TimeSeriesBook.pdf

I j 1 ψ i thus x t x μt t x j 1 u j x μt t x j 1

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i = j +1 Ψ i . Thus, X t = X 0 + μt + t X j =1 U j = X 0 + μt + t X j =1 Ψ(L) Z j = X 0 + μt + t X j =1 Ψ(1) + (L - 1) e Ψ(L) Z j = X 0 + μt + Ψ(1) t X j =1 Z j + t X j =1 (L - 1) e Ψ(L) Z j = X 0 + μt + Ψ(1) t X j =1 Z j + e Ψ(L) Z 0 - e Ψ(L) Z t . The only point left is to show that e Ψ(L) Z 0 - e Ψ(L) Z t is stationary. Based on Theorem 10.2, it is sufficient to show that the coefficient matrices are abso- lutely summable. This can be derived by applying the triangular inequality and the condition for integrated processes: X j =0 k e Ψ j k = X j =0 X i = j +1 Ψ i X j =0 X i = j +1 k Ψ i k = X j =1 j k Ψ j k < . The process { X t } can therefore be viewed as the sum of a linear trend, X 0 + μt , with stochastic intercept, a multivariate random walk, Ψ(1) t j =0 Z t , and a stationary process { V t } . Based on this representation, we can then define the notion of cointegration (Engle and Granger, 1987). Definition 16.3 (Cointegration) . A multivariate stochastic process { X t } is called cointegrated if { X t } is integrated of order one and if there exists a vector β R n , β 6 = 0 , such that { β 0 X t } , is integrated of order zero, given a corresponding distribution for the initial random variable X 0 . β is called the cointegrating or cointegration vector . The cointegrating rank is the maximal number, r , of linearly independent cointegrating vectors β 1 , . . . , β r . These vectors span a linear space called the cointegration space . The Beveridge-Nelson decomposition implies that β is a cointegrating vector if and only if β 0 Ψ(1) = 0. In this case the random walk component t j =1 Z j is annihilated and only the deterministic and the stationary com- ponent remain. 3 For some issues it is of interest whether the cointegration 3 The distribution of X 0 is thereby chosen such that β 0 X 0 = β 0 e Ψ(L) Z 0 .
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322 CHAPTER 16. COINTEGRATION vector β also eliminates the trend component. This would be the case if β 0 μ = 0. The cointegration vectors are determined only up to some basis transfor- mations. If β 1 , . . . , β r is a basis for the cointegration space then ( β 1 , . . . , β r ) R is also a basis for the cointegration space for any nonsingular r × r matrix R because (( β 1 , . . . , β r ) R ) 0 Ψ(1) = 0. 16.2.2 Vector Autoregressive (VAR) and Vector Error Correction Models (VECM) Although the Beveridge-Nelson decomposition is very useful from a theo- retical point of view, in practice it is often more convenient to work with alternative representations. Most empirical investigations of integrated pro- cesses start from a VAR(p) model which has the big advantage that it can be easily estimated: X t = c + Φ 1 X t - 1 + . . . + Φ p X t - p + Z t , Z t WN(0 , Σ) where Φ(L) = I n - Φ 1 L - . . . - Φ p L p and c is an arbitrary constant. Subtracting X t - 1 on both sides of the difference equation, the VAR model can be rewritten as: X t = c + Π X t - 1 + Γ 1 X t - 1 + . . . + Γ p - 1 X t - p +1 + Z t (16.3) where Π = - Φ(1) = - I n + Φ 1 + . . . Φ p and Γ i = - p j = i +1 Φ j . We will make the following assumptions: (i) All roots of the polynomial det Φ( z ) are outside the unit circle or equal to one, i.e.
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