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Page 1: Ols by hiron

Welcome to the Presentation

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Course Title: Econometrics II

Course No.: Econ 4203

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Ordinary Least Squares (OLS) Method

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Points to be discussed

What is OLS (In a word)

Why OLS is In Econometrics

Details on OLS Concept

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What is OLS (In a word)

OLS is an econometric method used to derive estimates (particular

numerical value obtained by the estimator) of the parameters of

economic relationships from statistical observations. It is the

technique used to estimate a line that will minimize the error (The

difference between the predicted and the actual values of a

dependent variable). The method of Ordinary Least Squares is

attributed to Carl Friedrich Gauss, a German mathematicians.

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Why OLS is in Econometrics

In econometric analysis we usually try to find out the causal

relationship between two sets of variables. Among them one is

dependent and another is independent. By ‘Regression’ analysis

we can draw the nature and intensity of relationship between two

sets of variable. In regression, the dependent variable is termed as

‘explained’ or ‘regressand’ variable and independent variable(s)

is/are termed as ‘explanatory’ or ‘regressor’ variable(s).

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Why OLS is in EconometricsA two variable regression function :

Where, = Dependent variable; = Independent variable; = Intercept;

= Slope; and = Stochastic disturbance term.

From the above function, it is clear that the relationship between X and Y is

mainly determined by , and the disturbance term . So it is

important to estimate the s.

There are some other methods like- Moment method, Maximum Likelihood

method. But having some desirable properties (property of linearity,

unbiasedness and minimum variance) we can apply OLS simply to

estimate the regression parameters , s.

But, Why we need to estimate s by OLS?6/21/2013 7

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Why we need to estimate s by OLS?

A two variable regression function explicits the relationship between the

explained and explanatory variables. In this case if we consider the whole

‘population’ then we get Population Regression Function (PRF). In PRF

there is no estimation error, because all the elements are considered.

But, in practice, we usually consider a ‘sample’ from the population to draw

a relationship between the variables. We get the form of relationship by

Sample Regression Function (SRF). By considering a portion of whole

elements, estimation errors (due to sampling fluctuation) are raised here.

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Why we need to estimate s OLS?

From SRF we need to approximate the PRF. But, there are some

estimation errors. OLS help to minimize (make least) the

estimation errors and help to find a close approximation of PRF

from SRF.

Due to the sampling fluctuation, the true parameters ( s of PRF)

varies significantly than the sampling parameters ( s of SRF). In

this case OLS is a simpler devise or method by which we can

estimate the least valued s to make the SRF as a best possible

mirror of PRF .

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Details on OLS Concept

To understand the method of Ordinary Least Squares, we first explain

the least squares principle.

We know , the two variable PRF:

However, the PRF is not directly observable (as noted earlier). We

estimate it from the SRF:

(1)

(2)

(3)

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Where is the estimated (of SRF) (conditional mean) value of .

But how is the SRF itself determined?

To see this, let us proceed as follows.

First, from equation (3), we get:

(4)

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Now given n pairs of observations on Y and X, we would like to

determine the SRF in such a manner that it is as close as possible

to the actual Y. To this end, we may adopt the following least

square criterion: Choose the SRF in such a way that the sum of

the squared residuals is as small as possible.

This criterion states that the SRF can be fixed in such a way that

is as small as possible, where are the squared residuals.

(5)

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It is obvious from equation (5) that

that is the sum of the squared residuals is some function of the

estimators and . For any given set of data, choosing different

values for and will give different .

Now which sets of values should we choose?

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Since s are the function of the squared residuals, we should choose those

s whice give us a lower (least) value of squared residuals.

And the method of Ordinary Least Squares (OLS) provide us a very shortcut

way to get the potential s to estimate the SRF as the best possible

approximation of PRF.

As an very essential additional concept, we will discuss shortly about the

‘goodness of fit’. It means that how ‘good’ the estimated least square

regression (SRF) line ( ) fits to the sample observations. That

is, what portion of change in dependent variable (X) can be explained by

the change of independent variable (Y). And, we also get known the

remained unexplained portion from it.6/21/2013 14

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The ‘goodness of fit’ is measured by coefficient of determination,

Higher the value of , more good to fit and vice versa.

From the following figure,Total variation inY (TSS) = Explained variation (ESS) + Unexplained variation (RSS)

r2

Total variation

Explained Variation

Unexplained Variation

Yi

0

Y

X

Estimated SRF

r20 r2 1

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If we expressed the variations by equations then we get,

Total variation inY=

Explained variation=

Unexplained variation=

And, by the function: we can compute the value of

‘goodness of fit’ of an estimated SRF; that is how good the estimated

SRF is close to the PRF.

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Please, recall that, is the summation of squared residuals. So

from the equation it is easily understandable that, if

the value of squared residuals is low then the value of will be

high, and the estimated SRF will be more close to PRF. And, the

OLS gives us the least value of . So, it is certainly realized that,

by following least squares method, we can find the best

approximated SRF of PRF. And, the concept of OLS has much

wider application in Econometrics.

r2

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References:

Gujarati,N.D.(2007) Basic Econometrics, 4th ed., Tata McGraw-Hill

Publishing Company Limited: New Delhi, pp. 1-118.

Hansen, E.B.(2010) Econometrics, University Press, University of

Wisconsin, pp. 1-64.

Koutsoyiannis, A.(2003) Theory of Econometrics, 2nd ed., PALGRAVE:

New York, pp. 48-62.

Nagler, J.(January, 2001). Notes on Ordinary Least Squares Estimates.

Retrived December 14, 2010, from http:/ www.

Economics.about.com/OLS/pdf.6/21/2013 18

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Queries?

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