Abstract
The identification of causal effects between two groups of time series has been an important topic in a wide range of applications such as economics, engineering, medicine, neuroscience, and biology. In this paper, a simplified causal relationship (called trimmed Granger causality) based on the context of Granger causality and vector autoregressive (VAR) model is introduced. The idea is to characterize a subset of “important variables” for both groups of time series so that the underlying causal structure can be presented based on minimum variable information. When the VAR model is specified, explicit solutions are provided for the identification of important variables. When the parameters of the VAR model are unknown, an efficient statistical hypothesis testing procedure is introduced to estimate the solution. An example representing the stock indices of different countries is used to illustrate the proposed methods. In addition, a simulation study shows that the proposed methods significantly outperform the Lasso-type methods in terms of the accuracy of characterizing the simplified causal relationship.
Highlights
We have explained how we could identify the important variables in two groups of time series so that a simplified Granger causal relationship can be presented based on the vector autoregressive (VAR) model
When the VAR model is specified, explicit conditions are provided for identifying the important variables in both groups of time series
When the parameters of the VAR model are unknown, a multiple hypothesis testing procedure along with two different search algorithms is introduced for estimating the important variables
Summary
The Lasso-penalized VAR modeling approach and its variants (Arnold et al [1], Song and Bickel [35], Davis et al [7], Basu and Michailidis [3]) provide a convenient tool for solving the desired variable-selection problem This type of approaches have the disadvantage that the accuracy of the estimated important variables (or the estimated trimmed Granger causality) can not be satisfactorily controlled based on finite samples (see Section 5.2 for numerical illustrations). To overcome this issue, we propose an alternative framework based on which the important variables are identified by validating a class of designated constraints on the VAR coefficients throughout a sequential hypothesis testing procedure. The “causality” discussed here is defined in terms of projection of an L2-space onto the Hilbert space with respect to a probability measure μ, where for any integrable L2-functions f and g, the “inner product” in the Hilbert space is defined as
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