Daniel Kuehn and I have been reading a lot of the major papers in the minimum wage debate. (I had asked Daniel if he would be willing to work through these papers with me, since this is his area and [given our different political perspectives] I wanted to make sure I was being fair to the guys arguing the case for raising the minimum wage.) Some of Daniel’s very helpful posts are here, here, here, and here.
I am going to be writing more on the broader debate in other outlets, but for the present post I want to spell out a crucial point that Meer and West raise in their 2013 working paper. Specifically, if the minimum wage doesn’t cause a sharp reduction in the level of employment, but instead permanently reduces the growth of employment, then a regression looking for a level effect might seriously understate the minimum wage’s influence.
In this post, let’s not worry about the context of the minimum wage, but instead focus just on the narrow econometric point, because at first glance it’s counterintuitive. So, consider the following two scenarios (taken from Appendix A in Meer & West):
The idea is that some “treatment” is applied to State A first, then after some time has passed the same treatment is applied to State B. (Here “state” means an actual territorial unit, i.e. one of the 50 states of the Union.) The time durations are chosen so that the pre-treatment period is exactly the same length as the post-treatment.
As the charts clearly illustrate, in case (b) the treatment causes a one-shot (but permanent) reduction in the level of the variable of interest, while in case (c) the treatment causes a permanent reduction in the growth of the variable of interest.
Now, if we run a “differences in differences” (DiD) regression and choose the level of the y-variable as our dependent variable, then in case (b) it will attribute a negative coefficient to the treatment variable, while in case (c) the regression output will report no effect of the treatment. In contrast, if we run a DiD regression and analyze the growth of the y-variable, then in case (b) it will show no effect of the treatment, while in case (c) it will correctly assign a negative coefficient to the treatment variable.
Thus, if you accept these claims for the sake of argument, you can see the relevance to the minimum wage debate: If raising the minimum wage primarily affects the growth of employment, then the typical regressions in the literature (which look at the impact on the level of employment) could be vastly understating the actual effect.
To repeat, at first this seems counterintuitive. After all, we can quite clearly see that the treatment in case (c) has made the level of employment lower than it otherwise would have been; why wouldn’t a regression looking at level effects pick this up?
One way to see it is to read the discussion in Meer and West’s appendix. It has to do with the fact that the DiD estimator uses an “indicator variable” that is the same for both states except in the middle time interval. This makes the common “time trend” variable soak up the actual work that the treatment is doing. (Obviously I’m putting this into my own words.)
I think the way to make this “click” intuitively is to look at a different graph, this one from Meer and West’s Appendix B:
Now in the chart above, the “control state” (for which there is no treatment) chugs along at the same rate of growth the whole time. (That’s why the logarithm of its level of employment is a straight line.)
In contrast, employment in the treated state was originally growing at the same rate (though it had a higher level, initially), but then after the treatment it suffered a permanent reduction in the rate of growth. (Be sure you are focusing on the solid line, not the dotted one.)
Now, the dotted line shows the trend of employment in the treated state, over the entire period. Before the treatment, the treated state grows “above trend,” while after treatment, it grows “below trend.” (Note that if we were to plot the control state’s time trend, it would overlap perfectly with the actual employment level; for any interval in this graph, the control state employment would be growing “exactly even with trend.”)
So Meer and West are here illustrating a huge potential pitfall in including “state time trends” as a control factor in these types of analyses. In the above example, a regression on the level of employment, with a state time trend included as a variable, would show no effect of treatment. Daniel made this “click” for me by pointing out that if you shift the dotted line in the diagram down just a tad, so that it lines up perfectly with the solid line, then it’s “obvious” that the treatment had no effect on the level of employment in the state: The dotted line and solid line end up at the same level, at the last time period.
UPDATE: In the comments, Daniel elaborates on what happens in the regression that corresponds to shifting the dotted line in that way: “the intercept is accounted for by the fixed effects (geographic fixed effects effectively give every geographic area their own intercept). So if you clean out all the differences in the intercept, you see what controlling for the trend does.” Also, to be clear, Daniel is not agreeing that the “revisionist” studies are wrong; he is just clarifying what Meer and West are claiming, and how their cute pictures relate to the econometric specifications in the minimum wage literature.
Now to be sure, these hypothetical examples don’t prove that the empirical studies finding little impact of the minimum wage are necessarily spurious. However, they do highlight the serious pitfalls in such undertakings, and in particular we can see how including geographical “trend” controls might end up obscuring the quite real impact of the policy.