321_09_slides8

321_09_slides8 - Inference Chapter 4 Econ 321 Introduction...

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Inference Chapter 4 Econ 321 Introduction to Econometrics Econ 321-Stéphanie Lluis 1
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Outline The Distribution of beta Test of single parameter Confidence intervals Test of linear combination of parameters The F-test Econ 321-Stéphanie Lluis 2
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Hypothesis Testing in a Regression Context Wish to know about population regression model (the true model) We do not observe the population, instead we use samples to make inferences about the population and the population regression. Samples are characterized by sampling variability sample regression coefficients do not exactly reproduce the population regression coefficients Sample regression coefficients do not exactly reproduce one another We know the first and second moment of For statistical inference we need the whole distribution
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Normality Assumption The sampling distribution of the OLS estimator depends on the underlying distribution of the errors We assume that the errors are normally distributed u ~ Normal(0, 2 ) What is the distribution of y | x ? y is a linear function of u Econ 321-Stéphanie Lluis 4
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Econ 321-Stéphanie Lluis 5 E(y|x 2 )= β 0 + β 1 x 2 V(y/X) = σ 2
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Why? Use of the Central limit theorem Limitations: u contains factors that can have different distributions The use of CLT assumes all the factors in u enter in an additive way which is restrictive Econ 321-Stéphanie Lluis 6
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Normality of betas But the variance of beta is a function of the unknown σ 2 Because we use σ and that it follows a chi- square with n-k-1 degrees of freedom The ratio of a normal over the square root of a chi-square follows a t-distribution (see review notes + appendix C) In sum: Econ 321-Stéphanie Lluis 7
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Reviewing the Steps in Hypothesis Testing We lay out what we believe as our alternative hypothesis β alternative We make what we do not believe our null hypothesis β null We draw our sample, and estimate the regression We form the t statistic as: We reject or cannot reject the null Using p-value or cutoffs for 5%, 1%
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Econ 321-Stéphanie Lluis 9
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Reviewing the Steps in Hypothesis Testing We compare our estimated sample t* to our t- critical points. If n > 30 and if we are doing one-tailed test these are: 10% 1.282 5% 1.645 1% 2.326 Using our critical points, we reject or do not reject the null
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Test of single parameter one-sided alternative Suppose that we believe that typically professionals earn more than $80 per week per year of education. We have the following set up:
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This note was uploaded on 07/11/2011 for the course ECON 321 taught by Professor Louis during the Fall '09 term at Waterloo.

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321_09_slides8 - Inference Chapter 4 Econ 321 Introduction...

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