ABSTRACT
Suppose we have proved, for each fixed p, local asymptotic normality of the one-dimensional model with transition distribution p(y-ax) dy. Suppose we have constructed an estimator not depending on p which is efficient in all of these one-dimensional models. Then the parameter a is adaptive, and the estimator is efficient in the model with both a and p unknown. It is not necessary to prove local asymptotic normality of this model. Kreiss (l987a) uses this approach for ARMA-models, and Gassiat (1993) for noncausal AR-models. 0
It can be treated in a similar way as the AR(l)-model above. A review of ARCH-models is Bollerslev et al. ( 1992). Efficient estimators in this model and generalized ARCH-models are constructed in Engle and Gonzalez-Rivera (1991), Linton (1993) and Drost and Klaassen (1997) under increasingly weaker assumptions. 0
Xi= 1110 (Xi-l) + v,(Xi-1) 1/2ci, where the ci are iid with a density p which has mean 0 and variance 1 as in Example 7. Efficient estimation in nonlinear autoregressive models is studied by Hwang and Basawa (1993), Drost et al. (1994), (1997), Jeganathan (1995) and Koul and Schick ( 1997). D
8. MODELING CONDITIONAL MOMENTS Let X0, .•• , X 11 be observations from a real-valued ergodic Markov chain with transition distribution Q(x, dy) and invariant distribution 1r(dx). Suppose we have a parametric model for the conditional mean, or autoregression function, but that the transition distribution is unspecified otherwise. A simple such specification is
J Q(x, dy)y = ax. (8.1) (The linear autoregressive model of Sec. 7 is a submodel, with a transition distribution of the form Q(x, dy) = p(y-ax) dy, where pis a mean 0 density.)
Such a model suggests a special class of martingale estimating equations. Note that the model can be described by saying that the innovations ci = Xi -aXi-l are martingale increments. In particular, "stochastic integrals" of the form
(8.2)
Len(Xi-I>Xi) = 0 as in Sec. 4, now with t(a) =a and ea(x,y) = w0 (x)(y-ax). As before, we write T11 for the corresponding estimator. The choice 11'0 (x) = x leads to the least squares estimator, Til = L xi-1 Xd xf_l. In general, to solve the estimating equation, we expand it as in Eq. (4.2),
(E Xll'(\ (X)) - 27r( VH'~J, (8.4) where v denotes the conditional variance
i.e.,
Itfollowsthatthetangentspaceistheunionoftheaffinespaces
Ha={hEH:JQ(x,dy)h(x,y)y=ax},aER. ByEq.(3.4),thecanonicalgradienth'ofa,viewedasafunctiontofQ,isin oneoftheseaffinespacesandsolves
n 112(t(Q 11,)- t(Q))=n112 (a11h-a)=a :b(h,h')=1r0Q(hh')foraER,hEHa-(8.5)
Ourcandidateistheinfluencefunctionoftheoptimalweightedleastsquares estimator,
g(x,y)=(Ev(X)- 1X 2)- 1v(x)- 1x(y-ax). ThisfunctionisclearlyinH,andsince
JQ(x,dy)v(xf 1x(y-ax)y=v(x)- 1xIQ(x,dy)(y-ax) 2=x, wealsohavegEHafora=(Ev(X)- 1X2f 1.Itremainstocheckwhether h'=gsolvesEq.(8.5).But
1r0Q(hg)=(Ev(X)- 1X 2)- 1I1r(dx)v(xf 1xIQ(x,dy)h(x,y)(y-ax)=a. Hencetheoptimalweightedleastsquaresestimatorisefficient.
