a. Using data from the data file toody 5 , estimate the following model. Comment on the results.

\[\text { YIELD }_{t}=\alpha_{1}+\alpha_{2} t+\beta_{1} \text { RAIN }_{t}+\beta_{2}\text{RAIN}_{t}^{2}+e_{t}\]

b. Plot the residuals from the model estimated in part (a) and check the residual correlogram. What do you observe?

c. Estimate the following model and comment on the results.

\[\text { YIELD }_{t}=\alpha_{1}+\alpha_{2} t+\alpha_{3} t^{2}+\beta_{1} \text { RAIN }_{t}+\beta_{2} R A I N_{t}^{2}+e_{t}\]

d. Plot the residuals from the model estimated in part (c) and check the residual correlogram. How do the properties of the residuals differ from those in part (b)?

e. Using a 5\% significance level, test whether YIELD, RAIN, and RAIN \({ }^{2}\) are trend stationary after subtracting out the quadratic trend.