A Maximum Likelihood Method for the Incidental Parameter Problem

No 13787, NBER Working Papers from National Bureau of Economic Research, Inc

Abstract: This paper uses the invariance principle to solve the incidental parameter problem. We seek group actions that preserve the structural parameter and yield a maximal invariant in the parameter space with fixed dimension. M-estimation from the likelihood of the maximal invariant statistic yields the maximum invariant likelihood estimator (MILE). We apply our method to (i) a stationary autoregressive model with fixed effects; (ii) an agent-specific monotonic transformation model; (iii) an instrumental variable (IV) model; and (iv) a dynamic panel data model with fixed effects. In the first two examples, there exist group actions that completely discard the incidental parameters. In a stationary autoregressive model with fixed effects, MILE coincides with existing conditional and integrated likelihood methods. The invariance principle also gives a new perspective to the marginal likelihood approach. In an agent-specific monotonic transformation model, our approach yields an estimator that is consistent and asymptotically normal when errors are Gaussian. In an instrumental variable (IV) model, this paper unifies asymptotic results under strong instruments (SIV) and many weak instruments (MWIV) frameworks. We obtain consistency, asymptotic normality, and optimality results for the limited information maximum likelihood estimator directly from the invariant likelihood. Our approach is parallel to M-estimation in problems in which the number of parameters does not change with the sample size. In a dynamic panel data model with N individuals and T time periods, MILE is consistent as long as NT goes to infinity. We obtain a large N, fixed T bound; this bound coincides with Hahn and Kuersteiner's (2002) bound when T goes to infinity. MILE reaches (i) our bound when N is large and T is fixed; and (ii) Hahn and Kuersteiner's (2002) bound when both N and T are large.

JEL-codes: C13 C23 C30 (search for similar items in EconPapers)
Date: 2008-02
New Economics Papers: this item is included in nep-ecm
Note: TWP
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Citations: View citations in EconPapers (1)

Published as Aizer, Anna “Neighborhood Violence and Urban Youth” chapter in Disadvantaged Youth , Jonathon Gruber, ed. (April 2007 ) also NBER Working Paper #13773
Published as Marcelo J. Moreira, 2009. "A maximum likelihood method for the incidental parameter problem," The Annals of Statistics, vol 37(6A), pages 3660-3696.

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