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Unformatted text preview: The Multiple Regression Model ECON 399 Neil Hepburn Contents 1 Introduction 1 2 Multiple Regression 1 3 Computing OLS Coefficients 5 4 Goodness of Fit 6 5 Properties of OLS Estimators in the Multiple Regression Model 7 6 The GaussMarkov Theorem 13 1 Introduction Introduction • In our look at the two variable model, we examined the relationship be tween house price and square footage • It is a bit unrealistic to think that square footage is the only determinant of house price • Multiple regression models allow us to add in other explanatory factors 2 Multiple Regression Multiple Regression • Recall the ceteris paribus assumption from ECON101 • How can we examine the effects of single factor on something like house price when there are so many other factors that influence it? • Multiple regression analysis is the tool to do this • The following example will illustrate the concept 1 Multiple Regression • Returning to the house price example (but with a different data set) • The following is the result of regressing house price on a number of other factors \ lprice = 1 . 34959 (0 . 65104) + 0 . 167819 (0 . 038181) llotsize + 0 . 707193 (0 . 092802) lsqrft + 0 . 0537962 (0 . 044773) colonial + 0 . 0268304 (0 . 028724) bdrms N = 88 ¯ R 2 = 0 . 6322 F (4 , 83) = 38 . 378 ˆ σ = 0 . 18412 (standard errors in parentheses) Multiple Regression  Overview • The log of the price is a function of the log of square footage, the log of the lot size, the number of bedrooms, and whether or not the house is colonial architectural style • To see what the effect of an increase in the number of bedrooms is, we look at the coefficient on bdrms and hold all of the others constant • An extra bedroom increases the price of the house by approximately 2.7% (why percent?) Multiple Regression  Interpretation • A one percent increase in the lot size increases the house price by about 0.16 percent • A one percent increase in square footage increases the price of the house by about 0.71 percent • A colonial style house, all else being equal, sells for about 5% more than a noncolonial style house Multiple Regression  Interpretation • In the previous example, in each case we were assuming that all of the other factors remained unchanged when we assessed the effect of one variable • This allows us to apply the ceteris paribus assumption • There are a couple of exceptions to this that we need to be aware of – nonlinear terms (such as quadratic) – interaction effects 2 NonLinear Terms • There are times when we want to allow the effect of a particular regressor to increase or decrease as the regressor increases • For example, suppose that as X increases, Y increases but at a decreasing rate (diminishing marginal returns) • We could model this with a quadratic term for X as follows Y i = β + β 1 X i + β 2 X 2 i + u i • If ˆ β 2 < 0, then as X increases, the effect on Y will diminish....
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 Spring '11
 Neil.H
 Econometrics, Regression Analysis, OLS estimators

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