硕士计量1 03 SimpleRegressionModel 2整理ppt.pptVIP

Recommended Homework 2.7, 2.9, C2.3 (data=Sleep75, 03年第一版无此题，请参考06年第三版), C2.5 (data=Rdchem, 03年第一版无此题，请参考06年第三版) Introduction to Multiple Regression Analysis Without specifying any functional forms, we can say that the rental price of a house in Beijing is a function of many factors Can you name a few? Prent =f(Size, #bed, #bath, Dists, Safty, Shopping, Entertainment , …) Or more general, Y=f(X1,X2,…) But for regression purpose, we assume: y = ?0 + ?1x1 + ?2x2 + . . .+?kxk + u Parallels with Simple Regression b0 is still the intercept b1 to bk all called slope parameters u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1,x2, …,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions Deriving the First Order Condition Let Setting the derivative of m wrt ?j to zero Meaning, for example ?j0 ^ Deriving the First Order Condition II From there, we can find, for example To have a compact expression for the ?j ‘s, without using matrix, we need to take a regression of each x’s on all other x’s, as shown next ^ If we regress the first x on all remaining x’s, then we can say that , where and ri,1’s are the said regression residuals The first order condition gave us earlier After substitution, it becomes ^ Let’s see what means Since xi1 are a linear function of all other x’s , so Furthermore, since ri1’s are the regression residuals, for all j1 also. Thus ^ ^ “Partialling Out” Interpretation This equation implies that regressing y on x1 and other x’s gives same effect of x1 as regressing y on residuals from a regression of x1 on other x’s This means only the part of xi1 that is uncorrelated with all other x’s are being related to yi so we’re estimating the effect of x1 on y after all other x’s has been “partialled out” In Matrix Form Since We have a nx1 matrix We want to find a (k+1)x1 vector b to minimize the sum squared re