Causality in Time Series - ClopiNet

Causal Search in SVARIn search for a more general specification of conditional independency, Chlaß andMoneta (2010) suggest a procedure based on nonparametric density estimation. Therein,neither the type of dependency between Y and X, nor the probability distributions ofthe variables need to be specified. The procedure exploits the fact that if two randomvariables are independent of a third, one obtains their joint density by the product of thejoint density of the first two, and the marginal density of the third. Hence, hypothesistest (17) translates into:f (Y, X,Z)H 0 : = f (YZ)f (XZ) f (Z) . (18)If we define h 1 (·) := f (Y, X,Z) f (Z), and h 2 (·) := f (YZ) f (XZ), we have:H 0 : h 1 (·) = h 2 (·). (19)We estimate h 1 and h 2 using a kernel smoothing approach (see Wand and Jones, 1995,ch.4). Kernel smoothing has the outstanding property that it is insensitive to autocorrelationphenomena and, therefore, immediately applicable to longitudinal or time seriessettings (Welsh et al., 2002).In particular, we use a so-called product kernel estimator:)︁ (︁1ĥ 1 (x,y,z;b) = KYi)︁ (︁−y KZi −z)︁}︁{︁∑︀ (︁ ni=1KZi)︁}︁−zp{︁∑︀ ni=1K (︁ X i −xN 2 b m+d b )︁ (︁ b bbZi −z)︁}︁{︁∑︀ ni=1 KZ K (︁ )︁ (︁Y i −y Zi)︁}︁−z (20)Kp ,1{︁∑︀ĥ 2 (x,y,z;b) = ni=1K (︁ X i −xN 2 b m+d bwhere X i , Y i , and Z i are the i th realization of the respective time series, K denotes thekernel function, b indicates a scalar bandwidth parameter, and K p represents a productkernel 2 .So far, we have shown how we can estimate h 1 and h 2 . To see whether these aredifferent, we require some similarity measure between both conditional densities. Thereare different ways to measure the distance between a product of densities:(i) The weighted Hellinger distance proposed by Su and White (2008):⎧ √︃⎫d H = 1 2n∑︁ ⎪⎨n ⎪⎩ 1 − h 2 (X i ,Y i ,Z i ) ⎪⎬a(X i ,Y i ,Z i ), (21)h 1 (X i ,Y i ,Z i ) ⎪⎭i=1where a(·) is a nonnegative weighting function. Both the weighting function a(·),and the resulting test statistic are specified in Su and White (2008).(ii) The Euclidean distance proposed by Szekely and Rizzo (2004) in their ‘energytest’:d E = 1 n∑︁ n∑︁||h 1i − h 2nj|| − 1 n∑︁ n∑︁||h 1i − h 12nj|| − 1 n∑︁ n∑︁||h 2i − h 22nj||, (22)i=1j=1i=1j=12. I.e. K p ((Z i − z)/b) = ∏︀ dj=1 K((Z ji − z j )/b). For our simulations (see next section) we choose thekernel: K(u) = (3 − u 2 )φ(u)/2, with φ(u) the standard normal probability density function. We use a“rule-of-thumb” bandwidth: b = n −1/8.5 .bbi=1j=1b119