The Mathematical History Behind the Granger–Johansen Representation Theorem
追溯了格兰杰-约翰森表示定理的数学发展史，从矩阵值函数奇点几何的角度解释该定理，并讨论其在时间序列和数学上的意义，适合对协整理论数学基础感兴趣的学者。
ABSTRACT When can a vector time series that is integrated once (i.e., becomes stationary after taking first differences) be described in error correction form? The answer to this is provided by the Granger–Johansen representation theorem. From a mathematical point of view, the theorem can be viewed as essentially a statement concerning the geometry of singularities of matrix‐valued functions. Historically, the study of this subject has been motivated primarily by applications in mechanical engineering, in particular vibrations of airplanes. The paper traces the development of formulations and proofs of the theorem, with attention to mathematical detail. The meaning of the theorem is discussed both from a mathematical perspective and from a time series perspective.