https://www.nber.org/books/hick72-1

This link may be usefull.

Hoa - you are asking a question that I think is more likely to be explained in a modern econometric textbook than in journal papers. I do not have any particular textbook in mind so will offer a brief explanation myself.

The topic that you are interested in is usually referred to as "cointegration / error correction" and the established theory applies to cases where variables are non-stationary ("integrated") and a VAR is an adequate model. Mathematically, a VAR can always be rewritten as a Vector Error Correction Model (VECM), which combines first differences with one or more error correction terms (ECTs). The ECTs are constructed from the levels of the variables. Each expresses an equilibrium relationship between the variables that will be satisfied in the long run and so the ECTs are often called the "long run" part of the model.

The Engle-Granger representation theorem tells us that the true coefficients on the ECTs will all be zero if the variables are not cointegrated. If the variables are not cointegrated then the correct model only has first difference terms. These are often called the "short run" part of the model.

If variables are not cointegrated then there does not exist any equilibrium relationship between them - their long-run levels are independent of each other. In this case the VECM does not have any ECTs - it is just a VAR in differences. This does not mean that we have no interest in the long run; it means that we believe we cannot explain what happens there - we can only explain the short run response of each variable's movements to recent movements of itself and the other variables.

You might argue that the long run is an accumulation of short runs so if we can explain the latter then we can explain the former. This argument fails because (i) in addition to the "short run" and "long run" parts of the model, a VAR also has a random error term in each equation and (ii) the variables are each individually integrated. Integration means that the random disturbances are permanently accumulated into the level of each variable so we cannot predict what the levels will be in the long run. If cointegration is present then we still cannot say what the long run levels will be but we can say what relationship(s) will exist between them.

Dear Hoa: I presume you are looking for documentation explaining Vector Error Correction Models (VECM), the methodology usually used to model short- and long-run relations simulatenously in time series data. The econometric theory of these models is pretty well established, so you better look at some book. One such book is "The Cointegrated VAR model: theory and applications", by Katerina Juselius. If you want something shorter, you can try Marno Verbeek's "A guide to modern econometrics."

You can look for thé econometric test book written by Greene or Hamilton

see links

On estimation of an econometric model of short-run bank behaviour

Dennis J. Aigner

https://booksc.xyz/book/2885035/97c488

An econometric model of the short-run demand for workers and hours in the U.S. auto industry

Julius C. Chang

https://booksc.xyz/book/8573494/8a92ac