When unable to conduct experiments with control groups to explain causality, the only method is post-hoc control — extracting this from the data afterwards.
Arranging data sequences vertically, using a collection of factors $\mathbf{X}$ (data matrix) to explain $\mathbf{y}$ (vector), there will inevitably be unexplained error $\mathbf{e}$. By decomposing the outcome $\mathbf{y}$ to extract the causal factors $\mathbf{X}$ (signal) for explanation, then $\mathbf{e}$ (noise) becomes the control group without explanatory power. Mathematically, $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{e}$, where $\boldsymbol{\beta}$ represents "how much is explained". These are all familiar concepts, but the most important core principle is that the treatment and control groups must be fundamentally different, meaning $\mathbf{X}'\mathbf{e} = \mathbf{0}$. Only then can we have $\mathbf{X}'\mathbf{y} = \mathbf{X}'\mathbf{X}\widehat{\boldsymbol{\beta}}$, which gives us $\widehat{\boldsymbol{\beta}} = \left( \mathbf{X}'\mathbf{X} \right)^{-1}\mathbf{X}'\mathbf{y}$.
