Agenda

Linear regressions

Simple linear model

Ordinary Least Squares

Estimations

Inferences

Applications

1

Financial Econometrics

Application of statistical techniques to problems in finance

Estimating and testing relationships between variables as

suggested by theories in finance, e.g. CAPM or APT

Modeling and forecasting asset returns

Evaluating the performance of portfolio management

Examining determinants of firms’ financial decisions

Steps involved in the formulation of econometric models

Economic or Financial Theory (Previous Studies)

Formulation of an Estimable Theoretical Model

Collection of Data

Model Estimation

Is the Model Statistically Adequate?

No

Reformulate Model

Yes

Interpret Model

Use for Analysis

2

Simple Regression: An Example

Suppose that we have the following data on the excess returns

on a fund manager’s portfolio (“fund XXX”) together with the excess returns on a market index:

Year, t

Excess return

= rXXX,t – rft

17.8

39.0

12.8

24.2

17.2

1

2

3

4

5

Excess return on market index

= rmt - rft

13.7

23.2

6.9

16.8

12.3

We have some intuition that the beta on this fund is positive,

and we therefore want to find whether there appears to be a relationship between x and y given the data that we have.

Graph (Scatter Diagram)

Excess return on fund XXX

45

40

35

30

25

20

15

10

5

0

0

5

10

15

20

25

Excess return on market portfolio

3

Finding a Line of Best Fit

We can use the general equation for a straight line,

y=α+βx to get the line that best “fits” the data.

However, this equation (y=α+βx) is completely deterministic.

Is this realistic? No. So what we do is to add a random

disturbance term, u into the equation. yt = + xt + ut where t = 1,2,3,4,5

Ordinary Least Squares

The most common method used to fit a line to the data is

known as OLS (ordinary least squares).

What we actually do is take each distance and square it and

minimise the residual sum of the squares (hence least squares). Tightening up the notation, let

yt denote the actual data point t denote the fitted value from the regression line

ˆ

ˆ ut denote the residual, yt - yt

ˆ

yt

4

Determining the Regression Coefficients

Choose and so that the (vertical) distances from the data

points to the fitted lines are minimised (so that the line fits the data as closely as possible): y yt

ˆ

ut

ˆ

yt

How OLS Works

5

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

So min. u 12 u 22 u 32 u 42 u 52 , or minimise u t2 . This is

known as the residual sum of squares.

t 1

But what was ut ? It was the difference between the actual

ˆ

point and the line, yt - yt .

ˆ

ˆ

u

with respect to

So minimising

2 t is equivalent to minimising and .

y

ˆ

yt

2

t

5

Deriving the OLS Estimator

ˆ

ˆ

ˆ

y t x t , so let L

ˆ

ˆ ˆ

( y t y t ) 2 ( y t xt ) 2 t i

Want to minimise L with respect to (w.r.t.) and

, so

differentiate L w.r.t. and

L

ˆ

ˆ

2 ( yt xt ) 0

ˆ

t (1)

L

ˆ

ˆ

2 xt ( yt xt ) 0

ˆ

t

(2)

`

From (1),

ˆ ˆ

ˆ ˆ

( y t x t ) 0 y t T x t 0 t We know y t T y and

x

t

.

Tx

Deriving the OLS Estimator (cont’d)

ˆ

So we can write T y T T x 0 or y ˆ ˆ x 0

ˆ

From (2),

ˆ

x t ( y t ˆ x t ) 0

(3)

(4)

t

ˆ

ˆ

From (3), y x

(5)

Substitute into (4) for from (5),

x t ( y t y ˆ x ˆ x t ) 0 t 2

x t y t y x t ˆ x x t ˆ x t 0 t 2

x t y t T y x ˆ T x 2 ˆ x t 0 t 6

Deriving the OLS Estimator (cont’d)

Rearranging for ,

ˆ ( T x

2

x t2 ) T y x x t y t

So