Hybrid processing in millimeter wave (mmWave) communication has been proposed as a solution to reduce the cost and energy consumption by reducing the number of radio-frequency (RF) chains. However, the impact of the inevitable residual transceiver hardware impairments (RTHIs), including the residual additive transceiver hardware impairments (RATHIs) and the amplified thermal noise (ATN), has not been sufficiently studied in mmWave hybrid processing. In this work, the hybrid precoder and combiner are designed, which include both digital and analog processing by taking into account the RATHIs and the ATN. In particular, a thorough study is provided to shed light on the degradation of the spectral efficiency (SE) of the practical system. The outcomes show the steady degradation of the performance by the ATN across all SNR values, which becomes increasingly critical for higher values of its variance. Furthermore, it is shown that RATHIs result in degradation of the system only in the high SNR regime. Hence, their impact in mmWave system operating at low SNRs might be negligible. Moreover, an increase concerning the number of streams differentiates the impact between the transmit and receive RATHIs with the latter having a more severe effect.