# ch04 - Multiple Regression Analysis y = 0 1x1 2x2 kxk u 2...

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Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 + . . . β k x k + u 2. Inference

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Economics 20 - Prof. Anderson 2 Assumptions of the Classical Linear Model (CLM) So far, we know that given the Gauss- Markov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x 1 , x 2 ,…, x k and u is normally distributed with zero mean and variance σ 2 : u ~ Normal(0, σ 2 )
Economics 20 - Prof. Anderson 3 CLM Assumptions (cont) Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows y| x ~ Normal( β 0 + β 1 x 1 +…+ β k x k , σ 2 ) While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality

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Economics 20 - Prof. Anderson 4 . . x 1 x 2 The homoskedastic normal distribution with a single explanatory variable E( y | x ) = β 0 + β 1 x y f( y|x ) Normal distribution s
Economics 20 - Prof. Anderson 5 Normal Sampling Distributions ( 29 [ ] ( 29 ( 29 ( 29 errors the of n combinatio linear a is it because normally d distribute is ˆ 0,1 Normal ~ ˆ ˆ that so , ˆ , Normal ~ ˆ s t variable independen the of values sample the on l conditiona s, assumption CLM Under the j β β β β β β β j j j j j j sd Var -

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Economics 20 - Prof. Anderson 6 The t Test ( 29 ( 29 1 : freedom of degrees the Note ˆ by estimate to have we because normal) (vs on distributi a is this Note ~ ˆ ˆ s assumption CLM Under the 2 2 1 j - - - - - k n t t se k n j j σ σ β β β
Economics 20 - Prof. Anderson 7 The t Test (cont) Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis For example, H 0 : β j =0 If accept null, then accept that x j has no effect on y , controlling for other x ’s

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Economics 20 - Prof. Anderson 8 The t Test (cont) ( 29 0 ˆ j H , hypothesis null accept the o whether t determine to rule rejection a with along statistic our use then will We ˆ ˆ : ˆ for statistic the" " form to need first e our test w perform To t se t t j j j β β β β
Economics 20 - Prof. Anderson 9 t Test: One-Sided Alternatives Besides our null, H 0 , we need an alternative hypothesis, H 1 , and a significance level H 1 may be one-sided, or two-sided H 1 : β j > 0 and H 1 : β j < 0 are one-sided H 1 : β j 0 is a two-sided alternative If we want to have only a 5% probability of rejecting H 0 if it is really true, then we say our significance level is 5%

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Economics 20 - Prof. Anderson 10 One-Sided Alternatives (cont) Having picked a significance level, α , we look up the (1 – α ) th percentile in a t distribution with n – k – 1 df and call this c ,
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