Consider the infinite lag representation \(y_{t}=\alpha+\sum_{s=0}^{\infty} \beta_{s} x_{t-s}+e_{t}\) for the ARDL model

a. Show that \(\alpha=\delta /\left(1-\theta_{1}-\theta_{3}\right), \beta_{0}=0, \beta_{1}=\delta_{1}, \beta_{2}=\theta_{1} \beta_{1}, \beta_{3}=\theta_{1} \beta_{2}\), and \(\beta_{s}=\theta_{1} \beta_{s-1}+\theta_{3} \beta_{s-3}\) for \(s \geq 4\).

b. Using quarterly data on U.S. inflation (INF), and the change in the unemployment rate ( \(D U)\) from 1955Q2 to 2016Q1, we estimate the following version of a Phillips curve

a. Show that \(\alpha=\delta /\left(1-\theta_{1}-\theta_{3}\right), \beta_{0}=0, \beta_{1}=\delta_{1}, \beta_{2}=\theta_{1} \beta_{1}, \beta_{3}=\theta_{1} \beta_{2}\), and \(\beta_{s}=\theta_{1} \beta_{s-1}+\theta_{3} \beta_{s-3}\) for \(s \geq 4\).

b. Using quarterly data on U.S. inflation (INF), and the change in the unemployment rate ( \(D U)\) from 1955Q2 to 2016Q1, we estimate the following version of a Phillips curve

c. Using the results in part (a), find estimates of the first 12 lag weights in the infinite lag representation of the estimated Phillips curve in part (b). Graph those weights and comment on the graph.

d. What rate of inflation is consistent with a constant unemployment rate (where \(D U=0\) in all time periods)?

e. Let \(\hat{e}_{t}=0.564 \hat{e}_{t-1}+0.333 \hat{e}_{t-3}+\hat{u}_{t}\) where the \(\hat{u}_{t}\) are the residuals from the equation in part (b), and the initial values \(\hat{e}_{1}, \hat{e}_{2}\), and \(\hat{e}_{3}\) are set equal to zero. The \(S S E\) from regressing \(\hat{u}_{t}\) on a constant, \(I N F_{t-1}, I N F_{t-3} D U_{t-1}, \hat{e}_{t-1}\), and \(\hat{e}_{t-3}\) is 47.619. Using a 5\% significance level, test the hypothesis that the errors in the infinite lag representation follow the \(\operatorname{AR}(3)\) process \(e_{t}=\theta_{1} e_{t-1}+\theta_{3} e_{t-3}+v_{t}\). The number of observations used in this regression and that in part (b) is 241 . What are the implications of this test result?

Transcribed Image Text:

y=8+0y-1+031-3 +81x1 + v