EC339: Lecture 7Chapter 4-5: Analytical Solutions to OLS

The Linear Regression Model Postulate: The dependent variable, Y, is a function

of the explanatory variable, X, or Yi = ƒ(Xi)

However, the relationship is not deterministic Value of Y is not completely determined by value of X

Thus, we incorporate an error term (residual) into the model which provides a statistical relationship Yi = ƒ(Xi) + ui

The Simple Linear Regression Model (SLR) Remember we are trying to predict Y for a

given X. We assume a linear relationship (in the parameters (i.e., the BETAS))

Ceteris Paribus—All else held equal To account for our ERROR in prediction, we

can add an error term to our prediction. If Y is a linear function of X then

ERRORS are typically written as u, or epsilon representing ANYTHING ELSE that might cause the deviation between actual and predicted values

We are interested in determining the intercept (0) and slope (1)

XY 1ˆ

uYY ˆ

uXY 1

SLR Uses Multivariate Expectations Univariate Distributions

Means, Variances, Standard Deviations Multivariate Distributions

Correlation, Covariance Marginal, Joint, and Conditional Probabilities

,|

1 1

( , )[ | ] ( | )

( )

m mX Y

j Y X j ij j X

f y xE Y x y f y x y

f x

Marginal ProbabilityDensityFn.

Joint ProbabilityDensity Fn.

Conditional ProbabilityDensity Fn.

Conditional Expectation

Joint Distributions Joint Distribution Probability Density Functions

Now want to consider how Y and X are distributed when considered together

INDEPENDENCE When outcomes of X and Y have no influence on one

another, the joint probability is equal to the product of the marginal probability density function

, ( , ) ( , )X Yf x y P X x Y y

, ( , ) ( ) ( )X Y X Yf x y f x f yThink about BINOMIAL DISTRIBUTIONS, each TRIAL is INDEPENDENTand has no effect on the subsequent trial. Also, think of marginal distributions much like a histogram of a single variable.

Conditional Distributions Conditional Probability Density Functions

Now want to consider how Y is distributed when GIVEN a certain value for X Conditional Probability of Y occurring given X, is equal to the joint

probability of X and Y, divided by the marginal probability of X occurring in the first place

INDEPENDENCE If X and Y are independent then the conditional distribution shows

these as marginal distributions. Just as if there is no new information.

,|

( , )( | )

( )X Y

Y XX

f x yf y x

f x

|

( ) ( )( | ) ( )

( )X Y

Y X YX

f x f yf y x f y

f x |

( ) ( )( | ) ( )

( )X Y

X Y XY

f x f yf x y f x

f y

A joint probability is like finding the probability of a “high school graduate” with an hourly wage between “$8 and $10” if looking at education and wage data.

Discrete Bivariate Distributions—Joint Probability Function

For example, assume we flip a coin 3 times, recording the number of heads (H) X = number of heads on the last (3rd) flip Y = total number of heads in three flips

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} X takes on the values {0,1} Y takes on the values {0,1,2,3}

There are 8 possible different joint outcomes (X = 0,Y = 0) (X = 0,Y = 1) (X = 0,Y = 2) (X = 0,Y = 3) (X = 1,Y = 0) (X = 1,Y = 1) (X = 1,Y = 2) (X = 1,Y = 3)

Attaching a probability to each of the different joint outcomes gives us a discrete bivariate probability distribution or joint probability function

Thus, ƒ(x,y) gives us the probability the random variables X and Y assume the joint outcome (x,y)

Properties of Covariance If X and Y are discrete

If X and Y are continuous

If X and Y are independent then cov(X,Y) = 0

1

2

,1 1

( , ) [ ] [ ] [ ]

If independent

[ ] [ ] [ ]

Since this expectation is a "function" of X

[ ( )] ( ) ( )

Remember, X is a "function" of X

[ ( , )] ( , ) ( , )

k

j X jj

k m

h j X Y h jh j

Cov X Y E XY E X E Y

E XY E X E Y

E g X g x f x

E g X Y g x y f x y

Properties of Conditional Expectations,

|1 1

( , )[ | ] ( | )

( )

Conditional Expectations, see Wooldridge

[ | ] 1.05 .45

This is a regression of Wages (Y) on Education (X)

Summing over all possible values of Y i

m mX Y

j Y X j ij j X

f y xE Y x y f y x y

f x

E WAGE EDUC EDUC

n the conditional

expectation. E[WAGE|EDUC=12] is the expected value of

a wage given that the years of education is 12 years, giving

a value of $6.45, much like the predictions we have seen.

The Linear Regression Model Ceteris Paribus—All else held equal Conditional Expectations can be linear or nonlinear

—We will only examine LINEAR functions here.

,|

1 1

( , )[ | ] ( | )

( )

m mX Y

j Y X j ij j X

f y xE Y x y f y x y

f x

The Linear Regression Model For any given level of X many possible values

of Y can exist If Y is a linear function of X then

Yi = 0 + 1Xi + ui

u represents the deviation between the actual value of Y and the predicted value of Y (or 0 + 1X1i)

We are interested in determining the intercept (0) and slope (1)

The Simple Linear Regression Model (SLR) Thus, what we are looking for is the Conditional Expectation of Y Given

values of X. This is what we have called Y-hat thus far. We are trying to predict values of Y given values of X. To do this we must hold ALL OTHER FACTORS FIXED (Ceteris Paribus).

XYXYE 1ˆ]|[

uYY

uYY

ˆ

ˆ

The Simple Linear Regression Model (SLR) LINEAR POPULATION REGRESSION FUNCTION

We can assume that the EXPECTED VALUE of our error term is zero. If the value were NOT equal to zero, we could alter this expected value to equal zero by altering the INTERCEPT to account for this fact.

This makes no statement about how X and the errors are related. IF u and X are unrelated linearly, their CORRELATION will equal zero! Correlation is not sufficient though, since they could be related

NONLINEARLY… Conditional probability gives sufficient conditions as it looks at ALL values

of u, given a value for X. This is zero conditional mean error.

0][ uE

0][]|[ uExuE

The Linear Regression Model: Assumptions Several assumptions must be made about the

random variable error term The mean error is zero, or E(ui) = 0

Errors above and below the regression line tend to balance out

Errors can arise due to Human behavior (may be unpredictable) A large number of explanatory variables are not in the

model Imperfect measuring of dependent variable

The Simple Linear Regression Model (SLR) Beginning with the simple linear regression, taking conditional

expectations, and using our current assumptions gives us the POPULATION REGRESSION FUNCTION (Notice, no hats over the Betas, and that y, is equal to the predicted value, plus an error).

xxyE

xuExxyE

uxy

10

10

]|[

0x]|E[u that assumption theusing and

]|[]|[

on x lconditiona value,expected taking

The Linear Regression Model The regression model asserts that the

expected value of Y is a linear function of X E(Yi) = 0 + 1X1i

Known as the population regression function

From a practical standpoint not all of a population’s observations are available Thus we typically estimate the slope and intercept

using sample data

The Simple Linear Regression Model (SLR) We can also make the following assumptions knowing that E[u|x]=0

0][)]([

0E[xu] that assumption theusing and

0][][

ˆ and 0E[u]

and eduncorrelat areu and that x assumption theUsing

][][][],[

10

10

xuExyxE

uExyE

uyy

uExExuEuxCov

WE NOW HAVE TWO EQUATIONS IN TWO UNKNOWNS!!(The Beta’s are the unknowns). This is how the Method of Momentsis constructed.

Additional assumptions are necessary to develop confidence intervals and perform hypothesis tests

The Linear Regression Model: Assumptions

Says that errors are drawn from a distribution with a constant variance (heteroskedasticity exists if this assumption fails)

ui and uj are independent One observation’s error does not influence

another observation’s error—errors are uncorrelated (serial correlation of errors exist if this assumption fails)

Cov(ui,uj) = 0 for all i j

ii all for )var(ui2

2 ui ~ N(0, ) Error term follows a normal distribution

The Linear Regression Model: Assumptions Cov(Xi,ui) = 0 for all i

Error term is uncorrelated with the explanatory variable, X

The Linear Regression Model: Assumptions Cov(Xi,ui) = 0 for all i

Error term is uncorrelated with the explanatory variable, X

Error term follows a normal distribution

),0(~ 2ui Nu

Ordinary Least Squares-Fit

SSRSSESST

Residuals Squared of Sum ,)ˆ(

Squares of Sum Explained ,)ˆ(

Squares of Sum Total ,)(

0ˆ

ˆˆˆ where,0ˆ

2

1

2

1

2

1

1

101

n

ii

n

ii

n

ii

n

iii

iii

n

ii

uSSR

yySSE

yySST

ux

xyuu

Ordinary Least Squares-Fit

eduncorrelat are valuespredicted and residuals since 0)ˆ(ˆ2 where

SSE )ˆ(ˆ2SSRSST showing

)ˆ()ˆ(ˆ2ˆ)(

)]ˆ(ˆ[)(

)]ˆ()ˆ[()(

SSRSSESST

1

1

1 1

2

1

22

1

2

1

2

1

2

1

2

1

n

iii

n

iii

n

i

n

iiii

n

ii

n

ii

i

n

ii

n

ii

i

n

iii

n

ii

yyu

yyu

yyyyuuyy

yyuyy

yyyyyy

Ordinary Least Squares-Fit

2

1

2

1

2

1

2

12

2

ii

2

)(

)ˆ(1

)(

)ˆ(

1

SSRSSESST

y andbetween x n correlatio down to boils this,t variableindependen one

on dependsonly prediction When .y and ybetween n correlatio the

of valuesquared the toequal is R same. eexactly th istion Interpreta

ncorrelatio squaresimply cannot you ,regression multipleIn

DGENERALIZE SQUARED-R

n

ii

n

ii

n

ii

n

ii

yy

u

yy

yyR

SST

SSR

SST

SSER

Estimation (Three Ways-We will not discuss Maximum Likelihood) We need a formal method to determine the

line that “fits” the data well Distance of the line from observations should be

minimized^ ^ ^ Let Yi = 0 + 1X1i

The deviation of the observation from the line is the estimated error, or residual (ui)

^ ui = Yi - Yi

Ordinary Least Squares Designed to minimize the magnitude of estimated

residuals Selecting an estimated slope and estimated intercept that

minimizes the sum of the squared errors Most popular method known as Ordinary Least Squares

Ordinary Least Squares—Minimize Sum of Squared Errors Identifying the parameters (estimated slope and

estimated y-intercept) that minimize the sum of the squared errors is a standard optimization problem in multivariable calculus Take first derivatives with respect to the estimated

slope coefficient and estimated y-intercept coefficient Set both equations equal to zero and solve the two

equations

Ordinary Least Squares

EQUATIONS. NORMAL the

called are These . parametersour for system thesolve tous allowsWhich

0)ˆˆ(2ˆ

ˆˆ

0)ˆˆ(2ˆ

ˆˆ

where,)ˆˆ(min)(min

.ˆˆˆˆ giving rulechain andon optimizati of calculus

usingfunction thisEstimate ).,( parameters

theoffunction a s themselveare which errors, squared theof value

theminimize want to weY and Xon data of sample a Using

110

1

10

110

0

10

1

210ˆ,ˆ

1

2

ˆ,ˆ

1010

1010

1010

n

iiii

n

iii

n

iii

n

ii

ii

ii

xyx)β,βQ(

xy)β,βQ(

xyu

)β,βQ(xy

Qxy

Ordinary Least Squares-Derived

2)1

(

1)(

2

1

1

1

2

1

2

1 111

1

2

1

21

1 11

1

21

1

21

1 11

1

21

1 111

11

1

21

1 11

11

1

21

1 11

0

11 1

101 1

101

10

1

21

1 10

1 1 1

210

110

)(

)(

][

ˆ

)][(ˆ

][ˆˆ

ˆˆ

ˆˆ1

ˆ)ˆ(

for value thisngSubstituti

ˆ]ˆ[1ˆˆˆ0)ˆˆ(2

equation derivativefirst theFrom

ˆˆ

ˆˆ0)ˆˆ(2

xn

i ix

n

i iyxix

xxn

yxxn

xxn

yxyxn

xxnyxyxn

xxnyxyxn

xnxxyxyxn

xxxxyn

yx

xxxyyx

xyxyn

xnyxy

xxyx

xxyxxyx

n

ii

n

iii

n

ii

n

ii

n

i

n

ii

n

iiii

n

ii

n

ii

n

i

n

ii

n

iiii

n

ii

n

ii

n

i

n

ii

n

iiii

n

ii

n

i

n

ii

n

ii

n

ii

n

iiii

n

ii

n

i

n

ii

n

ii

n

iiii

n

ii

n

i

n

iiii

n

i

n

iii

n

i

n

iii

n

iii

n

ii

n

i

n

iiii

n

i

n

i

n

iiiii

n

iiii

Ordinary Least Squares This results in the normal equations

Which suggests an estimator for the intercept. The means of X and Y are ALWAYS on the regression line.

Ordinary Least Squares

No other estimators will result in a smaller sum of squared errors

Which yields an estimator for the slope of the line

SLR Assumption 1 Linear in Parameters SLR.1

Defines POPULATION model

The dependent variable y is related to the independent variable x and the error (or disturbance) u as SLR.1

andare population parameters

uxy 10

SLR.1 Assumption

SLR Assumption 2 Random Sampling Use a random sample of

size n, {xi,yi): i=1,2,…,n} from the population model

Allows redefinition of SLR.1. Want to use DATA to estimate our parameters

andare population parameters to be estimated

},...,2,1:),{(

SLR.2 Assumption

niyx ii

n1,2,..., ,10 iuxy iii

SLR Assumption 3 Sample variation in

independent variable X values must vary. The

variance of X cannot equal zero

0])[(x

SLR.3 Assumption

2

1i

xEn

i

SLR Assumption 4 Zero Conditional Mean For a random sample,

implication is that NO independent variable is correlated with ANY unobservable (remember error includes unobservable data)

0]|[

SLR.4 Assumption

xuE

n1,2,...,i allfor

0]|[

Sample RANDOM aFor

ii xuE

SLR Theorem 1 Unbiasedness of OLS, estimators should equal the

population value in expectation

12

1

11

2

1

1

2

11

1

2

1

1 1110

1

2

1

1 11

10

1

2

1

110

1

102

1

11

1100

)(

)(

)(

)()(ˆ

)(

)()()(ˆ

)(

)()()(ˆ

numerator examining and)(

))((ˆ

ˆˆ and)(

)(ˆ

]ˆ[ and ,]ˆ[

1 Theorem

,

,

xx

uxx

xx

uxxxx

xx

uxxxxxxx

xx

uxxxxxxx

xx

uxxx

xyxx

yxx

EE

n

ii

i

n

ii

n

ii

i

n

ii

n

ii

n

ii

n

ii

n

iii

n

iii

n

ii

n

ii

n

iii

n

iii

n

ii

n

iiii

n

ii

n

iii

This holds because x and u are assumed to be uncorrelated.

Thus our estimator equals the actual value of Beta

SLR Theorem 1 Unbiasedness of OLS, estimators should equal the

population value in expectation

00

1100

1100

1100

11010

1100

]ˆ[

)]ˆ([]ˆ[

])ˆ([]ˆ[

)ˆ(ˆ

ˆˆˆ

]ˆ[ and ,]ˆ[

1 Theorem

E

xEE

uxEE

ux

xuxxy

EE

The expected value of theresiduals is zero.

Thus our estimator equals the actual value of Beta

SLR Assumption 5 Homoskedasticity The variance of the errors is INDEPENDENT of the

values of X.

2

10

2

)|(

]|[

thatalso implies valueThis

SLR.5 and , SLR.4 SLR.3, Rewriting

)|(

SLR.5 Assumption

xyVar

xxyE

xuVar

Method of Moments Seeks to equate the moments implied by a statistical

model of the population distribution to the actual moments found in the sample

Certain restrictions are implied in the population E(u) = 0

Cov(Xi,uj) = 0 i,j

Results in the same estimators as least squares method

Interpretation of the Regression Slope Coefficient The coefficient, 1, tells us the effect X has on

Y Increasing X by one unit will change the mean

value of Y by 1 units

Units of Measurement and Regression Coefficients Magnitude of regression coefficients depends

upon the units in which the dependent and explanatory variables are measured For example, using cents versus dollars will result

in smaller coefficients Changing both the Y and X variables by the

same amount will not affect the slope although it will impact the y-intercept

Models Including Logarithms For a log-linear model the slope represents the proportionate

(like percentage change) change in Y arising from a unit change in X The coefficients in your regression result in the SEMI-elasticity of Y

with respect to X For a log-log model the slope represents the proportionate

change in Y arising from a proportionate change in X The coefficients in your regression results in the elasticity of Y with

respect to X. This is the CONSTANT ELASTICITY MODEL. For a linear-log model the slope is the unit change in Y

arising from a proportionate change in X

Regression in Excel Step 1: Reorganize data so that variables are

right next to one another in columns Step 2: Data AnalysisRegression

Regression in Excel-Ex. 2.11

Regression in Excel

Regression in Excel

Regression in Excel

ceotensalary 0097.5055.6)log(

follows as isEquation EstimatedYour

T-statistics show that the coefficient on ceoten is insignificant at the 5% level. The p-value for ceoten is 0.128368 which is greater than .05, meaning that you could see this value about 13% of the time. You areInherently testing the null hypothesis that all coefficients are equal to ZERO. YOU FAIL TO REJECT THE NULL HYPOTHESIS HERE ON BETA-1.

Regression in ExcelX Variable 1 Line Fit Plot

y = 0.0097x + 6.5055

R2 = 0.0132

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 40

X Variable 1

Y

Y

Predicted Y

Linear (Y)

Regression in ExcelX Variable 1 Line Fit Plot

y = 0.0097x + 6.5055

R2 = 0.0132

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 40

X Variable 1

Y

Y

Predicted Y

Linear (Y)