In the search for the hypothetical missing link between a variety of atmospheric parameters and solar variability, we apply a Granger causality test to monthly averages of neutron counts (a proxy for cosmic rays), as measured at the Climax station, and stratospheric aerosol number densities, derived from extinction coefficients measured by the Stratospheric Aerosol and Gas Experiment II. A causal connection between the two time series is suggested. Further study of the cross-correlation coefficient confirms this finding, indicating that cosmic rays influence stratospheric aerosols through a process with a time response of a few months. Our results have important implications for the hypothesis of a physical mechanism that links both quantities.