This paper empirically tests housing market efficiency in the spatial dimension by using the spatial autoregressive conditional heteroskedastic (ARCH) and spatial quantile regression models. The tests were conducted in terms of both housing returns and squared returns (volatility). The sale price data used is from Cook County residential MLS for the years 2010-2016. The main findings are: housing returns are not spatially correlated but squared returns are spatially correlated, and the spatial dependence of squared returns seems to be stronger for higher squared return quantiles.