Chapter2 Econometrics Old

The Simple Regression Model

Võ Đức Hoàng Vũ

University of Economics HCMC

June 2015

Võ Đức Hoàng Vũ (UEH) Applied Econometrics June 2015 1 / 1

Some Terminology

In the simple linear regression model, where y = β0 + β1x + u,we typically refer to y as the

depedent variable, or

left-hand side variable, or

explained variable, or

regressand


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.


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 β0

to 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 implies

E (y |x) = β0 + β1x .


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


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 = β0 + β1xi + uu


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


Deriving OLS Estimates

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

Cov(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 , β0, andβ1, since u = y − β0 + β1x

E (y − β0 − β1x) = 0, and

E [x(y − β0 − β1x)] = 0

These are called moment restrictions.


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 − β0 − β1xi) = 0

1

n

∑ni=n xi(yi − β0 − β1xi) = 0


More Derivation of OLS

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

y = β0 + β1x or

β0 = y − β1x

n∑i=1

xi(yi − (y − β1x)− β1xi) = 0

n∑i=1

xi(yi − y) = β1

n∑i=1

xi(xi − x)

n∑i=1

(xi − x)(yi − y) = β1

n∑i=1

(xi − x)2


Solve for the OLS estimated slope is

β1 =

∑ni=1(xi − x)(yi − y)∑n

i=1(xi − x)2

provided that∑n

i=1(xi − x)2 > 0

β0 = y − β1x


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.


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


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) =∑n

i=1(yi − β0 − ˆβ1xi)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.

n∑i=1

(yi − β0 − ˆβ1xi) = 0

n∑i=1

xi(yi − β0 − β1xi) = 0


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

= 0∑ni=1 xi ui = 0

y = β0 + β1x


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


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


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


Unbiasedness of OLS

Assume the population model is linear in parameters asy = β0 + β1x + 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 = β0 + β1xi + ui

Assume E (u|x) = 0 and thus E (ui |xi) = 0

Assume 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

β1 =

∑(xi − x)yi

s2x

, where

s2x =

∑(xi − x)2


Unbiasedness of the OLS

∑(xi − x)yi =

∑(xi − x)(β0 + β1xi + ui)

=∑

(xi − x)β0 +∑

(xi − x)β1xi +∑

(xi − x)ui

= β0

∑(xi − x) + β1

∑(xi − x)xi +

∑(xi − x)ui

∑(xi − x) = 0∑(xi − x)xi = 0


Unbiasedness of OLS (cont)

so, the numerator can be written as β1s2x +

∑(xi − x)ui , and

thus

β1 = β1 +

∑(xi − x)ui

s2x

let di = (xi − x), so that

β1 = β1 + (1

s2x

)∑

diui , then

E (β1) = β1 + (1

s2x

)∑

diE (ui) = β1


Unbiasedness Summary

The OLS estimates of β1 and β0 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.


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 |) = σ2 (Homoskedasticity)


Variance of OLS (cont)

Var(u|x) = E (u2|x)− [E (u|x)]2

E (u|x |) = 0, so σ2 = E (u2|x) = E (u2) = Var(u)

Thus σ2 is also the unconditional variance, called the errorvariance

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

Can say: E (y |x) = β0 + β1x and Var(y |x) = σ2


Homoskedastic Case


Heteroskedastic Case


Variance of OLS (cont)

Var(β1) = Var(β1 +1

s2x

∑diui)

= (1

s2x

)2Var(∑

diui)

= (1

s2x

)2∑

d2i Var(ui)

= (1

s2x

)2∑

d2i σ

2 = σ2(1

s2x

)2∑

d2i

= σ2(1

s2x

)2 =σ2

s2x

= Var(β1)


Variance of OLS Summary

The larger the error variance, σ2, 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 larger sample size should decrease the variance ofthe slope estimate.

Problem that the error variance is unknown.


Estimating the Error Variance

We don’t know what the error variance, σ, is, because we don’tobserve the errors, ui .

What we observe are the residuals, ui .

We can use the residuals to form an estimate of the errorvariance.


Error Variance Estimate (cont)

ui = yi − β0 − β1xi

= (β0 + β1xi + ui)− β0 − β1xi

= ui − (β0 − β0)− (β1 − β1)

Then , an unbiased estimator of σ2 is

σ2 =1

(n − 2)

∑ui

2 = SSR/(n − 2)


Error Variance Estimate (cont)

σ =√σ2: Standard error of the regression recall that sd(β) =

σ

sx

if we substitute σ for σ then we have the standard error of β1.

se(β1) = σ/√∑

(xi − x)2


The Gauss-Markov Assumption (GM) for SimpleRegression

Assumption SLR. 1 Linear in Parameters

In the population model, the dependent variable, y , is related tothe independent variable, x , and the error (or disturbance), u, as

y = β0 + β1x + u

where β0 and β1 are the population intercept and slope param-eters, respectively.


The GM Assumption (cont)

Assumption SLR. 2 Random Sampling

We have a random sample of size n, {xi , i = 1, 2, . . . , n}, fol-lowing the population model in Assumption SLR. 1



Assumption SLR. 3 Sample Variation in the Explanatory

The sample outcomes on x , namely, {xi , i = 1, 2, . . . , n}, arenot all the same value.



Assumption SLR. 4 Zero Conditional Mean

The error u has the same variance given any value of theexplanatory variable. In other words,

E (u|x) = 0.



Assumption SLR. 5 Homoskedasticity

The error u has the same variance given any value of theexplanatory variable. In other words,

Var(u|x) = σ2.


Stata code

reg lwage exp wks occ

Source | SS df MS Number of obs = 4165

-------------+------------------------------ F( 3, 4161) = 266.12

Model | 142.774178 3 47.5913928 Prob > F = 0.0000

Residual | 744.130723 4161 .178834589 R-squared = 0.1610

-------------+------------------------------ Adj R-squared = 0.1604

Total | 886.904902 4164 .212993492 Root MSE = .42289

------------------------------------------------------------------------------

lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

exp | .0100694 .0006 16.78 0.000 .0088931 .0112456

wks | .0058775 .0012784 4.60 0.000 .0033711 .0083839

occ | -.311163 .0131532 -23.66 0.000 -.3369502 -.2853758

_cons | 6.360351 .062024 102.55 0.000 6.238751 6.481951

------------------------------------------------------------------------------


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