How can I compare degree of stability (or persistency) in two dynamic systems/models?

Let's say I have two dynamic VAR models for two countries. What is the best way to compare persistency or degree of stability in the two models. Use eigenvalues of the two systems?

• Can you provide more information? The structural form of the VAR would help provide context for the question – Brennan Jan 7 at 22:30
• Let say I have a bivariate VAR model for examining the effect of oil price shocks on GDP for two countries (Nigeria and Gabon). I want to compare the persistency of the oil shock for the two countries...or say the persistency of the response of GDP to oil shock for the two countries. – Emmanuel Ameyaw Jan 8 at 0:57
• ahh ok, im assuming you have done the necessary detrending, lag selection, and tests for stationarity. Would it not just be one VAR with e.g. the world price of oil with $GDP_N$ and $GDP_G$ as variables in the model? I would look at the impulse response functions from like a one SE positive shock on those two variables. R has a really nice package for plotting them. – Brennan Jan 8 at 1:08
• But how would you judge persistency from visually looking at IRFs? Does persistency have a clear definition after plotting IRFs? Maybe which impulse response returns to the zero line the fastest has the lowest persistency? Sometimes the impulse response returns to zero quickly and after that, it takes a long time to return to zero. Would you say the response has low or high persistency? – Emmanuel Ameyaw Jan 8 at 2:07
• Exactly. If the impact is short-lived, then it is less persistent relative to others. I would say that approach would only serve as a complement to @1muflon1's answer---the coefficient defines persistence. – Brennan Jan 8 at 5:21

$${\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,} \\ {\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}$$
Now, in very similar fashion as with AR models how persistent the variable is will be given by how much it depends on its own past (lags). For example, persistence of $$y_1$$ can be gauged by coefficient $$a_{1,1}$$. The larger the coefficient the more persistent the series is.
• Thanks. So if there is a shock in $e_{1,t}$, the persistency of the responses ($y_{1,t}$, $y_{2,t}$) depends on $a_{1,1}$ and $a_{2,2}$? How about $a_{1,2}$ and $a_{2,1}$? – Emmanuel Ameyaw Jan 8 at 2:17
• A shock in $e_{1,t}$ would show up in $y_{1,t}$, as shown above, and then in $y_{2,t+1}$ as this is a function of $y_{1,t}$. So yes, the coefficients define that relationship. Think about it in terms of the simple AR(1) model and how the shock persists and, if stationary, dies out over time. – Brennan Jan 8 at 5:24