This paper first establishes a selection of stylized facts for high-frequency cointegration processes in the European equity market. Empirical evidence is given by one minute-binned transaction data of all DAX 30 constituents as traded on Deutsche Börse's Xetra market in 2014. A methodology is introduced to simulate cointegrated stock pairs, following none, some or all of the discussed stylized facts. In particular, AR(1), AR(1)-GARCH(1,1) and multiple regime STAR(1)-GARCH(1,1) processes are used to model the cointegration relationship. Furthermore, this cointegration relationship is contaminated with jumps. Based on these processes, the power and size properties of ten contemporary cointegration tests are assessed. We provide an economic interpretation of our approach by relating cointegration to relative-value arbitrage strategies in near-efficient markets. Quintessentially, we find that in a high-frequency setting typical for stock price data, selected cointegration tests still exhibit high power. Especially the Phillips-Perron and the Pantula, Gonzalez-Farias and Fuller tests perform best at very limited size distortions.