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Assume you have k cointegrated time series $Y$. You want to estimate a VECM and have figured out a suitable lag order p, cointegration rank r and the normalized cointegrating vector $\beta'$. You specified a model (in matrix notation) $$\Delta Y_{t} = \alpha \beta'Y_{t-1} +\Gamma\Delta X + U_t.$$ The error term $U$ is some i.i.d mean zero random variable and $\Delta X$ are lagged $\Delta Y_{t-i}$'s for $i=1,..,p-1$. A more complete description of the model can be found in on p.286 in Lutkepohls book "New introduction to multiple time series" (2005).

I am interested in the $k\times r$ dimensional vector $\alpha$. I want to know the range of possible values we could expect the elements in $\alpha$ to take on, and when there are signs a certain $\alpha$ is "off". This term $\alpha$ is often referred to as the "loading matrix" or "speed of adjustment (to equilibrium) parameter".

Some thoughts to get us on our way; in this excellent post Hardy gives an argument that $\alpha$ can be both positive and negative. Freedom in the choice of normalization of $\beta$ seems to permit this. However, here and here claims are made that $\alpha$ can (1) not be positive, and (2) typically does not exceed 1 in absolute terms, since this would violate stationarity.

A thought experiment, consider the following VECM in equilibrium and imagine a unit shock in $y_{t-1}$ at t-1; $$\Delta y_{1,t}=\alpha_1(y_{1,t-1}-y_{2,t-1}) \\ \Delta y_{2,t}=\alpha_2(y_{1,t-1}-y_{2,t-1}).$$ When all else equal, what follows $$\Delta y_{1,t} = \alpha_1 \\ \Delta y_{2,t} = \alpha_2 \\ \Delta y_{1,t+1}=\alpha_1(\alpha_1-\alpha_2) \\ \Delta y_{2,t+1}=\alpha_2(\alpha_1-\alpha_2) \\ \Delta y_{1,t+2}=\alpha_1((\alpha_1-\alpha_2)^2) \\ \Delta y_{2,t+2}=\alpha_2((\alpha_1-\alpha_2)^2).$$ This seems to suggest there is a restriction on the difference or sum of alphas, since when the difference in alpha in this example is greater then one the error correction term will grow to infinity, which violates stationarity. But it would suggest that an individual element alpha can take on any value, and that only the collection of all elements of alpha together is restricted.

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  • $\begingroup$ See also this, this and this. $\endgroup$ Sep 28, 2020 at 17:54
  • $\begingroup$ Thank you for the additional links. I think we can consider this issue of positive adjustment coefficients resolved. Now the issue of the magnitude of elements in alpha and whether there are possible connections/restrictions when evaluating these elements as a whole. $\endgroup$ Sep 28, 2020 at 20:28

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