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Large Multiple Graphical Model Inference via Bootstrap

Large economic and financial networks may experience stage-wise change as a result of external shocks. To detect and infer a structural change, we consider an inference problem in the framework of multiple Gaussian graphical models, particularly when the number of graphs and the dimension of a graph expand with the sample size. In such a situation, two major challenges emerge as a result of the bias and uncertainty inherent in regularization, which is required treating such overparameterized models. To deal with these challenges, bootstrap is utilized to approximate the sampling distribution of a likelihood ratio test statistic. Theoretically, we show that the proposed method leads to correct asymptotic inference in a high-dimensional situation regardless of the distribution of the test statistic. Numerically, it compares favorably to its competitors through simulations. Finally, our statistical analysis of the network of 200 stocks reveals that the financial network exhibits a dramatic change in that the interacting units become more connected due to the financial crisis between 2007 and 2009. More importantly, certain units respond more strongly than others, and after the crisis, some alterations fade while others strengthen. Statistica Sinica: Preprint doi:10.5705/ss.202017.0141