In this paper, we consider a panel data model which allows for heterogeneous time trends at different locations. We propose a new estimation method for the panel data model before we establish an asymptotic theory for the proposed estimation method. For inferential purposes, we develop a bootstrap method for the case where weak correlation presents in both dimensions of the error terms. We examine the finite–sample properties of the proposed model and estimation method through extensive simulated studies. Finally, we use the newly proposed model and method to investigate rainfall, temperature and sunshine data of U.K. respectively. Overall, we find the weather of winter has changed dramatically over the past fifty years. Changes may vary with respect to locations for the other seasons.