  On the basis of detrended fluctuation analysis (DFA), we propose a new
bivariate linear regression model. This new model provides estimators of
multi-scale regression coefficients to measure the dependence between variables
and corresponding variables of interest with multi-scales. Numerical tests are
performed to illustrate that the proposed DFA-based regression estimators are
capable of accurately depicting the dependence between the variables of
interest and can be used to identify different dependence at different time
scales. We apply this model to analyze the PM2.5 series of three adjacent
cities (Beijing, Tianjin, and Baoding) in Northern China. The estimated
regression coefficients confirmed the dependence of PM2.5 among the three
cities and illustrated that each city has different influence on the others at
different seasons and at different time scales. Two statistics based on the
scale-dependent $t$-statistic and the partial detrended cross-correlation
coefficient are used to demonstrate the significance of the dependence. Three
new scale-dependent evaluation indices show that the new DFA-based bivariate
regression model can provide rich information on studied variables.
