25.2 Vector Error Correction Model (VECM)
\[ \Delta y_t = \mu + \alpha \left(\beta' y_{t-1}\right) + \sum_{i=1}^{p-1} \Gamma_i \Delta y_{t-i} + \varepsilon_t, \]
where:
\(y_t\) is a \(k \times 1\) vector of endogenous variables at time \(t\). \(\Delta y_t\) represents the first difference of \(y_t\), i.e., \(y_t - y_{t-1}\). \(\mu\) is a \(k \times 1\) vector of constants (intercepts). \(\alpha\) is a \(k \times r\) matrix of adjustment coefficients, indicating the speed of adjustment to long-run equilibrium. \(\beta\) is a \(k \times r\) matrix of cointegrating vectors, capturing the long-term equilibrium relationships. \(\beta' y_{t-1}\) is the error correction term, which represents deviations from the long-term equilibrium. \(\Gamma_i\) are \(k \times k\) coefficient matrices that capture short-term dynamics. \(p\) is the lag length of the model. \(\varepsilon_t\) is a \(k \times 1\) vector of error terms (white noise disturbances).
https://ideas.repec.org/a/kea/keappr/ker-20080630-24-1-08.html
http://fmwww.bc.edu/EC-C/S2013/823/EC823.S2013.nn10.slides.pdf
structural VECM (SVECM) https://www.econstor.eu/bitstream/10419/62675/1/724872310.pdf