That parameters 2 the for adjustment freedom of

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that parameters 2 the (for adjustment freedom of degrees a is 2 of Divisor 2 2 where 1 2 2 β - - = - = = = n n SSE n e s s SEE n i i
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18 Coefficient of determination ) regression linear simple in ( fit the better the unity to is Closer 1 0 : Note model regression the by explained is that variable dependent the in variation of proportion the measures 1 : Define 2 2 ˆ 2 2 2 2 2 XY Y Y r r R R R R SST SSE SST SSR R = = - = =
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OLS inference l Properties of OLS point estimators b 0 & b 1 l Can show they are unbiased  E ( b j) = β j l They are normally distributed as they are linear functions of Y i which are assumed to be normal l Even without normality of Y i can invoke CLT & assume b j will be asymptotically normal l So we need Var(b0) and Var(b1) in order to conduct inference 19
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20 OLS inference... l Just as we did when estimating means can define true & estimated variances l Panel below gives l True variances on the left, & l Estimated standard errors on the right l In latter, unknown σ is replaced by estimated s - = - = - = - = 2 2 2 1 2 2 2 2 2 0 ) ( 1 ) ( ) var( ) ( ) ( ) var( 1 0 X X s s X X b X X n X s s X X n X b i b i i i b i i σ σ
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21 OLS inference… [ ] ) 2 ( 0 0 2 2 2 2 0 0 0 ~ ) r( a ˆ v 2 by replacing by ) var( estimate to need unknown With . : under ) 1 , 0 ( ~ ) var( 1 , 0 , ) var( , ~ s ' about hypotheses testing for basis have Now - - = - - = = - = n b j j j j j i j j j j j j j j j t s b b b n e s b H N b b j b N b j β β σ σ β β β β β
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22 Executive salaries l Consider relationship between salaries of chief executive officers & firm performance l Data for 209 US CEO’s for 1990 l Assume regression model Y i = β 0 + β 1 X i + ε i l Y = salary in thousands of US dollars ( salary ) l X = average (over 3 years) return on equity ( roe ) l Range for salary from 223 to 14,822 ($223,000 to $14,822,000) with mean of 1,281 ($1,281,000) l roe ranges from 0.5% to 56.3% with mean of 17.2%
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23 Executive salaries… SUMMARY OUTPUT Regression Statistics Multiple R 0.115 R Square 0.013 Adjusted R Square 0.008 Standard Error 1366.555 Observations 209 ANOVA df SS MS F Significance F Regression 1 5166419.327 5166419.327 2.767 0.098 Residual 207 386566562.7 1867471.317 Total 208 391732982 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 963.191 213.240 4.517 0.000 542.790 1383.592 542.790 1383.592 roe 18.501 11.123 1.663 0.098 -3.428 40.431 -3.428 40.431
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24 Executive salaries… l Only a weak linear relationship between salary & roe l R 2 = 0.013  model explains only 1.3% of variation in CEO salaries l Also SEE = 1,367 compared with mean salary of 1,281 l Estimated effect of roe on salary ( b 1 = 18.5) l A unit increase (1%) in roe would on average result in an increase in salary of 18.5 ($18,500) l This point estimate has a standard error of 11.1 l p -value = 0.098  would not reject H 0: β 1 = 0 versus H 1: β 1 ≠ 0 for any α < 9.8% l Similarly 95% CI is -3.4 to 40.4 & hence covers β 1 = 0
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25 Executive salaries… l Is b 1 = 18.5 a big effect in an economic sense?
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