We study the exchange rate effects of monetary policy in a balanced macroeconometric two-country model for the US and UK. In contrast to the empirical literature on the `delayed overshooting puzzle', which consistently treats the domestic and foreign countries unequally in the modelling process, we consider the full model feedback, allowing for a thorough analysis of the system dynamics. The consequential inevitable problem of model dimensionality is tackled in this paper by invoking the approach by Aoki (1981) commonly used in economic theory. Assuming country symmetry in the long-run allows to decouple the two-country macro dynamics of country averages and country differences such that the cointegration analysis can be applied to much smaller systems. Secondly the econometric modelling is general-to-specific, a graph-theoretic approach for the contemporaneous effects combined with an automatic general-to-specific model selection. The resulting parsimonious structural vector equilibrium correction model ensures highly significant impulse responses, revealing a delayed overshooting of the exchange rate in the case of a Bank of England monetary shock but suggests an instantaneous response to a Fed shock. Altogether the response is more pronounced in the former case.