This paper introduces various consistent tests for the null of cointegration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests are analogs of Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, Econometrica 60, 119–143) canonical cointegrating regression (CCR) residuals to make the tests free of nuisance parameters in the limit. The asymptotic distributions of the tests are complex but expressed in unified manner by using standard vector Brownian motion. These distributions are tabulated by simulation for some practical cases. Furthermore, the rates of divergence of the tests are reported. Because there are methods for estimating cointegrating matrices other than CCR, it is illustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (1990, Review of Economic Studies 57, 99–125) fully modified ordinary least-squares (OLS) procedure is used. Also, is shown that difficulties arise when OLS residuals are used to formulate the tests. Small-scale simulation results are reported to examine the finite sample performance of the tests. The tests are shown to work reasonably wellin finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper can be a better testing strategy in terms of the finite sample size and power than applying univariate tests several times to each equation in a system of equations.

We obtain simple and generally applicable conditions for the existence of mixed moments E([X′ AX]″/[X′BX]U) of the ratio of quadratic forms T = X′ AX/X′BX where A and B are n × n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.

If u ≤ v, the precise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T; if u > v, a prerequisite for existence of the (u, v)th mixed moment is the existence of the 2(u − v)th moment of X When Xis not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2v.

Asymptotic distributions of some test statistics in near-integrated AR processes are studied. Some exact formulas for the distribution functions are given as well as approximative results obtained by saddlepoint approximation techniques.

The asymptotic variance matrix of the quantile regression estimator depends on the density of the error. For both deterministic and random regressors, the bootstrap distribution is shown to converge weakly to the limit distribution of the quantile regression estimator in probability. Thus, the confidence intervals constructed by the bootstrap percentile method have asymptotically correct coverage probabilities.

The definition of causation, discussed in Granger (1980) and elsewhere, has been widely applied in economics and in other disciplines. For this definition, a series yt is said to cause xt+l if it contains information about the forecastability for xt+l contained nowhere else in some large information set, which includes xt−j, j ≥ 0. However, it would be convenient to think of causality being different in extent or direction at seasonal or low frequencies, say, than at other frequencies. The fact that a stationary series is effectively the (uncountably infinite) sum of uncorrelated components, each of which is associated with a single frequency, or a narrow frequency band, introduces the possibility that the full causal relationship can be decomposed by frequency. This is known as the Wiener decomposition or the spectral decomposition of the series, as discussed by Hannan (1970). For any series generated by , where xt, and are both stationary, with finite variances and a(B) is a backward filter

with B the backward operator, there is a simple, well-known relationship between the spectral decompositions of the two series.

Many economic models imply that ratios, simple differences, or “spreads” of variables are I(0). In these models, cointegrating vectors are composed of 1's, 0's, and —1's and contain no unknown parameters. In this paper, we develop tests for cointegration that can be applied when some of the cointegrating vectors are prespecified under the null or under the alternative hypotheses. These tests are constructed in a vector error correction model and are motivated as Wald tests from a Gaussian version of the model. When all of the cointegrating vectors are prespecified under the alternative, the tests correspond to the standard Wald tests for the inclusion of error correction terms in the VAR. Modifications of this basic test are developed when a subset of the cointegrating vectors contain unknown parameters. The asymptotic null distributions of the statistics are derived, critical values are determined, and the local power properties of the test are studied. Finally, the test is applied to data on foreign exchange future and spot prices to test the stability of the forward–spot premium.

This paper presents theoretical results on the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.

This paper considers unit root tests based on M estimators. The asymptotic theory for these tests is developed. It is shown how the asymptotic distributions of the tests depend on nuisance parameters and how tests can be constructed that are invariant to these parameters. It is also shown that a particular linear combination of a unit root test based on the ordinary least-squares (OLS) estimator and on an M estimator converges to a normal random variate. The interpretation of this result is discussed. A simulation experiment is described, illustrating the level and power of different unit root tests for several sample sizes and data generating processes. The tests based on M estimators turn out to be more powerful than the OLS-based tests if the innovations are fat-tailed.

Pötscher (1991, Econometric Theory7, 163–181) has recently considered the question of how the use of a model selection procedure affects the asymptotic distribution of parameter estimators and related statistics. An important potential application of such results is to the generation of confidence regions for the parameters of interest. It is demonstrated that a great deal of care must be exercised in any attempt at such an application. We also consider the effect of model selection on prediction regions. It is demonstrated that the use of asymptotic results for the construction of prediction regions requires the same sort of care as the use of such results for the construction of confidence regions.

This paper provides weak and strong laws of large numbers for weakly dependent heterogeneous random variables. The weak laws of large numbers presented extend known results to the case of trended random variables. The main feature of our strong law of large numbers for mixingale sequences is the less strict decay rate that is imposed on the mixingale numbers as compared to previous results.

The asymptotic behavior of the sample paths of two popular statistics that test market efficiency are investigated when markets learn to have rational expectations. Two cases are investigated, where, should markets start out at a rational expectations equilibrium, both statistics would asymptotically generate standard Brownian motions. In a first case, where agents are Bayesian and payoffs exogenous, the statistics have identical sample paths, but they are not standard Brownian motions. Whereas the finite-dimensional distributions are Gaussian, there may be a bias if agents' initial beliefs differ. A second case is considered, where payoffs are in part endogenous, yet agents consider them to be drawn from a stationary, exogenous distribution, which they attempt to learn in a frequentist way. In that case, one statistic behaves as if the economy were at a rational expectations equilibrium from the beginning on. The other statistic has sample paths with substantially non-Gaussian finite-dimensional distributions. Moreover, there is a negative bias. The behavior of the two statistics in the second case matches remarkably well the empirical results in an investigation of the prices of six foreign currency contracts over the period 1973–1990.

This paper investigates through Monte Carlo simulation the finite sample properties of likelihood ratio tests for cointegrating ranks that were proposed by Johansen (1991, Econometrica 59, 1551–1580). We transform the model into a canonical form so that the experiment is well controlled without loss of generality and then conduct a comprehensive simulation study. As expected, the test performance is very sensitive to the value of the stationary root(s) of the process. We also find that the test performance depends crucially on the correlation between the innovations that drive the stationary and the nonstationary components of the process. We conclude that 100 observations are not sufficient to ensure reasonably good performance uniformly over the values of the nuisance parameters that affect the distributions of the test statistics.

In this comment, I first want to offer a few general remarks on the problems caused by nonuniformity in the convergence of coverage probabilities as pointed out in Kabaila's (1995) very stimulating paper. Second, I also argue that within the context of set estimation after model selection one typically will be more interested in conditional coverage probabilities than in unconditional ones. Concentrating on the last example of Section 2 in Kabaila, I then illustrate how a confidence region can be constructed such that the conditional coverage probabilities do not suffer from the nonuniformity problem discussed in Kabaila. As a by-product, I show that the confidence region considered in this example in Kabaila is not a very natural one and that it has undesirable conditional coverage properties, which I find actually more troublesome than the lack of uniformity in the unconditional coverage probability pointed out by Kabaila.

This paper presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1994, Econometrica 62, 43–72). The results allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., nonparametric regression based on residuals), and nonparametric regression on index functions.

The paper develops a statistical theory for regressions with integrated regressors of unknown order and unknown cointegrating dimension. In practice, we are often unsure whether unit roots or cointegration is present in time series data, and we are also uncertain about the order of integration in some cases. This paper addresses issues of estimation and inference in cases of such uncertainty. Phillips (1995, Econometrica 63, 1023–1078) developed a theory for time series regressions with an unknown mixture of 1(0) and 1(1) variables and established that the method of fully modified ordinary least squares (FM-OLS) is applicable to models (including vector autoregressions) with some unit roots and unknown cointegrating rank. This paper extends these results to models that contain some I(0), I(1), and I(2) regressors. The theory and methods here are applicable to cointegrating regressions that include unknown numbers of I(0), I(1), and I(2) variables and an unknown degree of cointegration. Such models require a somewhat different approach than that of Phillips (1995). The paper proposes a residual-based fully modified ordinary least-squares (RBFMOLS) procedure, which employs residuals from a first-order autoregression of the first differences of the entire regressor set in the construction of the FMOLS estimator. The asymptotic theory for the RBFM-OLS estimator is developed and is shown to be normal for all the stationary coefficients and mixed normal for all the nonstationary coefficients. Under Gaussian assumptions, estimation of the cointegration space by RBFM-OLS is optimal even though the dimension of the space is unknown.