SLN Multiple Regression Models Spring 2007

SLN Multiple Regression Models Spring 2007 - Multiple...

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Multiple Regression Models Supplementary Lecture Notes // Spring 2007 Multiple regression models are equations with multiple regressors. In other words models where the mean value of the dependent variable Y is a function of more than one independent X variable. Consider a model equation with 3 independent variables, X 1 , X 2 , X 3 . E(Y| X 1 = x 1 , X 2 = x 2 , X 3 = x 3 ) = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 This relationship can also be expressed in term of any random observation of Y. Y i = β 0 + β 1 X 1i + β 2 X 2i + β 3 X 3i + u i where Y is the dependent variable X 1 , X 2 , X 3 are the independent or explanatory variables.(aka regressor variables) u is the random error term (as usual, it represents a measure of all other factors or random forces that may have an influence on the dependent variable Y). We assume u ~ N( 0 , σ 2 ) β 0 is the intercept term = the mean value of Y when X 1 = 0, X 2 = 0, X 3 = 0 All the other betas are called Partial Slope Coefficients . β 1 = the expected change in Y per unit change in X 1 , holding X 2 and X 3 constant. β 2 = the expected change in Y per unit change in X 2 , holding X 1 and X 3 constant. β 3 = the expected change in Y per unit change in X 3 , holding X 2 and X 1 constant. In general, β j measures the rate of change in E(Y) per one unit change in X j , holding all other X variables in the model constant, i.e., all else equal or ceteris paribus. In other words, β j is a measure of the partial or marginal effect of Xj on the mean value of Y. () j jj EY XX β ∆∂ == Residuals: As usual, the ith residual is uY ˆ ˆ ii =− i Y 3 i X where YX 01 1 2 23 ˆˆ ˆ ˆ ˆ i X ββ =+ + + and ˆ j is the Least Squares estimate/estimator for j The model is estimated by obtaining estimates for the beta parameters using the method of Least Squares (LS). Recall this method chooses the betas that minimize SSR for a given sample of observations. Solving the minimization problem (which involves differential calculus) results in formulas (estimators) for each beta. The formulas change as the more X variables are added to the model. You don’t need to know the LS formulas of the beta coefficients for estimating multiple regression models. Multiple Regression models are usually estimated with the aid of a computer and statistical software. Important Note: The formulas for the intercept 0 ˆ and slope parameter 1 ˆ in multiple regression models are not the same as the formulas you have for estimating simple regression models. 1
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Illustrative Examples EXAMPLE I: Average Starting Pay for MBA graduates Model 1: E(ASP) = β 0 + β 1 GMAT where ASP = average starting pay ($) GMAT= test score form GMAT (points). 1 () EA SP GMAT β = = the expected change in ASP per unit change in GMAT score. Results from estimating Model 1 using EViews software. Dependent Variable: ASP Method: Least Squares Date: 04/01/04 Sample: 1 30 Included observations: 30 Variable Coefficient Std. Error t-Statistic Prob.
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SLN Multiple Regression Models Spring 2007 - Multiple...

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