What is the impact of a decrease of house prices on consumption? If you go by the wealth hypothesis, one should see that home owners would reduce their consumption significantly because their wealth has suffered, while young renters are barely affected. According to the common factor hypothesis, both agent classes should respond in the same way, because they respond to common factors, for example future income prospects. It should be easy to test one hypothesis against the other and settle this. It turns out that using the exact same dataset, John Campbell and João Cocco could not reject the first, while Orazio Attanasio, Laura Blow, Robert Hamilton and Andrew Leicester could not reject the second. Well, that is embarrassing.
Annalisa Cristini and Almudena Sevilla Sanz replicate these results and show that because the two models are not nested, they cannot test against each other. Worse, it all boils down to the specification: estimate an Euler equation (which uses consumption growth) and the wealth hypothesis wins; estimate a consumption function, with consumption in levels, and the common factors hypothesis wins. So I am afraid it is not sufficient to estimate a single equation, one needs to estimate the whole structural model jointly.