To understand the dynamic relationship of discrete time series processes, we adopted copula directional dependence via beta regression model applied with generalized autoregressive conditional heteroscedasticity (INGARCH) marginals. To validate the proposed method, we completed simulations of two INGARCH processes from asymmetric bivariate copula function with members such as Gaussian and Plackett copula functions. The simulations show that the proposed method is consistent for deriving directional dependent measurements regardless of the choice of the symmetric members. The proposed method is applied to the bivariate discrete time series data of the monthly counts of sandstorms and dust haze phenomena in Saudi Arabia.