By Robert F. Engle, Halbert White

This e-book is a suite of essays in honor of Clive Granger via the various world's best econometricians, all of whom have collaborated with or studied with Granger. It displays imperative issues in Granger's paintings with awareness to exams for unit roots and cointegration, checks of misspecification, forecasting versions and forecasting evaluate, and non-linear and non-parametric econometric suggestions.

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**Extra resources for Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive W.J. Granger**

**Example text**

00, then there exists X E L 2 such that X" ~X. This is indeed true but not so easy to prove as the completeness of IRk. 10. 3 (Complex L 2 Spaces). 2) E(XY). 3) 00 (with identification of functions f and g such that Jn If- gl 2 dJ1 becomes a Hilbert space if we define the inner product to be = 0), then D 48 2. u. u). 3). ) Remark 1. u) will be reserved for the respective real Hilbert spaces unless we state specifically that reference is being made to the corresponding complex spaces. 1 (Norm Convergence and the Cauchy Criterion).

And every t = (t 1, •• • , tnf such that 0 ~ t 1 < t 2 < · · · < tn, (c) B(t) - B(s) "' N(O, t - s) for t ;;::: s. To establish the existence of such a process we observe that conditions (a), (b) and (c) are satisfied if and only if, for every t = (t 1, ... 9) and so by Kolmogorov's theorem there exists a process with characteristic functions tfot( ·),or equivalently with the properties (a), (b) and (c). 2 (Brownian Motion with Drift). t + uB(t), and B(t) is standard Brownian motion. The existence of Brownian motion with drift follows at once from that of standard Brownian motion.

Leta= IIYII 2 ,b = l(x,y)landc = llxll 2 • Thepolarrepresentationof(x,y) is then (x,y) = bei8 forsomeOE(-n,n]. 4). The minimum value, c- b 2 ja, of c- 2rb + r 2 a is achieved when r = b/a. 4) then c- b 2 ja = 0. 6) we then obtain (x- yei 8 b/a,x- yei 8 bja) = 0, which, by property (e) of inner products, implies that x = yei8 b/a = y(x,y)j(y,y). 4). 2 (The Angle between Elements of a Real Inner-Product Space). 1, the angle between two non-zero vectors x and y is the angle in [0, n] whose cosine is L~=l xiyJ (llxiiiiYII).