While we were able to estimate the ATT through mean differences alone, we can also use linear regression for DID. A simple DID regression model predicts the outcome from the variables for treatment group and time, along with the interaction of treatment with time.

To simplify our output, we will first transform the state variable to a treatment indicator called treat (1 for California and 0 for Washington). Then we transform year to a time indicator called time (1 for 2017 and 0 for 2016).

# transform state to treat
wages2$treat <- ifelse(wages2$state=="California",1,0)
# transform year to time
wages2$time <- ifelse(wages2$year==2017,1,0)

We can print and inspect our dataset to make sure our transformations were done correctly.

       state treat year time  avg_wage
1 California     1 2016    0 13.311279
2 California     1 2017    1 16.000000
3 Washington     0 2016    0  9.728146
4 Washington     0 2017    1 10.000000

To create a DID regression for our student wage data, we run a model that predicts average student wages from the treatment, time, and the interaction of treatment and time. Note that in R, treat*time is equivalent to treat + time + treat:time.

did_mod <- lm(
  #include interaction
  avg_wage ~ treat*year,
  #use subsetted data 
  data = wages2 
)

When we print did_mod we get the following output. Note that the coefficient on the interaction term treat:year is exactly what we computed for the ATT by taking the difference of means. We estimate the impact of the minimum wage law on California student wages to be an increase of about $2.42.

Call:
lm(formula = avg_wage ~ treat * time, data = wages2)

Coefficients:
(Intercept)        treat         time   treat:time  
     9.7281       3.5831       0.2719       2.4169

What do all the other coefficients represent?

• Intercept indicates the expected value for pre-treatment average student wages for the control group (Washington 2016).
• treat is the difference between the control group and the treatment group at the pre-treatment time (California 2016 - Washington 2016).
• time is the difference between the pre-treatment and post-treatment times for the control group (Washington 2017 - Washington 2016).

Combinations of these coefficients give us back all four means from our dataset.