The covariance structure among other properties of the square of the purely diagonal bilinear time series model is obtained. The time series properties of these squares are compared with those of the linear moving average time series model. It was discovered that the square of a linear moving average process is also identified as a moving average process whereas, while the nonlinear purely diagonal bilinear process is identified as a linear moving average process, its square is identified as an autoregressive moving average process.