Lecture_9_Slides_486609574.pdf

Lecture_9_Slides_486609574.pdf - I NTRODUCTORY E...

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I NTRODUCTORY E CONOMETRICS Lecture 9 Spring 2017, Tsinghua University Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 1 / 60
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Hypothesis Test and Confidence Intervals Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 2 / 60
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O UTLINE 1 Rejection Region Method 2 p-value Approach 3 Choose between "Valid" Decision Rules 4 Confidence Interval Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 3 / 60
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H YPOTHESIS T EST : G ENERAL S TEPS We begin with a null H 0 and alternative hypothesis H 1 . The testing procedure begins with the assumption that H 0 is true. The goal is to determine whether there is enough evidence in the data to support H 1 . There are only two possible decisions: 1 Conclude that there is enough evidence to support H 1 . 2 Conclude that there is not enough evidence to support H 1 . The decision rule revolves around = Pr ( Type I error ) = Pr ( H 0 being rejected | H 0 is true ) Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 4 / 60
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T ESTING M ETHODS There are two general approaches: 1 The rejection region method. 2 The p-value approach. We will describe how each of these approaches works in both two-tail and one-tail hypothesis tests. Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 5 / 60
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In this course, we are primarily interested at testing about the regression coefficient β j ’s. Yet, we will start with tests about the population mean μ . Note: we will, at first, maintain the assumption that the population standard deviation σ is known. Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 6 / 60
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R EJECTION R EGION M ETHOD All tests about the population mean μ are based on looking at the sample average X . The rejection region method works by establishing a region of values such that we reject H 0 whenever X Falls inside this region. The features of the rejection region depend on: 1 The significance level we want to achieve. 2 Whether we have a one-tail or a two-tail test. Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 8 / 60
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R EJECTION R EGION M ETHOD WITH O NE - TAIL T ESTS First, we begin with a pre-specified significance level (we get to choose this). Recall that is defined as: Pr ( Rejecting H 0 when it is true ) . Now, suppose our test is H 0 : μ = μ vs H 1 : μ > μ . Intuitively: Since X is a consistent estimator of μ , we would have evidence in favor of H 1 if X - μ is “large”. According to this logic, we should reject H 0 in favor of H 1 if X - μ > x L Where x L is a cut-off value. How do we determine x L by making sure that we achieve the significance level that we pre-specified? For this, we invoke the Central Limit Theorem . Shengjie Hong (SEM, Tsinghua) [email protected] Lecture 9 9 / 60
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That is, we choose x L in order to satisfy Pr ( Rejecting H 0 when it is true ) = .
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