Generation and propagation of lower hybrid drift wave (LHDW) within and near the electron diffusion region (EDR) during guide field reconnection at the magnetopause is studied with data from the Magnetospheric Multiscale mission and a theoretical model. Inside the EDR where the electron beta is high (β ~ 5), the long-wavelength electromagnetic LHDW propagating obliquely to the local magnetic field is observed. In contrast, the short-wavelength electrostatic LHDW propagating nearly perpendicular to the local magnetic field is observed slightly away from the EDR, where β is small (~0.6). These observed LHDW features are explained by a local theoretical model only after including effects from the electron temperature anisotropy, finite electron heat flux and parallel current. The short-wavelength LHDW is capable of generating significant drag force between electrons and ions.