When building an Error Correction Model, is there any point in keeping p=1? I see a lot of equations where there t is influenced not only by t-1 but also t-2. If I only have t-1, is it useful at all to use the VECM framework? Or should I use a different framework? Initially I looked into it to address the non stationarity in my dataset.
Broadly speaking, lag length selection can be guided by
- subject-matter knowledge (your understanding how the system develops over time),
- goodness of fit (how well the model with a given lag approximates the observed process) and
- theoretical considerations (e.g. you need a model that includes lag $p$ if you want to test a hypothesis associated with some coefficient of lag $p$).
A VECM is flexible in terms of lag length as technically you can specify any lag length greater or equal to zero. There is no need to always stick to $p=1$ as in your example.
Note that the error correction term is included only once, typically with lag 1 or lag $p$. Including error correction term more than once (with different lags) will result in perfect multicollinearity in the model.