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Few approaches exist that explicitly use the uncertainty associated with the spread of climate model simulations in assessing climate change impacts. An approach that does so is second-order approximation (SOA). This incorporates quantification of uncertainty to ascertain its impact on the derived response using a Taylor series expansion of the model. This study uses SOA in a statistical downscaling model of monthly streamflow, with a focus on the influence of dependence in the uncertainty of multiple atmospheric variables. Uncertainty is quantified using the square root error variance concept with a new extension that allows the inter-dependence terms among the atmospheric variable uncertainty to be specified. Applying the model to selected point locations in Australia, it is noted that the downscaling results differ considerably from downscaling that ignores uncertainty. However, when the effects of dependence in uncertainty are incorporated, the results differ according to the regional variations in dependence structure.