hours constraints and unobserved heterogeneity in structural discrete choice models of labour supply...

Hours Constraints and Unobserved Heterogeneity in Structural Discrete Choice

Models of Labour Supply

Alan DuncanCPE, University of Nottingham and IFS

http://www.nottingham.ac.uk/~lezadhttp://www.nottingham.ac.uk/~lezad

Mark HarrisUniversity of Monash and CEU

IntroductionIntroduction

Much of the recent literature on modelling household labour supply applies discrete choice econometric modelling methods

Why? Because discrete choice methods … allow for a series of improvements on traditional continuous

methods of estimation (Hausman,1985)

offer the potential to model household decisions cope relatively well with non-linear taxes in estimation translate easily to ex ante evaluation of policy reform


What is the typical model of household labour supply?

Economic foundations direct estimation of well-defined preference function preferences expressed over discrete hours choices can model preferences at the level of the household

(Van Soest, JHR 1995) can accommodate fixed & search costs of work

(Blundell, Duncan, McRae & Meghir, FS 2000) can model welfare programme participation

(Moffitt & Keane, IER 1999)


What is the typical model of household labour supply?

Econometric specification issues stochastic errors added to each discrete hours choice errors are typically extreme value, leading to a classic

conditional logit specification potential problem: IIA some models additionally allow for random preference

heterogeneity random heterogeneity is usually MVN, leading to a

mixed logit specification (Duncan and Harris, ER 2002)

Motivation for paperMotivation for paper

PROBLEMS:

1) the typical model remains relatively unchallenged in terms of its underlying stochastic structure

choice-specific errors are not strictly necessary extreme value assumption is potentially significant distributions for random heterogeneity terms are

not always obvious or intuitive


PROBLEMS:

2) observed hours choices are generally equated to preferred hours choices – not always so.

people cannot always locate to their preferred hours choice (institutional constraints).

the range of hours alternatives from which people choose may depend on specific job characteristics

for some job types, the unconstrained choice of hours may simply not be available, or at least may be relatively unlikely


IMPLICATIONS:

1) estimated models of household labour supply are not typically able to replicate the distribution of observed hours choices

one often sees peaks or clusters in observed hours distributions (0, 20, 38-40)

most household labour supply models are smooth functions, with a continuous stochastic structure.

they are therefore unable to replicate accurately such bunching in hours choices.


IMPLICATIONS:

2) preference parameters may therefore be biased when institutional constraints are ignored

with optimising errors, one cannot assume that someone who is observed to work, say, 20 hours, would prefer 20 hours to any other choice;

they may have ideally liked to work, say, 27 hours, but labour market institutions prevent this

if ignored, estimated preference parameters in fact become a convolution of unconstrained tastes & labour market characteristics


BACKGROUND:

institutional constraints have been addressed in continuous studies of labour supply

Arrufat and Zabalza (E’trica 1986) observed an absence of bunching of hours at kink points in the tax system, despite theory requiring that such bunching should be observed in observed choices


BACKGROUND:


Their explanation: institutional constraints in the labour market prevented workers from adjusting their hours choices to suit important parameters of the tax system


BACKGROUND:


Their solution: include optimising errors alongside random preference heterogeneity in a labour supply model, to ‘smooth’ observed outcomes around tax kinks.

optimising errors were separately identified by exploiting the non-linearities in the tax system


BACKGROUND:

some unusual suggestions in discrete studies of labour supply

Van Soest, Das and Gong, 1999: define a discrete (Heckman-Singer) distribution for preference heterogeneity

The number of support points assessed empirically leads to them categorising a finite set of ‘taste types’

conditional on observed characteristics bunching is therefore caused by types of people rather than

labour market institutions


OUR APPROACH:

confront the presence of institutional constraints in discrete labour supply models directly

(1) model directly the degree of captivity to each observed hours alternative (DOGIT)

(2) attempt to control for optimising error in the conditional logit model by integrating the error over its (finite) empirical distribution (cf.Arrufat&Zabalza)

CLOE (Conditional Logit with Optimising Error)

an economic model of labour supply an economic model of labour supply

the basic model: preferences over hours h (or ‘leisure’ T-h) and net

income yh: U=U(h, yh | X)

budget constraint:

yh=w.h+I-T(wh,I; Zt)+B(w,h,I; Zb)

missing wages:

log(w)=Xwbw+ uw , uw has density f(uw)

discrete labour supply estimation discrete labour supply estimation a structural discrete model of labour supply: Assume that hours h(.) chosen from a set of J discrete alternatives:

h(.) = h1 if h <= h1

B

h(.) = h2 if h1

B <= h < h2B

……….

h(.) = hJ-1 if hJ-1

B <= h < hJB

h(.) = hJ if h > hJ

B

Household net incomes are calculated for each h(.) ={h1, h2,…, hK } as yh =w.h(.)+I-T(wh(.),I; Zt) + B(w,h(.),I; Zb)

a discrete choice seta discrete choice set

yh

h

yh

h


yh

h

h(.)*=maxh(.) U= U( h(.) , yh | X )


discrete labour supply estimationdiscrete labour supply estimation

a structural discrete model of labour supply: Define preferences over h(.)={h1, h2,…, hJ }:

Choice of h(.) {h1, h2,…, hJ } solves

maxh(.) U= U( h(.) , yh | X )

s.t. yh =w.h(.)+I-T(wh(.) ,I) + B(w,h(.) ,I)

Avoids the complexities of nonlinear functions T(.) & B(.) Problem? Introduces rounding errors through h =h(.). Needs testing…

functional form choice for U(.): We follow a number of authors in choosing a quadratic

direct utility:

Blundell, Duncan, McCrae and Meghir (2000)Duncan and Harris (2002)Keane and Moffitt (1998)

2 2( , )h yy h hh yh h y h hU h y y h y h y h

specifying preferencesspecifying preferences

unobserved preference heterogeneity Observed and unobserved heterogeneity enters through

preference terms. eg,

Unobserved heterogeneity in preferences is typically assumed multivariate normal.

2 2( , | , )

( ) ( )

h yy h hh yh h

y y h h h

U h y X v y h y h

v y v h

0 1

0 2

y y yx

h h hx

X

X

discrete choice:discrete choice:econometric estimationeconometric estimation

controlling for costs of employment fixed costs are FC are incurred for all choices which

involve work parameterise fixed costs in terms of a set of observed

characteristics and a stochastic element:

FC = Xfcfc + ufc

modify preferences for those in work, h(.) >0:

U=U(h(.) , yh - FC | X )


state-specific disturbances Introduce a stochastic component to preferences for

each discrete hours alternative:

By assuming a distribution for each uh , one can derive an expression for the probability Pr(h(.) = hj

| X, v, ufc).

* *( , | , , , )

( , | , , )

h h fc h

h fc h

U U h y X v u u

U h y X v u u


deriving likelihood contributions Pr(h(.) = hj

| X, v) = Pr[ Uhj* =max(Uhk

* for all k=1,..,J)].

If each uh is extreme value,

Ignoring random components, this is exactly analagous to the conditional logit specification (eg.McFadden,1984)

(.)

1

|

| )

exp[ ( , , )]Pr( | , )

exp[ ( , , ]

j

s

j

j h

J

s hs

U h y X vh h X v

U h y X v


deriving likelihood contributions Taking account of random components, we can integrate the

likelihood over the distributions of w and v Assuming independence of w and v ,

Stochastic structure makes this more akin to (non-IIA) mixed logit (eg. McFadden and Train, 2000)

(.)

1

|( ). ( ) .

| )

exp[ ( , , )]Pr( | )

exp[ ( , , ]

j

s

j w v

v w

j h

J

s hs

f w f v w vU h y X v

h h XU h y X v


the basic likelihood function

In estimation, this integral is approximated using simulation methods


1( )

1

log log Pr( | , ) ( ) ( ). .j

Jh h

j w vi jv w

L h h X v f w f v dwdv

The DOGIT model (Manski 1977) We use this to parameterises institutional constraints

explicitly in terms of the degree of ‘captivity’ to each discrete outcome in the set of hours alternatives

Captivity is parameterised in terms of a series of (non-negative) parameters J

(.)

1

1

1

| | )

1 . | )

exp[ ( , )] exp[ ( , ]Pr ( | )

exp[ ( , ]

j s

s

jDGT

j J

js

J

sj h hs

J

s hs

U h y X U h y Xh h X

U h y X

discrete choice:discrete choice:controlling for institutional constraints (1)controlling for institutional constraints (1)

The DOGIT model (Gaudry and Dagenais) We use this to parameterises institutional constraints

explicitly in terms of the degree of ‘captivity’ to each discrete outcome in the set of hours alternatives

Captivity is parameterised in terms of a series of (non-negative) parameters J

(.)

1

1

11 .

exp( ) exp( )Pr ( | )

exp( )

j s

s

jDGT

j J

js

J

h hs

J

hs

U Uh h X

U

discrete choice:discrete choice: controlling for institutional constraints (1)controlling for institutional constraints (1)

Characteristics of the DOGIT model Parameters J denote the ‘degree of captivity’

The Dogit reduces to the CLogit when j = 0 for all j If we denote

Then

(.)

(.)

Pr( | )

Pr ( | )j

j

CLh j

DGT DGTh j

P h h X

P h h X


1 11 1

j

jDGTh

j

CLh

J J

j js s

PP


Characteristics of the DOGIT model From manipulation of it can be shown that

That is, the Dogit model draws towards states where the CL probability is ‘small’ relative to the size of the corresponding captivity parameter

j

DGThP

1

if j j j

jDGT CL CLh h h J

js

P PP


Characteristics of the DOGIT model Fixed costs and random preference heterogeneity

can be introduced in the same manner as for the CLogit

Can test CLogit against DOGIT using standard LR methods

Possible drawbacks: (1) degree of captivity doesn’t depend on observed factors

(2) no background in economic theory


CLogit with optimising error An alternative approach to the Dogit is to specify

directly the distribution of the optimising error opt in a discrete choice framework

By analogy with the continuous approach of Arrufat and Zabalza (E’trica 1986), we ‘integrate out’ the optimising error over its discrete range

Result: a direct estimation of the finite J-point distribution of opt, similar in spirit to the approach of Heckman and Singer (E’trica,1984).


CLogit with optimising error Let discrete observed hours be hobs(.)={h1, h2,…, hJ }

Let discrete ‘desired’ hours choice be h*(.)={h1, h2,…, hJ }, defined

according to the CL max rule Then, hobs(.) = h*

(.)+ opt

Optimising error will be drawn from the set opt= { hk- hs , for k,s, =1,…, J }

Parameterise the finite probability distribution of opt as follows: Pr(opt= hk- hs)=ks

Probability: (.) kjPr( | )

k

CLobs j h

k

Ph h X


CLogit with optimising error Example: hobs(.)={0, 10, 20, 30, 40}

h*(.)={0, 10, 20, 30, 40}

0 10 20 30 40

10 0 10 20 30

20 10 0 10 20

30 20 10 0 10

40 30 20 10 0

opt

0

10

20

30

40

0 10 20 30 40


CLogit with optimising error Example: hobs(.)={0, 10, 20, 30, 40}

h*(.)={0, 10, 20, 30, 40}

0,0 0,10 0,20 0,30 0,40

10,0 10,10 10,20 10,30 10,40

20,0 20,10 20,20 20,30 20,40

30,0 30,10 30,20 30,30 30,40

40,0 40,10 40,20 40,30 40,40

Pr( ) opt

0

10

20

30

40

0 10 20 30 401

1

1

1

1


Characteristics of CLogit with optimising error Nests the standard CL: kk = 1, ks = 0 for k not equal s

Institutional constraints can vary depending on h*(.)

Identification must come from variation in tax / welfare payments (cf. Arrufat and Zabalza 1986)

Closer integration with economic theory

Data and identificationData and identification

Data: Family Resources Survey: 39,000 mothers from 1994 – 2000 Select single parent families only Exclude self-employed, students, pensioners,

sick/disabled, women on maternity leave Leaves sample of 10,665 single parents Period embraces substantial tax/welfare reform

Variation over time and across individuals age, number and age of children, housing tenure,

education

Data and identificationData and identification

Estimation: CML for GAUSS Six hours alternatives h={0,10,19,24,33,40} Simulated ML for random preference

heterogeneity specifications Constraints directly imposed on DOGIT &

CLOE parameters

Results: CL model

Coeff SE Coeff SEayy -0.0557 0.0088 fc(c) 1.6024 0.2227ahh 0.1131 0.0116 fc(yk02) 3.2614 0.7711ayh -0.0005 0.0009 fc(yk34) 1.7478 0.5646

fc(Lon) 1.2420 0.1351

by(c) 0.0676 0.0079 bh(c) -0.0352 0.0072by(yk02) -0.0586 0.0101 bh(yk02) -0.0268 0.0049by(yk34) -0.0467 0.0096 bh(yk34) -0.0296 0.0042by(yk10) -0.0171 0.0044 bh(yk10) -0.0236 0.0021by(nkids) 0.0090 0.0018 bh(nkids) -0.0097 0.0012by(age) -0.0099 0.0023 bh(age) 0.0058 0.0014by(educ) -0.0010 0.0006 bh(educ) 0.0060 0.0004by(ethnic) 0.0046 0.0034 bh(ethnic) 0.0039 0.0027

Log-likelihood -13373.3Number of cases 10665

Results summary: CL model Temporary simplifications:

Estimate wages first and draw random realisation

Fixed costs are deterministic

Preferences for income Increase with number of

children, age of youngest

Reduce with age and education

Fixed costs decrease with age of

youngest child vary by region

Distaste for work increases with number of

children reduces with age and

education

Results: CL & DOGIT compared

CL DOGIT CL DOGITayy -0.0557 -0.0652 fc(c) 1.6024 1.7118ahh 0.1131 0.1408 fc(yk02) 3.2614 3.0171ayh -0.0005 0.0006 fc(yk34) 1.7478 1.7689

fc(Lon) 1.2420 1.296

by(c) 0.0676 0.0773 bh(c) -0.0352 -0.0361by(yk02) -0.0586 -0.068 bh(yk02) -0.0268 -0.0292by(yk34) -0.0467 -0.0466 bh(yk34) -0.0296 -0.0326by(yk10) -0.0171 -0.0194 bh(yk10) -0.0236 -0.0252by(nkids) 0.0090 0.0141 bh(nkids) -0.0097 -0.0111by(age) -0.0099 -0.0144 bh(age) 0.0058 0.0073by(educ) -0.0010 -0.0027 bh(educ) 0.0060 0.0052by(ethnic) 0.0046 0.0052 bh(ethnic) 0.0039 0.0021

Log-likelihood -13373.3 -13281.6Number of cases 10665 10665

Results: DOGIT model

Coeff SE theta/sum(theta)theta(0) 0.1712 0.0349 0.6605theta(10) 0.0134 0.0091 0.0517theta(19) 0.0592 0.0064 0.2284theta(24) 0.0000 . 0.0000theta(33) 0.0011 0.0037 0.0042theta(40) 0.0143 0.0072 0.0552

h(.) Obs CL DOGIT0 0.561 0.576 0.562

10 0.071 0.079 0.06919 0.106 0.085 0.10124 0.051 0.069 0.05633 0.075 0.086 0.07440 0.137 0.105 0.138

Budget constraints in Budget constraints in the United Kingdomthe United Kingdom

a stylised viewa stylised view

-

5,000

10,000

15,000

20,000

25,000

-

1,0

00

2,0

00

3,0

00

4,0

00

5,0

00

6,0

00

7,0

00

8,0

00

9,0

00

10,

000

11,0

00

12,

000

13,

000

14,

000

15,

000

16,

000

17,

000

18,

000

19,

000

20,

000

21,

000

22,

000

23,

000

24,

000

25,

000

26,

000

27,

000

28,

000

29,

000

30,

000

Gross income (£)

Net

inco

me

(£)

Child Benefit Net income Income Support WFTC

Results summary: CL & DOGIT compared

Some adjustments in taste parameters:

marginal utility of income increases

marginal disutility of income increases

Captivity parameters pick up UK labour market institutions

‘gravity’ at 20, 40 hours and non-employment

Issues need really to explain

non-employment more in terms of market characteristics

clearly get some such explanation through fixed cost interactions

Results: CLOE model

h(.)obs 0 10 19 24 33 40h*(.)

0 0.84 0.02 0.08 0.00 0.01 0.0510 0.69 0.07 0.12 0.00 0.03 0.0919 0.67 0.04 0.15 0.00 0.04 0.1024 0.68 0.02 0.13 0.00 0.04 0.1333 0.58 0.02 0.14 0.00 0.06 0.2040 0.50 0.02 0.12 0.00 0.08 0.28

Estimated Pr(opt)

Results summary: CLOE & DOGIT compared

Taste parameters very similar:

small increase in marginal utility of income under CLOE

differences not significant

Predicted hours distribution under DOGIT and CLOE match observed frequencies very closely

unsurprising

Issues is there a formal

equivalence between CLOE and DOGIT under certain restrictions?

not clear yet, but if so, this offers the first grounding of DOGIT in economic theory

SummarySummary

Concerned with ‘received wisdom’ in modelling household labour supply using discrete methods Stochastic structure often unchallenged Failure to recognise some pertinent labour

market issues in estimation Possibility that preference estimates and

simulated policy responses are inaccurate

SummarySummary

Paper attempts to confront directly the effect of institutional constraints on household decisions DOGIT model deals with constraints by setting

up a parametric form of labour market ‘inertia’ CLOE attempts to integrate out the (discrete)

optimising error, using tax/welfare variation Both models suggest preference estimates do

adjust, with stronger ‘unconditional’ wage & income responses.

SummarySummary

Work is still early… How do DOGIT and CLOE methods relate one

to another? How do both methods respond to admitting

correlation between adjacent state-specific errors (OGEV, DOGEV)

And to other discrete choice methods (eg. using so-called Alternative-Specific Constants, ASCs, as in nested logit)

EXTRA SLIDESEXTRA SLIDES

hours constraints and unobserved heterogeneity in structural discrete choice models of labour supply...

Documents

hours constraints

discrete hours choices

unconstrained choice

observed hours distributions

household labour supply

discrete choice methods

range of hours alternatives

choicespecific errors