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Unformatted text preview: THE UNIVERSITY OF HONG KONG
School of Economics 81. Finance
20012002 1st Semester Examination
Economics: ECON0701 Introductory Econometrics
Dr P Lau December 2'7, 2001 9:3011:30 AM Candidates may use any self—contained, silent, batteryoperated and pocketsized calculator. The calcu
lator should have numeraldisplay facilities only and should be used only for the purposes of calculation. It
is the candidate’s responsibility to ensure that his calculator operates satisfactorily. Candidates must record the name and type of their calculators on the front page of their examination
scripts. Some critical values for the t—distribution and the Fdistribution are given in the last two pages. (If the
critical value you want is not shown, use the closest one.) Answer any four questions. (If you answer all ﬁve questions, only the ﬁrst four questions
answered will be graded.) All parts of a question are equally weighted unless speciﬁed otherwise. Please
give logical, concise and precise answers. Marks will be deducted for inadequacy as well as redundancy or
irrelevancy. Failure to answer one part will generally not prevent you from able to answer subsequent parts. 1. (25%) Give a brief answer (of a few sentences) to the following short questions: a What is the distinction between a ‘ o ulation regression line’ and a ‘sample re ession line’ ?
P P 8T (b) Do we typically observe signiﬁcant tvalues but an insigniﬁcant analue when colinearity is
present? (o) Is the slope term in a simple regression model more precisely estimated (by the least squares
method) when the values of the explanatory variable are less spread out? (d) According to the econometric software output for a twosided test of Ho : ,62 = 0 against H1 :
ﬁg # 0, the p value is 0.082. What is the meaning of p value? Based on the above p value, how would you decide, at a 5% level of signiﬁcance, regarding the onesided test of H0 : 32 = 0 against
H1 '. ﬁg :3 0? (e) Give an exarnple of a regression equation which is ‘nonlinear in parameter’ and an example of a
regression equation which is ‘nonlinear in variable but linear in parameter’. Does the term ‘linear
regression model’ refer to ‘Iinearity in parameter’ or ‘linearity in variable’ ? 2. (25%) For each of the following statements, indicate whether it is true or false, and give a brief
explanation. (a) The lease squares estimate for the slope coefﬁcient of a simple regression model is b2 = 0.25
for a particular sample. Since it is obtained by the least squares method, the above estimate is unbiased. (b) The requirement for a test statistic is that its probability distributions under both the null and
alternative hypotheses must be known. (c) The slope of the simple regression model 111(y) = 61 + £32 In (a) + s (where 1n represents natural
log) indicates the percentage change of the dependent variable when the explanatory variable
changes by one percent. (d) It is easy to identify the collinearity problem by checking the pairwise correlation between any
two explanatory variables. (e) John and Tom work together on a simple regression model involving an intercept dummy variable.
John is interested in the slope coeﬁcient and Tom in the coeﬁcient of the dununy variable. Since
the variables that John and Tom focus are different (one is a continuous variable and the other is
a discrete variable), the estimation and testing procedures they use are also different. 3,. A simple econometric model relating the quantity demanded of chocolate (q) to the price of chocolate
(p), the price of other goods and services (r) and income (is) is as follows: 1M4)=ﬁ1+52111®l+33111(rl+541niwl+51 (1) where in represents natural log. a is the random error term, and the standard assumrations of the
multiple regression model are satisﬁed. The estimation result based on a sample of 30 observations is
as follows: __...h.. ln(q) = —2.339  1.130 x 1n(p) + 0.195 x 111(r) + 0.733 x 111(11):),
(3.053) (0.215) (0.079) (0.374) with residual sum of squares = 0.0938, and standard deviation of dependent variable = 0.133. (a) (4%) What is the interpretation of 62 and [34 in (I). (b) (5%) Construct a 95% interval estimate for ﬁg. (c) (5%) Find R2, the coefﬁcient of determination. (d) (5%) Test the joint hypothesis H0 : [32 = (:33 = ii; = 0. What is your conclusion? (e) (6%) Describe how to test the null hypothesis of no money illusion versus the alternative hypothesis
that money illusion is present. 4. Consider the following multiple linear regression model: y=ﬁ1+ﬂ2$2+ﬁ3$3+5v (1)
where the standard assmnptions are satisﬁed. (a) Let be and b3 denote respectively the least squares estimators of [32 and 83. Express ear (b2  4b3)
in terms of the variances of ()2 and 63 and the covariance between them. (b) What is the tstatistic for testing the null hypothesis H0 : )6; —— 433 = 1 against H1 : ﬁg —4ﬁ3 at 1?
In what way is the test statistic related to your answer in part (a)? (0) Deﬁne 6 = ﬁg — 4:33. Transform Equation (1) into a regression equation involving parameter 6',
[31 and {33. (d) Describe how you would use the transformed regression equation in part (c) to directly test the
null hypothesis Ho : ,8? —— 4ﬁ3 = 1 against H; : A92  4,63 at 1. Specify the t—statistic and the
rejection region (at 5% level of signiﬁcance) for this hypothesis in terms of '5 (the least squares
estimator of 9) and its standard error. (e) Describe an alternative procedure to test the hypothesis in part (d). 5. It is common to observe that new releases of compact discs are priced lower than older CDs. One way
to represent the above relationship is: Price = a. + agree + a. where Price retail price of CD, Age = age of the recording. We have information on variables Price and Age of CDs sold through interact and through traditional ways. For 46 observations of ODs sold through traditional ways, we have the following estimation
result: P323 = 15.076 + 0.493 x Ago
(0.221) (0.098) where T = 46, R2 = 0.367, and residual sum of squares (RSSNI) = 41.71. For 72 observations of CDs
sold through internet, we have the following estimation result: FEE = 13.343 + 0.130 x Age
(0.284) (0.125) where T = 72, R2 = 0.015, and standard error of regression (3) = 1.574. (a) (4%) Calculate the residual sum of squares for the second sample (R551). (b) (6%) We are interested to know whether there is any systematic djﬁ‘erence in the pricing rela
tionship of CDs sold through internet versus traditional ways. If you are allowed to run only one
more regression, describe how you would do. State the null hypothesis and give the test statistic. (c) (5%) The hypothesis in part (b) can also be tested by a dummy variable approach. Describe how
you would carry out such a test. (d) (4%) Are the test procedures suggested in parts (b) and (c) equivalent? Ehcplain. (e) (6%) Test, at a 10% signiﬁcance level, the null hypothesis of equal error variances for the two
samples against the alternative that the error variances are diﬂ'erent. In light of your result,
comment on the method you suggested in part (b). ***** END OF PAPER ##3##“le Introductory Econometrics 20012002 1. Probability and statistics 5 if X is a discrete random variable that can take the values m1, ..., so, with probability density f (ml),
f (m2), ..., f (33“) respectively, then the expected value of X: EUQ = 2113f (ﬂ?) = 2155f ($21) = 931f($1) +$2f($2) + +$nf($a) 11’! i=1 9. if X is a discrete random variable and g (X) is a function of it, then E[9(X)l=29(ﬂ=)f(w) a. variance = expected value of the squared deviations around the expected value of X nor (X) = E [X — E (.74012 y if X is a normally distributed random variable with mean a and variance a”, symbolized as X m
N (a, :72), then its p.d.f. is expressed mathematically as: 1  (a: — :02] f(m)=W9xP[ 2J2 for—oo<s:<co 2. Simple linear regression model
9 = 131 + 132:1: + e AOLS AOLS I. the least squares estimators of £31 and ﬁg, denoted by {3, and ,62 (or simply, b1 and b2), are given
by:
T T T
b T (2:31 55:31:) — (2&1 mt) (2&1 gt) 231:1 (mt — E) (3;; ~— ":9")
2 = W'_‘ "—2 = T  2
T (2:1 11:?) * (2;, 33;) 21:1 (3t — 35')
_ 23:1 yt 23:1 ﬂit __ .. _,.
bl * T 52 T — y ‘ I32:17 I if the regression model assumptions (SLRI) to (SLR4) are correct, then the variances and covariance of .51 and be are:
W) __. as [__.1_.._] 23:1 (“it ‘* —)2 5 an unbiased estimator of 02 is o. if we make the additional assumption that the random errors at are normally distributed, then in and
b2 are normal: bj~N( j,ver(bj)), j: 1,2 c the random variable t
bi " ﬁj 36’ (ha) t = N It(tr—52);; where as (bi) is the standard error of hi I a (1 — or) x 100% conﬁdence interval for 55, is
[bj — tcse (bi) ,le + $536 lbs“ I: the least squares predictor of go is
so = b1 + 5255's
a the variance of the forecast error is
1 ($0 — E 2 wen—yo) =03 1+——+—~s—~——~—:——§
T Zt=1(mtm) : a (1 — or) x 100% conﬁdence interval (or prediction interval) for ya is
[is '— taBB (ifs  so) as?) + We (ﬁe  '90)] 3. Multiple linear regression model a coefﬁcient of determination (R2): where RSS is residual (or unexplained) sum of squares, ESS is explained sum of squares and TSS(=
ESS  RS S) is total sum of squares u an unbiased estimator of :72 is T 2
32 = Eta1 (ﬁt) —_...__I—n T — K
where K is the number of parameters being estimated I the random variable t bj " X35 .
ts 360,3.) N t(T——K): J — 1121*“:K
e the Futest statistic is
F __ (R353 — RSSU) /J N F
“ RSSU/ (T — K) ”'T‘K) where J is the number of restrictions, RSSU and R853 are the unrestricted residual sum of squared
deviation and restricted residual sum of squared deviation respectively o Goldfeld and Quandt test statistic: under the normality assumption and Ho, the ratio of the two
variance estimators follows an Fdistrihution with (T1 —— K1,,T2  K 2) degrees of freedom A 2
1
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DF a=.05 a=.025 a=.005
1 6.314 12.706 63.657
2 2.920 4.303 9.925
3 2.353 3.132 5.341
4 2.132 2.776 4.604
5 2.015 2.571 4.032
6 1.943 2.447 3.707
7 1.395 2.365 3.499
3 1.360 2.306 3.355
9 1.833 2.262 3.250
10 1.312 2.223 3.169
11 1.796 2.201 3.106
12 1.732 2.179 3.055
13 1.771 2.160 3.012
14 1.761 2.145 2.977
15 1.753 2.131 2.947
16 1.746 2.120 2.921
17 1.740 2.110 2.393
13 1.734 2.101 2.393
19 1.729 2.093 2.361
20 1.725 2.036, 2.345
21 1.721 2.030 2.331
22 1.717 2.074 2.319
23 1.714 2.069 2.307
24 1.711 2.064 2.797
25 1.703 2.060 2.737
26 1.706 2.056 2.779
27 1.703 2.052 2.771
23 1.701 2.043 2.763
.29 1.699 2.045 2.756
30 1.697 2.042 2.750
40 1.634 2.021 2.704
50 1676 2.009 2.673
60 1.671 2.000 2.660
70 1.667 1.994 2.643
30 16‘64 1.990 2.639
90 1:662 1.937 2.632
100 1.660 1.934 2.626
110 1.659 1.932 2.621
120 1.653 1.980 2.617
m 1.645 1.960 2.576 Source: This table. was generated using the 313.30 function TINV ...
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