Klein and Goldberger attempted to fit the following regression model to the U.S. economy:

Y_i = β₁ + β₂X_2i + β₃X_3i + β₄X_4i + u_i

where Y = consumption, X₂ = wage income, X₃ = nonwage, nonfarm income, and X₄ = farm income. But since X₂, X₃, and X₄ are expected to be highly collinear, they obtained estimates of β₃ and β₄ from cross-sectional analysis as follows:

β₃ = 0.75β₂ and β₄ = 0.625β₂. Using these estimates, they reformulated their consumption function as follows:

Y_i = β₁ + β₂(X_2i + 0.75X_3i + 0.625X_4i ) + u_i = β₁ + β₂Z_i + u_i

where Z_i = X_2i + 0.75X_3i + 0.625X_4i.

a. Fit the modified model to the data in Table 10.12 and obtain estimates of β₁ to β₄.

b. How would you interpret the variable Z?

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Xз Xз Year Year X2 43.41 X4 3.96 5.48 4.37 4.51 X2 28.26 27.91 X4 9.76 9.31 9.85 7.21 7.39 7.98 7.42 1946 1947 1948 76.73 75.91 1936 1937 1938 1939 1940 1941 1945* 62.8 17.10 95.7 18.65 17.09 19.28 23.24 65.0 63.9 67.5 71.3 76.6 46.44 44.35 47.82 98.3 100.3 1949 103.2 108.9 108.5 111.4 77.62 78.01 83.57 90.59 95.47 32.30 31.39 35.61 37.58 35.17 4.88 6.37 8.96 51.02 58.71 1950 1951 28.11 30.29 1952 86.3 87.69