Aswer key to problem set # ECON 34 J. Marcelo Ochoa Sprg, 009 Problem. For T cosder the stadard pael data model: y t x t β + α + ǫ t a Numercally compare the fxed effect ad frst dfferece estmates. b Compare the error varace estmates from the two methods. Soluto. The dfferece estmates are obtaed by takg the dfferece across tme perods to elmate the uobservable. Hece, for dvdual we have, y y x x β + ǫ ǫ y x β + ǫ Assumg that E x ǫ 0, the dfferece estmate of β s, N N ˆβ DE x x x y The fxed effects estmator s obtaed demeag each equato. Let ȳ y +y, x x +x, ad ǫ ǫ +ǫ the: the, the fxed effect estmate of β s, ˆβ FE Now, ote that, N y t ȳ x t x β + ǫ t ǫ, t, N x t x x t x t x t x x t x t x t x y t ȳ t x t x + x x t x + x t x x x x + x x x x x x
ad, x t x y t ȳ x t x + x y t y + y t x x y y x x y y + t x y Replacg these results to the FE estmator we have: Thus, ˆβ FE N x x N ˆβ FE ˆβ DE. The varace covarace matrx of ˆβ DE equals to, N x x ˆσ DE x y ˆβ DE where the resduals from the dfferece estmator, ê DE y x ˆβDE, are used to calculate, N ˆσ DE êde N k O the other had, varace covarace matrx of ˆβ FE equals to, N x t x x t x ˆσ FE t where the resduals from the fxed-effect estmator, are used to calculate, ê FE t y t ȳ x t x ˆβ FE t,, N êfe + ê FE ˆσ FE N k Usg the fact that ˆβ DE ˆβ FE : ê FE y y x x ˆβFE êde y + x ˆβDE ê FE y y x x ˆβFE y x ˆβDE êde
The, the sum of squared resduals from the fxed effect estmator s, Therefore, σ FE êfe + ê FE N k êde + ê DE 4N k êde N k σ DE N V ˆβ DE ˆσ DE ˆσ DE N σ FE V ˆβ FE x x x x N x t x x t x t Problem. Cosder the followg pael data model: y t α + x t β + z γ + ε t Let x x,...,x T, ad assume E ε t x, z, α 0. Let σ α V α ad σ ε V ε t. a Let c α + z γ. Fd V c ad compare t to σ α. b Compare the estmated varace of the uobserved effect whe estmatg the model by fxed effects to the estmated varace of the uobserved effect to f we estmated the model by radom effects.. Soluto. Gve the geeral assumptos we have: E c x, z E α + z γ x, z E α x, z +z }{{} γ 0 z γ E c x, z E α + z γ + α z γ x, z E α x, z + z γ + E α x, z z γ σ α + z γ 3
the, V c x, z σ α + z γ z γ σ α Usg the codtoal varace detty we have: V c V E c x, z + E V c x, z γ V z + σ α > σ α For b ote that whe you estmate a fxed effects model, the uobserved effect that s estmated s c, whle whe you estmate a radom effects model α s the uobserved effect as z ca eter as a explaatory varable. From a t follows that the estmated varace of the uobserved effect s larger whe t s estmated usg fxed-effects. Problem Hayash Aalytcal Exercse #. Soluto. a Followg the ht the book, let, M D I M DD D D the, we ca pre-multply the orgal model by ths ahlator matrx assocated to D ad obta, M D y 0 M D Dα + M D Fβ + M D η The estmate of β s, ˆβ M D M D F M D M D y To show that ths estmate equals the fxed effects estmator t must be true that M D F F whch s true f M D I Q, where Q I M ι M ι M ι M ι M, M D I M I ι M I ι M I ι M I ι M I I M I ι M I ι M I ι M I ι M I I M I ι M I I ι Mι M I ι M I I M I ι M I M I ι M I I M I ι M M I ι M I I M I ι M ι MM I I M ι M ι MM I Q as we wated to show. b We have that, ˆα D D D y D F ˆβ 4
the, D D I ι M I ι M I I ι M ι M MI D y I ι M y D F I ι M F The th elemet of ˆα ca be wrtte as, as we wated to show. ˆα M ι M y ι M F ˆβ ȳ M ι M F ˆβ c Gve the assumptos of the model, we have that: E η W E η F, D thus, the assumpto of strct exogeety holds. E η F, D 0 sce D s full of costats E η F by assumpto 0 by assumpto ad, E η η W E η η F, D E η η F σ η I M by assumpto E η η j W E η η j F, D E η E η j j F 0 by assumpto therefore, the resdual s sphercal, ad the assumptos of the classcal regresso hold. Problem Hayash Aalytcal Exercse #. Soluto. a To show ths s true cosder, 0 C 0 the, C I M 0 + + 0 + + 0 0 I the case where C s created from Q, the detty follows drectly. 5
b The model s, y F β + ι M b γ + ι mα + η whch multpled by C yelds, C y C F β + C ι M b γ + C ι m α + C η C y C F β + C η sce C ι M 0. d We have that, S xz C F x C I K s xy C I K y x F x Usg Ŵ CC x x we have, β GMM S xz xzŵs Sxz Ŵs xy F x C I K CC F x C I K CC QF Qy QF whch s the fxed effect estmator. x x x x x x C I K x x x x Qy x x C I K F x y x e I ths case the effcet weghtg matrx W S, wth S Eη η Ex x. Therefore the effcet weghtg matrx s gve by, Ŵ ˆΨ x x 6
The effcet GMM estmator s, β GMM x ˆΨ x ˆΨ ˆΨ F ˆΨ ỹ x x x x x x F ˆΨ F F x ỹ x x x x x x x ˆΨ ỹ To obta the asymptotc varace of the GMM estmator ote that, β GMM F ˆΨ F ˆΨ F β + η β + F ˆΨ F ˆΨ η Usg the samplg error equato we ca easly obta the asymptotc varace of ths estmator. f The proposed estmate of Ψ satsfes the codtos of Proposto 4.. Most mportatly, the resduals are calculated usg a cosstet estmate of β, ad the cross momet correlato betwee the regressors exsts ad s of colum full rak. Hece, t s a cosstet estmate of Ψ. g Sarga test ths case equals, J g ˆβŜ g ˆβ F ỹ F x ˆβ ˆΨ x x F ỹ F x ˆβ h The frst result that has to be verfed follows from the fact that, Ψ E η η E C η η C σ η C I M C σ η C C 7
hece, a cosstet estmator of Ψ s, ˆΨ ˆσ η C C Replacg ths value of the estmator foud e we obta, ˆβ ˆσ η C C F F C C F CC C C F QF Q QF Qy ˆσ η C C ỹ C C ỹ Q Qy CC C Cy whch f the fxed effects estmator. Ths was derved usg the fact that Q CC C C ad that Q s a dempotet matrx. Problem Hayash Aalytcal Exercse #4. Soluto. b Let y 0 be gve, the: y α + ρy 0 + η y α + ρα + ρy 0 + η + η. α + ρ + ρ y 0 + ρη + η y m α + ρ m ρ + ρm y 0 + η m + ρη,m + + ρ m η Multplyg ths last equato by η h ad takg expectatos we have: E y m η h E α η h + ρm ρ + ρm E y 0 η h + E η h η m + ρe η h η,m + + ρ m E η η h By assumpto we have that E α η h 0, E y 0 η h 0 ad E η h η,m j 0, the E y m η h 0. 8
c We use aga the recurso, ad multply ths last equato by η,m j, E y m η,m j E α η,m j + ρm ρ + ρm E y 0 η,m j + E η,m j η m + ρe η,m j η,m the, + + ρ m j E η,m j + + ρ m E η η h E y m η,m j ρ m j σ η 9