The object of this paper is to report, for a simple testing problem of a unit root hypothesis, some experience regarding the numerical problems involved by using a Bayesian encompassing test, i.e., a Bayesian procedure that treats the null and the alternative hypotheses as different models, the null one and the alternative one, that share a same sample space but with different parameter spaces. Numerical procedures and efficient simulations are discussed briefly, and the numerical results so obtained are used to evaluate the meaning of the prior specification and of the empirical evidence about a unit root inference.

Limit theory with stochastic integrals plays a major role in time series econometrics. In earlier contributions on weak convergence to stochastic integrals, the literature commonly uses martingale and semi-martingale structures. Liang, Phillips, Wang, and Wang (2016) (see also Wang (2015), Chap. 4.5) currently extended weak convergence to stochastic integrals by allowing for a linear process or a α-mixing sequence in innovations. While these martingale, linear process and α-mixing structures have wide relevance, they are not sufficiently general to cover many econometric applications that have endogeneity and nonlinearity. This paper provides new conditions for weak convergence to stochastic integrals. Our frameworks allow for long memory processes, causal processes, and near-epoch dependence in innovations, which have applications in a wide range of econometric areas such as TAR, bilinear, and other nonlinear models.

This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

Deviations of asset prices from the random walk dynamic imply the predictability of asset returns and thus have important implications for portfolio construction and risk management. This paper proposes a real-time monitoring device for such deviations using intraday high-frequency data. The proposed procedures are based on unit root tests with in-fill asymptotics but extended to take the empirical features of high-frequency financial data (particularly jumps) into consideration. We derive the limiting distributions of the tests under both the null hypothesis of a random walk with jumps and the alternative of mean reversion/explosiveness with jumps. The limiting results show that ignoring the presence of jumps could potentially lead to severe size distortions of both the standard left-sided (against mean reversion) and right-sided (against explosiveness) unit root tests. The simulation results reveal satisfactory performance of the proposed tests even with data from a relatively short time span. As an illustration, we apply the procedure to the Nasdaq composite index at the 10-minute frequency over two periods: around the peak of the dot-com bubble and during the 2015–2106 stock market sell-off. We find strong evidence of explosiveness in asset prices in late 1999 and mean reversion in late 2015. We also show that accounting for jumps when testing the random walk hypothesis on intraday data is empirically relevant and that ignoring jumps can lead to different conclusions.

This paper establishes conditions for the nonparametric identifiability of the mixed proportional hazards model with time-varying coefficients. Unlike the mixed proportional hazards model, a regressor with two distinct values is not sufficient to identify this model. An unbounded regressor, however, is sufficient for identification.

Consider a linear regression model y1 = x1β + u1, where the u1'S afe weakly dependent random variables, the x1's are known design nonrandom variables, and β is an unknown parameter. We define an M-estimator βn of) β corresponding to a smooth score function. Then, the second-order Edgeworth expansion for βn is derived. Here we do not assume the normality of (u1), and (u1) includes the usual ARMA processes. Second, we give the second-order Edgeworth expansion for a transformation T(βn) of βn. Then, a sufficient condition for T to extinguish the second-order terms is given. The results are applicable to many statistical problems.

I was very pleased to be invited to the Universidad Carlos III as visiting professor, to meet all my old Spanish friends again, and to have the opportunity to give this public lecture. My address will have two themes: a brief review of the history and current status of the London School of Economics, and then a discussion of the strong tradition which has developed there of the study of mathematical methods in the social sciences, particularly in economics, by building mathematical models of economic behaviour, and then estimating these models statistically. That briefly is how I define econometrics.

This paper develops an econometric framework for (i) estimating excess returns of the security process from high frequency derivative prices, (ii) testing for risk neutral pricing, and (iii) measuring premiums outside the no-arbitrage pricing model. The estimator is constructed by applying quasi-likelihood and Feynman–Kac theory to the risk neutral contingent claims pricing model to generate the optimal orthogonality restriction. The strong consistency and asymptotic normality of the estimator are established in the context of a nonstationary underlying state process. These results further imply that the estimator is robust to distributional assumptions on the underlying asset process. The proposed approach is applicable to any arbitrary derivative security, does not require estimation of the risk neutral probability measure, and has application to spot rate bond pricing models. A controlled diagnostic study based on generating the S&P500 index and calls verifies the ability of the estimators to correctly estimate security excess returns and test for risk neutral pricing. The estimator is invariant to call strikes, and larger samples constructed by cycling over shorter maturity options can be used to reduce its variance.

This paper examines the identification power of assumptions that formalize the notion of complementarity in the context of a nonparametric bounds analysis of treatment response. I extend the literature on partial identification via shape restrictions by exploiting cross-dimensional restrictions on treatment response when treatments are multidimensional; the assumption of supermodularity can strengthen bounds on average treatment effects in studies of policy complementarity. This restriction can be combined with a statistical independence assumption to derive improved bounds on treatment effect distributions, aiding in the evaluation of complex randomized controlled trials. Complementarities arising from treatment effect heterogeneity can be incorporated through supermodular instrumental variables to strengthen identification in studies with one or multiple treatments. An application examining the long-run impact of zoning on the evolution of urban spatial structure illustrates the value of the proposed identification methods.

We develop a new method for generating dynamics of conditional correlation matrices of asset returns. These correlation matrices are parameterized by a subset of their partial correlations, whose structure is described by a set of connected trees called “vine”. Partial correlation processes can be specified separately and arbitrarily, providing a new family of very flexible multivariate GARCH processes, called “vine-GARCH” processes. We estimate such models by quasi-maximum likelihood. We compare our models with DCC and GAS-type specifications through simulated experiments and we evaluate their empirical performances.

We consider censored structural latent variables models where some exogenous variables are subject to additive measurement errors. We demonstrate that overidentification conditions can be exploited to provide natural instruments for the variables measured with errors, and we propose a two-stage estimation procedure. The first stage involves substituting available instruments in lieu of the variables that are measured with errors and estimating the resulting reduced form parameters using consistent censored regression methods. The second stage obtains structural form parameters using the conventional linear simultaneous equations model estimators.

We show that the cumulated sum of squares statistic has a standard Brownian bridge–type asymptotic distribution in nonlinear regression models with (possibly) nonstationary regressors. This contrasts with cumulated sum statistics which have been previously studied and whose asymptotic distribution has been shown to depend on the functional form and the stochastic properties, such as persistence and stationarity, of the regressors. A recursive version of the test is also considered. A local power analysis is provided, and through simulations, we show that the test has good size and power properties across a variety of situations.

Testing for causality is of critical importance for many econometric applications. For bivariate AR(1) processes, the limit distributions of causality tests based on least squares estimation depend on the presence of nonstationary processes. When nonstationary processes are present, the limit distributions of such tests are usually very complicated, and the full-sample bootstrap method becomes inconsistent as pointed out in Choi (2005, Statistics and Probability Letters 75, 39–48). In this paper, a profile empirical likelihood method is proposed to test for causality. The proposed test statistic is robust against the presence of nonstationary processes in the sense that one does not have to determine the existence of nonstationary processes a priori. Simulation studies confirm that the proposed test statistic works well.

This paper discusses some asymptotic uniform linearity results of randomly weighted empirical processes based on long range dependent random variables. These results are subsequently used to linearize nonlinear regression quantiles in a nonlinear regression model with long range dependent errors, where the design variables can be either random or nonrandom. These, in turn, yield the limiting behavior of the nonlinear regression quantiles. As a corollary, we obtain the limiting behavior of the least absolute deviation estimator and the trimmed mean estimator of the parameters of the nonlinear regression model. Some of the limiting properties are in striking contrast with the corresponding properties of a nonlinear regression model under independent and identically distributed error random variables. The paper also discusses an extension of rank score statistic in a nonlinear regression model.

Several estimation procedures such as the efficient method of moments (EMM) of Gallant and Tauchen (1996, Econometric Theory 12, 657–681) and indirect inference procedure of Gouriéroux, Monfort, and Renault (1993, Journal of Applied Econometrics 8, S85–S118) involve two models, an auxiliary one and a model of interest. The role played by both models poses challenges and provides new opportunities for hypothesis testing beyond the usual Wald-, Lagrange multiplier–, and likelihood ratio–type tests. In this paper we present and derive the asymptotic distribution theory for various classes of tests for structural change. Some procedures are extensions of standard tests, whereas others are specific to the dual model setup and exploit its unique features.The first author gratefully acknowledges financial support from Fonds pour la Formation de Chercheurs et l'aide à la Recherche (FCAR). The second author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada through a grant to NCM2 (Network for Computing and Mathematical Modeling). We also thank Alastair Hall and Éric Renault for comments on an earlier draft of the paper.

This paper generalizes the intuition of Hausman and Taylor (1981, Econometrica 49, 1377–1398) and develops a method of dealing with a time-invariant regressor in nonlinear panel models with fixed effects. We illustrate the usefulness of our result by discussing the implication for some nonlinear models of social interactions.We are grateful to Dan Ackerberg and Jerry Hausman for helpful comments. The first author gratefully acknowledges financial support from NSF grant SES-0313651.

Research on model/procedure selection has focused on selecting a single model globally. In many applications, especially for high-dimensional or complex data, however, the relative performance of the candidate procedures typically depends on the location, and the globally best procedure can often be improved when selection of a model is allowed to depend on location. We consider localized model selection methods and derive their theoretical properties.This research was supported by U.S. National Science Foundation CAREER Grant DMS0094323. We thank three referees and the editors for helpful comments on improving the paper.

Let (X1) be a discrete multivariate Gaussian autoregressive process of order 1. The paper derives the exact finite-sample joint moment generating function (m.g.f.) of the three quadratic forms constituting the sufficient statistic of the process. The formula is then specialized to some cases of interest, including the m.g.f. of functional of multivariate Ornstein-Uhlenbeck processes that arise asymptotically from more general (X1) processes as well.