This paper investigates whether there are bubbles in stock prices. We do this using a previously studied structural vector autoregressive (SVAR) model claiming to distinguish fundamental and non-fundamental shocks to real stock prices. The SVAR model relies on an identification restriction in order to correctly label the shocks. We test this restriction by means of a Markov switching-SVAR (MS-SVAR) model in heteroskedasticity. Using data from France, Germany, Italy, Japan, the UK and the US we find that the restriction is rejected for Italy, supported at the 1% level for Japan and supported at least at the 5% level for the remaining countries. Several alternative specifications confirm the robustness of these findings. Using SVAR impulse responses and forecast error variance decompositions we further examine the structural shocks and confirm the shock labeling for Japan. Through historical decompositions we observe that stock prices tended to be undervalued throughout the 1970s and 1980s. This undervaluation corrects itself by the mid 1990s, after which stock prices tend to move in tandem with their fundamentals. We therefore find no evidence in favor of stock price bubbles in all the countries invested.