We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient $\sigma $ is continuously differentiable in y, but with a jump discontinuity at some level y, say $y=0$. We first study sign-constrained kernel estimators of functions of the left and right limits of $\sigma $ at $0$. These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.

This paper is concerned with the practical problem of conducting inference in a vector time series setting when the data are unbalanced or incomplete. In this case, one can work with only the common sample, to which a standard HAC/bootstrap theory applies, but at the expense of throwing away data and perhaps losing efficiency. An alternative is to use some sort of imputation method, but this requires additional modeling assumptions, which we would rather avoid. We show how the sampling theory changes and how to modify the resampling algorithms to accommodate the problem of missing data. We also discuss efficiency and power. Unbalanced data of the type we consider are quite common in financial panel data; see, for example, Connor and Korajczyk (1993, Journal of Finance 48, 1263–1291). These data also occur in cross-country studies.I thank Greg Connor, Esfandiar Maasoumi, Peter Phillips, Peter Robinson, Yoon Whang, and two referees for helpful comments.

This problem discusses an I(2) model in the VAR(1) case. The I(2) representation theorem of Johansen (1992) (JRT) holds also for VAR(1) processes. The I(2) model for VAR(k) processes has been discussed for k ≥ 2 in Johansen (1996, Ch. 9; 1997). We here discuss a parametrization for the I(2) case of VAR(1) that differs from the VAR(k) model.

We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

In this paper, we first transform a set of moment conditions into a set of transformed moment conditions, based on which the efficient partial generalized method of moments estimation for part of a parameter vector is defined. Given the set of transformed moment conditions, we then show that the conditions for partial redundancy of an additional set of moment conditions given an original set of moment conditions simply become the conditions for full redundancy of the second subset of transformed moment conditions given the first subset of transformed moment conditions. Thus the transformed moment conditions proposed in this paper unify partial redundancy of moment conditions with full redundancy of moment conditions. Using transformed moment conditions, we then straightforwardly derive necessary and sufficient conditions for partial redundancy of one or two subset(s) of moment conditions given the other when the large set of moment conditions consists of three subsets of moment conditions. The paper also provides several easily checkable sufficient conditions for partial redundancy of one set of moment conditions given other sets of moment conditions.

We develop theoretical finite-sample results concerning the size of wild bootstrap-based heteroskedasticity robust tests in linear regression models. In particular, these results provide an efficient diagnostic check, which can be used to weed out tests that are unreliable for a given testing problem in the sense that they overreject substantially. This allows us to assess the reliability of a large variety of wild bootstrap-based tests in an extensive numerical study.

We describe a (nonparametric) prediction algorithm for spatial data, based on a canonical factorization of the spectral density function. We provide theoretical results showing that the predictor has desirable asymptotic properties. Finite sample performance is assessed in a Monte Carlo study that also compares our algorithm to a rival nonparametric method based on the infinite $AR$ representation of the dynamics of the data. Finally, we apply our methodology to predict house prices in Los Angeles.

This study considers the estimation of spatial autoregressive models with censored dependent variables, where the spatial autocorrelation exists within the uncensored latent dependent variables. The estimator proposed in this paper is semiparametric, in the sense that the error distribution is not parametrically specified and can be heteroskedastic. Under a median restriction, we show that the proposed estimator is consistent and asymptotically normally distributed. As an empirical illustration, we investigate the determinants of the risk of assault and other violent crimes including injury in the Tokyo metropolitan area.

Correcting for heteroskedasticity of unspecified form.

In our paper published in vol. 22, no. 1, of Econometric Theory (2006, pp. 69–97) the proof of (22) in Lemma 3.1 is incorrect as given. The problem in the proof occurs after the first display on p. 93 where we attempted to deduce (22) from the already established relation (21). The argument given there is incorrect, because an existence-quantor was mistaken for an all-quantor. We show how the problem can be repaired. In the course of this, we also note a useful variant of another result in Lemma 3.1.

The purpose of this paper is to provide a systematic treatment of the problem of estimation in systems of linear stochastic equations where some of the disturbances are uncorrelated and where the identifying equations are equivalent to a linear system.

We derive a closed-form expression for the asymptotic distribution of the LIML estimator for the coefficients of both endogenous and exogenous variables in a partially identified linear structural equation. We extend previous results of Phillips (1989) and Choi and Phillips (1992), where the focus was on IV estimators. We show that partial failure of identification affects the LIML in that its moments do not exist even asymptotically.

Professor Davidson's Econometric Theory offers a clear, well written, graduate level econometrics textbook, which would be highly appropriate as the principal text for the second and/or third semester of a general Ph.D. econometrics sequence or for a specialized course on time series. It provides good reading material for anyone interested in deepening his or her understanding of econometrics and also a valuable reference source for econometricians. Although the book reviews the basics of multivariate regression, it is written at a level that requires either some degree of mathematical sophistication or prior familiarity with econometrics. As a result, it may be too advanced for a first semester Ph.D. course, except where students have strong preparation.

Trygve Haavelmo and the author were colleagues at the Cowles Commission for Research in Economics during the academic year 1945–46. The econometric analysis of simultaneous equation models and its uses in economic analysis were explored.