Structural causal models for macro-variables in time-series
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We consider a bivariate time series (X_t,Y_t) that is given by a simple linear autoregressive model. Assuming that the equations describing each variable as a linear combination of past values are considered structural equations, there is a clear meaning of how intervening on one particular X_t influences Y_t' at later times t'>t. In the present work, we describe conditions under which one can define a causal model between variables that are coarse-grained in time, thus admitting statements like `setting X to x changes Y in a certain way' without referring to specific time instances. We show that particularly simple statements follow in the frequency domain, thus providing meaning to interventions on frequencies.
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