The New Jersey-Pennsylvania Study on the Effect of Minimum Wages $\hat { \beta }$

Essay

The New Jersey-Pennsylvania study on the effect of minimum wages on employment mentioned in your textbook used a comparison in means "before" and "after" analysis. The difference-in-difference estimate turned out to be 2.76 with a standard error of 1.36.
The authors also used a difference-in-differences estimator with additional regressors of the type
ΔY_i = β₀ + β₁X_i + β₂W_1,t + ... + β_{1+ r}W_r,i + u_i
where i = 1, …, 410. X is a binary variable taking on the value one for the 331 observations in New Jersey. Since the authors looked at Burger King, KFC, Wendy's, and Roy Rogers fast food restaurants and the restaurant could be company owned, four W-variables were added.
(a)Given that there are four chains and the possibility of a company ownership, why did the authors not include five W-variables?
(b)OLS estimation resulted in $\hat { \beta }$ ₁ of 2.30 with a standard error of 1.20. Test for statistical significance and specify the alternative hypothesis.
(c)Why is this estimate different from the number calculated from Δ $\bar { Y } \text { treatment }$ - Δ $\bar { Y } ^ { \text {control } }$ = 2.76? What is the advantage of employing this estimator of the simple difference-in-difference estimator?