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ECON61001: Econometric Methods

创建日期：2024-04-24 16:02:07

所属分类：经济Economics

标签： ECON61001

Econometric Methods

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ECON61001: Econometric Methods
Instructions:
• You must answer all five questions in Section A and two out of the four ques-
tions in Section B. If you answer more questions than are required and do not
indicate which answers should be ignored, we will mark the requisite number of
answers in the order in which they appear in your answer submission: answers
beyond that number will not be considered.
• Your answers could be typed or hand-written (and scanned to a single pdf file
that can be submitted) or a combination of a typed answer with included images
of algebra or figures.
• Where relevant, questions include word limits. These are limits, not targets. Ex-
cellent answers can be shorter than the word limit. If you go beyond the word
limit the additional text will be ignored. Where a question includes a word limit
you HAVE to include a word count for your answer (excluding formulae).
• Candidates are advised that the examiners attach considerable importance to
the clarity with which answers are expressed.
• You must correctly enter your registration number and the course code on
your answer.

1. Suppose a researcher is interested in the following linear regression model
yi = x
′
iβ0 + ui, i = 1, 2, . . . N,
where xi = (1, x2,i, x3,i, x4,i)
′ and β0 = (β0,1, β0,2, β0,3, β0,4)
′ . Given the context, the
researcher is able to assume that {ui, x
′
i}
N
i=1 form a sequence of independently
and identically distributed random vectors with E[xix
′
i] = Q, a finite, positive
definite matrix of constants, E[ui|xi] = 0 and V ar[ui|xi] = σ
2
0 . Therefore, she
estimates the model via Ordinary Least Squares (OLS), obtaining the following
estimated equation
yˆi = 1.0213
(0.1416)
− 0.0020
(0.0005)
x2,i − 0.0208
(0.0080)
x3,i + 0.0095
(0.0088)
x4,i,
where the number in parenthesis is the conventional OLS standard error for the
coefficient in question. The OLS estimator of σ20 in this model is σˆ
2
N = 0.0081.
Given these results, the researcher concludes that β0,4 = 0 and so decides to
estimate the model via OLS with x4,i excluded, obtaining the following estimated
equation
yˆi = 1.1485 − 0.0020x2,i − 0.0211x3,i. (1)
Given that the sample size is N = 108 in both estimations, calculate the OLS
estimator of the error variance σ20 from the estimation in (1). Be sure to care-
fully explain your calculations. Hint: consider the F-statistic for testing β0,4 =
0. [8 marks]
2. Suppose it is desired to predict zt using zˆt = w
′
tγ where wt is a vector of ob-
servable variables and γ is a vector of constants that needs to be specified.
The choice of γ associated with the linear projection of zt on wt is γ0 where
E[(zt − w
′
tγ0)wt] = 0.
(a) What optimality property does zˆot = w
′
tγ0 possess? (Word limit: 50) [1mark]
(b) Now consider the regression model zt = w
′
tγ0 + vt. Is wt contempora-
neously exogenous or strictly exogenous for estimation of γ0 in this model?
Justify your answer. (Word limit: 150) [4 marks]
(c) Suppose the model in part (b) is dynamically complete. What optimal-
ity property does zˆot possess? Briefly justify your answer. (Word limit:
150) [3 marks]
Continued over
1
ECON61001
SECTION A continued
3.(a) Let A be n × n nonsingular symmetric matrix. Show that if A is positive definite
then A−1 is positive definite. Hint: AA−1 = In. [4 marks]
3.(b) Consider the classical linear regression model
y = Xβ0 + u, (2)
whereX is the T×k observable data matrix that is fixed in repeated samples with
rank(X) = k, and u is a T × 1 vector with E[u] = 0 and V ar[u] = σ20IT where σ
2
0
is an unknown positive finite constant. Let βˆT be the OLS estimator of β0 based
on (2) and βˆR,T be the Restricted Least Squares (RLS) estimator of β0 based on
(2) subject to the restrictions Rβ0 = r where R is a nr × k matrix of specified
constants with rank(R) = nr and r is a specified nr × 1 vector of constants.
Assuming the restrictions are correct, prove that βˆR,T is at least as efficient as
βˆT . Hint: you may quote the formula for the variance-covariance matrix of the
OLS and RLS estimators without proof; you may also take advantage of the
stated result in part (a). [4 marks]
4. Consider the linear regression model
y = Xβ0 + u,
where y and u are T × 1 vectors, X is T × k matrix, and β0 is the k × 1 vector
of unknown regression coefficients. Assume that X is fixed in repeated sam-
ples with rank(X) = k, and u ∼ N ( 0, σ20IT ) where σ
2
0 is an unknown positive
constant. Let θˆT denote the maximum likelihood estimator of the unknown pa-
rameter vector θ0 = (β
′
0, σ
2
0)
′. Derive the information matrix for this model. Hint:
you may state the form of the log likelihood function and score function for this
model without proof. [8 marks]
5. Consider the model
yi = x
′
iβ0 + ui, i = 1, 2, . . . , N,
where β0 is the k × 1 vector of unknown regression coefficients, {(x
′
i, ui)}
N
i=1
is a sequence of independently and identically distributed random vectors with
E[ui|xi] = 0, V ar[ui|xi] = σ
2
0, an unknown finite positive constant and E[xix
′
i] =
Q, a finite positive definite matrix of constants. Let σˆ2N be the OLS estimator of
σ20. Show that N
1/2(σˆ2N − σ
2
0)
d
→ N( 0, µ4 − σ
4
0) where µ4 = E[u
4
i ].
Hint: You may assume that: N−1
∑N
i=1 xix
′
i
p
→ Q; (ii) N−1/2
∑N
i=1 vi
d
→ N(0,Ω)
where vi = (u
2
i − σ
2
0, x
′
iui)
′, and Ω = V ar[vi] is a finite, positive definite
(k + 1) × (k + 1) matrix whose elements you must specify as needed to develop
your answer. [8 marks]
Continued over
2
ECON61001
SECTION B
6. Consider the regression model
yi = x
′
iβ0 + ui, i = 1, 2, . . . , N,
where β0 is the k × 1 vector of unknown regression coefficients, {(x
′
i, ui)}
N
i=1
is a sequence of independently and identically distributed random vectors with
E[ui|xi] = 0, V ar[ui|xi] = σ
2
0, an unknown finite positive constant and E[xix
′
i] =
Q, a finite positive definite matrix of constants. You may further assume that: (i)
N−1
∑N
i=1 xix
′
i
p
→ Q; (ii) N−1/2
∑N
i=1 xiui
d
→ N(0, σ20Q).
Let βˆR,N denote the RLS estimator based on the linear restrictionsRβ = r where
R is a nr × k matrix of pre-specified constants with rank equal to nr and r is a
nr × 1 vector of pre-specified constants, and let λˆN be the vector of Lagrange
Multipliers associated with this RLS estimation. Assuming Rβ0 = r, answer the
following questions.
(a) Show that N1/2(βˆR,N − β0)
d
→ N ( 0, VR ) where
VR = σ
2
0
(
Q−1 − Q−1R′(RQ−1R′)−1RQ−1
)
.
Hint: you may quote the formulae for βˆR,N and βˆN , the Ordinary Least
Squares estimator of β0, without proof. [10 marks]
(b) A colleague proposes testing H0 : Rβ0 = r versus H1 : Rβ0 6= r using the
decision rule of the form: reject H0 at the (approximate) 100α% significance
level if
λˆ′NMN λˆN > cnr(1− α)
where cnr(1− α) is the 100(1 − α)
th percentile of the χ2nr distribution. How-
ever, your colleague is unsure what the matrix MN should be in order that
this decision rule has the properties implied by the stated significance level.
Provide a suitable choice of MN , being sure to justify your choice carefully.
Hint: you may quote without proof: (i) the formulae for λˆN and βˆN ; (ii) that
both the OLS and RLS estimators of σ20 are consistent under the conditions
of the question. [20 marks]
Continued over
3
ECON61001
SECTION B continued
7.(a) Let {vt}
T
t=−3 be a weakly stationary time series process. Consider the following
statistic,
ρˆ4,T =
∑T
t=1 vtvt−4∑T
t=1 v
2
t
.
Let {εt}
∞
t=−∞ denote a sequence of independently and identically distributed
(i.i.d.) random variables with mean zero and variance σ2ε .
(i) Assume that vt = εt. Show that T
1/2ρˆ4,T
d
→ N(0, 1). [6 marks]
(ii) Assume that
vt = θ4vt−4 + εt,
where |θ4| < 1. What is the probability limit of ρˆ4,T as T → ∞? Be sure
to justify your answer carefully. Hint: vt has the following representation,
vt =
∑
∞
i=0 θ
i
4εt−4i. [9 marks]
7.(b) A researcher wishes to test the simple efficient-markets hypothesis in the foreign
exchange market. Let st = ln(St) and ft,n = ln[Ft,n], where St and Ft,n are the
levels of the spot exchange rate at time t and the n−period forward exchange
rate at time t. The simple efficient-markets hypothesis is that ft,n = E[st+n | It]
where It is the information set at time t which for the purposes of this question
can be taken to be It = {st, ft,n, st−1, ft−1,n, st−2, ft−2,n, . . .}. Using daily spot and
thirty-day forward exchange rate data for the US dollar UK pound exchange rate,
the researcher estimates the model,

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关键词一关键词二关键词三关键词四关键词五和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 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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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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