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The Simple Regression Model

y = 0 + 1x + u

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Some Terminology

In the simple linear regression model, where y = 0 + 1x + u, we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or

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Some Terminology

we typically refer to x as the Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Control Variables

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A Simple Assumption

The average value of u, the error term, in the population is 0. That is,

E(u) = 0

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We also assume

E(u|x) = 0

E(y|x) = 0 + 1x

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..

x1 x2

E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x)

E(y|x) = 0 + 1x

y

f(y)

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Ordinary Least Squares (OLS)

Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population

For each observation in this sample, it will be the case that

yi = 0 + 1xi + ui

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.

..

.

y4

y1

y2

y3

x1 x2 x3 x4

}

}

{

{

u1

u2

u3

u4

x

y

Population regression line, sample data pointsand the associated error terms

E(y|x) = 0 + 1x

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Basic idea of regression is to estimate the population parameters from a sample

Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible.

The residual, û, is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point

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.

..

.

y4

y1

y2

y3

x1 x2 x3 x4

}

}

{

{

û1

û2

û3

û4

x

y

Sample regression line, sample data pointsand the associated estimated error terms (residuals)

xy 10ˆˆˆ

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One approach to estimate coefficients

Given the intuitive idea of fitting a line, we can set up a formal minimization problem

That is, we want to choose our parameters such that we minimize the following:

n

iii

n

ii xyu

1

2

101

2 ˆˆˆ

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It could be shown that estimated coefficient is

1

12

1

ˆ

n

i ii

n

ii

x x y y

x x

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Summary of OLS slope estimate

The slope estimate is the sample covariance between x and y divided by the sample variance of x

If x and y are positively correlated, the slope will be positive

If x and y are negatively correlated, the slope will be negative

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Algebraic Properties of OLS: in English

The sum of the OLS residuals is zero

Thus, the sample average of the OLS residuals is zero as well

The sample covariance between the regressors and the OLS residuals is zero

The OLS regression line always goes through the mean of the sample

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Algebraic Properties of OLS: In mathematics:

xy

ux

n

uu

n

iii

n

iin

ii

10

1

1

1

ˆˆ

0ˆ

0

ˆ

thus,and 0ˆ

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More terminology

SSR SSE SSTThen

(SSR) squares of sum residual theis ˆ

(SSE) squares of sum explained theis ˆ

(SST) squares of sum total theis

:following thedefine then Weˆˆ

part, dunexplainean and part, explainedan of up

made being asn observatioeach ofcan think We

2

2

2

i

i

i

iii

u

yy

yy

uyy

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Notations Alert

The notation SSR (Sum of Squared Residuals) in this handout and my other lecture slides= ESS (Error Sum of Squares) in our textbook.

The notation SSE (Sum of Squared Explained) in this handout and my other lecture slides = RSS (Regressed Sum of Squares) in our textbook.

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Goodness-of-Fit

How do we think about how well our sample regression line fits our sample data?

Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression

R2 = SSE/SST = 1 – SSR/SST

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OLS regressions

Now that we’ve derived the formula for calculating the OLS estimates of our parameters, you’ll be happy to know you don’t have to compute them by hand

Regressions in GRETL are very simple.

Have you installed the software yet?

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Under some conditions, OLS esimated coefficients are unbiased.

22

1 1 2

21 1 1

. :

ˆ

ˆ 1 *

x i

i i

x

i ix

Define

S x x It could be shown that

x x u

S

E x x E uS

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Unbiasedness Summary

The OLS estimates of 1 and 0 are unbiased

Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter

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Variance of the OLS Estimators

Now we know that the sampling distribution of our estimated coefficient is centered around the true parameter

Want to think about how spread out this distribution is

Assume Var(u|x) =Var(u) = 2

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Variance of OLS estimators

2 is called the error variance

, the square root of the error variance is called the standard deviation of the error

E(y|x)=0 + 1x and Var(y|x) = 2

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Variance of OLS estimator

2

21̂x

VarS

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Variance of OLS Summary

The larger the error variance, 2, the larger the variance of the slope estimate

The larger the variability in the xi, the smaller the variance of the slope estimate

Problem that the error variance is unknown

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Estimating the Error Variance

We don’t know what the error variance, 2, is, because we don’t observe the errors, ui

What we observe are the residuals, ûi

We can use the residuals to form an estimate of the error variance

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Estimating the Error Variance

0 1

2

2 2

ˆ ˆˆ

an unbiased estimator of is

1ˆ ˆ / 2

2

i i i

i

u y x

u SSR nn

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Estimating Standard Error of coefficients Estimate

1

2 2

1̂ ˆ se / ix x