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Unformatted text preview: Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (SW Chapter 5) Object of interest: β 1 in, Y i = β + β 1 X i + u i , i = 1,…, n β 1 = ∆ Y / ∆ X , for an autonomous change in X ( causal effect ) The Least Squares Assumptions: 1. E ( u  X = x ) = 0. 2. ( X i ,Y i ), i =1,…, n , are i.i.d. 3. Large outliers are rare ( E ( X 4 ) < ∞ , E ( Y 4 ) < ∞ . General setup Null hypothesis and twosided alternative: H : β 1 = β 1,0 vs. H 1 : β 1 ≠ β 1,0 where β 1,0 is the hypothesized value under the null. Null hypothesis and onesided alternative: H : β 1 = β 1,0 vs. H 1 : β 1 < β 1,0 General approach : construct tstatistic, and compute pvalue (or compare to N (0,1) critical value) • In general: t = estimator  hypothesized value standard error of the estimator where the SE of the estimator is the square root of an estimator of the variance of the estimator. • For testing the mean of Y : t = ,0 / Y Y Y s n μ • For testing β 1 , t = 1 1,0 1 ˆ ˆ ( ) SE β β β , where SE ( 1 ˆ β ) = the square root of an estimator of the variance of the sampling distribution of 1 ˆ β Formula for SE( 1 ˆ β ) Recall the expression for the variance of 1 ˆ β (large n ): var( 1 ˆ β ) = 2 2 var[( ) ] ( ) i x i X X u n μ σ = 2 4 v X n σ σ , where v i = ( X i – μ X ) u i . SE ( 1 ˆ β ) = 1 2 ˆ ˆ β σ = the standard error of 1 ˆ β Summary: To test H : β 1 = β 1,0 v. H 1 : β 1 ≠ β 1,0 , • Construct the tstatistic t = 1 1,0 1 ˆ ˆ ( ) SE β β β = 1 1 1,0 2 ˆ ˆ ˆ β β β σ • Reject at 5% significance level if t > 1.96 • The pvalue is p = Pr[ t  >  t act ] = probability in tails of normal outside  t act ; you reject at the 5% significance level if the pvalue is < 5%. • This procedure relies on the large n approximation; typically n = 50 is large enough for the approximation to be excellent. Example: Test Scores and STR , California data Estimated regression line: test score = 698.9 – 2.28 × STR Regression software reports the standard errors: SE ( ˆ β ) = 10.4 SE ( 1 ˆ β ) = 0.52 tstatistic testing β 1,0 = 0 = 1 1,0 1 ˆ ˆ ( ) SE β β β = 2.28 0 0.52 = –4.38 • The 1 % 2sided significance level is 2.58, so we reject the null at the 1% significance level. • Alternatively, we can compute the pvalue… The pvalue based on the large n standard normal approximation to the tstatistic is 0.00001 (10 –5 ) Because the tstatistic for β 1 is N(0,1) in large samples, construction of a 95% confidence for β 1 is just like the case of the sample mean: 95% confidence interval for β 1 = { 1 ˆ β ± 1.96 × SE ( 1 ˆ β )} Confidence interval example : Test Scores and STR Estimated regression line: testscore= 698.9 – 2.28 × STR SE ( ˆ β ) = 10.4 SE ( 1 ˆ β ) = 0.52 95% confidence interval for 1 ˆ β : { 1 ˆ β ± 1.96 × SE ( 1 ˆ β )} = {–2.28 ± 1.96 × 0.52} = (–3.30, –1.26) A concise (and conventional) way to report regressions:...
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 Fall '09
 Li
 Normal Distribution, Standard Error, Interest, Yi, standard errors

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