Causality in Time Series - ClopiNet

Moneta Chlass Entner HoyerThe Wald statistic for testing vanishing partial correlations of any order is obtainedby applying the delta method, which suggests that if X T is a (r × 1) sequenceof vector-valued random-variables and if [ √ T(X 1T − θ 1 ),..., √ T(X rT − θ r )] −→ N(0,Σ)and h 1 ,...,h r are r real-valued functions of θ = (θ 1 ,...,θ r ), h i : R r → R, defined andcontinuously differentiable in a neighborhood ω of the parameter point θ and such thatthe matrix B = ||∂h i /∂θ j || of partial derivatives is nonsingular in ω, then:[ √ T[h 1 (X T ) − h 1 (θ)],..., √ T[h r (X T ) − h r (θ)]] −→ N(0, BΣB ′ )(see Lehmann and Casella 1998: 61).Thus, for k = 4, suppose one wants to test corr(u 1t ,u 3t |u 2t ) = 0. First, notice thatρ(u 1 ,u 3 |u 2 ) = 0 if and only if σ 22 σ 13 − σ 12 σ 23 = 0 (by definition of partial correlation).One can define a function g : R k(k+1)/2 → R, such that g(vech(Σ u )) = σ 22 σ 13 − σ 12 σ 23 .Thus,∇g ′ = (0, −σ 23 , σ 22 , 0, σ 13 , −σ 12 , 0, 0, 0, 0).Applying the delta method:√ dT[( ˆσ22 ˆσ 13 − ˆσ 12 ˆσ 23 ) − (σ 22 σ 13 − σ 12 σ 23 )] −→ N(0,∇g ′ Ω∇g).The Wald test of the null hypothesis corr(u 1t ,u 3t |u 2t ) = 0 is given by:ddT( ˆσ 22 ˆσ 13 − ˆσ 12 ˆσ 23 ) 2∇g ′ Ω∇g≈ χ 2 (1).Tests for higher order correlations and for k > 4 follow analogously (see also Moneta,2003). This testing procedure has the advantage, with respect to the alternative methods,to be straightforwardly applied to the case of cointegrated data, as will be explainedin the next subsection.2.3. Cointegration caseA typical feature of economic time series data in which there is some form of causaldependence is cointegration. This term denotes the phenomenon that nonstationaryprocesses can have linear combinations that are stationary. That is, suppose that eachcomponent Y it of Y t = (Y 1t ,...,Y kt ) ′ , which follows the VAR processY t = A 1 Y t−1 + ... + A p Y t−p + u t ,is nonstationary and integrated of order one (∼ I(1)). This means that the VAR processY t is not stable, i.e. det(I k − A 1 z − A p z p ) is equal to zero for some |z| ≤ 1 (Lütkepohl,2006), and that each component ∆Y it of ∆Y t = (Y t − Y t−1 ) is stationary (I(0)), that isit has time-invariant means, variances and covariance structure. A linear combinationbetween between the elements of Y t is called a cointegrating relationship if there is alinear combination c 1 Y 1t + ... + c k Y kt which is stationary (I(0)).112