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. For 1974 , when \(T R E N D=2.4\) and \(R A I N=4.576\), use a \(5 \%\) significance level to test the null hypothesis that extra rainfall will not increase expected yield against the alternative that it will increase expected yield.

c. Assuming rainfall is equal to its median value of \(3.8355 \mathrm{dm}\), find point and \(95 \%\) interval estimates of the expected improvement in wheat yield from technological change over the period 1960-1995.

d. There is concern that climate change is leading to a decline in rainfall over time. To test this hypothesis, estimate the equation RAIN \(=\alpha_{1}+\alpha_{2}\) TREND \(+e\). Test, at a \(5 \%\) significance level, the null hypothesis that mean rainfall is not declining over time against the alternative hypothesis that it is declining.

e. Using the estimated equation from part (d), estimate mean rainfall in 1960 and in 1995.

f. Suppose that \(T R E N D_{1995}=T R E N D_{1960}\), implying there had been no technological change from 1960 to 1995. Use the estimates from part

(e) to find an estimate of the decline in mean yield from 1960 to 1995 attributable to climate change.

g. Suppose that \(E\left(\operatorname{RAIN}_{1995}\right)=E\left(\operatorname{RAIN}_{1960}\right)\), implying there had been no rainfall change from 1960 to 1995. Find an estimate of the increase in mean yield from 1960 to 1995 attributable to technological change.

h. Compare the estimates you obtained in parts (c), (f), and (g).

Transcribed Image Text:

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