Considering the widely existed time-delayed causalities in economic realities, this paper constructs a time-delay accumulative multivariable grey mode with the time trend TDAGM($1,N,t$) based on the analysis of the linear trend of system behaviors and the time-delayed accumulative mode of driving factors. It is proved that the prediction model GM(1,1), GM(1,N), OGM(1,N), time-delay GM(1,N), and TDDGM(1,N) are all special forms of TDAGM($1,N,t$) with different coefficients. Furthermore, the direct solution method of the time response formula of TDAGM($1,N,t$) is proposed, in order to avoid the error in the transformation of differential form and difference form in the process of model solving. Aiming at the problem of identification and optimization of the time-delay parameters, the algorithm framework of model parameter estimation is presented based on the particle swarm optimization algorithm. The numerical experiments of the time-delay system show that TDAGM($1,N,t$) can solve the problem of predictive modeling for multivariable systems with time-delay characteristics. To verify the effectiveness of the development, the proposed model is applied to prediction of grain yield production in Henan Province. The result of the model is consistent with the development trend of grain production, and shows the effectiveness of the proposed model.