DEFINICIONES Y ABREVIATURAS

Sometimes program status is turned on when an individual reaches a certain age. Receipt of pension benefits is typically tied to reaching a particular age (see Edmonds (2004); Edmonds et al. (2005)), and in the United States eligibility for the Medicare program begins at age 65 (see Card et al. (2009a)), and young adults reach the legal drinking age at 21 (see Carpenter and Dobkin (2009)). Similarly, one is subject to the less punitive juvenile justice system until the age of majority (typically, eighteen) (see Lee and McCrary (2005)).

These cases stand apart from the typical RD designs discussed above because here assignment to treat- ment is essentially inevitable, as all subjects will eventually age into the program (or, conversely, age out of the program). One cannot, therefore, draw any parallels with a randomized experiment, which necessarily involves some ex ante uncertainty about whether a unit ultimately receives treatment (or the intent to treat). Another important difference is that the tests of smoothness in baseline characteristics will generally be uninformative. Indeed, if one follows a single cohort over time, all characteristics determined prior to reaching the relevant age threshold are by construction identical just before and after the cutoff.48 Note that in this case, time is the assignment variable, and therefore cannot be manipulated.

This design and the standard RD share the necessity of interpreting the discontinuity as the combined effect of all factors that switch on at the threshold. In the example of Thistlethwaite and Campbell (1960), if passing a scholarship exam provides the symbolic honor of passing the exam as well as a monetary award, the true treatment is a package of the two components, and one cannot attribute any effect to only one of the two. Similarly, when considering an age-activated treatment, one must consider the possibility that the age of interest is causing eligibility for potentially many other programs, which could affect the outcome.

48_{There are exceptions to this. There could be attrition over time, so that in principle, the number of observations could dis-}

continuously drop at the threshold, changing the composition of the remaining observations. Alternatively, when examining a cross-section of different birth cohorts at a given point in time, it is possible to have sharp changes in the characteristics of individ- uals with respect to birthdate.

There are at least two new issues that are irrelevant for the standard RD, but are important for the analysis of age discontinuities. First, even if there is truly an effect on the outcome, if the effect is not immediate, it generally will not generate a discontinuity in the outcome. For example, suppose the receipt of Social Security benefits has no immediate impact, but does have a long-run impact on labor force participation. Examining the labor force behavior as a function of age will not yield a discontinuity at age 67 (the full retirement age for those born after 1960), even though there may be a long-run effect. It is infeasible to estimate long-run effects because by the time we examine outcomes five years after receiving the treatment, for example, those individuals who were initially just below and just above age 67 will be exposed to essentially the same length of time of treatment (e.g. five years).49

The second important issue is that because treatment is inevitable with the passage of time, individu- als may fully anticipate the change in the regime, and therefore they may behave in certain ways prior to the time when treatment is turned on. Optimizing behavior in anticipation of a sharp regime change may either accentuate or mute observed effects. For example, simple life-cycle theories, assuming no liquidity constraints, suggest that the path of consumption will exhibit no discontinuity at age 67, when Social Se- curity benefits commence payment. On the other hand, some medical procedures are too expensive for an under-65-year-old, but would be covered under Medicare upon turning 65. In this case, individuals’ greater awareness of such a predicament will tend to increase the size of the discontinuity in utilization of medical procedures with respect to age (e.g. see Card et al. (2009a)).

At this time we are unable to provide any more specific guidelines for analyzing these age/time disconti- nuities, since it seems that how one models expectations, information, and behavior in anticipation of sharp changes in regimes will be highly context-dependent. But it does seem important to recognize these designs as being distinct from the standard RD design.

We conclude by emphasizing that when distinguishing between age-triggered treatments and a standard RD design, the involvement of age as an assignment variable is not as important as whether the receipt of treatment – or analogously, entering the control state – is inevitable. For example, on the surface, the analysis of the Medicaid expansions in Card and Shore-Sheppard (2004) appears to be an age-based discontinuity, since effective July 1991, U.S. law requires states to cover children born after September 30, 1983, implying a discontinuous relationship between coverage and age, where the discontinuity in July 1991 was around

49_{By contrast, there is no such limitation with the standard RD design. One can examine outcomes defined at an arbitrarily long}

8 years of age. This design, however, actually fits quite easily into the standard RD framework we have discussed throughout this paper.

First, note that treatment receipt is not inevitable for those individuals born near the September 30, 1983 threshold. Those born strictly after that date were covered from July 1991 until their 18th birthday, while those born on or before the date received no such coverage. Second, the data generating process does follow the structure discussed above. Parents do have some influence regarding when their children are born, but with only imprecise control over the exact date (and at any rate, it seems implausible that parents would have anticipated that such a Medicaid expansion would have occurred 8 years in the future, with the particular birthdate cutoff chosen). Thus the treatment is assigned based on the assignment variable, which is the birthdate in this context.

Examples of other age-based discontinuities where neither the treatment nor control state is guaranteed with the passage of time that can also be viewed within the standard RD framework include studies by Cascio and Lewis (2006), McCrary and Royer (2003), Dobkin and Ferreira (2009), and Oreopoulos (2006).