Hello,
How can we decide, which one is the most appropriate just-identified structure in the cointegrated VAR model?
When we get the cointegration vectors with the Johansen type of test, we need r2 restriction (normalisation included, and r is the number of cointegration vectors) to have just-identified long-run relations. Since this is only a rotation of the vector space and no "real" restrictions were imposed, one can get more than one just-identified structure. Which is the best practical approach to decide between them?
I would say, that if data structure coincide with theory, we're on the right track. Unfortunately, in a lot of cases, there is not much a priori information. My personal experience is, that when a "non-natural" structure is imposed, we can get wierd numbers immidiately or later when one aims to have an over-identified structure (this is rather a "data-driven" approach).
Many thanks,
Hi: I don't know but this might have something useful. http://pages.stern.nyu.edu/~dbackus/Identification/VARs/Watson_VARs_handbook_94.pdf
no problem. it's been a long time since i touched vars but the authors are good writers so hopefully it's discussed.
There is no way that statistics/econometrics/mathematics can help with your identification, The Johansen methodology only identifies the space of the cointegrating vectors. Potentially there can be an infinity of transformations of this cointegrating space all of which give rise to the same likelihood and therefore can not be distinguished. This is classic identification problem discussed in the Cowles Commission more than 80 years ago. Identification can only be achieved by the application of economic theory. you must derive your n(n-1) restrictions outside of the estimation process.
I am not sure about your statement about " not much a priori information ". If you wish to make "causal" or structural inferences you must at least have an an a priori model. In this context econometrics is the application of statistical.mathematical methods to an economic model. It is not valid to apply this methodology to a data set in the hope of discovering a new model.
The mechanics of identification often involve facilities such as zero restrictions , cross equation restrictions etc, These should be justified by your economic theory. You may also need to jointly test various over-identifying restrictions suggested by theory. If after applying your theory you do not get a sensible solution consider if there is a problem with your data (discontinuities, outliers - you might have detected these in your diagnostic tests). You might consider dropping some variables or alternatively adding variables.
If you are forecasting you do not need identification. I would stress that any "causal" or structural inferences that you make are conditional on the correctness of your underlying model. I would recommend Juselius (2007), The Cointegrated Var Model: Methodology and Applications, Oxford University Press. THis covers in great detail the matters that I have outlined above.
Hi John: Thank for you nice answer. One question ( I have that book so I'll look at it but it's quite thick !!!!! ) though. When you you're talking about identification and Cowles, you're referring to the need for restrictions in simultaneous equations models, right ? I ask because cointegration nor VARS weren't around 80 years ago. So, I think you're talking about the problem addressed in the book "The identification problem in econometrics" by Fisher but I wanted to make sure. The usually example that explains what I' m referring to is the well known supply demand example. Thanks for clarifying.
Mark, The two identification problems are very similar. The supply/demand system is discussed on page 121 of Juselius (2007), In a non-stationary system where there are two cointegrating vectors a variable that effect s supply only and a variable that effects demand only, using the appropriate zero restrictions one can interpret one of the cointegrating relationships as a supply function and the other as demand. This then is exactly what one would do in the simultaneous equation system.
There is more material on identifying the long run structure in Chapter 12 but it is not easy. Reading some of the text in the book will give one an appreciation of what is not covered in standard text books even if you find the math difficult.
There are several ECB working papers on money demand systems in the early 2000's which use this methodology. One such paper is https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp039.pdf?a1cb4280848b9c3557120a146468f3ab and there are others that can be found on the ECB web site in the Working Papers area. We even tried to estimate a similar system for Ireland but failed - probably due to changes in the definition of money variables over the period. This can be a very difficult topic
Hi John: much appreciated. I'm printing this out and saving as an email because I can't look at it now but hope to someday. thanks for clarification and suggestion of the Juselius book. I'm familar with that text series ( the blue books ) and yes, the material can be difficult. I generally slog through it and get to some kind of understanding. All the best.
Dear John. Thank you for your very detailed answer. I have Katarina Juselius' book, it is really remarkable. The "not much a priori information" part is about the situations, when one has a little theoretical guidance. In that case, I think, identification can be really hard (especially with 3 or more cointegration vectors).
To Mark: Basically they divide the CVAR (or VECM) into two parts. The first consists the long-run relations (betas), while the second consists the short-run part (the adjustment matrix alpha and the lagged variables' parameters). Since the variables in the long-run relations have the same time index, the identification process is very similar to the identification of the simultaneous equation systems (as John remarked).
About the identification:
Soren Johansen and Katarina Juselius (1994): Identification of the long-run and the short-run structure an application to the ISLM model. Journal of Econometrics. 1994, vol. 63, issue 1, 7-36 .
About the modelling approach:
Katarina Juselius (2018): Searching for a Theory That Fits the Data: A Personal Research Odyssey: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3239944
Hi Janos: I'm saving your wisdom also. The paper title sounds interesting also. Thanks.