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On the basis of detrended fluctuation analysis (DFA), we propose a newbivariate linear regression model. This new model provides estimators ofmulti-scale regression coefficients to measure the dependence between variablesand corresponding variables of interest with multi-scales. Numerical tests areperformed to illustrate that the proposed DFA-based regression estimators arecapable of accurately depicting the dependence between the variables ofinterest and can be used to identify different dependence at different timescales. We apply this model to analyze the PM2.5 series of three adjacentcities (Beijing, Tianjin, and Baoding) in Northern China. The estimatedregression coefficients confirmed the dependence of PM2.5 among the threecities and illustrated that each city has different influence on the others atdifferent seasons and at different time scales. Two statistics based on thescale-dependent $t$-statistic and the partial detrended cross-correlationcoefficient are used to demonstrate the significance of the dependence. Threenew scale-dependent evaluation indices show that the new DFA-based bivariateregression model can provide rich information on studied variables.

A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities