I may not know whether some \(I(1)\) variables are cointegrated, but if they are, I often have a very strong view about the likely number and nature of cointegrating combinations. Single-factor structure is common in many areas of economics and finance, so if cointegration is present in an \(N\)-variable system, for example, a natural benchmark is 1 common trend (\(N-1\) cointegrating combinations). And moreover, the natural cointegrating combinations are almost always spreads or ratios (which of course are spreads in logs). For example, log consumption and log income may or may not be cointegrated, but if they are, then the obvious benchmark cointegrating combination is \((ln C - ln Y)\). Similarly, the obvious benchmark for \(N\) government bond yields \(y\) is \(N-1\) cointegrating combinations, given by term spreads relative to some reference yield; e.g., \(y_2 - y_1\), \(y_3 - y_1\), ..., \(y_N - y_1\).
There's not much literature exploring this perspective. (One notable exception is Horvath and Watson, "Testing for Cointegration When Some of the Cointegrating Vectors are Prespecified", Econometric Theory, 11, 952-984.) We need more.