Lecture Notes 3 - Hypothesis Testing and Confidence...

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1 Hypothesis Testing and Confidence Interval for Simple Linear Regression 1. So far we have discussed the distributions of ) , ( 1 0 b b or ) ˆ , ˆ ( 1 0 β β , given by equations (10)-(11) [in lecture notes 2]. Note that, we had another estimator for the unknown error variance ° = - = n i i u n 1 2 2 ˆ 2 1 ˆ σ In this section we discuss the distribution of 2 ˆ σ since these will be used in hypothesis testing as well. First recall the following basic results: Definition of 2 χ (chi-squared) distribution: Let n Z Z Z , ........ , , 2 1 are IID N(0,1), i.e., each i Z follows a standard normal distribution and they are independent. Then ° = n i i Z 1 2 follows a 2 χ -distribution with n degrees of freedom (d.f). Note that, under the OLS assumptions we have, ) , 0 ( ~ 2 σ IIDN u i , i.e., ) 1 , 0 ( ~ IIDN u i ± ² ³ ´ µ σ Therefore, ° = ± ² ³ ´ µ n i n i u 1 2 2 ~ χ σ Using this concept we can write, ° = - ± ² ³ ´ µ n i n i u 1 2 2 2 ~ ˆ χ σ Note that, we loose 2 degrees of freedom because of estimation of ) , ( 1 0 b b , in i i i X b b Y u 1 0 ˆ - - = . Consider the random variable ° ° = = - = = ± ² ³ ´ µ n i n i i i n u u 1 2 2 2 1 2 2 ˆ ) 2 ( ˆ ˆ σ σ σ σ Which we can express as 2 2 2 2 ~ ˆ ) 2 ( - - n n χ σ σ
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2 2. Testing Intercept: Now let us put together all the tools to discuss the hypothesis testing problem 0 1 1 0 : β β = H , where 0 1 β is known.
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