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ECON3203 Econometric Theory and Methods

创建日期：2024-05-14 14:43:44

所属分类：经济Economics

标签： ECON3203

Econometric Theory and Methods

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ECON3203 Econometric Theory and Methods

Linear Regression
Table of content
1. Simple Linear Regression
2/22
Recommended reading
• Chapter 3, An Introduction to Statistical Learning with
Applications in R by James et al.: easy to read, sloppy
discussions sometimes, comes with R/Python code for
practice.
• Chapters 3, The Elements of Statistical Learning by Hastie et
al.: well-written, deep in theory, suitable for students with a
sound maths background.
3/22
Spending Data
The Spending Data contains information on 1000 customers in a
customer database for the company Direct Marketing (DM). The
data consist of
• AmountSpent ($): the amount spent by each customer in one
year on DM products.
• Salary ($)
• Catalogs: number of shopping catalogs sent to each
customer per year
• Children: number of children each customer has
• and many others
The company would like to build a model to explain/predict the
amount each customer spending on its products, based on the
other variables.
4/22
Spending Data
5/22
Spending Data
Questions might be asked
• Is there a relationship between the spending amount and the
number of catalogs, for example?
• How is the salary associate with the spending amount?
• How are the salary and the number of children interactively
associate with the spending amount?
• What is the best subset of the explanatory variables in terms
of predicting the spending amount?
• Given a particular customer, how much spending amount the
company would expect from he/she?
• ...
6/22
Linear Regression
Y = β0 + β1X1 + ...+ βpXp + ϵ
• Y : the response variable or dependent variable to be predicted
• X1, ..., Xp: potential predictors or covariates or independent
variables
• the intercept β0 and slope coefficients β1, ..., βp are unknown
population parameters to be estimated
• ϵ: error term or something that can’t be explained by the
model. Assume E(ϵ) = 0 and V(ϵ) = σ2, σ2 is also unknown
Caution: the model imposes many assumptions that need to be
checked/testified! Y should be continuous, not suitable for
categorical/binary response variables. The Xj can be both
continuous/discrete or categorical.
7/22
Linear Regression
Y = β0 + β1X1 + ...+Xp + ϵ
• Linear regression might sound like too simplistic, it’s hardly
true that Y is linearly dependent on the Xj ’s
• But linear regression is extremely useful and important. It
forms a basic framework for non-linear regression and many
advanced regression models
• The linearity is in terms of the coefficients βj ’s, not the Xj ’s.
E.g.,
Y = β0 + β1X1 + β2eX2 + β3X31 + β4X1X2 + ϵ
is still a linear regression model!
• Precisely, X1, X2 are called covariates. In this model, X1, X31 ,
eX2 and X1X2 are called predictors or features
8/22
Simple Linear Regression
Simple Linear Regression
Y = β0 + β1X + ϵ, E(ϵ) = 0, V(ϵ) = σ2
So the conditional mean of Y given X = x is a linear function of x
µY |X=x = E(Y |X = x) = β0 + β1x
Our working example
AmountSpent = β0 + β1Catalogs+ ϵ
9/22
Simple Linear Regression
Our working example
AmountSpent = β0 + β1Catalogs+ ϵ
Let β̂0 and β̂1 be estimates of β0 and β1 respectively. Then
µ̂Y |X=x = β̂0 + β̂1x
is a point estimate of E(Y |X = x). This is an estimate of the
average spending amount among all customers who are sent x
catalogs per year.
We also use ŷ = β̂0 + β̂1x to predict the spending amount of an
individual customer who is sent x catalogs.
10/22
Estimating the coefficients by the least squares method
• ŷi = β̂0 + β̂1xi is the prediction of the observation yi when
X = xi. So ei = yi − ŷi represents the ith residual
• We define the residual sum of squares (RSS) as
RSS(β̂0, β̂1) =
n∑
i=1
e2i =
n∑
i=1
(
yi − (β̂0 + β̂1xi)
)2
.
Note that {yi, xi, i = 1, ..., n} is the training dataset.
• The best β̂0 and β̂1 will be the ones that minimise
RSS(β̂0, β̂1).
• This is known as Least Squares Method. No probability
distribution of ϵ is needed.
11/22
Estimating the coefficients by the least squares method
• β̂0 = ...
• β̂1 = ...
12/22
Estimating the coefficients by the least squares method
β̂0 = y¯ − β̂1x¯
β̂1 =
∑(xi − x¯)(yi − y¯)∑(xi − x¯)2 = x¯y − x¯y¯x¯2 − (x¯)2
13/22
Estimation by the maximum likelihood method
• Assume that ϵi ∼ N (0, σ2), i = 1, ..., n. Note that
yi = β0 + β1xi + ϵi
Therefore yi ∼ N (β0 + β1xi, σ2).
• E.g., the amounts spent by customers who are sent x catalogs
are normally distributed with mean β0 + β1x and variance σ2.
Seems a reasonable assumption!
• The likelihood function is
p(y|β0, β1, σ2) =
n∏
i=1
1√
2πσ2
exp
(
−(yi − β0 − β1xi)
2
2σ2
)
• Maximising this likelihood with respect to β0 and β1 leads
exactly to the same least squares estimates β̂0 and β̂1
• Estimate of σ2: σ̂2= 1n
∑n
i=1
(
yi−(β̂0+β̂1xi)
)2
.
14/22
Brief introduction to Maximum Likelihood Estimation
Given data y = {y1, ..., yn}. In almost all areas of applications, we
then assume the data come from a statistical model, depending on
a vector of unknown parameters θ. This model allows us to write
down the density p(yi|θ) of yi.
Example. yi iid∼ N (µ, 1), i = 1, 2, . . . , n. The joint density of y is
p(y|µ) =
n∏
i=1
p(yi|µ) =
n∏
i=1
1√
2π
e−
(yi−µ)2
2
=
( 1√
2π
)n
exp
[
−12
n∑
i=1
(yi − µ)2
]
This function, considered as a function of µ, measures how likely a
value of µ is as the underlying parameter that generated the data
{y1, . . . , yn}
15/22
Brief introduction to Maximum Likelihood Estimation
Let y = {y1, . . . , yn} be a random sample from a distribution with
pdf p(y|θ). The likelihood function, as a function of θ, is defined as
p(y|θ) =
n∏
i=1
p(yi|θ).
• This likelihood function reflects the probability of observing
the data y if θ is the true parameter
• We wish to estimate the (unknown) true value of θ that
generated the data y by those values that maximise p(y|θ).
• The maximum likelihood estimator (MLE) of θ is a value that
maximises p(y|θ).
• MLE is one of the most popular estimation methods.
16/22
Maximum Likelihood Estimator
Exercise. Suppose that {x1 = 5, 0, 1, 1, 0, 3, 2, 3, 4, x10 = 1} are
n = 10 observations from the Poisson distribution with pdf
f(y|θ) = e
−θθy
y! .
• Write down the log-likelihood function for the sample.
• Find the MLE of θ
17/22
Spending data example

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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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