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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 31: Multiple Regression Analysis 1 More than one X Up to this point we have studied the simple linear regression model in which we try to use one measurement X to predict or explain Y. ost xamples, e an ink f ore an In most examples, we can think of more than one thing to measure that might be related to Y. In slr we used the example of Y is price of an Accord and X is the odometer. What other things might you measure besides mileage? 2 Multiple Linear Regression Multiple linear regression is very similar to simple linear regression except that the dependent variable Y is described by k independent variables X 1 , , X k X X X k k = + + + + + 1 1 2 2 3 Multiple Linear Regression s Intercept is the same s Slope b i is the change in Y given a unit change in X i while holding all other variables constant (more X X X k k = + + + + + 1 1 2 2 on this later) s SST, SSE, SSR, and R 2 are the same s s e is the same except now s e = sqrt( SSE / (nk1) ) s Slope coefficient C.I.s are the same s pvalues (one for each X i ) are the same 4 Example : Housing Data We have data on 15 randomly selected house sales from last year: price size age lotsize 89.5 20.0 5 4.1 79.9 14.8 10 6.8 83.1 20.5 8 6.3 56.9 12.5 7 5.1 66.6 18.0 8 4.2 price in $1000s size in 100 sqfeet 82.5 14.3 12 8.6 126.3 27.5 1 4.9 79.3 16.5 10 6.2 119.9 24.3 2 7.5 87.6 20.2 8 5.1 112.6 22.0 7 6.3 120.8 19.0 11 12.9 78.5 12.3 16 9.6 74.3 14.0 12 5.7 74.8 16.7 13 4.8 age in years lot size in 1000 sqfeet 5 How does selling price relate to the three variables ? s e 6 Interpretation: The relationship between house size and price is measured by b 1 = 4.146. This indicates that in this model, for each additional 100 square feet, the price of the house increases (on average) by $4,146 (assuming that the other independent variables are fixed). The coefficient b 2 = .236 specifies that for each additional year in the age of the house, the price decreases by an average of $236 (as long as the values of the other independent variables do not change). The coefficient b 3 = 4.831 means that for each additional 1000 sqfeet if lot size, the price increases by an average of $4831 (assuming that house size and age remain the same). 7 Rsquared and the Anova Table S S T = S S R + S S E ho hum as before, SSR 2 R SST = Analysis of Variance SOURCE DF SS MS F p Regression k SSR SSR/k ??? ??? Error nk1 SSE SSE/(nk1) Total n1 SST 8 confidence intervals are as before: 1.96 j j b b s and the hypothesis test: * ject : if Confidence Intervals and Hypothesis Tests reject : if j j H = * 1.96 j j j b b t s  = 9 The housing data : = = 2 : 0, and : H H Clearly we accept 1 2 3 b b b A confidence interval for 1 is: 4.145 +/ 1.96(.751) Clearly we reject = = 1 3 : 0, and : H H 10 Example...
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen
 Linear Regression, Regression Analysis

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