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103lect9hand6

# 103lect9hand6 - Multiple Regression Model General Multiple...

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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!) 0 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 0 , 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) ) 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. Estimation - I Under A1) – A4), the OLS estimators ( β 0 , β 1 ,….…, β k ), which minimize: are unbiased and consistent estimators of the parameters ( β 0 , 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” ( ) 2 0 1 1 1 ˆ ˆ ˆ ....... n i i k ki i Y X X β β β = + + + ( ) ( ) ( ) ( ) ˆ ˆ ˆ ,Var and 0,1 ˆ j j j j j j N N SE β β β β β β Estimation - II Predicted (expected) values and residuals are the same as before, i.e.

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