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The Lagrange Multiplier (LM) test is one of the principal tools to detect ARCH and GARCH effects in financial data analysis. However, when the underlying data are non-normal, which is often the case in practice, the asymptotic LM test, based on the χ2-approximation of critical values, is known to perform poorly, particularly for small and moderate sample sizes. In this paper we propose to employ two re-sampling techniques to find critical values of the LM test, namely permutation and bootstrap. We derive the properties of exactness and asymptotically correctness for the permutation and bootstrap LM tests, respectively. Our numerical studies indicate that the proposed re-sampled algorithms significantly improve size and power of the LM test in both skewed and heavy-tailed processes. We also illustrate our new approaches with an application to the analysis of the Euro/USD currency exchange rates and the German stock index. The Canadian Journal of Statistics 40: 405-426; 2012 © 2012 Statistical Society of Canada [ABSTRACT FROM AUTHOR]