Multiple regression analysis y=Bo B Bx+ Bx +.Bkk+u ◆2. Inference Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 2. Inference
Assumptions of the Classical Linear Model(CLm) So far, we know that given the Gauss Markov assumptions, ols iS BLUE e In order to do classical hypothesis testing we need to add another assumption(beyond the Gauss-Markov assumptions) ◆ Assume that u is independent ofxI,x2,…,xk and u is normally distributed with zero mean and variance 02: u- normal(0, 0) Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Assumptions of the Classical Linear Model (CLM) So far, we know that given the GaussMarkov 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 x1 , x2 ,…, xk and u is normally distributed with zero mean and variance s 2 : u ~ Normal(0,s 2 )
CLM ASsumptions(cont) o Under CLM, OLS is not only blue, but is the minimum variance unbiased estimator o We can summarize the population assumptions of ClM as follows ◆yx~ Normal(B0+Bx1+…+Bxha) e While for now we just assume normality clear that sometimes not the case e Large samples will let us drop normality Economics 20- Prof anderson
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(b0 + b1 x1 +…+ bk xk , s 2 ) While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality
The homoskedastic normal distribution with a single explanatory variable fylx E(x)=Bo+ Bx t Normal distributions x Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 . . x1 x2 The homoskedastic normal distribution with a single explanatory variable E(y|x) = b0 + b1x y f(y|x) Normal distributions
Normal Sampling distributions Under the clm assumption s conditiona l on the sample values of the independen t variable s B - NO orma 1 B, vare, l so that B 16) normal l(0,4) B is distribute d normally because it is a linear combinatio n of the errors Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Normal Sampling Distributions ( ) ( ) ( ) ( ) is a linear combinatio n of the errors is distribute d normally because it ˆ ~ Normal 0,1 ˆ ˆ ,so that ˆ ~ Normal , ˆ the sample values of the independen t variable s Under the CLM assumption s, conditiona l on b j b b b b b b j j j j j j sd Var −