For pedagogic purposes Hanushek and Jackson estimate the following model:

C_t = β₁ + β₂GNP_t + β₃D_t + u_i €¦€¦€¦€¦€¦€¦€¦€¦€¦.. (1)

where C_t = aggregate private consumption expenditure in year t, GNP_t = gross national product in year t, and D = national defense expenditures in year t, the objective of the analysis being to study the effect of defense expenditures on other expenditures in the economy.

Postulating that σ²_t = σ²(GNP_t )², they transform (1) and estimate

C_t/GNP_t = β₁ (1/GNP_t) + β₂ + β₃ (D_t/GNP_t) + u_t/GNP_t €¦€¦€¦.. (2)

The empirical results based on the data for 1946€“1975 were as follows (standard errors in the parentheses):

a. What assumption is made by the authors about the nature of heteroscedasticity? Can you justify it?

b. Compare the results of the two regressions. Has the transformation of the original model improved the results, that is, reduced the estimated standard errors? Why or why not?

c. Can you compare the two R2 values? Why or why not?

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Ĉ, = 26.19 (2.73) + 0.6248 GNP, - 0.4398 D; (0.0736) 0.4315 (D;/GNP,) (0.0597) R² = 0.999 (0.0060) C:/GNP, = 25.92 (1/GNP,) + 0.6246 (2.22) R² = 0.875 (0.0068)