Statistical test for detrending-moving-average-based multivariate regression model

The detrending-moving-average-based multivariate linear regression has been recently introduced as a new tool to probe into multiscale dependent behavior between multivariate data. It well solves the dependence among the nonstationary variates while classic multivariate linear regression dose not. The real-world applications demonstrate its versatility and robustness. However, the statistical significance of this new model needs to be further improved mathematically not just empirically. Hence, the main motivation of this paper is to provide a solid theoretical basis for the latest regression framework. Specifically, four propositions and three theorems are deduced theoretically to support the statistical significance test of detrending-moving-average-based mul-tivariate linear regression model. These propositions and theorems revolve around hypothesis testing of each partial regression coefficients and regression model. They can examine the statistical significance at every given scales. Moreover, we also resort to synthetic time series to verify the performances of the proposed statistical test procedure. By using those statistical tests, we investigate the multiscale interdependence between the crude oil price and the exchange rates among oil-importing and oil-exporting countries. Some interesting findings show that the dependent behavior between the oil price and exchange rates varies at various scales, which cannot be achieved from classical multivariate regression model.