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.

We show that it is possible to adapt to nonparametric disturbance autocorrelation in time series regression in the presence of long memory in both regressors and disturbances by using a smoothed nonparametric spectrum estimate in frequency–domain generalized least squares. When the collective memory in regressors and disturbances is sufficiently strong, ordinary least squares is not only asymptotically inefficient but asymptotically non–normal and has a slow rate of convergence, whereas generalized least squares is asymptotically normal and Gauss–Markov efficient with standard convergence rate. Despite the anomalous behavior of nonparametric spectrum estimates near a spectral pole, we are able to justify a standard construction of frequency–domain generalized least squares, earlier considered in case of short memory disturbances. A small Monte Carlo study of finite sample performance is included.

There is frequently interest in testing that a scalar or vector time series is I(0), possibly after first-differencing or other detrending, while the I(0) assumption is also taken for granted in autocorrelation-consistent variance estimation. We propose a test for I(0) against fractional alternatives. The test is nonparametric, and indeed makes no assumptions on spectral behaviour away from zero frequency. It seems likely to have good efficiency against fractional alternatives, relative to other nonparametric tests. The test is given large sample justification, subjected to a Monte Carlo analysis of finite sample behaviour, and applied to various empirical data series.