Econometrics Assignment
Section 1
1. List three of the least squares assumptions.
2. Explain the concepts of Homoscedasticity and Heteroscedasticity.
3. Give two statistics used for hypothesis testing and confidence intervals
4. In Hypothesis testing, what are the following representations called?
H1: H0:
5. What does the R2 tell us?
Section 2
Please write the formulas for the following:
1. Covariance
2. Correlation
3. The t statistic
4. The F statistic
Section 3
The expected sales of a product in a city are assumed to be affected by the per capita discretionary income and the population of the city. Per capita discretionary income will be referred to as PCDI in all the questions. In Questions 1-10 examine only the effect of per capita discretionary income on the mean sales. Thus the following model is hypothesized:
E(Y) = B0 + B1 X1 where
Y = Sales (in thousands of dollars)
X1 = Per Capita Discretionary Income (in dollars)
A sample of 15 cities, along with their sales, per capita discretionary income, and the population of the city (in thousands) is given in the attached printout. The 15 values and a printout follow:
|
|
OBS
|
INCOME
|
SALES
|
|
1
|
2450
|
162
|
|
|
2
|
3254
|
120
|
|
|
3
|
3802
|
223
|
|
|
4
|
2838
|
131
|
|
|
5
|
2347
|
67
|
|
|
6
|
3782
|
169
|
|
|
7
|
3008
|
81
|
|
|
8
|
2450
|
192
|
|
|
9
|
2137
|
116
|
|
|
10
|
2560
|
55
|
|
|
11
|
4020
|
252
|
|
|
12
|
4427
|
232
|
|
|
13
|
2660
|
144
|
|
|
14
|
2088
|
103
|
|
|
15
|
2605
|
212
|
|
|
16
|
2500
|
.
|
|
|
17
|
3500
|
.
|
|
|
Root MSE
|
49.51434
|
R-square
|
0.4087
|
|
Dep Mean
|
150.60000
|
Adj R-sq
|
0.3632
|
Parameter Estimates
Coefficient Standard T for H0:
Variable Estimate Error B=0 Prob
INTERCEP -10.207 55.147 -0.185 0.8560
INCOME 0.054 0.018 2.998 0.0103
Dep 95% LCL 95% UCL 95% LCL 95% UCL
Obs Actual Predicted Mean Mean Individual Individual
|
16
|
. 125.5
|
92.5
|
158.5 13.5
|
237.5
|
|
17
|
. 179.8
|
145.1
|
214.5 67.3
|
292.3
|
1. The 95% confidence interval for the mean sales of all cities with PCDI = 2500 is
A. 92.5 to 158.5
B. can not be calculated because of missing values C. 3500
D. 88.6 to 156.9
E. 13.5 to 237.5
2. When testing the null hypothesis that the slope equals to zero versus the alternative hypothesis that the slope does not equal to zero, the rejection region would be: reject the Null if
A. t > t(14, 0.025) or t < -t(14, 0.025)
B. t > t(13, 0.05)
C. F < F(1, 13, 0.05)
D. |t| > t(13, 0.025)
E. p-value > alpha
3. Given the p-value of the F-test is 0.0103, we can interpret this as
A. Given the null is true, there is a 1.03% chance of finding this value of the test statistic or something more extreme.
B. The percent of sample variability of Y explained by the independent variable is 1.03%
C. There is a 98.97% probability that the null hypothesis is right.
D. There is a 98.97% probability that the null hypothesis is wrong.
E. The probability of a type I error is 0.0103.
4. Does the PCDI help predict the sales of the product?
A. Yes, because 2.998 > the table value
B. No, because .8560 is greater than alpha
C. Yes, because 8.986 < the table value
D. Yes, because of MSE = 2451.66959
E. No, because 0.018 is less than the table value
5. What is the interpretation of the coefficient of determination?
A. Don't know and don't care (Hint, this is a wrong answer and best left unspoken within hearing of instructor).
B. 40.87 probability that sales is linearly related to PCDI.
C. 40.87 percent of the sample variability of sales can be attributed to changes in PCDI.
D. 40.87 percent of the variability of PCDI can be attributed to a linear relationship between mean PCDI and sales.
E. 40.87 percent of the sample variability of PCDI can be attributed to a linear relationship between mean PCDI and sales.
6. What table value would you use in the calculation of a 90% confidence interval for a value of Y given a value of X?
A. 1.645
B. 3.140
C. 1.771
D. 2.650
E. 2.998
7. How many estimated standard errors is the point estimate of the slope away from zero? Slope is the change in the mean sales for each dollar increase in PCDI.
A. 0.054
B. 0.4087
C. -10.207
D. 2.998
E. 0.018
8. You know that most cities have small PCDI and only a few have large PCDI. Is this a violation of any assumption?
A. Yes, because the variation of PCDI would then be unequal.
B. No, because sales has to be normally distributed but PCDI does not have to be.
C. Yes, this would violate the linear relationship between the mean sales and PCDI.
D. No, because the variance of sales has nothing to do with the problem.
E. Yes, a violation of normality.
9. What would be the change in the estimated mean sales for each one standard deviation increase in PCDI?
A. 0.3632 standard deviations
B. Cannot be calculated.
C. 0.4087 squared dollars
D. 0.6393 (square root of 0.4087) standard deviations
E. 0.0540 dollars
10. What is the possible equation for the above graph?
A. Y= 1.22 + 0.95x
B. Y = 1.22 - 0.95x
C. X = 1.22 + 0.95x
D. X = 1.22 - 0.95x
11. What is the possible value of the correlation between the variables?
A. -0.88
B. 0.00
C. 0.88
D. 1.00.