Differences-in-differences: Pre and Post, Treatment and Control
The fixed effects strategy requires panel data, that is, repeated observations on the same individuals (or firms or whatever the unit of observation might be). Often, however, the regressor of interest varies only at a more aggregate level such as state or cohort. For example, state policies regarding health care benefits for pregnant workers or minimum wages change across states but not within states. The source of omitted variables bias when evaluating these policies must therefore be unobserved variables at the state and year level.
To make this concrete, suppose we are interested in the effect of the minimum wage on employment, a classic question in Labor Economics. In a competitive labor market, increases in the minimum wage move us up a downward-sloping demand curve. Higher minimums therefore reduce employment, perhaps hurting the very workers minimum-wage policies were designed to help. Card and Krueger (1994) use a dramatic change in the New Jersey state minimum wage to see if this is true.
On April 1, 1992, New Jersey raised the state minimum from $4.25 to $5.05. Card and Krueger collected data on employment at fast food restaurants in New Jersey in February 1992 and again in November 1992. These restaurants (Burger King, Wendy’s, and so on) are big minimum-wage employers. Card and Krueger collected data from the same type of restaurants in eastern Pennsylvania, just across the Delaware river. The minimum wage in Pennsylvania stayed at $4.25 throughout this period. They used their data set to compute differences-in-differences (DD) estimates of the effects of the New Jersey minimum wage increase. That is, they compared the change in employment in New Jersey to the change in employment in Pennsylvania around the time New Jersey raised its minimum.
DD is a version of fixed-effects estimation using aggregate data. To see this, let
yist = fast food employment at restaurant i and period t if there is a high state minimum wage yoist = fast food employment at restaurant i and period t if there is a low state minimum wage
These are potential outcomes – in practice, we only get to see one or the other. Fort example, we see yiist in New Jersey in November of 1992. The heart of the DD setup is an additive structure for potential outcomes in the no-treatment state. Specifically, we assume that
E (yo ist |S;t)= 7s + Xt (5.2.1)
where s denotes state (New Jersey or Pennsylvania) and t denotes period (February, before the minimum wage increase or November, after the increase). This equations says that in the absence of a minimum wage change, employment is determined by the sum of a time-invariant state effect and a year effect that is common across states. The additive state effect plays the role of the unobserved individual effect in the previous subsection.
Let Dst be a dummy for high-minimum-wage states, where states are index by s and observed in period t. Assuming that E(y1ist — yoist|s, t) is a constant, denoted ft, we have:
y ist = 7 s + Xt + ftdst + "ist (5.2.2)
where E(sist |s, t) = 0. From here, we get
E[yist|s = PA, t = Nov] – E(уist|s = PA, t = Feb) = Xnov – XFeb
E(yist ^ = NJ;t = Nov) – E(yis! = NJ;t = Feb) = XNov – AFeb + ft – The population difference-in-differences,
[E(уist|s = PA;t = Nov) – E(yisfts = PA;t = Feb)]
– [E(yist|s = NJ, t = Nov) – E(уist|s = NJ, t = Feb)] = ft,
This is easily estimated using the sample analog of the population means.
Table 5.2.1: Average employment per store before and after the New Jersey minimum wage increase
Notes: Adapted from Card and Krueger (1994), Table 3. The table reports average full-time equivalent (FTE) employment at restaurants in Pennsylvania and New Jersey before and after a minimum wage increase in New Jersey. The sample consists of all stores with data on employment. Employment at six closed stores is set to zero. Employment at four temporarily closed stores is treated as missing. Standard errors are reported in parentheses
Table 5.2.1 (based on Table 3 in Card and Krueger, 1994) shows average employment at fast food restaurants in New Jersey and Pennsylvania before and after the change in the New Jersey minimum wage. There are four cells in the first two rows and columns, while the margins show state differences in each period, the changes over time in each state, and the difference-in-differences. Employment in Pennsylvania restaurants is somewhat higher than in New Jersey in February but falls by November. Employment in New Jersey, in contrast, increases slightly. These two changes produce a positive difference-in-differences, the opposite of what we might expect if a higher minimum wage pushes businesses up the labor demand curve.
How convincing is this evidence against the standard labor-demand story? The key identifying assumption here is that employment trends would be the same in both states in the absence of treatment. Treatment induces a deviation from this common trend, as illustrated in figure 5.2.1. Although the treatment and control states can differ, this difference in captured by the state fixed effect, which plays the same role as the unobserved individual effect in (5.1.3).
The common trends assumption can be investigated using data on multiple periods. In an update of their
original minimum wage study, Card and Krueger (2000) obtained administrative payroll data for restaurants in New Jersey and Pennsylvania for a number of years. These data are shown here in Figure 5.2.2, similar to Figure 2 in their follow-up study. The vertical lines indicate the dates when their original surveys were conducted, and the third vertical line denotes the increase in the federal minimum wage to $4.75 in October 1996, which affected Pennsylvania but not New Jersey. These data give us an opportunity to look at a new minimum wage "experiment".
Like the original Card and Krueger survey, the administrative data show a slight decline in employment from February to November 1992 in Pennsylvania, and little change in New Jersey over the same period. However, the data also reveal fairly substantial year-to-year employment variation in other periods. These swings often seem to differ substantially in the two states. In particular, while employment levels in New Jersey and Pennsylvania were similar at the end of 1991, employment in Pennsylvania fell relative to employment in New Jersey over the next three years (especially in the 14-county group), mostly before the 1996 change in Federal minimum. So Pennsylvania may not provide a very good measure of counterfactual employment rates in New Jersey in the absence of a policy change, and vice versa.
A more encouraging example comes from Pischke (2007), who looks at the effect of school term length on student performance using variation generated by a sharp policy change in Germany. Until the 1960s, children in all German states except Bavaria started school in the Spring. Beginning in the 1966-67 school year, the Spring-starters moved to start school in the Fall. The transition to a Fall start required two short school years for affected cohorts, 24 weeks long instead of 37. Students in these cohorts effectively had their time in school compressed relative to cohorts on either side and relative to students in Bavaria, which
,NJ _ «._ PA; 7 counties…….. PA; 14 counties
Figure 5.2.2: Employment in New Jersey and Pennsylvania fast-food restaurants, October 1991 to September 1997 (from Card and Krueger 2000). Vertical lines indicate dates of the original Card and Krueger (1994) survey and the October 1996 federal minimum-wage increase.
already had a Fall start.
Figure 5.2.3 plots the likelihood of grade repetition for the 1962-73 cohorts of 2nd graders in Bavaria and affected states (there are no repetition data for 1963-65). Repetition rates in Bavaria were reasonably flat from 1966 onwards at around 2.5%. Repetition rates are higher in the short-school-year states, at around 4 – 4.5% in 1962 and 1966, before the change in term length. But repetition rates jump up by about a percentage point for the two affected cohorts in these states, a bit more so for the second cohort than the first, before falling back to the baseline level. This graph provides strong visual evidence of treatment and control states with a common underlying trend, and a treatment effect that induces a sharp but transitory deviation from this trend. A shorter school year seems to have increased repetition rates for affected cohorts.
о см р
° 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973
School year ending
Figure 5.2.3: Average rates of grade repetition in second grade for treatment and control schools in Germany (from Pischke 2007). The data span a period before and after a change in term length for students outside of Bavaria.