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Unformatted text preview: 4/22/2010 1 Multiple Regression Model • General Multiple Regression Model . • k regressor variables: X 1 i , ……, X ki • k “slope” coefficients (parameters): β 1 ,….…, β k • Each “slope” coefficient β j measures the effect of a one unit change in the corresponding regressor X ji , holding all else (e.g. the other regressors) constant . • u i – still omitted variables (but hopefully there are less in here since we are including more regressors!) 1 1 2 2 = ...... i i i k ki i Y X X X u β β β β + + + + + Multiple Regression Assumptions  I • As in the simple regression model, we need to make some assumptions in order to estimate the coefficients β , β 1 ,….…, β k . The first 3 are very similar to our previous set of assumptions. • A1) Cov ( u i , X ji ) = 0 for every j (i.e. u i is uncorrelated with each of the k regressors) or A1b) E[ u i  X 1i = c 1 , X 2i = c 2 ,....., X ki = c k ] = E[ u i  X 1i , X 2i ,…., X ki ] = 0 (i.e. the expectation of u i is zero regardless of the values of the k regressors.) (Note the minor notational change in Assumption 1b) ) 4/22/2010 2 Multiple Regression Assumptions II • A2) ( X 1i , X 2i ,…., X ki , Y i ) are i.i.d. (again, this is true with random sampling) • A3) ( X 1i , X 2i ,…., X ki , Y i ) have finite fourth moments (again, this is generally true in economic data). • We also need a fourth assumption in the multiple regression model. This fourth assumption addresses how the various X ji ’s are related to each other. Multiple Regression Assumptions III • A4) The regressors ( X 1i , X 2i ,…., X ki ) are not perfectly multicollinear. This means that none of the regressors can be written as a perfect linear function of only the other regressors. For example: – If X 2i = 11 + 7 X 5i + X 4i 3 X 9i + 5.5 X 3i , then A4) is violated – If X 2i = 11 + 7 X 5i + X 4i 3 X 9i + 5.5 X 3i + W i (where W i is some other variable that is not one of the X ji ’s), then A4) is not violated • Assumption 4) is rarely violated in practice, and when it is, it is typically by accident. We will discuss A4) in more detail momentarily. 4/22/2010 3 Estimation  I • Under A1) – A4), the OLS estimators ( β , β 1 ,….…, β k ), which minimize: are unbiased and consistent estimators of the parameters ( β , β 1 ,….…, β k ). Moreover, the CLT implies that for each j , • The formulas for the OLS estimators (and their standard errors) are too complicated to write down (unless one uses matrix notation ☺ ), but in STATA the estimates (and their SEs) can be computed with the command (e.g. with 3 regressors): “regress y x1 x2 x3” or “regress y x1 x2 x3, robust” ( 29 2 1 1 1 ˆ ˆ ˆ ..........
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This note was uploaded on 06/17/2010 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.
 Spring '07
 SandraBlack
 Econometrics

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