notes03 - The Multiple Regression Model ECON 399 Neil...

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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 Gauss-Markov 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 = 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 non-colonial 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 non-linear terms (such as quadratic) interaction effects 2 Non-Linear 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 = + 1 X i + 2 X 2 i + u i If 2 < 0, then as X increases, the effect on Y will diminish....
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notes03 - The Multiple Regression Model ECON 399 Neil...

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