課程名稱︰計量
課程性質︰選修
課程教師︰
開課學院：劉錦添
開課系所︰經濟系
考試日期（年月日）︰2013.11.15
考試時限（分鐘）：120MINS
是否需發放獎勵金：YES
（如未明確表示，則不予發放）
試題 :
1.(30%)
The Office of the Registrar at UCSD took a random sample of 427 students and
obtained their grade point average in college (COLGPA), high school GPA
(HSGPA), verbal Scholastic Aptitude Test scores (VSAT), and the mathematics
scores in the SAT (MSAT). The following model was estimated (subscript t is
omitted for simplicity):
COLGPA = b1 + b2HSGPA + b3 VSAT + b4MSAT + u
The estimated coefficients and their standard errors are given below:
(standard errors in parentheses)
b1 0.423 (0.22)
b2 0.398 (0.061)
b3 0.0007375 (0.00028)
b4 0.001015 0.0002936
(a)
The unadjusted R^2 was 0.22. Because this is very low, we might suspect that
the model is inadequate. Test the model for overall goodness of fit (using a
1 percent level of significance). Be sure to state the null and alternative
hypotheses, the test statistic, its distribution, and the criterion for
acceptance or rejection. What is your conclusion?
(b)
Test each regression coefficient for significance at the 1 percent level
against the alternative that the coefficient is positive. Is any of them
insignificant?
(c)
Suppose a student took a special course to improve her SAT scores and
increased the verbal and math scores by 100 points each. On average, how much
of an increase in college GPA could she expect from this?
(d)
Suppose you want to test the hypothesis that the regression coefficients for
VSAT and MSAT are equal (but need not be equal to zero). Describe
step-by-step how you should do this. State the null and alternative
hypotheses, the regression(s) to be run, the test statistic to be computed,
its distribution, and the criterion for accepting or rejecting the null
hypothesis. What do you conclude?
(e)
List at least two other variables that should have been included in the
model. Explain why you think they belong in the model.
2.(30%)
The following Table presents estimates and related statistics for 4 models
relating the list price of an automobile to a number of characteristics using
82 observations.
PRICE=b1+b2WBASE+b3LENGTH+b4WIDTH+b5HEIGHT+b6WEIGHT+b7CYL
+b8LITERS+b9GASMPG+u
where
PRICE = List Price
WBASE = Wheelbase in inches
LENGTH = Length of car
WIDTH = Width of Car
HEIGHT = Height of car
WEIGHT = weight of car
CYL = number of cylinders
LITERS = engine displacement in liters
GASMPG = Estimated gas miles per gallon, averaged betweeen city and freway
friving.
Variable Model A Model B Model C Model D
CONSTANT 58.866 54.400 65.476 71.554
(27.33) (23.19) (20.1) (19.93)
WBASE 0.036
(0.28)
LENGTH 0.394 0.383 0.391 0.403
(0.14) (0.117) (0.117) (0.118)
WIDTH 0.104
(0.24)
HEIGHT 0.748 0.741 0.703 0.839
(0.46) (0.43) (0.43) (0.42)
WEIGHT 2.184 2.148 1.926 2.227
(0.47) (0.43) (0.36) (0.31)
CYL 0.959 1.046 1.095
(1.31) (0.691) (0.69)
LITTERS 0.264
(1.83)
GASMPG 0.196 0.194
(0.22) (0.20)
ESS 2303.75 2309.978 2337.952 2414.724
R^2 0.559 0.576 0.576 0.568
sigma^2 31.558 30.394 30.363 30.958
AIC 34.991 32.610 32.210 32.466
FPE 35.022 32.618 32.214 32.468
HQ 38.906 34.999 34.164 34.033
SCHWARZ 45.57 38.889 37.301 36.51
SHIBATA 34.262 32.293 31.989 32.321
GCV 35.449 32.794 32.335 32.546
RICE 35.996 33 32.472 32.631
(a)For model 1 only. I believe some of the coefficients are wrong in sign.
For each regression coefficient (ignore the constant), state the expected
sign.
(b) Test the joint hypothesis that the coeffieient for WBASE, WIDTH, CYL,
LITERS and GASMPG are all zero at 5%. State the null and alternative
hypothesis. Compute the test statistic and state its distributuon under the
null, and identify the criterion for rejection. State your conclusion.
(c) Which of the model is "best"? Explain the criterion you used.
(d) Test model a for overall significance at the 1% level. Compute the test
statistic and state its distributuon under the null, and identify the
criterion for rejection. State your conclusion. (You have all the information
you need for this.)
Note: ESS = SUM OF SQUARE RESIDUAL
3.(20%)
The following table contains the ACT scores and the GPA (grade point average)
for eight college students. Grade point average is based on a four-point scale and has
been rounded to one digit after the decimal.
Student GPA ACT
1 2.8 21
2 3.4 24
3 3.0 26
4 3.5 27
5 3.6 29
6 3.0 25
7 2.7 25
8 3.7 30
(a)
Estimate the relationship between GPA and ACT using OLS; that is, obtain the
intercept and slope estimates in the equation
GPA = b0 + b1 ACT
Comment on the direction of the relationship. Does the intercept have a
useful interpretation here? Explain. How much higher is the GPA predicted to be if
the ACT score is increased by 5 points?
(b)
Compute the fitted values and residuals for each observation, and verify that
the residuals (approximately) sum to zero.
(c)
What is the predicted value of GPA when ACT = 20?
(d)
How much of the variation in GPA for these eight students is explained by
ACT?
Explain.
4. (20%)
The following model is a simplied version of the multiple regression model
used by Biddle and Hamermesh (1990) to study the trade-off between time spent sleeping
and working and to look at other factors affcting sleep:
sleep = b0 + b1 totwrk + b2 educ + b3 age + u
where sleep and totwrk (total work) are measured in minutes per week and educ
(years of education) and age are measured in years.
(a) If adults trade o sleep for work, what is the sign of b1?
(b) What signs do you think b2 and b3 will have?
(c) Using the data in Biddle and Hamermesh (1990), the estimated equation is
Sleep = 3638-0.148TOTWRK -11.13EDUC+2.20AGE
n = 706, R^2 = 0.113
If someone works ve more hours per week, by how many minutes is sleep
predicted to fall? Is this a large trade-of?
(d) Discuss the sign and magnitude of the estimated coefficient on educ.
(e) Would you say totwrk; educ and age explain much of the variation in sleep?
What other factors might aect the time spent sleeping? Are these likely to be
correlated with totwrk?
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