Question

Ten years ago, the town of Easton decided to increase its annual spending on education so that its high school graduates would be able to earn higher wages. Now Easton has asked you to evaluate the effectiveness of the spending increase. Their data show that before the spending increase, the average annual salary of recent high school graduates was $25,000 and that now that average salary has risen to $28,500. Fortunately for your analysis, a neighboring community (Allentown) did not change its annual spending on education. Ten years ago, recent Allentown high school graduates earned an average of $22,500, and now that average is $23,750. a. Use a difference-in-differences estimator to determine whether Easton’s spending increase caused the wages of their high school graduates to increase. b. What underlying assumption do you have to make in order for your estimate to be valid? What might cause your underlying assumption to be invalid? c. This data set contains only two observations. Even if the underlying assumption in part b is met, how much confidence can you have in conclusions based on two observations?

Answer 1

A.

	Easton	Allentown	Change
t= 10	28500	23750	4750
t = 0	25000	22500	2250
Difference	3500	1250	2500

DID Estimator = 2500

B.

All the assumptions of the OLS model apply equally to DID. In addition, DID requires a parallel trend assumption.

Parallel assumption is most important and it requires that in the absence of treatment, the difference between the ‘treatment’ and ‘control’ group is constant over time and if it is not fulfilled then DID estimator will be biased.

C.

DID is a popular method in part because the data requirements are not particularly onerous – it requires data from only two points in time – and the results are robust to any possible confounder as long as it doesn’t violate the Parallel Paths assumption.

Proof:

The DID method can be implemented according to the table below, where the lower right cell is the DID estimator.

$y_{{st}}$	$s=2$	$s=1$	Difference
$t=2$	$y_{{22}}$	$y_{{12}}$	{ $y_{{12}}-y_{{22}}$
$t=1$	$y_{{21}}$	$y_{{11}}$	$y_{{11}}-y_{{21}}$
Change	$y_{{21}}-y_{{22}}$	$y_{{11}}-y_{{12}}$	$(y_{{11}}-y_{{21}})-(y_{{12}}-y_{{22}})$

Running a regression analysis gives the same result. Consider the OLS model

$y~=~\beta _{0}+\beta _{1}T+\beta _{2}S+\beta _{3}(T\cdot S)+\varepsilon$

where is a dummy variable for the period, equal to when $t=2$ , and is a dummy variable for group membership, equal to when $s=2$ . The composite variable $(T\cdot S)$ is a dummy variable indicating when {\displaystyle S=T=1} $S=T=1$ . Although it is not shown rigorously here, this is a proper parametrization of the model formal definition, furthermore, it turns out that the group and period averages in that section relate to the model parameter estimates as follows

${\displaystyle {\begin{aligned}{\hat {\beta }}_{0}&={\widehat {E}}(y\mid T=0,~S=0)\\[8pt]{\hat {\beta }}_{1}&={\widehat {E}}(y\mid T=1,~S=0)-{\widehat {E}}(y\mid T=0,~S=0)\\[8pt]{\hat {\beta }}_{2}&={\widehat {E}}(y\mid T=0,~S=1)-{\widehat {E}}(y\mid T=0,~S=0)\\[8pt]{\hat {\beta }}_{3}&={\big [}{\widehat {E}}(y\mid T=1,~S=1)-{\widehat {E}}(y\mid T=0,~S=1){\big ]}\\&\qquad {}-{\big [}{\widehat {E}}(y\mid T=1,~S=0)-{\widehat {E}}(y\mid T=0,~S=0){\big ]},\end{aligned}}}$

where ${\displaystyle {\widehat {E}}(\dots \mid \dots )}$ stands for conditional averages computed on the sample, for example, $T=1$ is the indicator for the after period, $S=0$ is an indicator for the control group. To see the relation between this notation and the previous section, consider as above only one observation per time period for each group, then

${\displaystyle {\begin{aligned}{\widehat {E}}(y\mid T=1,~S=0)&={\widehat {E}}(y\mid {\text{ after period, control}})\\[3pt]\\&={\frac {{\widehat {E}}(y\ I({\text{ after period, control}}))}{{\widehat {P}}({\text{ after period, control}})}}\\[3pt]\\&={\frac {\sum _{i=1}^{n}y_{i,{\text{after}}}I(i{\text{ in control}})}{n_{\text{control}}}}={\overline {y}}_{\text{control, after}}\\[3pt]\\&={\overline {y}}_{\text{12}}\end{aligned}}}$

and so on for other values of and , which is equivalent to

${\displaystyle {\hat {\beta }}_{3}~=~(y_{11}-y_{21})-(y_{12}-y_{22}).}$

But this is the expression for the treatment effect that was given in the formal definition and in the above table.