16-8. Our operating assumption in this chapter is that Y is a linear function of X

(plus an additive error term). What is important is not that the regression model is linear in the variables but that it must be linear in the parameters in order to be estimated. For example:

(a) Y = β0 + β1X−1 is nonlinear in X but linear in the parameters; and

(b) Y = β0Xβ1 is nonlinear in both X and in the parameters yet lnY =

ln β0 + β1 lnX is linear in the parameters.

For case (a):
1. Show that the slope is everywhere negative and decreases in absolute value as X increases.
2. Graph the function for X > 0. Verify that as X → 0,Y →∞;
and as X →∞,Y → β0.
3. What is the interpretation of β0?
4. Convert Y into a linear estimating equation by introducing a new variable Z = X−1. Estimate β0 and β1 via least squares using the data in Table A.
For case (b):
1. Show that if β1 > 0, the slope is always positive and Y →∞as X →∞. If β1 > 1, the slope is monotonically increasing as X increases; if 0 < β1 < 1, the slope is monotonically decreasing as X increases. If β1 < 0, the slope is always negative as X increases.
2. Graph the function for β1 = −1 and X > 0. What is this type of expression called?
3. Verify that this function has a constant elasticity equal to β1.
4. Using the data in Table B, estimate this expression via least squares by defining new variables W = lnY and Z = lnX.
How is β0 obtained?
A B X Y X Y 1 14 1 2 2 10 2 5 3 8 3 8 4 6 4 12 5 5.8 5 17 6 5.5 6 25 7 5.3 7 37 8 4.9 8 52