A concept used in macroeconomics is Okun's Law, which states that the change in unemployment from one period to the next depends on the rate of growth of the economy relative to a "normal" growth rate:

where \(U_{t}\) is the unemployment rate in period \(t, G_{t}\) is the growth rate in period \(t\), the "normal" growth rate \(G_{N}\) is that which is required to maintain a constant rate of unemployment, and \(0

a. Show that the model can be written as \(D U_{t}=\beta_{1}+\beta_{2} G_{t}\), where \(D U_{t}=U_{t}-U_{t-1}\) is the change in the unemployment rate, \(\beta_{1}=\gamma G_{N}\), and \(\beta_{2}=-\gamma\).

b. Estimating this model with quarterly seasonally adjusted U.S. data from 1970 Q1 to 2014 Q4 yields

Use the estimates \(b_{1}\) and \(b_{2}\) to find estimates \(\hat{\gamma}\) and \(\hat{G}_{N}\).

c. Find standard errors for \(b_{1}, b_{2}, \hat{\gamma}\), and \(\hat{G}_{N}\). Are all these estimates significantly different from zero at a \(5 \%\) level?

d. Using a 5\% significance level test the null hypothesis that the natural growth rate is \(0.8 \%\) per quarter against the alternative it is not equal to \(0.8 \%\).

e. Find a \(95 \%\) interval estimate for the adjustment coefficient.

f. Find a 95\% interval estimate for \(E\left(U_{2015 Q 1} \mid U_{2014 Q 4}=5.7991, G_{2015 Q 1}=0.062\right)\).

g. Find a \(95 \%\) prediction interval for \(U_{2015 Q 1}\) given \(U_{2014 Q 4}=5.7991\) and \(G_{2015 Q 1}=0.062\). Explain the difference between this interval and that in (f).

Transcribed Image Text:

= U-U-Y(G,- GN)