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# lecture4 - ECON 103 Lecture 4 Statistical Inference Maria...

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ECON 103, Lecture 4: Statistical Inference Maria Casanova April 9th (version 1) Maria Casanova Lecture 4

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Requirements for this lecture: Chapter 3 of Stock and Watson Maria Casanova Lecture 4
0. Introduction In this lecture we will be doing statistical inference about the population mean ( μ ). We will cover: Hypothesis tests p-values Conﬁdence intervals Maria Casanova Lecture 4

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1. Hypothesis tests Are the data consistent with a hypothesis we have about the true (unknown) value of the parameter of interest, e.g. E ( X ) = μ ? The hypothesis to be tested or null hypothesis H 0 : μ = μ 0 [e.g. H 0 : μ = 0] The null hypothesis is compared against the alternative hypothesis Two-sided alternative hypothesis: H 1 : μ 6 = μ 0 [e.g. H 1 : μ 6 = 0] One-sided alternative hypothesis: H 1 : μ < μ 0 [e.g. H 1 : μ < 0] The signiﬁcance level α is the probability that you will reject the null when it is true. Conversely, the conﬁdence level (1 - α ) × 100% is the probability that the hypothesis is true and you will not reject it. Maria Casanova Lecture 4
1. Hypothesis tests Suppose we want to test H 0 : μ = 6 against H 1 : μ 6 = 6 with α = 0 . 05 Case 1: X i i.i.d. N ( μ, σ 2 ) with μ unknown, σ 2 known. We need an estimator for the parameter of interest, e.g. ˆ μ . Remember that the standardized version of the sample mean had a N (0 , 1) distribution for X i i.i.d normal: ˆ μ - μ p σ 2 / n N (0 , 1) The value of the standardized sample mean under the null is the t-statistic or t-ratio: t = ˆ μ - 6 p σ 2 / n N (0 , 1) Maria Casanova Lecture 4

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1. Hypothesis tests Figure: N (0 , 1) distribution 0 .1 .2 .3 .4 y -4 -2 0 2 4 x Maria Casanova Lecture 4
Based on the chosen α , we can determine a value c such that P ( - c t c ) = (1 - α ). c is known as the critical value . In the example α = 0 . 05, so we want the area under each tail to have probability 0.025. The corresponding critical value is 1.96: Pr ( - 1 . 96 t 1 . 96) = 0 . 95 To test H 0 we compute the value of t in our sample. Then: - if t < - 1 . 96 we reject the null hypothesis. -

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lecture4 - ECON 103 Lecture 4 Statistical Inference Maria...

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