Preliminary and Incomplete

Explaining and Forecasting Results of

The Self-Sufficiency Project

Christopher Ferrall

Department of EconomicsQueen’s University

http://qed.econ.queensu.ca/pub/faculty/ferrall

February 11, 2002

Abstract:

The Self-Sufficiency Project is a controlled randomized experiment which studies how long-term re-

cipients of income assistance (IA, i.e. welfare) respond to a large subsidy to earnings on full-time jobs.

This paper develops a dynamic programming model of the SSP experiment and estimates the model’s

parameters using the data generated by the experiment. The model includes stochastic evolution of

labor market skill, job opportunities, and the value of non-labor market time. The evolution of these

constraints and opportunities which single parents face allows the model to quantify common notions of

welfare dependency and the welfare trap. Parameter estimates are used to study the impact of the SSP

and of similar hypothetical welfare reforms on the target populations. The parameters are estimated

without access to the last 18 months of data generated by the experiment. Forecasts for the remaining

months’ data are reported and upon release will be compared to the forecasts.

Access to the SSP data was made possible through a research contract with Social Research andDemonstration Corporation (SRDC). Research support from the Social Sciences and Humanities Re-search Council of Canada is also gratefully acknowledged. Preliminary versions of the model werepresented at the following meetings and seminars: CERF (Ottawa), The EALE/SOLE (Milan), “TheEconometrics of Strategy and Decision Making” (Yale), NHH (Bergen), IES (Stockholm), and Upsala.

I. Introduction

The Self-Sufficiency Project, a controlled randomized experiment taking place in two Canadian

provinces, is designed to study whether long-term recipients of income assistance (IA) respond to

earnings subsidies. The main SSP treatment group consists of single parents who have received income

assistance (i.e. welfare) for at least one year. They are offered a large supplement to earnings if they

acquire a full-time job within one year and go off IA. Approximately 35 percent of parents in the

treatment groups qualified for the supplement, and at the peak they experienced a 100 percent increase

in full-time work and a 70 percent increase in earnings relative to parents in the control groups (Card

and Robbins 1996).

This paper builds a forward-looking model of single parent households that predicts labor market

outcomes before, during, and after the SSP experiment. The model accounts for the sampling design of

the SSP, basic features of the Canadian welfare system, and most details of the experimental treatment.

The model’s parameters are estimated using Generalized Method of Moments (GMM) on month-by-

month mean outcomes within treatment and demographic groups. The estimated model is thus nested

by a atheoretic impact analysis on the same outcomes. The estimates are to predict the outcomes of

hypothetical alternatives treatments on both the selected population and a larger at-risk population.

The model is also used to forecast outcomes generated by the SSP at the end of the experiment beyond

the data available for estimation purposes. When the remaining data become available an appendix will

(or has been attached) to the paper that reports the empirical outcomes alongside the out-of-sample

predicted outcomes.

Because the SSP is designed to study and to influence sequential labor market decisions, the

model is a discrete-choice dynamic program.1A sophisticated model is required when the goal is to use

experimental results to address the perceived problems with income assistance policies. These policies

impose a high implicit tax rate on earnings, making it costly for single parents to take low-paying jobs.

By discouraging low-paid work, IA keeps recipients from acquiring skills that would allow them to move

to higher paying jobs and would make leaving IA attractive. Their skills may also deteriorate while

outside the labor force.

1 Using estimates of a dynamic programming model as the basis for policy analysis is becomingcommon. Since surveyed by Wolpin(1996) the number and variety of papers has increased substantially.

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The incentive created by IA to abandon or to never develop skills is often called the welfare trap.

2 The welfare trap resists direct measurement because it relates a person’s current skills to their

labor market history.3 Nearly all empirical models of wages rely on Mincer’s earnings equation which

proxies skill with potential labor market experience. The Mincer equation does not keep track of when

experience was acquired, and so nearly all the empirical literature on wage dynamics assumes that

skills do not to deteriorate while not working. Only recently have optimizing models of labor market

dynamics replaced potential experience with actual experience.4

To quantify the welfare trap the single-parent household in this model chooses a vector of four

actions each month: active job search if unemployed, part-time or full-time hours if employed, receipt

of IA, and receipt of outside support that disqualifies the household from IA. In turn the value of each

action vector depends on stochastic state variables that govern skills, job opportunities, work hour

flexibility, the opportunity cost of labor market time, and support available from sources other than

IA. Full-time earnings depend on current skills, a job-specific value that the parent searches for, and the

legislative minimum wage. Job opportunities are summarized by whether a job is currently available,

which depends on active search while unemployed, an exogenous layoff probability, and the parent’s

own decision to quit into unemployment. Work-hour flexibility is based on an unobserved maximum

amount of hours that, like the initial wage offer, is specific to the job. The opportunity cost of time

evolves so that in any given month the parent may favor full-time, part-time, or no work, all else equal.

Finally, outside support is modeled as an unobserved source of income worth an amount that changes

over time. Taking this support rules out any IA.

Starting from this model of household behavior the SSP experiment is modeled as the unexpected

interruption of the status quo environment. The reaction to the interruption is modeled in the simplest

possible way given the assumption that behavior of single parents will be modeled as forward-looking

and rational. Namely, households in the treatment group immediately switch to optimal behavior

within the experiment given their state at the time of random selection. Because several of the state

2 A forward-looking agent does not fall unwittingly into a welfare trap. Instead a government setsits own trap by choosing policy parameters without accounting for skill appreciation and depreciation.

3 The incentives created by U.S. welfare programs have long been discussed in great detail (Moffitt1992), but only recently have the dynamic incentives been addressed by Miller and Sanders (1997),Swan (1998), Kennan and Walker (2000) and Keane and Wolpin (2000).

4 Examples include: Eckstein and Wolpin (1989), Imai (2000), Keane and Wolpin (1997), Altug andMiller (1998), and French (2000).

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variables and all parameters of the objective function are unobserved and varying over parents, a full

distribution of reactions to the experiment is generated.

II. The Environment

The model is an example of a social experiment carried out on a social environment as defined in

Ferrall (2001). The next section summarizes the notation necessary to define the social experiment.

Then a mostly verbal description is provided of the environment developed to explain and forecast SSP

results. Details of the model in symbolic form are provided in the Appendix.

II.A Individual States

In each month the household is in a state θ, an element of the state space Θ. The state vector θ is

a concatenation of five sub-vectors:

θ ≡(

θclock θexp θend θexog θpol

). (1)

The sub-states are listed in decreasing variability (loosely speaking), and their roles are defined as

are needed to describe the model. The ‘end’ vector contains typical state variables endogenous to the

parent’s decision making process that pertain to both the world inside and outside of the experiment.

The parent chooses an action vector α from a space of available actions A ( θ ). An action/state pair

(α, θ) is referred to as an outcome. Utility of an outcome is U(α, θ), and the transition between outcomes

to states next period is governed by P θ′ |α , θ . The value of an outcome satisfies Bellman’s equation

with discount factor δ. Choice probabilities conditional on the state are smoothed using a logistic kernel

with parameter ρ over the value of outcomes.

In the model of the SSP a household’s situation outside the experiment is described by the endoge-

nous state variables:

θendog ≡ ( l n x b s h d k ) . (2)

These are indices for the following aspects of the parent’s situation: the parent lost their previous

job; initial wage offer in current job, skill level based on previous experience, upper bound on working

hours in current job, level of outside support, opportunity cost of time outside the household, observed

demographic group, and unobserved type.

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The values of the demographic index d and the unobserved type index k do not vary over time

for a parent. Their roles are to determine which exogenous and policy parameters contained inside

θpol and θexog apply to the individual. Two measured characteristics of the household are treated as

demographic variables, indicators for the province of residence and whether the parent has more than

one child or not. This defines D = 4 demographic groups indexed by d. Table 1 lists the number of

subjects in each demographic group. Each demographic group has a vector of policy instruments that

apply to the outside world.

Ψp[d] ≡ ( IAB[d] SA[d] CB[d] MW[d] ) , (3)

These are, in order, the maximum level of income assistance benefits, the income set-aside before

benefits are clawed back, the claw-back rate on benefits, and full-time earnings at the minimum wage.

The vector of all policy instruments is therefore

θpol = (Ψp[1] Ψp[2] Ψp[3] Ψp[4] ) . (4)

Table 2 lists the values contained in the policy vector.

The exogenous vector for a subject’s problem is written

Γ ≡ (Υ Π δ ρ ) , (5)

which includes δ and ρ already defined. The parameters shift the utility are contained in the vector

Υ and the parameters that shift the transition are contained in Π. These are defined in the Appendix.

Exogenous parameters are type-specific: Γ = Γ[k]. The number of unobserved types is set to K = 4.

There are no exogenous preference parameters associated with the demographic groups (for example

no province dummy in the wage-offer distribution and no coefficient for the number of children in

the opportunity cost of time). Instead, associated with each demographic group is a distribution over

the K distinct types of single parents. The proportion of type k within a demographic group is also

a exogenous parameter, denoted λ[k, d] satisfying λ[4, d] = 1 − λ[1, d] − · · · − λ[3, d]. The values λ[k, d]

are collected in a 12 × 1 vector Λ. The extent to which welfare policy affects fertility and place of

residence is still an important empirical question. Although the SSP and the model developed here

are not well-suited for studying these issues, allowing full unobserved heterogeneity and demographic

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variables makes it possible to incorporate lessons learned from other approaches about fertility and

mobility. Policy experiments that hold the whole vector of endogenous variables constant assume that

the policy change would not affect fertility and residency. However, if other evidence is available about

this response, the mixing proportions can be adjusted appropriately along with the policy parameters.

All parameters to estimate from the results into the exogenous vector of size K(D+N) = 4(4+17) =

84:

θexog = (Λ[1] · · · Λ[4] Γ[1] · · · Γ[4] ) . (6)

The decision horizon is infinite, and there exists an ergodic or stationary distribution over the

endogenous variables, P∞θ, where the initial and invariant proportions within demographic and latent-

type groups are given by λ[k, d]. For the remaining endogenous variables the ergodic distribution is

derived by combining optimal choice probabilities and the primitive transition. An experiment draws

households from the ergodic distribution. Once sampled, the distribution of states continues to evolve

according to the same transition for those individuals assigned to the control groups. Eventually the

distribution over states for the control groups converges back to the ergodic distribution. The transition

for treatment groups is different, but ultimately their distributions over states will return to the ergodic

distribution as well. So the impact of a treatment is relative not to the ergodic distribution but to a

non-stationary distribution that is converging back to the ergodic outcomes at a different (presumably

slower) rate.

II.B Actions

The parent has three separate choice variables each month:

α ≡ (m a i ) . (7)

These are, in order, indices for the parent’s choice of labor market hours, active job search, and

acceptance of income assistance. The set of feasible actions places two restrictions on α. First, job

search while working is ruled out. Second, in any month the parent faces an upper bound on work

hours. When the parent has no job the upper bound is zero. When they have a part-time job they can

work at most PT hours, which is a SSP policy instrument. PT could correspond to any definition of

part-time work, but the critical value in the SSP is 3/4 of full-time hours.

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II.C Utility

Utility equals income plus transfers minus the opportunity cost of labor market time:

U(α, θ) = Income(α, θ) + OS(α, θ)−C(α, θ). (8)

Income is composed of labor market earnings, income assistance payments, and the SSP supplement

(defined in section IV below):

Income(α, θ) ≡ IA(α, θ) + Earn(α, θ) + SUP(α, θ). (9)

The cost of labor market time depends on working hours and search time, converted to work time

by an exogenous parameter κ. The cost of full-time work is also an exogenous parameter, and the

curvature of the cost function is determined by the endogenous variable h, which evolves from month

to month to reflect changing child care arrangements and other demands on the parent’s time. If the

value of h jumps to a new value, a working parent may change hours or quit and drop out of the labor

market for at least one month. Either change may induce a change in welfare receipt (see Figure 1). A

non-working parent may respond to a change in h by beginning or ending active search.

IA has various eligibility requirements, and many apparently eligible people do not take up IA. IA

rules limit support accepted from earnings and from other people (for example, through cash or shared

living arrangements).5 The value of IA is determined by demographic variables and reported earnings,

a fraction of actual earnings. The reporting fraction β is an exogenous parameter. It is assumed that

the parent fully reports earnings in the SSP survey. The middle term in (8) captures both outside

support and any additional utility (expressed in dollars) that the parent gets from forgoing IA. The

transfer component of OS is support that, if accepted, disqualifies the parent for IA. Without loss of

generality OS, is expressed as a fraction of maximum IA benefits, IAB. The fraction available to the

parent each month is determined by the endogenous variable s, which like the variable h that determines

preferences for work arrangements, evolves over time. If the value s jumps up the parent may go off

welfare and rely on other sources of support with or without any change in labor market status. A

drop in s may push the parent back on welfare (again Figure 1 with h = 0).

5 There have been surprisingly few attempts to model either meeting welfare requirements and oradditional elements of utility (welfare ’stigma’ or ’dependence’). Moffitt (1983) and Miller and Sanders(1997) present static and dynamic models, respectively.

6

Figure 1 summarizes the static utility (8). Shifting preferences for work hours is represented by

indifference curves indexed by the state variable h that cross each other. Shifts in the level of outside

support are represented by parallel budget lines in black. The rules of the IA system, including the

setaside and disallowance of outside support, is represented by the green budget line.

II.D Job Search and Wages

Finding a job takes time in two senses. Job search itself takes time that cannot be spent in the

home or working productively, and this opportunity cost is already embodied in the presence of a

in the choice vector and the cost of time in the utility. In addition, active job search is not always

immediately successful. The probability that a job offer is received as a function of the current outcome

is pj(α, θ), which is zero when a = 0. Otherwise it is equal to an exogenous parameter πj, except in

one experimental group, SSP+, that was offered additional job search help. The effect of the search

assistance is therefore treated as an unobserved exogenous parameter π+ that increases the probability

of getting an offer.

Since long-term welfare recipients tend to earn low wages, it is important to take into account

minimum wage legislation. MW is the earnings made in a month by someone working full time at the

provincially-set hourly minimum wage. The model allows for an interaction between MW, skills, the

distribution initial earnings offers, and the subsequent growth of earnings. To describe the relationship,

begin with the hypothetical case where there is no minimum wage. If MW were zero the earnings

collapse to a standard log-linear form:

lnW0 ( θ ) ≡ lnw0 + η lnχ. (10)

Full-time equivalent earnings on a currently available job is denoted W0 ( θ ). It depends on two en-

dogenous outcomes: the inherent earnings offer w0 and the parent’s level of skill χ.

The inherent wage is indexed by the endogenous variable n and is the random outcome of the

parent’s search for jobs. The value of w0 associated with n is specific to the parent-firm match, as is the

upper bound on work hours indexed by the endogenous variable b. The proportion of job offers that

are full-time (upper bound of 1) is equal to the exogenous parameter πf . The index b changes from one

month to the next for the following reasons: a non-working parent finds a job with probability pj(α, θ),

which will either have an upper bound on m of 1 or PT; a working parent loses a job permanently with

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probability πl (indicated by l = 1 for the period following the lay off), so that the upper bound on m

goes to 0 for the next month; or a working parent chooses to quit a job by setting the choice m to 0. A

parent who quits or is laid-off can immediately engage in job search (setting a = 1), but offers received

that month are scheduled to begin the next month. So leaving or losing a job is matched to cases in the

data where the person experiences at least one month not working. Job-to-job transitions are treated

as the same job and any wage growth would be attributed to skill acquisition.

A parent’s current skill level χ is indexed by the endogenous variable x. It is expressed as a fraction

of a ‘fully’ skilled worker, so w0 in (10) can be interpreted as the earnings offered by the firm to a fully-

skilled worker (lnχ = 0). The exogenous parameter η is the return skill. The marginal return to skill is

decreasing (increasing) when the sign of η−1 is greater (less) than one. A parent’s level of skill is related

to past labor market experience only indirectly and stochastically. From month to month x can remain

unchanged or can increase or decrease by one unit with a probability that depends on labor market

status. While working, skills increase with a probability proportional to the amount of time working,

mπi. While not working, skills decrease randomly with probability πd. When πi = πd = 0 endogenous

skill accumulation and depreciation are eliminated. Otherwise, the skill accumulation process allows

for welfare dependency and a welfare trap. The longer the parent is out of work, perhaps taking IA,

the more likely skills have fallen. Job offers then become less valuable relative to time spent in the

household (dependency), because the wage offer distribution shifts to the left. If a job were taken, skills

would begin to develop again and the increase in wages would buffet the parent’s labor supply against

future household shocks. But this may not happen because current job offers are too low and/or wages

grow too slowly (the trap).

With MW = 0, (10) is in effect and each wage offer is equally likely next period for a parent actively

searching for a job. Otherwise, the wage offer does not change next period. When MW > 0 wage offers

are not equally likely. Instead, wage offers that result in a minimum wage are equally likely to each

other as are offers above the minimum wage. Under the distribution given in the Appendix, all the

wage offers made to parents with maximal skill are above MW. On the other hand, almost all the

wages offers made to those with minimal skill are minimum wage offers. In general, an increase in

skill decreases the proportion of minimum wage offers and it shifts the distribution of offers above MW

to the right. Once the parent acquires additional skill, through an endogenous but stochastic change

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in x, they will experience a wage increase only if the value of the match has a high enough index n.

Those holding jobs with lower value of n will have to wait for additional increases to skill to get a raise.

Thus, jobs differ in their growth potential and parents with skills below the maximum may reject some

minimum wages but not others.

Because the SSP is essentially a proportional subsidy to wages, it may encourage the treatment

group to be more selective of minimum wage jobs than the control group. In a bargaining model of

wages and job search the minimum wage is a side constraint placed on the feasible bargaining set,

leading to complicated effects (Flinn 1999). A parent-firm match with W0 ( θ ) < MW may still form

if the parent accepts conditions such as slower wage growth or working unreported hours. The SSP

subsidize and endogenous skill accumulation complicates the determination of job rejection and wages

in a bargaining model even further, and it is beyond the scope of this framework to endogenize wage

offers as a function of MW. In the model the reason for being selective about minimum wage offers is

the exogenous relationship among minimum wage laws, job offers, and skill that mimics the endogenous

outcomes of a bargaining model.

II.E Exogenous Parameters

To recap,

Υ ≡ (β η κ µ ν σ ζ ) (11)

Π ≡ (πd πf πh πi πl πs π+ ) ,

where: β is the rate of income-reporting, η is the curvature in skill; κ converts job search time into an

equivalent work time; ν is the income-equivalent cost of full-time work; µ and σ determine the location

and spread of wage offers (and are defined only in the Appendix); ζ determines the variance in the

curvature of time-costs over time; πd is the probability that skills decline while not working; πf is the

proportion of job offers that are full-time jobs; πh is probability that the curvature in time-costs change;

πi is the probability that skills increase while working; πl is the probability that a working parent loses

their job exogenously; πs is the probability that the level of outside support available to the parent

changes; and π+ is the parameter that determines the effectiveness of the SSP Plus treatment.

III. The SSP Experiment

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Ferrall (2001) defines a social experiment carried out on a social environment as a program of

treatment and a set of treatment groups. Each treatment group has associated with it a sample

proportion, an initial experimental state, and a sequence of feasible histories. Each true treatment

group is paired with a control group (which may be shared with another treatment group).

Within each province two different samples were drawn that entered the experiment at different

points. The “main” sample selected parents that had been on IA for at least one year, and the treatment

group immediately had the chance to qualify for the SSP supplement. The “applicant” sample included

parents initiating (applying for) a new IA spell. The applicant treatment group could qualify for the

SSP only after a year on IA. Finally, a separate “SSP Plus” treatment in New Brunswick offered

job-search and employment services in addition to the chance at the SSP supplement.

θexp = ( e p g ) .

The treatment itself is defined by a matrix of experimental instruments

Ψx ≡ (F R(F ) PT TB UL ) , (12)

which includes the number of phases, F=6, and the maximum duration for each phase, R(f) = 12

for 0 < f < 6. PT is the limit on part-time hours for SSP eligibility, TB is the tax-back rate on the

SSP subsidy, and UL is the upper limit to earnings eligible for supplement (as a fraction of full-time

earnings at the minimum wage, MW). In the experiment PT = 3/4, TB = 1/2, and UL = 3.90. Thus

the supplement equals half the difference between actual earnings and nearly four times minimum wage

earnings. The supplement enters into utility defined in (8) through income defined in (9).

The phases f = 0 and f = 6 = F are the real, non-experimental world before and after random as-

signment, respectively. The control groups transit immediately from phase 0 to phase 6. The treatment

groups transit from phase 0 to the initial phase for their treatment group, but ultimately reach/return

to phase 6 as well. Phase 1 is the entry phase, where a parent must remain on IA for twelve months to

get a chance to qualify for the SSP treatment. Phase 2 is the qualification period in which the parent

becomes eligible for the SSP supplement if and when they begin a full-time job. They remain eligible

for the supplement during phases 3 to 5 as long as they decline income assistance (i = 0).

The number of months a parent spends in any phase of treatment is endogenous to their choices. In

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addition, the traditional method of evaluating an experiment is to compare outcomes in the treatment

and control groups month-by-month after random assignment. To make results generated by the model

compatible with this approach a data clock is introduced. The variable t equals the calendar months

before or after the beginning of the qualification phase (f = 2). This associates t = 0 with the latest

entry point of any experimental group into the program of treatment. The entry group has entry point

t = −11. For example, one parent may take 8 months to leave phase 2 while another may take only

4 months. In the month t = 7 the first parent’s clock would read (72) and the second (33). Parents

are oblivious to t, but not to the experimental clock (rf). One way to see this is to note t is properly

defined and tracked in the control groups, but it can have no effect on utility or future states since

these parents remain in the real world phase at all times.

Figure 2 summarizes the selection, assignment, and transition rules in the SSP. The ellipse repre-

sents the state space in the real world both before and after the experiment (phases 0 and 6). The

discrete state spaces can be separated into those where the parent receives IA and those where they do

not. The main sample requires the parent to receive IA for at twelve months or more, represented by

the path contained in the on IA side of the state space. Assignment to the control group is represented

by the green line which leads back to the same state. Assignment to the treatment group at time 0

and the start of phase 1 is represented by the red line leading to the phases. The applicant sample

requires a movement from off IA to on IA. The treatment group gets place at the beginning of phase 1

(blue line). A parent exits the treatment if they go of IA during phase 1, go on IA during phases 3-5,

or make it to the end of phase 5.

IV. Explaining Results

IV.A Data

Define as Y (α, θ) the vector of all measured results related to the complete outcome (α, θ) for a

parent in a month. The moments chosen for study are listed in Table 3a. These moments correspond

closely to the moments chosen by Card and Robbins (1996) in their summary of the data generated by

the experiment. Table 3b lists some other values which are used to fit the model (and in the case of

utility and present discounted value not observable) but are of interest to report from simulations.

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As Table 1 summarizes, a total of 9294 people participate in the SSP experiment. Roughly two-

thirds are in the main (Year One) sample. The sample is split evenly between British Columbia and New

Brunswick except for the 292 people in the SSP+ sub-sample in New Brunswick alone. By coincidence

the proportion of households with one child is also 50%. Release of results form the SSP experiment

is on-going. The results explained here include 36 post-assignment months (t = 1 to t = 36) of data in

the main sample and 30 months for the applicant sample (t = −11 to t = 16).

Figures 4 to 7 summarize four selected variables across demographic and experimental groups. The

variables are receipt of IA, earnings, supplement payments, and working full time. Figure 4a shows

that by month 16 roughly 25% of the main (Year1) sample control group (marked with a ‘C’) has gone

off IA on their own. In the treatment groups (marked with ‘SSP’ and unmarked for the SSP+ group)

the proportion falls to about 60%. Both changes are larger for parents in New Brunswick (NB) and

with one child. In the applicant sample many people go off welfare right away and the proportion

receiving IA at t = 0 is at a level slightly below the main SSP sample in each demographic group.

The difference between the treatment and control groups is small but slowly widens leading up to

the month at which the treatment group becomes eligible for the subsidy. This small effect indicates

that conditioning on one year of IA receipt limits the entry effect of the subsidy. Within the model it

indicates that the combination of the discount factor and transition probabilities (job losses and offers)

make it unattractive to hold out for eligibility.

Figure 4b shows the average monthly earnings (in hundreds of dollars). Earnings rise over time for

all the groups, although the main control group sees only slight changes to earnings over time. Earnings

growth in the SSP also levels off after the twelve month deadline for qualifying. Figure 4c shows the

supplement payments received by the treatment groups. As with earnings the payments level off after

month twelve in New Brunswick and they fall in British Columbia. These patterns closely reflect the

pattern in full-time employment shown in Figure 4d.

IV.B Econometric Objective

The exogenous vectors Θ introduced in (6) contain all the parameters to estimate from the experi-

mental data, including the type proportions within demographic groups. Let Y (θcond) denote the mean

value of the model vector in the data associated with the conditioning variables in the experiment. The

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model predicts that this vector equals:

E[Y (θcond)] =∑

θ cond

λ[k, d]Ω( θ , tmax + 1 )Y (θcond). (13)

There are several ways to estimate the parameters of the model. Estimates reported here are based on

Generalized Method of Moments estimates, which solve

Θexog = arg minΘexog

W (Θexog) ≡∑

θcond

(Y (θcond)− E

[Y (θcond)

])′Σ(θcond)

(Y (θcond)− E

[Y (θcond)

]),

(14)

and where Σ(θcond) is a weighting matrix for each moment vector. This estimator is based on the same

information as the traditional impact analysis (given the choice of moments/outcomes to fit/evaluate).

From a comparability perspective, this is attractive. Averaging the data and the predictions also avoids

inconsistencies between the experiment and the model which are costly to resolve (see the Appendix).

However, GMM based on (14) is not efficient. First, the optimal weighting matrix Σ(θcond) is not being

used. Second, the model can predict the joint vector of outcomes for an individual throughout the

experiment after random assignment, but (14) does not exploit these restrictions. Eliminating these

sources of statistical inefficiency would require considerable increases in computational time. Any

econometric objective function chosen would have to balance the realism in the model, the dimensions

of behavior explained, comparability with atheoretic approaches, and statistical efficiency.

IV.C Identification

Since the model’s predictions can only be computed numerically it is not possible to provide an

analytical proof that the estimated parameters are identified from the data generated by the experiment.

Instead, a heuristic argument must suffice before estimation of the model determines whether the

parameters are at least identified from the generated data. First, consider that in the model job loss

occurs exogenously. The SSP survey instrument records reasons why a parent stop working in a job

(which can be grouped into losses and quits). Thus the sample proportion of parents’ losing a job is

identified directly from the data (although this proportion enters into all other aspects of the model).

In short, job loss data identifies the job-loss probability πl.

The rate at which parents go off and on IA with no change in labor market status identifies the jump

probability for outside support, πs. Parents in the control group receiving IA do not quit jobs unless the

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convexity parameter c(h) changes value. Observed quit rates identify the jump probability for h. How

the quit rate correlates with labor market earnings helps identify the distribution of c(h) and thus ζ.

Accepted wage data (converted to full-time earnings) identify the wage offer distribution. Wage growth

and duration dependency in accepted starting wages identify the skill accumulation and depreciation

parameters. The correlation between income and welfare benefits helps identify the implicit tax rate

on income.

In a stationary model estimated on non-experimental data, the job search parameters (cost of

search, offer probability, proportion of full-time jobs) would have to be identified through the reservation

wage (that truncates the offer distribution) and the proportion of households working part-time. It is

not guaranteed that they would be identified in such data. The SSP experiment, however, includes

exogenous variation in the value of job search and the value of keeping a full-time job. For example, the

proportion of people working part-time in the first month of the SSP that switch to full-time work right

away would pick up the proportion of jobs that are potentially full-time. The time-varying incentives

in the SSP provide leverage for identifying these parameters.

Even more leverage is provided by a comparison between the outcomes of the applicant sample

(e = 0) and the main sample (e = 1). When using the standard impact methodology, the behavior

between the treatment and matched control group are the focus of attention. For the applicant sample,

this consists of those who know the SSP subsidy exists and can anticipate becoming eligible for it (i.e.

they are in phase f = 1), and those in the control group who cannot (f = 6). The model makes clear

predictions between the behavior of those two groups. The value of taking a job and/or leaving IA

changes with the time spent in phase 1. As r approaches T (1) the higher the value of continued receipt

of IA becomes among the treated.

The situation is identical for the treated groups of both the applicant and main samples once they

reach the qualifying phase of the experiment (f = 2). From that point on, any difference between the

behavior of the eligible households within the two groups is, within the model, forced to come from

the difference in household states conditional upon reaching phase 2. In the year-one sample reaching

phase 2 is exogenous to the SSP and unexpected, whereas for the applicant group it is completely

endogenous and can be expected one year in advance. Thus, the two samples provide experimental

variation in unobserved household states caused by lagged decisions made while anticipating different

14

future opportunities. Comparison of the two SSP treatment groups provides fairly direct evidence

about the discount factor δ, which determines how far-sighted the household is in making decisions.

These and other extra experimental variations in states and opportunities across SSP samples come at

little or no extra cost in terms of free model parameters. In contrast, a standard impact analysis uses

a new parameter for each individual comparison, across groups, time, and measured variables.

IV.D Estimates

Table 4 reports parameter estimated based on using (14) to fit the model to the data currently

available.

V. Forecasting Further Results

VI. Conclusion

VII. Technical Appendix

VII.A Notation

2 ≡ 0,1

q′ = q?(q, πj, Qj) ↔ P q′|α, θ =J∑

j

πjB[q′ ∈Qj

]/|Qj |+ (1−

∑

j

πj)B [q′ = q] . (15)

Arguments of the jump process can depend on the current outcome (α, θ), but the default value

and the jump sets Qj are pre-determined.

VII.B Outcomes

Endogenous Variables.

1. Unobserved Type: k ∈K = 1,2,3,4

¦ Role: index into Γ and and the mixing distribution within a demographic group, λ[k, d].

¦ Transition: k′ = k

2. Observed Type: d ∈D = 1,2,34

15

¦ Role: index into the policy vector θpol, and and the exogenous vector to retrieve λ[d].

¦ Transition: d′ = d

3. Household Time Cost: h ∈H = 1,2,??

¦ Role: determines the curvature of the time-cost function.

¦ Auxillary Equations

c(h) = −ζ ln(1− h + 1

H

). (16)

The right hand-side is the inverse exponential distribution with decay rate 1/ζ > 0. The value of

c(h) determines the convexity of costs for activity less than full-time. For values of c(h) < 1 the cost

function is concave for feasible labor market time, creating a tendency to prefer part-time work. For

example, when children are in school part-time work has lower opportunity cost than full-time work

that would require additional child-care. On the other hand, costs are convex when c > 1, which

creates a tendency either to stay at home or work full time. It captures cases where commuting or

part-time day care are expensive.

¦ Equation in Text:

¦ Transition: h′ = h?(πh, h,H)

4. Outside Support Opportunities: s ∈ S = 1,2, ??

¦ Role: determines the cash-equivalent amount of support available to the parent that, if accepted,

disqualifies the parent from IA.

¦ Auxillary Equations: OS(α, θ) ≡ (1− i)(sS)IAB.

¦ Equations in Text:

¦ Transition: s′ = s?(πs, s,S)

5. Upper bound on working hours: b ∈B = 0,1,2

¦ Role: constraint on work hours in current job

¦ Auxillary Equations m(b) =

0 if b < 2PT if b = 2M − 1 if b = 3

¦ Equations in Text:

¦ Transition:

16

6. Accumulated Skill: x ∈X = 1,2,??

¦ Role: level of earnings and future growth potential

¦ Auxillary Equations

χ =x− 1X − 1

¦ Equations in Text:

¦ Transition:

x′ = x?

(mπi + B [m = 0]πd , x , X ′

)(17)

where

X ′ ≡

x , min [x + 1, X − 1]

if m > 0max [x− 1,0] , x

if m = 0.

(18)

7. Wage Offer: n ∈N = 1,2, ??

¦ Role: search-sensitive component of wage

¦ Auxillary Equations The

w0(n′) = exp

µ + σΦ−1(

n′ + 1N + 1

)(19)

For example, if N = 3 then the values of w0(n′) correspond to quartiles of the normal distribution.

Define

φ ( θ ) = Φ[ln(MW)− η ln (x+1

X )− µ

σ

](20)

as the probability of receiving an offer below the actual minimum wage under the MW=0 distribu-

tion. Then

W ( θ ) =

MW n < N − 1− x

exp(

µ + σΦ−1(

φ ( θ ) +(1−φ( θ ))(n+1)min(x+1,N)+1

))( xX+1)

ηn ≥ N − x− 1. (21)

¦ Equations in Text:

¦ Transition: With MW=0, each wage offer is equally likely next period if one is searching for a job.

Otherwise, the wage offer does not change next period.

n′(α, θ) = n?(pj , n,N)

With MW¿0

n′(α, θ) = n?(pj , n, 1,2, . . . , X − x′ − 1) + n?(pj , n, X − x′)

17

P n′ = η =

φ ( θ′ ) /(X − x′) η < X − x′ − 1

(1− B [x′ < X − 1])φ ( θ′ ) /(x′ + 1) η ≥ X − x′ − 1.(22)

Notice that the distribution of n′ depends on the contemporaneous state ( θ′ ), but only through the

value of skill x′. So between periods x′ must be determined before n′.

8. Job Loss: l ∈ 2

¦ Role: exogenous loss of job.

¦ Transition: l′ = l?(B [m > 0]πl,0, 1)

Actions.

A1. Labor market hours m ∈M = 1,2,??

¦ Auxillary Equations:

C(α, θ) = ν [m/(M − 1) + κj]c(h)

, (23)

The exogenous parameter ν is the opportunity cost of full-time labor market activity.

¦ Equations in Text:

A2. Active Job Search a ∈ 2

¦ Auxillary Equations: Probability of receiving a job offer

pj(α, θ) = a [πa + B [g = 0] (1− πa)ψ] . (24)

The parameter πa is the efficiency of active search. The effectiveness of job search assistance is

determined by ψ. A fully effective assistance program (ψ = 1) makes the probability of a job offer

equal to one every month of active search.

¦ Equations in Text:

A3. Accept Income Assistance i ∈ 2

¦ Auxillary Equations: hours = m/(M − 1)

¦ Equations in Text:

A4.

18

A ( θ ) ≡(m a i )∈

M×2×2

: m < m(b)&mj = 0

. (25)

IA(α, θ) ≡ imax

IAB− βCBmin(

m

M − 1

)W ( θ )− SA , 0

,0

Earn(α, θ) ≡(

m

M − 1

)W ( θ ) + SUP(α, θ). (26)

Several of the policy instruments introduced earlier enter (26). Under Canadian law the clawback

rate is CB=100% of reported income. The estimated parameter β allows for under-reporting of income.

VII.C Experiment

Phases of Treatment

The transition rules for the state variables f and r capture the design of the SSP experiment.

frifQ

1− 06

1 < 1211

11202

2 < 12f + Q(α, θ)

212Q(α, θ)(f + 1) + (1−Q(α, θ))6

33 + B [r = 12]

44 + B [r = 12]

55 + B [r = 12]

r′(α, θ) =1 + rB [f < 6&f ′(α, θ) = f ] . (27)

Time residing in the current phase is incremented each month unless the phase is changing.

Conditional on (α, θ) the transitions are not random, so they can be determined sequentially.

Treatment Groups

19

Program of Treatment

Q(α, θ) = B [i = 1&2 ≤ f ≤ 5&m > PT&W ( θ ) ≥ MW] (28)

SUP(α, θ) = Q(α, θ)max

0 , (1− TB)[UL×MW −Earn(α, θ)]

(29)

In the experiment TB = 1/2, and UL = 3.90. Thus the supplement equals half the difference between

actual earnings and nearly four times minimum wage earnings.

VII.D Selection

The selection process occurs during a range of months that depends on the entry group e, tmin(e) ≤

t ≤, tmax(e). The set to condition choices on is i?(t, e), and the initial experimental state is clockmax(g, e).

Let IA(ia) ≡ α : i ∈ ia be the set of actions for which i takes on the values in some set ia ⊆ I. The

main sample (e = 1) selects parents from the ergodic distribution who have been on welfare for twelve

months or more. They can qualify for the SSP immediately so the selection process begins at t = −11

and ends at t = 0:

tmin(1) = −11 tmax(1) = 0

i?(t,1) =1 t ∈ [tmin, tmax](1)I t > tmax(1).

(30)

clockmax(3,1) = (6,1) clockmax = (2,1) = (2,1)) clockmax(1,1) = (2,1)

The process is shorter and simpler for the applicant sample (e = 0): off IA for at least one month and

then back on for one month. They enter the experiment at t = −11:

tmin(0) = −12 tmax(0) = −11

i?(t,1) =

0 t = −121 t = −11I t > −11

(31)

clockmax(3,0) = (6,1) clockmax = (2,1) = (1,1)

20

VIII. References

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Review of Economic Studies .

Card, David and Philip K. Robins. 1996. Do Financial Incentives Encourage Welfare Recipients to

Work? Initial 18-Month Findings from the Self-Sufficiency Project, Social Research and Demon-

stration Corporation, Vancouver.

Fortin, B. and G. Lacroix 1997. “Welfare Benefits, Minimum Wage Rate and the Duration of Welfare

Spells: Evidence from a Natural Experiment in Canada,” working paper 9708, Department of

Economics, University of Laval.

Ferrall, C. 2001. “Estimation and Inference in Social Experiments,” working paper, http://-

highway61.econ.queensu.ca/papers/experiments.pdf.

Kennan, J. and Walker 2000.“ ,” Cowles Commission Conference presentation, http://www.econ.yale.edu/

seminars/sdm/sdm/???.pdf.

Miller, R. A. and S. Sanders 1997. “Human Capital Development and Welfare Participation,” Carnegie-

Rochester Conference Series on Public Policy 46.

Moffitt, R. 1983. “An Economic Model of Welfare Stigma,” American Economic Review 73, 5, 1023-35.

Moffitt, R. 1992. “Incentive Effects of the U.S. Welfare System: A Review,” Journal of Economic

Literature , 1-61.

Swan, Christopher. 1998. “Welfare Reform When Agents are Forward-Looking,” http://

ms.cc.sunysb.edu / cswann/prof/papers/forward looking.pdf.

Wolpin, K. I. 1996. ”Public Policy Uses of Discrete Choice Dynamic Programming Models,” American

Economic Review 86, 3, May.

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