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

code: https://www.r-econometrics.com/timeseries/vecintro/