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This is to be distinguished from the conditional covariance, which will in general be a random variable, though later we note that for Gaussian variables the notions coincide.
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This is to be distinguished from the conditional covariance, which will in general be a random variable, though later we note that for Gaussian variables the notions coincide.
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17
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77951102201
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While our analysis may be extended to continuous time we focus here on the discrete-time case.
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While our analysis may be extended to continuous time we focus here on the discrete-time case.
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18
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77951126725
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Here and in the remainder of this paper we assume, without loss of generality, that all random vectors and random processes have zero mean; thus constant terms are omitted in all linear regressions.
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Here and in the remainder of this paper we assume, without loss of generality, that all random vectors and random processes have zero mean; thus constant terms are omitted in all linear regressions.
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The analysis carries through for the nonstationary case, but for simplicity we assume here that all processes are stationary.
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The analysis carries through for the nonstationary case, but for simplicity we assume here that all processes are stationary.
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77951142596
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This might be more familiar as conditional G-causality, with Z as the conditioning variable. In practice it is the more useful form; for the nonconditional version, Z may simply be omitted.
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This might be more familiar as conditional G-causality, with Z as the conditioning variable. In practice it is the more useful form; for the nonconditional version, Z may simply be omitted.
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77951112230
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- will generally be multivariate (at least if p,q>1); hence they are written in bold type.
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- will generally be multivariate (at least if p, q > 1); hence they are written in bold type.
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We remark that for significance testing of G-causality it is quite common to use the appropriate F statistic for regressions rather than FY→X Z itself; the quantities are in any case related by a monotonic transformation.
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We remark that for significance testing of G-causality it is quite common to use the appropriate F statistic for regressions rather than F Y → X Z itself; the quantities are in any case related by a monotonic transformation.
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If the predictee variable has a continuous (multivariate) distribution, we note that the Jacobian determinants in the standard change-of-variables formula for entropy calculation cancel out.
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If the predictee variable has a continuous (multivariate) distribution, we note that the Jacobian determinants in the standard change-of-variables formula for entropy calculation cancel out.
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35
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77951113343
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- ) which appears in FY→X (using a finite number of lags will generally result in an overestimate of FY→X since residual errors will be larger than for the exact regression). As applied to empirical data, it is in any case good practice to choose "sufficient" lags for all regressions so as to model the data adequately without overfitting.
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-) which appears in F Y → X (using a finite number of lags will generally result in an overestimate of F Y → X since residual errors will be larger than for the exact regression). As applied to empirical data, it is in any case good practice to choose "sufficient" lags for all regressions so as to model the data adequately without overfitting.
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In it is stated that partial G-causality may in some circumstances be negative; the justification for this is unclear.
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In it is stated that partial G-causality may in some circumstances be negative; the justification for this is unclear.
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In, under the section headed "partial complex Granger causality," the quantity developed appears to be (the trace version of) what is conventionally referred to as conditional G-causality rather than partial G-causality as introduced in and referenced in this section.
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In, under the section headed "partial complex Granger causality," the quantity developed appears to be (the trace version of) what is conventionally referred to as conditional G-causality rather than partial G-causality as introduced in and referenced in this section.
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This is the "weighted" version of causal density. An unweighted and [0,1] bounded alternative can be defined as the fraction of all pairwise conditional causalities that are statistically significant at a given significance level.
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This is the "weighted" version of causal density. An unweighted and [0,1] bounded alternative can be defined as the fraction of all pairwise conditional causalities that are statistically significant at a given significance level.
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