shen: Are you willing to die to find the truth?
PO: You bet I am! … Although, I’d prefer not to.
Rung Fu Panda 2
With the repeal of federal alcohol Prohibition in 1933, U. S. states were free to regulate alcohol. Most instituted an MLDA of 21, but Kansas, New York, and North Carolina, among others, allowed drinking at 18. Following the twenty-sixth amendment to the constitution in 1971, which lowered the voting age to 18 in response to agitation sparked by the Vietnam War, many states reduced the MLDA. But not all: Arkansas, California, and Pennsylvania are among the states that held the line at 21. In 1984, the National Minimum Drinking Age Act punished youthful intemperance by withholding federal aid for highway construction from states with an age-18 MLDA. By 1988, all 50 states and the District of Columbia had opted for an MLDA of 21, though some had taken the federal highway hint more quickly than others.
As with much American policymaking, the interaction of federal and state law produces a colorful and oft-changing quilt of legal standards. This policy variation is a boon to masters of ’metrics: variation in state MLDA laws is easily exploited in a DD framework. In efforts to uncover effects of alcohol policy, this framework provides an alternative to the RD approach detailed in Chapter 4.-
Alabama lowered its MLDA to 19 in 1975, but alphabetically and geographically proximate Arkansas has had an MLDA of 21 since Prohibition’s repeal. Did Alabama’s indulgence of its youthful drinkers cost some of them their lives? We tackle this question by fitting a regression DD model to data on the death rates of 18-20-year-olds from 1970 to 1983. The dependent variable is denoted Yst, for death rates in state s and year t. With a sample including only Alabama and Arkansas, the regression DD model for Yst takes the form
Ysl = a + p TREATS + у POST,
+ 6г00(ТИЕАТ, x POST,) + eit, (5.4)
where TREATS is a dummy variable indicating Alabama, POSTt is a dummy indicating years from 1975 onward, and the interaction term TREATS x POSTt indicates Alabama observations from low-drinking-age years. The coefficient SrDD captures the effect of an age-19 MLDA on death rates.
Equation (5.4) parallels the regression DD model for Mississippi’s two Federal Reserve Districts. But why look only at Alabama and Arkansas? There’s more than one MLDA experiment in the legislative record. For example, Tennessee’s MFDA fell to 18 in 1971, then rose to 19 in 1979. A complicating but manageable consequence of differences in the timing of MFDA reductions in Alabama and Tennessee is the absence of a common posttreatment period. When combining multiple MFDA experiments in a DD framework, we swap the single POSTt dummy for a set of dummies indicating each year in the sample, with one omitted as a reference group. The coefficients on these dummies, known as time effects, capture temporal changes in death rates that are common to all states.-
Our multi-MFDA regression DD procedure should also reflect the fact that there are many states driving causal comparisons. Instead of controlling only for the difference between, say, the Sixth and Eighth Federal Reserve Districts as in the Mississippi experiment of Section 5.1. or the difference between Alabama and Arkansas in the example above, the multistate setup controls for the differing death rates in each of many states. This is accomplished by introducing state effects, a set of dummies for every state in the sample, except for one, which is omitted as a reference group. A regression DD analysis of data from Alabama, Arkansas, and Tennessee, for example, includes two state effects. State effects replace the single TREATs dummy included in a two-state (or two – group) analysis.
A final complication in this scenario is the absence of a common treatment variable that discretely switches off and on. The MFDA runs from age 18 to age 21, generating treatment effects for legal drinking at ages 18, 19, or 20. Masters of ’metrics simplify such things by reducing them to a single measure of exposure to the policy of interest, in this case, access to alcohol. Our simplification strategy replaces TREATd x POSTt with a
variable we’ll call LEGALst. This variable measures the proportion of 18-20-year-olds allowed to drink in state s and year t. In some states, no one under 21 is allowed to drink, while in states with an age-19 MFDA, roughly two-thirds of 18-20-year-olds can drink,
and in states with an age-18 MLDA, all 18-20-year-olds can drink. Our definition of LEGALst also captures variation due to within-year timing. For example, Alabama’s age – 19 MLDA came into effect in July 1975. LEGAL ^}1975 is therefore scaled to reflect the fact that Alabama’s 19-20-year-olds were free to drink for only half that year.
The multistate regression DD model looks like
Yt[ — a + SrDDLEGALIt
+ £ PtSTATEts + £ (5.5)
Don’t let the big sums in this equation scare you. This notation describes models with many dummy variables compactly, just as in the models with college selectivity group dummies in Chapter 2. Here every state but one (the reference state) gets its own dummy variable, indexed by the subscript к for state k. The index s keeps track of the state supplying the observations. The kth state dummy, STALEks equals one when an observation is from state k, meaning s = k, and is zero otherwise. Observations from California, for example, have STATEca s switched on, and all other state dummies switched off.
The state effects, (3k, are the coefficients on the state dummies. For example, the California state effect, j6CA is the coefficient on STATECAs. Every state except the reference state, the one omitted when constructing state dummies, has a state effect in equation Г5.5У Because there are so many of these, we use summation notation,
coefficients on the year dummies, YEARjt. These switch on when observations in the data come from year j, that is, when t = j. We therefore also call them year effects. The 1975 year effect, Yi975> is the coefficient on YEAR1975> t. Here, too, every year in the sample except the reference year has a year effect, so we use summation notation to write these out compactly. –
Our multistate MLDA analysis uses a data set with 14 years and 51 states (including the District of Columbia), for a total of 714 observations. This data structure is called a state- year panel. The state effects in equation (Б.51 control for fixed differences between states (for example, fatal car accidents are more frequent, on average, in rural states with high average travel speeds). The time (year) effects in this equation control for trends in death rates that are common to all states (due, for example, to national trends in drinking or vehicle safety). Equation Г5.5І attributes changes in mortality within states to changes in LEGALst. As we’ll see shortly, this causal attribution turns on a common trends assumption, just as in our analysis of Caldwell-induced bank failures in the previous section.
Estimates of 6rDD in equation 15.51 suggest that legal alcohol access caused about 11 additional deaths per 100,000 18-20-year-olds, of which seven or eight deaths were the result of motor vehicle accidents. These results, reported in the first column of Table 5.2. are somewhat larger than but still broadly consistent with the RD estimates reported in
Table 4.1 in Chapter 4. The MVA estimates in Table 5.2 are also reasonably precise, with standard errors of about 2.5. Importantly, as with the RD estimates, this regression DD model generates little evidence of an effect of legal drinking on deaths from internal causes. The regression DD evidence for an effect on suicide is weaker than the corresponding RD evidence in Table 4.1. At the same time, both strategies suggest any increase in numbers of suicides is smaller than for MVA deaths.
Regression DD estimates of MLDA effects on death rates
Notes: This table reports regression DD estimates of minimum legal drinking age (MLDA) effects on the death rates (per 100,000) of 18-20-year-olds. The table shows coefficients on the proportion of legal drinkers by state and year from models controlling for state and year effects. The models used to construct the estimates in columns (2) and (4) include state-specific linear time trends. Columns (3) and (4) show weighted least squares estimates, weighting by state population. The sample size is 714. Standard errors are reported in parentheses.
Samples that include many states and years allow us to relax the common trends assumption, that is, to introduce a degree of nonparallel evolution in outcomes between states in the absence of a treatment effect. A regression DD model with controls for state – specific trends looks like
YS! = a 4- Sr[)[)LEGALst
+ e E vA£ARj>
ik= Alaska j=1971
+ E & * 0 + ?sf (5$)
it = Alaska
This model presumes that in the absence of a treatment effect, death rates in state к deviate from common year effects by following the linear trend captured by the coefficient 9k.
Heretofore and hitherto we’ve been sayin’ that DD is all about common trends. How can it be, then, that we’re now entertaining models like equation Г5.6У which relax the key common trends assumption? To see how such models work, consider a sample of two states: The first, Allatsea, reduced the MLDA to 18 in 1975, while neighboring Alabaster held the line at 21. As a baseline, Figure 5.4 sketches the common trends story. Deaths per 100,000 move in parallel until 1975 (most things got worse in the 1970s, so we show death rates increasing). Death rates also jump above trend in Allatsea in 1975, when that state lowered its MLDA. Given the parallelism and the timing, it seems fair to blame Allatsea’s lower MLDA for this jump.
Figure 5.5 sketches a scenario with a steeper trend in Allatsea than in Alabaster. As with the data plotted in the previous figure, simple regression DD estimation in this case generates estimates implicating the MLDA (the post-minus-pre contrast in Allatsea is larger than the post-minus-pre contrast in Alabaster). In this case, however, the resulting DD estimate is spurious: the difference in state trends predates Allatsea’s MLDA liberalization and must therefore be unrelated to it.
Luckily, such differences in trend can be captured by the state-specific trend parameters, вк, in equation 15.61. In models that control for state-specific trends, evidence for MLDA
effects comes from sharp deviations from otherwise smooth trends, even where the trends are not common. Figure 5.6 shows how regression DD captures treatment effects in the face of uncommon trends. Death rates in Allatsea increase more steeply than in Alabaster throughout the sample period. But the Allatsea increase is especially steep from 1974 to 1975, when Allatsea lowered its MLDA. The coefficient on LEGALst in equation (5.6) picks this up, while the model allows for the fact that death rates in different states were on different trajectories from the get-go.
An MLDA effect in states with parallel trends
A spurious MLDA effect in states where trends are not parallel
Models with state-specific linear trends provide an important check on the causal interpretation of any set of regression DD estimates using multiperiod data. In practice, however, empirical reality may be considerably mushier and harder to interpret than the stylized examples laid out in Figures 5.4-5.6. The findings generated by a regression model like equation (5.6) are often imprecise. The sharper the deviation from trend induced by a causal effect, the more likely we are to be able to uncover it. On the other hand, if treatment effects emerge only gradually, estimates of equations like Г5.6І may fail to distinguish treatment effects from differential trends, with the end result being an imprecise and therefore inconclusive set of findings.
Happily for a coherent causal DD analysis of MLDA effects, introduction of state – specific trends has little effect on our regression DD estimates. This can be seen in column (2) of Table 5.2. which reports regression DD estimates of MLDA effects from the model described by equation Г5.6У The addition of trends increases standard errors a little, but the loss of precision here is modest. The findings in column (2) support a causal interpretation of the more precise MLDA effects reported in column (1) of the table.
State policymaking is a messy business, with frequent changes on many fronts. DD estimates of MLDA effects, with or without state-specific trends, may be biased by contemporaneous policy changes in other areas. An important consideration in research on
alcohol, for example, is the price of a drink. Taxes are the most powerful tool the government uses to affect the price of your favorite beverage. Many states levy a heavy tax on beer, which we measure in dollars per gallon of alcohol content. Beer taxes range from just pennies per gallon to more than a dollar per gallon in some Southern states. Beer taxes change from time to time, mostly increasing, much to the dismay of the Beer Institute (with a tax rate of 2 cents per gallon since 1935, Wyoming is beer bliss). It stands to reason that states might raise tax rates at the same time that they increase their ML DA, perhaps as a part of a broader effort to reduce drinking. If so, we should control for time – varying state tax rates when estimating ML DA effects.
Regression DD models that include controls for state beer taxes generate MLDA estimates similar to those without such controls. This can be seen in Table 5.3. which reports both the estimated coefficients on LEGALst and the estimated coefficients on state beer taxes in models for the four death rates examined in Table 5.2. Columns (1) and (2) of Table 5.3 show beer tax and MLDA effects estimated using a single regression without controls for state-specific trends, while those in columns (3) and (4) come from another regression including controls for state-specific trends. Beer tax effects are estimated less precisely than MLDA effects, most likely because beer taxes change less often than the MLDA. The beer tax estimates from models that include state trends are especially noisy. Still, the Beer Institute will be pleased to learn that these results don’t speak in favor of further beer tax increases. We’re likewise pleased to know that our MLDA estimates are robust to the inclusion of a beer tax control; we’ll share a beer to celebrate!
Regression DD estimates of MLDA effects controlling for beer taxes
Notes: This table reports regression DD estimates of minimum legal drinking age (MLDA) effects on the death rates (per 100,000) of 18-20-year-olds, controlling for state beer taxes. The table shows coefficients on the proportion of legal drinkers by state and year and the beer tax by state and year, from models controlling for state and year effects. The fraction legal and beer tax variables are included in a single regression model, estimated without trends to produce the estimates in columns (1) and (2) and estimated with state-specific linear trends to produce the estimates in columns (3) and (4). The sample size is 700. Standard errors are reported in parentheses.
The estimates of equations Г5.51 and Г5.61 in columns (1) and (2) of Table 5.2 give all observations equal weight, as if data from each state were equally valuable. States are not created equal, however, in at least one important respect: some, like Texas and California, are bigger than most countries, while others, like Vermont and Wyoming, have populations smaller than those of many American cities. We may prefer estimates that reflect this fact by giving more populous states more weight. The regression procedure that does this is called weighted least squares (WLS). The standard OLS estimator fits a line by minimizing the sample average of squared residuals, with each squared residual getting equal weight in the sum.- Just as the name suggests, WLS weights each term in the residual sum of squares by population size or some other researcher-chosen weight.
Population weighting has two consequences. First, as noted in Chapter 2. regression models of treatment effects capture a weighted average of effects for the groups or cells represented in our data. In a state-year panel, these groups are states. OLS estimates of models for state-year panels produce estimates of average causal effects that ignore population size, so the resulting estimates are averages over states, not over people. Population weighting generates a people-weighted average, in which causal effects for states like Texas get more weight than those for states like Vermont. People-weighting may sound appealing, but it need not be. The typical citizen is more likely to live in Texas than Vermont, but changes in the Vermont MLDA provide variation that may be just as useful as changes in Texas. You should hope, therefore, that regression estimates from your state-year panel are not highly sensitive to weighting.
Population weighting may also increase the precision of regression estimates. With far fewer drivers in Vermont than in Texas, MVA death rates in Vermont are likely to be more variable from year to year than those in Texas (this reflects the sampling variation discussed in the appendix to Chapter 11. In a statistical sense, the data from Texas are more reliable and therefore, perhaps, worthy of higher weight. Here too, however, the case for weighting is not open and shut. As a matter of econometric theory, masters of ’metrics can claim that weighted estimates are more precise than unweighted estimates only when a number of restrictive technical conditions are met.— Once again, the best scenario is a set of findings (that is, estimates and standard errors) that are reasonably insensitive to weighting.
Columns (3) and (4) in Table 5.2 report WLS estimates of equations Г5.51 and Г5.6У These correspond to the OLS estimates shown in columns (1) and (2) of the table, but the WLS estimator weights each observation by state population aged 18-20. Happily for our understanding of MLDA effects, weighting here matters little. It would seem once again that teetotaling masters have been rewarded for their temperance.
master stevefu: Wrap it up for me, Grasshopper.
grasshopper: Treatment and control groups may differ in the absence of treatment, yet move in parallel. This pattern opens the door to DD estimation of causal effects.
master stevefu: Why is DD better than simple two-group comparisons?
grasshopper: Comparing changes instead of levels, we eliminate fixed differences between groups that might otherwise generate omitted variables bias.
master stevefu: How is DD executed with multiple comparison groups and multiple years?
grasshopper: I have seen the power and flexibility of regression DD, Master. In a state-year panel, for example, with time-varying state policies like the ML DA, we need only control for state and year effects.
master stevefu: On what does the fate of DD estimates turn?
grasshopper: Parallel trends, the claim that in the absence of treatment, treatment and control group outcomes would indeed move in parallel. DD lives and dies by this. Though we can allow for state-specific linear trends when a panel is long enough, masters hope for results that are unchanged by their inclusion.