计量经济学 第六章 Chapter 6 Multiple Linear Regression.ppt

Chapter 6 Multiple Linear Regression Xi’An Institute of Post Telecommunication Dept of Economic Management Prof. Long * * Two Explanatory Variables yt = b1 + b2xt2 + b3xt3 + et ?yt ?xt2 = b2 ?xt3 ?yt = b3 xt‘s affect yt separately But least squares estimation of b2 now depends upon both xt2 and xt3 . 6.1 Correlated Variables yt = output xt2 = capital xt3 = labor Always 5 workers per machine. If number of workers per machine is never varied, it becomes impossible to tell if the machines or the workers are responsible for changes in output. yt = b1 + b2xt2 + b3xt3 + et 6.2 The General Model yt = b1 + b2xt2 + b3xt3 +. . .+ bKxtK + et The parameter b1 is the intercept (constant) term. The “variable” attached to b1 is xt1= 1. Usually, the number of explanatory variables is said to be K-1 (ignoring xt1= 1), while the number of parameters is K. (Namely: b1 . . . bK). 6.3 1. E(et) = 0 2. var(et) = s2 3. cov(et , es) = 0 for t?s? 4. et ~ N(0, s2) Statistical Properties of et 6.4 1. E (yt) = b1 + b2xt2 +. . .+ bKxtK 2. var(yt) = var(et) = s2 3. Cov(yt ,ys) = Cov(et , es) = 0 for t?s 4. yt ~ N(b1+b2xt2 +. . .+bKxtK, s2) Statistical Properties of yt 6.5 Assumptions 1. yt = b1 + b2xt2 +. . .+ bKxtK + et 2. E (yt) = b1 + b2xt2 +. . .+ bKxtK 3. var(yt) = var(et) = s2 4. cov(yt ,ys) = cov(et ,es) = 0 t?s 5. The values of xtk are not random 6. yt ~ N(b1+b2xt2 +. . .+bKxtK, s2) 6.6 Least Squares Estimation yt = b1 + b2xt2 + b3xt3 + et S o S(b1, b2, b3) = S(yt - b1 - b2xt2 - b3xt3)2 t = 1 T Define: yt = yt - y * xt2 = xt2 - x2 * xt3 = xt3 - x3 * 6.7 b1 = y - b1 - b2x2 - b3x3 b3 = (Syt xt3)(Sxt2 ) - (Syt xt2)(Sxt3xt2) * * * * * * * 2 (Sxt2 )(Sxt3 ) - (Sxt2xt3) * * * * 2 2 2 b2 = (Syt xt2)(Sxt3 ) - (Syt xt3)(Sxt2xt3) * * * * * * * 2 (Sxt2 )(Sxt3 ) - (Sxt2xt3) * * * * 2 2 2 Least Squares Estimators 6.8 Dangers of Extrapolation Statistical models generally are good only “within the relevant range”. This m