What’s a stationary VAR?

  • What is a stationary VAR (vector autoregression)?
  • Can a VAR with non-stationary variables be stationary?
  • How do you test whether a VAR is stationary or non-stationary? (Example in R language if possible/applicable).

Answer

  1. VAR is actualy an equation. We say that process Xt is VAR when it satisfies the following equation:

Xt=α+Φ1Xt1+...+ΦpXtp+εt,

where Φi are matrices and εt is white noise process. If the process satistfying this equation is stationary we say that the VAR is stationary. Given matrices Φi you can test whether the solution is stationary or not. If the roots of the following equation are in modulo greater than 1, then the solution is stationary:

|IλΦ1...λpΦp|=0,

where |A| is the determinant of matrix A.

  1. Stationarity (both in weak and strong sense) is a property of a process. Whether it is a vector valued or scalar valued. For example the process is called stationary in weak sense, if it satisfies two conditions: EXt=c and Cov(Xt,Xs)=r(ts), where c is a constant and r is apropriate function. The definition is the same for vector and scalar processes. For vector processes it immediately implies that each individual element of the vector is stationary. Now if one of the elements is not stationary, then it is also immediately clear, that whole vector cannot be stationary too. The same reasoning applies for stationarity in strong sense. So the answer to the question would be no. You cannot get a stationary solution for VAR equation if one of the elements is not stationary.

  2. It is usual to test non-stationarity, or to be more precise unit-root non-stationarity for individual variables and then estimate VAR. Estimation assumes that you have either stationarity or cointegration. If the process is non-stationary and not cointegrated the estimation is not possible (it is possible to argue differently, but it is safe to assume that this holds for all the usual cases).

Attribution
Source : Link , Question Author : Jase , Answer Author : mpiktas

Leave a Comment