This paper develops an asymptotic theory of Bayesian inference for time series. A limiting representation of the Bayesian data density is obtained and shown to be of the same general exponential form for a wide class of likelihoods and prior distributions. Continuous time and discrete time cases are studied. In discrete time, an embedding theorem is given which shows how to embed the exponential density in a continuous time process. From the embedding we obtain a large sample approximation to the model of the data that corresponds to the exponential density. This has the form of discrete observations drawn from a nonlinear stochastic differential equation driven by Brownian motion. No assumptions concerning stationarity or rates of convergence are required in the asymptotics. Some implications for statistical testing are explored and we suggest tests that are based on likelihood ratios (or Bayes factors) of the exponential densities for discriminating between models.

This paper studies the asymptotic properties of instrumental variable (IV) estimates of multivariate cointegrating regressions and allows for deterministic and stochastic regressors as well as quite general deterministic processes in the data-generating mechanism. It is found that IV regressions are consistent even when the instruments are stochastically independent of the regressors. This phenomenon, which contrasts with traditional theory for stationary time series, is a beneficial artifact of spurious regression theory whereby stochastic trends in the instruments ensure their relevance asymptotically. Problems of inference are also addressed and some promising new theoretical results are reported. These involve a class of Wald tests which are modified by semiparametric corrections for serial correlation and for endogeneity. The resulting test statistics which we term fully-modified Wald tests have limiting X2 distributions, thereby removing the obstacles to inference in cointegrated systems that were presented by the nuisance parameter dependencies in earlier work. Some simulation results are reported which seek to explore the sampling behaviour of our suggested procedures. These simulations compare our fully modified (semiparametric) methods with the parametric error-correction methodology that has been extensively used in recent empirical research and with conventional least squares regression. Both the fully-modified and errorcorrection methods work well in finite samples and the sampling performance of each procedure confirms the relevance of asymptotic distribution theory, as distinct from super-consistency results, in discriminating between statistical methods.

A limit theory for Wald tests of Granger causality in levels vector autoregressions (VAR's) and error correction models (ECM's) is developed, which allows for stochastic trends and cointegration. Earlier work is extended to the general case, thereby characterizing when these Wald tests are asymptotically valid as 'x'(superscript 2) criteria. Our results for inference from unrestricted levels VAR are not encouraging: the limit theory often involves nuisance parameters and nonstandard distributions, a situation offering no satisfactory statistical basis for these tests. Granger causality tests in ECM's also suffer from nuisance parameter dependencies asymptotically and in some cases nonstandard limit theory. Both these results are somewhat surprising in light of earlier research.

Band spectral regression with both deterministic and stochastic trends is considered. It is shown that trend removal by regression in the time domain prior to band spectral regression can lead to biased and inconsistent estimates in models with frequency dependent coefficients. Both semiparametric and nonparametric regression formulations are considered, the latter including general systems of two-sided distributed lags such as those arising in lead and lag regressions. The bias problem arises through omitted variables and is avoided by careful specification of the regression equation. Trend removal in the frequency domain is shown to be a convenient option in practice. An asymptotic theory is developed and the two cases of stationary data and cointegrated nonstationary data are compared. In the latter case, a levels and differences regression formulation is shown to be useful in estimating the frequency response function at nonzero as well as zero frequencies.

Our subject is estimation and inference concerning long-run economic equilibria in models with stochastic trends. An asymptotic theory is provided to analyze a menu of currently existing estimators of cointegrated systems. We study in detail the single-equation ECM (SEECM) approach of Hendry. Our theoretical results lead to prescriptions for empirical work, such as specifying SEECM's nonlinearly and including lagged equilibrium relationships rather than lagged differences of the dependent variable as covariates. Simulations support these prescriptions, and point to problems of overfitting not encountered in the semiparametric approach of Phillips and Hansen (1990).

A new methodology is proposed to estimate theoretical prices of financial contingent claims whose values are dependent on some other underlying financial assets. In the literature, the preferred choice of estimator is usually maximum likelihood (ML). ML has strong asymptotic justification but is not necessarily the best method in finite samples. This paper proposes a simulation-based method. When it is used in connection with ML, it can improve the finite-sample performance of the ML estimator while maintaining its good asymptotic properties. The method is implemented and evaluated here in the Black-Scholes option pricing model and in the Vasicek bond and bond option pricing model. It is especially favored when the bias in ML is large due to strong persistence in the data or strong nonlinearity in pricing functions. Monte Carlo studies show that the proposed procedures achieve bias reductions over ML estimation in pricing contingent claims when ML is biased. The bias reductions are sometimes accompanied by reductions in variance. Empirical applications to U.S. Treasury bills highlight the differences between the bond prices implied by the simulation-based approach and those delivered by ML. Some consequences for the statistical testing of contingent-claim pricing models are discussed.

Prices of interest rate derivative securities depend crucially on the mean reversion parameters of the underlying diffusions. These parameters are subject to estimation bias when standard methods are used. The estimation bias can be substantial even in very large samples and much more serious than the discretization bias, and it translates into a bias in pricing bond options and other derivative securities that is important in practical work. This article proposes a very general and computationally inexpensive method of bias reduction that is based on Quenouille's (1956; Biometrika, 43, 353–360) jackknife. We show how the method can be applied directly to the options price itself as well as the coefficients in the models. We investigate its performance in a Monte Carlo study. Empirical applications to U.S. dollar swap rates highlight the differences between bond and option prices implied by the jackknife procedure and those implied by the standard approach. These differences are large and suggest that bias reduction in pricing options is important in practical applications.

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill-posed inverse problems. In functional cointegrating regressions where the regressor is an integrated time series, it is shown here that inverse and ill-posed inverse problems do not arise. Remarkably, nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving simple useable asymptotics in practical work. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series. The methods may be applied to a wide range of empirical models where functional estimation of cointegrating relations is required.

This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak or uninformed) instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter, and conditions for consistent GMM estimation are given. A general framework for the GMM limit distribution theory is developed based on epiconvergence methods. Some illustrations are provided, including consistent GMM estimation of a panel model with time varying individual effects, consistent limited information maximum likelihood estimation as a continuously updated GMM estimator, and consistent IV structural estimation using large numbers of weak or irrelevant instruments. Some simulations are reported. Copyright The Econometric Society 2006.

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay (1998)). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate , slower than , when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.