Quz - Lear Regreo Aaly (Baed o Lecture -4). I the mple lear regreo model y = β + βx + ε, wth Tme: Hour Ε ε = Ε ε = ( ) 3, ( ), =,,...,, the ubaed drect leat quare etmator ˆβ ad ˆβ of β ad β repectvely, are where ˆ ˆ ˆ (A) β = y βx, β = ˆ ˆ 3, ˆ = = 3 (B) β y βx β ˆ ˆ, ˆ = = 3 (C) β y βx β ˆ ˆ ˆ (D) β = y βx 3, β = x x y y x x x x y y = ( )( ), x = ( ), =, =. = = = = Awer: (C) Soluto: Mmzg ε wth repect to = β ad β gve Now expre ε ( β β ) = = S = = y x S S, ˆ = = β =, ˆ β = y ˆ β x. β β = ky k = = ˆ β where Ε ( ˆ β ) = kε ( β + β x + ε ) = Ε( ˆ β 3) = β. = + β + 3 x x ˆ β 3= 3 a ubaed etmator of β. So
Next Ε ( ˆ β ) =Ε( y ˆ β x) =Ε ( β + β x + ε ˆ β x) = β + β x + 3 x( β + 3) = β. ˆ β = a ubaed etmator of β. So. I the mple lear model y = β + β x + ε Ε ε = Ε ε = = ad aume that, ( ), ( ),,,...,, kow. The bet lear ubaed etmator of β = (A) ( x x)( y y) = ( x x) (B) (C) (D) = = ( x x)( y y) = = ( x x) ( x x)( y y) ( x x) x x y y = = = x x = = Awer: (D)
Soluto: The Gaur Markoff theorem tell that the bet lear ubaed etmator of β b = = ( x x)( y y) whe ( x x) = Ε cotat ad depedet of. So traform the model ( ε ) y = + x + Ε = Ε = = a β β ε, ( ε), ( ε ),,,...,, y β βx ε = + + or y = β + β x + ε, * * * * where y x ε y =, x =, ε =. I the traformed model, we have * * * * * ( ε ), ( ε ) Ε = Ε =. Thu the bet lear ubaed etmator of β x x y y ˆ β. * * * * ( x x )( y y ) = = = = = = * * ( x x ) x x = = = 3. Coder the multple lear regreo model y = Xβ + ε where y a vector of obervato o repoe varable, X a K matrx of obervato o each of the K explaatory varable, β a K vector of regreo coeffcet ad ε a vector of radom error wth Ε ( ε ) = ad Ε ( εε ') quare etmator of β =Ω. The covarace matrx of the ordary leat (A) (B) (C) (D) ( X ' X) ( X ' X) X ' Ω X( X ' X) ( X ' ΩX) X ' X( X ' Ω X) ( X ' X) X ' Ω X( X ' X) Awer: (B) 3
Soluto: The ordary leat quare etmator of β ˆ ( X ' X) X ' y ( X ' X) X '( X ). β = = β + ε The etmato error of ˆβ ˆ ( ' ) = X X X ' β β ε ad the covarace matrx of ˆβ Cov( ˆ β) =Ε( ˆ β β)( ˆ β β)' = E X X X εε X X [( ' ) ' ' ( 'X) ] = ( X ' X) X ' Ε( εε ') X( X ' X) = X X X ΩX X X ( ' ) ' ( ' ). 4. The value of coeffcet of determato the two multple lear regreo model are.4 ad.7. Let y be the tudy varable, ad X ' ( =,, 3) be the explaatory varable. Thee value of coeffcet of determato belog to whch of the two poble ftted model out of followg four poble ftted model, y = 3+ 3 X + 3 X + 3 X I. 3 y =. +.3 X +.4 X +.5 X II. 3 III. y = 3 X 3 X 3 X3 y =.3 X.4 X.5 X IV. 3 (A) (I) ad (II) (B) (III) ad (IV) (C) (I) ad (III) (D) (II) ad (IV) Awer: (A) Soluto: The coeffcet of determato defed oly whe the tercept term preet the lear model. Sce oly the lear model (I) ad (II) are havg tercept term, o the coeffcet of determato ca be defed oly thee two poble model. 4
5. The obervato o tudy ( y) ad explaatory varable X ' a uual multple lear regreo model wth four explaatory varable are tadardzed,.e., every obervato codered a devato from t mea ad dvded by t tadard devato. Whch of the followg two model repreet the poble ftted model wth tadardzed obervato: I. y = + X+ X + X3+ X4 X X X 3 X II. y = + 3 4 3 4 4 III. y = + X X + X3 X4 IV. y =.X+.X +.3X3 +.4X4 (A) (I) ad (III) (B) (I) ad (IV) (C) (II) ad (IV) (D) (II) ad (III) Awer: (C) Soluto: The tercept term become zero whe the obervato o tudy ad explaatory varable are tadardzed ay multple lear regreo model. Sce the model (II) ad (IV) do ot have tercept term, o they repreet the two poble ftted model wth tadardzed obervato. 5