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Unformatted text preview: 13.2 In deviation term the true rnedel ean be written as:
Jr"; : 1813:; + [Ht _ E)
New Theret'ere, EH38] J = ,8] , l'IlElkiﬂg use et‘ the varieus preperties et' in and sj. That is, even if we intreduee the unneeded intercept in the
seeend rnedel, the slepe eeet‘t‘ieient rernains unbiased. This is
as per theery. The variances et‘ the twn estirnatnrs are: .3 2. A D. h {I am a = " 2 and tartar): E X. — X] I which are net the same. 13.3 We know that
. E,ij EXftarﬂsLarlX. +13.) 3s

 LXE EX;
aoEXr EWX.‘
= —..+c;tfl+—
EXT TX;
. A or EX.
Therefore, £1,531]: 0—2! +63]
TX. Here. the slope estin'iator in the incorrect model gives a biased
estimator of the true slope coefficient. The variances are as given
in Esercise 13.2. 13.4 (a) Recall the follcna'ingigr formula from Chapter 7:
E .3 _ . . .
R2 : ha +ha "ha’ra’aa '2
1 _ r23
Since X3 is irrelevant, r13 = U, which reduces the preceding t'onnula [Di R2: '2 1— if}. Typically, then, the addition of X3 will increase the R‘ value. However, if r23 is zero, the R2 value will remain unchanged. (in Yes, they are unbiased for reasons discussed in the chapter.
This can be easily proved from the multiple regression formulas given in Chapter 'r', noting that the true ,83 is zero. A (c) The variances of ’33 in the tvvo models are: '3‘
vma: “ 2 (true mod el) H'E' f {T
T '} A O". .
varﬁ2 : {incorrect model)
E x; (1 _ £33) Thus the variances are not the same (a) As discussed in the ehapter, emitting a relevant yariahle will lead to biased estirnatien. Henee lit/{3’1}: a1 and H33] :t (32. The deriyatiens using sealar algebra leads to unwieldy espressiens.
They can be easily derived using matris algehra. But if you want to
preeeed, estimate the parameters of the "ineerreet" model and then put the true model in the estimated parameters, talte
espeetatiens, and find out of if espeeted yalues ef the parameters
from the ineerreetly speeit‘ied rnedel equal their true values. If
they do net, then there is bias. (b) If L3 is an irrelevant yariahle, then the estimates remain
unhiased, eseept that they have larger yarianees due tn the presenee
of the "nuisanee" yariahle Lg. .J'H. : .J'H. ﬁﬁﬁmel : (c) The intercept coefficient will be unbiased but the elcpe
coefficient will be biased and inccl’ieietent. ...
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 Spring '13

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