Labour Supply: Empirical Literature and Methods

Some sources:

- Berndt, Ch. 11 provides a good discussion of most issues. - Any general econometrics book will have a chapter dealing with Probit and Tobit models

models e.g. Wooldridge Ch. 7 (section 7.5) and 17 of the 4th edition (see website) . - B. Heim (2007) 'The Incredible Shrinking Elasticities' Journal of Human Resources XLII (a recent

study employing many of the methods).

Estimation of a Labour Force Participation Model for Individuals

- Microdata: - survey data- individual person is the unit of observation.

- Dependent variable (Labour force dummy: Yi where ‘i’ denotes individual i):

- only two values: participant (=1), non-participant (=0)

- sometimes an employment dummy is used instead of a labour force dummy.

- Example specification:

Yi = b0 + b1 x1i + b2 x2i + ei

- explanatory variables (x): suggested by theorye.g. indicators of tastes, value of home time,

non-labour income, wages.

- residual: ei= residual for observation i

Why a residual? relationship is inexact: measurement errors, reporting errors, unobserved or omitted variables.

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Estimation of a Model with a Dummy Dependent Variable

(a) Could estimate using Ordinary Least Squares (OLS):

- OLS with a dummy dependent variable is the Linear Probability Model (see: Wooldridge discussion Ch. 7)

- choose b0, b1, b2 to minimize the sum of the squared deviations of the fitted line from actual Yi.

- OLS: the equation estimated by OLS goes through sample means.

Y = b0 + b1 x1 + b2 x2 (^ for OLS estimate, bar denotes mean)

Note that where N is number of observations:

Mean of Yi =Y = ∑i Yi/N = share of the sample with Yi=1.

i.e. an estimate of the probability Yi=1 ( Pr(Yi=1) )

- interpretation of coefficients?

b1 = effect on the share participating of a change in x1

i.e. measures the effect of the variable on the probability of participation.

(b1=∂ Pr (Y i=1)

∂ x1 )

e.g. from Wooldridge (LF=labour force dummy NLI non-labor income, EDUC=years of education, EXPER=years of work experience)

LF = .586 -.0034 NLI +.038 EDUC +.039 EXPER-.0006 EXPER2-.016 AGE-.262 KIDSLT6

A one year rise in education raises labour force participation probability by .038 (3.8%)Having children under age 6 (KIDSLT6=1) reduces the probability of participation by .262

(-26.2%)

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- Problems with OLS when the dependent variable is a dummy variable?

- nonsense results are possible:

- predicted Yi can be outside the 0-1 interval: nothing prevents this.

- marginal effects: Pr(Yi=1)/x1, Pr(Yi=1)/x2 - constant regardless of starting value of Pr(Yi=1)- could take you outside the 0-1 interval.

- ei are not normally distributed

- only two values are possible for given xi

Yi=1: ei = 1 -b0 - b1 x1i - b2 x2i Yi =0: ei = -b0 - b1 x1i - b2 x2i

- raises questions for statistical inference (usual t and F tests assume normality).

- heteroskedastic errors: variance differs with values of x1, x2

Var(ei) = (b0 + b1 x1i + b2 x2i) (1-b0 - b1 x1i - b2 x2i)

- estimates will be consistent but not efficient.

- How serious are these problems?

- linear probability estimates are often not too different from approaches below.

- if interest is in average effects of x’s results may be reasonable.

- bias is more serious at the extremes.

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(b) Alternative: Specify the probability of participation (non-participation) as a function of xi.

- Let “F” be the cumulative distribution function (CDF) then.

- Probit model: F() is the cumulative normal- Logit model: F() is the cumulative logistic.

- Let: Ii * = b0 + b1 x1i + b2 x2i + ei

I i * = unobservable, "latent" variable reflecting propensity to participate

with: Ii *> 0 then Yi =1 (participation)

Ii * 0 then Yi =0 (non-participation)

- Participation decision and Labour Supply theory:

- theory: participate if reservation wage (w*) < market wage (w)

- So I* has a natural interpretation:

Ii* = (wi-wi*) > 0 participate

= (wi-wi*) 0 non-participant

Now:

Prob(Yi=1) = Prob( b0 + b1 x1i + b2 x2i + ei >0 )

= Prob(ei > -b0 - b1 x1i - b2 x2i )

= F(b0 + b1 x1i + b2 x2i ) for a symmetric distribution

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Prob(Yi=0) = Prob(b0 + b1 x1i + b2 x2i + ei 0)

= Prob(ei -b0 - b1 x1i - b2 x2i )

= 1- F(b0 + b1 x1i + b2 x2i )

if: ei distributed std. normal: probit model ei distributed logistic: logit model

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- Estimation? - "maximum likelihood": choose b0, b1, b2 to maximize the

likelihood (probability) of observing the actual sample.

Likelihood of observing the sample (assume outcomes independent across i):

F(b0 + b1 x1i + b2 x2i) [ 1-F(b0 + b1 x1i + b2 x2i) ] iYi=1 iYi=0

where: product of following term, taken over all with Yi=1iYi=1

product of following term, taken over all with Yi=0iYi=0

- estimate parameters by maximizing the likelihood with respect to b0 , b1 , b2

Equivalently maximize the log-likelihood (take logs of the likelihood function):

log F(b0 + b1 x1i + b2 x2i) + log [ 1-F(b0 + b1 x1i + b2 x2i) ] iYi=1 iYi=0

- Advantage?

- Estimating ordinate of the CDF: predicted probabilities are constrained to the 0-1 interval.

- Avoids the statistical problems of the linear probability model.

- Disadvantage?- Computationally more demanding.

- Interpretation of coefficients more complicated.

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- Marginal Effects: effect of a change in x on Pr(Yi=1):

- Linear probability:

b1 = Prob(Yi=1)/x1i

- simple: b1 is the effect of x1 on the probability of Yi=1.

- Probit or logit function:

- If x1 is a continuous variable:

Prob(Yi=1)/ x1i = b1 f(b0 + b1 x1i + b2 x2i)

- f() is the probability density function

- coefficients have a less straightforward interpretation

- marginal effect on probability of participation depends on values of the person’s x’s.

i.e. highest for f(0), lowest for f() evaluated at extremes.

- If x1 is a dummy explanatory variable the marginal effect is:

F(b0 + b1 ∙1 + b2 x2i) - F(b0 + b2 x2i)

i.e. probability of participation with x1=1 vs. x1=0.

(again values depends on where it is evaluated)

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- Conventions for reporting marginal effects?

- could evaluate marginal effect at mean values of x's (often done).

- could calculate the marginal effect for each observation and then take the sample average of the individual measured effects.

- could evaluate the effect starting at a given Prob(Y=1).

(STATA: a number of options available under the "margins" command)

- Logit vs. Probit (normal)- very similar distributions: symmetric, logit fatter in tails- logit computationally easier (closed form for integrals)- probit has become more common with increased computer power.

Standard specifications of the participation model (choice of Xs):

- What explanatory variables should be included in the participation model?

- Non-labour income

- would include all income other than the person’s own wages.

- should income of other family members be included? (model of family and this)

- Determinants of value of time (reservation wage: w*)

- indicators for preferences or value of home time: age, education, social/cultural characteristics, family status variables.

- Wage rate (w):

- key variable according to theory

- problem? not observed for non-participants- solutions to this lack of wage data?

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(1) predict the wage based on observed variablese.g., education, work experience

- to identify wage effect requires a variable that affects wage but has no other effect on participation (like z1 below).

e.g. Yi = b0 + b1 x1i + b2 x2i + b3 wi + ei

say w is predicted using: wi = a0 + a1 x1i + a2 z1i + ni

(x1 and z1 must be observed for participants and non-participants)

substitute the predicted wage wi=a0 + a1 x1i +a2 z1i for wi in Yi:

now: Yi = b0 + b1 x1i + b2 x2i + b3 (a0 + a1 x1i + a2 z1i) + ei

collect terms:

Yi = (b0+ b3 a0) + (b1+ b3 a1) x1i + b2 x2i + (b3 a2) z1i + ei

estimation of this equation does allows you to identify b3 (wage effect on participation) as coefficient on z1i in participation equation divided by coefficient on z1i in the wage equation.

(notice if wage equation did not contain z1i as a regressor b3 could not be identified)

(2) reduced form approach:- acknowledge role of wage- treat as if wage function has been substituted into the participation

equation.- interpret coefficients accordingly i.e. some may at least partly

capture wage effects.

- Heim (2007): adopts both of these solutions in his study of US women.

- Examples of participation equation estimates: - Heim (2007) US data (see above)- Gunderson (1998) Women in the Canadian Labour Market uses Canadian Census data.- Meyer and Rosenbaum (2001) discussed below adopts a similar specification with

employment as the dependent variable.

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Estimation of Hours of Work Equations

- Key interest: value of the elasticity of labour supply to wages and non-labour income.

- key economic variables of the model

- policy applications: taxes, income support programs.

- Dependent variable? Some measure of hours of work

- Usually: Annual Hours of Work

- Alternatives? Weekly or monthly hours.

Estimation Issues: Elasticities and functional form

- Log-log version: coefficient estimates can be interpreted as elasticities.

ln(Hi) = a + b ln(wi) + …

b = ln(Hi)/ln(wi) is an elasticity.- elasticity here is constant for all Hi, wi.

- Semi-log version: Hours in levels, wages in logs

Hi = a + b ln(wi) + …

b = Hi/ln(wi) - divide by Hi to obtain an elasticity estimate- elasticity estimate varies with value of Hi.

- Hours and wages in levels:

Hi = a + b wi + …

b = Hi/wi - multiply by (w/H) to get elasticity.- elasticity estimate varies with w and H.- often evaluated at sample means of H and w.

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- Choice between functional forms?

- Statistical criterion: - which best fits the data.

- tests to determine best choice: e.g., Reset test, LM tests of Box-Cox transformation.

- In practice: studies tend to follow past practice in the literature (log-log and semi-log are common)

Estimation Issues: Treatment of Non-Participants

Option 1: Include them with Hi=0

- Common in older studies.

- Limit value on H will likely result in biased estimates.

- Hours relationship breaks down at Hi=0.

e.g. Hi = a + b Xi + ei , say b>0 for diagrams and discussion

- Xi a regressor, ei = error term

- consider fitting this to a scatterplot of Hi, Xi obs.

- assume many Hi=0 observations.

- OLS is likely to underpredict |b|

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- Inconsistent with theory:

- theory suggests separate participation and hours relationships.

- participation: w>w* or not.

- hours relationship: H = H(w, I, tastes) given that are a participant (w>w*).

Option 2: Ignore Non-participants (delete them from the sample)

- There is a “sample selection” problem giving results much like Option 1.

- censoring (deleting) non-participants selects the sample in a non-random way.

- censoring rule:

Hi = a + bXi + ei > 0

Implies:

ei > -a –bXi

(Technically: deleting the lower tail of the distribution of e; if e is normaland is the std. deviation the mean of e on the selected sample is: f[(-a-bXi)/] / [1-F[(-a-bXi)/]] = f[(a+bXi)/] / F[(a+bXi)/]; the

ratio f/F is the "inverse Mills ratio" often just called λ)

- Omitting H=0 observations implies that the error term will be correlated with Xi.

- OLS estimates of the equation will be biased: b partly reflects b and partly cov(ei, Xi).

- Options 1 and 2 are both problematic!

- Need methods that account for the problems above.

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Solution 1: Tobit Model Estimation

- The Tobit model is a standard solution to “limited dependent variable” problems of this kind.

- explicitly allows for the presence of a limit value.

- specify probabilities of observing each type of observation:(1) those at the limit value;(2) those at any other values.

- use maximum likelihood to obtain parameter estimates.

- Tobit specification of the labour supply model:

Hi* = a + b Xi + ei ei - normally distributed error term

Hi* is a "latent" variable (“tendency to supply labour hours”) with:Hi=Hi* if Hi*>0Hi=0 if Hi*0

Likelihood of observing Hi=0 observation:

Prob(Hi*0) = Prob(ei -a-bXi)

= F[(-a-bXi)/] F – cumulative normal. – std. deviation of ei

Likelihood of observing Hi=H0

Prob(Hi=H0) = f(e0/) f – normal densitye0 = H0 – a - bXi

Likelihood of the sample:

f(ei/) F[(-a-bXi)/] iH>0 iH=0

maximizing this with respect to a, b and gives the maximum likelihood estimates of a, b and .

- This avoids the problems mentioned earlier.

- Interpretation of the tobit coefficient estimates and calculation of marginal effects:13

- "b" estimates the effect of X on H* not H.

- the effect of X on H (including H=0 observations) is measured by: b F((a+bX)) (see Wooldridge eqn. 17.31)

- the effect of X on H for those with H>0? - Note: E(Hi|Hi>0)=a+bXi +

- So: E(Hi|Hi>0)/Xi = b + Xi = b∙(1-λ(a+bXi)- λ2)

(see Wooldridge eqn. 17.26)

= the inverse Mill's ratio f((a+bXi)/)/F((a+bXi)/)

- There are many economic applications of tobit:

- Any dependent variable with limit values and many observations at the limit.

- Earnings functions with top-coded data or minimum wages.

- Consumer durables spending, charitable giving.

- The tobit model example above has a lower limit on H.

- version with an upper limit follows the same logic.

- easily generalized to versions with:- both an upper and lower limit - that has limit values that differ across observations.

- truncated regression model is quite similar (rather than having observations at some limit value, observations below or above some value are not observed).

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Solution 2: Two-step Sample Selection Correction Procedures

- A common solution to sample selection problems.

Step 1: Estimate the censoring rule (rule determining if at limit value)

- Generally censoring rule (selection equation) is a dummy dependent variable:= 1 if in sample of interest= 0 if not in sample interest.

- Participation equation is the selection rule in the labour supply model:Participant = 1Non-participant = 0

- Estimate this using probit.

Step 2: Estimate the Hours equation including an estimate of “Heckman’s lambda” as a regressor.

- If you estimate:

Hi = a + bXi + ei

- the expected value of e is non-zero on the censored sample:

E(ei | Hi>0) = E(ei | a + bXi + ei >0) 0

- The error term is correlated with Xi due to the form of the censoring rule.

- Estimates of a and b ignoring this will be biased.

- Heckman’s solution?

- Adjust the error term to remove the non-zero expectation.

- Include an estimate of E(ei | Hi>0) as a regressor.

- Estimate is obtained from the Step 1 model.

- How?- In hours sample if: Hi = a + bXi + ei > 0

ei > - a - bXi

- so: e is in the upper tail of the error distribution.

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- A result from statistics for truncated normal distribution:

(truncated? without one of the tails).

- mean of a “truncated” normal distribution (λ)

f( (-a-bXi)/ ) = f( (a+bX)/ ) 1- F[(-a-bXi)/] F[(a+bXi)/]

- the step 1 probit model can provide an estimate of (λ). (inverse Mills’s ratio)

- then at step two you estimate:

Hi = a + bXi + cλ i + i

- on the Hi>0 sample; now asymptotically E(i | Hi>0)=0 .

(see Heim (2007) for a recent example using this method)

- Two-step approach in other applications: a difference

- In this application the selection rule (participation equation) and the hours equation both reflect labour supply behaviour.

(so: same regressors, closely related coefficient estimates and error terms)

- In other applications the two relationships may reflect different behavior.

e.g. estimation of a wage equation. Wi=b0 + b1 Xi + b2 Yi+ ei

- 1st step: censoring rule: wage earner or not. - 2nd step: estimate a wage equation with λ estimate included.

- logic is the same as above: ignoring selection non-zero expected value of wage equation error term.

- correction is as above: estimate Wi=b0 + b1 Xi + b2 Yi+b3λ i + ei

- technical difference: non-zero expected value of error term in the wage equation reflects the correlation between the error term of the

selection equation and the error term of the wage equation).

E(ei | wage earner) = i

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captures correction between the wage and censoring rule error terms

- Tobit vs. Two-step?

- Two-step has become more common.

- Objection to the tobit?

- assumes a continuous relationship between hours and explanatory variable right up to Hi=0.

- discontinuities could exist: - fixed cost - income support programs.

- Two-step is less restrictive.

- Identification issues with two step:

- is there an explanatory variable in the censoring rule that is not an explanatory variable in the stage 2 regression?

- If not λ i is just a non-linear combination of the stage 2 explanatory variables.

- is it capturing selection effects or non-linearity?

- see earlier discussion of wage effects in participation equation.

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Estimation Issues: Identifying Wage Effects

- Wage rate: - a key variable in the theoretical model- policy analysis: wage elasticity a key parameter.

Participation model (see earlier discussion p. 9)

- Problem? - Wage is not observed for non-participants.

- Solution? - Predict the wage rate (w).- Include the predicted wage as a regressor in the participation model (see above

p.9)

- If predicting wages using: wi = a0 + a1 x1i + a2 z1i + ni

x1i and z1i must be observed for both participants and non-participants.

- To identify wage effects on participation (i.e. to estimate b3) in:

Yi = b0 + b1 x1i + b2 x2i + b3 wi + ei , where Y is participation dummy.

– there must be at least one variable in the wage regression that affects participation only through its effect on wages.

- this can be problematic: are there good candidates for wage equation regressors that don’t affect participation directly?

(Heim: uses higher order terms in age and education)

- Sample selection bias.

- Wage equation is estimated on the same non-random sample as the hours equation.

- Selection issues arise in estimating the wage equation.

e.g. say only see w if H>0. Then likely that: E(ni| Hi>0) 0

- this is likely to be correlated with wage equation regressors.

(remember Hi>0 implies wi>wi* )

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- possible solution? - Heckman two stage procedure.

Hours of Work Equation and Wage Effects

- Say: Hi = a + bXi + cWi + ei

- Here the wage is observed for participants.

- But as above say that:

wi = a0 + a1 x1i + a2 z1i + ni

- likely that e and nwill be correlated.

i.e. unobserved characteristics that boost wages could also affect hours worked. (e.g. say “work ethic” is unobserved)

- so if just include actual wi as a regressor wi and ei correlated: estimates of hours equation will be biased.

- Solution? - use an instrument for W

e.g. an estimate of the wage equation (as in the probit discussion).

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Estimation Issues: Kinked Budget Constraints (time permitting)

- Berndt discusses this.

- Moffitt (1990) “The Econometrics of Kinked Budget Constraints” Journal of Economic Perspectives provides more detail.

- Taxes and income support programs the usual focus.

- New Problem? - choice of H determines, tax bracket and post-tax wage.

H = H( w(1-t) )

- choice of H depends on post-tax wage.

w(1-t) = f(H)

- simultaneity!

- same issue arises with “virtual income”.

Virtual income?

- hypothetical income associated with each budget segment.

- OLS estimates ignoring this will be biased.

e.g.,

- Shape of budget set (with progressive taxes) implies:

- Low-post tax wage associated with high H.

- High-post tax wage associated with low H.

- Labour supply elasticity estimates tend to be biased downward because of this.

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- Solutions?

- Instrumental variables:- a standard solution for simultaneity problems.

- replace: w(1-t) = f(H)

- with an instrument correlated with w but which is not determined by the choice of H.

e.g., w(1-t) = f(H*)

H* = an estimate of H.

- Advantage: computationally easy.

- Disadvantage:- estimated post-tax wage inaccurate.- ignores kinks.

- Maximum likelihood specification recognizing the endogeneity problem.

- specify the likelihood that someone with observed characteristic X could end up on each segment of the budget set or at each kink.

- maximize the likelihood of the observed sample by choosing the hours equation coefficients and other parameters.

- Advantage: formally allows for complexity of the choice problem.

- Disadvantage:- computationally difficult.

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Hours Equation Estimates: Literature

- Literature Summaries / overviews :- Heim (2007) for US women.- Mroz (1987) – discussed in Berndt.- Heckman (1993) “What has been learned about labour supply in the last 20 years?”- Blundell and MaCurdy (1999)

- General conclusions:

- Distinction between the hours and participation decision is important. - Participation more elastic with respect to wages, non-labour income.- Implication: tobit model of labour supply too restrictive.

- Sample selection issues are empirically important.

- Size of estimated elasticities:

Older literature: - men small wage and income elasticities.- women labour supply somewhat more elastic.- old studies: mixed participation and hours estimates.

Newer literature: small hours elasticities for men and women (larger for women).(close to 0)

participation is more responsiveHeim: US female elasticities are growing smaller.

Tax effects: 1980s studies with large elasticities appear to be sensitive to minor changes in estimation methods.

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Experimental literature and Natural Experiments:

- Some studies use experimental data to study labour supply behaviour.

e.g., 1970s : Negative Income Tax experiments.

1990s: U.S.: unemployment insurance experimentsCanada: “Self-sufficiency project” (handout)

- Experimental approach:

(1) use random assignment to divide experimental sample into:

- treatment (experimental, program) group

- control group

(2) observe initial labour supply outcomes of the two groups.

(3) subject the treatment group to the “treatment”

e.g., some event of interest, a policy change like an income supplement or minimum wage rise.

(4) observe labour supply outcomes of the two groups after “treatment”.

(5) measure effect of the “treatment” as the difference in behaviour changes of the treatment and control groups.

i.e. (4)-(2) for the treatment group minus (4)-(2) for the control group.

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- “Natural Experiments”

- Not real experiments: conducted on non-experimental data.

- Event occurs, e.g., policy change

- one group is obviously affected by the change (“treatment group”).

- there is another “similar” group not affected by the change.

- Card suggests using this latter group as a control group.

- Measure effects of the change via the “differences in differences”.

i.e., compare difference in before and after behaviour of the group affected with that of the “control” group.

- Say y is the measure of interest where:

For person i in the treatment group T the difference in measured behaviour between time t-1 and t is:

Δy iT= y it

T− y i , t−1T

For person i in the control group the difference in behaviour is:

Δy iC= y it

C− y i , t−1C

The difference-in-differences estimator (D) is then:

D=Δ y iT−Δ y i

C

where the bars denote mean of the differences for the treatment and control groups.

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- Key assumptions of the Difference-in-Differences (DID) estimator:

- Control group is a good proxy for treatment group behaviour

i.e. changes in its behaviour reflect how treatment group behaviour would have changed if the treatment had not occurred.

- checking this? Did behaviour of the two groups change in similar ways before the “treatment”.

- So treatment and control groups must experience same time effects

(independent of the “treatment “event or policy)

- otherwise if they are subject to different trends the DIDestimator will in part capture these different trends.

- Composition of the treatment and control groups must remain stable before and after the event under study.

- otherwise estimate mixes in composition effects.

- Interval between measurements must be sufficient to capture lagged responses or lead responses.

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Meyer and Rosenbaum (2001) “Welfare, EITC and the Labor Supply of Single Mothers” Quarterly Journal of Economics.

- A labour supply example of the natural experiment approach.

- Event studied? - US tax, welfare policy changes.

- Data:Current Population Survey 1984-1996.

- Whether the person works or not the focus:- participation model underlying this.- focus on employment and employment rates (ignore unemployment).

- Treatment group: single mothers- Policy changes should have major effects on them (in theory).

- Control group: single women without children.

- Differences-in-differences estimator: - difference between treatment and control in the change in employment rates.- see Table II. - National Tax Journal version of the paper: several control groups experimented with.

- Table III: extending the method by controlling for other factors

- Probit model of employment status E=1 for person i, year t:

Pr(Eit=1) = [ aXit + bt YEARt +ct (YEARt ANYKIDSt) ]

- X includes: age, education, race, state unemployment rate, family variables.- Year dummies - Year dummy multiplied by kids dummy. - Difference-in-differences estimates calculated from “year-anykids” coefficients.

- Also a version with policy controls and controls for some personal characteristics:- Taxes- Welfare and Medicaid payments if working- Welfare and Medicaid payments if not working- Welfare participation control (stigma, transactions costs)

Based on: 26

Pr(Eit=1) = [ (E(Yw)-Ynw) + (E(Lw)-Lnw) – E((Pw)-Pnw) + X’ ]

- employment decision depends on: - differences in income working (Yw) and income not working (Ynw)- differences in non-market time working (Lw) and not working (Lnw)- indicator of welfare participation if working (Pw) and not working (Pnw)- X other controls.

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