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7/29/2019 Ch04 Lectu
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Eco 205: Econometrics
Any questions?
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population data point
Y
X
Observed Value
of Y for X3
X3
Population Linear Regression Model
Yi = b 0 + b 1X i + u
u
b 0X11
Y11
u11
pop slope = b 1
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chosen in sample
not chosen in sample
estimated error for X3(residual)
Y
X
Y3
Yi
=b0
+b1
Xi
X3
estimated slope =estimated
intercept =
Sample Regression Equation
u
u
b 0
b1
pop slope = b 1
b 0
Y3
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The OLS estimator solves
b 0 , b 1min[Yi - ( b 0
i=1
n
+ b 1Xi)]2
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CaliforniaTest Score/Class Size data
Interpretations
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Predicted values & residuals:
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OLS regression: STATA output
regress testscr str, robust
Regression with robust standard errors Number of obs = 420
F( 1, 418) = 19.26
Prob > F = 0.0000
R-squared = 0.0512
Root MSE = 18.581
-------------------------------------------------------------------------| Robust
testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
--------+----------------------------------------------------------------
str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671
_cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057
-------------------------------------------------------------------------
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Measures of Fit
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TheStandard Error of theRegression (SER)
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Root Mean Squared Error (RMSE)
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R2 and SER Example
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The Least Squares Assumptions
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LSA #1: E(u|X = x) = 0
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LSA #2: (Xi,Yi), i= 1,,n are i.i.d.
LSA #3: E(X4) < and E(Y4) <
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OLS can be sensitive to an outlier
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Sampling Distribution of1b
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Some Preliminary Algebra
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b 1 - b 1 =
(Xi - X)
i=1
n
(ui - u )
(Xi - X)2
i=1
n
=
(Xi - X)
i=1
n
ui
(Xi - X)2
i=1
n
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Now we can calculate E( ) and var( )1b1b
E[ b 1 ] = E[b 1] + E (Xi-
X)i=1
n
ui
(Xi - X)2
i=1
n
E[ b 1 ] = b 1 + E(X
i- X)E[
i=1
n
u iXi ]
(Xi - X)2
i=1
n
= ?
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Next calculate var( )1
b
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The larger the variance of X, thesmaller the variance of
1
b
There are the same number of black and blue dots using whichwould you get a more accurate regression line?
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What is the sampling distribution of ?1
b
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We are now ready to turn to hypothesis tests & confidence
intervals