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Unformatted text preview: UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF ECONOMICS
ECON 2206/ECON 3290(ARTS)
INTRODUCTORY ECONOMETRICS
FINAL EXAMINATION SESSION 1, 2005 INSTRUCTIONS 1. TIME ALLOWED  2 Hours. TOTAL NUMBER OF QUESTIONS  5. ANSWER ALL QUESTIONS. TOTAL MARKS FOR THE EXAMINATION  70. THE NUMBER OF MARKS AWARDED TO EACH QUESTION IS INDICATED. THE MARKS AWARDED TO EACH PART OF A QUESTION ARE
INDICATED. CANDIDATES MAY BRING THEIR OWN CALCULATORS TO THE EXAM. STATISTICAL TABLES AND FORMULAS ARE PROVIDED AT THE END OF
THE EXAM. ALL ANSWERS MUST BE WRITTEN IN PEN. PENCILS MAY BE USED ONLY
FOR DRAWING, SKETCHING OR GRAPHICAL WORK. ANSWER ALL FIVE QUESTIONS REMINDER: When performing statistical tests, always state the null and alternative
hypotheses, the test statistic and its distribution under the null hypothesis, the level
of signiﬁcance and the conclusion of the test. Question 1 (15 marks) The following is a model for the weekly revenue of individual restaurants in Hungry Jims
a chain of hamburger restaurants; tr=ﬂ0 +,[31price+ﬂ2adv+,[33(adv)2 +u (1,1) where tr is weekly total revenue (thousands of dollars) for a restaurant, price is the price
of a hamburger (in dollars) charged by a restaurant and adv is weekly advertising
expenditure (thousands of dollars) by a restaurant. (i) Explain what it means for the OLS estimator of .31 in model (1.1) to be unbiased. What assumptions about the error term u are required for OLS to be unbiased?
(2 marks) (ii) Explain what it means for the OLS estimator of ,8, in model (1.1) to be consistent. What assumptions about the error term u are required for OLS to be consistent?
(2 marks) Using data from a sample of 78 restaurants the following estimates are obtained using
OLS. The numbers in (.) brackets are OLS standard errors. t; = 110.46 10.198price+3.362adv—0.0268(adv)2 (1.2)
(93.74) (2.58) (0.42) (0.01) n = 78, SSR=2,592.301 (iii) What is the interpretation of the estimated intercept coefﬁcient in (1.2)? Does this
make economic sense? (I mark)
(iv) Is the intercept coefﬁcient statistically signiﬁcant at the 10 % level of signiﬁcance?
(2 marks)
(v) Interpret the estimated coefﬁcient on price. .
(1 mark) (vi) Use the results given by (1.2) to test the hypothesis that [31 = —5 against the
alternative that A < —5. (Use a 1% signiﬁcance level.)
(2 marks) (vii) Using the estimates given by (1.2) construct a 99 percent conﬁdence interval for ,8] .
(1 mark) (viii) According to model (1.2), after what point does additional spending on advertising
reduce a restaurant’s total revenue? (2 marks) (ix) Given the following information, test the hypothesis that advertising has no effect on
total revenue. t? =121.76—15.234price (1.3)
(92.34) (2.25) n = 78, SSR=3,067.331
(2 marks) Question 2 (15 marks) Consider the following population regression model for the salary paid to chief executive
ofﬁcers (CEOs), log(salary) = ﬂo + ,8x log(sales) + ,6er3 + ﬂ3ceoten + ,B4ceoterz2 + u (2.1) where salary is CEO salary ($ per annum), sales is ﬁrm sales ($_per annum), ros is return
on the ﬁnn’s stock (percent per annum) and ceoten is the tenure of the CEO (in years). (i) Explain how you could test the joint hypothesis that A =1 and ,32 = 0. (Be sure to write down the restricted model that you would estimate.)
(3 marks) The following estimates of (2.1) were obtained by OLS log(saiary) = 4.32 + 0.28010g(sales) + 0.0174ros + 0.053ce0ten — 0.024ceoten2
(2.01) (0.041) (0.008) (0.013) (0.010) n=120, R2 =0.274, R2 =0.238 (ii) What are the estimated marginal effects of each of the explanatory variables on a
CEO’s salary? (3 marks) (iii) Test the hypothesis that all of the slope coefﬁcients in model (2.1) are jointly
insigniﬁcant (i.e. all the slope coefﬁcients are equal to zero) using a 5% signiﬁcance level.
(2 marks) (iv) What is measured by theR2 in the above regression model? How does the 1—2— 2differ from the R2 ?
(2 marks) (V) Suppose you wish to test for heteroskedasticity in model (2.1). Write down the null
hypothesis that you would test and describe one formal test that you could use to test this
hypothesis. (2 marks) (vi) Suppose that you reject the null hypothesis of homoskedasticity. Explain why this is
a problem and indicate how you would respond. (3 marks) Question 3 (15 marks) The following model is used to explain house prices in a particular suburb of Sydney, price = ,60 + ﬂlbdrms + ﬂzlotsize + [J’sarea + [3417001 + u (3.1) where price is house price ($0005), bdrms is number of bedrooms, lotsize is the size of the
allotment (in square metres) area is the ﬂoor area of the house (in square metres) and pool
is a dummy variable that takes the value 1 if the house has a swimming pool. (i) Write out the hypothesis that lotsize and area have an identical effect on price and
explain how you could test this hypothesis using a regression model. (3 marks)
The above model was estimated and the following results were obtained. prfce = —24.016 + 11.059bdrms + 0.0023lotsize + 0.122area +13.716pool (3.2)
(29.60) (3.52) (0.0006) (0.012) (6.24) n=88, R2 =0.6758, §2=0.6602, SSR=2,975,580 (ii) What is the estimated effect of having a swimming pool on the price of a house? Is this
effect statistically signiﬁcant at the 1% level of signiﬁcance?
(2 marks) (iii) What is the estimated effect on the price of a house of one additional bedroom?
(I mark) (iv) Suppose you wished to test the hypothesis that the value of an additional bedroom was
the same for all houses regardless of whether they have a swimming pool. Explain what
model you would estimate to test this hypothesis. (2 marks) (v ) If the dependent variable price in (3.2) were measured in dollars instead of thousands
of dollars how would the results reported above change?
(2 marks) (vi) Suppose we wish to test for any differences in the regression equation, price = ﬂo + ﬂlbdrms + ﬂzlotsize + ﬂ3area + u for houses with pools and houses without pools. Write down the null hypothesis that is to
be tested and describe two ways in which you could perform the test. (5 marks)
Question 4 (15 marks) The following model for the growth rate of prices was estimated using quarterly data for
the period 1960:1 to l999:4 gpﬁce, = —o.00093+ 0.119gw, +0.097gw,_1 +0.040gw,_2 +0.038gw,_3 —0.081gw,_4 (4.1)
(0.0057) (0.052) (0.039) (0.038) (0.028) (0.030) n = 160, R2 = 0.317' where gprz'ce = A log( price) , price is the consumer price index , gw = A log(wage) and
wage is an index of hourly earnings. OLS standard errors are reported in parentheses (.). (i) What is the estimated long run propensity (or multiplier) of gw on gprz'ce? What is the
interpretation of the long run propensity in this case?
(2 marks)
(ii) What regression would you run to obtain the standard error of the long run propensity
(3 marks)
(iii) If the error term in the population model underlying (4.1) displays serial correlation
what are the consequences for the OLS estimators of the coefﬁcients and standard errors? (2 marks) (iv) Suppose that the error term for the population model underlying (4.1) is believed to
follow an AR(1) process: u, = plut—l +82 Explain in detail how to test whether the error term follows an AR(1) process. Explain
how your answer would change if (4.1) contained a lagged dependent variable?
(3 marks) (v) Suppose that you ﬁnd evidence of ﬁrst—order serial correlation in the OLS residuals of
equation (4.1). Explain how you would proceed. (5 marks) Question 5 (10 marks) This question requires you to use the regression output from SHAZAM on pages
7 and 8. The following is a simple model for interest rates on home loans,
hloan, = .30 +ﬂ1bbill, +11, (5.1) where hloan is the standard housing loan rate for banks (percent per annum) bbz‘ll is the
interest rate on 90 day bank accepted bills (percent per annum). Available data on the two
variables consists of 281 monthly observations from January 1980 to May 2003. (i) What problems for estimation can arise if the two interest rate series in (5.1) contain
unit roots?
(2 marks) Use the SHAZAM output on pages 7 and 8 to answer following questions. (ii) Perform a test on the two series hloan and bbill to see if they have unit roots.
(2 marks) (iii) Test for the presence of a cointegrating relationship between the two interest rates.
What do you conclude? (3 marks) (iv) In light of your ﬁndings in part (iii) what is the most appropriate dynamic model for
the housing loan interest rate? Brieﬂy explain your answer.
(3 marks) The following SHAZAM output is relevant for Question 5. _sample 1 281
_read bbill hloan _sample 1 281 _genr bbilll = lag(bbill,l) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
_genr dbbill = bbill  bbilll ~genr dbbilll = lag(dbbill,l) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
_genr hloanl = lag(hloan,l) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
_genr dhloan = hloan — hloanl _genr dhloanl = lag(dhloan,l) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
_sample 3 281 _ols dbbill bbilll dbbilll REQUIRED MEMORY IS PAR: 32 CURRENT PAR: 2000
OLS ESTIMATION
279 OBSERVATIONS DEPENDENT VARIABLE: DBBILL
...NOTE..SAMPLE RANGE SET TO: 3, 281
RSQUARE = 0.0230 RSQUARE ADJUSTED = 0.0159
VARIANCE OF THE ESTIMATE—SIGMA**2 : 0.80484E01
STANDARD ERROR OF THE ESTIMATE—SIGMA = 0.28370 SUM OF SQUARED ERRORS—SSE: 22.214
MEAN OF DEPENDENT VARIABLE = 0.92473E—02
LOG OF THE LIKELIHOOD FUNCTION = 42.8782 3 VARIABLES AND 281 OBSERVATIONS STARTING AT OBS VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 276 DF PVALUE CORR. COEFFICIENT AT MEANS
BBILLl 0.21965E02 0.5289E02 —0.4153 0.678—0.025 0.0248 2.5941
DBBILLl 0.15135 0.5967E01 2.537 0.012 0.151 0.1514 0.1514
CONSTANT 0.16140E01 0.6025E01 0.2679 0.789 0.016 0.0000 1.7454 i_ols dhloan hloanl dhloanl REQUIRED MEMORY IS PAR: 32 CURRENT PAR: 2000
OLS ESTIMATION
279 OBSERVATIONS DEPENDENT VARIABLE: DHLOAN
...NOTE..SAMPLE RANGE SET TO: 3, 281 RSQUARE = 0.0789 RSQUARE ADJUSTED = 0.0722
VARIANCE OF THE ESTIMATESIGMA**2 : 0.47813
STANDARD ERROR OF THE ESTIMATESIGMA = 0.69147
SUM OF SQUARED ERRORSSSE: 131.96
MEAN OF DEPENDENT VARIABLE = —0.1B925E01
LOG OF THE LIKELIHOOD FUNCTION = —291.444 VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 276 DF PVALUE CORR. COEFFICIENT AT MEANS
HLOANl 0.13294E'01 0.887BE02 1.497 0.1350.090 0.0869 7.0612
DHLOANl 0.27520 0.5801EOl 4.744 0.000 0.275 0.2752 0.2731
CONSTANT 0.11988 0.9847E01 1.217 0.224 0.073 0.0000 6.3344 The following SHAZAM output is relevant for Question 5. 1‘sample 1 281 yuols hloan bbill / resid=uhat REQUIRED MEMORY IS PAR: 32 CURRENT PAR: 2000
OLS ESTIMATION
281 OBSERVATIONS DEPENDENT VARIABLE: HLOAN
...NOTE..SAMPLE RANGE SET TO: 1, 281
RSQUARE = 0.7451 RSQUARE ADJUSTED = 0.7442
VARIANCE OF THE ESTIMATESIGMA**2 = 5.6144
STANDARD ERROR OF THE ESTIMATESIGMA = 2.3695
SUM OF SQUARED ERRORSSSE: 1566.4
MEAN OF DEPENDENT VARIABLE = 10.033 LOG OF THE LIKELIHOOD FUNCTION = —640.127 VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 279 DF P~VALUE CORR. COEFFICIENT AT MEANS BBILL 1.2531 0.4388E01 28.56 0.000 0.863 0.8632 1.3613 CONSTANT —3.6245 0.4987 7.268 0.0000.399 0.0000 ~0.3613 _genr uhacl = lag(uhat,1) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
_genr duhat = uhat  uhatl \_genr duhatl = lag(duhat,l) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
\_sample 3 281 _ols duhat uhatl duhatl REQUIRED MEMORY IS PAR: 41 CURRENT PAR: 2000
OLS ESTIMATION ' 279 OBSERVATIONS DEPENDENT VARIABLE: DUHAT
...NOTE..SAMPLE RANGE SET TO: 3, 281 RSQUARE = 0.0642 RSQUARE ADJUSTED = 0.0574
VARIANCE OF THE ESTIMATESIGMA**2 = 0.55475
STANDARD ERROR OF THE ESTIMATESIGMA = 0.74482
SUM OF SQUARED ERRORSSSE= 153.11
MEAN OF DEPENDENT VARIABLE = —0.73370E02
LOG OF THE LIKELIHOOD FUNCTION = 312.l78 VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 276 DF PVALUE CORR. COEFFICIENT AT MEANS UHATl —0.63821E01 0.1909E—01 —3.344 0.0010.197 —0.1972 0.0752 DUHATl 0.19307 0.5896E01 3.275 0.001 0.193 0.1931 0.1893 CONSTANT 0.64996E02 0.4459E—01 0.449 0.650 0.000 0.0000 0.0000 FORMULAS _ (R:,—R3)/q
<1—R3,>/(n—k1) F _ (SSR, — 55R", ) / q
53R“, /(n — k — 1) Signiﬁcance 1%
Level Critical Value As m . totic Critical Values for Unit Root ttest : Linear Time Trend Signiﬁcance 1% 2.5% 5% 10%
Level Critical Value Signiﬁcance
Level Critical Value Signiﬁcance 1%
Level Critical Value 4.03 10 Appendix G Statistical Tables 817 TABLE 6.2 Critical Values of the t Distribution Signiﬁcance Level lTailed:
2Tailed: . 1 . .
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 ‘ 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
D 8 1.397 1.860 2.306 2.896 3.355
e 9 1.383 1.833 2.262 2 821 3.250
g 10 1.372 1.812 2 228 2 764 3.169
2 11 1.363 1.796 2.201 2 718 3.106
12 1.356 1.782 2.179 2 681 3.055
e 13 1.350 1.771 2.160 2 650 3.012
S 14 1.345 1.761 2.145 2 624 2.977 5—
‘JI Examples: The 1% criﬁcal value for a onetailed test with 25 dfis 2.485. The 5% critical for a twotailed test
with large (> 120) dfis 1.96.
Source: This table was generated using the 818121 1unclion 1mm Ch! 4' Appendix G Statistical Tables 819 TABLE 6.313
5% Critical Values of the F Distribution Numerator Degrees of Freedom IIIIIIIIIIII w
4 96 4. 10 3 71 3. 48 3. 33 3. 22 3.07 3.02 2.98 11 4.84 3.98 3.59 3.36 3.20 3.09 . 2.95 2.90 2.85 1e) 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75
n 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67
o 14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.65 2.60
"in 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 . 2.59 2.54
n 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 . 2.54 2.49
f 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 _. 2.49 2.45
0 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 . 2.46 2.41
r 19 4.38 3.52 3.13 2.90 2.74 2.63 2.42 2.38
1) 20 4.35 3.49 2.35
e 21 4.32 3.47 2.32
g 22 4.30 3.44 2.30
e 23 4.28 3.42 2.27
e 24 4.26 3.40 2.25
S 25 4.24 3.39 2.24
o 26 4.23 3.37 2.22
f 27 4.21 3.35 2.20
F 28 4.20 3.34 2.19
r 29 4.18 3.33 2.18
E 30 4.17 3.32 2.16
d 40 4.08 3.23 2.08
:1). 60 4.00 3.15 1.99
90 3.95 3.10 1.94 120 3.92 3.07 . . . , . . . . 1.91 Example: The 5% critical value for numerator df = 4 and large denominator df (no) is 2.37.
Source: This table was generated using the Stata® function invfprob. ...
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 Three '11
 berrick
 Econometrics

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