In this paper we propose a new test procedure with more general steady state information to test the convergence hypothesis for a specific economy. We consider a model where demeaned per capita output of an economy is a function of time trend and then set the convergence hypothesis as negative average slope of that model. Applying the new procedure to 22 OECD countries we find strong evidence of convergence for 20 countries towards their average level. We also consider the per capita output of USA as a common steady state level for OECD countries. Then using the per capita output gap from USA we test the convergence hypothesis for an individual economy. This approach also shows strong evidence in favour of convergence towards the USA for most economies. France and Iceland do not converge towards the average level of OECD countries although they are converging towards USA. Australia and New Zealand are showing the opposite pattern as they are converging towards the average level but moving away from USA. This study also points out why using standard unit root tests with Bernard and Durlauf's (1995) definition of convergence is inappropriate.