SUMMARYThis paper develops a simple sequential multiple-horizon non-causation test strategy for trivariate VAR models (with one auxiliary variable). We apply the test strategy to a rolling window study of money supply and real income, with the price of oil, the unemployment rate and the spread between the Treasury bill and commercial paper rates as auxiliary processes. Ours is the first study to control simultaneously for common stochastic trends, sensitivity of test statistics to the chosen sample period, null hypothesis over-rejection, sequential test size bounds, and the possibility of causal delays. Evidence suggests highly significant direct or indirect causality from M1 to real income, in particular through the unemployment rate and M2 once we control for cointegration.

In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by B. Hill (1975) for dependent, heterogeneous processes. We develop new extremal dependence measures that characterize a massive array of linear, nonlinear, and conditional volatility processes with long or short memory. We prove the Hill-estimator is weakly and uniformly weakly consistent for processes with extremes that form mixingale sequences, and asymptotically normal for processes with extremes that are Near Epoch Dependent on the mixing extremes of some arbitrary process. The extremal persistence assumptions in this paper are known to hold for mixing, Lp-NED and some non-Lp-NED processes, including ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear distributed lags like random coe¢ cient and regime switching autoregressions.Finally, we deliver a simple nonparametric estimator of the asymptotic variance of the Hill-estimator, and prove consistency for processes with NED extremes. 1.INTRODUCTION This paper develops an asymptotic theory for the popular distribution tail index estimator due to B. Hill (1975) under general conditions. Many time series in …nance, macroeconomics and meteorology exhibit extreme values that appear to cluster (Leadbetter, Lindgren and Rootzén 1983, Embrechts, Klüppelberg, andMikosch 1997). In order to deliver a Gaussian limit theory that is robust to the nature of persistence and heterogeneity in extremes, we introduce new extremal dependence measures and develop an associated weak and uniform limit theory for dependent, heterogeneous tail arrays.Denote by fX t g = fX t : 1 < t < 1g a stochastic process on some probability measure space, write F t (x) := P (X t x) and assume F t has support on [0; 1). Assume F t (x) := P (X t > x) is regularly varying at 1: for all > 0 and each t (1) limI would like to thank Enno Mammen for insight into the extremal dependence properties developed here, Oliver Linton for discussions concerning the strong-GARCH case, and Holger Drees for comments on an earlier version. In particular, I kindly thank three anonymous referrees, Co-Editor Bruce Hansen and Editor Peter C. B. Phillips for expert commentary that lead to substantial improvements. All errors, of course, are mine alone. y

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