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ECON0125 functions Assessment

创建日期：2024-07-06 16:58:31

所属分类：数学mathematics

标签： ECON0125

functions Assessment

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Summative Assessment 2, ECON0125

March 5, 2024

This homework is the second of the two that will determine the coursework part of your grade. Your grade will depend only on the maximum mark out of the two homeworks. Each question is split up into multiple parts. Some parts have been deliberately made substantially more difficult than others, so do not worry if you are not able to complete all of them.

Question 1 (Properties of Penalized Estimators)

Let Θ be some set of functions. Suppose for all θ ∈ Θ, P en(θ) ≥ 0. For any λ ≥ 0 and c ≥ 0 let us define estimators and as below, where we assume that both optimization problems have unique global minima.

i. Consider some fixed λ ≥ 0. Define c(λ) = P en(). Show that (λ) = .

ii. Consider some fixed c ≥ 0. Suppose there exists a λ(c) so that P en((c)) = c. Show that = (c) .

iii. In light of your answers to parts i. and ii. discuss how you can apply results like Proposition 1.5 for estimators that incorporate penalization.

Now let satisfy the following for some sample loss function which need not be the sum of squared residuals:

iv. Show that P en() is weakly decreasing in λ.

v. Suppose that (θ) ≥ 0 for all θ ∈ Θ, Θ contains the zero function θ(x) = 0, ∀x (we will simply write θ = 0 in this case), and suppose P en(0) = 0. Show that for all λ > 0:

vi. Under the conditions of part v., what happens to P en() as λ → ∞?

Question 2 (IV Revisited)

Consider the LATE theorem on page 6 of the second set of lecture notes. Throughout this question, assume all of the conditions of that theorem hold.

i. Show that the following two equalities hold, where the random variable T is defined as on page 6.

P(T = a) = P(X = 1|Z = 0)

P(T = n) = P(X = 0|Z = 1)

ii. Show that the following equalities hold:

E[Y (1)|T = a] = E[Y |X = 1, Z = 0]

E[Y (0)|T = n] = E[Y |X = 0, Z = 1]

iii. Prove the following:

P(X = 1|Z = 1) = P(T = c) + P(T = a)

P(X = 0|Z = 0) = P(T = c) + P(T = n)

iv. Show that the following two equalities hold:

v. Using the results in parts i.-iv. solve for E[Y (0)|T = c] and E[Y (1)|T = c] in terms of directly observable quantities. (Note: you must show how your so-lution follows from the equalities in the previous parts of this question. Do not simply copy the formula in the lecture notes).

vi. In this setting, the ATE is in general not point-identified. Suppose Y is binary. Using the results above, identify scalars a > 0 and b < 1 so you can achieve upper and lower bounds on the ATE as follows:

a ≤ E[Y (1) − Y (0)] ≤ b

Try and find such an a and b so that b − a is as small as possible. When are your bounds more/less informative (i.e., when is b−a small/large)? Discuss.

Question 3 (Neyman Orthogonality)

Let Yi ∈ R be a measure of an individual i’s wealth and Xi ∈ a potentially high-dimensional vector of individual characteristics that includes a constant. Assume is non-singular. Let us consider a linear model for wealth with true parameters β0 ∈ . To be precise, we consider the model below:

Note that, because Xi includes a constant = 0. We are interested in asymmetry in the distribution of residual variation in wealth. More precisely, we wish to make inferences about the skewness of the residual in the model above. That is, the parameter θ0 given by . We intend to estimate the high-dimensional parameters β0 using penalized regression. We then plan to use these first-stage estimates to obtain an estimate of the scalar parameter θ0 in a second stage.

i. What is ‘Neyman orthogonality’? Why might it be relevant in the setting described above?

ii. Express β0 as the solution to a vector of moment conditions. That is, write down a score function ϕ so that E[ϕ(Yi , Xi ; β)] = 0 if and only if β = β0.

iii. Express θ0 as the solution to a simple moment condition that depends on β0. That is, write down a score function g so that E[g(Yi , Xi ; θ, β0)] = 0 if and only if θ = θ0.

iv. Is your moment condition in part iii. Neyman orthogonal with respect to β0? Does your answer depend on whether = 0?

v. Construct Neyman orthogonal moments condition for θ0. That is, find a score and a vector of additional nuisance parameters µ0 so that:

if and only if θ = θ0. And moreover:

Note that there is more than one possible correct answer here.

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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 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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 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COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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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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