We consider a class of multivariate processes which, when differenced enough, yield covariance stationary processes whose determinants of the Wold representation have I as their only root on the unit circle. A representation theorem is proved for this class of processes that generalizes the Granger representation theorem.

Deciding the order of differencing is an important part in the specification of an autoregressive integrated moving average (ARIMA) mode. In most, though not all, cases this means deciding whether to use the original observations or their first differences. Common test procedures used in this context are some variants of autoregressive unit root tests. In these tests, one tests the null hypothesis that the order of differencing is one against the alternative that it is zero. The null hypothesis thus states that the original series is nonstationary and integrated of order one, whereas the alternative assumes that it is stationary. In this paper the situation is reversed so that our null hypothesis states that the original series is stationary, whereas the alternative states that it is integrated of order one. In our approach the use of a differenced series thus means overdifferencing and, consequently, a model with a moving average unit root. Testing for this moving average unit root is the topic of this paper. As discussed by Saikkonen and Luukkonen [26] and Tanaka [31], test procedures obtained for this null hypothesis can also be used to test the null hypothesis that a multivariate time series is cointegrated with a given theoretical cointegrating vector. Since the null hypothesis of cointegration is often of interest and cannot be naturally tested by autoregressive unit root tests, this connection provides an important motivation for the test procedures of this paper.

The asymptotic distribution of the least-squares estimators in the random walk model was first found by White [17] and is described in terms of functional of Brownian motion with no closed form expression known. Evans and Savin [5,6] and others have examined numerically both the asymptotic and finite sample distribution. The purpose of this paper is to derive an asymptotic expansion for the distribution. Our approach is in contrast to Phillips [12,13] who has already derived some terms in a general expansion by analyzing the functionals. We proceed by assuming that the errors are normally distributed and expand the characteristic function directly. Then, via numerical integration, we invert the characteristic function to find the distribution. The approximation is shown to be extremely accurate for all sample sizes ≥25, and can be used to construct simple tests for the presence of a unit root in a univariate time series model. This could have useful applications in applied economics.

Although considerable attention has recently been paid to the behavior of the maximum likelihood estimator of simple moving average models, little progress has been made in finding a good approximation to its distribution in cases where the process is close to being noninvertible. In this paper a method is produced that gives an excellent approximation to the distribution function, even in the case where the process is strictly noninvertible. Also studied is the related problem of the distribution of the maximum likelihood estimator of the signalto-noise ratio in the local level model.

The central limit theorem in Davidson [2] is extended to allow cases where the variances of sequence coordinates can be tending to zero. A trade-off is demonstrated between the degree of dependence and the rate of degeneration. For the martingale difference case, it is sufficient for the sum of the variances to diverge at the rate of log n.

This paper investigates the asymptotic distribution of the maximum likelihood estimator in a stochastic frontier function when the firms are all technically efficient. For such a situation the true parameter vector is on the boundary of the parameter space, and the scores are linearly dependent. The asymptotic distribution of the maximum likelihood estimator is shown to be a mixture of certain truncated distributions. The maximum likelihood estimates for different parameters may have different rates of stochastic convergence. The model can be reparameterized into one with a regular likelihood function. The likelihood ratio test statistic has the usual mixture of chi-square distributions as in the regular case.

Under more general assumptions than those usually made in the sequential analysis literature, a variable-sample-size-sequential probability ratio test (VPRT) of two simple hypotheses is found that maximizes the expected net gain over all sequential decision procedures. In contrast, Wald and Wolfowitz [25] developed the sequential probability ratio test (SPRT) to minimize expected sample size, but their assumptions on the parameters of the decision problem were restrictive. In this article we show that the expected net-gain-maximizing VPRT also minimizes the expected (with respect to both data and prior) total sampling cost and that, under slightly more general conditions than those imposed by Wald and Wolfowitz, it reduces to the one-observation-at-a-time sequential probability ratio test (SPRT). The ways in which the size and power of the VPRT depend upon the parameters of the decision problem are also examined.

This paper proposes a general framework for specification testing of the regression function in a nonparametric smoothing estimation context. The same analysis can be applied to cases as varied as testing for omission of variables, testing certain nonlinear restrictions in the regressors, and testing the correct specification of some parametric or semiparametric model of interest, for example, testing a certain type of nonlinearity of the regression function. Furthermore, the test can be applied to i.i.d. and time-series data, and some or all of the regressors are allowed to be discrete. A Monte Carlo simulation is used to assess the performance of the test in small and medium samples.

This paper studies the qualitative robustness properties of the Schwarz information criterion (SIC) based on objective functions defining M-estimators. A definition of qualitative robustness appropriate for model selection is provided and it is shown that the crucial restriction needed to achieve robustness in model selection is the uniform boundedness of the objective function. In the process, the asymptotic performance of the SIC for general M-estimators is also studied. The paper concludes with a Monte Carlo study of the finite sample behavior of the SIC for different specifications of the sample objective function.

It is shown that in a first-order mixed autoregressive moving average model, a Lagrange multiplier test for the autoregressive unit-root hypothesis can be inconsistent against stationary alternatives.

This note proves that the estimator of panel data censored regression models proposed in Honoré [2] is median unbiased when only one parameter is estimated. This result is obtained without parametric assumptions about the distribution of the error terms.

In this paper we consider the small sample properties of the coefficient of determination in a linear regression model with multivariate t errors when proxy variables are used instead of unobservable regressors. The results show that if the unobservable variable is an important variable, the adjusted coefficient of determination can be more unreliable in small samples than the unadjusted coefficient of determination from both viewpoints of the bias and the MSE.