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# ppt05 - Chapter 5 Regression with a Single Regressor...

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Chapter 5 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

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2 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (SW Chapter 5) Overview Now that we have the sampling distribution of OLS estimator, we are ready to perform hypothesis tests about β 1 and to construct confidence intervals about β 1 Also, we will cover some loose ends about regression: Regression when X is binary (0/1) Heteroskedasticity and homoskedasticity (this is new) Efficiency of the OLS estimator (also new) Use of the t -statistic in hypothesis testing (new but not surprising)
3 But first… a big picture view (and review) We want to learn about the slope of the population regression line, using data from a sample (so there is sampling uncertainty). There are four steps towards this goal: 1. State precisely the population object of interest 2. Derive the sampling distribution of an estimator (this requires certain assumptions) 3. Estimate the variance of the sampling distribution (which the CLT tells us is all you need to know if n is large) – that is, finding the standard error ( SE ) of the estimator – using only the information in the sample at hand! 4. Use the estimator ( 1 ˆ β ) to obtain a point estimate and, with its SE , hypothesis tests, and confidence intervals.

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4 Object of interest: β 1 in, Y i = β 0 + 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 ) < . The Sampling Distribution of 1 ˆ β : Under the LSA’s, for n large, 1 ˆ β is approximately distributed, 1 ˆ β ~ 2 1 4 , v X N n σ β σ , where v i = ( X i μ X ) u i
5 Hypothesis Testing and the Standard Error of (Section 5.1) The objective is to test a hypothesis, like β 1 = 0, using data – to reach a tentative conclusion whether the (null) hypothesis is correct or incorrect. General setup Null hypothesis and two-sided alternative: H 0 : β 1 = β 1,0 vs. H 1 : β 1 h β 1,0 where β 1,0 is the hypothesized value under the null. Null hypothesis and one-sided alternative: H 0 : β 1 = β 1,0 vs. H 1 : β 1 < β 1,0 1 ˆ β

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6 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 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
7 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 .

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