Chapter 6: Extensions of The Two-Variable Regression Model

6.1. Consider the regression model yi = β1 + β2xi + ui where yi = (Yi − Y¯ ) and xi = (Xi − X¯ ). In this case, the regression line must pass through the origin. True or false? Show your calculations.

6.2. The following regression results were based on monthly data over the period January 1978 to December 1987:
Yˆt = 0.00681 + 0.75815Xt
se = (0.02596) (0.27009)
t = (0.26229) (2.80700)
p value = (0.7984) (0.0186) r2 = 0.4406
Yˆt = 0.76214Xt
se = (0.265799)
t = (2.95408)
p value = (0.0131) r2 = 0.43684
where Y = monthly rate of return on Texaco common stock, %, and X = monthly market rate of return,%.*
a. What is the difference between the two regression models?
b. Given the preceding results, would you retain the intercept term in the first model? Why or why not?
c. How would you interpret the slope coefficients in the two models?
d. What is the theory underlying the two models?
e. Can you compare the r 2 terms of the two models? Why or why not?
f. The Jarque–Bera normality statistic for the first model in this problem is 1.1167 and for the second model it is 1.1170. What conclusions can you draw from these statistics?
g. The t value of the slope coefficient in the zero intercept model is about 2.95, whereas that with the intercept present is about 2.81. Can you rationalize this result?

6.3. Consider the following regression model:
1/Yi = β1 + β2 (1/Xi) + ui
Note: Neither Y nor X assumes zero value.
a. Is this a linear regression model?
b. How would you estimate this model?

c. What is the behavior of Y as X tends to infinity?
d. Can you give an example where such a model may be appropriate?

6.4. Consider the log-linear model: ln Yi = β1 + β2 ln Xi + ui

PlotY on the vertical axis and X on the horizontal axis. Draw the curves showing the relationship between Y and X when β2 = 1, and when β2 > 1, and when β2 < 1.

6.5. Consider the following models:
Model I: Yi = β1 + β2Xi + ui
Model II: Y*
i = α1 + α2X*
i + ui
where Y* and X* are standardized variables. Show that αˆ2 = βˆ2(Sx/Sy) and hence establish that although the regression slope coefficients are independent of the change of origin they are not independent of the change of scale.

6.6. Consider the following models:
ln Y*
i = α1 + α2 ln X*
i + u*i
ln Yi = β1 + β2 ln Xi + ui
where Y*
i = w1Yi and X*
i = w2Xi , the w’s being constants.
a. Establish the relationships between the two sets of regression coefficients and their standard errors.
b. Is the r2 different between the two models?

6.7. Between regressions (6.6.8) and (6.6.10), which model do you prefer? Why?

6.8. For the regression (6.6.8), test the hypothesis that the slope coefficient is not significantly different from 0.005.

6.9. From the Phillips curve given in (6.7.3), is it possible to estimate the natural rate of unemployment? How?

6.10. The Engel expenditure curve relates a consumer’s expenditure on a commodity to his or her total income. Letting Y = consumption expenditure on a commodity and X = consumer income, consider the following models:
Yi = β1 + β2Xi + ui
Yi = β1 + β2(1/Xi ) + ui
ln Yi = ln β1 + β2 ln Xi + ui
ln Yi = ln β1 + β2(1/Xi ) + ui
Yi = β1 + β2 ln Xi + ui
Which of these model(s) would you choose for the Engel expenditure curve and why? (Hint: Interpret the various slope coefficients, find out the expressions for elasticity of expenditure with respect to income, etc.)

6.11. Consider the following model:
Yi =(e^{β₁+β₂X_i})/(1 + e^{β₁+β₂X_i})
As it stands, is this a linear regression model? If not, what “trick,” if any, can you use to make it a linear regression model? How would you interpret the resulting model? Under what circumstances might such a model be appropriate?

<See Tutorial Attachment for the full answer.>