the simple linear regression model: specification and estimation econ 4550 econometrics memorial...
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ECON 4550Econometrics Memorial University of Newfoundland
Adapted from Vera Tabakova’s notes
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2.1 An Economic Model 2.2 An Econometric Model 2.3 Estimating the Regression Parameters 2.4 Assessing the Least Squares Estimators 2.5 The Gauss-Markov Theorem 2.6 The Probability Distributions of the Least
Squares Estimators 2.7 Estimating the Variance of the Error Term
FIRST
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First:2.1 An Economic Model2.2 An Econometric Model
FIRST
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Figure 2.1a Probability distribution of food expenditure y given income x = $1000
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Figure 2.1b Probability distributions of food expenditures y given incomes x = $1000 and x = $2000
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The simple regression function
E y x xy x( | ) | 1 2
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Figure 2.2 The economic model: a linear relationship between average per person food expenditure and income
Slide 2-7
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Slope of regression line
“Δ” denotes “change in”
2
E y x
x
dE y x
dx
( | ) ( | )
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Figure 2.3 The probability density function for y at two levels of income
Slide 2-9Principles of Econometrics, 3rd Edition
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Slide 2-10Principles of Econometrics, 3rd Edition
1 2|E y x x
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Slide 2-11Principles of Econometrics, 3rd Edition
2var |y x
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Slide 2-12Principles of Econometrics, 3rd Edition
cov , 0i jy y
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Slide 2-13Principles of Econometrics, 3rd Edition
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Slide 2-14Principles of Econometrics, 3rd Edition
y~N 1 2X, 2
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Slide 2-15Principles of Econometrics, 3rd Edition
1 2( | )E y x x
2var( | )y x
cov , 0i jy y
21 2~ ,y N x
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2.2.1 Introducing the Error Term The random error term is defined as
Rearranging gives
y is dependent variable; x is independent variable
Slide 2-16Principles of Econometrics, 3rd Edition
1 2( | )e y E y x y x
y x e 1 2
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The expected value of the error term, given x, is
The mean value of the error term, given x, is zero.
Slide 2-17Principles of Econometrics, 3rd Edition
1 2| | 0E e x E y x x
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Figure 2.4 Probability density functions for e and y
Slide 2-18Principles of Econometrics, 3rd Edition
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Slide 2-19Principles of Econometrics, 3rd Edition
1 2y x e
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Slide 2-20Principles of Econometrics, 3rd Edition
( ) 0E e
1 2( )E y x
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Slide 2-21Principles of Econometrics, 3rd Edition
2var( ) var( )e y
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Slide 2-22Principles of Econometrics, 3rd Edition
cov( , ) cov( , ) 0i j i je e y y
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Slide 2-23Principles of Econometrics, 3rd Edition
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Slide 2-24Principles of Econometrics, 3rd Edition
e~N0, 2
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Slide 2-25Principles of Econometrics, 3rd Edition
y x e 1 2
( ) 0E e 1 2( )E y x 2var( ) var( )e y
cov( , ) cov( , )e e y yi j i j 0
e N~ ( , )0 2
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Figure 2.5 The relationship among y, e and the true regression line
Slide 2-26Principles of Econometrics, 3rd Edition
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Slide 2-27Principles of Econometrics, 3rd Edition