For the Tobit model with independent observations, Amemiya [1] has established the strong consistency and asymptotic normality of a stationary point, 9, of the log-likelihood. The likelihood for dependent observations may be computationally intractable, so the behavior of 9 in the presence of serially correlated observations is of interest. Under a relaxation of Amemiya's assumption of independence, we prove that 9 is strongly consistent and asymptotically normal, and give an expression for the limiting covariance matrix.

In a number of semiparametric models, smoothing seems necessary in order to obtain estimates of the parametric component which are asymptotically normal and converge at parametric rate. However, smoothing can inflate the error in the normal approximation, so that refined approximations are of interest, especially in sample sizes that are not enormous. We show that a bootstrap distribution achieves a valid Edgeworth correction in the case of density-weighted averaged derivative estimates of semiparametric index models. Approaches to bias reduction are discussed. We also develop a higher-order expansion to show that the bootstrap achieves a further reduction in size distortion in the case of two-sided testing. The finite-sample performance of the methods is investigated by means of Monte Carlo simulations from a Tobit model.