Dynamic models models with variables dated in different periods

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1 Panel Daa Economercs Semnar 4: Ocober 3,7 Soluon proposal by: Aleksander Sahnoun (7h semeser Problem 4.: Auoregressve models Problem 4. A: Dynamc models models wh varables daed n dfferen perods Panel daa handles dynamc models beer han cross-secon daa and me seres daa. Pure me seres daa can handle ndvdual-specfc heerogeney (can handle some me specfc heerogeney f has o some exend be parameerzed. h panel daa s possble o handle ndvdual-specfc heerogeney and me-specfc heerogeney whn he same framework. In auoregressve models, wh laggng LHS varables as explanaory varables, panel daa offers he possbly o handle me-lags beween me ndexed varables and ndvdualspecfc heerogeney. Problem 4. : ( y, c y, γ x, α u, u IID o σ,..., ;,...,,, (, Cross-secon daa: Assume ha y,, y,, and x, are observed varables. y c y γ x α u u IID( o, σ,,..., ; (, * * (' y y, x u α γ α α c e have equaon bu coeffcens, so s no possble o esmae all of hem. e are no able o esmae he ndvdual-specfc nercep erms from cross secon daa. Remark: Gven ha he model specfcaon s correc, and snce he dsurbance erm s IID, OLS on ('' y α y, δ x u ould be MVLUE.

2 Problem 4. C: I assume access o panel daa, * (3 y, α y, γ x, u u, x, y are ndependen for all (, α α γ < * c u IID o σ, (,,..., ;,...,, a hn-ndvdual esmaon explos nformaon from observaons whn each ndvdual over me. If y, were srcly exogenous we could have esmae OLS on (3. he OLS esmaor would concde wh he whn-ndvdual esmaor and would have gven MVLUE. s *, γ, γ α, s, s s y y ( x u If he process, for each ndvdual, have sared an nfnely long dsance back n me and γ <, follows ha α y x u x * s * s, γ ( α, s, s γ (, s u s s γ s, e can ransform (3 no an equaon n deparures from he ndvdual-specfc means, * * (4 ( y, y, ( α α ( y, y, γ ( x, x, ( u, u,,..., ; ( y, y, γ ( x, x, ( u, u,,...,, and apply OLS. y, s uncorrelaed wh u,, bu correlaed wh u, so we ge esmaors ha are conssen. (nconssen f s fne and are asympocally negavely based (larger han ½ when he number of ndvduals goes o nfny.. e can nerpre ˆ γ, ˆ w w as a weghed mean of ndvdual-specfc esmaors, all whch are conssen when goes o nfny. ( y y ( y y,,,, ˆ 45, 459 Y, Y Y, Y 98 ( y, y, γ ( y y ( x x,,,, ˆ 53,547 X, X X, Y 5 ( x, x, b ha do src exogeney requre?

3 e can ransform (3 no an equaon where we subrac he ndvdual specfc mean wh he global mean. e hen explo he nformaon beween ndvduals. (5 ( y y ( α α ( y y γ ( x x ( u u,,,,,..., ;,...,, If we assume ha he ndvdual specfc nerceps are uncorrelaed wh he RHS varables, we can wre he esmaors as: ( y y( y y,, ˆ 33,37 Y, Y Y, Y ( y, y γ ( y y ( x x,, ˆ 4,9449 X, X X, Y 54 ( x, x e can see ha he esmae of γ s volang our assumpon γ <. If he ndvdual specfc coeffcens are correlaed wh he RHS varables, he esmaors are : ( α α( y, y ˆ γ γ ( y, y ˆ ( α α( x,, ( x x x c rng he equaon n frs dfference form (5 ( y, y, ( y, y, γ ( x, x, ( u, u,,..., ; y y γ x u,...,,,,,, In (5 he lagged RHS varable y, s correlaed wh u, (5 ( y y ( y γ ( x x ( u u,, y,,,,,, Always based and nconssen?

4 ˆ γ ˆ,,,,, OLS ( y, y,, OLS ( y y ( y y ( y ( y,, G G Y, Y Y, Y ( y, ( y y ( x,,, ( x x,, ( y ( x x,,, G G X, X X, Y,57 ( x, 9, 6,56 8,44 5,33,54 he OLS esmaors are nconssen regardless of wheher, or boh go o nfny even hough he equaon s pu on dfference form. e can alernavely ransform he equaon, or esmae by nsrumen varables and GMM. he ndvdual nercep erm, In (a * ˆ ˆ * ˆ α y y λ x, ˆ α ˆ α c,,, * * *, c j j ˆ α ˆ α ˆ α ˆ α e have x equaons and unknown, so we don need any addonal nformaon. In (b e have unknown, bu we have jus lnear ndependen equaons, so we need some addonal knowledge/resrcons o esmaeα. 3 In (c he nercep vansh when we ake he equaon on dfference form. 3 Addonal condons?

5 Problem 4. D : * (6 y, α y, γ y, γ x, x, u,,..., ;,...,, (6' y, y, γ y, γ x, x, u, u, x, y are ndependen for all (, α * α c,..., ;,...,, u IID o σ, (, Snce we have a Panel Daa se,. e have seen n problem C ha we have o esmae he coeffcens wh IV o ge conssen and unbased esmaors. Snce x, or x, are correlaed wh y, y,, or y,, y, and uncorrelaed wh u,, u,, hey are vald nsrumens. he equaons: y,4 y,3γ y,γ x,4 x,3 u,4 y,5 y,4γ y,3γ x,5 x,4 u,4 M y y γ y γ x x u,,,,,, Vald nsrumens: z ( y, x, x for ( y, y, x, x,3,,4,3,,3,4,3 ( snce cov( y,, u, u, whle cov( y,, u, u, z,4 ( y,, y,, x,5, x,4 for ( y,3, y,4, x,5, x,4 M z ( y, y,..., y x, x for ( y, y, x, x n he (-3'h equaon,,,, 3,,,,,, e are able o esmae he coeffcens when 4 be used as IV s (uncorrelaed wh u, 4. hen we have enough varables ha can 4 hen are hey conssen and unbased?

6 Problem 4. E: α γ δ * (7 y, y, x, z, u, Assume: u, s sochascally ndependen of x,, y, z α, * α c for all (, u IID o σ, (,,..., ;,...,, he mnmzaon problem ( whn-ndvdual esmaon: mn ( y k y ˆ γ x ˆ z ˆ δ ˆ α k, γδα,,, foc :,,,. ˆ ˆ ˆ ( y ˆ ˆ, k y, γ x, zδ α k. ( y ˆ ˆ ˆ ˆ ˆ, k y, γ x, zδ α y, ˆ γ 3. ( y kˆ y ˆ γ x ˆ,,, ˆ z ˆ δ ˆ α x, 4. ( y ˆ ˆ ˆ ˆ ˆ, k y, γ x, zδ α z ˆ δ 5. ( y ˆ ˆ ˆ ˆ ˆ, k y, γ x, zδ α,..., ˆ α If he las equaons are sasfed, hen he frs and he fourh also hold. e hus only have lnear ndependen equaons, from whch we canno deermne 4 unknowns. From he las equaons s follows ha: kˆ ˆ ˆ ˆ ˆ zδ α y, γ x,,..., Insered n FOC (,3 ( ˆ,, ˆ y y γ γ x ˆ ˆ, ( zδ ˆ α kˆ y, ( y y ˆ γ x ˆ y ˆ γ x ˆ y,,,,,, ( ˆ ˆ,,, ( ˆ ˆ ˆ, ˆ y y γ x zδ α k x ( ˆ ˆ ˆ ˆ y, y, γ x, y, γ x, x, hch we can solve for ˆ γ and ˆ. he compose erm: kˆ z ˆ δ ˆ α y y,, ˆ λ x ˆ,

7 z e can only esmae he compose parameers k ˆ ˆ ˆ zδ α and ˆ, ˆ λ. e are unable o separae he fxed ndvdual-specfc effec ˆ α from he effec from he ndvdual specfc varable snce we have a mulcollneary problem. he esmaor of he wodmensonal varable x, concdes wh he one we could have obaned whou z. Problem 4.: Model wh sochasc slope coeffcens Problem 4. A: he model: ( y α x u u IID( o, σ,..., ;,...,, Indvdual-specfc nercep: α α ε ε IID( o, σ ε,..., ; Indvdual-dependen coeffcen δ δ IID( o, σ δ,..., ; ( ( y α ε δ x u α x δ x ε u v δ x ε u - Compose dsurbance ' ( y α x v,..., ;,...,, Assume ha x, ε, u, δ are sochascally ndependen for all and. Properes of he compose dsurbance: E( v x E( δ x ε u x E( δ x x E( ε x E( u x ( δ ε Ev ( x E x u x E( δ x x E( ε x E( u x E( δ x ( ε u x E( u ε x x σ δ σ ε σ σ σ σ x δ ε ( ( Evv ( s xx s E δx ε u δxs ε us xs E( δ x x x x E( ε x x E( u u x x s s s s s xxσ s δ xx sσ δ σ ε s

8 Varance covarance marx x σδ σε σ for j, s x xsσδ σε for j, s cov( vv js xx js, j,..., ; s,,...,, for j, s for j, s he compose dsurbance s heeroskedasc boh across ndvduals and perods (dfferen varance across ndvduals and perods. Ev ( x x σδ σε σ Furher, he dsurbance s auocorrelaed over perods, bur no across ndvduals (correlaon beween dfferen perods, bu no beween ndvduals. Evv ( s xx s xx sσ δ σ ε s he compose dsurbance erm s uncorrelaed wh he equaon s rgh hand sde varable. Usng he law of eraed expecaons: cov( v, x E( v x E E ( v x x E x E ( v x [ ] [ ] x v x v u he equaon ( RHV s no sochasc ndependen of he compose dsurbance, snce he condonal and margnal dsrbuon dffers. Usng he law of eraed expecaons: Ev ( x x σ σ σ δ ε E[ E( v x] Ev ( Ex ( σ σ σ Ex ( σ σ σ x v δ ε δ ε Ev ( Ex ( σ σ σ δ ε ogeher wh: 3 E( v x Ex[ Ev ( v x x ] Ex[ xev ( v x ] E x ( σδ σε σ E( x σδ E( x σε E( x σ 3 Evx ( Ex ( σ Ex ( σ Ex ( σ δ ε hs mples ha: cov( v, x Evx ( Evx ( Ev ( Ex ( 3 Ex ( σδ Ex ( σε Ex ( σ Ex ( Ex ( σδ σε σ ( 3 ( Ex ( Ex ( Ex ( δ ( Ex ( Ex ( ε ( Ex ( Ex ( σ σ σ cov( x, x σ δ So σ δ >, x s uncorrelaed wh v, bu correlaed wh v In marx noaon ( can be wren as:

9 (3 y eα x v v eε xδ u,..., ; ' ' j ' Σ eeσε σδ σ xx I (4 E( v, vj x, xj,j,...,, Σ j j e can also wre (3 as: (5 y X v here ( α, he varance covarance marx s hen: Σ K Σ Σ L ' (6 E( vv X M O M Snce he varance covarance marx has nonzero elemens ousde he dagonal GLS wll be MVLUE. (he GLS esmaor s more effcen han he OLS esmaor. CASE If he varances of he sochasc elemens are know ( σδ, σε, σ,..., σ esmaor s:, he GLS * ' ' XΣ X XΣ y he esmaor s hen weghed by he varance and covarances. * * ' - ' - Le denoe he ndvdual GLS esmaor XΣ X XΣ y,..., CASE If σδ, σε, σ,..., σ are unknown, we can esmae hem from her emprcal counerpar ˆΣ and hen esmae by GLS usng ˆΣ. hs s he mehod of FGLS/EGLS/wo Sep. Frs esmae ˆ and ˆ α by OLS when hey are consdered as unknown consans. hen fnd he resdual vecor y e ˆ α x ˆ. uˆ Form ˆ ε ˆ ˆ α α j and esmae j And symmerc for σ ˆδ. Form ˆ δ ˆ ˆ j and esmae j σ ε by s emprcal counerpar σ δ by s emprcal counerpar ˆ σ ˆ σ ˆ ε α j j ˆ δ j j

10 Problem 4. y α x u (7 where, α are consans. u IID( o, σ,..., ;,...,, σ for j, s for j, s cov( uu js xx s for j, s for j, s e can esmae he whn esmaor of. ˆ ' ' ( X X ( X y XX XY XX XX XY XY e use he varance whn he ndvdual o esmae ˆ. he GLS esmaor n A of s: ˆ GLS ' ' ' ' X ( I θ ( X X I X X ( I y θx ( I y XX θ XX XY θ XY σ here : θ σ ( σ σ ε δ In problem A we esmae he common slope coeffcen. he GLS esmaor s a weghed esmaor, weghed wh he nverse of he varance covarance marx. If α, are consans, we exrac he whn varance (and sesθ.

11 Problem 4. C es of heerogeney n slope coeffcen. Assumng unknown and consan α, (FE. Full ndvdual heerogeney: H : α,..., α and,..., are unresrced A Indvdual nercep heerogeney, homogeneous slopes: H : α,..., α and are unresrced Full homogeney: H : α α α and are unresrced C e make he assumpon u II(,, σ I ( replacng u IID(,, σ I II > Independen, dencally normally dsrbued. hs normaly assumpon ensure ha he F-es s vald. he sum of squared resduals are: SSR uˆ for model H A SSR uˆ for model H SSR uˆ for model H C A C e wan o es he null hypohess homogeneous slope coeffcen - agans he alernave hypohess heerogeneous slope coeffcen. he es can n general be wren as SSR SSR umber of resrcons under H F SSR umbers of degrees of freedom under H If we assume ndvdual nercep, he es would be: SSR SSRA K( SSR SSRA ( K FA SSR A SSR A K( ( K hch s F-dsrbued wh K(- and (-K- degrees of freedom under. H If we assume a random effec model, we mus use he reusch-pagan ess 5. 5 Lecure noe 7

12 Problem 4. D y α x ε δ x u (8 u IID( o, σ,..., ;,...,, α α ε δ u x, ε, δ ε (, σ,..., ; IID o ε δ IID( o, σ δ,..., ; Expandng he model allowng for correlaon beween α, and x,,...,. hs may reflec ha a laen ase varable for household - ε, δ-, s correlaed wh household s ncome - he observable explanaory varable x. Frs we formalze a correlaon beween ε and x. x Ex ( λ w w IID( o, σ [ ] ε If δ s correlaed wh x hen can be δ IID( o, σ δ, so we do he same formalzaon for δ : δ [ x E( x ] ϕ η η IID( o, σ μ (9 y ( α μλ x x xλ ϕxx w xη u where E( x μx k x x λ ϕx x Q where Q w x η u, k α μ λ x e have now conrolled for he problem wh laen heerogeney, bu we now have a problem wh he compose dsurbance erm. So we ransform he equaon as we dd n problem 4. a o ge a covarance-varance marx ha has zero ousde he dagonal. Σ K ' E( vv X M O M Σ Σ L he ransformed model can be wren: y% k% % x x % % x x Q% ( λ ϕ w OLS on wll gve he MVLUE esmaors.

13 Problem 4.3 Measuremen error models Problem 4.3 A ( ( y k ξ u x ξ η (3 ξ, u, η are ndependen for all, u η IID o σ (, IID o σ (, η,..., ;,...,, ˆ ( ( ( ( ( ( x x y y x ˆ x y y C x x x x (a plm ( x x( y y x x y y lm ˆ p plm ( ( ( x x Slusky plm ( x x plm ( x (( ( ( x xx η η u u plm ( x x plm (( x x( xx ( x ( ( ( x η η x x u u plm ( x x plm ( x x( x x plm (( x x ( plm ( x x ( u η η u plm ( x x p lm (( ξ η ξ η ( η η plm ( x x plm ( ξ ξ ( η η plm ( η η( η η plm ( x x p lm ( ξ (, ξ η η ξ ξ η η snce plm ( η η( η η plm ( η ηη η ( E( η E( ηη E( η, by assumpon E( η E( ηη E( η

14 b plm ( ξ ( lm ( ( ξ η η p lm ˆ η η η η ση p * plm ( ξ ξ plm ( η η plm ( ξ ξ ( η η b σ η * b plm ( ξ ξ snce: plm η ( ( lm ( η η η p η ηη η E( η E( η E( η E( η E( η ση,(snce s fne and by assumpon E( η c plm ( x x( y y x x y y lm ˆ p plm C ( ( ( x x Slusky plm ( x x plm ( x x( x x plm (( x x ( plm ( x x ( u η η p lm ( x x plm ( ξ ( plm ( ( ξ η η η η η η plm ( x x u d plm ( ξ ( lm ( ( ξ η η p lm ˆ η η η η ση p C * plm ( ξ ξ plm ( η η plm ( ξ ξ ( η η c σ η * c plm ( ξ ξ

15 Problem 4.3 ( ( (3 ξ, u, η, φ, ψ, α, γ are ndependen for all, (4 y k ξ α γ u x ξ φ ψ η, y k x α γ u φ ψ η u η IID o (, σ IID o σ (, η α φ IID o (, σα IID o σ (, φ ψ γ,..., ;,...,, IID o (, σψ IID o σ (, γ a plm ( x x( y y x x y y lm ˆ p plm ( ( ( x x Slusky plm ( x x plm ( x (( ( ( ( x x x α α u φ φ η η plm ( x x plm x x plm ( x x plm ( x ( lm ( ( x x x p x x η, plm ( x x plm ( x x u plm, ( ( x lm x φ p x x, α, plm ( x x plm (( ξ ξ η η φ φ ( η η plm (( ξ ξ η η φ φ φ plm ( x x p lm ( η η( ηη lm ( ( p φ φ φ φ plm ( x x So: p lm ˆ, (( ( x x ( φ φ ( x x ( α α( x x ( x x ( η η ( x x u

16 plm ( x x( y y x x y y lm ˆ p plm C ( ( ( x x Slusky plm ( x x plm ( x x( x x plm (( x x ( plm ( x x ( u η η p lm ( x x plm ( x ( lm (( ( x γ γ p x x ψ ψ plm ( x x plm (( η η ( η η plm ( ( ψ ψ( ψ ψ plm ( x x so : p lm, ˆC b s fne: u plm ( φ φ ( φ φ plm ( ( lm ˆ η η η η σ σ p * plm ( ξ ξ plm ( φ φ plm ( η η b * b plm ( ξ ξ p lm ˆ C ( η φ ( ση σφ c s Fne: p lm ˆ

17 plm ( x x( y y x x y y lm ˆ p plm C ( ( ( x x Slusky plm ( x x plm ( x x( x x plm (( x x ( η η plm ( x x ( u u p lm ( x x plm ( x ( lm (( ( x γ γ p x x ψ ψ plm ( x x plm (( η η ( η η plm ( ( ψ ψ( ψ ψ ( ση σψ * plm ( ξ ξ plm ψ ( p lm ( c ( ψ η ψ η σ σ η * c plm ( ξ ξ