1. Use the t- and F-tables to determine the appropriate critical value for conducting the stated hypothesis based on the following OLS regression result:
yˆ = β^0 + β^1x1 + β^2x2 + ? + β^kxk
If multiple restrictions are given, then the k refers to the unrestricted model. For b, c, d and f, if the answer differs from the critical value in the prior hypothesis, explain why it is larger or smaller.
t-Tests
a. H0: β1 = 0 n = 29
H1: β1 ≠ 0 k = 6
5% significance level
b. H0: β1 = 1 n = 26
H1: β1 ≠ 1 k = 3
5% significance level
c. H0: β1 = 0 n = 350
H1: β1 > 0 k = 10
10% significance level
d. H0: β1 = 0 n = 200
H1: β1 < 0 k = 5
1% significance level
F-Tests
e. H0: β1 = β2 = β3 = 0 n = 66
H1: H0 is not true k = 5
5% significance level
f. H0: β3 = β4 = β5 = 0 n = 66
H1: H0 is not true k = 5
1% significance level
2. Consider the following estimation results of a model studying the effects of skipping class on college GPA. Standard errors are in parentheses below the parameter estimates
(colGPA)^ = 1.39 + 0.412 hsGPA + 0.15 ACT - 0.083 skipped
(0.33) (0.22) (0.011) (0.026)
n = 141, R2 = 0.234
a. Construct the 95% confidence interval for βhsGPA.
b. Can you reject the hypothesis H0: βhsGPA = 0 against the two-sided alternative at the 5% level? Explain, or show your work.
c. Can you reject the hypothesis H0: βhsGPA = 1 against the two-sided alternative at the 5% level? Explain, or show your work.
d. Can you reject the hypothesis H0: βhsGPA = 0 against the alternative that βhsGPA > 0 at the 5% level? Explain, or show your work.
3. Qualitative variables (Dummies).
a. Given the following fictional data on home prices, fill in the six empty columns based on the variable descriptions given.
|
price
|
sqft
|
beds
|
type
|
beds1
|
beds2
|
beds3
|
condo
|
house
|
sqft_condo
|
sqft_house
|
|
193,943
|
1000
|
1
|
Condo
|
|
|
|
|
|
|
|
|
253,966
|
1200
|
2
|
House
|
|
|
|
|
|
|
|
|
190,159
|
900
|
3
|
House
|
|
|
|
|
|
|
|
|
227,882
|
1150
|
3
|
Condo
|
|
|
|
|
|
|
|
|
167,404
|
800
|
1
|
House
|
|
|
|
|
|
|
|
|
261,975
|
1300
|
1
|
Condo
|
|
|
|
|
|
|
|
|
149,846
|
800
|
2
|
Condo
|
|
|
|
|
|
|
|
price price of the home in dollars
sqft floor area of home in square feet
beds number of bedrooms
type type of home
beds1 Dummy variable indicating one bedroom homes
beds2 Dummy variable indicating two bedroom homes
beds3 Dummy variable indicating three bedroom homes
condo Dummy variable indicating condos
house Dummy variable indicating houses
sqft_condo Interaction term indicating the floor area of a condo, 0 otherwise
sqft_house Interaction term indicating the floor area of a house, 0 otherwise
b. Using the 7 observations in the table above, is it possible to estimate the following model? Explain briefly.
price = β0 + β1sqft + β2condo + β3house + u
c. Using the 7 observations in the table above, you obtain the following OLS regression results. What is the interpretation of βˆ2?
(price)^ = β^0 + β^1sqft + β^2beds2 + β^3beds3
d. In the following OLS regression results. For which type of home, a condo or a house, is Δprice/Δsqft greater? Explain briefly.
(price)^ = -18914 + 230 sqft - 15.7 sqft_condo
(6404) (6.65) (2.26)
4. Suppose that you are interested in estimating the ceteris paribus relationship between y and x1. For this purpose you also collect data on two control variables, x2 and x3. Let β˜ be the simple regression estimate from y on x1 and let βˆ be the multiple regression estimates from y on x1, x2, x3.
a. If x1 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial effects on y, would you expect β˜1 and βˆ1 to be 0similar or very different? Explain.
b. If x1 is almost uncorrelated with x2 and x3, but x2 and x3 are highly correlated, will β˜1 and βˆ1 tend to be similar or very different? Explain.
c. If x1 is highly correlated with x2 and x3, and x2 and x3 have small partial effects on y would you expect se(β˜1) or se(βˆ1) to be smaller? Explain.
d. If x1 is almost uncorrelated with x2 and x3, and x2 and x3 have large partial effects on y, and x2 and x3 are highly correlated, would you expect se(β˜1) or se(βˆ1) to be smaller? Explain.
5. Testing multiple linear restrictions.
Consider the following model of how the price of a car relates to its age, transmission type, and color:
price = β0 + β1age + β2automatic + β3red + β4black + β5white + u
price - price of the car in $
age - age of car in years
automatic - dummy equal to one if car has automatic transmission, and equal to 0 if the car has manual transmission
red - dummy equal to one if car is red, 0 otherwise
black - dummy equal to one if car is black, 0 otherwise
white - dummy equal to one if car is white, 0 otherwise
|
------------------------------------------------------------
|
|
(1)
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(2)
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(3)
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price
|
price
|
price
|
|
------------------------------------------------------------
|
|
age
|
-979.0
|
-979.1
|
-980.9
|
|
|
(21.66)
|
(21.47)
|
(21.37)
|
|
automatic
|
|
855.9
|
888.3
|
|
|
|
(198.2)
|
(197.4)
|
|
red
|
|
|
789.9
|
|
|
|
|
(264.6)
|
|
black
|
|
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283.3
|
|
|
|
|
(264.5)
|
|
white
|
|
|
818.7
|
|
|
|
|
(264.7)
|
|
_cons
|
26384.1
|
25820.6
|
25339.7
|
|
|
(188.7)
|
(228.1)
|
(281.4)
|
|
-------------------------------------------------------------
|
|
R2
|
0.672
|
0.678
|
0.682
|
|
N
|
196
|
196
|
196
|
In the following sub questions, write down and test the appropriate null and alternative hypotheses using the results in the above table of regression output.
a. Test whether a car's color is significant at the 5% level. (Hint: This is a joint hypothesis.)
b. Test whether a car's color and transmission type are jointly significant at the 5% level.
6. Using data on 32 chemical plants you plan to estimate the relationship between firm size measure by total sale in thousand dollars (sales) and amount contributed to research and development (rdintens) also measured in thousand dollars. The following equation was estimated by OLS to determine the relationship:
(rditens)^ = 2.613 + .00030 sales - .0000000070 sales2
(.429) (.00014) (.0000000037)
n = 32, R2 = .1484.
a. The model estimated allows sales to have a positive but diminishing effect on rdintens. Do the estimated parameters show that? Explain.
b. At what point does the marginal effect of sales on rdintens become negative?
c. Should you keep the quadratic term in the model? Explain. (Hint: Consider statistical significance).