Hours Constraints and Unobserved Heterogeneity in Structural Discrete Choice Models of Labour Supply Alan Duncan CPE, University of Nottingham and IFS Mark Harris University of Monash and CEU

Introduction Much of the recent literature on modelling household labour supply applies discrete choice econometric modelling methods Why? Because discrete choice methods … l allow for a series of improvements on traditional continuous methods of estimation (Hausman,1985) l offer the potential to model household decisions l cope relatively well with non-linear taxes in estimation l translate easily to ex ante evaluation of policy reform

Introduction What is the typical model of household labour supply? Economic foundations l direct estimation of well-defined preference function l preferences expressed over discrete hours choices l can model preferences at the level of the household (Van Soest, JHR 1995) l can accommodate fixed & search costs of work (Blundell, Duncan, McRae & Meghir, FS 2000) l can model welfare programme participation (Moffitt & Keane, IER 1999)

Introduction What is the typical model of household labour supply? Econometric specification issues l stochastic errors added to each discrete hours choice l errors are typically extreme value, leading to a classic conditional logit specification l potential problem: IIA l some models additionally allow for random preference heterogeneity l random heterogeneity is usually MVN, leading to a mixed logit specification (Duncan and Harris, ER 2002)

Motivation for paper PROBLEMS: 1) the typical model remains relatively unchallenged in terms of its underlying stochastic structure l choice-specific errors are not strictly necessary l extreme value assumption is potentially significant l distributions for random heterogeneity terms are not always obvious or intuitive

Motivation for paper PROBLEMS: 2) observed hours choices are generally equated to preferred hours choices – not always so. l people cannot always locate to their preferred hours choice (institutional constraints). l the range of hours alternatives from which people choose may depend on specific job characteristics l for some job types, the unconstrained choice of hours may simply not be available, or at least may be relatively unlikely

Motivation for paper IMPLICATIONS: 1)estimated models of household labour supply are not typically able to replicate the distribution of observed hours choices l one often sees peaks or clusters in observed hours distributions (0, 20, 38-40) l most household labour supply models are smooth functions, with a continuous stochastic structure. l they are therefore unable to replicate accurately such bunching in hours choices.

Motivation for paper IMPLICATIONS: 2)preference parameters may therefore be biased when institutional constraints are ignored l with optimising errors, one cannot assume that someone who is observed to work, say, 20 hours, would prefer 20 hours to any other choice; l they may have ideally liked to work, say, 27 hours, but labour market institutions prevent this l if ignored, estimated preference parameters in fact become a convolution of unconstrained tastes & labour market characteristics

Motivation for paper BACKGROUND: institutional constraints have been addressed in continuous studies of labour supply l 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

Motivation for paper BACKGROUND: institutional constraints have been addressed in continuous studies of labour supply l Their explanation: institutional constraints in the labour market prevented workers from adjusting their hours choices to suit important parameters of the tax system

Motivation for paper BACKGROUND: institutional constraints have been addressed in continuous studies of labour supply l Their solution: include optimising errors alongside random preference heterogeneity in a labour supply model, to ‘smooth’ observed outcomes around tax kinks. l optimising errors were separately identified by exploiting the non-linearities in the tax system

Motivation for paper BACKGROUND: some unusual suggestions in discrete studies of labour supply l Van Soest, Das and Gong, 1999: define a discrete (Heckman-Singer) distribution for preference heterogeneity l The number of support points assessed empirically l leads to them categorising a finite set of ‘taste types’ conditional on observed characteristics l bunching is therefore caused by types of people rather than labour market institutions

Motivation for paper OUR APPROACH: confront the presence of institutional constraints in discrete labour supply models directly l (1) model directly the degree of captivity to each observed hours alternative (DOGIT) l (2) attempt to control for optimising error in the conditional logit model by integrating the error over its (finite) empirical distribution (cf.Arrufat&Zabalza) l CLOE (Conditional Logit with Optimising Error)

an economic model of labour supply the basic model: l preferences over hours h (or ‘leisure’ T-h ) and net income y h : U=U(h, y h | X) l budget constraint: y h =w.h+I-T(wh,I; Z t )+B(w,h,I; Z b ) l missing wages: log(w)=X w b w + u w, u w has density f(u w )

discrete labour supply estimation a structural discrete model of labour supply: l Assume that hours h (.) chosen from a set of J discrete alternatives: h (.) = h 1 if h <= h 1 B h (.) = h 2 if h 1 B <= h < h 2 B ………. h (.) = h J-1 if h J-1 B <= h < h J B h (.) = h J if h > h J B l Household net incomes are calculated for each h (.) ={h 1, h 2,…, h K } as y h =w.h (.) +I-T(wh (.),I; Z t ) + B(w,h (.),I; Z b )

a discrete choice set yhyh h

yhyh h

yhyh h h (.) * =max h(.) U= U( h (.), y h | X ) a discrete choice set

discrete labour supply estimation a structural discrete model of labour supply: l Define preferences over h (.) ={h 1, h 2,…, h J } : l Choice of h (.)  {h 1, h 2,…, h J } solves max h(.) U= U( h (.), y h | X ) s.t. y h =w.h (.) +I-T(wh (.),I) + B(w,h (.),I) l Avoids the complexities of nonlinear functions T(.) & B(.) l Problem? Introduces rounding errors through h =h (.). l Needs testing…

functional form choice for U(.): l 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) specifying preferences

unobserved preference heterogeneity l Observed and unobserved heterogeneity enters through preference terms. eg, l Unobserved heterogeneity in preferences is typically assumed multivariate normal. discrete choice: econometric estimation

controlling for costs of employment l fixed costs are FC are incurred for all choices which involve work l parameterise fixed costs in terms of a set of observed characteristics and a stochastic element: FC = X fc  fc + u fc l modify preferences for those in work, h (.) >0 : U=U(h (.), y h - FC | X ) discrete choice: econometric estimation

state-specific disturbances l Introduce a stochastic component to preferences for each discrete hours alternative: l By assuming a distribution for each u h, one can derive an expression for the probability Pr(h (.) = h j | X, v, u fc ). discrete choice: econometric estimation

deriving likelihood contributions l Pr(h (.) = h j | X, v) = Pr[ U hj * =max(U hk * for all k=1,..,J)]. l If each u h is extreme value, l Ignoring random components, this is exactly analagous to the conditional logit specification (eg.McFadden,1984) discrete choice: econometric estimation

deriving likelihood contributions l Taking account of random components, we can integrate the likelihood over the distributions of w and v l Assuming independence of w and v, l Stochastic structure makes this more akin to (non-IIA) mixed logit (eg. McFadden and Train, 2000) discrete choice: econometric estimation

the basic likelihood function l In estimation, this integral is approximated using simulation methods discrete choice: econometric estimation

The DOGIT model (Manski 1977) l 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 l Captivity is parameterised in terms of a series of (non- negative) parameters    J discrete choice: controlling for institutional constraints (1)

The DOGIT model (Gaudry and Dagenais) l 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 l Captivity is parameterised in terms of a series of (non- negative) parameters    J discrete choice: controlling for institutional constraints (1)

Characteristics of the DOGIT model l Parameters    J denote the ‘degree of captivity’ l The Dogit reduces to the CLogit when  j = 0 for all j l If we denote l Then discrete choice: controlling for institutional constraints (1)

Characteristics of the DOGIT model l From manipulation of it can be shown that l That is, the Dogit model draws towards states where the CL probability is ‘small’ relative to the size of the corresponding captivity parameter

discrete choice: controlling for institutional constraints (1) Characteristics of the DOGIT model l Fixed costs and random preference heterogeneity can be introduced in the same manner as for the CLogit l Can test CLogit against DOGIT using standard LR methods l Possible drawbacks: (1) degree of captivity doesn’t depend on observed factors (2) no background in economic theory

discrete choice: controlling for institutional constraints (2) CLogit with optimising error l An alternative approach to the Dogit is to specify directly the distribution of the optimising error  opt in a discrete choice framework l By analogy with the continuous approach of Arrufat and Zabalza (E’trica 1986), we ‘integrate out’ the optimising error over its discrete range l 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).

discrete choice: controlling for institutional constraints (2) CLogit with optimising error l Let discrete observed hours be h obs(.) ={h 1, h 2,…, h J } l Let discrete ‘desired’ hours choice be h * (.) ={h 1, h 2,…, h J }, defined according to the CL max rule l Then, h obs(.) = h * (.) +  opt l Optimising error will be drawn from the set  opt = { h k - h s, for k,s, =1,…, J } l Parameterise the finite probability distribution of  opt as follows: Pr (  opt = h k - h s )=  ks l Probability:

discrete choice: controlling for institutional constraints (2) CLogit with optimising error l Example: h obs(.) ={0, 10, 20, 30, 40} h * (.) ={0, 10, 20, 30, 40}

discrete choice: controlling for institutional constraints (2) CLogit with optimising error l Example: h obs(.) ={0, 10, 20, 30, 40} h * (.) ={0, 10, 20, 30, 40}

discrete choice: controlling for institutional constraints (2) Characteristics of CLogit with optimising error l Nests the standard CL:  kk = 1,  ks = 0 for k not equal s l Institutional constraints can vary depending on h * (.) l Identification must come from variation in tax / welfare payments (cf. Arrufat and Zabalza 1986) l Closer integration with economic theory

Data and identification l Data: Family Resources Survey: 39,000 mothers from 1994 – 2000 l Select single parent families only l Exclude self-employed, students, pensioners, sick/disabled, women on maternity leave l Leaves sample of 10,665 single parents l Period embraces substantial tax/welfare reform l Variation over time and across individuals l age, number and age of children, housing tenure, education

Data and identification l Estimation: CML for GAUSS l Six hours alternatives h={0,10,19,24,33,40} l Simulated ML for random preference heterogeneity specifications l Constraints directly imposed on DOGIT & CLOE parameters

Results: CL model

Results summary: CL model l Temporary simplifications: l Estimate wages first and draw random realisation l Fixed costs are deterministic l Preferences for income l Increase with number of children, age of youngest l Reduce with age and education l Fixed costs l decrease with age of youngest child l vary by region l Distaste for work l increases with number of children l reduces with age and education

Results: CL & DOGIT compared

Results: DOGIT model

Budget constraints in the United Kingdom a stylised view

Results summary: CL & DOGIT compared l Some adjustments in taste parameters: l marginal utility of income increases l marginal disutility of income increases l Captivity parameters pick up UK labour market institutions l ‘gravity’ at 20, 40 hours and non-employment l Issues l need really to explain non-employment more in terms of market characteristics l clearly get some such explanation through fixed cost interactions

Results: CLOE model Estimated Pr(  opt )

Results summary: CLOE & DOGIT compared l Taste parameters very similar: l small increase in marginal utility of income under CLOE l differences not significant l Predicted hours distribution under DOGIT and CLOE match observed frequencies very closely l unsurprising l Issues l is there a formal equivalence between CLOE and DOGIT under certain restrictions? l not clear yet, but if so, this offers the first grounding of DOGIT in economic theory

Summary l Concerned with ‘received wisdom’ in modelling household labour supply using discrete methods l Stochastic structure often unchallenged l Failure to recognise some pertinent labour market issues in estimation l Possibility that preference estimates and simulated policy responses are inaccurate

Summary l Paper attempts to confront directly the effect of institutional constraints on household decisions l DOGIT model deals with constraints by setting up a parametric form of labour market ‘inertia’ l CLOE attempts to integrate out the (discrete) optimising error, using tax/welfare variation l Both models suggest preference estimates do adjust, with stronger ‘unconditional’ wage & income responses.

Summary l Work is still early… l How do DOGIT and CLOE methods relate one to another? l How do both methods respond to admitting correlation between adjacent state-specific errors (OGEV, DOGEV) l And to other discrete choice methods (eg. using so-called Alternative-Specific Constants, ASCs, as in nested logit)

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