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ECONM1022 ECONOMETRICS

创建日期：2024-04-24 16:07:20

所属分类：经济Economics

标签： ECONM1022

ECONOMETRICS

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ECONM1022
ECONOMETRICS
Time allowed: SEVEN DAYS
Answer ALL questions
Candidates must show the basis of all calculations.
Part A [25 marks]
Consider the linear regression model y = β0 + β1x + β2w + u, where y, x and w are
observed variables, u is an error term and β0, β1 and β2 are parameters. We denote
β = (β0, β1, β2)
′. We assume exogeneity of the regressors E(u|1, x, w) = 0. We have
an iid sample with n observations: {yi, xi, wi}ni=1. Regressing y on a constant, x and w
by OLS yields the estimate β̂ =
(
β̂0, β̂1, β̂2
)′
1. (a) Write down the first-order conditions characterizing the OLS estimator β̂.
[3 marks]
(b) Derive the formal expression of β̂ as a function of data matrices that you
need to specify. [3 marks]
(c) Show that β̂ is an unbiased estimator of β. [3 marks]
2. (a) Describe a test for the hypothesis β1 = β2 against β1 6= β2 [3 marks]
(b) Assume, for this question only, that the test does not reject β1 = β2 and
consider an OLS regression of y on a constant and the regressor x−w. How
would the coefficient of determination from this regression compare with the
one from the OLS regression of y on a constant, x and w? [2 marks]
(c) Consider the following test of the hypothesis β1 = β2. We regress y on
a constant x, w and x − w by OLS and then test whether the coefficients
of x and w are jointly equal to 0. Would you recommend this approach?
[2 marks]
3. (a) Show that the first-order conditions characterizing β̂ (derived in 1.a) can
be rewritten as a system of three equations involving only β̂0, β̂1, β̂2 and
sample variances and covariances of y, x and w. (Hint: recall that the
sample variance is the sample mean of the square minus the square of the
sample mean and that the empirical covariance is the sample mean of the
product minus the product of the sample means) [4 marks]
(b) Show that if the covariance of x and w were equal to 0 then regressing y on
a constant and x by OLS would yield a consistent estimator of β1. [5 marks]
Part B [25 marks]
Consider the linear regression model y = β0 + β1x + β2w + u, where y, x and w
are observed scalar variables, u is an error term and β0, β1 and β2 are parameters.
We denote β = (β0, β1, β2)
′. We assume that E(u|1, w) = 0 but for the moment
we do not know whether x is exogenous or not. We also assume that there is an
observed scalar variable z that is uncorrelated with the error term u. We have an iid
sample with n observations: {yi, xi, wi, zi}ni=1. Lastly, we assume that the matrices
E[(1, w, x)′(1, w, x)] and E[(1, w, z)′(1, w, z)] are full column rank.
1. In this first section, we will use z (together with the constant and w) as instrumen-
tal variables.
(a) Write down the IV identifying assumptions. [3 marks]
(b) Describe the two steps of the 2SLS estimation of β and derive the expres-
sion of the estimator, denoted as β̂2sls [4 marks]
(c) Which validity condition can we test? How would this test work? [4 marks]
(d) The 2SLS estimation gives β̂2sls1 = 1.24 and V̂
(
β̂2sls1
)
= 0.05. Is β1 different
from 1 at the 5% significance level? [3 marks]
2. From now on, in this second section, we will assume that x is uncorrelated with
u and we will use x (together with the constant and w) as instrumental variables.
We will not use the variable z.
(a) Are the IV identifying assumptions verified? [2 marks]
(b) Write down the moment conditions following from the exogeneity of the three
instruments (the constant, w and x). [3 marks]
(c) Show that the IV estimator is then equal to the estimator obtained by re-
gressing y on a constant, x and w by OLS. [4 marks]
(d) Would you rather use this IV estimator (using x, w and the constant as
instruments) or the 2SLS estimator from the first section (derived in question
1.b and using z, w and the constant as instruments)? [2 marks]
Part C [25 marks]
We have an iid sample {Yit, X ′it}N,Ti=1,t=2000 of N = 417 wines bottled from 1980 to 1995,
where Yit is the log price for wine i during the year t ∈ {2000, 2005, 2010, 2015}, and
Xit = (aromait, f lavorit, intenit)
′
is a vector of covariates measuring aromatic -aromait-, gustatory -flavorit-, and color
intensity -intenit- characteristics. We posit the linear specification:
Yit = X
′
itβ + ηi + ξit,
where ηi is the terroir of wine i, and ξit is an idiosyncratic disturbance term.
1. Derive the First-Difference (FD) Estimator βˆfd. [5 marks]
2. Below is a tentative proof of consistency of the FD estimator. It contains mistakes.
Identify and correct those mistakes. [6 marks]
The starting point is
βˆfd =
( T∑
t=2
N∑
i=1
∆Xit∆X
′
it
)−1 T∑
t=2
N∑
i=1
∆Xit∆Yit.
Replacing ∆Yit = ∆X ′itβ + ∆ξit,
βˆfd − β =
( T∑
t=2
N∑
i=1
∆Xit∆X
′
it
)−1 T∑
t=2
N∑
i=1
∆X ′it∆ξit.
Multiplying by N/N ,
βˆfd − β =
( T∑
t=2
N−1
N∑
i=1
∆Xit∆X
′
it
)−1 T∑
t=2
N∑
i=1
N−1∆X ′it∆ξit.
Since the data are iid, on the proviso that
∑T
t=2 E(∆X ′it∆Xit) is invertible, the Law of
Large Numbers and the Continuous Mapping Theorem imply( T∑
t=2
N−1
N∑
i=1
∆Xit∆X
′
it
)−1
→
P
T∑
t=2
E(∆X ′it∆Xit). (A)
Since the data are iid, the central limit theorem and the restriction E(ξit|Xit) = 0 imply
T∑
t=2
N∑
i=1
N−1∆X ′it∆ξit = E(∆Xit∆ξit) = 0. (B)
Combining (A) and (B) using the Slutzky Lemma, one has that βˆfd converges in prob-
ability to β.
3. What test would you use to differentiate between the random-effect and the fixed-
effect models? Write down the null and alternative hypotheses, the test statistic and its
asymptotic distribution under the null, and the decision rule. [6 marks]
4. Abraham, a wine expert, argues that aromatic and gustatory characteristics of a
wine are both related to its price. Write down a test based on the FD estimator (null
and alternative hypotheses, test statistic and its asymptotic distribution under the null,
critical values, and the decision rule) that could be employed to confirm Abraham’s ar-
gument. [4 marks]
5. Abraham also argues that the information on the wine bottle label is related with
the average price of the wine. Could we use an Asymptotic T-test based on the FD
estimator to confirm this argument? Justify. [4 marks]

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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