In this paper, we propose a new noncausal vector autoregressive (VAR) model for non-Gaussian time series. The assumption of non-Gaussianity is needed for reasons of identifiability. Assuming that the error distribution belongs to a fairly general class of elliptical distributions, we develop an asymptotic theory of maximum likelihood estimation and statistical inference. We argue that allowing for noncausality is of particular importance in economic applications that currently use only conventional causal VAR models. Indeed, if noncausality is incorrectly ignored, the use of a causal VAR model may yield suboptimal forecasts and misleading economic interpretations. Therefore, we propose a procedure for discriminating between causality and noncausality. The methods are illustrated with an application to interest rate data.

This paper studies the asymptotic theory of least squares estimation in a threshold moving average model. Under some mild conditions, it is shown that the estimator of the threshold is n-consistent and its limiting distribution is related to a two-sided compound Poisson process, whereas the estimators of other coefficients are strongly consistent and asymptotically normal. This paper also provides a resampling method to tabulate the limiting distribution of the estimated threshold in practice, which is the first successful effort in this direction. This resampling method contributes to threshold literature. Simultaneously, simulation studies are carried out to assess the performance of least squares estimation in finite samples.

We study the problem of identifying a forecaster’s loss function from observations on forecasts, realizations, and the forecaster’s information set. Essentially different loss functions can lead to the same forecasts in all situations, though within the class of all continuous loss functions, this is strongly nongeneric. With the small set of exceptional cases ruled out, generic nonparametric preference recovery is theoretically possible, but identification depends critically on the amount of variation in the conditional distributions of the process being forecast. There exist processes with sufficient variability to guarantee identification, and much of this variation is also necessary for a process to have universal identifying power. We also briefly address the case in which the econometrician does not fully observe the conditional distributions used by the forecaster, and in this context we provide a practically useful set identification result for loss functions used in forecasting binary variables.

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest multivariate GARCH model, namely, the multivariate ARCH(1) model of the Baba, Engle, Kraft, and Kroner form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the log-likelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this nonstandard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sufficient for the stationarity of the score when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

We analyze fast procedures for conducting Monte Carlo experiments involving bootstrap estimators, providing formal results establishing the properties of these methods under general conditions.

In this paper we extend fixed-b asymptotic theory to the nonparametric Phillips–Perron (PP) unit root tests. We show that the fixed-b limits depend on nuisance parameters in a complicated way. These nonpivotal limits provide an alternative theoretical explanation for the well-known finite-sample problems of the PP tests. We also show that the fixed-b limits depend on whether deterministic trends are removed using one-step or two-step detrending approaches. This is in contrast to the asymptotic equivalence of the one- and two-step approaches under a consistency approximation for the long-run variance estimator. Based on these results we introduce modified PP tests that allow for asymptotically pivotal fixed-b inference. The theoretical analysis is cast in the framework of near-integrated processes, which allows us to study the asymptotic behavior both under the unit root null hypothesis and for local alternatives. The performance of the original and modified PP tests is compared by means of local asymptotic power and a small finite-sample simulation study.

We propose a consistent kernel-based specification test for conditional density models when the dependent variable is categorical/discrete. The method is applicable to popular parametric binary choice models such as the logit and probit specification and their multinomial and ordered counterparts, along with parametric count models, among others. The test is valid when the conditional density function contains both categorical and real-valued covariates. Theoretical support for the test and for a bootstrap-based version of the test is provided. Monte Carlo simulations are conducted to assess the finite-sample performance of the proposed method.

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

Cai, Li, and Park (Journal of Econometrics, 2009) and Xiao (Journal of Econometrics, 2009) developed asymptotic theories for estimators of semiparametric varying coefficient models when regressors are integrated processes but the smooth coefficients are functionals of stationary processes. Using a recent result from Phillips (Econometric Theory, 2009), we extend this line of research by allowing for both the regressors and the covariates entering the smooth functionals to be integrated variables. We derive the asymptotic distribution for the proposed semiparametric estimator. An empirical application is presented to examine the purchasing power parity hypothesis between U.S. and Canadian dollars.