Your Textbook Gives a Graphical Example Of $\widehat { \beta } _ { 1 } ^ { \text {diffs-in-diffs } }$

Essay

Your textbook gives a graphical example of $\widehat { \beta } _ { 1 } ^ { \text {diffs-in-diffs } }$ , where outcome is plotted on the vertical axis, and time period appears on the horizontal axis. There are two time periods entered: " $\mathrm { t } = 1$ " and " $\mathrm { t } = 2$ ." The former corresponds to the "before" time period, while the latter represents the "after" period. The assumption is that the policy occurred sometime between the time periods (call this " $\mathrm { t } = \mathrm { p }$ "). Keeping in mind the graphical example of $\widehat { \beta } _ { 1 } ^ { \text {diffs-in-diffs } }$ , carefully read what a reviewer of the Card and Krueger (CK) study of the minimum wage effect on employment in the New Jersey-Pennsylvania study had to say:

"Two assumptions are implicit throughout the evaluation of the 'natural experiment:' (1) $\left[ \widehat { \beta } _ { 1 } ^ { \text {diffs-in-diffs } } \right]$ would be zero if the treatment had not occurred, so a nonzero $\left[ \widehat { \beta } _ { 1 } ^ { \text {diffs-in-diffs } } \right]$ indicates the effect of the treatment (that is, nothing else could have caused the difference in the outcomes to change), and (2) ... the intervention occurs after we measure the initial outcomes in the two groups. ... Three conditions are particularly relevant in interpreting CK's work: (1) $[ \mathrm { t } = 1 ]$ must be sufficiently before $[ \mathrm { t } = \mathrm { p } ]$ that [the treatment group] did not adjust to the treatment before $[ \mathrm { t } = 1 ] -$ otherwise $\left[ \bar { Y } ^ { \text {treatment,before } } - \bar { Y } ^ { \text {conrol,hefofor } } \right]$ will reflect the effect of the treatment; (2) $[ \mathrm { t } = 2 ]$ must be sufficiently after $[ \mathrm { t } = \mathrm { p } ]$ to allow the treatment's effect to be fully felt; and (3) we must be sure that the same difference $\left[ \bar { Y } ^ { \text {treatment } , b \text { before } } - \bar { Y } ^ { \text {control, byfore } } \right]$ would have been observed at $[ \mathrm { t } = 2 ]$ if the treatment had not been imposed, that is, [the control group must be good enough] that there is no need to adjust the differences for factors other than the treatment that might have caused them to change." Use a figure similar to the textbook to explain what this reviewer meant.