Over a wide range of the scattering vector q, inelastic neutron scattering measurements have been performed to investigate the local structural relaxation of a polymer glass, in terms of the Debye-Waller factor having two characteristic parameters: the mean square displacement for hydrogen motion α and the non-Gaussian parameter A(t = ~). The studied polymer is a random copolyester which is composed of binary aromatic groups, and is highly aligned forming a spatially anisotropic glassy phase. In the case when q is normal to the molecular axis (q ), A(t = ~) increases as the temperature T decreases in a glassy phase, while in the case when q is parallel (q ), A(t = ~) is independent of T and negligible. The dynamical heterogeneity, thus manifested through the non-Gaussian parameter A(t = ~) for q , originates from a spatially anisotropic cage existing within a time domain determined by the energy resolution of the spectrometer (> 10 - 1 2 s).