Lieberman and Phillips (2017; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and standard UR coefficient tests show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise. The results are applied to study the relationships between stock returns and bond spread changes.

Consistency, asymptotic normality, and efficiency of the maximum likelihood estimator for stationary Gaussian time series were shown to hold in the short memory case by Hannan (1973, Journal of Applied Probability 10, 130–145) and in the long memory case by Dahlhaus (1989, Annals of Statistics 34, 1045–1047). In this paper we extend these results to the entire stationarity region, including the case of antipersistence and noninvertibility.

We consider the stochastic process , t = 2, …, n, where sw(xt, xi) is a similarity function between the tth and the ith observations and {εt} is a random disturbance term. This process was originally axiomatized by Gilboa, Lieberman, and Schmeidler (2006, Review of Economics and Statistics 88, 433–444) as a way by which agents, or even nature, reason. In the present paper, consistency and the asymptotic distribution of the quasi-maximum likelihood estimator of the parameters of the model are established. Connections to other models and techniques are drawn. In its general form, the model does not fall within any class of nonstationary econometric models for which asymptotic theory is available. For this reason, the developments in this paper are new and nonstandard.

The saddlepoint approximation as developed by Daniels [3] is an extremely accurate method for approximating probability distributions. Econometric and statistical applications of the technique to densities of statistics of interest are often hindered by the requirements of explicit knowledge of the c.g.f. and the need to obtain an analytical solution to the saddlepoint defining equation. In this paper, we show the conditions under which any approximation to the saddlepoint is justified and suggest a convenient solution. We illustrate with an approximate saddlepoint expansion of the Durbin-Watson test statistic.

It is common for an applied researcher to use filtered data, like seasonally adjusted series, for instance, to estimate the parameters of a dynamic regression model. In this paper, we study the effect of (linear) filters on the distribution of parameters of a dynamic regression model with a lagged dependent variable and a set of exogenous regressors. So far, only asymptotic results are available. Our main interest is to investigate the effect of filtering on the small sample bias and mean squared error. In general, these results entail a numerical integration of derivatives of the joint moment generating function of two quadratic forms in normal variables. The computation of these integrals is quite involved. However, we take advantage of the Laplace approximations to the bias and mean squared error, which substantially reduce the computational burden, as they yield relatively simple analytic expressions. We obtain analytic formulae for approximating the effect of filtering on the finite sample bias and mean squared error. We evaluate the adequacy of the approximations by comparison with Monte Carlo simulations, using the Census X-11 filter as a specific example

A typical statistic encountered can be characterized as a ratio of polynomials of arbitrary degree in a random vector. This vector may possess any admissible cumulant structure. We provide in this paper general formulae for the effect of nonnormality on the density and distribution functions of this ratio. The results appear in terms of generalized cumulants, a theory developed by McCullagh (1984, Biometrika 71, 461–476). With the aid of suitable notation, the expressions are applied to the distributions of tests for heteroskedasticity and autocorrelation, the least-squares estimator of the autoregressive coefficient in a dynamic model, and tests for linear restrictions.

This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poor's 500, Nikkei, and Hang Seng indexes.

In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter . The error of the (s − 2)-order expansion is shown to be o(n−(s−2)/2)—the usual independent and identically distributed rate—for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of the spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a theorem by Fox and Taqqu (1987, Probability Theory and Related Fields 74, 213–240) concerning the asymptotic behavior of Toeplitz matrices.

Lieberman, Rousseau, and Zucker (2003, Annals of Statistics 31, 586–612) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n−1). The results given here improve upon those of Lieberman et al. in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as Lieberman et al. do, but the error of the expansion is shown to be o(n−(s−2)/2), not o(n−1), for a broad range of models.

We consider the Gaussian ARFIMA(j,d,l) model, with spectral density and an unknown mean . For this class of models, the n−1-normalized information matrix of the full parameter vector, (μ,θ), is asymptotically degenerate. To estimate θ, Dahlhaus (1989, Annals of Statistics 17, 1749–1766) suggested using the maximizer of the plug-in log-likelihood, , where is any n(1−2d)/2-consistent estimator of μ. The resulting estimator is a plug-in maximum likelihood estimator (PMLE). This estimator is asymptotically normal, efficient, and consistent, but in finite samples it has some serious drawbacks. Primarily, none of the Bartlett identities associated with are satisfied for fixed n. Cheung and Diebold (1994, Journal of Econometrics 62, 301–316) conducted a Monte Carlo simulation study and reported that the bias of the PMLE is about 3–4 times the bias of the regular maximum likelihood estimator (MLE). In this paper, we derive asymptotic expansions for the PMLE and show that its second-order bias is contaminated by an additional term, which does not exist in regular cases. This term arises because of the failure of the first Bartlett identity to hold and seems to explain Cheung and Diebold's simulated results. We derive similar expansions for the Whittle MLE, which is another estimator tacitly using the plug-in principle. An application to the ARFIMA(0,d,0) shows that the additional bias terms are considerable.Research on this topic commenced during 2000–2002, while the author was visiting the Cowles Foundation for Research in Economics at Yale University. The author is most grateful to the Cowles Foundation for their generous hospitality and to Donald Andrews and Peter Phillips for numerous helpful comments.

The maximum likelihood estimator (MLE) of the fractional difference parameter in the Gaussian ARFIMA(0,d,0) model is well known to be asymptotically N(0,6/π2). This paper develops asymptotic expansions to the distribution of this statistic under the assumption of a known unit variance. The correction term for the density is shown to be independent of d, so that the MLE is second-order pivotal for d. This feature of the MLE is unusual, at least in time series contexts. Simulations show that the normal approximation is poor and that the expansions can make a significant improvement in accuracy provided the correction terms are computed without further asymptotic approximation.This paper was commenced and revised while Lieberman was visiting the Cowles Foundation during 2000–2002. Lieberman thanks the Cowles Foundation for support and hospitality during this visit. Phillips thanks the NSF for support under grants SBR 97-30295 and SES 0092509.

Modern time series econometrics involves a diversity of models. In addition to the more traditional vector autoregressive (VAR) and autoregressive moving average (ARMA) systems, cointegration and unit root models are in widespread use for macroeconomic data, nonlinear and non-Gaussian models are popular for financial data, and long memory models are becoming more common in both macroeconomic and financial applications. Much econometric thought relates to issues of estimation and hypothesis testing, and so, in the absence of a usable finite sample theory (as is the case for the models just mentioned), an enormous amount of effort has been given to developing adequate asymptotics for statistical inference. There is often a lag between the introduction of a new model and the development of an asymptotic theory. In consequence, applied econometricians sometimes have to estimate time series models for which no asymptotic theory is available. For instance, multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models have been in use in empirical research for a while, and practitioners have been using asymptotic normality of estimators in this model even though a theoretical justification is not available.

We prove in this paper the validity of an Edgeworth expansion to the joint distribution of the sample autocorrelations of a stationary Gaussian long memory process. The method of proof relies on a verification of the suitably modified conditions for the validity of a multivariate Edgeworth expansion of Durbin (1980, Biometrika 67, 311–333). A simulation study proves the expansion to be useful and accurate.

The autoregressive fractionally integrated moving average (ARFIMA) model has become a popular approach for analyzing time series that exhibit long-range dependence. For the Gaussian case, there have been substantial advances in the area of likelihood-based inference, including development of the asymptotic properties of the maximum likelihood estimates and formulation of procedures for their computation. Small-sample inference, however, has not to date been studied. Here we investigate the small-sample behavior of the conventional and Bartlett-corrected likelihood ratio tests (LRT) for the fractional difference parameter. We derive an expression for the Bartlett correction factor. We investigate the asymptotic order of approximation of the Bartlett-corrected test. In addition, we present a small simulation study of the conventional and Bartlett-corrected LRT's. We find that for simple ARFIMA models both tests perform fairly well with a sample size of 40 but the Bartlett-corrected test generally provides an improvement over the conventional test with a sample size of 20.