On the Sensitivity of Granger Causality to Errors‐In‐Variables, Linear Transformations and Subsampling

Abstract

This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical spectral factors and Wold decompositions, we give general conditions under which additive noise or filtering distorts Granger‐causal properties by inducing (spurious) Granger causality, as well as conditions under which it does not. For the errors‐in‐variables case, we give a continuity result, which implies that: a ‘small’ noise‐to‐signal ratio entails ‘small’ distortions in Granger causality. On filtering, we give general necessary and sufficient conditions under which ‘spurious’ causal relations between (vector) time series are not induced by linear transformations of the variables involved. This also yields transformations (or filters) which can eliminate Granger causality from one vector to another one. In a number of cases, we clarify results in the existing literature, with a number of calculations streamlining some existing approaches.

Re-Evaluation of World Population Figures: Politics and Forecasting Mechanics

“…Since population is perceived to maintain trend over time period, differencing may be required (Raman, Sathianandan, Sharma, & Mohanty, 2017). The study used Augmented Dickey-Fuller test, Philip-Perron test of stationarity alongside Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) that indicate stationarity/non-stationarity through rapid/weak dampening in the spikes (Anderson, Deistler, & Dufour, 2019;Petrova, 2019).…”

Granger Causality of Gaussian Signals from Quantized Measurements

“…and If there exists some q ∈ (m, k] such that the matrix C x Q →z Q (m, q) (8) involving the covariances of quantized data is full-rank with smallest singular value σ min (C x Q →z Q (m, q)) > Γ (m, q) , where Γ (m, q) is defined in (9), then the unquantized Gaussian signal x Granger causes the unquantized Gaussian signal z.…”

“…[19]. In [6,11,45], time starts from negative infinity while Granger [23] considers the starting time k = 1. In any case, it makes no practical difference because when we infer Granger causality, we need a large number of data points.…”

“…A more restrictive version of feedback-free random processes having block-diagonal innovation covariance is introduced in [6] and its equivalence to strictly causal linear systems is discussed. The effect of sampling rate on Granger causality is investigated in [7]- [9] while the impact of filtering is discussed in [8], [9] and the effects of noise are addressed in [7], [9]- [11].…”

“…The relation between Granger non-causality and a linear time-invariant state space representation with a star graph structure as a network topology is considered in [29], and it was shown that Granger noncausality is equivalent to the existence of such a representation. Granger causality between processes corrupted by additive noise or affected by filtering or sampling has also been investigated [4,6,20,37,44,45].…”