lecturenotes3 - 1 Economics 620 Lecture 3 Simple Regression...

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1 Economics 620, Lecture 3: Simple Regression II ^ ° and ^ ± are the LS estimators ^ y i = ^ ° + ^ ±x i are the estimated values The Correlation Coe¢ cient: r = P ( x i ° ° x )( y i ° ° y ) q P ( x i ° ° x ) 2 P ( y i ° ° y ) 2 : R 2 = (squared) correlation between y and ^ y Note: ^ y is a linear function of x . So corr ( y; ^ y ) = j corr ( y; x ) j : Prof. N. M. Kiefer, Econ 620, Cornell University, Lecture 3. Copyright (c) N. M. Kiefer.
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2 Correlation Proposition : ° 1 < r < 1 r 2 = ( P ( x i ° ° x )( y i ° ° y ) ) 2 P ( x i ° ° x ) 2 P ( y i ° ° y ) 2 : Use Cauchy-Schwartz ( P x i y i ) 2 ± P x 2 i P y 2 i ) r 2 ± 1 ) ° 1 ± r ± 1 Proposition : ± and r have the same sign. Proof: ^ ± = P ( x i ° ° x ) y i P ( x i ° ° x ) 2 = r q P ( y i ° ° y ) 2 q P ( x i ° ° x ) 2 Prof. N. M. Kiefer, Econ 620, Cornell University, Lecture 3. Copyright (c) N. M. Kiefer.
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3 Correlation cont°d. P e 2 i = P ( y i ° ° y ) 2 ° ^ ± 2 P ( x i ° ° x ) 2 SSR = TSS - SS explained by x Proposition : r 2 = 1 ° SSR TSS = 1 ° P e 2 i P ( y i ° ° y ) 2 Proof: P e 2 i P ( y i ° ° y ) 2 = 1 ° ^ ± 2 P ( x i ° ° x ) 2 P ( y i ° ° y ) 2 = 1 ° r 2 ) r 2 = 1 ° P e 2 i P ( y i ° ° y ) 2 Prof. N. M. Kiefer, Econ 620, Cornell University, Lecture 3. Copyright (c) N. M. Kiefer.
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4 Warning: Correlation 6 = Dependence Variables are completely dependent, correlation is zero. Correlation is a measure of linear dependence. Prof. N. M. Kiefer, Econ 620, Cornell University, Lecture 3. Copyright (c) N. M. Kiefer.
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5 The Likelihood Function A complete speci°cation of the model Conditional distribution of observables Conditional on regressors x ±exogenous variables² - vari- ables determined outside the model Conditional on parameters P ( y j x; °; ±; ² 2 ) Previously, speci°ed only mean and maybe variance Incompletely speci°ed = ±semiparametric² Point estimate: MLE ³intuition Details, asy. justi°cation lecture 9.
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