Based on the time-lag effect in the realistic behavior system and the need to distinguish the old from the new information and the uncontrollable prediction trend in the multivariable grey prediction system, by introducing an improved damping accumulated generation operator and time-lag coefficient, a multivariable time-lag damping accumulated grey model(TLDAGM(1,$ N $)) and its extended form are proposed, which can theoretically achieve compatibility with traditional multivariable grey prediction models. The parameter estimation method and solution method of the model are discussed, the parameter optimization method and specific model steps of the model are given. Finally, the model is applied to the prediction of the output value of high-tech enterprises in China and the grain yield in Henan province and compared with the traditional multivariate grey prediction model. The results show that the simulation accuracy and prediction accuracy of the proposed model are significantly better than those of the traditional multivariable grey prediction model. The model is able to better identify the time-lag, validity, significance and trend factors of the time-series contained in the data generated by different factors in the multivariable system at different times. The results of the empirical analysis verify the rationality, applicability and validity of the proposed model.