International stock price index numbers play a key role in the analysis of financial convergence and financial integration at the international level. In the recent literature, convergence and integration processes are usually tested through the analysis of the existence of cointegration among price index numbers. Traditional cointegration tests, however, are developed under the assumption of constant volatility and their use may be not appropriate for variables characterized by time-varying volatility, such as international stock price index data. We propose to analyze the long term interrelations among stock price index numbers by means of a novel methodology, where non stationarity in the second moments of the observed variables is allowed. In contrast to previous researches, we find no evidence of cointegration among international financial markets, hence showing that country-specific shocks may in fact have a permanent effect on the long-run performance of stock market price index numbers.