Keywords: Labor supply, taxation, quasiexperimental methods.
Abstract:
JEL Codes: H2, J2
In estimating and interpreting the labor supply response to taxation, the researcher is confronted with two types of challenges  first, to identify exogenous variation in marginal tax rates, and second, to choose a suitable theoretical framework to interpret the resulting estimates. Exploiting ageeligibility rules for claiming a dependent on a tax return provides an opportunity to estimate households' labor supply responses to anticipated changes in the marginal tax rate.^{1} Ageeligibility rules permit a quasirandom assignment of households to different tax schedules around the ageeligibility cutoffs. Barely ineligible households, whose children have "aged out" of being claimed as a dependent, are the counterfactual for what would happen to barely eligible households if their children were to "age out" in the subsequent year. Any ensuing differences are attributable to facing a tax schedule by which a household is allowed to exempt more income while facing a different set of statutory and effective marginal tax rates. A comparison of these households' labor supply behavior yields an estimate of the response to anticipated changes in tax rates.
Exploiting the ageeligibility rules and applying the regression discontinuity design allow for testing two strong predictions of the lifecycle labor supply model (MaCurdy, 1981). First, because the loss of an ageeligible dependent corresponds to predictable reductions in many households' aftertax incomes but without accompanying marginal tax rate changes, the lifecycle model predicts that many barely eligible and ineligible households supply the same amount of labor. It is possible to compare the labor supply decisions of barely eligible households whose marginal tax rates would not change if they were to lose an ageeligible dependent to those of barely ineligible households whose marginal tax rates would also not change if they were to gain a dependent. This comparison provides a test of the lifecycle model's prediction that anticipated changes in income have no effect on labor supply decisions, independent of aftertax wage changes. Second, the lifecycle model predicts that households respond to changes in their tax liability if the loss of a dependent corresponds to changes in their marginal tax rates. By comparing the decisions of barely eligible households whose marginal tax rates would change if they were to gain a dependent to their barely ineligible counterparts, I assess the extent to which changes in the marginal tax rate account for differences in households' labor supply. If the lifecycle model is an appropriate approximation of households' labor supply behavior, this second comparison yields an estimate of the intertemporal labor supply elasticity.
A preview of this paper's results suggests, first, that the ageeligibility rules have a reduced form effect on mothers' annual hours but not on fathers' labor supply. I find that barely ineligible mothers, who face higher tax rates, work about 40 hours less per year than their barely eligible counterparts. The ageeligibility rules have little to no effect on mothers' (or fathers') labor force participation decisions. Second, mothers' labor supply responses are only weakly related to the variation in marginal tax rates arising from the ageeligibility of a dependent. I provide evidence that the dominant mechanism by which mothers adjust their labor supply is the average, rather than the marginal, tax rate. I also show that attributing mothers' labor supply responses solely to marginal tax rates results in an implausibly large estimate of mothers' intertemporal labor supply elasticity. Given that the average decrease in the netoftax rate is 0.5 percentage points, or 0.3%, the finding for mothers initially suggests an intertemporal labor supply elasticity of 12, which is an order of magnitude larger than previous estimates of no larger than one (Blundell and MaCurdy, 1999). Anticipated changes in income, therefore, have statistically significant effects on mothers' labor supply.
The remainder of this paper is organized as follows: Section two formally revisits the lifecycle labor supply model while Section three provides the policy context and analyzes what we can infer from previous research on the labor supply response to taxation. Section four describes the institutional details concerning the ageeligibility rules of the U.S. tax code, and develops the estimation procedure using these features. Sections five and six present the data and results. Section seven interprets the results in the context of a lifecycle labor supply model, and Section eight concludes with policy implications.
In a lifecycle labor supply model, the responses to unanticipated and anticipated changes are distinct. Because unanticipated changes in the marginal tax rate correspond to revisions of lifetime income and to intertemporal substitution effects that affect labor supply in opposite directions, for
a given change, the labor supply response is larger if it is anticipated than if the change is unanticipated.^{2} To see this more formally, consider the behavior
of a utilitymaximizing household () with the following lifetime utility function and intertemporal budget constraint in each period:
The wellknown firstorder conditions to this problem are:
For a given change in the marginal tax rate that occurs in period , an anticipated change elicits a larger response than an unanticipated change. Following the mechanics and notation
presented in Card (1990), let be the elasticity of hours with respect to the marginal utility of wealth, and let be the intertemporal labor supply elasticity.^{3} By loglinearizing the expression
for labor supply and substituting a firstorder condition, it is possible to express the change in hours between period and as:
In this paper, the ageeligibility rules for claiming an eligible dependent on a tax return provides a source of exogenous variation in the netoftax rate. By treating barely ineligible households as a counterfactual for what would happen to barely eligible households if the latter's children were to "age out" and become ineligible in the subsequent year, I compare equivalent households that are at similar stages in the lifecycle, have identical levels of lifetime wealth, and face the same economic environment. If the lifecycle model governs households' behavior, these similarities enable me to identify , the intertemporal labor supply elasticity. If, however, the lifecycle model is not a good approximation for households' behavior, the variation in the netoftax rate between barely eligible and ineligible households may not explain differences in their labor supply behavior. The regression discontinuity design, unlike other empirical strategies, uncovers whether the lifecycle model provides an appropriate characterization of the labor supply response to taxation.
In general, the empirical research design necessary to assess whether the effects of anticipated changes in the marginal tax rate are consistent with a lifecycle model must meet three requirements. First, to ascertain the causal effect of taxation, researchers must isolate random variation in tax rates that is exogenous to other factors that may also influence labor supply decisions, such as assets or ability. As discussed in Triest (1998), at any given point in time, individuals face the same tax system, and their tax rates and liabilities are correlated with a number of economic factors, which can conflate the effects of tax rates. Crosssectional variation in tax rates and liabilities is correlated with observable and unobservable characteristics of households. This means that, in general, it is difficult to isolate to a single causal factor that explains differences in behavior among these kinds of households. Second, researchers must compare identical individuals who are similar in all respects except for the tax regime that they face. That is, researchers must compare individuals who are at similar stages in the lifecycle and have similar preferences for leisure, so as not to conflate the effect of taxes with other effects. Although this requirement is always satisfied with random assignment to a tax schedule, tax reforminspired natural experiements may not compare identical individuals. Third, researchers must identify anticipated temporal variation in taxes or tax policy changes. Using unanticipated tax changes is inappropriate for testing some aspects of the lifecycle model because they also correspond to revisions to lifetime wealth. With these revisions, it is difficult to disentangle the effect of the tax system per se.
Few empirical studies have met these three requirements for both a consistent estimate of the labor supply response to an anticipated tax change and a test of the lifecycle model. Given the difficulty of finding a research design that simultaneously meets these requirements, most studies are but able to meet a subset of them. As such, it is difficult to both consistently estimate a labor supply elasticity and learn about the predictions of the lifecycle model from existing estimates. Here, I outline the key issues to elaborate upon this point.
Though participants were randomly assigned to various tax regimes, the Negative Income Tax (NIT) experiments of the 1970's produce results that do not inform us of the labor supply response to anticipated changes in tax liabilities or marginal tax rates (Burtless and Hausman, 1978). At the crux of this difficulty are issues in the design of the experiment. Although researchers were interested in assessing the consequences of implementing a permanent NIT, the participants were assigned a treatment that involved a temporary reduction in marginal tax rates and a temporary income guarantee, relative to a control group that did not receive these benefits. In addition, participants did not anticipate the changes in the tax regime prior to being in the experiment. One can perhaps conjecture that the participants' small labor supply responses to marginal tax rates are understatements of the intertemporal labor supply elasticity, but it is difficult to gauge by how much.
In fact, it is difficult to specify the parameters of a largescale randomized controlled trial designed to test whether anticipated changes in taxes are consistent with the predictions of the lifecycle model. Randomizing individuals into groups facing different tax schedules during a calendar year corresponds to a temporary and unanticipated change in tax rates. Any ensuing behavioral responses do not correspond to an intertemporal labor supply elasticity. Randomizing individuals into groups facing higher and lower tax liabilities over the lifecycle may possibly circumvent the issue of unanticipated tax changes, but it introduces the difficulty of having to observe individuals and control their economic environment over an entire lifetime. Furthermore, it is difficult to determine at what point in the lifecycle the randomization ought to occur, as well as when it ought to be announced, particularly in light of any equity or distributional concerns one might have. It may be possible to manipulate the timing of tax payments over the lifecycle to learn about the intertemporal substitution effects of taxation, but this would also entail observing individuals and controlling their economic environment over an entire lifetime. Even the most savvy social planner would have a difficult time designing a randomized controlled trial to test the predictions of the lifecycle model.
Nonexperimental studies examining the labor supply effects of taxation, while recognizing the pitfalls of basing inference on crosssectional estimates, rely on tax policy changes to generate exogenous variation in tax schedules (Moffitt and Wilhelm, 2000; Eissa, 1995; Hausman and Poterba, 1987; Eissa and Liebman, 1996). These "natural experiments" are attractive in that researchers need not parameterize the tax system, a process that is both difficult and sensitive to specification errors. The behavioral effects of taxation are identified in these studies if either the groups affected and unaffected by the policy change are good comparisons for one another (either in a crosssection or in terms of the trends they experience over time). However, the Tax Reform Act of 1986 (TRA86), the policy change most often studied to estimate the labor supply response, changed marginal tax rates more for higher income individuals, the "treated" group (Auerbach and Slemrod, 1997). The estimates from these studies, accordingly, are subject to the difficulty of assuming that lower income individuals are an appropriate counterfactual for their higher income counterparts, either in terms of their characteristics or the economic trends they experience. In light of the documented increases in income inequality during the 1980's, this identifying assumption is tenuous, at best (Saez, 2003; Gruber and Saez, 2002).
The primary difficulty, though, with estimates of the labor supply response to taxation from "natural experiments" is that most policy changes contain an element of surprise in them, and thus cannot be used to test the lifecycle model. For example, TRA86, though intended to be revenue neutral for each income class, is likely to have affected individual levels of lifetime wealth. The ensuing labor supply responses inform us of the direction and magnitude of the effect of a particular tax reform, but the research design is not wellsuited to neither conceptually nor statistically reject the predictions of the lifecycle model.
More recent studies isolate the labor supply response to anticipated changes in taxation via novel methods (Saez, 2003; Looney and Singhal, 2006). Both of these studies report labor earnings elasticities, and the shared spirit of their respective research designs is to identify predictable changes in marginal tax rates to estimate an intertemporal labor supply elasticity.^{4} Both studies, however, assume away income effects associated with the variation in tax rates by relying on the lifecycle model's prediction that anticipated changes in income have no effect on behavior. This reliance on the lifecycle model to identify an intertemporal elasticity is at odds with the vast literature on the "excess sensitivity" puzzle and consumption's response to predictable changes in income (Carroll, Hall, and Zeldes, 1992; Flavin, 1981; Zeldes, 1989; Campbell and Mankiw, 1987; Deaton, 1991). Whether labor supply responds to anticipated changes in income induced by variation in taxes is closely tied to the "excess sensitivity" puzzle in that the forces driving the latter may, at least theoretically, also explain the former.^{5}
The two main contributions of this paper are to identify households' labor supply response to an anticipated change in tax rates and liabilities in a lifecycle setting, and to assess the lifecycle model's prediction that these changes have no effect on behavior. The first contribution is important in a methodological sense as identifying random variation in anticipated tax rates, which is necessary for estimating an intertemporal elasticity, is particularly challenging. The second contribution addresses whether the lifecycle model is a suitable framework for interpreting the estimated elasticity. This aspect of this paper distinguishes it from previous work exploiting ageeligibility rules for the dependent exemption and EITC (Looney and Singhal, 2006). Consistent with the "excess sensitivity" puzzle, many arguments for why the lifecycle model is not a good approximation for behavior abound (Courant, Gramlich, and Laitner, 1984). If, for example, individuals "schmedule" and conflate average and marginal tax rates, one might observe responses to anticipated changes in income (Liebman and Zeckhauser, 2004). Indeed, Feldman and Katuscak (2005) find empirical support for "schmeduling" and show that taxpayers' average tax rates influence their labor income. Alternatively, individuals may not be forwardlooking (with respect to responding to taxation) and a static labor supply model may be a better approximation.^{6}
The ageeligibility rules for qualifying a child as a dependent provide an opportunity to, first, consistently estimate a labor supply response to taxation and, second, to assess whether the mechanisms by which households respond are attributable to variation in marginal tax rates. I estimate a labor supply response by exploiting that eligibility for a child primarily hinges on whether s/he is strictly under 19 (or 24 if s/he is a fulltime student) as of the end of the calendar year. The benefits of having an eligible dependent are that a household is allowed to exempt more income and to face a tax schedule with, on the whole, lower statutory and effective marginal tax rates due to the Earned Income Tax Credit.^{7} The December 31st cutoff date lends itself nicely to an experiment entailing the quasirandom assignment of households to a particular tax schedule (see Figures (1) and (2)).^{8} Households with a child turning 19 (or 24) on December 31st of a calendar year are not allowed to claim him/her as an eligible dependent whereas those with a child turning 19 on the subsequent January 1st are allowed.^{9}
Because only a subset of households experience changes in the marginal tax rate due to a dependent (see Figures (1) and (2)), whereas all households face average tax rate changes, I identify the mechanisms by which fathers and mothers respond to the tax schedule. Observing that households experiencing marginal tax rate changes adjust their labor supply, while the other households do not, means that the lifecycle model is an appropriate approximation of behavior. If, on the other hand, those experiencing only average tax rate changes also respond to the tax schedule, this finding presents a puzzle for the lifecycle model.
The key identifying assumption in the regression discontinuity design is that households with similarly aged children are also similar in many observed and unobserved ways. Most notably, within a narrow range of the December 31st cutoff for having an ageeligible child, households are assumed to be at similar stages of the lifecycle and to have similar levels of lifetime income. This assumption allows for a causal estimate of the effect of the ageeligibility rule on labor supply. An attractive feature of the regression discontinuity design is that, like in a randomized controlled trial, this identifying assumption is testable (Lee, 2005). The identification strategy also entails households' knowing when their children turn 19 (or 24) and being aware that they can no longer claim their dependent. In this sense, the changes in the tax schedule that households experience upon losing an ageeligible dependent are fully anticipated. A final advantage of the regression discontinuity design is that I am able to isolate exogenous marginal and average tax changes that households near the cutoff would experience if they were to lose or gain an eligible dependent. By doing so, I can whether solely marginal tax incentives provide intertemporal labor supply incentives.
Consider a unitary decisionmaking household that has one child, and suppose its labor supply in period () can be expressed as the following structural equation:
Other equations of interest include the firststage and reducedform equations:
Identification of all the parameters in (4) to (6) rests on four conditions (McCrary and Royer, 2005; Hahn, Todd, and van der Klauuw, 2001; Lee, 2003). First, the household's labor supply as a function of the child's age must be smooth in the absence of the ageeligibility rules. Second, households around the ageeligibility cutoffs must be exchangeable so that barely ineligible households are an appropriate counterfactual for barely eligible households. Third, there must be a discontinuity in the treatment assignment mechanism, and ageeligibility rules must influence whether households claim an ageeligible dependent on their tax returns. Fourth, the ageeligibility rule can only influence households' treatment in a monotone way. Taken together, the first three conditions ensure that the estimated effect of the ageeligibility rule on household labor supply does not capture extraneous effects, such as those of households' levels of lifetime wealth or preferences for leisure. The fourth condition allows for a local average treatment effect (LATE) interpretation of (Imbens and Angrist, 1994).
An attractive feature of the regression discontinuity design is that these conditions also correspond to a large number of testable predictions, which can increase the confidence that the estimates are valid. In the next section, I present evidence that, based on implications of the first three conditions, there is not sufficient evidence to reject the validity of the research design. There may, however, be a concern that the monotonicity condition may not be met. Thus far, I have defined the "treatment" facing a different tax schedule due to having an eligible dependent. In this case, the monotonicity assumption is trivially satisfied in that an ageeligible dependent corresponds to a different tax schedule and lowers tax liabilities for all households. For a treatment defined in terms of changes in the marginal tax rate, which is useful for computing labor supply elasticities, due to the nonincreasing structure of marginal tax rates as a function of taxable income, the monotonicity assumption may not hold (see Figures (1) and (2)). That is, it is possible for the ageeligibility rule to both increase and decrease a household's marginal tax rate, depending on the household's characteristics. In the next subsection, I discuss in further detail the mechanisms for the effect of the ageeligibility rule in light of its nonmonotonic effect on the marginal tax rate. Also, when interpreting the results from estimation, I discuss the difficulty of interpreting as a LATE parameter.
To see the basis of the identification strategy more formally, consider households with a child over 19 by an amount ( ), who are ineligible to claim a dependent, and households with a child under 19 ( ), who are eligible. In general, it is likely that for large values of , (and therefore ) is correlated with having an ageeligible dependent, the household's tax schedule, and labor supply.^{11} As such, the simplest comparison of comparing the labor supply behavior of all households with and without an ageeligible dependent fails to produce a consistent estimate of . The Appendix describes why this comparison is invalid in greater algebraic detail.
To consistently estimate , consider a sequence of estimators for different values of . As becomes smaller, the two types of families become increasingly comparable in all ways, except in whether they can claim a dependent. In the limit as
, the average observables for the two types of families are the same:
To estimate the the parameters of this model, it is helpful to graph the outcome of interest against the age of the child to, first, visually detect the effect of the ageeligibility rule, and to, second, estimate the magnitudes of and . In the data, which I further discuss in the next section, age is measured in months and is therefore "binned" and not continuous. To estimate the reducedform effects of the ageeligibility rule, I plot the agespecific averages of fathers' and mothers' labor supply, and fit a flexible global polynomial through the agespecific averages (DiNardo and Lee, 2004; Lee and Card, 2004). Fitting a global polynomial entails making a parametric assumption about the underlying function, so in discussing the results, I present alternative fits as a robustness check. If there is an effect of the ageeligibility rule on fathers' or mothers' labor supply, the polynomial ought to detect a break in the agespecific averages around the ageeligibility cutoffs. The discontinuity in the agespecific averages around the ageeligibility thresholds ought to be graphically visible as well. The magnitude of this break represents the effect of the ageeligibility rule.
To estimate the magnitude of this break, I fit thirdorder global polynomials through the agespecific local averages of fathers' and mothers' labor supply, allowing the slope and intercept to vary according to whether the child's age is under 19, between 19 and 24, or 24 and older. I also fit
first and secondorder polynomials through the local averages to show the sensitivity of the estimated break to assumptions about functional form. I normalize the age of the child to equal zero at ages 19 and 24 in order to readily compute the reduced form parameters,
and
, at each ageeligibility cutoff. Dropping the subscripts, I estimate the
following equations through the local averages:
To understand whether the variation in marginal tax rates of barely eligible and ineligible households drives their labor supply responses to the ageeligibility rule, I adapt a "simulated instrument" to the regression discontinuity framework (Rothstein, 2007; Leigh, 2004; Dahl and Lochner, 2005). All tax rates and liabilities depend on taxable income, which in turn depends on a household's deductions, exemptions, and pretax income. Furthermore, tax rates affect pretax income via labor supply and other incentives. Using demographic variables to predict pretax income breaks the simultaneity with tax rates, and using this predicted measure of pretax income allows for estimates of marginal and average tax rates, as well as liabilities, that are independent of (unobserved) labor supply effort.
Mechanically, it is possible to fit polynomials through agespecific averages of these instrumented marginal tax rates in order to obtain an estimate of in equation (5). This simply entails combining a household's predicted pretax income and their deductions and exemptions to compute their tax rates and liability. As discussed earlier, however, because the ageeligibility rule has a nonmonotonic effect on marginal tax rates, this estimate does not yield the structural estimate of the effect of marginal tax rates on households' labor supply (Hahn, Todd, and van der Klauuw, 2001). Furthermore, the elasticity implied by dividing the reduced form parameter by the firststage parameter, , is neither consistent nor a LATE estimate.
Combining the instrumented tax variables with the ageeligibility rule allows for the identification of households who are at "kink" points in the tax schedule. For ageeligible households, it is possible to compute , which denotes a vector of tax variables corresponding to their tax rates and liabilities taking into account their ageeligible dependents. It is also possible to construct for these households, which denotes their tax rates and liabilities if they were to lose an ageeligible dependent. When , household is at a "kink" point. Ageineligible households also have analogously defined tax vectors.^{13} Here, an example helps clarify the purpose of these tax vectors and their components. Among ageeligible households means that they are at "kink" in the marginal tax rate schedule such that if they were to lose a dependent, their marginal tax rate would rise by 10 percentage points. The counterfactual for these households are the barely ineligible households who would experience a decrease in their marginal tax rate by 10 percentage points if they were to gain an eligible dependent.
Near the eligibility thresholds, ageineligible households whose "kink" size equals are thus a counterfactual for ageeligible households whose "kink" size equals . Defining accordingly for eligible and ineligible households, I augment (8) to:^{14}
According to the lifecycle model, , and the ageeligibility rule does not have an independent effect on fathers' and mothers' labor supply. Instead, the ageeligibility rule affects labor supply through the disincentives of higher marginal tax rates. If it is indeed the case that higher marginal tax rates cause work disincentives, then should be negative. In other words, the ageeligibility rule should only matter to the extent that it coincides with changes in the marginal tax rate. Also, in a lifecycle model, the coefficients on the interactions of and the average tax rate ( ) and tax liability ( ) "kinks" should be zero under the null hypothesis of lifecycle behavior.
If, on the other hand, the lifecycle model is not a good approximation of behavior, it is possible to observe the ageeligibility rule having an independent effect on labor supply or the ageeligibility rule affecting labor supply decisions through average tax rates. Finding that the coefficient on the interaction of and the average tax rate is negative ( ) would be consistent with the "schmeduling" hypothesis of decisionmaking (Liebman and Zeckhauser, 2004). If households are myopic and a static labor supply model characterizes their behavior, the substitution effect in static labor supply model suggests that while is consistent with a negative income effect.
Thus far, I have assumed that it is possible to randomly sample from households near the ageeligibility rules. This is a sufficient condition for being able to consistently estimate the reduced form and structural estimates of the effect of facing a different tax schedule on household labor supply. It is, however, possible for the agedependent exemption to affect household formation. That is, children may delay leaving their parents, as well as other transitions to adulthood, in which case households observed around the ageeligibility cutoffs comprise selfselected groups.
Empirically, one can assess the sample selection problem by estimating the effect of the ageeligibility rules on the probability of being observed.^{16} Alternatively, testing for discontinuities in the distribution of the running variable provides evidence on the extent to which sample selection is an issue (McCrary, 2004). In either case, if the estimated effect is zero, it may be possible to consistently estimate the reduced form and structural parameters. If this effect is not zero, the resulting estimates of the reduced form and structural parameters are an admixture of the treatment effect and a selection effect. It is not possible to consistently estimate the parameters of the model, however one may be able to generate consistent bounds on this effect (Lee, 2003).
The data for this paper are from the 19901993, 1996, and 2001 panels of the Survey of Income and Program Particiapation (SIPP). These nationally representative panels of households span the calendar years 19901994, 1996, 1997, 1998, 2001, and 2002. Unlike the publicuse versions of the Census or the Current Population Survey (CPS), the SIPP variables include the month of an individual's birth rather than just the year, quarter, or age at last birthday. This means that there is greater flexibility than otherwise to finely separate the older 18yearolds from their younger counterparts, and similarly the younger 19yearolds from the older ones. This is an advantage of the SIPP to the extent that there is seasonality in births that would be more pronounced with less precise data on births (such as with quarters) (Bound, Jaeger, and Baker, 1995). In addition, the identification strategy requires observable and unobservable similarity among households around the December 31st cutoff, and this condition is more likely to hold in a narrower range of dates.
Each observation in the dataset is a "child" sampled in the SIPP, where the term "child" means those who can be linked to a parent. I further limit this set to children under 40 living at home with a parent during part or all of a calendar year, which yields 373,966 childparent observations. Low socioeconomic status households are oversampled in the SIPP, and though the Census Bureau provides sampling weights, all estimates in this paper are unweighted.^{17} A set of descriptive characteristics for the sample as a whole are in Table (1). The sample is 67% white, and parents, on average, have 12 years of education. Fiftythree percent of children under 40 are from twoparent households. The columns of Table (1) suggest that there are statistically significant differences in the parental characteristics of households with older (ineligible) and younger (eligible) children. For example, older children are less likely to be a part of a twoparent household and more likely to have less educated parents. These differences lend support for the necessity of a research design that addresses this heterogeneity.
Data for marginal and average tax rates, as well as liabilities, are from the NBER's tax simulator (TAXSIM) program (Feenberg, October 7, 2005; Feenberg and Coutts, 1993). For each childparent observation, TAXSIM computes marginal and average tax rates using the parents' marital status, the number of eligible dependents, and predicted measures of pretax earnings, interest income, and other income items appearing on the IRS Form 1040. Constructing the counterfactual tax changes entails the same TAXSIM calculation but instead using the counterfactual number of dependents. Because the core waves of the SIPP do not report charitable and other itemized deductions, I assume that all households are nonitemizers in executing the TAXSIM calculations.^{18}
To test the validity of the research design, I present a set of baseline characteristics of the parents around the ageeligibility cutoffs in Table (2). The third and fifth columns report the estimated differences in these characteristics at the age19 and age24 eligibility cutoffs, respectively. The same characteristics are graphically represented in Figures (4) to (6). The results from Table (2) and these figures suggest that around the age 19 and age 24 ageeligibility cutoffs, parents are equally likely to be white, while the average education levels and ages of the fathers and mothers are also similar around the cutoffs. Though these findings do not prove that the unobservable characteristics of households are the same around the ageeligibility cutoffs, they do not provide any a priori grounds for rejecting the research design.
In this section, I discuss the effect of the ageeligibility rule on a child's eligibility status, and present the reduced form results on the effect of the ageeligibility rule on households' labor supply outcomes. In the next section, I interpret these results in light of the structural equation in (4). The labor supply outcomes analyzed include fathers' and mothers' annual hours of work, as well as the corresponding employment participation rates. I examine these measures of labor supply, rather than labor earnings, in order to isolate a real behavioral response, rather than one related to income reporting or income shifting (Slemrod, 1990).
At the age19 and age24 cutoffs in Figure (3), the ageeligibility rule has a noticeable effect on whether a household is eligible to claim a child as a dependent. The top panel shows that the likelihood that an 18yearold child is eligible is 1 while this probability drops to around 0.6 for 19yearold children. In determining eligibility, I assume that the other eligibility tests, such as the support test, do not apply.^{19} This probability does not drop to zero at the age19 threshold because parents can claim a child if they are a fulltime student, and in the SIPP sample, around 60% of 19yearold children are fulltime students. The bottom panel of Figure (3) shows how the total number of eligible dependents in a child's household is discontinuous at ages 19 and 24. Based on the discontinuities observed in Figure (3), the ageeligibility rule provides variation in the tax schedule that households face around the age19 and age24 cutoffs.^{20}
Estimates of the reducedform effects of the ageeligibility rules are in table (3). Here, I present estimates of in equation (7), which I estimate by fitting first, second, and thirdorder polynomials through the covariate unadjusted and adjusted local averages. Figures (7) and (8) display the covariateadjusted local averages for annual hours and participation among fathers and mothers.
In Figure (7), the curvature in the local averages leads one to reject the linear specification for the conditional expectations function. For fathers' annual hours, the sizeable estimate given by the quadratic specification belies the visual smoothness of the local averages, particularly around the age19 cutoff. The thirdorder polynomial, plotted in the figure, appears to best approximate the conditional expectations formulate for the effect of the ageeligibility rule.
Across all three polynomial approximations, the ageeligibility rule results in fewer annual hours among mothers at the age19 cutoff. The thirdorder polynomial suggests that the estimated effect of having an eligible dependent is to increase annual labor supply by 40 hours. Alternatively, mothers are likely to work 40 hours less upon losing the age exemption. The result for mothers around the age24 cutoff deserves a more careful interpretation. First, the amount of variability in this region is significantly greater than elsewhere. This is due to the fact that there are not many "older" children living with their parents, and the ones who do are likely to differ from households with similarly aged parents but without "children" living at home. Second, as I discuss in greater detail in the next section, the magnitude and direction of the induced marginal tax rate changes is different at the age24 cutoff than at the age19 one.
For both fathers and mothers, the ageeligibility rule has little to no effect on the participation decision. This result is not surprising for fathers, who work, on average, over 1,800 hours per year (see Table (1)) and among whom there is no effect of the ageeligibility rule on annual hours. The result for mothers differs from the previous literature's finding that women's labor supply responses to taxation are larger on the extensive margin than on the intensive margin (Meyer and Rosenbaum, 2001; Eissa and Liebman, 1996). Mothers' nonresponse in participation, however, is consistent with the lifecycle model. In contrast to previous studies that exploit unanticipated tax policy changes as a source of variation in tax rates, the tax changes induced by the ageeligibility rule are are anticipated and do not correspond to revisions to lifetime wealth. Furthermore, with the regression discontinuity design, I arguably compare households with similar lifetime wealth and unobserved preferences for nonmarket time, which are determinants of the participation decision. These two explanations may account for the relatively larger intensive margin response among mothers.
To deconstruct the intensive labor supply response, I uncover where in the distribution of hours the ageeligibility rule has the largest effect by estimating the following sequence of regressions for annual hours worked ():
In this section, I interpret the findings above in the context of a standard lifecycle labor supply model in order to assess the magnitude of households' (particularly mothers') labor supply responses, and to infer households' intertemporal labor supply elasticities. Recall that barely ineligible households are a counterfactual for barely eligible households in the subsequent year. I first discuss a LATE interpretation of the labor supply response to the ageeligibility rule. This interpretation points in the direction of sizeable (and implausible) intertemporal labor supply elasticity estimates for mothers. The estimates for fathers are statistically insignificant from zero, but the point estimates are well within the range of previous estimates. I then provide an alternative explanation of the labor supply response, and present evidence suggesting that the dominant mechanism for the labor supply response may not be through marginal tax rates. Overall, I argue that the effect of losing an ageeligible dependent appears not to work through the concomitant changes in the marginal tax rate, which suggests that it may be tenuous to assume that predictable changes in income have no effect on labor supply (holding aftertax wages constant).
Quantifying how tax rates change when households lose dependents is necessary to know the incentives they face. With an ageeligible dependent, households face a different tax schedule than otherwise that enables them to exempt more income and face a different set of statutory and effective marginal tax rates. As discussed earlier, Figures (1) and (2) illustrate how the presence of dependents changes the relationships among marginal tax rates, tax liabilities, and pretax income for married and heads of households, respectively. The presence of an ageeligible dependent lowers tax liabilities at all levels of income. The effect of an ageeligible dependent on a household's marginal tax rate, however, is less clear. At high and very low levels of income, having a dependent lowers a household's marginal tax rate whereas between $20,000 and $40,000 (in 2001 $ for married households) of pretax income, marginal tax rates are higher with an ageeligible dependent. For some ranges of income, an ageeligible dependent does not change a household's marginal tax rate.
As discussed earlier, estimating how the ageeligibility rule affects a household's marginal tax rates presents a challenge because the latter is a nonlinear and nonmonotonic function of taxable income. In turn, taxable income depends on the household's deductions as well as the behavioral response to being (in)eligible for the dependent exemption and the EITC (Triest, 1998). To address this simultaneity between a household's taxable income and its exemptions, I construct a "simulated instrument." I use all childparent observations to run a Tobit regression of pretax income on a vector of predetermined characteristics.^{21} I then predict pretax income. With this prediction and using households' actual dependenteligibility status, I use the NBER's TAXSIM program to produce a set of predicted tax variables for each childparent observation in the data.
Table (4) presents the results from constructing a "simulated instrument" and fitting a thirdorder global polynomial (see Equation (8)) through the local averages of the tax variables (analogous to Table (3)). According to this table, the effects of the ageeligibility rule on tax rates and liabilities differ depending on whether one is looking at the age19 or age24 cutoffs. This is largely due to the fact that the distribution of households' predetermined characteristics are correlated with the age of the child over large age ranges. In addition, the nonlinearities in the tax schedule (seen in Figures (1) and (2), mean that the direction of the ageeligibility rule's effect depends on the distribution of predetermined characteristics. At the age19 threshold, I estimate that barely ineligible households face a netoftax rate that is 0.5 percentage points (or 0.3%) lower than their barely eligible counterparts. Barely ineligible households with a 24yearold child face a netoftax rate that is 0.4 percentage points (or 0.6%) higher than those barely eligible.
While neither of the netoftax rate estimates at the age19 threshold is statistically significant from zero, we can nonetheless discuss the implied labor supply elasticities. The point estimates imply a labor supply elasticity of around 0.4 for fathers but an implausibly large elasticity of about 12 for mothers.^{22} This estimate for mothers vastly exceeds the usual estimates found for women in the literature, which suggest an intertemporal elasticity of no larger than one. Because the ageeligibility rule can both increase or decrease a household's marginal tax rate, the estimated intertemporal elastiticities for mothers and fathers cannot have a typical LATE interpretation. It is, however, possible to sign the bias in the estimated effect from the true LATE parameter. Assuming that the the response in hours due to changes in the marginal tax rate are in the direction predicted by the lifecycle model, these estimates are a lower bound on the true parameters.^{23}
In the context of the ageeligibility rule affecting the entire tax schedule or the tax liability that a household faces, the subpopulation likely to be affected is relatively broad in this case, and captures the majority of households.^{24} Hence the monotonicity assumption that is needed for a LATE interpretation is unlikely to be violated. Indeed, as the bottom half of Table (4) shows, the ageeligibility rule increases the average tax rate by about 1 percentage point (or 4.3%) among those near the age19 cutoff. The corresponding increase in households' tax liabilities is $430. The results are in the same direction but somewhat more muted at the age24 cutoff as the number of households affected is considerably less.^{25}
It is possible to identify whether marginal tax rates or average tax rates (or neither) comprise the mechanism through which ageineligible households are incentivized to decrease their labor supply. For each childparent observation, I construct a set of counterfactual tax variables. For barely eligible households, I use TAXSIM to to compute their tax rates and liability if they were to become ineligible. Similarly, TAXSIM produces a set of tax variables for barely ineligible households. Table (5) shows that around 30% of those around the age19 cutoff, and 40% of those around the age24 cutoff, would experience a change in their marginal (and hence their netof) tax rate if they were to lose or gain a dependent. According to this exercise, virtually all households would experience a change in their average rate and their tax liability upon losing or gaining a dependent.
Another goal of this particular TAXSIM exercise is to identify households who are at similarly sized kinks on both sides of the ageeligibility rule. In Table(5), around 30% of households around the age19 cutoff, and 40% around the age24 cutoff, would experience changes in their marginal tax rates if they were to lose or gain an eligible dependent. Table (5) also reports the average magnitude of the tax changes that households would experience if they were to gain or lose a dependent. Around the age19 cutoff, the average increase is about half a percentage point whereas around the age24 cutoff, households' marginal tax rates would decrease by 2.5 percentage points if they were to lose a dependent.^{26} This identification enables a test of the lifecycle prediction that the ageeligibility rule alters labor supply incentives through the netoftax rate if the effect of the ageeligibility rule is identified among those who would experience changes in their marginal tax rates. In addition, households who are not at a netoftax rate "kink" should not be affected by the ageeligibility rule, according to the lifecycle hypothesis.
Table (6) presents the results from estimating (9). In Columns (1) and (3), I only interact the magnitude of the marginal tax rate "kink" with the ageeligibility rule whereas in Columns (2) and (4), I also include interactions with the average tax rate "kink." From this table, at best, the variation in marginal tax rates can only partially account for the labor supply response of fathers and mothers. Barely eligible fathers facing higher marginal tax rates are likely to increase their labor supply upon losing an ageeligible dependent (Columns (1) and (2)), which is at odds with the lifecycle model. In Column (3), barely eligible mothers, on the other hand, adjust their labor supply consistent with lifecycle labor supply behavior, and are likely to reduce their annual hours when faced with higher marginal tax rates around the age19 cutoff. The statistically significant coefficient on the age19 dummy in Column (3) suggests that the ageeligibility rule has an independent effect on mothers' labor supply. Indeed, when controlling for the average tax rate changes facing households in Column (4), the independent effect of the ageeligibility rule disappears. And not only does the direction of the effect of the marginal tax rate incentives reverse itself for mothers in Column (4), differences in the average tax rate between barely eligible and ineligible households explain the negative effect of the ageeligibility rule, consistent with the "schmeduling" hypothesis.
As alluded to earlier, a key issue for identifying the effect of the ageeligibility rule is whether the sample of childparent observations around the age19 and age24 thresholds comprise selfselected groups. Nonrandom selection around each side of the ageeligibility cutoffs can occur if children delay leaving their parents' home or if, around the time of birth, households manipulate the timing of their children's births (Chandra and DickertConlin, 1999). To obviate the concern that the sample of children living with their parents is a selfselected group around the ageeligibility cutoffs, I present the age distribution of this sample in Figure (11). If there were selection into the sample and the probability of being observed were different on each side of the ageeligibility rule, one would expect to see a break in the age density at the thresholds. Using the methods in McCrary (2004), I cannot reject that the distribution is continuous at ages 19 and 24.^{27} Hence I am reasonably confident that mothers' labor supply responses to the ageeligibility rule are not instead due to issues concerning household formation.
Another potential concern about the effect of the ageeligibility rule, particularly for mothers, is that the result is a falsepositive. If the location of the discontinuity were instead elsewhere and not at the December/January cutoff, it may be possible that the polynomial would detect a discontinuity at the age19 threshold when in fact the "true" experiment is either earlier or later in the calendar year. Table (7) presents estimates of the effect of the ageeligibility rule at various cutoffs. In column (2), I shift the "experiment" to the July when the child is 18 by redefining the age19 dummy in (8) to equal 1 for children aged 18 and 1/2 or older. Similarly in column (3), the "experiment" occurs in the July when the child is 19. In both cases, the effect of the ageeligibility rule for mothers decreases (in absolute value) and becomes statistically insignificant. For mothers, the standard errors are also somewhat higher in columns (2) and (3) than in column (1), which represents the relevant estimates from Table (3). Because the estimates in columns (2) and (3) are not statistically different from those in column (1), these placebo tests merely suggest that the experiment is not occurring at another point in time.
The effect of an anticipated increase in a household's tax liability on a household's labor supply is difficult to reconcile with the predictions of a lifecycle labor supply model. Though it is difficult to detect a response among fathers, mothers supply about 40 fewer hours per year upon losing an ageeligible dependent. They are equally likely to participate in the labor market as their barely eligible counterparts. Though the implied intertemporal labor supply elasticity estimates for mothers is 12, this estimate is implausibly large and further analysis shows that average tax rates have an independent effect, controlling for the accompanying changes in households' marginal tax rates. This finding presents a puzzle for the standard lifecycle labor supply model, which predicts that anticipated changes in income do not influence the householdspecific lifecycle variation in hours.^{28}
Though it seems that the behavior of households is inconsistent with the predictions of a lifecycle model, there are nonetheless important lessons from the results of the quasiexperiment studied. First, the tax system, through policies like the dependent exemption and the EITC, can encourage households' labor supply decisions, even if this process occurs through a mechanism that economists have yet to precisely identify. This finding is particularly important in light of the federal government's increasing reliance on the tax system to achieve its redistributional goals. Second, for mothers, the effects of these policies is largest for parttime and fulltime workers than it is for those entering the labor market. This suggests that intertemporal tax incentives are not as effective at inducing nonworking mothers into the labor market as they are at encouraging working mothers to work longer hours.
All  19 to  Over 24  

Father's Annual Hours  1,818  1,897  1,745  1,161 
Father's Annual Hours (standard error)  (2.2)  (2.4)  (6.4)  (8.6) 
Mother's Annual Hours  1,078  1,070  1,271  885 
Mother's Annual Hours (standard error)  (1.8)  (2.1)  (5.3)  (6.3) 
Father's Participation  .926  .944  .916  .769 
Father's Participation (standard error)  (.001)  (.001)  (.002)  (.003) 
Mother's Participation  .8  .803  .845  .708 
Mother's Participation (standard error)  (.001)  (.001)  (.002)  (.003) 
Percent White  .667  .662  .686  .671 
Percent White (standard error)  (.001)  (.001)  (.002)  (.002) 
Father's Education  12.4  12.5  12.3  11.3 
Father's Education (standard error)  (.008)  (.008)  (.023)  (.029) 
Mother's Education  12.1  12.3  12.1  11.2 
Mother's Education (standard error)  (.006)  (.006)  (.018)  (.022) 
Father's Age  42  39  49  58 
Father's Age (standard error)  (.021)  (.019)  (.042)  (.066) 
Mother's Age  39  36  47  56 
Mother's Age (standard error)  (.019)  (.016)  (.032)  (.051) 
From 2Parent HH  .525  .672  .492  .161 
From 2Parent HH (standard error)  (.001)  (.001)  (.002)  (.001) 
Percent of observations from: 1990  .07  .074  .077  .056 
Percent of observations from: 1990 (standard error)  (0)  (.001)  (.001)  (.001) 
Percent of observations from: 1991  .113  .119  .122  .092 
Percent of observations from: 1991 (standard error)  (.001)  (.001)  (.001)  (.001) 
Percent of observations from: 1992  .103  .11  .109  .08 
Percent of observations from: 1992 (standard error)  (0)  (.001)  (.001)  (.001) 
Percent of observations from: 1993  .119  .129  .12  .094 
Percent of observations from: 1993 (standard error)  (.001)  (.001)  (.001)  (.001) 
Percent of observations from: 1994  .061  .065  .061  .049 
Percent of observations from: 1994 (standard error)  (0)  (.001)  (.001)  (.001) 
Percent of observations from: 1996  .055  .061  .051  .043 
Percent of observations from: 1996 (standard error)  (0)  (0)  (.001)  (.001) 
Percent of observations from: 1997  .107  .113  .105  .092 
Percent of observations from: 1997 (standard error)  (.001)  (.001)  (.001)  (.001) 
Percent of observations from: 1998  .115  .103  .109  .149 
Percent of observations from: 1998 (standard error)  (.001)  (.001)  (.001)  (.001) 
Percent of observations from: 2001  .119  .112  .112  .14 
Percent of observations from: 2001 (standard error)  (.001)  (.001)  (.001)  (.001) 
Percent of observations from: 2002  .138  .114  .134  .204 
Percent of observations from: 2002 (standard error)  (.001)  (.001)  (.002)  (.001) 
Source: SIPP 19902001. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Estimated Difference at 19 
Estimated Difference at 24 


Percent White  .678  .009  .699  .021 
Percent White (standard error)  (.006)  (.01)  (.007)  (.012) 
Father's Education  12.5  .06  12.1  .245 
Father's Education (standard error)  (.057)  (.125)  (.099)  (.156) 
Mother's Education  12.2  .044  11.8  .074 
Mother's Education (standard error)  (.045)  (.083)  (.072)  (.155) 
Father's Age  46.8  .07  52.1  .057 
Father's Age (standard error)  (.107)  (.156)  (.175)  (.226) 
Mother's Age  44.0  .125  49.6  .217 
Mother's Age (standard error)  (.082)  (.086)  (.128)  (.17) 
Source: SIPP 19902001. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Estimated differences at ages 19 and 24 calculated by fitting a thirdorder polynomial through agespecific local averages. See text for further details.
Dependent Variable  Without Covariates: Age19 
Without Covariates: Age24 
With Covariates: Age19 
With Covariates: Age24 

Father's Annual Hours: Linear  8  38  24^{*}  43^{*} 
Father's Annual Hours: Linear (standard error)  (14.0)  (23.9)  (13.2)  (22.9) 
Father's Annual Hours: Quadratic  42^{**}  33  40^{**}  22 
Father's Annual Hours: Quadratic (standard error)  (18.8)  (32.7)  (19)  (33.7) 
Father's Annual Hours: Cubic  0.2  37  2  38 
Father's Annual Hours: Cubic (standard error)  (21.6)  (40)  (21.8)  (45) 
Mother's Annual Hours: Linear  49^{***}  34^{**}  16^{*}  57^{***} 
Mother's Annual Hours: Linearb (standard error)  (10.4)  (16.9)  (9.2)  (16) 
Mother's Annual Hours: Quadratic  30^{**}  32  29^{**}  27 
Mother's Annual Hours: Quadrati (standard error)  (11.8)  (20.1)  (11.4)  (18.6) 
Mother's Annual Hours: Cubic  39^{***}  49^{**}  40^{***}  39^{*} 
Mother's Annual Hours: Cubic (standard error)  (14.3)  (23.6)  (13.4)  (22.4) 
Father's Participation: Linear  0.01^{***}  0  0.02^{***}  0 
Father's Participation: Linear (standard error)  (.003)  (.006)  (.003)  (.005) 
Father's Participation: Quadratic  0.01^{**}  0.01  0.01  0 
Father's Participation: Quadratic (standard error)  (.003)  (.009)  (.004)  (.004) 
Father's Participation: Cubic  0.006  0.01  0.01  0.01 
Father's Participation: Cubic (standard error)  (.004)  (.011)  (0.005)  (.009) 
Mother's Participation: Linear  0.01^{**}  0.01  0.01^{***}  0.04^{***} 
Mother's Participation: (standard error)  (.003)  (.006)  (.004)  (.006) 
Mother's Participation: Quadratic  0  0  0.02^{***}  0.01 
Mother's Participation: (standard error)  (.004)  (.009)  (.005)  (.009) 
Mother's Participation: Cubic  0.01  0.01  0  0.01 
Mother's Participation: (standard error)  (.005)  (.011)  (.007)  (.011) 
Source: SIPP 19902001. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Covariates include calendar year, white, fathers' and mothers' education, fathers' and mothers' age, and marital status.
* denotes statistical significance at the 10% level, twotailed test.
** denotes statistical significance at the 5% level, twotailed test.
*** denotes statistical significance at the 1% level, twotailed test.
Dependent Variable  Age19  Age24 

0.45  0.35  
(standard error)  (.409)  (.303) 
0.3%  0.6%  
(standard error)  (.432)  (.339) 
Predicted Average Tax Rate  0.97^{***}  0.14 
Predicted Average Tax Rate (standard error)  (.178)  (.224) 
Predicted ATR  4.3%^{***}  2.5% 
Predicted ATR (standard error)  (1.26)  (1.56) 
Predicted Tax Liability  430^{***}  64 
Predicted Tax Liability (standard error)  (123)  (102) 
In Percentage Points for Levels, Percent for Logs
Source: SIPP 19902001 and NBER TAXSIM. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Effect of ageeligibility rule calculated using thirdorder global polynomial. Results for combined federal and state tax rates and liabilities reported. Tax rates and liabilities are calculated using a prediction for pretax income, which is based on a Tobit regression of actual pretax income on marital status, race, and flexible functions of schooling and age. Tax rates and liabilities depend on actual ageeligibility status. See text for further details.
* denotes statistical significance at the 10% level, twotailed test.
** denotes statistical significance at the 5% level, twotailed test.
*** denotes statistical significance at the 1% level, twotailed test.
(Lose) 
(Gain) 
(Lose) 
(Gain) 


% would experience MTR change  .306  .297  .384  .434 
% would experience MTR change (standard error)  (.006)  (.006)  (.008)  (.008) 
Avg. size of MTR change  .462  .656  2.5  2.4 
Avg. size of MTR change (standard error)  (.107)  (.099)  (.143)  (.145) 
% would experience ATR change  .986  1  1  .999 
% would experience ATR change (standard error)  (.002)  (0)  (0)  (0) 
Avg. size of ATR change  2.9  2.2  2.7  2.7 
Avg. size of ATR change (standard error)  (.076)  (.078)  (.074)  (.087) 
% would experience tax liability change  .987  .943  .966  .968 
% would experience tax liability change (standard error)  (.002)  (.003)  (.003)  (.003) 
Avg. size of tax liability change  874  721  715  915 
Avg. size of tax liability change (standard error)  (14.2)  (7.8)  (8.2)  (9.5) 
5723  5819  3857  3940 
Source: SIPP 19902001 and NBER TAXSIM. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Households determined to be at "kink" in tax schedule in the following way. Among households with children under 19 or 24, counterfactual tax schedule corresponds to one less dependent, holding all else constant. Among households with children greater than 19 or 24, counterfactual tax schedule has one more dependent. If tax rates and/or liabilities differ between the actual and counterfactual tax schedules, household is said to be at a "kink" and would thus experience a change.
Fathers' Hours (1) 
Fathers' Hours (2) 
Mothers' Hours (3) 
Mothers' Hours (4) 


Age  4.2  35.6^{*}  34.2^{**}  16.6 
Age (standard error)  (20.66)  (21.1)  (14.92)  (15.34) 
Age  3.38^{***}  2.71^{*}  1.37^{*}  1.84^{**} 
Age (standard error)  (1.45)  (1.51)  (.79)  (.81) 
Age  1.67  9.31^{***}  
Age (standard error)  (2.56)  (1.22)  
Age  56.8  89.6  25.2  41^{*} 
Age (standard error)  (45.44)  (47.72)  (22.59)  (25) 
Age  2.46  .32  1.85  .53 
Age (standard error)  (1.69)  (1.83)  (.98)  (1.11) 
Age  2.08  2.75^{*}  
Age (standard error)  (2.89)  (1.49) 
Source: SIPP 19902001 and NBER TAXSIM. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Estimates are from fitting a thirdorder global polynomial through the covariateadjusted agespecific local averages of fathers' and mothers' labor supply.
* denotes statistical significance at the 10% level, twotailed test.
** denotes statistical significance at the 5% level, twotailed test.
*** denotes statistical significance at the 1% level, twotailed test.
Dependent Variable  "Location of Experiment": Dec./Jan. (Actual) (1) 
"Location of Experiment": Pre19 July (2) 
"Location of Experiment": Post19 July (3) 

Fathers' Annual Hours  2  5  13 
Fathers' Annual Hours (standard error)  (21.8)  (20)  (28) 
Mothers' Annual Hours  40^{***}  24  22 
Mothers' Annual Hours (standard error)  (13.4)  (18.1)  (19.9) 
Fathers' Participation  .01  0  0.01 
Fathers' Participation (standard error)  (.005)  (.007)  (.006) 
Mothers' Participation  0  0.01  0 
Mothers' Participation (standard error)  (.007)  (.007)  (.006) 
Source: SIPP 19902001 and NBER TAXSIM. Sample consists of "children" under 40 living with their parent(s) for part or all of the calendar year.
Notes: Effect of ageeligibility rule calculated using thirdorder global polynomial. Outcomes are adjusted for covariates.
* denotes statistical significance at the 10% level, twotailed test.
** denotes statistical significance at the 5% level, twotailed test.
*** denotes statistical significance at the 1% level, twotailed test.




Among households with a child older than younger than 19 by an amount , the conditional expectation of the reduced form is:
The simplest comparison of comparing the labor supply behavior of all households with and without an ageeligible dependent fails to produce a consistent estimate of . This naive
estimate fails to consider that these types of households are different in both observable and unobservable dimensions. In general, for large values of ,
A similar argument applies for the firststage parameter, .
The extent to which this sample selection issue poses a problem for estimating the reduced form and structural estimates is best seen in a standard Heckman sample selection framework (Heckman, 1976). Consider the average labor supply response among barely eligible and
ineligible households:
A standard approach to this problem involves modelling the selection process and including a selectivity regressor that is a function of the probability of being observed. If is
distributed normally, the Inverse Mills ratio augments the estimation, and is identified if there is an additional (exogenous) variable affecting the probability of selection. In the
context of this paper's research design, however, the selection problem merely introduces another regressor, which is a continuous function of the probability of being observed in the sample (Das, Newey, and Vella, January 2003). As long as the ageeligibility rule
has no causal effect on the probability of being observed, then taking progressively smaller windows around the ageeligibility cutoff will also eliminate this additional term. To make this clear, consider the case where is normal so that we can use the standard selection framework and notation, and the probability of being observed can be written as a function of the child's age( is observed )). In this case, if the observable characteristics balance, is identified around the ageeligibility threshold: