<p>The most straightforward way is to examine the coefficients of your VAR model, since the persistence is broadly understood in econometrics as a system dependence on its own past (see Verbeek (2008): a guide to modern econometrics).  A simple VAR model is given:</p>
<p><span class="math-container">$${\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,} \\
{\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}$$</span></p>
<p>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 <span class="math-container">$y_1$</span> can be gauged by coefficient <span class="math-container">$a_{1,1}$</span>. The larger the coefficient the more persistent the series is.</p>

CC BY-SA 4.0
2021-01-08T02:17:23.557Z
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
CC BY-SA 4.0
2021-01-08T05:24:07.727Z
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
CC BY-SA 4.0
2021-01-08T05:51:28.483Z
Oh ok. All the coefficients matter then. Thanks!!
Emmanuel Ameyaw
CC BY-SA 4.0
2021-01-08T16:40:54.083Z
@EmmanuelAmeyaw persistence by definition is the relation of present realization of values to their past realization so they are governed by those coefficients.
1muflon1
CC BY-SA 4.0
2021-01-08T01:20:10.327Z
2021-01-08T01:20:10.327Z
1muflon1
<p>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?</p>

CC BY-SA 4.0
2021-01-07T22:30:22.423Z
Can you provide more information? The structural form of the VAR would help provide context for the question
Brennan
CC BY-SA 4.0
2021-01-08T00:57:39.533Z
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
CC BY-SA 4.0
2021-01-08T01:08:20.253Z
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
CC BY-SA 4.0
2021-01-08T02:07:43.060Z
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
CC BY-SA 4.0
2021-01-08T05:21:12.507Z
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
<macroeconomics><applied-econometrics><eigenvalue>
How can I compare degree of stability (or persistency) in two dynamic systems/models?
Emmanuel Ameyaw
# How can I compare degree of stability (or persistency) in two dynamic systems/models?

(Asked by: username_0 on 2021-01-07)

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?

**Tags:** macroeconomics, applied-econometrics, eigenvalue

> **Comments:**
> **username_1**: Can you provide more information? The structural form of the VAR would help provide context for the question  
> **username_0**: 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.  
> **username_1**: 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.  
> **username_0**: 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?  
> **username_1**: 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 @username_2's answer---the coefficient defines persistence.  

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## 1 answers

### Answer by username_2 on 2021-01-08. Score: 1

The most straightforward way is to examine the coefficients of your VAR model, since the persistence is broadly understood in econometrics as a system dependence on its own past (see Verbeek (2008): a guide to modern econometrics). A simple VAR model is given:

$${\displaystyle y\_{1,t}=c\_{1}+a\_{1,1}y\_{1,t-1}+a\_{1,2}y\_{2,t-1}+e\_{1,t}\,} \\
{\displaystyle y\_{2,t}=c\_{2}+a\_{2,1}y\_{1,t-1}+a\_{2,2}y\_{2,t-1}+e\_{2,t}.\,}$$

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.

> **Comments:**
> **username_0**: 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 $a_{2,1}$?  
> **username_1**: 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.  
> **username_0**: Oh ok. All the coefficients matter then. Thanks!!  
> **username_2**: @username_0 persistence by definition is the relation of present realization of values to their past realization so they are governed by those coefficients.  

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