A. The Mechanics of the Tax Credits

In the Taxpayer Relief Act of 1997, two important and permanent tax credits were enacted: the HTC and the TCLL. Both credits reduce the amount that a household owes in taxes, dollar for dollar, when the household spends a sufficient amount on qualifying postsecondary tuition and fees. A household does not need to itemize deductions to take these credits, but the HTC and TCLL are nonrefundable so that these credits can only reduce a household’s tax liability to zero. Thus, a household with no tax liability cannot benefit from the credits and a household with little tax liability may benefit only to a limited extent.

The HTC gives a household a credit equal to 100% of its first $1,200 and 50% of its second $1,200 of expenditure on each student’s qualifying tuition and fees.³Thus, the maximum amount of the credit is $1,800 per student, and this maximum can be reached only by households that spend at least $2,400 per student. For his tuition and fees to qualify, a student must be enrolled at least half time in the first two years of a postsecondary education that could lead to some degree or certificate. In 2008, the HTC phased out between $48,000 and $58,000 of modified adjusted gross income for single tax filers. The phase-out range was $96,000 to $116,000 for married joint filers. (Hereafter, we use “income” to denote modified adjusted gross income.)

Independent students (mainly students age 24 and over) receive the credit themselves. However, if a student is a dependent of another tax filer, as most full-time students under age 24 are, then the HTC goes to the filer—typically a parent. In theory, there is no limit on the number of dependent students a family could have who qualify for the HTC. In practice, it is most common for there to be only one credit per filer and it is rare for there to be more than two because each child is eligible only during her first two years of college.

It is important to note that the credit is for expenses paid in the relevant tax year. For instance, if a student entered college in the 2007/08 school year, his family would typically pay for the fall term in the summer of 2007 and pay for the spring term in December of 2007 or January of 2008. If the family paid for both fall and spring in calendar year 2007, their expenditures on the 2007/08 school year would generate a credit on the taxes due on April 15, 2008. However, if the family paid for spring in January 2008, they would only receive the credit with their 2008 tax filing, due in April 2009.

The household itself must spend the money for tuition and fees. If some or all of a student’s college expenses are paid by a tax-free scholarship, fellowship, grant, employer assistance, or veterans’ assistance, the qualifying tuition and fees are reduced commensurately.

The TCLL is a nonrefundable credit equal to 20% of a taxpayer’s first $10,000 of expenditure on tuition and fees.⁴Thus, the maximum credit is $2,000. Unlike the HTC, the credit is per taxpayer, not per student. The TCLL can be generated by the taxpayer’s expenditure on virtually any postsecondary coursework: undergraduate, graduate, or courses that improve job skills (even if they are not part of a degree or certificate program). The other major features of the TCLL—timing, the phase-out ranges—are the same as for the HTC.

Although both the HTC and TCLL are permanent tax credits, the HTC has been in abeyance since the enactment of AOTC in 2009 because the AOTC is more generous than the HTC on all dimensions and expenses cannot qualify for both. Thus, so long as the AOTC is in effect, the HTC is effectively suspended. The AOTC was passed as a temporary measure as part of the American Recovery and Reinvestment Act.⁵Due to expire in 2012, it was extended for an additional five years by the American Taxpayer Relief Act and will expire at the end of 2017 under current law.

The AOTC is equal to 100% of the first $2,000 plus 25% of the next $2,000 of a student’s qualifying tuition and fee expenditures. Thus, the maximum credit is $2,500 per student, but $4,000 per student must be spent to reach that maximum. Unlike the HTC, the AOTC can be claimed for all four of a student’s first four years of postsecondary education.

It is important for its distributional consequences that, unlike the HTC and TCLL, the AOTC is partially refundable. Specifically, a taxpayer receives a minimum of 40% of what he would receive, per student, had his taxes owed not been taken into account. Thus, a taxpayer who owes zero taxes receives a check for $1,000 per student if each student spends at least $4,000 on qualifying tuition and fees. Also important for its distributional consequences are the substantially higher income thresholds before the AOTC begins to phase out: the range is $80,000 to $90,000 for single filers and $160,000 to $180,000 for married joint filers. In other words, the AOTC makes eligible many taxpayers who were previously ineligible for higher education tax credits because their incomes were either too low or too high. Table 1 summarizes key parameters of the HTC, TCLL, and AOTC from 1999 through 2012.

Figures 1 through 6 show the tax credits available to a student by income and spending on qualified tuition and fees. In order to emphasize the change in the law between 2008 and 2009, these figures are for an individual who, if he were to attend college, would be a dependent of a married couple filing jointly. Figures 1 and 2 show the tax credits available to him if he were to spend $10,000, the tuition at which the TCLL is maximized. Figures 3 and 4 show the credits if he were to spend $4,000, the tuition at which the AOTC is maximized. Figures 5 and 6 show the credits if he were to spend $2,400, the tuition at which the HTC is maximized.

There are a few things to take away from these figures. First, for students with less than $9,000 in qualified tuition, the HTC is always preferable to the TCLL. Above $9,000, the TCLL is always preferable to the HTC. Thus, up through 2008, many students would take the HTC in their first two years and only take the TCLL in the third and high years. Second, the AOTC is more generous at every income and tuition level than the HTC or TCLL. Thus, from 2009 to today, all students in their first four years of postsecondary school should take the AOTC in preference to the TCLL. Third, each figure shows that the tax credits phase out rather sharply. The HTC and TCLL always phase out in the same income range, which is much lower than the AOTC’s phase-out range.

Fourth, on the left-hand side of each figure, there is what appears to be a phase-in range for the HTC and TCLL. This is not a true phase in but, rather, the empirical evolution of federal tax liability, which limits the HTC and TCLL because they are nonrefundable. That is, owing to exemptions, deductions, brackets, and certain noneducation credits (the credits for child care expenses and foreign taxes), households with fairly low incomes have negative, no, or small positive tax liability before the education tax credits are considered. Thus, the tax credits grow with income in the lower ranges, but not because of income per se. Rather, a whole host of tax provisions affect a household’s tax liability in the lower range and it is the combination of all these provisions that makes the education credits tend to grow with income as an empirical matter. Put another way, the apparent phase in would look different if we made different assumptions about households’ children under age 17, deductions, allocation of earnings, childcare expenses, and so on.

We make further use of figures 1 through 6 below when we exploit the changes in the tax law between 2008 and 2009 to analyze the causal effects of the tax credits.

B. The Fiscal Cost of the Tax Credits

Even prior to the enactment of the AOTC, the higher education tax credits were the single-largest education-related tax expenditure. They represented a very important component of federal support for students’ higher education expenses. Total tax expenditures on the tax credits were $25.1 billion in 2011.

Table 2 and figure 7 show how the fiscal cost of the tax credits grew from their inception. Actual credits claimed are shown for 1998 through 2012. Estimated tax expenditures are shown for 2013 through 2015. The data show the fiscal cost of the higher education tax credits grew gradually in real dollars until 2008 when they totaled $8.3 billion in 2014 dollars. With the advent of the AOTC, the tax expenditures on the credits more than doubled in a single year: the 2009 total was $20.3 billion. Much of the growth in expenditures was due to the introduction of the refundable component of the credit, which cost $8.4 billion in 2009 and grew to $12.1 billion in 2011, the last year for which we have nonpreliminary actual numbers available. However, the cost of the nonrefundable component of the tax credits also grew dramatically—by 42% between 2008 and 2009.

C. The Manner in Which the AOTC is Computed

Here, it is useful to note a curious feature of how the refundable credits are computed because it affects interpretation of table 2, figure 7, and all of our distributional calculations. One might think that the legislation would have individuals calculate the AOTC they could take as a nonrefundable credit and then take the partially refundable credit only if the latter were greater than the nonrefundable credit. This is not, however, what is done. Instead, everyone eligible for the AOTC takes the partially refundable credit first (40% of qualified tuition and fees after imposing the phase out). Only then is the remaining 60% of tuition and fees considered for the nonrefundable credit.⁶

The result is that many filers who do not need the AOTC to be refundable—that is, they have more than enough taxes owed—are reported as taking a refundable credit. To see why this matters, consider a filer with a single child for whom she spends $10,000 on tuition in 2008. If she has tax liability of at least $2,000, the filer can get a $2,000 TCLL. With the same circumstances in 2009, the filer can get a refundable credit of $1,000 and a nonrefundable credit of $1,500. Although her total tax credits have risen by $500 and the refundability of the AOTC is irrelevant to her, her nonrefundable credit will be reported as falling by $500.

In short, the enormous rise in refundable credits from 2009 onward and the smaller rise in nonrefundable credits does not mean that most of the tax expenditure on the AOTC has gone to low-income households. Most of the increased tax expenditure on the AOTC has gone to households that were already eligible for the HTC and TCLL or to households with incomes above the HTC/TCLL phase-out range. Later, we show the distributional consequences of the AOTC’s refundability where we define it as refundable only for those filers who would get a different credit were it nonrefundable.

D. The Intended Effects of the Tax Credits

It is always difficult to say what legislators intended a policy to do, but there is good documentation of the origins and debate surrounding the enactment of the HTC, TCLL, and AOTC. In particular, see Lederman (1997). The majority of the justifications for the tax credits suggest that they were intended to cause an increase in students’ investments in higher education.

For instance, in the Princeton University commencement address in which he initially proposed tax credits that would become the HTC and TCLL, President Clinton said:

Similarly, President Obama proposed a fully refundable higher education tax credit when campaigning late in 2007:

In contrast, Lederman (1997) indicates that, when pushing their passage, various policymakers argued that the higher education tax credits were simply a well-targeted middle-class tax cut. For instance, he quotes the then head of the National Economic Council, Gene Sperling, as saying, “This is a middle-class tax break, first and foremost.”

It would be hard to justify the higher education tax credits purely as a method of cutting the taxes of middle-income households because they require substantially more paperwork than would a reduction in the tax rates applied to middle incomes. However, we may speculate that the tax credits were intended to direct tax relief to middle-income households who invest in higher education rather than, say, spend money on consumption. Such intentions would be consistent with optimal tax logic which, as rule, suggests that income that is invested should be treated differently than income that is consumed. For instance, fundamental tax reform proposals often provide for income that is invested being pretax (not subject to tax) while income that is consumed is post-tax.⁹ Like the simpler ROI arguments, optimal tax-based arguments require that (a) the credits have causal effects on families’ spending on higher education, and (b) families’ college spending is actually an investment with positive expected returns. The latter requirement is something we evaluate in other studies, but that is beyond the scope of this study.

Alternatively, policymakers may have preferred the higher education tax credits to a simpler rate cut for middle-income households because the credits were less transparent. That is, policymakers may have been willing to accept more paperwork and a lack of causal effects in return for a tax cut that was politically more feasible because it appeared to be an education program, not a tax relief program.

A. Using Regression Kink Analysis at the Boundaries of the Phase Ranges to Identify the Causal Effects of the Tax Credits

In this section, we identify the causal effects of the tax credits by exploiting the fact that the relationship between the credits and income changes at each “edge” of the phase-out range. The estimates contained in this subsection are highly credible because it is very unlikely that any other factors that affect college-going also just happen to change at exactly the same income numbers. We are aware that the regression kink method has limitations. These are discussed below.

The logic of regression kink analysis is easy to see in figures. Suppose that there is an underlying relationship between a household’s income and its members’ propensity to attend college. This relationship could be the result of many factors: richer parents might find it easier to pay for tuition, but it might also be that they are more educated themselves and therefore make more effort to ensure that their children get a postsecondary education. The numerous factors that combine to generate the relationship do not matter since all that the regression kink method requires is that the combination of the other factors generate a relationship with income that changes smoothly, not sharply, at the exact incomes that form the edges of the phase-out range.

Figure 8 shows a stylized example. Panel (a) shows what the college attendance-income relationship might look like in the absence of the tax credits. That is, panel (a) shows the effects of many factors—parents’ ability to pay for tuition, parents’ education, secondary school quality— that affect college attendance and that are correlated with parents’ income. Panel (b) shows the statutory relationship between the maximum value of the tax credits and income. This statutory relationship is flat until income reaches the first edge of the phase-out range. At that first kink, the slope of the credit-income relationship becomes negative. At the end of the phase-out range, the relationship abruptly levels out, producing a second kink.

If the tax credits have a causal effect on college attendance then we will see kinks in the attendance-income relationship at exactly the income levels at which the phase out begins and ends. Moreover, the changes in the slope of the attendance-income relationship will mimic the changes in the slope of the tax credit-income relationship at those two points. This is shown in panel (c), which shows how the attendance-income relationship will appear if the credits have positive causal effects on attendance. If, instead, the credits have no causal effect, the attendance-income relationship will exhibit no change in slope at the edges of the phase-out range.

Of course, many students do not qualify for the maximum tax credit because they spend an insufficient amount on qualifying tuition and fees. We might expect (and, in fact, we will see) that throughout much of the eligibility range, higher-income students tend to qualify for higher credits because they spend more on tuition. If so, the credit-income relationship will look like that in figure 9, panel (a). It is upward sloping until it hits the first edge of the phase-out range. It is then downward sloping until it hits the second edge of the phase-out range. After that, the credit-income relationship is flat at zero credit. Once again, we have a first kink at which the slope changes abruptly in a negative direction and a second kink at which the slope changes abruptly in a positive direction. Once again, if tax credits have a causal effect, the attendance-income relationship will exhibit a negative change in slope at the first edge of the phase-out range and a positive change in slope at the second edge. (See figure 9, panel [b].)

The regression kink method requires that we estimate two equations. In a first stage, we estimate

From the above equations, the regression kink estimate of the causal effect of tax credits on attendance is:

We employ similar equations to estimate causal effects at U, the upper limit of each phase-out range:

So far, we have used stylized figures to illustrate how the regression kink method works. Now consider how data actually appear. Figures 10 and 11 show the empirical credit-income relationship in 2011 and in 2007 for students who are dependents in households that file jointly. In both years, credits rise smoothly with income until the first edge of the phase-out range. This sweeping curve reflects higher-income students’ tendency to pay more tuition. At the first edge, the relationship abruptly becomes very negative and nearly linear, indicating that the phase-out formula is dictating the relationship. At the second edge, the slope of the credit-income relationship abruptly changes again from negative to flat. Thus, the top panels of figures 10 and 11 look very much like the stylized example in panel (a) of figure 9.

Although we have not shown it for reasons of conciseness, the credit-income relationship looks very similar for every year from the introduction of the credits though 2012. The relationship also looks similar for single filers, as opposed to married couples filing jointly. There is always a sharp negative change in the slope of the credit-income relationship at the lower edge and a sharp positive change in the slope at the upper edge. Only the height of the credit and location of the phase-out range changes.

The bottom panels of figures 10 and 11 show close-ups of the kinks in the empirical credit-income relationship. These close-ups may seem unnecessary for displaying the kinks because they are very obvious. However, the regression kink method can depend on small changes in slopes at the edges of the phase-out range. Therefore, close-ups are useful for examining the empirical relationships between outcomes, like attendance and income.

We have seen that the credit-income relationship exhibits sharp kinks just as in the stylized example. Do outcome-income relationships similarly match what we would expect if the credits have causal effects? Figures 12 through 17 show that the answer is no. The students in figures 12 and 13 correspond to those on whom figures 10 and 11 are based, yet we see no change whatsoever in the slope of their college attendance-income relationship at the edges of the phase-out range. (The close-ups are helpful here.) Perhaps, however, students are not changing whether they attend college but are changing where they attend college. The tax credit might, for instance, allow them to attend a four-year college rather than a two-year college. It might allow them to attend a college with higher instructional resources. We look for evidence of such “upgrading” in figures 14, 15, and 16, but we see none. At the edges of the 2011 phase-out range, we do not perceive changes in the slope of the four-year college attendance-income relationship, the two-year college attendance-income relationship, or the instructional resources-income relationship. Perhaps the tax credit changes the grants and scholarships a student receives rather than his attendance? Figure 17 suggests not. It shows no perceptible changes in the slope of the grants-income relationship at the edges of the phase-out range.¹⁵

Of course, we should not rely exclusively on our visual ability to discern changes in slopes at the edges of the phase-out range. Therefore, we show the results of formal regression kink analysis in tables 13 and 14.

Table 13 shows estimates of the first-stage equations—that is, estimates of β₁ from equations (1) and (4) above. Each equation is estimated for 19- and 20-year-olds in the year mentioned using a bandwidth of plus or minus $3,000 around the location of the kink mentioned. The results shown are for a cubic polynomial following the guidance given by Ganong and Jäger (2014). Although we do not show alternative specifications, the first-stage estimates are, in fact, highly robust to our (a) changing the degree of the polynomial, (b) expanding the bandwidth around each kink as far as plus or minus $10,000, (c) focusing on older students, and (d) estimating the equations on years other than 2007 and 2011.¹⁶

Each estimated coefficient in table 13 should be interpreted as the change in the slope of the credit-income relationship that occurs at the kink mentioned. Thus, for instance, –0.107 in the first row tells us that the slope of the credit-income relationship decreases by about 11 cents per dollar at the lower edge of the phase-out range for joint filers in 2011. This makes a great deal of sense and corresponds closely to what we saw in the figure 10 close-up. The credit-income relationship is almost flat at a credit of about $2,200 as it approaches the lower edge. The relationship then falls fairly linearly so that the entire $2,200 goes to zero smoothly over a $20,000 income interval. This means that, just from the figure, we expect the slope to fall from approximately zero to approximately –0.11. This is exactly what it does. All of the estimated coefficients in table 13 can be similarly interpreted and line up similarly with what we expect based on the credit formula.¹⁷ For context, we have included a column showing the coefficient we would expect if credit-income relationship were flat at the maximum possible credit when approaching the lower edge of the phase-out range. Of course, students are not generally at the maximum credit and the relationship need not be flat approaching the phase-out range. Nevertheless, these expected coefficients are a reminder of the sign we expect and the maximum magnitude consistent with the formula.

We use selected data from the population of individuals to produce the estimates in table 13. Therefore, standard errors and p-values should not be interpreted using a conventional sampling framework. Moreover, the regression kink method employs a specification that imposes very little economic modeling, so it is not easy to interpret the standard errors as misspecification of the model as proposed by Abadie et al. (2013).

Robust-to-heteroskedasticity standard errors, which have been employed in numerous studies with regression kink analysis have been shown to be far too small. This is because regression kink estimates can be highly sensitive to curvature in the underlying relationship between the outcome and the assignment variable—income, in our case. Therefore, Ganong and Jäger (2014) propose a permutation test based on estimating the regression kink equations at placebo kinks where no kink in the relationship should exist.¹⁸ We do this using as placebos 100 randomly selected incomes that are 3,000 to 7,000 dollars plus or minus each actual kink.¹⁹ Armed with 100 placebo estimates for each kink, we examine whether our estimate based on the actual kink falls outside the 95% confidence interval generated by the estimates from the placebos. If the estimate based on the actual kink passes this test, we show a check mark below it in table 13.

All of the first stage estimates very easily pass the Ganong and Jäger test. Thus, we are confident that the phase-out range gives us a strong environment for testing the causal effects of the tax credits.

Table 14 shows estimates of the second stage equations—that is, estimates of γ₁ from equations (2) and (5) above. Again, each equation is estimated for 19- and 20-year-olds in the year mentioned using a bandwidth of plus or minus $3,000 and a cubic polynomial. Each estimated coefficient in table 14 should be interpreted as the change in the slope of the outcome-income relationship that occurs at the kink mentioned. Thus, for instance, –0.000007 in the upper-left-hand cell should be interpreted as saying that the slope of the attending-income relationship decreases by about 0.0007% per dollar at the lower edge of the phase-out range for joint filers in 2011.

However, there is little point in interpreting the estimated coefficients in table 14 because none of them passes the Ganong and Jäger permutation test. That is, there are no discernable causal effects of the tax credits on any of the following outcomes: attending postsecondary school at all, attending at least half time, attending a four-year college, attending a two-year college, instructional resources, core educational resources, the “list” tuition and fees of the student’s college, the grant and scholarships the student receives, and the tuition the student pays. (The last eight outcomes are all conditional on attending at all.)

We cannot rule out effects of the tax credits that are too small to be discernable. For instance, if a $1,000 tax credit were to increase attending by 0.1%, we would be unable to distinguish this effect from random noise. Generally, though, we interpret the estimates in table 14 as strong confirmation of what the figures suggest: the tax credits have no or at most extremely small causal effects on college-related outcomes in the vicinity of the phase-out ranges.

The advantage of regression kink analysis is that, so long as the assumptions of the method are met, it produces estimates that are very credibly causal. The disadvantage is that the estimates are relevant only for households with incomes near the phase-out regions. Owing to the changes in the phase-out ranges, especially the major rise in the ranges that accompanied the AOTC, we have estimates for married joint filers that cover a rather wide array of incomes: from about $107,000 to $180,000 in 2013 dollars. For single filers, we have regression kink estimates than span incomes from about $53,000 to about $90,000 in 2013 dollars. We find no causal effects of the tax credits over these fairly wide regions.

However, the phase-out range never occurs in the lower to middling percentiles of the income distribution and, therefore, we cannot extrapolate from the regression kink estimates to say that removing or reducing the tax credits would have no or only a slight effect on college attendance by students whose families have earnings at, say, the 50th percentile for their filing status. There is no reason to think that the effects on them will be as modest as they are around the phase-out range. Indeed, a policymaker who is trying to minimize the causal effects of withdrawing a tax credit might choose to phase it out among precisely those households who are likely to be unaffected. This does not imply that all households would be similarly unaffected.

B. Using the Introduction of the AOTC to Identify the Causal Effects of Tax Credits across All Eligible Incomes

Because we cannot extrapolate our regression kink estimates to lowand middle-income households, we turn to the introduction of the AOTC which, as we have seen, sharply increased the generosity of the education tax credits for some households. In this section, we employ the classic method of analyzing how the change in college-going behavior, from cohort to cohort, is related to the change in the tax credits they experience. Because the introduction of the AOTC affected students across a very wide range of incomes, including poor students, the estimates from this method have broad application.

This method can be embodied in the simple equation:

Because the tax credits that the household actually takes may reflect its response to the tax credits (for instance, receiving a larger credit because the credit causes one to attend a more expensive school) and not just the policy-driven change in the credits, we construct a simulated instrument for each tax credit. A properly constructed simulated instrument embodies the policy-driven change in the credit, holding the household’s behavior constant. Thus, our simulated instrument is the credit the household would receive if its college-going choices were typical for a household of its income in a base year (which we set to be 2008). These typical choices are run through the laws of the actual year so that the instrument reflects the changes in the tax credit parameters and only the tax credit parameters. An especially simple example would be the following. Suppose that households with $50,000 of income and one child aged 18 have a 50% probability of sending a child to college in such a way as to fulfill the requirements of the HTC (and therefore the AOTC): at least half-time enrollment, enrollment in a degree or certificate program, and so on. Suppose that, conditional on its child enrolling, such a household typically spends $4,000 on qualifying tuition and fees. Then, the simulated instrument for the 2008 credit would be 50% × $1,800, and the simulated instrument for the 2009 credit would be 50% × $2,500.

Of course, computing the simulated instruments is more complicated, but the essential elements are clear. For each level of income, we need the tax credit qualifying variables: spending on tuition and fees, halftime enrollment, taxes owed before the education credits, and so on.²⁰

With the simulated instruments, we have a simple first stage equation:

When estimating equations (6) and (7), it is important to cluster the standard errors to account for the fact that the tax-credit policy is not individual specific. For instance, consider two households, each of which spends $4,000 on tuition in 2008. If both households have incomes below $96,000 and owe at least $2,500 in taxes before credits, these households will experience the same change in credits due to the introduction of the AOTC even if they have quite different incomes—$60,000 and $95,000, say. To generate the clusters, we run each household through the tax credit formula in 2008 and 2009. We put households in the same cluster if they receive identical tax credits when they make the same college attendance and payment choices.

The weakness of the simulated instrumental variables method, which is often used to assess tax reforms, is that its estimates may reflect other events that occur at the same time as the change in the tax credits and that affect the same people who experience a change in tax credits. Given that the AOTC was introduced as part of the stimulus package, one might worry that certain households were either affected by the events that triggered the stimulus (the financial crisis, the rise in unemployment) or by other components of the stimulus bill. For instance, college enrollment has often been found to be anticyclical owing to the decrease in opportunity costs (that is, labor earnings) during downturns. So, we might expect enrollment to rise in 2009 regardless of the credits, or families who lost home equity in 2008 may have found it harder to finance college.

Despite these circumstances, we are fairly confident that our estimates based on the introduction of the AOTC bear a causal interpretation. Our confidence is based on three things. First, the AOTC increased tax credits not just for low-income or high-income households but for both types of households. Indeed, as we show below, the changes in the tax credits are not a simple function of household income. Some “control” (unaffected or hardly affected) households have high incomes, and some have relatively low incomes. Some of the most affected households have fairly high incomes, and some have low incomes. Our second reason for confidence is that trends in college-going behavior were parallel before and after the law change, not just for students as a whole, but when we define groups who were more and less affected by the change in tax credits. Our third reason for confidence is that this method produces results that closely match the regression kink results for households in the income ranges covered by those results.

Figures 18, 19, and 20 demonstrate our point that the changes induced by the AOTC were not a simple function of income. To construct these figures, we subtract each of the 2008 figures from Section II from its 2009 counterpart. This shows us the change in the tax credit available to a household with a given amount of spending on tuition, for each level of income. For instance, figure 19 shows the change in the tax credit available to a household that files jointly, that has no dependent other than the prospective student, and that would spend $4,000 on qualified tuition and fees if its child were to attend college. In zone (a) of the figure, we see that households with incomes up to about $35,000 ($30,000 for third and fourth year students) experience a large increase of about $1,000 in tax credits owing to the AOTC’s refundability. In zone (b), households with incomes from about $40,000 to $96,000 experience a modest increase of $500 if their student is in his first or second year. They experience a very large increase of $1,700 if their student is in his third or fourth year. In zone (c), households with incomes from $116,000 to about $160,000 experience a tremendous increase of $2,500 in their tax credit regardless of the year of their student. This is because they are above the phase-out range for the TCLL and HTC, but not for the AOTC. Finally, households in zone (d) with incomes of $180,000 and above experience zero change in the tax credits they could take.

Figures 18 and 20 show similar change in tax credit graphs for households who spend, respectively, $10,000 and $2,400 on tuition on fees. One can see that the most and least affected households are somewhat different than in figure 19. Indeed, there is variation in treatment not only as we scan the figures horizontally (across incomes), but also as we scan them vertically—across the second versus third year of college. Below, we refer to zones (a) through (d) in figures 18 through 20 so we encourage readers to fix them in their minds.

In econometric terms, it is helpful to have all this variation in the location of the most affected and least affected households because it means that the treatment and control groups have fairly common support, in terms of the income distribution. Put another way, we can compare high-income households who are greatly treated because they are just inside the top end of the new phase-out range with other high-income households who experience no treatment because they are just outside the top end of the phase-out range. We need not compare high-income households only to low-income households. Similarly, we can compare upper-middle-income households who are greatly treated because they are just outside of the upper limit of the old phase-out range with households who are only modestly treated because they are just inside the upper limit of the old phase-out range, and so on.

In addition, we gain a great deal of econometric common support from the “vertical” differences due to the AOTC (but not the HTC) being available to third and fourth year students.

Although the variation in tax credits between 2008 and 2009 is too complex to be reasonably described as differences-in-differences, the logic of our empirical strategy is essentially that of differences-in-differences. That is, we are simultaneously exploiting differences across cohorts and differences within cohorts in the tax credits that apply to them. An important test of such a identification strategy is whether the groups who are differentially treated by the law change are on parallel trends prior to the law change. If they are on trends that are fairly parallel, the method is usually reliable. If they are on diverging trends, the estimates may spuriously reflect the divergence in their preexisting trends.

In figures 21 through 25, we show trends in college attendance and other outcomes by income group, where each group is set up to be as close as possible to the edge of one of the zones mentioned above. Thus, in each figure, the panel for 19-year-olds has a group from the top of zone (a) ($25,000–35,000, experienced a moderate increase in credits) whom one would naturally compare to the bottom of zone (b) ($40,000–50,000, experienced only a slight increase in credits). It also has a group from the top of zone (b) ($86,000–96,000, experienced only a slight change) whom one would naturally compare to the bottom of zone (c) ($116,000–126,000, experienced an extremely large increase). It finally contains a group from the top of zone (c) ($150,000–160,000, experienced an extremely large increase) whom one would naturally compare to the bottom of zone (d) ($180,000–190,000, entirely unaffected by the introduction of the AOTC).

In all of figures 21 through 25, the pairs whom we need to be on parallel trends are on parallel trends as we approach 2008, both for 19-year-olds and 23-year-olds. All income groups are slightly affected by the business cycle, but the effects tend to be parallel. Another way to see the parallel trends is to group people by the change in tax credits that they experience if they behave as they do in the base year. That is, the next figure is set up to show the variation that will, through the simulated instruments, drive the estimates. For instance, for 19-year-olds, the groups are 0 change, a $1 to $500 increase in the simulated tax credit, a $501 to $750 increase, a $751 to $900 increase, and a $901 and greater increase. Figure 26 shows the trends for attendance and other outcomes for 19-year-olds when individuals are grouped that way. The figure demonstrates that people who were more and less affected by the tax credit changes were not previously on divergent attendance trends prior to the introduction of the AOTC. (We can show similar figures for other outcomes and people of other ages.)

Indeed, figures 21 through 26 not only show parallel trends prior to the AOTC. They show parallel trends right through the introduction of the AOTC. That is, the figures provide little evidence that people who experienced much larger increases in tax credit generosity had college outcomes that were any different as a result.

In tables 15 and 16 we formalize this finding by showing estimates of equations (6) and (7). Each regression contains a quintic polynomial in adjusted gross income as well as a full set of year effects.²¹To maximize statistical power, we use the three years before and three years after the introduction of the AOTC: 2006 through 2011. However, the results are very similar if we use only 2008 and 2009. We show results for 19-year-olds and 23-year-olds. The results for individuals of age 20 through 22 and age 24 through 26 are similar. As we have seen, older people are much less affected by the AOTC, so it is not worthwhile analyzing them.

Table 15 shows the first stage regressions in which the actual tax credits a person takes are regressed on her simulated instrument. There are multiple columns only because the individuals over whom the first stage is estimated varies slightly with the outcomes under study. Most of the coefficients for 19-year-olds are around 0.38. This suggests that, for every extra dollar that a student would receive as a result of the AOTC’s enactment if he were entirely typical of his income group, a student actually receives about 38 cents. This makes sense because the simulated instrument deliberately excludes variation in actual tax credits that is potentially endogenous. (That is, the coefficient on the simulated instrument is expected to be between 0 and 1, but well below 1 unless all individuals act in the same way as others with the same income.) The crucial thing for the method is not, however, the coefficient but the statistical power of the instrument. This it has in abundance. The t-statistic for the coefficient on the simulated instrument is 1,190 to 1,512 for regressions based on 19-year-olds. It is 533 to 669 for 23-year-olds. In other words, the simulated instrument is such a strong predictor of actual tax credits that we can estimate the effect of the tax credits with precision.

Table 16 demonstrates that enacting the AOTC did not affect any college-related outcome we study to a degree that is statistically significantly different from zero. All of the t-statistics are far from conventional levels of statistical significance such as 1.96. Moreover, about half of the coefficient estimates that are not statistically significantly different from zero have an unexpected sign. (For instance, the point estimates suggest that the tax credits make a student less likely to attend a four-year college or a college with generous instructional resources.) Thus, there is no pattern that might tempt us to interpret the statistically insignificant point estimates.

We interpret the evidence in table 16 as confirmation of figures 21 through 26. They all suggest that the introduction of the AOTC generated very little change in college attendance or other college-related outcomes.

Using the same simulated instruments method, we have generated separate estimates for relatively low-income households (from $10,000 below the upper edge of zone [a] to $10,000 above the lower edge of zone [b]), upper-middle-income households (from $10,000 below the upper edge of zone [b] to $10,000 above the lower edge of zone [c]), and upper-income households (from $10,000 below the upper edge of zone [c] to $10,000 above the lower edge of zone [d]). Purely for reference (since we include all households in the analysis), these income ranges are, for 19-year-olds in joint-filing households, $25,000 to $50,000, $86,000 to $126,000, and $150,000 to $190,000.

In these regressions that allow effects to vary with income, we find results that are fully consistent with those reported in tables 15 and 16. That is, the first stage of the method is very strong: the introduction of the AOTC greatly increased tax credits taken. The second stage estimates consistently suggest that the tax credits have no causal effect on college-related outcomes. Part of the reason we do not present these results in tables is that they are already fairly obvious in figures 21 through 26. The reader will observe that the two lowest income groups in those figures tend to track one another through the enactment of the AOTC. Similarly, the two middle groups track one another and the two highest groups track one another.

Finally, it is noteworthy that the regression kink method and simulated instruments method confirm one another’s results for households in the vicinity of the phase-out ranges. That is, both methods suggest no or very small causal effects of the tax credits on college outcomes.