Evaluating Alternative Representations of the Choice Set In Models of Labour Supply Rolf Aaberge,...
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Evaluating Alternative Representations of the Choice Set In
Models of Labour Supply
Rolf Aaberge, Ugo Colombino and Tom Wennemo
Workshop on Discrete Choice Labour Supply Models 8-9 December 2005, Galway
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Purpose
To evaluate alternative methods of representing the choice set in models of labour supply
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Background
• Accounting for complicated (non linear, non convex etc.) constraints in modelling behaviour
• Heckman 1974• Hausman 1979, Moffit 1986 etc.• The ”Hausman Approach”• Problems with the ”Hausman Approach”
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The ”Hausman Approach”
• Choice is characterized by K.T. conditions for constrained utility maximization
• Complicated and computationally cumbersome with non-convex budget constraints and more than one decision maker or more than two goods
• The problems above contributed to the popularity of a different approach
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The “Random Utility” Approach
• McFadden 1974• Dagsvik 1994• Van Soest 1995• Aaberge, Dagsvik and Strøm 1995
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The “Random Utility” Approach
• Any opportunity set – however complicated - can be represented by a set of points hi, Ci
• Specify a random utility v(hi, Ci)+i with i i.i.d. Type I Extreme Value
• Then the p.d.f. of choosing (hi, Ci) is
exp(vi)/kexp(vk)
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Alternative ways of implementing the “Random Utility” Approach
• Fixed vs Sampled Opportunities
• Number of Opportunities
• Uniform vs Non-Uniform Availability of Opportunities
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Fixed opportunities vs Sampled opportunities
• Potentially too many opportunities (e.g. 0, 1,..., 8760 hours of work for each person in the household)
• So, choose a subset of opportunities, but how?
• And, how many?• Most common practice: choose a subset
of fixed opportunities (e.g. 0, 100, 200 etc.)
• More appropriate procedure: sample a subset of opportunities according to some probabilistic rule (McFadden 1978)
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Fixed opportunities vs Sampled opportunities (cont.)
• Sampling appears as a more appropriate procedure, however it’s more time consuming
• In the labour supply literature, the fixed opportunities procedure is by far the most commonly used one
• Very little analysis and no systematic testing has been done of the empirical implications of the different procedures (Ben-Akiva and Lerman 1985)
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Availability of alternatives:uniform vs non-uniform density
• Some opportunities are more easily accessible than others
• The accessibility of opportunities might vary also across individuals
• How do we account for that?
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Availability of alternatives:uniform vs non-uniform density
(cont.)
• Most contributions to the labour supply literature do not account at all for heterogenous accessibility
• Some studies simply assume that only a few opportunities are available, in the same way for everybody (e.g. non participation, part-time, full-time)
• Some other studies assume a non uniform density of opportunities, in general differing across individuals
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Accounting for non-uniform density of opportunities
Let p(h,C) be the density in the choice set of opportunities of type (h,C).
The p.d.f. of a choice (h,C) can be written as follows:
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,
exp( ( , )) ( , )exp( ( , )) ( , )
x y
v h C p h Cv x y p x y
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The simulation exercise
• Generate a sample based on a “true” model
• Estimate various models with different choice-set representations
• Evaluate the estimation and simulation performance of the models
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The “true” model
• As the “true” model we take a model estimated on a sample of married/cohabitating females from the Norwegian 1995 Survey of Level of Living.
• It belongs to a class of various models we estimated for Norway, Sweden and Italy.
• Using the parameters estimated for married females, we generate a sample of 10000 observations.
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The estimated models
16 models defined according to:• Generation of opportunities: fixed vs
sampled• Number of opportunities: 6 vs 24• Availability of part-time and full-time
jobs: uniform vs non-uniform• Availability of jobs: uniform vs non-
uniform
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The estimated models (cont.)
• The “true” model assumes that choice sets vary across individuals and that availability of opportunities is not uniform
• Therefore it is expected that the models accounting for non uniform availability of opportunities perform better
• The issue we want to address is “how much better” and under what circumstances.
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Evaluating the models
210
1
( )
kj jk
j j
y yz
y
210
1
( )
kj jk
j j
y yz
y
210
1
( )
kj jk
j j
y yz
y
210
1
( )= relative squared error of model
= prediction of the "true" model for the j-th income decile
= prediction of model for the j-th income decile
In this exercise
kj j
kj j
j
kj
y yz k
y
where
y
y k
we predict income.
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Evaluating the models (cont.)
210
1
( )
kj jk
j j
y yz
y
210
1
( )
kj jk
j j
y yz
y
210
1
( )
kj jk
j j
y yz
y
0 1 1 2 2 3 3 4 4 34 3 4
1
2
3
4
( * )
= 1*(sampled alternatives)
= 1*(24 alternatives)
= 1*(non uniform jobs-availability )
= 1*(nonuniform part-time and full-time jobs-availabil
k k k k k k k
k
k
k
k
z x x x x x x
where
x
x
x
x
ty
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Contributions to the prediction error:prediction of net income under the
current tax regime
Basic model
Sampled opportunities
24 opportunities
Non-uniformJobs availability
Non-uniformPartime and Fulltime availability
Interaction
Estimate .021 -.002 .007 -.009 -.002 -.003
Standard deviation
.008 .004 .004 .006 .006 .008
Even the basic model performs very well. More complicated representations of the choice set do not significantly contribute to a better fit
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Contributions to the prediction error:prediction of net income after a flat tax
reform Basic model
Sampled opportunities
24 opportunities
Non uniformJobs -availability
Non uniformPartime and Fulltime -availability
Interaction
Estimate .128 -.019 -.008 -.029 -.039 .036
Standard deviation
.006 .005 .005 .007 .007 .010
Sampled opportunities, a larger choice set and accounting for heterogeneous availability of opportunities significantly contribute to reduce the prediction error
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Evaluating the models: Relative
computation time (estimation)
210
1
( )
kj jk
j j
y yz
y
210
1
( )
kj jk
j j
y yz
y
6 opportunities 24 opportunities
Fixed opportunities
1 4.62
Sampled opportunities
6.70 8.46
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Conclusions
• Random Utility models fit very well observed values, even the basic versions with fixed opportunities, and a very small choice set
• When it comes to predicting policy effects, models using sampled opportunities and accounting for heterogeneous availability of opportunities perform significantly better (at the price of larger computational costs)
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Heckman J. (1974): Effects of Child-Care Programs on Women’s Work Effort, Journal of Political Economy 82, 2-II, 136-163.
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Hausman, J.A. (1979): "The Econometrics of Labour Supply on Convex Budget Sets", Economic Letters, 3.
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Moffitt, Robert (1986): "The Econometrics of Piecewise-Linear Budget Constraints: A Survey
and Exposition of the Maximum Likelihood Method," Journal of Business & Economic
Statistics, American Statistical Association, vol. 4(3), pages 317-28.
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McFadden, D. (1974): "Conditional Logit Analysis of Qualitative Choice Behavior", in P. Zarembka (ed.), Frontiers in Econometrics, Academic Press, New York.
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Dagsvik, J.K. (1994): "Discrete and Continuous Choice, Max-stable Processes and Independence from Irrelevant Attributes", Econometrica, 62, 1179-1205.
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van Soest, A. (1995): "Structural Models of Family Labor Supply: A Discrete Choice Approach”, Journal of Human Resources, 30, 63-88.
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Aaberge, R.., J.K. Dagsvik and S. Strøm (1995): "Labor Supply Responses and Welfare Effects of Tax Reforms", Scandinavian Journal of Economics, 97, 4, 635-659.
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McFadden, D. (1978): "Modelling the Choice of Residential Location" in A. Karlquist, L. Lundquist, F. Snickard and J.J. Weilbull (eds.): Spatial Interaction Theory and Planning Models, Amsterdam, North-Holland.
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Ben-Akiva, M., and Lerman, S.R. (1985): Discrete choice analysis, (MIT Press, Cambridge).
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References
• Aaberge, R.., J.K. Dagsvik and S. Strøm (1995): "Labor Supply Responses and Welfare Effects of Tax Reforms", Scandinavian Journal of Economics, 97, 4, 635-659.
• Aaberge, R., U. Colombino and S. Strøm (1999): “Labor Supply in Italy: An Empirical Analysis of Joint Household Decisions, with Taxes and Quantity Constraints”, Journal of Applied Econometrics, 14, 403-422.
• Aaberge, R., U. Colombino and S. Strøm (2000): “Labour supply responses and welfare effects from replacing current tax rules by a flat tax: empirical evidence from Italy, Norway and Sweden”, Journal of Population Economics, 13, 595-621.
• Aaberge, R., U. Colombino and S. Strøm (2004): "Do More Equal Slices Shrink the Cake? An Empirical Investigation of Tax-Transfer Reform Proposals in Italy“, Journal of Population Economics, 17