Verification of regression models is primarily based on analysis of error terms and constitutes one of the most important steps in applied regression analysis. In cross-sectional models, the error terms are typically heteroskedastic, while in time series regressions the errors are often affected by serial correlation. Consequently, in this paper, we focus on Monte Carlo simulations applied to explore the power of several tests of homogeneity and tests for presence of autocorrelation. In the past decades, the computational power has increased significantly to allow the benefit of simulation from exact distributions, which are not defined explicitly. We will discuss 1) testing of homogeneity for a given number of components in the exponential mixture approximated by subpopulations and 2) simulation of power in several commonly used tests of autocorrelation. For the first case, we consider exact likelihood ratio test (ELR) and exact likelihood ratio test against the alternative with two-component subpopulation (ELR2). In the second case, we consider the Durbin-Watson, Durbin h, Breusch-Godfrey, Box-Pierce and Ljung-Box tests of 1st order serial correlation and the runs test of randomness in two different types of linear regression models.