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lecture_4_powerpoint2007

# lecture_4_powerpoint2007 - Lecture 4 Introduction to...

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Click to edit Master subtitle style 8/5/11 Lecture 4

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8/5/11 The Simple Linear Model A simple linear regression model can be expressed as: t t t X Y ε β + + = 1 0 (4.1) where: Yt = dependent variable at time t Xt = independent (explanatory) variable at time t 0 = intercept (constant term) β1 = slope for the coefficient Xt εt = random error in Yt for observation at time t.
8/5/11 The OLS Estimates A simple way to obtain the estimators is to minimize the sum of the squared errors, such that: [ ] + - = - = 2 1 0 2 2 ) ˆ ˆ ( ) ˆ ( t t t t t X Y Y Y Minimize β ε (4.2) where t = errors at observation t Yt = the actual value of observation at time t Xt = value of X for observation at time t Ŷ t = estimated value of Y for observation t the estimated value of β 0 = 0 ˆ the estimated value of β 1 = 1 ˆ

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8/5/11 OLS (Cont’d) By setting the first order condition equal to zero and solving β 1 and 0 simultaneously, we obtain: = = - - - = n t t n t t t X X Y Y X X 1 2 1 ) ( ) )( ( 1 ˆ - - = 2 2 1 ) ( ˆ t t t t t t X X n Y X Y X n , ˆ 1 0 X Y - = or (4.3) (4.4) (4.5)
8/5/11 OLS (Cont’d) The estimated equation is: t t X Y 1 0 ˆ ˆ ˆ β β+ = On the basis of this Equation, the h -period-ahead forecast equation is: h t h t X Y + + + = ˆ ˆ ˆ ˆ 1 0 (4.6) (4.7)

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lecture_4_powerpoint2007 - Lecture 4 Introduction to...

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