The Covariance Matrix of the Error Vector
Assumption (iv) of the linear regression model claims the covariance matrix of the error vector ɛ to be Cov(ɛ) = σ2In with an unknown parameter σ2 ∈ (0, ∞). This chapter discusses the estimation of σ2 in detail, and introduces situations under which it ap
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Assumption (iv) of t he linear regression mod el claims t he covariance mat rix of t he error vector e to be Cov(e ) = (J"2 I n wit h an unknown paramet er (J"2 E (0, 00) . T his chapter discusses t he est ima tion of (J"2 in det ail , and introduces sit uations und er which it appears t o be reasonable t o extend assum ption (iv) to COV(e) = (J" 2y for some symmet ric posit ive/ non negative definit e matrix
v ¥= t ;
5.1 Estimation of the Error Variance Section 2.2.4 introduces t he least squa res variance est imato r
a
2
1 ~ ~ (y - X f3 Y (y - X f3 ) n- p
= -
as an unbi ased est imator for (J"2 and ment ions t he existence of ot her possible est imators like a? and a~L" This sect ion deals with t he esti mation of (J"2 in more det ail. 5.1.1 The Sample Variance
In t he linear regression mod el y = X f3 + e , the random variables Y1, . . . , Yn can also be seen as a sample of size n , which, however , does not meet t he requirements of a simp le random sample [79, Definit ion 2.2.2], since t he ind ividu al Yi are not identically distribut ed as long as not all independ ent variables are const ant. T he appropriate measure for t he dispersion in a simple random sample is t he sample varianc e
~2 (J"y
1 ~( - )2 = n _ 1 L... Yi - Y . i= l
It coincides wit h t he least squa res variance est imator for (J"2 in t he special case t hat t he linear regression model is t he simple mean shift model, describ ed by the equation
J. Groß, Linear Regression © Springer-Verlag Berlin Heidelberg 2003
260
5 The Covariance Matrix of the Error Vector
where In denot es th e n x 1 vect or whose every element is equal to 1. In th at case t he ordinary least squares est imator for /1 is ji = ( l /n) I~ y, and t he vecto r of ordinary least squares residuals is given by g = Cy , where 1
,
C=In--InI n n is t he cente ring matrix. T hen 1 y'c y a~2 = - n-l
is identi cal to O'~. Hence, und er t he simple mean shift model, the estimator O'~ is unbi ased for a 2 . If we consider O'~ as an est imator for a 2 under th e linear regression mod el y = Xf3 + e with assumptions (i) to (iv) , then ~ 2 ) _ 2 t r (C ) E ( a y - a n- 1
+
f3'X'CXf3
n- l '
where tr (C ) = n -1. Hence, we have E(O'~) = a 2 for every a 2 E (0, 00) and every f3 E IRP if and only if C X = o. Th e latter mean s t hat t he column space of X is cont ained in t he column space of In which can only happen if each column of X consists of ident ical elements. Hence, the sample variance as an estimator for a 2 is usually biased upwards in t he linear regression model, meaning t hat the actua l est imate must be expected to be greater t han t he t rue a 2 . Therefore, wit h respect to t he bias , t he least squa res est imator 0'2 is mor e favor able t han the sa mple variance a y . This is not sur prising resul t , since und er the linear regression model the random variables Yl, . . . , Yn do not have ident ical expectations (unless all independ ent variables are constant ), t hus cont radict ing t he usual assumptions for a reasonabl e applica tio n of th e sa
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