Vector AutoRegressions (VARs) have now become the most popular tool of Time Series analysis amongst econometricians. Unfortunately, little is known about the analytic finite-sample properties of parameter estimators for such systems. The asymptotic analysis of VARs published to date does not address questions regarding the influence of the number and nature of the system's variates on parameter estimates. Clearly, both questions will have repercussions on the way VARs are used, and we intend to address them here.We consider the implications of varying the dimensions of VARs on the biases of Maximum Likelihood and Least Squares Estimators (MLE and LSE, respectively). In the purely nonstationary case (k-dimensional random walk), estimator biases are approximately equal to the dimension of the system (k) times the univariate bias, even when the variates are generated independently of each other. We show that the variance too increases with the dimension of the system, hence also raising the Mean Squared Error (MSE) of the estimator. When some stable linear combinations exist, the biases are generally smaller and are asymptotically proportional to the sum of the characteristic roots of the VAR. One source of such combinations is meaningful economic relations that are represented by the cointegration of some of the components of the VAR. Adding economically-irrelevant variables to a VAR is thus shown to have more serious negative consequences in integrated time series than in classical ergodic or cross section analyses. The findings strengthen the case for parsimonious modelling and for the reduction step of the general-to-specific marginalization method. They also support the use of seasonally unadjusted data whenever possible.

The paper by Kiefer, Vogelsang and Bunzel (2000), KVB henceforth, provides an interesting unconventional application of functional limit theory to a conventional problem. In this note, we point out that the limiting distribution of the t^{∗} test proposed by KVB turns out to be equivalent to the asymptotic distribution of one of the statistics analysed by Abadir and Paruolo (1997), AP henceforth. The mixed-Normal random variables studied in AP and KVB are different, but they have identical distributions. The purpose of this note is to prove this equivalence analytically.

WE WARN OF A CLASS of problems that can occur when inverting confluent characteristic functions (CF's). The term confluence is often used in Mathematics in connection with analysis and/or dynamic (difference, differential, and integral) equations; for example, see the classic text by Whittaker and Watson (1927). A confluence (of singularities) is a joint degeneracy that occurs within a function; here, the CF. When one is dealing with the CF of a k-dimensional variate where k > 1, these joint degeneracies can distort the derivation of the marginal density of some lower-dimensional combination of the k components. The distortions are both analytical and numerical. In this note, we first express the distributional problem in the simplest bivariate case, then clarify it with examples from a simple autoregressive (AR) model. Let R, S be two continuous (for simplicity) variates based on a sample of n observations, with joint CF pn(u,v) =E[euR+ivS], i = , and Pr{S > 0} = 1. The joint density h,jr, s) of R and S is expressed by means of the inversion formula as

We derive explicitly the exact density functions of two key mixed normal variates that arise from cointegration analysis. We also plot them, and analyze their analytical features and implications.