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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
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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)
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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 y^i
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
y^i= ^0+^1xi y^i 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
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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.
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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)
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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.
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