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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.
In many applications it is desirable to infer coarse-grained models from observational data. The observed process often corresponds only to a few selected degrees of freedom of a high-dimensional dynamical system with multiple time scales. In this wo
Causal models with unobserved variables impose nontrivial constraints on the distributions over the observed variables. When a common cause of two variables is unobserved, it is impossible to uncover the causal relation between them without making ad
We consider the problem of finding confidence intervals for the risk of forecasting the future of a stationary, ergodic stochastic process, using a model estimated from the past of the process. We show that a bootstrap procedure provides valid confid
Prediction for high dimensional time series is a challenging task due to the curse of dimensionality problem. Classical parametric models like ARIMA or VAR require strong modeling assumptions and time stationarity and are often overparametrized. This
We study high-dimensional linear models with error-in-variables. Such models are motivated by various applications in econometrics, finance and genetics. These models are challenging because of the need to account for measurement errors to avoid non-