In this paper, we analyze the Lax-Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The
method was originally proposed by Guo et al., where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax-Wendroff time discretization. We show that, under the standard CFL
condition $\tau\leq \lambda h$ (where $\tau$ and $h$ are the time step and the maximum element length respectively and $\lambda>0$ is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear
elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the $L^2$ norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional
problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution $u$ and its first order time
derivative $u_t$ in one dimension, and numerical examples are given to validate our analysis.