Time Series Predictability -- Is Cointegration a Statistical Fluke?

In this story we will shed some light on something quite unsettling regarding time series conintegration, that is model overfitting. Usually, we associate overfitting with other kinds of models, but not with conintegration, after all, if we find a stationary linear combination isn't that enough? It turns out that it is not. When we increase the number of time series used to find cointegration relationships, in most cases, we see that the resulting time series keeps getting lower Dickey-Fuller test values, so it becomes more and more likely that such a time series is stationary. For instance, if we use several Brownian motions (AR(1) unit root processes) it is easy to get Dickey-Fuller results of about -10.