The VAR and the VEC representation are simultaneous and do not by definition involve exogenous variables except as a result of some reduction of the system. Long-run equations computed by regression give rise to parameter estimates that are super consistent when the series are non-stationary (For a discussion of this see the book by Davidson and MacKinnon (2004), this is referred to Burke and Hunter (2005), Non-Stationary Economic Time Series, Palgrave). Please notice that the VAR can be estimated equation by equation by OLS regression and that these estimations of the short-run parameters are consistent when the dynamic is correctly identified.

Endogeneity is understood in a long-run and a short-run context. Endogeneity is also broken down into weak, strict, strong and super. In chapter 5 of Burke and Hunter (2005) there is a well developed introduction to these concepts especially in a long-run context. The book by Ericsson and Irons (1994), Exogeneity, OUP draws together some very good articles and excellent editorial chapter and a broad range of examples.

Endogeneity is easiest to test in the long-run. In particular, the long-run can be conditioned on variables that are weakly exogenous. There are a maximum of n-r weakly exogenous variables in an n variable system with r long-run relations. The long-run relations do not involve any error correction terms and the long-run can be explained by these variables. Granger non-causality in the VECM requires an absence of long-run relations in addition to dynamics for the forced variable in the forcing variable equations. This gives rise to dynamic equations that are block triangular so the endogenous variables are caused by everything and endogenous variable coefficients are all zero in the exogenous variable equations.

In the long-run the concept that leads to a block triangular matrix of long-run parameters (pi) is termed cointegrating exogeneity (Hunter (1992) is reprinted in Ericsson and Irons (1994) and the link on my RG page provides details of where this may be downloaded). CE is a pre-requirement for granger causality non-causality in a short-run sense.

VARs and VECMs can be used to define complete systems and so do not require limited information methods. It is possible to reduce these systems to compute and investigate both long-run and short-run behaviour. This may have advantages, but this requires a number of weakly exogenous variables either in a long-run or short-run sense. This is likely to obviate the need to estimate using limited information methods such as GMM (though GMM can also be applied to systems). OLS is used equation by equation on unrestricted VARs, but reduced blocks of equations with current variables require weak exogeneity for estimation to be efficient and consistent by OLS.

If the parameters of interest purely relate to the long-run, then OLS is generally reliable, though to appropriately compute the long-run for the reduced system weak exogeneity may be required.

The contribution by John Hunter is certainly very competent, although a bit detailed. I have seen so many bad applications of VECM that I would like to add some short hints.

The first recommendation is to discard inappropriate language: in a VAR/VECM, there are no "independent" variables, there are variables, even if some long-run exogeneity holds.

The second recommendation is to see the long-run relationship, the cointegrating vector, as static. As a static equilibrium relation, it does not express causality: a one-one relation between two interest rates per se can be viewed as the long rate depending on the short rate or the other way round, even if it was estimated by a regression method, which I would not recommend anyway.

In the VECM, causality is expressed by dynamics. Variables adjust to deviations from the equilibrium, and the variable that bears the main burden of this adjustment is a slave, while the one that does not adjust at all (zero loading coefficient) is the master, the long-run exogenous variable. All remaining short-run dynamics are rather about the inertia of motion. Thus, if the first difference of the short rate has a significant coefficient on the error-correction term (long minus short rate lagged one time period), and the long rate has an insignificant one, then the long rate is long-run exogenous, as some economists think it should be.

Finally, if you are using the Johansen method, the option of my personal preference, the software will generate the entire model estimates in one go, jointly with the test statistics, whether Stata or EViews or R. Do not re-estimate by regressing on error-correction variables "by hand". The outcome is an inefficient estimate, and I have rejected several papers where this was going on. Avoid direct regression of the "EC-two-step" type, this was OK in 1990, but it is outdated now.

In VAR modeling, all variables are dependent and independent. This is a "black box". In most cases, the coefficients are difficult to interpret, especially when the lags are large. The coefficients can be interpreted as short-term (causality) or long term (cointegration) relationships.

Although I see merit in Robert's answer, it is still important to understand that the short-run Granger causality result with I(1) series relies on the sub division of the cointegrating relations.

In the case with a single cointegrating relation, then short-run causality relies on the long-run exogenous variable/variables being weakly exogenous.

With more than r=1 cointegrating relations, then the pi matrix being triangular is necessary and sufficient for cointegrating exogeneity a long-run analogue of strong exogeneity (Hunter(1992) - a link to a google books download can be found on my RG pages). This requires more than a triangular matrix of long-run parameters in the Johansen formulation the matrix beta'.

In relation to Jamal, it is possible to undertake a more sophisticated interpretation of long-run relations. However, it is important to understand that any causal analysis is dependent on the system within which it is considered.

Subject to a particular set of variables, the reduced rank condition facilitates identification as it reduces the problem to more manageable size. For example, when r=1 there is a single long-run relation that is almost trivially identified (in a generic sense) to a normalisation. However, care must be taken that the problem is not normalized on a variable that is long-run excluded. This is problematic as once the vector is orientated to a normalisation the sensitivity of the analysis to the selection of the normalised variable is not directly testable. However, inflation of the other coefficients in the vector is a sign of an inappropriate normalisation (this is effectively division by zero).

I would further suggest testing for weak exogeneity as normalisation on a variable on which a vector can be validly conditioned also makes little sense (the article by Hunter and Menla Ali (2014) does this in relation to an r-1 case - an earlier version is available to download on RG). The finding of long-run exogenous variables determines how the system might be orientated. The effective dimension of cointegrating vector is reduced by finding long-run excluded variables the same result arises in conventional systems identification as a common restriction across all equations in a long-run or short-run relation gives rise to identifying restrictions that break the rank condition. The long-run relations can then be identified by the normalisation rule and this is unique when there are exactly n-r weakly exogenous variables. Irrespective of the n-r case the conditioning is still valid, but there will be a number of alternative ways to represent the long-run relations related to alternative normalisations.

Typically Engle-Granger type test and estimation are to be avoided unless the dataset is limited. It is also important to realise that the example in Engle and Granger is very particular as the model is bivariate and income is weakly exogenous, this makes no difference when testing cointegration using a Dickey-Fuller test on the residual from a cointegrating regression.

Further, the theoretical result that permits one to move from the Wold form to the VECM only follows when there is a single common stochastic trend as the difference on the left hand side of the Wold form cancels with the single unit root extracted from vector moving average polynomial. That lead Engle and Granger to make the simplifying assumption that in their notation d(L)=1 for more complicated cases. As can be observed from Johansen (1995) and the results in Burke and Hunter (2005) Chapter 4, the Engle and Granger (1987) result does not generalise in the way they suggest.

Interesting discussion.

please refer this paper, may be helpful for you....

Study on the relationship among Chinese unemployment rate, economic growth and inflation

by

Changshuai Li, Zi-juan Liu

You can refer to paper :

THE INTERPRETATION OF COEFFICIENTS OF THE VECTOR

AUTOREGRESSIVE MODEL  by Elcyon Caiado Rocha Lima

link

http://www.sbe.org.br/dated/ebe25/092.pdf

why is the long run estimates from Johansen VECM dole  multiplied by +1.

Very interesting discussions. The coefficients or the result of the VAR model can equally be treated as causal relationship in the short period, and co-integration relationship in the long period. However, all depends on how the analysis is carried out and the objective of the research as explained by Robert above.

VAR/VECM, there are no "independent" variables, there are variables, even if some long-run exogeneity holds.

The second recommendation is to see the long-run relationship, the cointegrating vector, as static. As a static equilibrium relation, it does not express causality: a one-one relation between two interest rates per se can be viewed as the long rate depending on the short rate or the other way round, even if it was estimated by a regression method, which I would not recommend anyway.

In the VECM, causality is expressed by dynamics. Variables adjust to deviations from the equilibrium, and the variable that bears the main burden of this adjustment is a slave, while the one that does not adjust at all (zero loading coefficient) is the master, the long-run exogenous variable. All remaining short-run dynamics are rather about the inertia of motion. Thus, if the first difference of the short rate has a significant coefficient on the error-correction term (long minus short rate lagged one time period), and the long rate has an insignificant one, then the long rate is long-run exogenous, as some economists think it should be.

Finally, if you are using the Johansen method, the option of my personal preference, the software will generate the entire model estimates in one go, jointly with the test statistics, whether Stata or EViews or R. Do not re-estimate by regressing on error-correction variables "by hand". The outcome is an inefficient estimate, and I have rejected several papers where this was going on. Avoid direct regressio

How do you interpret VEC and VAR models coefficients? - ResearchGate. Available from: https://www.researchgate.net/post/How_do_you_interpret_VEC_and_VAR_models_coefficients [accessed Nov 20, 2016].

Can one use empirical examples to show me the difference between ECM and VECM?

1. Estimation and

2. interpretation

The interpretation of the estimated coefficients of the VAR or VECM model is actually done in terms of the influence on nature (positive or negative effect) dynamic (short term and long term) between the endogenous variables taken together and especially with their own. historical information that we finally manage to identify through the student's significance test. Not to mention also the examination of the signs obtained from the coefcients estimated by the model with those already known in the literature. The most important and ample information to emerge in these vector studies is of course the residual analysis of the model. In this sense, it is enough to understand the decomposition of the variance in order to know more about the contribution of the variance of the error for each variable selected or to determine those which participate more in the determination of another variable and on the other hand, impulse analysis to identify the short- and long-term response of one variable following an economic and involuntary shock on another variable. In general, it is a question of studying magnitudes in a dynamically double perspective and to trace the effects between them over time. We also have in this perspective, the test of causality in the Granger sense (and in the Sims sense, instantaneous causality). To say more, it is only an instrument of precision for the predictable choice of each variable. But before that, it is unmistakable to test the stationarity of the variables of the model. Once again, the stationarity of the variables and that of the model must be distinguished by means of its residual. The first suggests an order of integration greater than an I (1) for all variables in order to estimate a VECM or ECM. As reported in Granger's (1983) theorem, the link between cointegration and error correction model. The order of integration or the rank of cointegration suggests the possibility of estimating these types of models. It is supposed to be smaller than the order of each of the variables and greater than 0. In addition, the two-variable and small-sample case, the use of the two-step method Engle and Granger (1987) leads to biases. estimation (Banerjee, Dolado, Hendry and Smith (1986)). To remedy this, it is enough to solicit that one step Hendry. By having the dynamic of short terms and that of long terms at the same time, a sort of autoregressive and phased delay model (ARDL). In order to consider these models, the significance of the restoring force coefficient or the catching up of the long-term equilibrium is needed. Although we admit the divergence of evolution in the short term. Johensen (1988) proposes estimation with the likelihood maximulm method. Indeed, the method of ordinary least squares is not consistent when we have several vectors of wedgegrations. The Trace test determines for itself, the number of possible integration, it is nothing more than the rank of the matrix of wedgegration. Nowadays, we also distinguish several categories of stationarity and cointegration, especially with the presence of a constant and trendy term, without constant and tendency or only with constant. The distinction in type is appreciable.

I can summarize by stressing above all stationarity, then the optimal order of shifts for the variables by the Schwartz (SC) and Akaike (AIC) criteria to finally examine the meaning of the causality before approaching the VAR model. . In the case of VECM, it is a question of verifying the possibility of cointegrating the variables and of controlling, on the other hand, the stationarity of the residues of the model to be able to reassure itself on its reliability. In general, the aim is to study the interdependence between variables in a dynamic plane through the decomposition of the variable and the impulse response function in a framework of innovative shocks.

PS: I forgot to mention also the exogeneity weak or strong according to the correlation between the exogenous variables and the error term of the estimated model. This is also a step that should not be overlooked in the process of validating the estimators obtained by the vector models.

Indeed, dear friend Robert Kunst , the research continues, many things are up to date, but it is very regrettable that the dissemination of information and its availability is an obstacle to this day. But also to say that with the new advanced techniques, we always recognize a defect, a limitation, a disadvantage, which is always present in each of the econometric or statistical methods. This lets us think of inventing a future an imaginary model as the complex number to remedy this. More seriously, I later consider a more successful econometric model than those already known today.

Coefficients of VECM and VAR are not for interpretation. What is informative and interpretable in the system of equations are the coefficients of the error correction terms for VECM, and Granger causality tests, impulse response function, variance and historical decomposition for both VECM and VAR.

In VEC and VAR models, coefficients represent the short-term and long-term relationships between variables. Short-term coefficients in VAR models show the immediate impact of one variable on another, while in VEC models, they indicate the speed of adjustment towards the long-term equilibrium. Long-term coefficients in both models indicate the equilibrium relationship between variables.