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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 29: More on Hypothesis Testing 1 Simple Linear Regression Model Relationship Between Variables Is a Linear Function Y intercept Slope Random Error 2 i i i X Y + + = 1 Dependent (Response) Variable Independent (Explanatory) Variable and 1 are unknown, therefore, are estimated from the data. Review : 3 Step Plan 1) Model: Y X N = + + 1 2 ~ ( , ) inexact relationship Noise 2) Data: ( , ),( , ), ,( , ) X Y X Y X Y n n 1 1 2 2 K 3) Estimate: 1 1 , , , , b b s Truth Guesses 3 Hypothesis Tests for the Regression Model We will discuss tests about 1 . Tests on 0 work in exactly the same way. Suppose you want to test whether 1 equals a roposed value: proposed value: For example, if we want test whether X affects Y, we would test whether 1 =0. Huh?? H 1 1 : * = H a : * 1 1 Null Alternative 4 Decision Rules for Testing the Slope: 1 1 1 1 : = If   1.96 : o a H T reject H H * * > 1 * 1 1 b s b T  = 1 1 1 1 1 1 1 1 : If 1.64 : < : If 1.64 : > o a o a H T reject H H H T reject H H * * * * <  > 5 In the simplest sense, the market model assumes that 1 t t t Stockreturn Indexreturn = + + he finance people call eta (go figure). Market Model (again) The finance people call 1 Beta (go figure). Beta=0 : cash under the mattress Beta=1 : same risk as the market Beta<1 : safer than the market Beta >1: riskier than the market 6 We will examine the market model for the stock Abercrombie & Fitch (ANF), using the S&P 500 as a proxy for the market. The returns are monthly from the last three years. .2 .4 7.4.2 ANF.2.1 .1 SP500 From Stata, We test the hypothesis that the slope is zero; that is, are ANF returns related to the market ? 8 The test statistic is 1 1 1.61 5.41 .297 b b t s = = = 1 1 : : a H H = and > 5.41 1.96 so we reject the null hypothesis. 9 Note: the hypothesis that the slope equals zero is tested so often that Stata automatically prints out the appropriate t statistic. The t for testing the intercept equal to 0 is also printed. b s b 1 1 1 : H = b s b : H = 10 We now test the hypothesis that ANF has the risk as the market, that is, the slope equals 1. The t statistic is 1 1 : 1 : 1 a H H = 1.61 1 2.05 .297 t = = Now 2.05 is greater than 1.96 so we reject the null hypothesis; ANF does not have the same risk as the market. 11 What is the confidence interval for the ANF beta? 1.61 +\ 1.96(.297) = [1.02,2.19] What does this interval tell us about our level of certainty about the beta for ANF? 12 BTW, We Agree with Yahoo 13 Recap: Regression Modeling So Far s Start with data where you think a linear relationship exists; verbal SAT vs. Height 700 800 14 400 500 600 v_sat 50 60 70 80 height Examine the Regression Output s What is the value of Rsquared? Is it low or high?...
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen
 Linear Regression

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