Electric: Statistical Hypothesis Testing and...

Zhaoxia Xu

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

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