Reconsider Example 6.20 where a logistic growth curve for the share of U.S. steel produced by electric arc furnace (EAF) technology was estimated. The data are stored in the data file steel. The curve is given by the equation

\[y_{t}=\frac{\alpha}{1+\exp (-\beta-\delta t)}+e_{t}\]

a. Plot the series \(y_{t}=E A F_{t}\). Does it give the appearance of being stationary or nonstationary? Does the logistic growth curve appear to be a good model for modeling its trend?

b. Using a \(5 \%\) significance level, test the series \(y_{t}=E A F_{t}\) for a unit root.

c. Estimate the equation by nonlinear least squares and plot the residuals. Do the residuals appear to be stationary. Test the residuals for a unit root.

d. Using a \(5 \%\) significance level, test the series \(\Delta y_{t}=\Delta E A F_{t}\) for a unit root.

e. Estimate a first-differenced version of the model and plot the residuals. Do the residuals appear to be stationary. Test the residuals for a unit root.

f. Based on your answers to the previous parts of this question, do you think \(y_{t}=E A F_{t}\) is trend stationary? Compare the estimates from parts (c) and (e). Do you think the nonlinear least-squares estimates in part (c) are reliable?