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Unformatted text preview: Announcements • Midterm 2 tomorrow Monday April 19 7pm, Foellinger • Some office hours still on Monday, check Compass • Tutoring hours too... Feel free to raise questions as we go along, and we’ll have time for Q and A, or looking at old midterm problems at end... Break down Excel output Top portion: Regression Statistics Multiple R .632 R Square .400 Adjusted R Square .381 Standard Error .589 Observations 100 Top Portion of Excel output • Multiple R • Multiple R = √ R 2 • Multiple R =  r  in simple regression • To figure out sign of r using Multiple R, look at sign on b 1 or test stat on b 1 • R Square • Also: Coefficient of Determination • R 2 = cov ( x , y ) 2 s 2 x s 2 y = SSR SST = 1 SSE SST • Proportion of variation in y that is explained by the variation in the independent variables • Between 0 and 1 • Ajdusted R Square • adj. R 2 = 1 SSE / ( n k 1) SST / ( n 1) = 1 SSE * ( n 1) SST * ( n k 1) • Proportion of variation in y that is explained by the variation in the independent variables, adjusting for degrees of freedom • Always less than R 2 , unless equal to 1. Negative values definitely possible Top Portion of Excel output • Standard Error • Standard error of the estimate, s e , s ε , standard error of the regression line • s ε = q SSE n k 1 = √ MSE • Can use empirical rule to describe ‘spread’ of points around regression line • 95% of data expected to be within ± 2 * s ε around regression line • 68% of data expected to be within ± 1 * s ε around regression line • Observations (we’ll skip this ...) Break down Excel output Middle portion of output: ANOVA df SS MS F Significance F Regression k SSR MSR F test stat pvalue Residual nk1 SSE MSE Total n1 SST • SSR + SSE = SST • MSR = SSR k • MSE = SSE n k 1 • F test stat = MSR MSE • pvalue is on overall F test • Don’t confuse with oneway ANOVA formulas on Midterm 1 Middle Portion of Excel output This portion of output can obviously be used to find things like R 2 , s ε ,... But reason it is included is overall F test: H : β 1 = β 2 = ... = β k = 0 At least one β i 6 = 0 F distribution, with k and n k 1 degrees of freedom Onetailed, right hand side test So α is put entirely on the right Reject H if test stat > critical value or pvalue < α implies Model is valid Do not reject H if test stat < critical value or pvalue > α implies Model is not valid Break down Excel output Bottom portion of output: Coeff. Standard Error t Stat Pvalue Lower 95% Intercept b s b b s b 2 tail p b t ( . 025 , n k 1) s b X1 b 1 s b 1 b 1 s b 1 2 tail p b 1 t ( . 025 , n k 1) s b 1 . . . (Got rid of Upper 95% for space) So Excel always assumes we are doing a twotailed test for β i = 0 vs β i 6 = 0. Bottom portion of output So we have formulas for simple regression to find b and b 1 b 1 = cov ( x , y ) s 2 x = ∑ x i y i n ¯ x ¯ y ∑ x 2 i n (¯ x ) 2 b = ¯ y b 1 ¯ x Formulas for multiple regression  too complicated for 203.Formulas for multiple regression  too complicated for 203....
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 Spring '09
 PETRY
 Economics, Regression Analysis, sε, qq q qq

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