clear all
use https://sscc.wisc.edu/~rdimond/pa871/ck.dta4 Difference in Differences
There’s been a lot of work on DID models recently, and anything beyond simple “before and after” turns out to be more complicated than we thought–and is somewhat unsettled yet. So if you want to do more than that some time in the future, you’ll want to read up on the latest thinking.
So we’re going to stick with something simple, Card & Krueger’s landmark paper on the effect of minimum wages on unemployment.
Card & Krueger collected employment data from fast food restaurants near the border between Pennsylvania and New Jersey shortly before and shortly after New Jersey raised their minimum wage.
tab state time, sum(emptot)
Means, Standard Deviations and Frequencies of emptot
| time
state | 0 1 | Total
-----------+----------------------+----------
PA | 23.331169 21.165584 | 22.248377
| 11.856283 8.276732 | 10.248645
| 77 77 | 154
-----------+----------------------+----------
NJ | 20.439408 21.027429 | 20.7325
| 9.1062391 9.2930238 | 9.1973191
| 321 319 | 640
-----------+----------------------+----------
Total | 20.998869 21.054293 | 21.026511
| 9.7498049 9.0944527 | 9.4227458
| 398 396 | 794The simple way to turn this into a regression model is to interact time and state.
reg emptot state##time, cluster(store)
Linear regression Number of obs = 794
F(3, 409) = 1.80
Prob > F = 0.1462
R-squared = 0.0074
Root MSE = 9.4056
(Std. err. adjusted for 410 clusters in store)
------------------------------------------------------------------------------
| Robust
emptot | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
state |
NJ | -2.891761 1.439546 -2.01 0.045 -5.721593 -.0619281
1.time | -2.165584 1.218025 -1.78 0.076 -4.559954 .2287855
|
state#time |
NJ#1 | 2.753606 1.306607 2.11 0.036 .1851025 5.322109
|
_cons | 23.33117 1.346536 17.33 0.000 20.68417 25.97816
------------------------------------------------------------------------------Note that we can recover the group means with margins.
margins state#time
Adjusted predictions Number of obs = 794
Model VCE: Robust
Expression: Linear prediction, predict()
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
state#time |
PA#0 | 23.33117 1.346536 17.33 0.000 20.68417 25.97816
PA#1 | 21.16558 .9400009 22.52 0.000 19.31775 23.01342
NJ#0 | 20.43941 .5090522 40.15 0.000 19.43872 21.44009
NJ#1 | 21.02743 .5211146 40.35 0.000 20.00303 22.05183
------------------------------------------------------------------------------This model decomposes the differences into three componets:
- The effect of being in New Jersey (
NJ, -2.89). Note that Pennsylvania is the reference state. - The effect of being in the seond time period (
1.time, -2.17) - The additional effect of being in New Jersey in the second time period (
NJ#1, 2.75)
Since the increased minimum wage only applies in New Jersey in the second time period, 2.75 is the effect of the minimum wage. Basic supply and demand principles says it should be negative, so this was a shocking result.
An alternative way of running the same model is to create an explicit treated variable that identifies the observations in the treated state and time period.
gen treated = (state==1 & time==1)reg emptot state time treated, cluster(store)
Linear regression Number of obs = 794
F(3, 409) = 1.80
Prob > F = 0.1462
R-squared = 0.0074
Root MSE = 9.4056
(Std. err. adjusted for 410 clusters in store)
------------------------------------------------------------------------------
| Robust
emptot | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
state | -2.891761 1.439546 -2.01 0.045 -5.721593 -.0619281
time | -2.165584 1.218025 -1.78 0.076 -4.559954 .2287855
treated | 2.753606 1.306607 2.11 0.036 .1851025 5.322109
_cons | 23.33117 1.346536 17.33 0.000 20.68417 25.97816
------------------------------------------------------------------------------The results are identical either way.
Card & Krueger tried adding some covariates to the DID model. That could potentially help with the parallel trends assumption. One model put in a fixed effect for each chain and an indicator for co-ownership, and a second added regional indicators.
reg emptot state##time i.chain co_owned, cluster(store)
Linear regression Number of obs = 794
F(7, 409) = 36.87
Prob > F = 0.0000
R-squared = 0.1964
Root MSE = 8.4843
(Std. err. adjusted for 410 clusters in store)
------------------------------------------------------------------------------
| Robust
emptot | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
state |
NJ | -2.376608 1.274861 -1.86 0.063 -4.882705 .1294889
1.time | -2.223565 1.224901 -1.82 0.070 -4.631452 .1843222
|
state#time |
NJ#1 | 2.845067 1.31287 2.17 0.031 .2642518 5.425881
|
chain |
KFC | -10.45339 .736662 -14.19 0.000 -11.90151 -9.005273
Roys | -1.624999 1.066468 -1.52 0.128 -3.721442 .4714437
Wendy's | -1.063709 1.113181 -0.96 0.340 -3.25198 1.124562
|
co_owned | -1.168545 .7785634 -1.50 0.134 -2.699031 .3619398
_cons | 25.95118 1.375668 18.86 0.000 23.24691 28.65544
------------------------------------------------------------------------------reg emptot state##time i.chain co_owned pa1 northj southj, cluster(store)
Linear regression Number of obs = 794
F(10, 409) = 29.81
Prob > F = 0.0000
R-squared = 0.2214
Root MSE = 8.3674
(Std. err. adjusted for 410 clusters in store)
------------------------------------------------------------------------------
| Robust
emptot | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
state |
NJ | -.9000803 1.931382 -0.47 0.641 -4.696755 2.896594
1.time | -2.211851 1.227688 -1.80 0.072 -4.625217 .201515
|
state#time |
NJ#1 | 2.814908 1.317374 2.14 0.033 .2252384 5.404578
|
chain |
KFC | -10.058 .7711267 -13.04 0.000 -11.57387 -8.542135
Roys | -1.693393 1.041944 -1.63 0.105 -3.741627 .3548421
Wendy's | -1.064952 1.078432 -0.99 0.324 -3.184913 1.055009
|
co_owned | -.7163097 .8685631 -0.82 0.410 -2.423715 .9910951
pa1 | .923862 1.684432 0.55 0.584 -2.387363 4.235087
northj | -.0078834 1.263233 -0.01 0.995 -2.491122 2.475356
southj | -3.709644 1.398393 -2.65 0.008 -6.458578 -.9607098
_cons | 25.32051 1.621324 15.62 0.000 22.13334 28.50768
------------------------------------------------------------------------------Adding these predictors just makes the effect of the minimum wage increase slightly larger.
The problem with this model (and our own John Kennan identified it at the time) is that the big change from “before” to “after” is a decrease in employment in Pennsylvania. DID always relies heavily on the parallel trends assumption, but in this case we’re assuming that New Jersey would have seen a similar decrease and was saved from that by the increase in the minimum wage. Since Card & Krueger only have data for one time period before the treatment, there’s no way to assess the validity of the parallel trends assumption.
The paper has other evidence that I actually find more convincing than their DID results, such as finding that restaurants in New Jersey who had lower wages before the minimum wage increase, and thus had to increase their wages by more, saw a bigger increase in total employment.