Causality in Time Series - ClopiNet

Causal Search in SVARwhere X ′ t = [Y′ t−1 , ...,Y′ t−p ], which has dimension (1 × kp) and Π′ = [A 1 ,...,A p ], whichhas dimension (k × kp). In case of stable VAR process (see next subsection), the conditionalmaximum likelihood estimate of Π for a sample of size T is given byMoreover, the ith row of ˆΠ ′ is⎡ ⎤⎡⎤T∑︁ T∑︁ˆΠ ′ = ⎢⎣ Y t X ′ t⎥⎦⎢⎣ X t X ′ t⎥⎦t=1t=1t=1⎤⎡⎤T∑︁ T∑︁ˆπ ′ i⎡⎢⎣ = Y it X ′ t⎥⎦⎢⎣ X t X ′ t⎥⎦which coincides with the estimated coefficient vector from an OLS regression of Y it onX t (Hamilton 1994: 293). The maximum likelihood estimate of the matrix of varianceand covariance among the error terms Σ u turns out to be ˆΣ u = (1/T) ∑︀ Tt=1 û t û ′ t , whereû t = Y t − ˆΠ ′ X t . Therefore, the maximum likelihood estimate of the covariance betweenu it and u jt is given by the (i, j) element of ˆΣ u : ˆσ i j = (1/T) ∑︀ Tt=1 û it û jt . Denoting by σ i jthe (i, j) element of Σ u , let us first define the following matrix transform operators: vec,which stacks the columns of a k × k matrix into a vector of length k 2 and vech, whichvertically stacks the elements of a k × k matrix on or below the principal diagonal intoa vector of length k(k + 1)/2. For example:[︃ ]︃σ11 σvec 12σ 21 σ 22⎡=⎢⎣t=1−1−1σ 11[︃ ]︃σ 21 σ11 σ, vech 12σ 12 σ⎤⎥⎦21 σ 22σ 22.,⎡= ⎢⎣σ 11σ 21σ 22⎤⎥⎦ .The process being stationary and the error terms Gaussian, it turns out that:√ dT [vech( ˆΣ u ) − vech(Σ u )] −→ N(0, Ω), (11)where Ω = 2D + k (Σ u ⊗ Σ u )(D + k )′ , D + k ≡ (D′ k D k) −1 D ′ k , D k is the unique (k 2 × k(k +1)/2) matrix satisfying D k vech(Ω) = vec(Ω), and ⊗ denotes the Kronecker product (seeHamilton 1994: 301). For example, for k = 2, we have,√T⎡⎢⎣ˆσ 11 − σ 11ˆσ 12 − σ 12⎤⎥⎦ˆσ 22 − σ 22d⎛⎡−→ N ⎜⎝ ⎢⎣000⎤⎥⎦ , ⎡⎢⎣2σ 2 112σ 11 σ 12 2σ 2 122σ 11 σ 12 σ 11 σ 22 + σ 2 122σ 12 σ 222σ 2 122σ 12 σ 22 2σ 2 22Therefore, to test the null hypothesis that ρ(u it ,u jt ) = 0 from the VAR estimated residuals,it is possible to use the Wald statistic:⎤⎞⎥⎦ ⎟⎠T ( ˆσ i j ) 2ˆσ ii ˆσ j j + ˆσ 2 i j≈ χ 2 (1).111