Lecure 8 :IV, Nonlinear Models Le Z, be an rx funcion of a kx paraeer vecor, r > k, and a rando vecor Z, such ha he r populaion oen condiions also called esiain equaions EZ, hold for all, where is he rue value of. For exaple, in he linear case we considered earlier, Z [y x w ] and Z, w *y -x, where r is equal o he diensionaliy of w. Anoher exaple nonlinear reression odel y fx, + ε Ew ε In his case, Z [y x w ] and Z, w *y -fx,
Ye anoher exaple Hansen and Sinleon Econoerica, 982 In he consupion-based asse pricin odel developed by Lucas Econoerica, 978, he represenaive household chooses a consupion plan in period o axiize expeced discouned uiliy subec o a sequence of bude consrains. In addiion o purchasin a consupion ood he nueraire ood, he aen can purchase N asses which differ accordin o he nuber of periods unil auriy,2,,n. hese providin he fuure incoe ha suppor fuure consupion. Wih a consan relaive risk aversion uiliy funcion, his opiizaion proble urns ou o produce he followin Euler equaions for he aen s opial consupion pah E[ δ r / p c / c γ, +, + I ] for,,n, where c is period consupion, r +, is he period + payoff fro a uni of asse purchased in period, and p, is he period price of a uni of asse. δ and γ are he paraeers of he odel δ he discoun rae, γ he paraeer ha
specifies he paricular CRRA uiliy funcion. I denoes he aen s inforaion se in period. We are ineresed in esiain he paraeers δ and γ. We can rewrie he Euler equaions as E[ δ r / p c / c I ] γ, +, + and, applyin he law of ieraed expecions, E[ δ r / p c / c w ] γ, +, + where w is an n-diensional vecor conained in I. r / p,, c / c E..,,, or any oher acroeconoic and/or financial variables ha ih be in I.
I follows ha he followin n oen condiions us hold for all : E[ w δ r, + / p, c + / c γ ] which is of he for EZ, Wha is he alernaive o o esiae his odel? MLE. Bu MLE requires specifyin he sequence of oin condiional disribuions for r / p, c / c,, + + hen axiizin he lo likelihood funcion subec o he sequence of Euler equaions. Excep in special cases e.., lonoral disribuion, his is a very essy proble copuaionally. [However, subec o he correc specificaion of he relevan disribuions, he MLE will be ore efficien asypoically han. is seiparaerically efficien bu he MLE is a fully paraeric esiaor and is ore efficien han.] he CAPM exaple and oher econoic exaples which ive rise o oen condiions of his for are provided in Chaper of Hall s exbook
he saple analoues of he r populaion oens EZ, are he r saple oens Z, he efficien esiaor of is: ar in Q where Q is he weihin arix Q ] ~ ~ / [ where ~ is any consisen esiaor of.
Noes. his esiaor will enerally have o be calculaed nuerically. See Ch. 3 of Hall s book for ore discussion of his. 2. o obain a preliinary consisen esiaor of we can apply inefficien. ha is, apply wih any p.d. Q, e.., Q I. An ieraive version of his esiaor ay work beer in applicaions alhouh here will be no asypoic ain o ierain.
Under suiable reulariy condiions, Ω G G N D where E and E G Ω Consisen esiaors of G and Ω: Ω and G
Overidenificaion es Assue r > k Under H : E, J where Χ D 2 r k J J Q Noe ha and Q are evaluaed here a.
esin Linear and/or Nonlinear Resricions on Consider a se of q linear and/or nonlinear resricions on. Le J J saisic fro he unresriced reression J,R J saisic fro he resriced s.. he sae Q arix used in forin J is also used in forin J,R. hen under he null hypohesis ha he q resricions are correc J 2, R J Χ q D