Following the approach proposed by Gregoir and Laroque (1993, Econometric Theory 9, 329–342), we consider a class of multivariate processes that, when differenced enough, yields covariance stationary processes whose determinant of the matrix series associated with their Wold representation has various unit roots with various orders of multiplicity we restrict to be integers. A representation theorem is provided that involves different polynomial error correction terms at each frequency associated with each unit root. An identification criterion for each set of error correction terms is proposed.

This paper extends the statistical results obtained by Gregoir and Laroque (1994, Journal of Econometrics 63, 183–214). It develops statistical tools to analyze multivariate time series that can be represented under an autoregressive equation of finite order with various polynomial error correction terms at various frequencies with possibly a non-null deterministic part as introduced by Gregoir (1999, Econometric Theory 15, 435–468). We propose an estimation procedure that proceeds through repeated applications of principal component analysis and a specification test for the omission of a polynomial relation of cointegration at each frequency.

This paper studies efficient detrending in cointegrating regression and develops modified tests for cointegration that use efficient detrending procedures. Asymptotics for these tests are derived. Monte Carlo experiments are conducted to evaluate the detrending procedures in finite samples and to compare tests for cointegration based on different detrending procedures. The limit theory allows for increasingly remote initial condition effects as the sample size goes to infinity.

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

Vector valued autoregressive models with fractionally integrated errors are considered. The possibility of the coefficient matrix of the model having eigenvalues with absolute values equal or close to unity is included. Quadratic approximation to the log-likelihood ratios in the vicinity of auxiliary estimators of the parameters is obtained and used to make a rough identification of the approximate unit eigenvalues, including complex ones, together with their multiplicities. Using the identification thus obtained, the stationary linear combinations (cointegrating relationships) and the trends that induce the nonstationarity are identified, and Wald-type inference procedures for the parameters associated with them are constructed. As in the situation in which the errors are independent and identically distributed (i.i.d.), the limiting behaviors are nonstandard in the sense that they are neither normal nor mixed normal. In addition, the ordinary least squares procedure, which works reasonably well in the i.i.d. errors case, becomes severely handicapped to adapt itself approximately to the underlying model structure, and hence its behavior is significantly inferior in many ways to the procedures obtained here.

This paper considers spurious regression between integrated processes with stable errors. Our results show that the t-ratios diverge at the rate of √T, which is identical to what Phillips (1986, Journal of Econometrics 33, 311–340) has obtained for the Gaussian case. Therefore, it is the long memory in the dependent variable and regressors, instead of the moment conditions of the error terms, that causes the spurious regression.

On June 4, 1999 G.S. Maddala (popularly and affectionately known as GS) passed away in Columbus, Ohio at the age of 66. A leading figure in the econometrics profession for more than three decades, he held the University Eminent Scholar Professorship in the Department of Economics at Ohio State University at the time of his death. GS is survived by his wife Kameswari, “Kay,” and several members of his immediate family: his daughter, Tara, of Houston; his son, Vivek, of San Francisco; and two sisters who live in India.