Multiple Regression Analysisfin.bus.ku.ac.th/01135534 Financial Modeling/Lecture... ·...

I. Analysis of Data

KULKUNYA PRAYARACH, PH.D.

Multiple Regression Analysis

II. Hypothesis Testing III. Dummy Variable IV. Research & Group Work

1

I. Analysis of Data




OUTLINE

Analysis of Data and Model

Hypothesis Testing

Dummy Variables

Research in Finance

2

I. Analysis of Data




Trend

Seasonal Variation

Cyclical Variation

Irregular Variation

Time Series data Cross-Sectional data

1-dimensional Data set

Observing many subjects

(size, company, counties,

etc) at the same time

Panel data

Multi-dimensional data set

Time-Series + Cross-

Sectional Data

MULTIPLE REGRESSION

ANALYSIS: Types of Data

3

I. Analysis of Data




Trend Component• Persistent, overall upward or downward pattern

• Due to population, technology etc.

• Several years duration

Mo., Qtr., Yr.

Response

© 1984-1994 T/Maker Co.

I. Analysis of Data




Trend Component• Overall Upward or Downward Movement

• Data Taken Over a Period of Years

Sales

Time

I. Analysis of Data




Cyclical Component

• Repeating up & down movements

• Due to interactions of factors influencing economy

• Usually 2-10 years duration

Mo., Qtr., Yr.

Response

Cycle

I. Analysis of Data




Cyclical Component

• Upward or Downward Swings

• May Vary in Length

• Usually Lasts 2 - 10 Years

Sales

Time

I. Analysis of Data




Seasonal Component

• Regular pattern of up & down fluctuations

• Due to weather, customs etc.

• Occurs within one year

Mo., Qtr.

Response

Summer

© 1984-1994 T/Maker Co.

I. Analysis of Data




Seasonal Component

• Upward or Downward Swings

• Regular Patterns

• Observed Within One Year

Sales

Time (Monthly or Quarterly)

I. Analysis of Data




Irregular Component

• Erratic, unsystematic, ‘residual’

fluctuations

• Due to random variation or unforeseen

events

– Union strike

– War

• Short duration & nonrepeating

I. Analysis of Data




Time Series Data

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Ap

r-7

5

May

-76

Jun

-77

Jul-

78

Au

g-7

9

Sep

-80

Oct

-81

No

v-8

2

Dec

-83

Jan

-85

Feb

-86

Mar

-87

Ap

r-8

8

May

-89

Jun

-90

Jul-

91

Au

g-9

2

Sep

-93

Oct

-94

No

v-9

5

Dec

-96

Jan

-98

Feb

-99

Mar

-00

Ap

r-0

1

May

-02

Jun

-03

Jul-

04

Au

g-0

5

Sep

-06

Oct

-07

No

v-0

8

Dec

-09

Jan

-11

Feb

-12

Mar

-13

Ap

r-1

4

May

-15

Jun

-16

Jul-

17

SET Index

I. Analysis of Data




Cross Sectional Data

I. Analysis of Data




Pool (Panel) Data

I. Analysis of Data




Least Square Estimator Maximum Likelihood Estimator

𝑌𝑖 = 𝛽1 + 𝛽2𝑋1𝑖 + 𝛽3𝑋2𝑖 + 𝑢𝑖

ANALYSIS: Type of Estimator

14

I. Analysis of Data




Linear model Non Linear Model

𝑌𝑡 = 𝐴𝐼𝑆 𝑅𝐸𝑇𝑈𝑅𝑁𝐷𝑇𝐴𝐶𝑡 = 𝛼 + 𝛽1𝑋1𝑡 + 𝛽2 𝑋2𝑡 + 𝜀𝑡

ANALYSIS: Type of Model

15

I. Analysis of Data




Y = a + b x

Time series Panel Model

Pooled or Panel Model

Fixed-Effect Model

Random-Effect Model

Time-Series with Condition

ARCH/GARCH Multiple Regression ARMA/ ARIMA

X ~ regressor

independent variable

explanatory variable

predictor Variable

Y ~ regressand var

response var

dependent var

observed var

ANALYSIS: Fitted Regression on Model

16

I. Analysis of Data




Y = a + b x

Logit Model Probit ModelY is discrete


17

I. Analysis of Data




Y = a + b x

Vector Auto Regression

(VAR)

Error Correction

Model (ECM)

Y and X are Dynamic


18

I. Analysis of Data




FITTED REGRESSION MODEL

Y = a + b x

ANALYSIS: Expansion from Simple Regression to Multiple Regression

19

I. Analysis of Data




simple linear regression

• x is the independent variable

• y is the dependent variable

• The regression model is

• The model has two variables, the independent or explanatory

variable, x, and the dependent variable y, the variable whose

variation is to be explained.

• The relationship between x and y is a linear or straight line

relationship.

• Two parameters to estimate – the slope of the line β1 and the y-

intercept β0 (where the line crosses the vertical axis).

• ε is the unexplained, random, or error component. Much more on

this later.

xy 10

I. Analysis of Data




Regression line

• The regression model is

• Data about x and y are obtained from a sample.

• From the sample of values of x and y, estimates b0 of β0

and b1 of β1 are obtained using the least squares or

another method.

• The resulting estimate of the model is

• The symbol is termed “y hat” and refers to the

predicted values of the dependent variable y that are

associated with values of x, given the linear model.

xy 10

xbby 10ˆ

y

I. Analysis of Data




Income hrs/week Income hrs/week

8000 38 8000 35

6400 50 18000 37.5

2500 15 5400 37

3000 30 15000 35

6000 50 3500 30

5000 38 24000 45

8000 50 1000 4

4000 20 8000 37.5

11000 45 2100 25

25000 50 8000 46

4000 20 4000 30

8800 35 1000 200

5000 30 2000 200

7000 43 4800 30

I. Analysis of Data




Summer Income as a Function of Hours Worked

0

5000

10000

15000

20000

25000

30000

0 10 20 30 40 50 60

Hours per Week

Inc

om

e

I. Analysis of Data




xy 2972461ˆ R2 = 0.311

I. Analysis of Data




I. Analysis of Data




I. Analysis of Data




Outliers

• Rare, extreme values may distort the outcome.

– Could be an error.

– Could be a very important observation.

• Outlier: more than 3 standard deviations from the

mean.

I. Analysis of Data




GPA vs. Time Online

0

2

4

6

8

10

12

50 55 60 65 70 75 80 85 90 95 100

GPA

Tim

e O

nlin

e

I. Analysis of Data




GPA vs. Time Online

0

1

2

3

4

5

6

7

8

9

50 55 60 65 70 75 80 85 90 95 100

GPA

Tim

e O

nlin

e

I. Analysis of Data




U-Shaped Relationship

0

2

4

6

8

10

12

0 2 4 6 8 10 12

X

Y

Correlation = +0.12.

OMITTED VARIABLE

I. Analysis of Data




• F-Test is of interest to test more than one

coefficient simultaneously.

F-Test

Conditional to Reject H0:

Significant if p-value < 0.05

TESTING MULTIPLE HYPOTHESIS: F-test

31

I. Analysis of Data




• t-Test is of interest to test ONLY one coefficient

t-Test

Conditional to Reject H0:

Significant if p-value < 0.05

Oh my gosh!!!! It fails to reject H0, what does it mean?

What I should do? Cut it or leave it?

TESTING MULTIPLE HYPOTHESIS: t-test

32

I. Analysis of Data




TMB

RP1BBLNPLFRNJASDJ

NIKKEI

1990M01 2011 M12

Example I: Stock Asset Price Regression

33

I. Analysis of Data




Dependent Variable : Y ~ Rental Values DefinitionsExample II: Hedonic Pricing Model

34

I. Analysis of Data




• R2 is desirable to answer how well regression model

actually fits the data

• In other words, R2 is desirable to answer how well does

the model containing the explanatory variables

R2 = 1 0 < R2 < 1

0 ≤ R2 ≤ 1

TESTING MULTIPLE HYPOTHESIS: Goodness of Fit Testing R2

35

I. Analysis of Data




• Cannot compare R2 of two models with same X but change Y

• R2 never falls if more regressors are added to the regression

• R2 can take values of 0.9 or higher for time series regressions,

and hence it is not good at discrimanating between models

R22 R1

2

TESTING MULTIPLE HYPOTHESIS: Problem with using R2

36

I. Analysis of Data




• If an extra regressor is added to the model, k increases

and unless R2 increases by a more than off-setting

amount, will actually fall.

• If model contains a lot of significant and insignificant

variables, can be negative

TESTING MULTIPLE HYPOTHESIS: Adjusted R2

37

I. Analysis of Data




Dummy is variables that assume such 0 and 1 values

If a model contains M categories, then only M-1 dummy

variables should be created. Otherwise, multicollinearity

Problem

Category for which no dummy variable is assigned is

known as base, benchmark

2 types of dummy variables: Intercept vs. slope change

dummy

DUMMY VARIABLE: How to Create Dummy

38

I. Analysis of Data




Slop = Β3 + β4D

I. Different Intercept

JAN is dummy = 1 if January

= 0 otherwise

II. Different Slope

𝑅𝑡 − 𝑅𝑓= 𝛼 + 𝛽1 𝑅𝑀 − 𝑅𝑓 + 𝛽2𝑆𝑀𝐵

+ 𝛽3𝐻𝑀𝐿 + 𝛽4𝐽𝐴𝑁

X

Y

α

β4Regression for Other months

Regression for JAN

𝑅𝐸𝑁𝑇𝑡= 𝛼 + 𝛽1𝐿𝑁𝐴𝐺𝐸 + 𝛽2𝑁𝑂𝑅𝑂𝑂𝑀+ 𝛽3𝐷𝐼𝑆𝑇 + 𝛽4𝐷𝐷𝐼𝑆𝑇

D is dummy = 1 if Safe Area

= 0 Otherwise

DISTANT

RENT

Regression for Criminal Area

Regression for Safe Area

α

DUMMY VARIABLE: 2 Type of Dummy Variables

39

α+β4

I. Analysis of Data




STEP BY STEP

Quantitative Analysis (Multiple Regression)

1. Conceptual Framework

2. Choose Type of regression (Linear vs. Non Linear)

3. Group Variables

4. Analyze Data (Take logarithm or not)

5. Look at the sign of estimated parameters.

6. Test Hypothesis

7. Take a look at Adjust R2

40

I. Analysis of Data




• Three Factor Model (Fama and French (1992))

Kenneth R. FrenchEugene Fama

RESEARCH PAPER: THREE FACTOR MODEL

41

I. Analysis of Data




42

I. Analysis of Data




43

I. Analysis of Data




44

WORK SHOP

#1

I. Analysis of Data




WORK ORDERS : Multiple Regression

(1) Using Three Factor Model to regress Multiple Regression on your group assignment

(2) Interpret F-test, and T-Test.(3) Explain Adjusted R2

(4) Create Dummy variables o Monthly Data : (1) Window Dressing in June and (2)

End-Year Effect. o Annual Data : (1) Asian Crisis during 1997-1999,

(2) Subprime Crisis during 2008-2010, (3) Europe Debt crisis during 2008-2012.

(5) Redo Work Orders (1) – (4) with new model

45

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