Gaussian normal error assumption is a basic assumption for co-integration tests. Ordinary Least Squares (OLS) based regression techniques are also widely used together with the normality assumption. To consider the heavy-tailed structure observed in many economic and financial time series, new residual-based co-integration tests are developed and analyzed via Monte Carlo simulations. The new tests are based on Least Absolute Deviation (LAD) regressions, whose error structure follows the infinite-variance stable distribution. Empirical applications on Forward Rate Unbiasedness Hypothesis (FRUH) and Purchasing Power Parity (PPP) verify the need to make use of the infinite-variance stable distributions as the error distributions.