Differences-in-Differences coefficients meaning (from Mostly Harmless Econometics)

Question

In Mostly Harmless Econometrics, section 5.2.1 (Regression DD), pages 233-234, equation (5.2.3) defines $Y_{ist}=\alpha + \gamma NJ_{s}+\lambda d_{t}+\delta (NJ_{s}.d_{t})+\epsilon_{ist}$, where $NJ_{s}$ is a dummy denoting observations from New Jersey and $d_{t}$ is a dummy for observations obtained in November. Considering $E[\epsilon_{ist} \vert s,t]=0$, the books states that:

$\alpha=E[Y_{ist} \vert s=PA, t=Feb] = \gamma_{PA} + \lambda_{Feb}$ $\gamma=E[Y_{ist} \vert s=NJ, t=Feb] - E[Y_{ist} \vert s=PA, t=Feb] = \gamma_{NJ} - \gamma_{PA}$

$(...)$

Although, in my opinion:

$E[Y_{ist} \vert s=NJ, t=Feb]=\alpha+\gamma$, so that $E[Y_{ist} \vert s=NJ, t=Feb] - E[Y_{ist} \vert s=PA, t=Feb] =\alpha+\gamma -\alpha=$ $=(\gamma_{PA}+\lambda_{Feb}+\gamma_{NJ})-(\gamma_{PA}+\lambda_{Feb})=\gamma_{NJ}$, where $\gamma_{NJ} \equiv \gamma$

Which has a different meaning from what the book stated (cited above). Could anyone explain what I am missing or if there is a mistake in the book? Found nothing about it at the blog.

Answer 1

The book is correct, but it is easier to see if you insert the corresponding values of the dummies and check what happens to the regression equation. Let's go through the possible values together.

• $NJ_s = 0$ is Pennsylvania
• $NJ_s = 1$ is New Jersey
• $d_t = 0$ is February
• $d_t = 1$ is November

So your baseline regression is $NJ_s = 0$ and $d_t = 0$, i.e. the control group in the pre-treatment period which gives the regression: $$Y_{ist} = \alpha + \epsilon_{ist}$$ So $\alpha$ captures the employment level for state Pennsylvania in February: $$\alpha = E[Y_{ist}|s=PA, t=Feb] = \gamma_{PA} + \lambda_{Feb}$$ Now if you "switch on" the $NJ_s$ dummy to $NJ_s = 1$, the regression becomes: $$Y_{ist} = \alpha + \gamma NJ_s + \epsilon_{ist}$$ in which $\gamma $ gives you the difference between the two states at time $d_t = 0$ (February), and it can be written as: $$\gamma= E[Y_{ist}|s=NJ, t=Feb]- E[Y_{ist}|s=PA, t=Feb] = \gamma_{NJ} - \gamma_{PA}$$ In chapter 2, section 2.3, they explain how these differences work out. In this case $NJ_s = 1$ measures the deviation from the baseline, which we said was Pennsylvania in February. So if this deviation is positive and significant with $\gamma > 0$, then employment levels in February were higher in New Jersey than in Pennsylvania.

Finally, let's also switch the time dummy $d_t$ to one, then the regression is the full model you've described in the question. The coefficient $\beta $ then gives the difference in employment levels between two differences (which is your difference in differences):

• New Jersey in November minus New Jersey in February
• Pennsylvania in November minus Pennsylvania in February