1 2 3Total lifetime earnings, £millions

Graduates

Non-Graduates

Females

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 X’s– 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 11. Estimate rich ARMA model on BHPS2. Estimate same ARMA model on data simulated from

copula model

Strategy 21. 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

, , ,

, , 1 ,

2 2 2, ,~ 0, , ~ 0, , ~ 0,

i t i i t i t

i t i t i t

i i t i t u

w z

z z u

N N u N

Copula Sims BHPS Data Copula Sims BHPS Datarho 0.963 0.931 0.907 0.928V(fixed effect) 0.056 0.077 0.018 0.143V(persistent 0.011 0.041 0.026 0.050V(transitory) 0.027 0.039 0.018 0.040

Graduates Non-Graduates

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

ModelT-Copula Gauss T-Copula Gauss

Male Graduates 5218 5220 1.19 1.19Male Non-graduates 7001 7003 1.37 1.37

Maximized Likelihood Aikaike Information

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