You Have Collected Quarterly Data for the Unemployment Rate (Unemp)in

You have collected quarterly data for the unemployment rate (Unemp)in the United States, using a sample period from 1962:I (first quarter)to 2009:IV (the data is collected at a monthly frequency, but you have taken quarterly averages).
a. Does economic theory suggest that the unemployment rate should be stationary?
b. Testing the unemployment rate for stationarity, you run the following regression (where the lag length was determined using the BIC; using the AIC instead does not change the outcome of the test, even though it chooses 9 lags of the LHS variable):
$\begin{array} { c } \widehat { \text { Unemp } _ { \mathrm { t } } } = 0.217 - 0.035 \text { Unemp } _ { \mathrm { t } - 1 } + 0.689 \Delta \text { Unemp } _ { \mathrm { t } - 1 } \\\left( \begin{array} { l l l } ( 0.01 ) & 0.0012&(0.054\end{array} \right)\end{array}$
Use the ADF statistic with an intercept only to test for stationarity. What is your decision?
c.The standard errors reported above were homoskedasticity-only standard errors. Do you think you could potentially improve on inference by allowing for HAC standard errors?
d. An alternative test for a unit root, the DF-GLS, produces a test statistic of -2.75. Find the critical value and decide whether or not to reject the null hypothesis. If the decision is different from (c), is there any reason why you might prefer the DF-GLS test over the ADF test?