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Unformatted text preview: Regression Analysis Harry Tsang Department of Economics University of Illinois, UrbanaChampaign April 11, 2008 Linear Regression: We Explain The Variation In Y, dependent variable, By Using Explanatory Variables X, independent variable I Simple Linear Regression  One Independent Variable X Y = β + β 1 × X 1 + I Multiple Linear Regression  More Than One Independent Variable X Y = β + β 1 × X 1 + β 2 × X 2 + ... + β k × X k + We Break The Total Variability In Y Into Two Different Components, SSR and SSE I SST (Sum of Squares Total): Total Amount of Variation, deg. of freedom = n  1 SST = n X i ( y i ¯ y ) 2 I SSR (Sum of Squares Regression): Total Amount of Variation Explained By Model, deg. of freedom = k (# of Independent Variables) SSR = n X i (ˆ y i ¯ y ) 2 I SSE (Sum of Squares Error): Total Amount of Variation Unexplained By Model, deg. of freedom = n  k  1 SSE = n X i ( y i ˆ y i ) 2 How Good Is Our Model At Explaining the Variability In Y? I Coefficient of Determination, R 2 ,: What proportion of the variation in y is explained by our model? R 2 = SSR SST = 1 SSE SST I Ftest For Overall Model Validity, d.f. = k, nk1 H : β 1 = β 2 = ... = β k = 0 H 1 : At least one β is not equal to zero F = MSR MSE = SSR k SSE n k 1 I ttest For Significance Of Slope Coefficient, β i , d.f. = n  k 1 H : β i = 0 , H 1 : β i 6 = 0 , H 1 : β i < , H 1 : β i > t = b i β i s b i Multiple Linear Regression Y = β + β 1 × X 1 + β 2 × X 2 + ... + β k × X k + I Standard Error of The Estimate s = r SSE n k 1 I Standard Error of The Slope Coefficient: s b i = Given in Excel output I Adjusted R 2 Adj R 2 = 1 SSE n k 1 SST n 1 = 1 SSE SST × n...
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 Spring '08
 Petry
 Economics, Regression Analysis, Yi, independent variables, slope coefficient, Sum of Squares Regression

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