Lecture 5 lecture notes

# Lecture 5 lecture notes - Lecture 5 A Multiple Linear...

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Lecture 5. A Multiple Linear Regression Model Summary of previous lectures on MLR Here we consider the case p explanatory variables Y t = β 0 + β 1 X t, 1 + ... + β p X t,p + ² t . (5.1) This can be expressed more compactly in matrix notation as Y = + ², (5.2) where Y = ( Y 1 ,...,Y n ) T , β = ( β 0 ,...,β p ) T , ² = ( ² 1 ,...,² n ) T ; X is an n × ( p + 1)-design matrix X = 1 X 1 , 1 ... X 1 ,p 1 X 2 , 1 ... X 2 ,p . . . . . . . . . . . . 1 X n, 1 ... X 1 ,p Here the historical data for the dependent variable consist of the observations Y 1 ,...,Y n ; the historical data for the independent variable consist of the observations in the matrix X . Minimizing SSE = ( Y - X β ) T ( Y - X β ) yields the least squares solutions ˆ β = ( X T X ) - 1 X T Y (5.3) for nonsingular X T X . The forecast of a future value Y k is then given by ˆ Y k = x T k ˆ β, (5.4) where x k is a (column) vector of regressors’ values for the k -th case. Under the OLS assumptions (recall them!) we obtain Var ˆ β j · = σ 2 X T X · - 1 jj , (5.5) where the X T X · - 1 jj denotes the j -th diagonal element of X T X · - 1 . 1

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This yields s . e . ˆ β j · = σ * r ( X T X ) - 1 jj . (5.6) Note that here the degrees of freedom (d.f.) are n - ( p + 1) = n - p - 1. (In fact, since the number of estimated parameters, i.e. p parameters for the independent variables and one for the intercept, is p + 1.) Under the OLS assumptions, a 100(1 - α )% C.I. for the parameter β j ,j = 0 , 1 ,...,p , is given by ˆ β j ± t α/ 2 ,n - ( p +1) s.e. ˆ β j · = ˆ β j ± t α/ 2 ,n - ( p +1) s r ( X T X ) - 1 jj , (5.7) where s = SSE/ ( n - p - 1). Typically s . e . ( β * j ) is read directly oﬀ the R output and so the calculations 5.7 should not be done manually. Under the OLS assumptions it can be shown that Var ˆ Y t - Y t · = σ 2 ± x T t X T X · - 1 x t + 1 , (5.8) yielding a 100(1 - α )% P.I. for Y t : x T t ˆ β ± t α/ 2 ,n - ( p +1) s q x T t ( X T X ) - 1 x t + 1 . (5.9) We usually never perform these calculations by hand and will use the corresponding software functions, e.g. predict.lm in R (see the example for the code template). What else can you get from the regression output?
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Lecture 5 lecture notes - Lecture 5 A Multiple Linear...

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