Lecture 10 Estimating Nonlinear Regression Models

Transcription

1 Lecure 0 Esimaing Nonlinear Regression Models References: Greene, Economeric Analysis, Chaper 0

2 Consider he following regression model: y = f(x, β) + ε =,, x is kx for each, β is an rxconsan vecor, ε is an unobservable error process and f is a ( sufficienly well-behaved ) funcion - f: R k xr r R. So, each y is a (fixed) funcion of x and β plus an addiive error erm, ε. Example: y β x β 2 = + ε he esimaion problem: given f, y,,y, and x,,x, esimae β. he soluion: esimae β by LS (NLS), ML, or GMM. he problem? In conras o he linear regression case, he FOCs are nonlinear and so, in general, numerical mehods mus be applied o obain (consisen) poin esimaes. Also, he avar marice of β-ha will have a slighly more complicaed form. Nonlinear models are commonly encounered in applied economics largely because advances in compuaional mahemaics and deskop/lapop compuer echnology have made solving nonlinear opimizaion problems more feasible and more reliable.

3 Nonlinear Leas Squares (NLS) Choose β-ha o minimize he SSR FOCs SSR( ˆ) β = ( y f (, ˆ)) β = g ( ˆ) β = ( y x 2 f ( x, ˆ)) β f ( x = β, ˆ) β = 0 which form a se of r nonlinear equaions in he r unknowns, ˆ,..., β ˆ β r. [In he case where f is linear in he β s, he derivaive vecor df/dβ = [ x x r ], r = k.] Example: y = 2 β β + ε x g ( ˆ) β = = ( y ˆ β x ˆ β ˆ ˆ 2 β2 )[ ˆ ˆ β2 x β β x ]' = 2 0

4 Compuing he NLS Esimaor In general, hese FOCs mus be solved numerically o find he NLS esimaor of β, βˆnls. (E.g., he Gauss- Newon procedure described in Greene, ) Some issues - choice of algorihm - selecing an iniial value for β-ha - convergence crieria - local vs. global min

5 Asympoic Properies of he NLS Esimaor If he x s are weakly exogenous he errors are serially uncorrelaed and homoskedasic he funcion f is sufficienly smooh he {x,ε } process is sufficienly well-behaved hen ( ˆ β 2 β ) N(0, σ Q / 2, NLS D where σ 2 = var(ε ) Q = plim(/ ) Q = [ f ( x = Q, ˆ β ) / β ][ f ( x, ), ˆ β ) / β '] he NLS esimaor is (under appropriae condiions), consisen, asympoically normal and asympoically efficien. Inference: For large samples ac as hough ˆ 2 ~ (, ˆ NLS N β σ Q β 2 2 σˆ = (/ ) ˆ ε )

6 If he disurbances are heeroskedasic and/or serially correlaed he NLS esimaor will be consisen bu no asympoically efficien. Also, he correc form of he asympoic variance marix of he NLS esimaor requires a heeroskedasiciy and/or auocorrelaion correcion. Heeroskedasiciy and HAC esimaors of he variance-covariance marix of ε can be used if he exac forms of he heeroskedasiciy and auorcorrelaion are no know. If he form of he heeroskedasiciy and/or serial correcion is known up o a small number of parameers (e.g., ε is known o be an AR() process wih unknown ρ) hen nonlinear GLS or (quasi)- maximum likelihood will be asympoically efficien esimaors. Example GNLS Suppose E(εε ) = Σ. hen he GNLS esimaor of β is he value of βˆ ha minimizes he weighed SSR: [y-f(x, βˆ )] Σ - [y-f(x, βˆ )] If Σ hen i can be replaced wih a consisen esimaor o obain he FGNLS esimaor. (Wha consisen esimaor of Σ?)

7 If he regressors are correlaed wih he errors, none of hese esimaors is consisen (even if he errors are homoskedasic and serially uncorrelaed). A consisen, semi-paramerically efficien esimaor ha does no rely on knowledge of he form/exisence of heeroskedasiciy/auocorrelaion and allows for endogenous regressors: Nonlinear GMM In addiion, GMM provides a semi-parameric alernaive o MLE for nonlinear models ha do no fi he nonlinear regression forma.

8 GMM in he nonlinear regression model Consider he populaion momen condiions: E[w (y f(x,β))] = 0 for all where w is an insrumen vecor. he GMM esimaor: choose βˆ corresponding sample momens o make he w ( y f ( x ˆ, β )) close o zero. As in he linear case, his will involve minimizing an opimally weighed quadraic form in hese momens.

9 GMM in a more general nonlinear seing - Hansen and Singleon s (Economerica, 982) Consumpion-Based Asse Pricing Model A he sar of each ime period, a represenaive agen chooses consumpion and saving o maximize expeced discouned uiliy: i c E[ δ U ( c+ i ) Ω ], ( c ) = γ i= 0 γ U, 0 < γ < A he sar of, he agen can allocae income o purchase he consumpion good or N asses wih mauriies,2,,n according o he sequence of budge consrains c N N + = p, q, = = r, q +, where p, = price of a uni of asse (i.e., an asse ha maures in +) in period q, = unis of asse purchased in period r, = payoff in period of asse purchased in - w = labor income in period Unknown parameers in his model- δ,γ w

10 he opimal consumpion pah mus saisfy he sequence of Euler equaions: E[ δ ( r γ, + / p, )( c+ / c ) Ω ] = 0 Le z be any vecor in Ω. hen he Euler equaions imply he following se of momen condiions which form he basis for esimaing δ and γ by GMM E[ δ ( r γ, + / p, )( c + / c ) z ] = 0 for all and =,...N GMM: Choose δ and γ o make he sample momens δ ( r, + / p, )( c + / c ) γ z, =,,N close o zero. he alernaive MLE: specify he oin disribuion for {(r,+ /p,+,c + /c ), =,2,,N} hen maximize he corresponding likelihood funcion subec o he Euler equaions. (See, e.g., Hansen and Singleon, JPE, 983).