I have attached my reply in the form of a pdf file. The file is also set out below but due to the limitations of the Quora system, the pdf is easier to read

You will find what I am saying here easier to understand if you first consult the chapters in your economics textbook dealing with simultaneous equations and identification. Most introductory econometrics texts contain this material. It is probably better that you consult a book with which you are familiar. Say you have a model with its likelihood determined by L(x, θ) where x may be a set of observations that are vector-valued and θ may be vector-valued. If you find that θ1 and θ2 give rise to the exact same maximum likelihood then the system can not be estimated. It is possible that in some cases there may be many other values of θ that also give the same value for the maximum likelihood. Such a model is not identified. If you have a model that is not identified you can not use statistics/econometrics to identify it. You must use economic theory to impose sufficient restrictions on the model to ensure that the model has a unique solution. You have 6 variables in your analysis, 1. LSNP, log of S&P 500 ? 2. CPI, a price index ? 3. IPI, Industrial production index ? 4. X, exchange rate foreign price of domestic currency or domestic currency price of foreign currency 5. M, M? supply ? 6. Yield 10 year bond yield ? I presume that the variables were entered in your program in that order. It is likely that the logs of items 2 to 5 are I(1). I suspect that item 6 might be stationary. Consider changing this to a bond price using some form of integration to make it I(1). You then have 6 I(1) variables and assuming your economic theory you can estimate your model using the VECM methodology. When there are two or more cointegrating relationships the estimates of the two cointegrating relationships are not identified in the sense that any

independent linear combinations of the estimated cointegrating relationships are also cointegrating relationships. Thus without restrictions imposed by economic reasoning or common sense, the system can not be identified. Software implementation of the Johansen VECM methodology must impose a set of identifying restrictions on the estimated ECMs when there 2 or more ecms. When there are two cointegrating relationships, the following restrictions are ususlly imposed. 1. In the first ecm the coeffcient on the first variable in the list of variables submitted to the routine is assumed to be 1 and that of the second is 0 2. In the second ecm the coefficient on the second is assumed to be 1 and on the first zero. This appears to be the case in your results. Your analysis gives the following two equilibrium structural equations

LSNP = 0.019IP I + .0061X + .10M + .21Y IELD (1)

CP I = −.12IP I − .42X + 9.3M + 13.6Y IELD (2)

Note that if you enter your variables in a different order you will get two different ecm's. If you wish to use this form of zero coefficients you must use economic arguments to decide which ordering of your variables is relevant. If you don't believe me try some more of the 720 possible orders in which you can enter your variables. They will give different answers. Identification is achieved by omitting variables in the equations. There are many other patterns of omitted variables, cross equation restrictions, etc. that will also provide identification. You need this information before you do your estimation. If, for example, the identification in your software is appropriate, (I think that it is not) the two estimated equations need to be interpreted. I would have preferred you to have used logs of IPI, X and M. As you have it configured a change of one unit in M has the same effect on the S&P regardless of the level of M. If you had used log M then the relationship would be an elasticity that would be better.

Your identification implies that the CPI has no effect on the long term value of the S&P. This I would doubt. I would also doubt the economics of your second variable as the equilibrium determinant of inflation. The first equation in your VECM system is

∆LSNPt = −0.04(0.019IP I − .0061X − .10M − .21Y IELD)t−1 + 0.0013(CP I + .12IP I + .42X − 9.3M − 13.6Y IELD)t−1 + lags + deterministic terms + u (3)

and similar equations for the other 5 variables. In the second paragraph of your question, I think that you are not reading your computer output correctly. The explanations above should clarify this point I think that the VECM proposed above and a reconsideration of the economics underlying your analysis should help you to specify a model and then estimate it. Your interpretation of the results of your econometric analysis should be made primarily in terms of the model you are estimating

John C Frain : Thank you very much for your comprehensive and very helpful reply to my questions. It definitely helped me a lot.

Best regards.

Before using econometric methods, one should always try to find an (economic!) specification of a model, i.e. think over how variables influence others. One should decide which functional relations to take and which transformations of the variables seem adequate. If you are sure (or economic theory says), that a percentage change is depandent on an abolute variable, it is ok. But I think that this is only seldom the case. Therefore, I am with John, that, in general, one should choose the same or "similar" transformations of the variables. Different from him, I do not like logs or dln, but I prefer percentage changes for time series. Before using methods like ARDL and ECM, I would look at diagrams and the correlations between pairs of variables.

Questions: How do you know, whether M3 is money supply or demand? What exchange rate to take? Which variables are endogenous or exogenous? etc. etc.