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lec29v3_1up

# lec29v3_1up - Stat 104 Quantitative Methods for Economists...

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Stat 104: Quantitative Methods for Economists Class 29: More on Hypothesis Testing 1

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Simple Linear Regression Model Relationship Between Variables Is a Linear Function Y intercept Slope Random Error 2 i i i X Y ε β β + + = 1 0 Dependent (Response) Variable Independent (Explanatory) Variable β 0 and β 1 are unknown, therefore, are estimated from the data.
Review : 3 Step Plan 1) Model: Y X N = + + β β ε ε σ 0 1 2 0 ~ ( , ) ( , ),( , ), ,( , ) X Y X Y X Y inexact relationship Noise 2) Data: n n 1 1 2 2 K 3) Estimate: β β σ 0 1 0 1 , , , , b b s Truth Guesses 3

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Hypothesis Tests for the Regression Model We will discuss tests about β 1 . Tests on β 0 work in exactly the same way. Suppose you want to test whether β 1 equals a proposed value: For example, if we want test whether X affects Y, we would test whether β 1 =0. Huh?? H 0 1 1 : * β β = H a : * β β 1 1 Null Alternative 4
Decision Rules for Testing the Slope: 0 1 1 1 1 : = If | | 1.96 : : If 1.64 o a H T reject H H H T reject H β β β β β β * * * > < - 1 * 1 1 b s b T β - = 0 1 1 1 1 0 1 1 1 1 : < : If 1.64 : > o a o a H H T reject H H β β β β β β * * * > 5

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In the simplest sense, the “market model” assumes that 0 1 t t t Stockreturn Indexreturn β β ε = + + The finance people call β Beta (go figure). Market Model (again) 1 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 0 ANF -.2 -.1 0 .1 SP500

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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 0 1.61 5.41 .297 b b t s - = = = 0 1 1 : 0 : 0 a H H β β = and > 5.41 1.96 so we reject the null hypothesis. 9

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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 0 1 : 0 H β = b s b 0 0 0 0 : 0 H β = 10
We now test the hypothesis that ANF has the risk as the market, that is, the slope equals 1. The t statistic is 1.61 1 - 0 1 1 : 1 : 1 a H H β β = 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

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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

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Recap: Regression Modeling So Far square6 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 square6 What is the value of R-squared? Is it low or high? square6 If we used this model for predictions, how accurate would we be? +/- 2s = 15

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Do we even need “x” in the model?
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lec29v3_1up - Stat 104 Quantitative Methods for Economists...

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