asb4408lecture1a

Structure of this lecture

• Basic info on myself and this module

• Essential of matrix arithmetic

• Multiple regression models

• Statistical inference and hypothesis testing

Main learning outcomes:

- Basic understanding of matrix algebra

- Basic understanding of classical linear regression model assumptions

- Basic understanding of how to construct and interpret a hypothesis test

Dr Enrico Onali

Division: Financial Studies

Location: Room 2.07, Alun Building

Telephone: 01248 38 3650

Email: e.onali@bangor.ac.uk

Office hours:

Monday 11.30-12.30

Monday 13.30-14.30

or by appointment

My contact details

Who am I?

• Senior Lecturer in Finance

• Italian

• Experience in Italian retail bank

• Current research interests: Banking and financial regulation, dividend policy, market efficiency, applied econometrics

• Profile at Google Scholar:http://scholar.google.co.uk/citations?user=cro4BU0AAAAJ&hl=it&oi=ao

Links to my publications (access from BU intranet needed):http://www.sciencedirect.com/science/article/pii/S0278425414000696

http://onlinelibrary.wiley.com/doi/10.1111/jbfa.12057/abstract

http://0-www.sciencedirect.com.unicat.bangor.ac.uk/science/article/pii/S1057521912000579

http://0-onlinelibrary.wiley.com.unicat.bangor.ac.uk/doi/10.1111/j.1467-646X.2010.01037.x/full3

Financial econometrics What is it? Why is it useful?

• Financial econometrics: Basically, a branch of applied mathematics

• Dissertation: This module will be very helpful for those of you who decide to write a dissertation

• Employability: employers like people with numeracy skills, although returns to skills vary with the country

http://www.economist.com/blogs/freeexchange/2014/01/economic-value-skills

Those of you who plan to work in the academia, central banks, think-tanks, will find this module extremely useful.

Short course overview

• Technical background (today)

• Regression analysis using time series data

• Univariate modelling of stationary time series variables

• Modelling non-stationary time series variables

• Multivariate modelling of stationary time series variables

• Cointegration and multivariate modelling of non-stationary time series

• Modelling volatility in financial time series data

• Panel data econometrics

Textbook and readings

Main textbook:

C. Brooks. Introductory econometrics for finance, 2nd edition. Cambridge University Press, 2008.

C. Brooks. Introductory econometrics for finance, 3rd edition. Cambridge University Press, 2014.

Other references:

B.H. Baltagi. Econometric analysis of panel data, 4th ed., Wiley 2008.

W. Enders. Applied econometric time series, 2nd ed., Wiley, 2005.

W.H. Greene. Econometric analysis, 6th edition, Prentice Hall, 2002.

G. Koop. Analysis of financial data, Wiley, 2006.

M. Verbeek. A guide to modern econometrics, 3rd ed., Wiley, 2008.

C. Cameron and P. K. Trivedi. Microeconometrics Using Stata, Revised Edition. Stata Press, 2010.

1. TECHNICAL BACKGROUND

Essentials of matrix arithmetic

(Several of the examples used here are taken from Brooks pp.611-615,

2nd edition or pp. 41-48, 3rd edition).

Notation and special types of matrix

A matrix is a collection or array of numbers. In econometrics, it is often

convenient to use matrices to represent data. Using matrices, we can write

down concise formulae for estimators or test statistics in terms of the data

from which they are computed.

The dimensions of a matrix are quoted as (RC), where R = number of

rows, C = number of columns.

The dimensions of a matrix are quoted as (RC), where R = number of

rows, C = number of columns.

Each element of a matrix is referred to using subscripts. For the matrix A,

ai,j refers to the element in the i’th row and the j’th column of A. If the

dimensions of A are (23), we can write:

3,22,21,2

3,12,11,1

Essentials of matrix arithmetic (cont'd)

A matrix with only one row, or dimensions of (1C), is known as a row

vector.

A matrix with only one column, or dimensions of (R1), is known as a

column vector.

vector.

column vector.

A square matrix has the same number of rows and columns.

vector.

column vector.

A square matrix has the same number of rows and columns.

A diagonal matrix is a square matrix with non-zero values on the main

diagonal, and 0’s in all other elements. For example:

An identity matrix, denoted I or Im (where m denotes the number of rows

and columns), is a square matrix with 1’s in all elements on the main

diagonal, and 0’s in all other elements.

I2 = I3 =

I2 = I3 = I4 =

A symmetric matrix is a square matrix that is symmetric around the main

diagonal, so that ai,j=aj,i for all i and j. For example:

Addition and subtraction

Two matrices can be added or subtracted if (and only if) both matrices

have the same dimensions. If so, the corresponding elements of the two

matrices are added or subtracted one by one.

If A = and B =

7.01.0

6.03.0

1.02.0

If A = and B =

A + B = =

7.01.0

6.03.0

1.02.0

3.07.001.0

1.06.02.03.0

5.05.0

If A = and B =

A + B = =

A – B = =

7.01.0

6.03.0

1.02.0

3.07.001.0

1.06.02.03.0

5.05.0

3.07.001.0

1.06.02.03.0

4.01.0

7.01.021

Multiplication

(i) Scalar multiplication

To multiply a matrix by a scalar (a single number), every element of the

matrix is multiplied by that number.

If A = then 2A = 2 =

7.01.0

6.03.0

7.01.0

6.03.0

4.12.0

2.16.0

Multiplication

(i) Scalar multiplication

To multiply a matrix by a scalar (a single number), every element of the

matrix is multiplied by that number.

If A = then 2A = 2 =

(ii) Matrix multiplication

Two matrices can be multiplied together if (and only if) the number of

columns in the 1st matrix equals the number of rows in the 2nd matrix.

The number of rows in the 1st matrix determines the number of rows in

the product matrix, and the number of columns in the 2nd matrix

determines the number of columns in the product matrix.

7.01.0

6.03.0

7.01.0

6.03.0

4.12.0

2.16.0

Multiplication (cont'd)

In other words, A and B can be multiplied together if A is (mn) and B is

(np). The dimensions of C=AB are (mp).

The element in the i’th row and j’th column of the product matrix is

obtained by summing the products of the corresponding elements in the

i’th row of the 1st matrix and the j’th column of the 2nd matrix.

Let A = and B =

The dimensions of A are (32) and the dimensions of B are (24).

Therefore the dimensions of C=AB are (34).

Let C =

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

Multiplication (cont'd) A = B =

c1,1 is the sum of the products of the 1st row of A and the 1st column of B

c1,1 = 10 + 26 = 12

c1,2 is the sum of the products of the 1st row of A and the 2nd column of B

c1,2 = 12 + 23 = 8

c1,1 = 10 + 26 = 12

c1,2 is the sum of the products of the 1st row of A and the 2nd column of B

c1,2 = 12 + 23 = 8

c1,3 is the sum of the products of the 1st row of A and the 3rd column of B

c1,3 = 14 + 20 = 4

c1,1 is the sum of the products of the 1st row of A and the 1st column of B:

c1,1 = 10 + 26 = 12

c1,2 is the sum of the products of the 1st row of A and the 2nd column of B:

c1,2 = 12 + 23 = 8

c1,3 is the sum of the products of the 1st row of A and the 3rd column of B:

c1,3 = 14 + 20 = 4

c1,4 is the sum of the products of the 1st row of A and the 4th column of B

c1,4 = 19 + 22 = 13

c1,1 = 10 + 26 = 12

c1,2 = 12 + 23 = 8

c1,3 = 14 + 20 = 4

c1,4 is the sum of the products of the 1st row of A and the 4th column of B:

c1,4 = 19 + 22 = 13

c2,1 is the sum of the products of the 2nd row of A and the 1st column of B

c2,1 = 70 + 36 = 18

c1,1 = 10 + 26 = 12

c1,2 = 12 + 23 = 8

c1,3 = 14 + 20 = 4

c1,4 is the sum of the products of the 1st row of A and the 4th column of B:

c1,4 = 19 + 22 = 13

c2,1 is the sum of the products of the 2nd row of A and the 1st column of B:

c2,1 = 70 + 36 = 18

c2,2 is the sum of the products of the 2nd row of A and the 2nd column of B

c2,2 = 72 + 33 = 23

and so on.

The full product matrix C is as follows:

C = AB = =

2142036

69282318

134812

C = AB = =

Note that in this case the product matrix AB exists, but the product matrix

BA does not exist, because the number of columns in B the number of

rows in A.

2142036

69282318

134812

C = AB = =

rows in A.

Transposition

The transpose of an (RC) matrix A, written A', is the (CR) matrix

obtained by transposing (switching) the rows and columns of the matrix

2142036

69282318

134812

C = AB = =

rows in A.

Transposition

The transpose of an (RC) matrix A, written A', is the (CR) matrix

obtained by transposing (switching) the rows and columns of the matrix

If A = A' =

2142036

69282318

134812

The rank of any matrix is the number of linearly independent rows or

columns. Loosely speaking, two rows are linearly independent if it is not

possible to express one of the rows as a multiple of another.

Rank (cont'd)

Suppose A = and B = and C =

rank(A) = 2, because row 2 cannot be expressed as a multiple of row 1 (or

vice versa). Therefore A has two linearly independent rows.

Rank (cont'd)

rank(B) = 1, because row 1 is 1.5 times row 2. Therefore rows 1 and 2 are

not linearly independent: B has only one linearly independent row.

Rank (cont'd)

rank(C) = 0, because a square matrix containing zeros only has no

independent rows.

Rank (cont'd)

rank(C) = 0, because a square matrix containing zeros only has no

independent rows.

A square matrix A with dimensions (mm) is said to have full rank if

rank(A) = m.

Matrix inversion

For any square matrix that has full rank, an inverse matrix can be defined

such that the product of the original matrix and the inverse matrix is an

identity matrix. In this case, the order in which the two matrices are

multiplied does not matter.

Matrix inversion

If the original matrix is A, the inverse matrix is denoted A–1.

Matrix inversion

If A is (mm), AA–1 = A–1A = Im

Matrix inversion

If A is (22), the formula for A–1 is relatively simple.

Matrix inversion

If A is (22), the formula for A–1 is relatively simple.

If A = then A–1 =

Matrix inversion (cont'd)

Suppose A =

Suppose A = A–1 =

8.02.0

6.04.0

Suppose A = A–1 =

Check: AA–1 = = = I2

8.02.0

6.04.0

8.02.0

6.04.0

Suppose A = A–1 =

Check: AA–1 = = = I2

If A is (22) as above, the quantity (ad – bc) is known as the determinant

of matrix A, also denoted det(A) or |A|.

8.02.0

6.04.0

8.02.0

6.04.0

The determinant can be interpreted as a summary measure of the

‘information’ contained in the matrix, in the form of a single numerical

value.

If A is (mm) where m>2, the formulae for det(A) and A–1 is more

complex. det(A) and A–1 would not usually be calculated manually.

value.

The inverse of an (mm) matrix A only exists if rank(A)=m.

value.

The inverse of an (mm) matrix A only exists if rank(A)=m.

Trace and eigenvalues of a square matrix

Two other useful summary measures of the ‘information’ contained in a

square matrix are the trace and the eigenvalues. The trace of any square

matrix is the sum of the main-diagonal elements (from top-left to bottom-

right).56

Trace and eigenvalues of a square matrix (cont'd)

The eigenvalues of an (mm) square matrix A are the m solutions

(1,2, ... ,m) to the m’th order polynomial in that is formed by the

expression det(A–Im)=0 or |A–Im|=0, listed in descending order of

absolute magnitude.

For A =

A – Im = – = – =

absolute magnitude.

For A =

A – Im = – = – =

det(A – Im) = (4 – )(2 – ) – 3 1 = 8 – 6 + 2 – 3 = 2 – 6 + 5

absolute magnitude.

For A =

A – Im = – = – =

det(A – Im) = (4 – )(2 – ) – 3 1 = 8 – 6 + 2 – 3 = 2 – 6 + 5

det(A – Im) = 0 2 – 6 + 5 = 0

absolute magnitude.

For A =

A – Im = – = – =

det(A – Im) = (4 – )(2 – ) – 3 1 = 8 – 6 + 2 – 3 = 2 – 6 + 5

det(A – Im) = 0 2 – 6 + 5 = 0

( – 5)( – 1) = 0

absolute magnitude.

For A =

A – Im = – = – =

det(A – Im) = (4 – )(2 – ) – 3 1 = 8 – 6 + 2 – 3 = 2 – 6 + 5

det(A – Im) = 0 2 – 6 + 5 = 0

( – 5)( – 1) = 0

1 = 5 and 2 = 1 are the eigenvalues of the matrix A.

Why are eigenvalues useful?

There are important properties of eigenvalues, in particular:

1. The sum of the eigenvalues is the trace of the matrix

2. The product of the eigenvalues is the determinant

3. The number of non-zero eigenvalues is the rank

Thus, eigenvalues are important to understand whether a matrix

is of full rank, and whether all rows are linearly independent

(this is important, in particular, for the concept of co-

integration)

asb4408lecture1a

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