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7/25/2019 Session 1-Upload ET

1/19

PGPEX

Session 1

2016

Understanding EconometricsA case of Simple Regression


2/19

Data types we face in Real World

Cross Section: bank wagesbankwages.csv

Time series: Stock return data

Panel data

1/28/2016 2Understanding Econometrics
http://bankwages.csv/http://stockreturn.csv/http://paneldata.csv/http://paneldata.csv/http://stockreturn.csv/http://bankwages.csv/


3/19

Graphical illustration of the correlation coefficient

X

Y

v

vv

v

1

4

2

3

x

y

Quadrant yi -y xi - x (yiy) (xi x)

1 + + +

2 + - -

3 - - +

4 - + -

Algebraic Signs of the Quantities (yiy ) and (xi x )



4/19

Fail to capture

The relationship: Y=50-X^2

The data:

Y X

1 -7

14 -6

25 -5

34 -4

41 -3

46 -2

49 -1

50 0

49 1

46 2

41 3

34 425 5

14 6

1 7 0

10

20

30

40

50

Y

-10 -5 0 5 10X



5/19

Flaw in Correlation CoefficientY1 X1 Y2 X2 Y3 X3 Y4 X4

8.04 10 9.14 10 7.46 10 6.58 8

6.95 8 8.14 8 6.77 8 5.76 8

7.58 13 8.74 13 12.74 13 7.71 8

8.81 9 8.77 9 7.11 9 8.84 8

8.33 11 9.26 11 7.81 11 8.47 8

9.96 14 8.1 14 8.84 14 7.04 8

7.24 6 6.13 6 6.08 6 5.25 8

4.26 4 3.1 4 5.39 4 12.5 19

10.84 12 9.13 12 8.15 12 5.56 84.82 7 7.26 7 6.42 7 7.91 8

5.68 5 4.74 5 5.73 5 6.89 8


use http://www.ats.ucla.edu/stat/stata/examples/chp/p025b, clear

Correlation=.82


6/19

Flaw in Correlation Coefficient

4

6

8

10

12

4 6 8 10 12 14X1

Y1 Fitted values

2

4

6

8

10

4 6 8 10 12 14X2

Y2 Fitted values

4

6

8

10

12

4 6 8 10 12 14X3

Y3 Fitted values

6

8

10

12

14

5 10 15 20X4

Y4 Fitted values



7/19

A flowchart illustrating the dynamic iterative regression process

Start

Formulate the problem

Fit the model

Validate assumptions

Evaluate the fitted

model

Choose a set of variables

Choose form of model

Choose method of fitting

Specify assumptions

Use method of fitting

Residual plots

Outliers detection

Sensitivity analysis

OK?

Ok?

Goodness of fit tests

Use the model for the

intended purposeStop

Yes

Yes

No

No



8/19

Bank wage

Sctatter: A starting point

9.5

10

10.5

11

11.5

12

y

5 10 15 20EDUC

Correlation: 0.6967



9/19

Simple Linear Regression

Although we could fit a line "by eye" e.g. using atransparent ruler, this would be a subjectiveapproach and therefore unsatisfactory.

An objective, and therefore better, way of

determining the position of a straight line is touse the method of least squares.

Using this method, we choose a line such that thesum of squares of distances of all points from the

line is minimized.



10/19

Least-squares or regression line

These distances, i.e., the distance between yvalues and their corresponding estimated

values on the line are called residuals

The line which fits the best is called theregression line or, sometimes, the least-

squares line

The line always passes through the pointdefined by the mean of Y and the mean of X



11/19

Steps involved

1. Statement of theory: Labour

Economics: Salary depends on

education

Step 2: Econometric Model

Step 3: data : bank wage

)(XfY=

11

ueducationysalary ++== )log( 21

1/28/2016 Understanding Econometrics


12/19

Econometric model Building Step

Why error term is appearing?.

1. Omission of other explanatory variables

examples?

Note that there can be many x variables:Multiple regression model

2. Measurement Error & Model Misspecification

3. Purely random

4. Linear approximation



13/19

Statistical Inference(step 4)

4. Next task is to estimate the parameters of the

model so that we can say show relative increase in

salary due to one year of additional education.

To obtain a fitted line

Where y=log(salary) and X=education

XbbY 21

+=



14/19Understanding Econometrics

DERIVING LINEAR REGRESSION COEFFICIENTS: method of OLS

XXnX1

Y

b1

XbbY

uXY

21

21

:lineFitted

:modelTrue

1211 XbbY

1Y

b2

nY

nn XbbY

21

14

1/28/2016 14

DERIVING LINEAR REGRESSION COEFFICIENTS


15/19

Understanding Econometrics


XXnX1

Y

b1

XbbY

uXY

21

21

:lineFitted

:modelTrue

nnnnn XbbYYYe

XbbYYYe

21

1211111

.....

1211 XbbY

1Y

b2

nY

1e

ne

nn XbbY

21

16

1/28/2016 15


16/19

9.5

10

10.5

11

11.5

12

5 10 15 20EDUC

y Fitted values

Bank wage Data



17/19

Actual and Fitted Model

1/28/2016 Understanding Econometrics 17



18/19

Understanding Econometrics

iiiiii

nnnnnn

nnn

XbbYXbYbXbnbY

XbbYXbYbXbbY

XbbYXbYbXbbY

XbbYXbbYeeRSS

2121

22

2

2

1

2

2121

22

2

2

1

2

12111211

2

1

2

2

2

1

2

1

2

21

2

1211

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222

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...

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)(...)(...


19

02220 211

=

ii XbYnb

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RSS

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02220 12

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iiii XbYXXb

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RSS

1/28/2016 18


19/19

19

=

=

22

21

)(

))((

xx

xxyyb

xbyb

i

ii


222XnX

YXnYXb

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ii

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