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Unformatted text preview: Economics 20  Prof. Anderson 1 Multiple Regression Analysis y = β + β 1 x 1 + β 2 x 2 + . . . β k x k + u 2. Inference 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 GaussMarkov 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 + 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 Economics 20  Prof. Anderson 4 . . x 1 x 2 The homoskedastic normal distribution with a single explanatory variable E( y  x ) = β + β 1 x y f( yx ) 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 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 : β j =0 If accept null, then accept that x j has no effect on y , controlling for other x ’s Economics 20  Prof. Anderson 8 The t Test (cont) ( 29 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: OneSided Alternatives Besides our null, H , we need an alternative hypothesis, H 1 , and a significance level H 1 may be onesided, or twosided H 1 : β j > 0 and H 1 : β j < 0 are onesided H 1 : β j ≠ 0 is a twosided alternative If we want to have only a 5% probability of rejecting H if it is really true, then we say our significance level is 5% Economics 20  Prof. Anderson 10 OneSided Alternatives (cont) Having picked a significance level, α...
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This note was uploaded on 03/03/2009 for the course ECON 382 taught by Professor Sun during the Spring '08 term at CUNY Queens.
 Spring '08
 Sun
 Economics, Econometrics

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