Two-step M-estimators deals with M-estimation problems that require preliminary estimation to obtain the parameter of interest. Two-step M-estimation is different from usual M-estimation problem because asymptotic distribution of the second-step estimator generally depends on the first-step estimator. Accounting for this change in asymptotic distribution is important for valid inference.
Description The class of two-step M-estimators includes Heckman's sample selection estimator, weighted non-linear least squares, and ordinary least squares with generated regressors. To fix ideas, let { W i } i = 1 n ⊆ R d {\displaystyle \{W_{i}\}_{i=1}^{n}\subseteq R^{d}} be an i.i.d. sample. Θ {\displaystyle \Theta } and Γ {\displaystyle \Gamma } are subsets of Euclidean spaces R p {\displaystyle R^{p}} and R q {\displaystyle R^{q}} , respectively. Given a function m ( ; ; ; ) : R d × Θ × Γ → R {\displaystyle m(;;;):R^{d}\times \Theta \times \Gamma \rightarrow R} , two-step M-estimator θ ^ {\displaystyle {\hat {\theta }}} is defined as:
θ ^ := arg max θ ∈ Θ 1 n ∑ i m ( W i , θ , γ ^ ) {\displaystyle {\hat {\theta }}:=\arg \max _{\theta \in \Theta }{\frac {1}{n}}\sum _{i}m{\bigl (}W_{i},\theta ,{\hat {\gamma }}{\bigr )}}
where γ ^ {\displaystyle {\hat {\gamma }}} is an M-estimate of a nuisance parameter that needs to be calculated in the first step. Consistency of two-step M-estimators can be verified by checking consistency conditions for usual M-estimators, although some modification might be necessary. In practice, the important condition to check is the identification condition. If γ ^ → γ ∗ , {\displaystyle {\hat {\gamma }}\rightarrow \gamma ^{*},} where γ ∗ {\displaystyle \gamma ^{*}} is a non-random vector, then the identification condition is that E [ m ( W 1 , θ , γ ∗ ) ] {\displaystyle E[m(W_{1},\theta ,\gamma ^{*})]} has a unique maximizer over Θ {\displaystyle \Theta } .
Asymptotic distribution Under regularity conditions, two-step M-estimators have asymptotic normality. An important point to note is that the asymptotic variance of a two-step M-estimator is generally not the same as that of the usual M-estimator in which the first step estimation is not necessary. This fact is intuitive because γ ^ {\displaystyle {\hat {\gamma }}} is a random object and its variability should influence the estimation of Θ {\displaystyle \Theta } . However, there exists a special case in which the asymptotic variance of two-step M-estimator takes the form as if there were no first-step estimation procedure. Such special case occurs if:
E ∂ ∂ θ ∂ γ m ( W 1 , θ 0 , γ ∗ ) = 0 {\displaystyle E{\frac {\partial }{\partial \theta \partial \gamma }}m(W_{1},\theta _{0},\gamma ^{*})=0}
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