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# It is not about the probability of β 1 being any

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It is not about the probability of β 1 being any particular value. β 1 is not a random variable. It is some unknown number. The data is what is random. In particular, the p-value is not the probability that H 0 is false given the data. Hypothesis testing: we must make a decision (usually reject or fail to reject H 0 ) Choose significance level α (usually 0.05 or 0.10) Construct procedure such that if H 0 is true, we will incorrectly reject with probability α Reject null if p-value less than α Simple Regression January 8, 2018 57 / 74
Growth and GDP Simple Regression January 8, 2018 58 / 74
Caution: economic significance 6 = statistical significance Simple Regression January 8, 2018 59 / 74
Other issues: Reporting Results There are two formats to report results 1 Table Appropriate for multiple Specifications Refer ”Growth and GDP” table 2 Equation \ Growth = 0 . 42 (0 . 96) + 0 . 09 (0 . 25) yearsschool n = 65 , R 2 = 0 . 11 , Adj - R 2 = 0 . 1 Simple Regression January 8, 2018 60 / 74
Other issues: Regression Interpretations 1 Semi-logarithmic form log( wage ) = β 0 + β 1 educ + Percentage change in Y associated with unit change in X β 1 = log( wage ) ∂educ = log( wage ) wage ∂educ 2 Log-logarithmic form log( salary ) = β 0 + β 1 log( sales ) + Percentage change in Y associated with percentage change in X β 1 = log( salary ) log( sales ) = log( salary ) salary log( sales ) sales Simple Regression January 8, 2018 61 / 74
Estimating σ 2 I Recall that Var( ˆ β | x 1 , ..., x n ) = σ 2 n i =1 ( x i - ¯ x ) 2 = σ 2 n d Var( x ) σ 2 unknown We estimate σ 2 using the residuals, ˆ σ 2 = 1 n - 2 n X i =1 ˆ 2 i |{z} =( y i - ˆ β 0 - ˆ β 1 x i ) 2 If SLR.1-SLR.5, E[ˆ σ 2 ] = σ 2 Using 1 n - 2 instead of 1 n makes ˆ σ 2 unbiased ˆ i depends on 2 estimated parameters, ˆ β 0 and ˆ β 1 , so only n - 2 degrees of freedom Simple Regression January 8, 2018 62 / 74
Estimating σ 2 II Estimate Var( ˆ β 1 | x 1 , ..., x n ) by d Var( ˆ β 1 | x 1 , ..., x n ) = ˆ σ 2 n i =1 ( x i - ¯ x ) 2 = 1 n - 2 n i =1 ˆ 2 i n i =1 ( x i - ¯ x ) 2 Standard error of ˆ β 1 is q d Var( ˆ β 1 | x 1 , ..., x n ) If SLR.1-SLR.6, t-statistic with estimated d Var( ˆ β 1 | x 1 , ..., x n ) has a t ( n - 2) distribution instead of N (0 , 1) t = ˆ β 1 - β 1 q d Var( ˆ β 1 | x 1 , ..., x n ) t ( n - 2) Simple Regression January 8, 2018 63 / 74
Confidence intervals I ˆ β 1 is random d Var( ˆ β 1 ) , p-values, and hypthesis tests are ways of expressing how random is ˆ β 1 Confidence intervals are another A 1 - α confidence interval, CI 1 - α = [ LB 1 - α , UB 1 - α ] is an interval estimator for β 1 such that P( β 1 CI 1 - α ) = 1 - α ( CI 1 - α is random; β 1 is not) Recall: if SLR.1-SLR.6, then ˆ β 1 N β 1 , Var( ˆ β 1 ) Simple Regression January 8, 2018 64 / 74
Confidence intervals II Implies P ˆ β 1 < β 1 + q Var( ˆ β 1 - 1 ( α/ 2) = α/ 2 P ˆ β 1 - q Var( ˆ β 1 - 1 ( α/ 2) < β 1 ) = α/ 2 and P ˆ β 1 > β 1 + q Var( ˆ β 1 - 1 (1 - α/ 2) = α/ 2 P ˆ β 1 - q Var( ˆ β 1 - 1 (1 - α/ 2) > β 1 = α/ 2 Simple Regression January 8, 2018 65 / 74
Confidence intervals III so P ˆ β 1 + q Var( ˆ β 1 - 1 ( α/ 2) < β 1 β 1 < ˆ β + q Var( ˆ β 1 - 1 (1 - α/ 2) = =1 - P ˆ β 1 + q Var( ˆ β 1 - 1 ( α/ 2) < β 1 - - P ˆ β 1 + q Var( ˆ β 1 - 1 (1 - α/ 2) > β 1 =1 - α For α = 0 . 05 , Φ - 1 (0 . 025) ≈ - 1 . 96 ,

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