It is known that an electrothermal instability arises in nonequilibrium partially ionized plasmas subjected to crossed electric and sufficiently high magnetic fields. A linear theory is applied to investigate the properties of instability waves in the vicinity of the critical condition, considering the damping factor due to electronic heat conduction and radiative transfer, and also the phase shift between the variations in the electron temperature and density. There are two kinds of electrothermal instabilities: One is the normal ionization instability, and the other is a radiative instability, which can arise even in the case of no magnetic field. In the former, the largest growth rate occurs for a wave with the wavelength comparable to the characteristic length of the device. The latter has sufficiently long wavelengths compared with the length of the device. The critical Hall parameter below which instabilities do not arise increases as the size of the device containing plasmas becomes larger or the plasma pressure rises.