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ECON6300/7320 Econometrics
Creation date：2024-06-12 17:16:25
Belonging category：经济Economics
Econometrics

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ECON6300/7320

Course Information
▶ The course will focus on estimation and inference methods
that are widely used in applied microeconomics.
▶ The course has a topics-based structure, and theory and
applications are closely integrated.
▶ Whenever you learn a method theoretically (lecture), there
will be an R exercise with data for the method (practical)
▶ The course will provide econometric skills that could be
used in quantitative research at the Graduate level.
3 / 54
Course Meetings: Lectures and Tutorials
▶ Lectures
▶ Go through theory, illustrating examples, perhaps some
proofs, and Q& A
▶ Tutorials
▶ See the Course Timetable posted on Blackboard for the
time and location.
▶ Focus is on practical implementation in R
▶ Start from week 2 (Introduction to R programming)
4 / 54
Assessment
▶ Two assignments + Final problem set.
▶ Analytical and empirical problems.
▶ Assignments 1-2 each cover 1/3 topics of the course (30%
each).
▶ Final problem set is comprehensive and covers the entire
course (40%).
5 / 54
Learning Resources
▶ Recommended Textbooks:
▶ Hansen, E.B. (2022). Econometrics. Princeton University
Press. (H)
▶ Cameron, A.C., and P.K. Trivedi (2005).
Microeconometrics: Methods and Applications. Cambridge
University Press. (CT)
▶ There are many other useful texts, e.g., Greene,
Wooldridge, etc. See ECP.
▶ As the semester progresses, lecture notes, slides,
datasets, problem sets, etc. will be provided. It is however
strongly encouraged that you read the relevant part of the
textbook and other references.
6 / 54
This week: math review
▶ You are EXPECTED to have a basic grasp of
mathematics/statistics. If most of the concepts we cover today are
unfamiliar to you, this course may be unsuitable (unless it is
compulsory (!!!)).
▶ Random Variables, Expectation, Variance, Covariance
▶ Point Estimation, Hypothesis Testing, Interval Estimation
▶ Linear Algebra: vector and matrix and their operations
7 / 54
Math Review random variable
▶ Informally, a Random Variable (RV) takes on numerical values
determined by an experiment.
▶ Example: Consider an experiment in which a fair coin is tossed. Then,
the possible outcomes are Head and Tail, i.e., {H,T}. Then, we define
a random variable X as follows;
X =
{
0 if the outcome is T
1 if the outcome is H
The X could be either 0 or 1 whenever the coin is tossed.
Each time, the realisation of X would be different.
The uncertainty can be summarised as Pr(X = 0) = 0.5.
8 / 54
Math Review probability functions
▶ The uncertainty of a random variable, say X , is represented by its
cumulative distribution function (CDF), defined as
FX (x) := Pr(X ≤ x)
▶ When FX (x) is a continuous function, X is said to be a continuous
random variable. For a continuous random variable, the probability
density function (PDF), denoted by fX (x), is a function that satisfies
Pr(a < X < b)︸ ︷︷ ︸
=FX (b)−FX (a)
=
∫ b
a
fX (t)dt
▶ If FX (x) is differentiable, we have
fX (x) =
d
dx
FX (x)
▶ If FX (x) is a step function, X is discrete, which we do not cover today.
9 / 54
Math Review expectation
▶ A function of a random variable is also a random variable. For example,
exp(X ), log(X ), etc. More generally, g(X ) with some function g(·).
▶ Suppose X have the PDF fX (x). The expectation of g(X ) is
E [g(X )] :=
∫ ∞
−∞
g(t) · fX (t)dt
which represents the central tendency of the random variable, g(X ).
▶ As a special case with g(X ) = X ,
E [X ] :=
∫ ∞
−∞
t · fX (t)dt
which is the expectation of X , or the expected value or the mean
▶ Consider another case with g(X ) = (X − E [X ])2.
V (X ) := E [(X − E [X ])2]
which is called the variance that represents the variability of X . Note
that
√
V (X ) is the standard deviation of X .
10 / 54
Math Review expectation
▶ E [X ] and V (X ) are the population parameters, or theoretical moments.
You need to know the distribution to compute E [X ] and V (X ).
▶ E [X ] must be distinguished from the sample mean (average). For
example, when you observe some numbers x1, . . . , xn, its average is
xn :=
1
n
n∑
i=1
xi
which is NOT the expectation of the distribution
▶ Similarly, you must distinguish V (X ) and the sample variance
1
n
n∑
i=1
(xi − xn)2
11 / 54
Math Review expectation
1. E [c] = c, for any constant c
2. E [aX + b] = aE [X ] + b for any constants a and b
3. a1, . . . , an are constants, and X1, . . . ,Xn are RV’s. Then,
E
[
n∑
i=1
aiXi
]
=
∑
i=1
aiE [Xi ].
As a special case, we have
E
[
n∑
i=1
Xi
]
=
∑
i=1
E [Xi ].
4. V (c) = 0, for any constant c.
5. V (aX + b) = a2Var(X ) for any constants a and c.
12 / 54
Math Review joint distribution
▶ Suppose X and Y are jointly distributed with the joint PDF fXY (x , y).
▶ The marginal PDF of X can be obtained by integrating Y out,
fX (x) =
∫ ∞
−∞
fXY (x , y)dy
▶ X and Y are independent if and only if for all x and y
fXY (x , y) = fX (x)fY (y)
▶ The expectation of g(X ,Y ) for some function g(X ,Y ) is
E [g(X ,Y )] =
∫ ∞
−∞
∫ ∞
−∞
g(x , y)fXY (x , y)dxdy
13 / 54
Math Review covariance
▶ If g(X ,Y ) = (X − E [X ])(Y − E [Y ]), then we have the covariance,
C(X ,Y ) = E [(X − E [X ])(Y − E [Y ])]
▶ If C(X ,Y ) > 0, the X and Y move in the same direction.
If C(X ,Y ) < 0, they move in the opposite direction.
▶ |C(X ,Y )| ≤√V (X )√V (Y ). So, the correlation coefficient
ρXY :=
C(X ,Y )√
V (X )
√
V (Y )
is always between -1 and 1.
▶ If X and Y are independent, C(X ,Y ) = 0. But, the converse is not
generally true
14 / 54
Math Review conditional distribution
▶ Suppose X and Y are jointly distributed with the joint PDF fXY (x , y).
Then, the conditional PDF of Y given X = x is
fY |X (y |x) := fXY (x , y)fX (x)
which summarises the distribution of Y when X takes a value x .
▶ If X and Y are independent, fY |X (y |x) = fY (y) and fX |Y (x |y) = fX (x)
15 / 54
Math Review conditional expectation
▶ The conditional expectation of Y given X = x is
E [Y |X = x ] =
∫ ∞
−∞
yfY |X (y |x)dy
▶ The conditional variance of Y given X = x is
V (Y |X = x) = E
[
(Y − E [Y |X = x ])2
∣∣∣X = x]
▶ E [Y |X = x ] and V (Y |X = x) are functions of x .
▶ When x is not specified, E [Y |X ] and V (Y |X ) are random.
Case category
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FN1024 MATH0094 MANG 6554 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 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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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