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:
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
Related Documents: Electric: Statistical Hypothesis Testing and Significance Approach Essay example
253 Chapter 6 Evaluation Research Design: Literature Reviews & Synthesis Frequently a research question or hypothesis can be answered through secondary research, i.e., a literature review or synthesis. Both strategies requires the researcher to mine existing data sources; ―pull out‖ relevant data or information; summarize it; logically analyze and/or statistically treat it; and report results. In many instances, the issue, problem, question, etc. which prompted the idea for an evaluation study is…
teacher and the client as a student whose task it is to substitute adaptive patterns of behavior for maladaptive patterns. Cognitive therapist see the therapist-client relationship as that of two collaborators on relatively equal footing, using “hypothesis-testing” techniques to correct irrational or inaccurate beliefs that might be triggering the client’s emotional pain or dysfunction. Therapy Theoretical Perspective Founders Key Concepts Psychoanalysis Psychoanalysis Sigmund Freud Disorders caused by…
psychology, this point of view emphasizes the physical and biological bases of behavior. Sociocultural approach- based on the idea that society and culture shape cognition. Social customs, beliefs, values, and language are all part of what shapes a person's identity and reality. According to this approach, what a person thinks is based on his or her socio-cultural background. A socio-cultural approach takes into account more than the individual in attempting to understand cognitive processes.…
only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.…