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lecture_4_slides - 5-1Regression with a Single Regressor...

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Unformatted text preview: 5-1Regression 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 β1and to construct confidence intervals about β1•Also, we will cover some loose ends about regression: oRegression when Xis binary (0/1) oHeteroskedasticity and homoskedasticity (this is new) oEfficiency of the OLS estimator (also new) oUse of the t-statistic in hypothesis testing (new but not surprising) 5-2But first… abig picture view (and review) Wewantto 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 nis large) – that is, finding the standard error (SE) of the estimator – using only the information in the sample at hand!4.Use the estimator (1ˆβ) to obtain a point estimate and, with its SE, hypothesis tests, and confidence intervals. 5-3Object of interest: β1in, Yi= β+ β1Xi+ ui, i= 1,…, n β1= ΔY/ΔX, for an autonomous change in X(causal effect) The Least Squares Assumptions: 1.E(u|X= x) = 0. 2.(Xi,Yi), i=1,…,n, are i.i.d. 3.Large outliers are rare (E(X4) < ∞, E(Y4) < ∞. The Sampling Distribution of 1ˆβ: Under the LSA’s, for nlarge, 1ˆβis approximately distributed, 1ˆβ~ 214,vXNnσβσ, where vi= (Xi– μX)ui5-4Hypothesis Testing and the Standard Error of 1ˆβ(Section 5.1) The objective is to test a hypothesis, like β1= 0, using data – to reach a tentative conclusion whether the (null) hypothesis is correct or incorrect. General setup Null hypothesis and two-sidedalternative: H: β1= β1,0vs. H1: β1≠β1,0where β1,0is the hypothesized value under the null. Null hypothesis and one-sidedalternative: H: β1= β1,0vs. H1: β1< β1,05-5General approach: construct t-statistic, and compute p-value (or compare to N(0,1) critical value) •In general: t= estimator - hypothesized valuestandard error of the estimatorwhere the SEof the estimator is the square root of an estimator of the variance of the estimator. •For testing the mean of Y:t= ,0/YYYsnμ-•For testing β1, t= 11,01ˆˆ()SEβββ-, where SE(1ˆβ) = the square root of an estimator of the variance of the sampling distribution of 1ˆβ5-6Formula for SE(1ˆβ) Recall the expression for the variance of 1ˆβ(large n): var(1ˆβ) = 22var[()]()ixiXXunμσ-= 24vXnσσ, where vi= (Xi–μX)ui. The estimator of the variance of 1ˆβreplaces the unknown population values of 2νσand 4Xσby estimators constructed from the data: 12ˆˆβσ= 2221estimator of (estimator of )vXnσσ×= 212211ˆ121()niiniivnnXXn==-×-∑∑where ˆiv= ˆ()iiXXu-. 5-712ˆˆβσ= 212211ˆ12...
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This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.

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lecture_4_slides - 5-1Regression with a Single Regressor...

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