# Lecture_4 - Lecture Week 4 Today we will do develop...

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Lecture Week 4 1 Today we will do develop inference for a single linear regression slope estimate Last week reference: SW Chapter 4 This week: SW Chapter 5

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2 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals Overview Now that we have the sampling distribution of OLS estimator, we are ready to perform hypothesis tests about 1 and to construct confidence intervals about 1 Also, we will cover some additional topics: Regression when X is binary (0/1) Heteroskedasticity and homoskedasticity
3 Review: Y i = 0 + 1 X i + u i , i = 1,…, n 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 The Sampling Distribution of 1 ˆ : Under the LSA‟s, for n large, 1 ˆ is approximately distributed, 1 ˆ ~ 2 1 4 , v X N n , where v i = ( X i X ) u i

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4 Hypothesis Testing and the Standard Error of The objective is to test a hypothesis, like 1 = 0, using data – to reach a probabilistic conclusion whether the (null) hypothesis is supported by the data or not General setup Null hypothesis and two-sided alternative: H 0 : 1 = 1,0 vs. H 1 : 1 1,0 where 1,0 is the hypothesized value under the null. Null hypothesis and one-sided alternative: H 0 : 1 = 1,0 vs. H 1 : 1 < 1,0 1 ˆ
5 General approach : construct t -statistic, and compute p -value (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 sn For testing 1 , t = 1 1,0 1 ˆ ˆ () SE , where SE ( 1 ˆ ) = the square root of an estimator of the variance of the sampling distribution of 1 ˆ

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6 Formula for SE( ) 1 ˆ Recall the expression for the variance of 1 ˆ (large n ): var( 1 ˆ ) = 22 var[( ) ] () i x i X Xu n = 2 4 v X n , where v i = ( X i X ) u i . The estimator of the variance of 1 ˆ replaces the unknown population values of 2 and 4 X by estimators constructed from the data: 1 2 ˆ ˆ = 2 1 estimator of (estimator of ) v X n = 2 1 2 2 1 1 ˆ 1 2 1 n i i n i i v n n XX n where ˆ i v = ˆ ii X X u
7 Summary: To test H 0 : 1 = 1,0 v. H 1 : 1 1,0 , Construct the t -statistic t = 1 1,0 1 ˆ ˆ () SE = 1 1 1,0 2 ˆ ˆ ˆ The p -value is p = Pr[| t | > | t act |] = probability in tails of normal outside | t act |; you reject at the 5% significance level if the p -value is < 5%. This procedure relies on the large- n approximation; typically n = 50 is large enough for the approximation to be excellent.

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8 Example: Test Scores and STR , California data Estimated regression line: TestScore = 698.9 – 2.28 STR Regression software reports the standard errors: SE ( 0 ˆ ) = 10.4 SE ( 1 ˆ ) = 0.52 t -statistic testing 1,0 = 0 = 1 1,0 1 ˆ ˆ () SE = 2.28 0 0.52 = –4.38 We now compute the p -value…
9 The p

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Lecture_4 - Lecture Week 4 Today we will do develop...

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