ECO 401 Econometrics

SI 2021
Week 1, 7 September

Dr Syed K Abbas
Office Location: BS304
Email: Syed.Abbas@xjtlu.edu.cn
 Agenda

1. Introduction
2. What would be covered in this unit?
3. Important tips
4. Expectations
5. Econometrics Concepts based on Chapter 1,2 and appendix A, PP450
(Verbeek)
At the end of this session, you should be able to understand:
Why do we study econometrics?
How do we collect data and evaluate the research using econometric
tools? (This is also relevant for your dissertation in S3.)
What is Ordinary Least Squares (OLS)?
How do we derive normal equations for estimation?
 Introduction

My name is Dr Syed Kanwar Abbas. I joined XJTLU on the 13th September 2017.
Education: Deakin University, Australia (PhD)
University of Melbourne, Australia (GCTT)
Experience
Deakin University, 2009-2017
Box Hill Institute, Faculty of Business, ICT and Library, 2013-2017
Ministry of Finance, 2006-2009
Research area: Macroeconomics, Applied Econometrics
Key publications
Energy Economics, Economic Record, Empirical Economics ,International
Review of Economics and Finance, Economic Modelling and EconModels
Contact Details
Email address: syed.abbas@xjtlu.edu.cn
Office number: 0512-88161670
Room number: BS304
  Delivery Schedule
Lecture room: 1-4: Online 5-6, 8-14:SC140
Lecture time: Tuesday 1:00-03:00PM
Tutorial room (onsite): BS109 and ES211
Tutorial times (onsite): Tuesday (6:00-8:00PM)-Syed (Room
BS109)
Wednesday (12:00-1:00PM)-Shuyang (Room BS109)
Thursday (9:00-11:00)-Shuyang (Room ES211)

Room number and office hours (without appointment): BS304, 15:30-17:30 (Tuesday)
 Learning Outcomes of this unit

• Be able to discuss, critically evaluate and apply a range of mathematical and

statistical techniques necessary for understanding and using econometric
methodology.

• Be able to formulate, estimate and test a wide range of models used in

empirical analyses.

• Be able to use Stata in real applications.

 Aims

• Acquire training in the use, presentation and interpretation of economic data.

• Understand aspects of the theories and principles of econometric analysis in

economics and finance.

• Aware of a range of inferential techniques commonly employed in

econometrics.

• Understand the limitations of such techniques under different circumstances.

 Access to teaching and learning material

• Each week, lecture slides, tutorial questions and data would be posted online,
Learning Mall before the lecture.

• Correct answers for tutorial questions will be uploaded after the tutorial.

• The tutorials will be based on an econometric software, Stata.

• You can also buy a personal student copy via

https://www.stata.com/order/new/edu/gradplans/student-pricing/
Text Book-Compulsory

Marno Verbeek

Erasmus University Rotterdam

ISBN: 978-1-1199-47211-7

Ch 1-5 + 10.1+10.2

9
 How to get your Textbook?
Wiley E-Text: Powered by VitalSource

I would distribute you the code once I get it.

Wiley E-Text: Powered by VitalSource gives students the opportunity to
benefit from the portability and following functionality: search,
highlighting, note-taking, and sharing. Users have the ability to read and
access their E-Text anytime, anywhere and multiple devices.

10
 VitalSource Bookshelf on a web browser,
computer, or mobile device
• http://www.vitalsource.com/redeem
• To open your book in Bookshelf on your computer, you first must have a
Bookshelf account, and then do the following:
• Download Bookshelf for Mac or PC.
(http://www.vitalsource.com/downloads)

• Once Bookshelf is installed, launch Bookshelf.

• Sign in with your Bookshelf account email address and password.
• Click on "All Titles" in the collection pane to view all the books in your account.
• Double click on the title to download the book to your computer. Once downloaded, double-click
again to open the book.

11
 Bookshelf for iOS and Android
• Download Bookshelf for iOS or Android.
(http://www.vitalsource.com/downloads)
• Once Bookshelf is installed, launch Bookshelf.
• Sign in with your Bookshelf account email address and password.
• Click on "All Titles" in the collection pane to view all the books in
your account.
• Tap on the title to download the book to your mobile device. Once
downloaded, double-click again to open the book.

12
Supplementary textbooks

13
Graduate level text Book-for those who have no
background

14
 What would be covered in this unit?

Details of topics covered in ECO 401 Econometrics (from book):

 Ch 1 Introduction
 Ch 2 Linear Regression
 Ch 3 Interpreting and Comparing Models
 Ch 4 Heteroskedasticity and Autocorrelation
 Ch 5 Endogenous Regressors, Instrumental Variables, and GMM
 Ch 10 Panel Data (sections 10.1-10.3; not 10.2.2; 10.2.6-9)

This unit is applied in nature. You would learn how to test theory using econometric
techniques. The real world datasets would be used.
 Syllabus & Teaching Plan

Tutorial Schedule
Week Number and/or
Lecture/Seminar Topic/Theme/Title Pre-reading
Date
Ch 1-2,
Introduction to module
Week 1, 7 September Lecture 1 Appendix A of
Vectors/Matrices, OLS regression
course text
Week 2, 14 September Lecture 2 OLS regression, Goodness of fit Ch 2
Hypothesis testing and model
Week 3, 23 September Lecture 3 Ch 2
interpretation
Student Group Date
Week 4, 28 September Lecture 4 Functional forms Ch 3
Group 1,2 (Syed) 12/10/21 - 19/10/21m 2/11/21 - 16/11/21
4 October-8 October (National Day, University closed day)
Group 3 (Shuyang) 13/10/21 - 20/10/21, 3/11/21 - 17/11/21
Week 5, 12 October Lecture 5 Testing for a structural break Ch 3 14/10/21 - 21/10/21, 4/11/21 - 18/11/21
Group 4, 5 (Shuyang)
Week 6, 19 October Lecture 6 Heteroskedasticity Ch 4
Week 7 Midterm 25 November-29 November
Week 8, 2 November Lecture 7 Heteroskedasticity/ Autocorrelation Ch 4
Week 9, 9 November Lecture 8 Autocorrelation Ch 4
Week 10, 16 November Lecture 9 Instrumental Variables Ch 5
Week 11, 23 November Lecture 10 Instrumental Variables Ch 5
Moments; GMM; Empirical
Week 12, 30 November Lecture 11 Ch 5/10
examples/ Panel Data
Week 13, 7 December Lecture 12 Panel Data Ch 10
Week 14, 14 December Lecture 13 Review and Exam discussions

16
Assessment
Study Tips/Expectations
A. Organize yourself-Time management
B. Only write down the key points of lecture
C. Ask Questions for clarity of ideas.
D. Do not disturb your fellows in the class (no mobile use)
E. Participate in the class discussion
F. Complete the tutorial questions and short tests
What is Econometrics?
 What is Econometrics?
“Econometrics is what econometricians do”
Kennedy (1996)

“Econometrics is the study of the application of statistical methods to the analysis of economic phenomena”
Tintner (1953)

“The application of statistical and mathematical methods to the analysis of economic data, with a purpose of
giving empirical content to economic theories and verifying them or refuting them.”
Maddala (1992)

“Econometrics is the art and science of using statistical methods for the measurement of economic relations.”
Chow (1985)
Econometrics is about how we can use theory and data from economics, business, and the social sciences,
along with tools from statistics, to answer ‘‘how much’’ questions.
Carter Hill et al. (2011)

(c) John Wiley and Sons, 2017 20

Why Study Econometrics?
 Why Study Econometrics?

“Econometrics fills a gap between being a ‘student of economics’ and being a ‘practicing economist’”. Carter
Hill et al (2011)

It lets you tell your employer:

“I can predict the sales of your product”

“I can estimate the effect on your sales if your competition lowers its price by
$1 per unit”
“I can test whether your new ad campaign is actually increasing your sales”
 Econometrics is about 'drawing the line'

Suppose that you are interested in finding the relationship between

two variables. First step is to plot the variables.
12 Observations on 𝑥𝑥 and 𝑦𝑦

error 1
variable y

error 2
this line?

or this line?

0
0 10
variable x
China Inflation and output gap
 Which distance to use?
- deviation? plus and minus cancel
12 - absolute value? difficult to work with
- squared value? standard use

observations
variable

objective: draw straight line such that the sum of distance

from observations (dots) to straight line is 'minimized'

0
0 variable Page 26 10
 1.2
What is
Econometrics
About

In economics we express our ideas about relationships between

economic variables using the mathematical concept of a function

Consumption = f (Income)
(
Q d = f P, P s , P C , INC )
Qs = f (P, P C
,Pf )

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 27

 Every day, decision-makers face ‘‘how much’’: for example

A public transportation council must decide how an increase in fares for public transportation
(trams, trains, and buses) will affect the number of travelers who switch to car or bike, and the
effect of this switch on revenue going to public transportation.

Any other example?

Econometrics focuses on how much a change in one variable affects

another variable.

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 28

 Linear regression model

We would like to draw conclusions about what happens if a variable changes

and how it affects the dependent variable

We would like to predict things that are not yet observed

We would like to see/test the economic relationships

The model is a statistical model and has an “error term” 𝜀𝜀𝑖𝑖 which contains all
influences that are not included explicitly in the model/relationship
 Linear regression model
We start with a linear relationship between 𝑦𝑦 & 𝑥𝑥1 ≡ 1 (a constant) and 𝑥𝑥2 to 𝑥𝑥𝐾𝐾 , assumed to be generally valid:
𝑦𝑦𝑖𝑖 = 𝛽𝛽1 + 𝛽𝛽2 𝑥𝑥𝑖𝑖𝑖 + …….+𝛽𝛽𝐾𝐾 𝑥𝑥𝑖𝑖𝑖𝑖 + 𝜀𝜀𝑖𝑖 (2.24)

In vector form, we write: 𝑦𝑦𝑖𝑖 = 𝒙𝒙′𝑖𝑖 𝜷𝜷 + 𝜀𝜀𝑖𝑖 (2.25)

where 𝒙𝒙′𝑖𝑖 = 1 𝑥𝑥2𝑖𝑖 … 𝑥𝑥𝐾𝐾𝐾𝐾 and 𝜷𝜷′ = 𝛽𝛽1 𝛽𝛽2 … 𝛽𝛽𝐾𝐾

We have systematic part and a random and unpredictable component 𝜀𝜀𝑖𝑖 that we will call a random error

We have a series of 𝑁𝑁 observations on the variable 𝑦𝑦 (which we call the endogenous variable) and the explanatory
(or exogenous) variables 𝑥𝑥𝑖𝑖𝑖𝑖

𝜷𝜷 is a vector of unknown parameters characterizing the population. The unknown coefficients 𝜷𝜷 have a meaning
and measure how we expect 𝑦𝑦 to change if 𝑥𝑥𝑘𝑘 changes (and all other 𝒙𝒙 values remain the same).
 Ordinary Least Squares (OLS)
Our econometric model is 𝑦𝑦𝑖𝑖 = 𝒙𝒙′𝑖𝑖 𝜷𝜷 + 𝜀𝜀𝑖𝑖 , 𝑖𝑖 = 1, . . , 𝑁𝑁

Estimation techniques (like OLS) produce an estimate for the unknown population parameter
vector 𝜷𝜷.

Coefficients in this approximation can be determined by Ordinary Least Squares (OLS),

which minimizes the sum of squared differences between y and the linear combination.

𝑁𝑁
� ≡ �(𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽)
𝑆𝑆(𝛽𝛽) � 2
𝑖𝑖=1
 Simple Linear Regression (SLR)

SLR means we have only one explanatory variable. Let us use OLS to derive
the normal equations in this case. Remember, OLS minimizes the residual
sum of squares.
N
~ ~ ~ ~
S ( β1 , β 2 ) = ∑ ( yi − β1 − β 2 xi ) 2 ( 2.12)
i =1

~ ~
FOC, ∂S ( β1 , β 2 ) N
~ ~
~ = −2∑ ( yi − β1 − β 2 xi ) =0 (2.13)
∂β1 i =1
~ ~
∂S ( β1 , β 2 ) N
~ ~
~ = −2∑ xi ( yi − β1 − β 2 xi ) =0 (2.14)
∂β 2 i =1
 Simple Linear Regression (SLR)
b1 = y − b2 x (2.15)
From (2.13), you get
N

~ ∑x y i i − Nx y
β 2 = b2 = i =1
N

∑ i
x 2

i =1
− N x 2

• Or ∑ ( x − x )( y − y )
i i
b2 = i =1
N
(2.16)
∑ i
( x
i =1
− x ) 2
 Ordinary Least Squares
𝑁𝑁
� ≡ �(𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽)
𝑆𝑆(𝛽𝛽) � 2
𝑖𝑖=1
First order condition (FOC=0)
𝑁𝑁
� =0
−2 � 𝑥𝑥𝑖𝑖 (𝑦𝑦𝑦𝑦 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽)
𝑖𝑖=1
An analytical expression for the OLS can be derived as:
−1
𝑁𝑁 𝑁𝑁

𝑏𝑏 = � 𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖′ � 𝑥𝑥𝑖𝑖𝑦𝑦𝑖𝑖
𝑖𝑖=1 𝑖𝑖=1

The second order condition (SOC ≥0) verifies that b correspond to minimum
𝑁𝑁

2 � 𝑥𝑥𝑖𝑖2 > 0
𝑖𝑖=1
The resulting linear combination of 𝑥𝑥𝑖𝑖 is then given by
𝒚𝒚�𝒊𝒊 = 𝒙𝒙′𝒊𝒊 𝒃𝒃
 12
Our example
y = 0.8717x + 2.7758
R² = 0.8307
variable y

∑𝑁𝑁
𝑖𝑖=1 𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ 𝑦𝑦𝑖𝑖 − 𝑦𝑦
�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑏𝑏2𝑂𝑂𝑂𝑂𝑂𝑂 =
∑𝑁𝑁
𝑖𝑖=1 𝑥𝑥𝑖𝑖 − 𝑥𝑥̅
2

= sample cov(x,y)/sample var(x)

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑏𝑏1𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑦𝑦� − 𝑏𝑏2𝑂𝑂𝑂𝑂𝑂𝑂 𝑥𝑥̅

0
0 10
variable x
 Error terms and residuals

Note that we write, yi = xi′ β + εi (2.25)

and yi = xi′ b + ei (2.8)
• We call εi the error term/ disturbance term and ei the residual.
• εi resembles to population relationship where as ei would resemble the
difference between the observed and the approximated value.
• The error term is unobserved, the residual is constructed (after estimation)
using the estimate b.
• By virtue of the first order conditions of OLS, the residual (ei) is mean zero
and uncorrelated with xi.
 Ordinary Least Squares

• OLS produces the ‘best’ linear approximation of y from x2 to xK and a constant.

– Best refers to the fact that sum of squared differences between the observed values yi and
fitted values yi cap is minimal for least squares solution b.

The result for a given sample is called an estimate. An estimate is a random variable,

because the sample is randomly drawn from a larger population.

because the data are generated by some random process.
Fitted line and observation points

• The best linear approximation of y from x

and a constant is obtained by minimizing
the sum of squared residuals which is equal
to the vertical distances between an
observation and the fitted value.

• All fitted values are on a straight line

called regression line.
 Is OLS a good estimator?
• The answer to this question depends upon the assumptions we are willing
to make.
• The most standard and most convenient ones are given by the Gauss-
Markov assumptions.
• Note that these assumptions are very strong and often not satisfied.
• Under the Gauss-Markov assumptions, the OLS estimator has nice
properties.
 The Gauss-Markov assumptions

(A1) Error terms have mean zero: E{εi}=0

(A2) All error terms are independent of all x
variables: {ε1 ,… εN} is independent of {x1,… xN}
(A3) All error terms have the same variance
(homoskedasticity): V{εi} = σ2.
(A4) The error terms are mutually uncorrelated (no
autocorrelation): cov{εi,,εj} = 0, i ≠ j.

Under (A2) we can treat the explanatory variables as fixed (deterministic).

Later, we shall discuss how essential the Gauss-Markov assumptions are
and how they can be relaxed.

40
 1.3
The Econometric Model

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 41

 1.3
The Econometric
Model

An econometric model consists of a systematic part and a random

and unpredictable component e that we will call a random error

=Q d f P, P s , P c , INC + e ( )
( s c
)
f P, P , P , INC =β1 + β 2 P + β3 P + β 4 P + β5 INCs c

Q =β1 + β 2 P + β3 P + β 4 P + β5 INC + e
d s c

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 42

 1.3
The Econometric
Model

The coefficients β1, β2, …, β5 are unknown parameters of the model

that we estimate using economic data and an econometric technique

– The functional form represents a hypothesis about the relationship

between the variables

– In a particular problem, one challenge is to determine a functional

form that is compatible with economic theory and the data.

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 43

 1.3
The Econometric
Model

The systematic portion is the part we obtain from economic theory,

and includes an assumption about the functional form

The random component represents a ‘‘noise’’ component, which

obscures our understanding of the relationship among variables, and
which we represent using the random variable e

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 44

 1.3
The Econometric
Model

We use the econometric model as a basis for statistical inference

The ways in which statistical inference are carried out include:
1. Estimating economic parameters, such as elasticities, using
econometric methods
2. Predicting economic outcomes, such as the enrollment in two-
year colleges in the United States for the next ten years
3. Testing economic hypotheses, such as the question of whether
newspaper advertising is better than store displays for increasing
sales

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 45

 1.4
How are Data Generated?

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 46

 1.4
How Are Data
Generated?

We must have data

Economists and other social scientists work in a complex world in

which data on variables are ‘‘observed’’ and rarely obtained from a
controlled experiment.

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 47

 1.5
Economic Data Types

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 48

 Economic Data
Types

– Data may be collected at various levels of aggregation: Micro or

Macro
– Data may also represent a flow or a stock:
• Flow: measured over time
• Stock: measured at a particular point in time
– Data may be quantitative or qualitative:
• Quantitative: expressed as numbers
• Qualitative: expressed as an ‘‘either-or’’ situation

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 49

 Economic Data Types

A time-series is data collected over discrete intervals of time

– The key feature of time-series data is that the same economic
quantity is recorded at a regular time interval

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 50

 Economic Data
Types
Table 1.1 Annual GDP of Real 2005 Dollars

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 51

 Economic Data
Types

Cross-Section Data

A cross-section of data is collected across sample units in a

particular time period

– The ‘‘sample units’’ are individual entities and may be firms,

persons, households, states, or countries

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 52

 Economic Data
Types
Table 1.2 Cross Section Data: CPS August 2009

Cross-Section Data

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 53

 1.5
Economic Data
Types

A ‘‘panel’’ of data, also known as ‘‘longitudinal’’ data, has

observations on individual micro-units who are followed over time
– The key aspect of panel data is that we observe each micro-unit
over time periods

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 54

 1.5
Economic Data
Types
Table 1.3 Panel Data from Two Rice Farms

1.5.3
Panel or
Longitudinal Data

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 55

 1.6
The Research Process

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 56

 1.6
The Research
Process

Econometrics is ultimately a research tool

– Students of econometrics plan to do research or they plan to read

and evaluate the research of others, or both

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 57

 1.6
The Research
Process

Steps in the research process:

1. Use economic theory to think about the problem
2. Develop a working economic model leading to an econometric model
3. Obtain sample data and choose a desirable method of statistical analysis
based on initial assumptions and an understanding of how the data were
collected
4. Estimate the unknown parameters with the help of a statistical software
package, make predictions, and test hypotheses
5. Perform model diagnostics to check the validity of assumptions
6. Analyze and evaluate the economic consequences and the implications of
the empirical results

Principles of Econometrics, 4th Edition Chapter 1: An Introduction to Econometrics Page 58

 Vectors and matrices

Vectors and matrices notation are often used in econometrics.

Read Appendix A of your textbook.
 Vectors and matrices
• A vector is an ordered set of numbers arranged either in a row or a column; our default setting is to use
column vectors
• A row vector consists of 1 row only; 𝐛𝐛′ = 3 8 5 is 1 × 3
8
• A column vector consists of 1 column only; 𝐚𝐚 = 3 is 3 × 1
7
• A matrix is a set of real or complex numbers arranged in rows and columns to form a rectangular
array.
• In other words, a matrix with single row is called a row vector and a matrix with a single column is
called a column vector.
• A matrix having n rows and k columns is called an 𝑛𝑛 × 𝑘𝑘 matrix and is referred to as having order
(dimension) 𝑛𝑛 × 𝑘𝑘
• A scalar is a regular (real or complex) number; it is a 1 × 1 matrix
 Vectors and matrices

A square matrix has the same # rows and columns: 𝑛𝑛 = 𝑘𝑘

A square matrix is symmetric if 𝑨𝑨 = 𝑨𝑨′

A diagonal matrix is a square matrix with all elements zero except on the leading diagonal:
𝑎𝑎𝑖𝑖𝑖𝑖 = 0, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖 ≠ 𝑗𝑗

The identity matrix is a diagonal matrix in which the elements of the leading diagonal are all of
value 1

A null matrix has all its elements equal to zero

1 0 0
6 0 0
0
0 0
0  1
 2
0 0 0 1
0 9
 Vectors and matrices

The transpose of a matrix 𝑨𝑨 (dimension 𝑛𝑛 × 𝑘𝑘) is denoted as 𝑨𝑨′ (dimension 𝑘𝑘 × 𝑛𝑛); the columns of 𝑨𝑨
𝑎𝑎11 𝑎𝑎12 𝑎𝑎13 𝑎𝑎11 𝑎𝑎21 𝑎𝑎31
are the rows of 𝑨𝑨′ and vice versa; 𝑨𝑨 = 𝑎𝑎21 𝑎𝑎22 𝑎𝑎23 implies 𝑨𝑨′ = 𝑎𝑎12 𝑎𝑎22 𝑎𝑎32
𝑎𝑎31 𝑎𝑎32 𝑎𝑎33 𝑎𝑎13 𝑎𝑎23 𝑎𝑎33

Two matrices of the same dimension can be added or subtracted simply by adding or subtracting
each
3 −5 1 2 −3 4 5 −8 5
element: 8 −4 9 + 1 6 5 = 9 2 14 and
−1 2 −6 −2 3 −4 −3 5 −10

3 −5 1 2 −3 4 1 −2 −3
8 −4 9 − 1 6 5 = 7 −10 4
−1 2 −6 −2 3 −4 1 −1 −2

Note: 𝑨𝑨 + 𝑩𝑩 = 𝑩𝑩 + 𝑨𝑨 ; 𝑨𝑨 + 𝑩𝑩 ′ = 𝑨𝑨′ + 𝑩𝑩′; and 𝑨𝑨′ ′

= 𝑨𝑨
 Vectors and matrices

• If 𝑘𝑘 is a scalar and 𝑨𝑨 is a matrix, then the product 𝑩𝑩 = 𝑘𝑘𝑨𝑨 multiplies each element of
𝑨𝑨 by 𝑘𝑘: 𝑏𝑏𝑖𝑖𝑖𝑖 = 𝑘𝑘𝑎𝑎𝑖𝑖𝑖𝑖 , 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖, 𝑗𝑗
• If 𝒂𝒂 and 𝒃𝒃 are two vectors of the same length 𝑛𝑛 we can calculate the inner product
𝒂𝒂′ 𝒃𝒃 = ∑𝑛𝑛𝑖𝑖=1 𝑎𝑎𝑖𝑖 𝑏𝑏𝑖𝑖
• Note: 𝒂𝒂′ 𝒃𝒃 = 𝒃𝒃′ 𝒂𝒂
• If the inner product of two vectors is zero they are said to be orthogonal: 𝒂𝒂′ 𝒃𝒃 = 0
• If matrix 𝑨𝑨 has dimension 𝑛𝑛 × 𝑘𝑘 and matrix 𝑩𝑩 has dimension 𝑘𝑘 × 𝑚𝑚 they can be
multiplied to generate a new matrix 𝐶𝐶 of dimension 𝑛𝑛 × 𝑚𝑚, where element 𝑐𝑐𝑖𝑖𝑖𝑖 consists
of the inner product of the 𝑖𝑖-th row of 𝑨𝑨 and the 𝑗𝑗-th column of 𝑩𝑩: 𝑐𝑐𝑖𝑖𝑖𝑖 = ∑𝑘𝑘𝑟𝑟=1 𝑎𝑎𝑖𝑖𝑖𝑖 𝑏𝑏𝑟𝑟𝑟𝑟 ;
1 5
1 3 2 15 20
for example: 2 1 =
4 6 5 36 56
4 6
 Next Week, we would start OLS in detail
Do you want to pass this course? Some suggestions.
If you want to pass this course, 8 important tips:
1. Read the relevant chapter at home before the lecture
2. Attend all lectures – evidence follows
3. Make the relevant exercises at home before the tutorial
4. Attend all tutorials – evidence follows
5. Work throughout the semester – make assignments early
6. Prepare well for presentation and discussion
7. Sleep – make sure you are well-rested before exams
8. If you work in the last week only: it is too late!

64
The End