Marginal vs. conditional models of vector autoregression (VAR) I have a vector autoregression, VAR(1). All random variables are weakly stationary and our white noises are all iid:
$$\begin{pmatrix}X_{1,t}\\X_{2,t}\\\vdots\\X_{k,t}\end{pmatrix}=\begin{pmatrix}\alpha_{11}X_{1,t-1}+...+\alpha_{1,k}X_{k,t-1}\\\alpha_{21}X_{1,t-1}+...+\alpha_{2,k}X_{k,t-1}\\\vdots\\\alpha_{k,1}X_{1,t-1}+...+\alpha_{k,k}X_{k,t-1}\end{pmatrix}+\begin{pmatrix}\varepsilon_{1,t}\\\varepsilon_{2,t}\\\vdots\\\varepsilon_{k,t}\end{pmatrix}$$
It follows that
$$X_{1,t}=\alpha_{11}X_{1,t-1}+...+\alpha_{1,k}X_{k,t-1}+\varepsilon_{1,t}$$
$$X_{2,t}=\alpha_{21}X_{1,t-1}+...+\alpha_{2,k}X_{k,t-1}+\varepsilon_{2,t}$$
$$\vdots$$
$$X_{k,t}=\alpha_{k1}X_{1,t-1}+...+\alpha_{kk}X_{k,t-1}+\varepsilon_{k,t}$$
Questions:


*

*I am wondering if my equations represent the marginal models of each of my random variables? I was thinking the equations represent conditional models. 

*If indeed they are conditional models, how could I go about obtaining marginal models?

 A: One particular equation of a VAR model targets the conditional mean (and the conditional distribution, if you are taking on distributional assumptions on model errors) of a univariate time series $X_{i,t}$ for $i \in 1,\dotsc,k$ and the conditioning is on past values of all the $k$ time series. 
In the context of the $k$-variate random process, this is also a marginal model because it targets only one time series out of $k$. 
So it is both a conditional and a marginal model - from certain perspectives. 
(You could always pick some other variable or a different lag that is not in the model, then with respect to that particular variable/lag the model would not be conditional.)
A: like any of these problems you need to first calculate the moving average representation; this will give you the steady state joint distribution, then you integrate out  k-1 variables to get the marginal distribution,however, if the errors are multivariate normal(MVN) then the joint will be MVN(some regularity conditions apply) and hence you only need to know the mean and variance to specify the marginal distribution these also come off the MA representation
