“Two-step M-estimators”的意思、由来-开放百科全书

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,^[1] weighted non-linear least squares, and ordinary least squares with generated regressors.^[2]

To fix idea, let

be an i.i.d. sample.

and

are subsets of Euclidean spaces

and

, respectively. Given a function

, two-step M-estimator

is defined as:

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 identification condition.^[2] If

where

is a non-random vector, then the identification condition is that

has a unique maximizer over

Under regularity conditions, two-step M-estimators have asymptotic normality. An important point to note is that 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.^[3] This fact is intuitive because

is a random object and its variability should influence the estimation of

. 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:

where

is the true value of

and

is the probability limit of

.^[3] To interpret this condition, first note that under regularity conditions,

since

is the maximizer of

. So the condition above implies that small perturbation in γ has no impact on the First-Order condition. Thus, in large sample, variability of

does not affect the argmax of the objective function, which explains invariant property of asymptotic variance. Of course, this result is valid only as the sample size tends to infinity, so the finite-sample property could be quite different.

References

1. ^Heckman, J.J., The Common Structure of Statistical Models of Truncation, Sample Selection, and Limited Dependent Variables and a Simple Estimator for Such Models, Annals of Economic and Social Measurement, 5,475-492.
2. ^¹Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
3. ^¹Newey, K.W. and D. McFadden, Large Sample Estimation and Hypothesis Testing, in R. Engel and D. McFadden, eds., Handbook of Econometrics, Vol.4, Amsterdam: North-Holland.