Consider the ARDL model

Assume that \(y_{t}\) and \(x_{t}\) are I(1) and cointegrated. Let the cointegrating relationship be described by the equation \(y_{t}=\beta_{1}+\beta_{2} x_{t}+e_{t}\).

a. Show that \(\beta_{1}=\delta /\left(1-\theta_{1}-\theta_{2}-\theta_{3}\right)\) and \(\beta_{2}=\sum_{r=0}^{5} \delta_{r} /\left(1-\theta_{1}-\theta_{2}-\theta_{3}\right)\).

b. Consider the corresponding error correction model \[
\Delta y_{t}=-\alpha\left(y_{t-1}-\beta_{1}-\beta_{2} x_{t-1}\right)+\phi_{1} \Delta y_{t-1}+\phi_{2} \Delta y_{t-2}+\sum_{r=0}^{4} \eta_{r} \Delta x_{t-r}+v_{t}
\]
Show that \(\delta=\alpha \beta_{1}, \theta_{1}=1-\alpha+\phi_{1}, \theta_{2}=\phi_{2}-\phi_{1}, \theta_{3}=-\phi_{2}, \delta_{0}=\eta_{0}, \delta_{1}=\alpha \beta_{2}-\eta_{0}+\eta_{1}\), \(\delta_{2}=\eta_{2}-\eta_{1}, \delta_{3}=\eta_{3}-\eta_{2}, \delta_{4}=\eta_{4}-\eta_{3}\), and \(\delta_{5}=-\eta_{4}\).

c. Using the data in usdata5, set \(y_{t}=B R_{t}\) and \(x_{t}=F F R_{t}\) and find least-squares estimates of the parameters in equation (XR12.18).

d. Use nonlinear least squares to estimate equation (12.32) in Example 12.9.

e. Substitute the parameter estimates of equation (12.32) obtained in part (d) into the expressions given in part (b) and compare the estimates you get with those obtained in part (c). What conclusion can you draw from this comparison?

Data From Equation 12.32:-

Transcribed Image Text:

3 5 y=8+0y+8x + v s=1 1=0 (XR12.18)