Local to unity limit theory is used in applications to construct confidence intervals (CIs) for autoregressive roots through inversion of a unit root test (Stock (1991)). Such CIs are asymptotically valid when the true model has an autoregressive root that is local to unity (ρ = 1 + c/n), but are shown here to be invalid at the limits of the domain of definition of the localizing coefficient c because of a failure in tightness and the escape of probability mass. Failure at the boundary implies that these CIs have zero asymptotic coverage probability in the stationary case and vicinities of unity that are wider than O(n−1/3). The inversion methods of Hansen (1999) and Mikusheva (2007) are asymptotically valid in such cases. Implications of these results for predictive regression tests are explored. When the predictive regressor is stationary, the popular Campbell and Yogo (2006) CIs for the regression coefficient have zero coverage probability asymptotically, and their predictive test statistic Q erroneously indicates predictability with probability approaching unity when the null of no predictability holds. These results have obvious cautionary implications for the use of the procedures in empirical practice.

The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions that are commonly encountered in applications and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation-based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequence-based delta method in the stationary case and a sequence-based implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than (partial) bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE, but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem.

Some new tools for analyzing spurious regressions are presented. The theory utilizes the general representation of a stochastic process in terms of an orthonormal system and provides an extension of the Weierstrass theorem to include the approximation of continuous functions and stochastic processes by Wiener processes. The theory is applied to two classic examples of spurious regressions: regressions of stochastic trends on time polynomials and regressions among independent random walks. It is shown that such regressions reproduce in part and in whole the underlying orthonormal representations.

Our general subject is model determination methods and their use in the prediction of economic time series. The methods suggested are Bayesian in spirit but they can be justified by classical as well as Bayesian arguments. The main part of the paper is concerned with model determination, forecast evaluation, and the construction of evolving sequences of models that can adapt in dimension and form (including the way in which any nonstationarity in the data is modelled) as new characteristics in the data become evident. The paper continues some recent work on Bayesian asymptotics by the author and Werner Ploberger (1995), develops embedding techniques for vector martingales that justify the role of a class of exponential densities in model selection and forecast evaluation, and implements the modelling ideas in a multivariate regression framework that includes Bayesian vector autoregressions (BVAR's) and reduced rank regressions (RRR's). It is shown how the theory in the paper can be used: (i) to construct optimized BVAR's with data-determined hyperparameters; (ii) to compare models such as BVAR's, optimized BVAR's, and RRR's; (iii) to perform joint order selection of cointegrating rank, lag length, and trend degree in a VAR; and (iv) to discard data that may be irrelevant and thereby reset the initial conditions of a model.

Fully modified least squares (FM-OLS) regression was originally designed in work by Phillips and Hansen (1990) to provide optimal estimates of cointegrating regressions. The method modifies least squares to account for serial correlation effects and for the endogeneity in the regressors that results from the existence of a cointegrating relationship. This paper provides a general framework which makes it possible to study the asymptotic behavior of FM-OLS in models with full rank I(1) regressors, models with I(1) and I(0) regressors, models with unit roots, and models with only stationary regressors. This framework enables us to consider the use of FM regression in the context of vector autoregressions (VAR's) with some unit roots and some cointegrating relations. The resulting FM-VAR regressions are shown to have some interesting properties. For example, when there is some cointegration in the system, FM-VAR estimation has a limit theory that is normal for all of the stationary coefficients and mixed normal for all of the nonstationary coefficients. Thus, there are no unit root limit distributions even in the case of the unit root coefficient submatrix (i.e., I n-r , for an n-dimensional VAR with r cointegrating vectors). Moreover, optimal estimation of the cointegration space is attained in FM-VAR regression without prior knowledge of the number of unit roots in the system, without pretesting to determine the dimension of the cointegration space and without the use of restricted regression techniques like reduced rank regression. The paper also develops an asymptotic theory for inference based on FM-OLS and FM-VAR regression. The limit theory for Wald tests that rely on the FM estimator is shown to involve a linear combination of independent chi-squared variates. This limit distribution is bounded above by the conventional chi-squared distribution with degrees of freedom equal to the number of restrictions. Thus, conventional critical values can be used to construct valid (but conservative) asymptotic tests in quite general FM time series regressions. This theory applies to causality testing in VAR's and is therefore potentially useful in empirical applications.

The author derives some exact finite sample disbibutions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating coefficients in error correction models. The reduced rank regression estimator has a distribution with Cauchy-like tails and no finite moments of integer order. The maximum likelihood estimator of the coefficients in a particular triangular system representation has matrix t-distribution tails with finite integer moments to order T - n + r, where T is the sample size, n is the total number of variables, and r is the dimension of cointegration space. This helps explain some recent simulation studies where extreme outliers occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures based on triangular representation.

This paper characterizes empirically achievable limits for time series econometric modeling and forecasting. The approach involves the concept of minimal information loss in time series regression and the paper shows how to derive bounds that delimit the proximity of empirical measures to the true probability measure (the DGP) in models that are of econometric interest. The approach utilizes joint probability measures over the combined space of parameters and observables and the results apply for models with stationary, integrated, and cointegrated data. A theorem due to Rissanen is extended so that it applies directly to probabilities about the relative likelihood (rather than averages), a new way of proving results of the Rissanen type is demonstrated, and the Rissanen theory is extended to nonstationary time series with unit roots, near unit roots, and cointegration of unknown order. The corresponding bound for the minimal information loss in empirical work is shown not to be a constant, in general, but to be proportional to the logarithm of the determinant of the (possibility stochastic) Fisher–information matrix. In fact, the bound that determines proximity to the DGP is generally path dependent, and it depends specifically on the type as well as the number of regressors. For practical purposes, the proximity bound has the asymptotic form (K/2)log n, where K is a new dimensionality factor that depends on the nature of the data as well as the number of parameters in the model. When ‘good’ model selection principles are employed in modeling time series data, we are able to show that our proximity bound quantifies empirical limits even in situations where the models may be incorrectly specified. One of the main implications of the new result is that time trends are more costly than stochastic trends, which are more costly in turn than stationary regressors in achieving proximity to the true density. Thus, in a very real sense and quantifiable manner, the DGP is more elusive when there is nonstationarity in the data. The implications for prediction are explored and a second proximity theorem is given, which provides a bound that measures how close feasible predictors can come to the optimal predictor. Again, the bound has the asymptotic form (K/2)log n, showing that forecasting trends is fundamentally more difficult than forecasting stationary time series, even when the correct form of the model for the trends is known.

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.