Transcription of Chapter 2: Simple Linear Regression
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Chapter 2: Simple Linear Regression
Chapter 2: Simple Linear Regression1 The modelThesimple Linear regressionmodel fornobser-vations can be written asyi= 0+ 1xi+ei, i= 1,2, ,n.(1)The designationsimpleindicates that there is onlyone predictor variablex, andlinearmeans thatthe model is Linear in 0and 1. The intercept 0and the slope 1are unknown constants, andthey are both calledregression coefficients;ei sare random errors. For model (1), we have thefollowing (ei) = 0fori= 1,2, ,n, or, equiva-lentlyE(yi) = 0+ var(ei) = 2fori= 1,2, ,n, or, equiva-lently, var(yi)) = (ei,ej) = 0for alli6=j, or, equivalently,cov(yi,yj) = Ordinary Least Square EstimationThemethod of least squaresis to estimate 0and 1so that the sum of the squares of the differ-ence between the observationsyiand the straightline is a minimum, , minimizeS( 0, 1) =n i=1(yi 0 1xi) = XE(Y|X=x) 1=Slope 0=Intercept1 Figure 1: Equation of a straight lineE(Y|X=x) = 0+ least-squares estimators of 0and 1, say 0and 1, must satisfy 2n i=1(yi 0 1xi) = 0(2) 2n i=1(yi 0 1xi)xi= 0(3)Simplifying these two equations yieldsn 0+ 1n i=1xi=n i=1yi 0n i=1xi+ 1n i=1x2i=n i=1yixi(4)Equations (4) are called theleast-squares nor-mal equations.
1 The model The simple linear regression model for nobser- vations can be written as yi= β 0 +β 1xi+ei, i= 1,2,··· ,n. (1) The designation simple indicates that there is only one predictor variable x, and linear means that the model is linear in β 0 and β 1.The intercept β 0 and the slope β 1 are unknown constants, and
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