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.

Many quantities of interest in economics and finance can be represented as partially observed functional data. Examples include structural business cycle estimation, implied volatility smile, the yield curve. Having embedded these quantities into continuous random curves, estimation of the covariance function is needed to extract factors, perform dimensionality reduction, and conduct inference on the factor scores. A series expansion for the covariance function is considered. Under summability restrictions on the absolute values of the coefficients in the series expansion, an estimation procedure that is resilient to overfitting is proposed. Under certain conditions, the rate of consistency for the resulting estimator achieves the minimax rate, allowing the observations to be weakly dependent. When the domain of the functional data is K(>1) dimensional, the absolute summability restriction of the coefficients avoids the so called curse of dimensionality. As an application, a Box–Pierce statistic to test independence of partially observed functional data is derived. Simulation results and an empirical investigation of the efficiency of the Eurodollar futures contracts on the Chicago Mercantile Exchange are included.

We prove the long standing conjecture of Ding and Granger (1996) about the existence of a stationary Long Memory ARCH model with finite fourth moment. This result follows from the necessary and sufficient conditions for the existence of covariance stationary integrated AR(∞), ARCH(∞), and FIGARCH models obtained in the present article. We also prove that such processes always have long memory.

The paper considers testing parametric assumptions on the conditional mean and variance functions for nonlinear autoregressive models. To this end, we compare the kernel density estimate of the marginal density of the process with a convolution-type density estimate. It is shown that, interestingly, the latter estimate has a parametric $\left( {\sqrt n } \right)$ rate of convergence, thus substantially improving the classical kernel density estimates whose rates of convergence are much inferior. Our results are confirmed by a simulation study for threshold autoregressive processes and autoregressive conditional heteroskedastic processes.

Here, we derive optimal rank-based tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one direction causality hypotheses are tested. Using the characterization of noncausality in the VAR context, the local asymptotic normality (LAN) theory described in Le Cam (1986, Asymptotic Methods in Statistical Decision Theory) allows for constructing locally and asymptotically optimal tests for the null hypothesis of noncausality in one or both directions. These tests are based on multivariate residual ranks and signs (Hallin and Paindaveine, 2004a, Annals of Statistics 32, 2642–2678) and are shown to be asymptotically distribution free under elliptically symmetric innovation densities and invariant with respect to some affine transformations. Local powers and asymptotic relative efficiencies are also derived. The level, power, and robustness (to outliers) of the resulting tests are studied by simulation and are compared to those of the Wald test. Finally, the new tests are applied to Canadian money and income data.

This paper reviews the contributions of Rex Bergstrom to the development of continuous time dynamic disequilibrium macroeconomic modeling since the early 1960s. The models provide an elegant integration of economic theory with analysis of steady state and stability properties. The subsequent contributions of his Ph.D. students, spawned by Bergstrom’s work over the years, is also reviewed. It was Bergstrom’s early pioneering vision 40 years ago of formulating and estimating continuous time models that underlies much of the research in that area of econometrics and finance today.

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.

We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH(1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparametric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH(1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with nonparametric long-run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

Let g(λ) be the spectral density of a stationary process and let fθ(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator of θ is defined that minimizes a distance between , where is a nonparametric spectral density estimator based on n observations. It is known that is asymptotically Gaussian efficient if g(λ) = fθ(λ). Because there are infinitely many candidates for the distance function , this paper discusses higher order asymptotic theory for in relation to the choice of D. First, the second-order Edgeworth expansion for is derived. Then it is shown that the bias-adjusted version of is not second-order asymptotically efficient in general. This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically second-order asymptotically efficient after a suitable bias adjustment (e.g., Ghosh, 1994, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semiparametric estimation it does not hold in general that “first-order efficiency implies second-order efficiency.” The paper develops verifiable conditions on D that imply second-order efficiency.This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor. The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly improved the paper.

Elliott, Rothenberg, and Stock (1996, Econometrica 64, 813–836) derive a class of point-optimal unit root tests in a time series model with Gaussian errors. Other authors have proposed “robust” tests that are not optimal for any model but perform well when the error distribution has thick tails. I derive a class of point-optimal tests for models with non-Gaussian errors. When the true error distribution is known and has thick tails, the point-optimal tests are generally more powerful than the tests of Elliott et al. (1996) and also than the robust tests. However, when the true error distribution is unknown and asymmetric, the point-optimal tests can behave very badly. Thus there is a trade-off between robustness to unknown error distributions and optimality with respect to the trend coefficients.This paper could not have been written without the encouragement of Thomas Rothenberg. This is based on my dissertation, which he supervised. I also thank Don Andrews, Jack Porter, Jim Stock, and seminar participants at the University of Pennsylvania, the University of Toronto, the University of Montreal, Princeton University, and the meetings of the Econometric Society in UCLA. Comments of three anonymous referees greatly improved the exposition of the paper. I owe special thanks to Gary Chamberlain for helping me to understand these results.

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.

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.

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.

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.

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 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.

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.