Using the data in the file toody 5 , estimate the model

where \(Y_{t}=\) wheat yield in tons per hectare in the Toodyay Shire of Western Australia in year \(t\); \(T R E N D_{t}\) is a trend variable designed to capture technological change, with observations \(0,0.1\), \(0.2, \ldots, 4.7 ; 0\) is for the year 1950, 0.1 is for the year 1951, and so on up to 4.7 for the year 1997; RAIN \(_{t}\) is total rainfall in decimeters (dm) from May to October (the growing season) in year \(t\) ( 1 decimeter \(=4\) inches).

a. Report your estimates, standard errors, \(t\)-values, and \(p\)-values in a table.

b. Are each of your estimates significantly different from zero at a (i) \(5 \%\) level, (ii) \(10 \%\) level?

c. Do the coefficients have the expected signs? Why? (One of the objectives of technological improvements is the development of drought-resistant varieties of wheat.)

d. Find point and 95\% interval estimates of the marginal effect of extra rainfall in (i) 1959 when the rainfall was \(2.98 \mathrm{dm}\) and (ii) 1995 when the rainfall was \(4.797 \mathrm{dm}\). Discuss the results.

e. Find point and \(95 \%\) interval estimates for the amount of rainfall that would maximize expected yield in (i) 1959 and (ii) 1995. Discuss the results.

Y = + TREND, + BRAIN, + BRAIN + B (RAIN, TREND) + e,