The Simple Regression Model

V c Hong V

University of Economics HCMC

June 2015

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Some Terminology

In the simple linear regression model, where y = 0 + 1x + u,we typically refer to y as the

depedent variable, or

left-hand side variable, or

explained variable, or

regressand

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Some Terminology (cont)

In the simple linear regression of y on x , we typically refer x asthe

independent varialbe, or

right-hand side variable, or

explanatory variable, or

regressor, or

covariate, or

control variable.

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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

This is not a restrictive assumption, since wa can always use 0to normalize E (u) to 0.

We need to make a crucial assumption about how u and x arerelated.

We want it to be the case that knowing something about x doesnot give us any information about u, so that they are completelyunrelated. That is, that

E (u|x) = E (u) = 0, which impliesE (y |x) = 0 + 1x .

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

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Ordinary Least Squares

Basic idea of regression is to estimate the population parametersfrom a sample.

Let {xi , yi)i = 1, . . . , n} denote a random sample of size n fromthe population.

For each observation in this sample, it will be the case thatyi = 0 + 1xi + uu

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Population regression line, sample data points and the associatederror terms.

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Deriving OLS Estimates

to derive the OLS estimates we need to realize that our mainassumption of E (u|x) = E (u) = 0 also implies thatCov(x , u) = E (xu) = 0

Why? Remenber from basic probability thatCov(X ,Y ) = E (XY ) E (X )E (Y )We can write our 2 restrictions just in terms ofx , y , 0, and1, since u = y 0 + 1xE (y 0 1x) = 0, andE [x(y 0 1x)] = 0These are called moment restrictions.

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Deriving OLS using M.O.M

The method of moments approach to estimation impliesimposing the population moment restrictions on the samplemoments.

What does this mean? Recall that for E (X ), the mean of apopulation distribution, a sample estimator of E (X ) is simplythe arithmetic mean of the sample.

We want to choose values of the parameters that will ensurethat the sample versions of our moment restrictions are true1

n

ni=1(yi 0 1xi) = 0

1

n

ni=n xi(yi 0 1xi) = 0

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More Derivation of OLSGiven the definition of a sample mean, and properties ofsummation, we can rewrite the first condition as follows

y = 0 + 1x or

0 = y 1x

ni=1

xi(yi (y 1x) 1xi) = 0n

i=1

xi(yi y) = 1n

i=1

xi(xi x)n

i=1

(xi x)(yi y) = 1n

i=1

(xi x)2

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Solve for the OLS estimated slope is

1 =

ni=1(xi x)(yi y)n

i=1(xi x)2

provided thatn

i=1(xi x)2 > 0

0 = y 1x

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Summary of OLS slope estimate

The slope estimate is the sample covariance between x and ydivided 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.

Only need x to vary in our sample.

Intuitively, OLS is fitting a line through the sample points suchthat the sum of squared residuals is as small as possible, hencethe term least squares.

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

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Sample regression line, sample data points and the associatedestimated error terms.

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Alternative approach to derivationGiven the intuitive idea of fitting a line, we can set up a formalminimize problemThat is, we want to choose our parameters such that weminimize the following:n

i=1(u2) =

ni=1(yi 0 1xi)2

If one uses calculus to solve the minimization problem for thetwo parameters you obtain the following first order condtions,which are the same as we obtained before, multiplied by n.

ni=1

(yi 0 1xi) = 0

ni=1

xi(yi 0 1xi) = 0

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Algebraic Properties of OLS

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 OLSresiduals is zero

The OLS regression line always goes through the mean of thesample.n

i=1 ui = 0 and thus

ni=1 uin

= 0ni=1 xi ui = 0

y = 0 + 1x

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More terminology

We can think of each observation as being made up of anexplained part, and an unexplained part, yi = yi + ui . We thendefine the following:n

i=1(yi y)2 is the total sum of squares (SST)ni=1(yi y)2 is the explained sum of squares (SSE)ni=1 ui

2 is the residual sum of squares (SSR)

Then SST = SSE + SSR

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Proof that SST = SSE + SSR

(yi y)2 =

[(yi yi) + (yi y)]2

=

[ui + (yi y)]2

=

ui2 + 2

ui

2(yi y) +

(yi y)2

= SSR + 2

ui2(yi y) + SSE

and we know that

ui(yi y) = 0

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Goodness-of-fit

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

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

R2 =SSE

SST= 1 SSR

SST

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Unbiasedness of OLS

Assume the population model is linear in parameters asy = 0 + 1x + u

Assume we can use a random sample of size n,{(xi , yi) : i = 1, 2, . . . , n}, from the population model. Thus wecan write the sample model yi = 0 + 1xi + ui

Assume E (u|x) = 0 and thus E (ui |xi) = 0Assume there is variation in the xi

In order to think about unbiasedness, we need to rewrite ourestimator in term of the population parameters.

Start with a simple rewrite of the formula as

1 =

(xi x)yis2x

, where

s2x =

(xi x)2V c Hong V (UEH) Applied Econometrics June 2015 19 / 1

Unbiasedness of the OLS

(xi x)yi =

(xi x)(0 + 1xi + ui)

=

(xi x)0 +

(xi x)1xi +

(xi x)ui= 0

(xi x) + 1

(xi x)xi +

(xi x)ui

(xi x) = 0(xi x)xi = 0

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Unbiasedness of OLS (cont)

so, the numerator can be written as 1s2x +

(xi x)ui , andthus

1 = 1 +

(xi x)ui

s2xlet di = (xi x), so that1 = 1 + (

1

s2x)

diui , then

E (1) = 1 + (1

s2x)

diE (ui) = 1

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Unbiasedness Summary

The OLS estimates of 1 and 0 are unbiased

Proof of unbiasedness depends on our 4 assumptions - if anyassumption fails, then OLS is not neccessarily unbiased

Remember unbiasedness is a description of the estimator - in agiven sample we may be "hear" or "far" from the trueparameter.

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Variance of the OLS Estimators

Now we know that sampling distribution of our estimate iscentered around the true parameter

We want to think about how spead out this distribution is

much easier to think about this variance under an additionalassumption, so

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

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Variance of OLS (cont)

Var(u|x) = E (u2|x) [E (u|x)]2E (u|x |) = 0, so 2 = E (u2|x) = E (u2) = Var(u)Thus 2 is also the unconditional variance, called the errorvariance

, the square root of the error variance is called the standarddeviation of the error

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

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Homoskedastic Case

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Heteroskedastic Case

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Variance of OLS (cont)

Var(1) = Var(1 +1

s2x

diui)

= (1

s2x)2Var(

diui)

= (1

s2x)2

d2i Var(ui)

= (1

s2x)2

d2i 2 = 2(

1

s2x)2

d2i

= 2(1

s2x)2 =

2

s2x= Var(1)

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Variance of OLS Summary

The larger the error variance, 2, the larger the variance of theslope estimate.

The larger the variability in the x , the smaller the variance of theslope estimate.

As a result, a