The investment-consumption correlation

In the analysis of business cycles, investment-specific technology (IST) shocks are the new rage. A few papers now have shown that they explain better business cycles than the classical shocks to total factor productivity ("technology"). Specifically, IST shocks are perturbations to how efficiently investment translates into productive capital. There is one problem, though, and it is a major one: in response to IST shocks, investment and consumption move in opposite directions, while the data clearly indicate they are positively correlated.

Francesco Furlanetto and Martin Seneca study under which circumstances a positive correlation could be obtained. They find that you need the following: non-separability of consumption and leisure in utility and nominal price rigidity (à la Calvo, sigh...). Remove any of the two, and the result crumbles. Thus it seems essential to make a point whether these two assumptions make sense. Regular readers know that I do not believe Calvo pricing is an appropriate way of looking at price rigidity, provided that it even is significant to matter economically in the first place. It is not clear that another pricing mechanism would yield the same result. And regarding non-separability of leisure and consumption, I have no evidence whether this is a valid assumption, and the authors do not help me. And what about other shocks? Those typically lead to positive correlations between consumption and investment, and include such shocks may help relaxing the requirements identified by Furlanetto and Seneca.