M-Estimation The Model –min E(q(z,  –∑ i=1,…,n q(z i,  )/n → p E(q(z,  –b = argmin ∑ i=1,…,n q(z i,  ) (or divided by n) –Zero-Score: ∑ i=1,…,n  q(z i,b  ′ = 0 –Positive Definite Hessian: ∑ i=1,…,n   q(z i,b  ′ 

NLS, ML as M-Estimation Nonlinear Least Squares –y = f(x,  ) +  z =[y,x] –S(β) = ε′ε = ∑ i=1,…,n (y i -f(x i,  )) 2 –q(y i,x i,  ) = (y i -f(x i,  )) 2 –b = argmin S(β) Maximum Log-Likelihood –ll(β) = ∑ i=1,…,n ln pdf(y i,x i,  ) –q(y i,x i,  ) = ln pdf(y i,x i,  ) –b = argmin -ll(β)

The Score Vector s(z,  ) =  q(z,  ′ –E(s(z,  )) = E(  q(z,  ′) = 0 ∑ i=1,…,n s(z i,  )/n → p E(s(z,  )) = 0 –V = Var(s(z,  )) = E(s(z,  )s(z,  )′) = E((  q(z,  ′) (  q(z,  ) ) ∑ i=1,…,n [s(z i,  )s(z i,  )′]/n → P Var(s(z,  )) ∑ i=1,…,n s(z i,  )/n → d N(0,V)

The Hessian Matrix H(z,  ) = E(  s(z,  )/  ) = E(   q(z,  ′  –∑ i=1,…,n  s(z i,  )/  n → p H(z,  ) Information Matrix Equality does not necessary hold in general: - H ≠ V –More general than NLS and ML

The Asymptotic Theory  n(b-  ) = {∑ i=1,…,n  s(z i,b)/  n} -1 [∑ i=1,…,n s(z i,b)/n]  n(b-  ) → d N(0,H -1 VH -1 ) V = E(s(z,  )s(z,  )′) = E[(  q(z,  )′(  q(z,  )] H = E(  s(z,  )/  ) = E(   q(z,  ′  –∑ i=1,…,n [s(z i,b)s(z i,b)′]/n → p V –∑ i=1,…,n  s(z i,b)/  n → p H

Asymptotic Normality b → p  b ~ a N( ,Var(b)) Var(b) = H -1 VH -1 –H and V are evaluated at b: H = ∑ i=1,…,n [  2 q (z i,b)/  ′  V = ∑ i=1,…,n [  q(z i,b)/  ][  q(z i,b)/  ′]

Application Heteroscedastity Autocorrelation Consistent Variance-Covariance Matrix –Non-spherical disturbances in NLS Quasi Maximum Likelihood (QML) –Misspecified density assumption in ML –Information Equality may not hold

Example Nonlinear CES Production Function ln(Q) =  1 +  4 ln[  2 L  3 +(1-  2 )K  3 ] +  –NLS (homoscedasticy) –NLS (robust covariances) –ML (normal density) –QML (misspecified normality)

Example Nonlinear CES Production Function Parameter Estimate M-Min Robust s.e. M-Min OIM s.e. M-Min OPG s.e. NLS s.e.     SSE Note: OIM, OPG, and NLS s.e. are under homo. & no auto.

Example Nonlinear CES Production Function Parameter Estimate M-Max Robust s.e. M-Max OIM s.e. M-Max OPG s.e. ML s.e     Log-Lik Note: OIM, OPG, and NLS s.e. are under homo. & no auto.