according to a normal rule of thumb procedure. Under this rule of thumb, the band- width used is optimal in a RMSE sense if the data are generated from a normal dis-
tribution. In segments of the distribution with fewer observations, the bandwidths are adjusted to be wider using the adaptive bandwidth rule of Silverman 1986. In con-
trast, the bandwidths are adjusted to be narrower in ranges where there are many observations to allow for sharper fluctuations in the estimated density in those ranges
Härdle 1991. Because the peak of the family income-to-needs distribution is typi- cally near one the poverty line, this technique increases the accuracy of the kernel
procedure in the area that may be of greatest interest.
We also apply these methods to analyze the densities of changes in income-to- needs, in which case we study subsamples of observations based on Year 1 income-
to-needs. To accommodate the widely differing sample sizes that result from this disaggregation, we pool the data for initial bandwidth selection following Marron and
Schmitz’s 1992 approach. This keeps the level of smoothing equal for the analyses of families in different initial income-to-needs categories, whereas standard rules
would result in more smoothed estimates for smaller sample sizes.
B. Contaminated Treatment and Control Groups
Because previous research has found that the effects of minimum wages are often stronger at a lag of one year see Neumark and Wascher 1992 and 2002, and Baker,
Benjamin, and Stanger 1999, we are interested in estimating both contemporaneous and lagged minimum-wage effects on the densities of family income-to-needs.
However, introducing lagged effects into our analysis creates complications because the observations for the treatment group or the control group may be contaminated
by the effects of minimum-wage increases not directly captured by the difference-in- difference estimator. For example, when we estimate f
2,MW = 1
I for the treatment group for the lagged effect, there could also be a contemporaneous effect in Year 2.
Similarly, when we estimate the density for the treatment group for the contempora- neous effect, there could be a lagged effect from a contemporaneous increase in Year
1. Of course, we could drop all of the observations in which the treatment is con- taminated. But as Table 1 shows, that would entail the loss of many observations.
Instead, we employ a procedure that uses all of the observations and distributes the overall effects into “pure” contemporaneous and “pure” lagged effects correcting for
the incidence of contaminated treatment and control groups. This procedure, which is explained in Appendix 1, turns out to have only a modest effect on the estimates.
C. Controlling for Other Differences Between Treatment and Control Groups
Also potentially confounding our analysis are factors that generate state-specific or year-specific shifts in the income-to-needs distribution and that are potentially corre-
lated with the incidence of minimum-wage increases. In particular, we would like to avoid attributing to minimum wages shifts in income-to-needs distributions that are
common to states or years, but correlated with minimum-wage changes. A prime example, as noted earlier, is the aggregate business cycle. But other possibilities
include changes in federal policies affecting the income distribution, aggregate influ- ences on wages or incomes other than the business cycle, or persistent state-specific
The Journal of Human Resources 874
differences in movements of families through the income-to-needs distribution, per- haps also stemming from policy differences.
We first take a crude approach to eliminating the spurious influence of the business cycle on the estimates. In particular, the years in our sample in which the federal mini-
mum wage was raised coincided with the recession in the early 1990s and thus with rel- atively sharp increases in state unemployment rates. As it seems unlikely that the federal
minimum-wage increases actually caused the recession, we also present estimates in which we exclude the years influenced by a federal minimum-wage increase 1990 and
1991 as well as 1992, in which the lagged increase from 1991 occurred. This restric- tion avoids confounding the influence of the recession with minimum-wage effects,
although it also likely goes too far in eliminating useful variation in the data.
An alternative and more informative approach is to mimic a regression model that includes fixed year and state effects. To do this, we first estimate the median propor-
tional change in income-to-needs by state across all years. We then adjust each fam- ily’s income-to-needs in Year 2 so that this common state change is taken out of the
change in the family’s income-to-needs from Year 1 to Year 2. We make a parallel adjustment for the median proportional change by year across all states.
6
The dif- ference between the adjusted data on income-to-needs for each family in Year 2 and
Year 1 is the deviation around the average state change over all years in the sample and the average year change over all states in the sample. We then perform the basic
difference-in-difference analysis described above using these adjusted data.
D. State versus Federal Increases and Biases from Migration
