Preface to First Edition - lib

100 SIMPLE AND MULTIPLE LINEAR REGRESSIONregression equation ŷ i = ˆβ 0 + ˆβ 1 x i is minimised, leading to the estimates;n∑(y i − ȳ)(x i − ¯x)i=1 ˆβ 1 = n∑(x i − ¯x) 2i=1ˆβ 0 = ȳ − ˆβ 1¯xwhere ȳ and ¯x are the means of the response and explanatory variable, respectively.The predicted values of the response variable y from the model are ŷ i =ˆβ 0 + ˆβ 1 x i . The variance σ 2 of the error terms is estimated asˆσ 2 = 1n − 2n∑(y i − ŷ i ) 2 .The estimated variance of the estimate of the slope parameter isVar( ˆβ 1 ) =i=1ˆσ 2n∑,(x i − ¯x) 2i=1whereas the estimated variance of a predicted value y pred at a given value ofx, say x 0 is√ √√√√Var(y pred ) = ˆσ 2 1n + 1 + (x 0 − ¯x) 2n∑.(x i − ¯x) 2In some applications of simple linear regression a model without an interceptis required (when the data is such that the line must go through the origin),i.e., a model of the formy i = β 1 x i + ε i .In this case application of least squares gives the following estimator for β 1n∑x i y ii=1 ˆβ 1 = n∑. (6.1)x 2 ii=1i=16.3 Multiple Linear RegressionAssume y i represents the value of the response variable on the ith individual,and that x i1 , x i2 , ...,x iq represents the individual’s values on q explanatoryvariables, with i = 1, ...,n. The multiple linear regression model is given byy i = β 0 + β 1 x i1 + · · · + β q x iq + ε i .© 2010 by Taylor and Francis Group, LLC