A famous article $by$ two $of2021$ the Nobel Laureates studied the effect $of$ minimum wages $on$ employment using panel data with $t=2.$ The state $of$ New Jersey $(NJ)in$ the $US$ raised its minimum wage from $$4\mathrm{.}25to$ $$5\mathrm{.}05on1$ April $1992$ while the minimum wage $in$ neighbouring state $of$ Pennsylvania $(PEN)$ remained unchanged. The data used $in$ this article consists $on$ wages and employment $in65$ fast$-$food restaurants $in$ Pennsylvania and $284$ fastfood restaurants $inin$ New Jersey the year before and the year after the change $of$ the $NJ$ minimum wage.

The data set contains the following variables:

$*$ restaurant $isan$ identifier for each restaurant;

$*nj$ equals $1if$ the restaurant $isinNJ,$ and $0ifitisin$ PEN.

$*$ after equals $1if$ the observation $is$ after the legislation change, and $0ifitis$ before.

$*nj$after $isan$ interaction terms between $nj$ and after.

$*$ FTE $is$ the number $of$ full$-$time employees.

$*$ DFTE $is$ change $in$ FTE after the legislation change.

The following table summarizes the mean $of$ FTE $bynj$ and after

Table $1$: Mean Full$-$time Employees $in$ New Jersey and Pennsylvania

$i.$ The researchers wanted $to$ know the effect $of$ the change $in$ the minimum wage $on$ employment. They started with the following regression:

DFTE$={}_0+{}_{1}nj+u$ What $do{}_0$ and ${}_1$ capture? Using the information $in$ the table above, calculate these coefficients. Provide $an$ interpretation $of{}_1.$

$ii.$ When estimating the regression $(1)by$ OLS, the non$-$robust standard error $of{}_{1}is1\mathrm{.}71,$

while the robust standard error $is1\mathrm{.}57.$ Why could this $be$ the case?

iii. The researchers then estimated the following regression:

FTE$={}_0+{}_{1}nj+{}_2$ after $+{}_{3}nj$after$+u$

How are the coefficients $of$ this regression related $to$ the coefficients $of$ regression $(1)?$

Calculate the coefficients $in$ regression $(2).$

$iv.$ When estimating the regression $(2)by$ OLS, the standard errors $of{}_2$ and ${}_3$ are $1\mathrm{.}78$ and $1\mathrm{.}93,$ respectively; while the clustered standard errors $on$ restaurant are $1\mathrm{.}39$ and $1\mathrm{.}47,$respectively. Why could this $be$ the case?