641 lecture midterm review

8/2/2019 641 Lecture Midterm Review

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Econ 641Review for the Midterm

1/31/2012

Office hour for Exam:Today: after lecture,

Wednesday: 2:00-3:00

Thursday: 10:00-11:00

1Leila Farivar, OSU, Econ 641


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The Exam

In essay format:

Theoretical/conceptual question (Theorems, Assumptions, Graphs,

Intuitions, Explanations,)

Calculation questions (Good example would be the problems in the

HW and Quiz, and the solved examples in the textbook.) Proofs (similar to those done in lecture, quizzes or problem sets)

You can use calculators

You can bring and use a formula sheet

Size: Standard or A4 paper It must be handwritten

It must be one-sided

You must turn it in (otherwise youll lose credit from your exam)

Leila Farivar, OSU, Econ 641 2


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Steps in Empirical Economic Analysis

An empirical analysisuses data to test a theoryor to estimate a relationship.

First step : Careful formulation of the question ofinterest.

Second step: Specify an economic model

Third Step: Construct an econometric model

Fourth Step: Using the model, various hypothesesof interest can be stated in terms of the unknownparameters



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What weve learnt so far:

Econometric Model

y=0+1x+u Simple(only one indep var) Linear model

u is a random variable, representing all the

factors that affect y, besides x.

0 : the intercept parameter

1 : the slope parameter



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Assumptions on Error term

(1) E(u)=0

(2) E(u|x)=E(u)

Combining assumptions (1) and (2): E(u|x)=0



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Systematic vs. Unsystematic parts of y

y= 0+1x +u 0+1x : Systematic part of y. The part of y

explained by independent variable(s).(Deterministic part of y)

u: Unsystematic part of y. The part of y notexplained by independent variable(s). (Stochasticpart of y)

Given E(u|x)=0 E(y|x)= 0+1x

E(y|x) : Systematic part of y

u : Unsystematic part of y



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E(y|x)= 0+1x



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Estimating 0 & 1

Use the assumptions E(u)=0 and E(u|x)=E(u)

1) E(u)=0 E(y- 0-1x)=0

2) E(u|x)=E(u) E[x(y- 0-1x)]=0



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Estimating 0 & 1

We call these Ordinary Least Squares (OLS) Estimates.



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Fitted Value yî

The value the model predicts for y when x=xi

To get the fitted value,

Substitute ^0

& ^1

for

0&

1in the

deterministic part of the model

Evaluate at x=xi

yî= ^0+^1xi yî is the predicted (by model) part of yi

The left over part of yi, is called the residual.



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Residual u^i

The residual for observation iis the difference

between the actual yi and its fitted value.

u^i=y

i-y^

i=yi- ^0-^1xi

Ordinary Least Squares is a technique that

estimates ^0 and ^1 by minimizing the sum

of squares of these residuals.



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SST, SSE, SSR

Define Total Sum of Squares (SST or TSS) as:

Define Explained Sum of Squares (SSE or ESS):

Define Residual Sum of Squares (SSR or RSS):



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Goodness of Fit

It can be proved (in Problem Set 2) that

SST=SSE+SSR

Need for a measure to say how well the OLSline fits the data:

Coefficient of determination (R-squared)

R2=SSE/SST

R2 is the fraction of sample variation in y thatis explained by x(s).



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Incorporating Nonlinearities



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Expectation and Variance of^

The parameters 0 and 1 are derived from

the population and are unique

The statistics ^0

and ^1

are derived from

sample, and are NOT unique. For each

different sample, we get a new set of^s.

^s are random variables.

^s have distribution, expected values, and

variance.Sampling distribution



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Bias

Bias of an estimator: The difference between

the Expected value of the estimator and the

true (popuation) value of the parameter.

Consider ^ as a general estimator for the

parameter ,

Bias(^)=E(^)-

If Bias(^)=0, then ^ is called an unbiasedestimator.



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SLR Assumptions

(Gauss-Markov Assumptions)

Assumption SLR.1 (Linear in Parameters)

Assumption SLR.2 (Random Sampling)

Assumption SLR.3 (Sample Variation in theExplanatory Variable)

Assumption SLR.4 (Zero Conditional Mean)

Assumption SLR.5 (Homoskedasticity)



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Theorem: Unbiasedness of OLS

Given assumptions SLR1-SLR4

E(^0)= 0 and E(^1)= 1

In other words:

distribution of^0 is centered around 0.

distribution of^1 is centered around 1.



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Sampling variance of OLS estimators


Theorem: Under assumptions SLR1-SLR5, andconditioned on the sample values of {x1,,xn},


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Estimating the Error Variance (^2)

2 (error variance) is a population parameter,

and thus often unknown to us.

we need an estimator for it: ^2

Use residuals and estimated their variance

The proposed estimator:



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Unbiasedness of^2

Theorem: Under assumptions SLR1-SLR5, ^2is an unbiased estimator for 2.

E(^2 )=2



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Another interpretation of ^1 inMLR

(Partialling out) Consider the MLR model of

y^= ^0+ ^

1x1 + ^

2x2

An alternative formula for ^1 is

Where r^1are the residual from the simpleregression of x1 on x2



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Partialling out

To put this into simple steps:1. Regress the one independent variable, x1, on the other

independent variable, x2.

2. Obtain the residuals r^1 (The y plays no role here).

3. Do a simple regression of y on r^

1to obtain ^

1. r^1 is the part of x1 that is uncorrelated with x2

r^1 is x1after the effects of x2has been partialled out.

Thus ^1 here, measures the relationship between y and

x1 after the effect of x2 has been taken care of



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Connection bw 2 & 3 variable OLS

Consider these two models:3 variables model: y^= ^0 +

^1x1+

^2x2

2 variables model: y~ = ~0 + ~

1x1

Define ~ as the slope estimate in the auxiliary

regression of x2 on x1:

x~2 = ~

0 + ~

1x1 We want to compare the estimators of 1 in these

two models. The relationship bw the two estimators of 1 is:

~1 = ^

1+ ^

2 . ~

1



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MLR Assumptions

Assumption MLR.1 Linear in Parameters

Assumption MLR.2 Random Sampling

Assumption MLR.3 No Perfect Collinearity

Assumption MLR.4 Zero Conditional Mean

Assumption MLR.5 Homoskedasticity

Assumption 1-5 are collectively known as

Gauss-Markov Assumptions for cross-sectionalanalysis



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Thm: Sampling Variance of OLS Slopes

Under Assumptions MLR. 1 through MLR. 5,the sampling variance of ^ is

j= 1, 2, , k

SSTj= ( xij _ xjbar)2 R2j is the R-squared from regressing xj on all other

independent variables (including the intercept).



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Estimating 2

2 is a population parameter, and thus it is

unknown to us.

An estimator for 2 is

n-k-1 is degree of freedom (df)

= (number of observation)-(number of estimated

parameters)



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Estimating 2

Thm: Under the 5 classical assumptions,

E(^2 )= 2

Standard Deviationof ^ : square root ofvariance of ^

Standard Errorof ^ : square root of variance of^, when 2 is unknown and we use ^2 instead



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Efficiency of OLS

Efficiency of an estimator= It having a smaller

variance.

Gauss-Markov Thm (next slide) shows: In the

class oflinear unbiased estimators, OLS

estimator have the least variance. They are

most efficient (Best).



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Gauss-Markov Theorem

Under Gauss-Markov assumptions of

Assumption MLR.1 Linear in Parameters

Assumption MLR.2 Random Sampling

Assumption MLR.3 No Perfect Collinearity Assumption MLR.4 Zero Conditional Mean

Assumption MLR.5 Homoskedasticity

OLS estimators for , are the Best LinearUnbiased Estimators . OLS is BLUE.



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Single Parameter Test(Simple hypothesis)

We learned how to do hypothesis testing on

just a single parameter at a time (Ho: j = jHo)

when the population variance is unknown to

us. t-Test

This is called Simple Hypothesis Test.

The hypothesis only involves a single

parameter of the model.

Simple=one statement in Ho



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Confidence Interval

An interval that contains the true value of theparameter, with some certain confidence level.

Under the classical assumptions, we can

construct a confidence Interval (C.I.) for thepopulation parameter (j).

The confidence level = 1-

A 95% C.I. for j :

j + - c.se( j)

Where c is the 97.5% percentile in a tn-k-1 distribution



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Testing single linear combination ofs

Example: Ho: 1= 2 Ho: 12 =0

t= (^1 - ^2)/se(^1 - ^2 )

Example: Ho: 1+ 2=1

t= (^1+ ^2 -1)/se(^1+ ^2 )

Example: Ho: 1+ 22=0

t= (^1

+2 ^2

)/se(^1

+ 2^2

)

Once you get the t-statistic, the rest of the test is like

before.


MLR example: Consider the estimation output of the MLR model of log(wage)


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MLR example: Consider the estimation output of the MLR model of log(wage)


Included observations: 6763

Variable Coefficient Std. Error t-Statistic Prob.

C 1.705508 0.023571 72.35610 0.0000JC 0.063786 0.006636 9.611468 0.0000

UNIV 0.071702 0.002269 31.59688 0.0000

EXPER 0.004041 0.000159 25.33776 0.0000

FEMALE -0.213527 0.010636 -20.07523 0.0000

HISPANIC -0.015440 0.024178 -0.638570 0.5231

R-squared 0.266218 Mean dependent var 2.248096Adjusted R-squared 0.265675 S.D. dependent var 0.487692

S.E. of regression 0.417917 Akaike info criterion 1.093817

Sum squared resid 1180.139 Schwarz criterion 1.099867

Log likelihood -3692.744 Hannan-Quinn criter. 1.095906

F-statistic 490.2903 Durbin-Watson stat 1.967646

Prob(F-statistic) 0.000000

Variance MatrixC JC UNIV EXPER FEMALE HISPANIC

C 0.000556 -1.81E-05 -1.82E-05 -3.45E-06 -0.000123 -4.33E-05

JC -1.81E-05 4.40E-05 1.85E-06 -9.68E-09 1.55E-06 -8.27E-07

UNIV -1.82E-05 1.85E-06 5.15E-06 4.86E-08 2.70E-06 5.68E-06

EXPER -3.45E-06 -9.68E-09 4.86E-08 2.54E-08 4.79E-07 2.41E-08

FEMALE -0.000123 1.55E-06 2.70E-06 4.79E-07 0.000113 4.33E-06HISPANIC -4.33E-05 -8.27E-07 5.68E-06 2.41E-08 4.33E-06 0.000585

Considering the estimation output of the MLR model of


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Considering the estimation output of the MLR model of

log(wage) in the last slide, answer the following questions.(For each hypothesis test, draw a distribution graph, and dont forget to do so for the exam as well.)

1. What is the estimated wage of a single Hispanic woman with 2 years of

education in JC who has no prior job experience?2. Is the effect of job experience statistically significant?

3. Does being female have a role in how much one person earns?

4. Does being Hispanic have a role on how much one person earns?

5. Do you want to reconsider your answer to (1)?

6. Do you agree to this statement: The effect of 1 additional year in junior

college balances the negative effect of being female?

7. What is the 99% confidence interval on the effect of junior college?

8. What are the SST, SSR, and SSE of regression?

9. Is the effect of being Hispanic statistically positive at 10%? What is thepvalue of this test?

10. You want to know if gender discrimination would decrease the wage of the

woman by more than 10%. Then State and test the relevant hypothesis that

answers this question.

Leila Farivar OSU Econ 641 36

641 lecture midterm review

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