We consider wireless sensor networks (WSNs) used for distributed estimation of unknown parameters. Due to the limited bandwidth, sensor nodes quantize their noisy observations before transmission to a fusion center (FC) for the estimation process. In this letter, the correlation between observations is exploited to reduce the mean-square error (MSE) of the distributed estimation. Specifically, sensor nodes generate local predictions of their observations and then transmit the quantized prediction errors (innovations) to the FC rather than the quantized observations. The analytic and numerical results show that transmitting the innovations rather than the observations mitigates the effect of quantization noise and hence reduces the MSE.