Question 1- You are given the following data:
| Y |
X2 |
X3 |
| 1 |
1 |
2 |
| 3 |
2 |
1 |
| 8 |
3 |
-3 |
Based on these data, estimate the following regressions (Note: Do not worry about estimating the standard errors):
a. Yi = A1 + A2X2i + ui
b. Yi = C1 + C3X3i + ui
c. Yi = B1 + B2X2i + B3X3i + ui
d. Is A2 = B2? Why or why not?
e. Is C3 = B3? Why or why not?
What general conclusion can you draw from this exercise?
Question 2- You are given the following data based on 15 observations:
Y-= 367.693; X2- = 402.760; X3- = 8.0; Σyi2 = 66,042.269
Σx2i2 = 84,855.096; Σx3i2 = 280.0; Σyix2i = 74,778.346
Σyix3i = 4,250.9; Σx2ix3i = 4,796.0
where lowercase letters, as usual, denote deviations from sample mean values.
a. Estimate the three multiple regression coefficients.
b. Estimate their standard errors.
c. Obtain R2 and R2-.
d. Estimate 95% confidence intervals for B2 and B3.
e. Test the statistical significance of each estimated regression coefficient using α = 5% (two-tail).
f. Test at α = 5% that all partial slope coefficients are equal to zero. Show the ANOVA table.
Question 3- A three-variable regression gave the following results:
| Source of variation |
Sum of squares (SS) |
d.f. |
Mean sum of squares (MSS) |
| Due to regression (ESS) |
65,965 |
|
|
| Due to residual (RSS) |
|
|
|
| Total (TSS) |
66,042 |
14 |
|
a. What is the sample size?
b. What is the value of the RSS?
c. What are the d.f. of the ESS and RSS?
d. hat is R2? And R2-?
e. Test the hypothesis that X2 and X3 have zero influence on Y. Which test do you use and why?
f. From the preceding information, can you determine the individual contribution of X2 and X3 toward Y?
Question 4- To explain what determines the price of air conditioners, B. T. Ratchford24 obtained the following regression results based on a sample of 19 air conditioners:
Yi^= - 68.236 + 0.023X2i + 19.729X3i + 7.653X4iR2 = 0.84
se = (0.005) (8.992) (3.082)
where Y = the price, in dollars
X2 = the BTU rating of air conditioner
X3 = the energy efficiency ratio
X4 = the number of settings
se = standard errors
a. Interpret the regression results.
b. Do the results make economic sense?
c. At α = 5%, test the hypothesis that the BTU rating has no effect on the price of an air conditioner versus that it has a positive effect.
d. Would you accept the null hypothesis that the three explanatory variables explain a substantial variation in the prices of air conditioners? Show clearly all your calculations.
Question 5- Based on the U.S. data for 1965-IQ to 1983-IVQ (n = 76), James Doti and Esmael Adibi25 obtained the following regression to explain personal consumption expenditure (PCE) in the United States.
Yt^= -10.96 + 0.93X2t - 2.094X3t
t = (-3.33) (249.06) (-3.09) R2 = 0.9996
F= 83,753.7
where Y = the PCE ($, in billions)
X2 = the disposable (i.e., after-tax) income ($, in billions)
X3= the prime rate (%) charged by banks
a. What is the marginal propensity to consume (MPC)-the amount of additional consumption expenditure from an additional dollar's personal disposable income?
b. Is the MPC statistically different from 1? Show the appropriate testing procedure.
c. What is the rationale for the inclusion of the prime rate variable in the model? A priori, would you expect a negative sign for this variable?
d. Is b3 significantly different from zero?
e. Test the hypothesis that R2 = 0.
f. Compute the standard error of each coefficient.
Questing 6- Table 4-7 (found on the textbook's Web site) gives data on child mortality (CM), female literacy rate (FLR), per capita GNP (PGNP), and total fertility rate (TFR) for a group of 64 countries.
a. A priori, what is the expected relationship between CM and each of the other variables?
b. Regress CM on FLR and obtain the usual regression results.
c. Regress CM on FLR and PGNP and obtain the usual results.
d. Regress CM on FLR, PGNP, and TFR and obtain the usual results. Also show the ANOVA table.
e. Given the various regression results, which model would you choose and why?
f. If the regression model in (d) is the correct model, but you estimate (a) or (b) or (c), what are the consequences?
g. Suppose you have regressed CM on FLR as in (b). How would you decide if it is worth adding the variables PGNP and TFR to the model? Which test would you use? Show the necessary calculations.
Attachment:- table 4-7.rar