We provide simulation evidence that shed light on several size and power issues in relation to lag selection of the augmented (nonlinear) KSS test. Two lag selection approaches are considered - the modified AIC (MAIC) approach of Ng and Perron (1995) and a sequential General to Specific (GS) testing approach proposed by Ng and Perron (2001). Either one of these approaches can be used to select the optimal lag based on either the augmented linear Dickey Fuller test or the augmented nonlinear KSS test, resulting in four possible selection methods, namely, MAIC, GS, NMAIC and NGS. The evidence suggests that the asymptotic critical values of the KSS test tends to result in over-sizing if the (N)GS method is used and under-sizing if the (N)MAIC method is utilised. Thus, we recommend that the critical values should be generated from finite samples. We also find evidence that the (N)MAIC method has less size distortion than the (N)GS method, suggesting that the MAIC-based KSS test is preferred. Interestingly, the MAIC-based KSS test with lag selection based on the linear ADF regression is generally more powerful than the test with lag selection based on the nonlinear version.