This article studies how to design the encoder and decoder in the context of dynamic remote state estimation with a strategic sensor. The cost of the remote estimator is the estimation error covariance, whereas the cost of the self-interested strategic sensor includes an additional term related to its private information. A Stackelberg game is employed to model the interaction between the strategic sensor and the remote estimator, where the leader (strategic sensor) first designs the encoder, and the follower (remote estimator) then determines the decoder. We derive the optimal encoder and decoder based on the mismatched cost functions, and characterize the equilibrium for some special cases. One interesting result is that the equilibrium can be achieved by transmitting nothing under certain conditions. The main results are illustrated by numerical examples.