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Consider VAR(1) process $$y_t = A_1y_{t-1} + u_t.$$

I am looking for a reference or proof that if all entries in $A_1$ are less than one in modulus then $y_t$ is stable. I am sure that last semester a professor told that, but I can't find a reference.


Edit1: In response to @Jarle Tufto answer, I believe that this is a sufficient condition. To support my argument I did a simulation in R where I simulate 20 different series (100 steps ahead) for the case when entries in $A_1$ are uniformly distributed between $0$ and $1$. All series converge to zero.

enter image description here

The R code

n <- 20 #fix the number of series
I = diag(n)
A <- matrix(runif(n^2), ncol = n, nrow = n) #Create matrix A with 0 < a_ij < 1
y = matrix(0, nrow = n, ncol = 100) #create vector of predictions (100 steps ahead)
y[, 1] = rnorm(n) #set initial values
for (i in 2:ncol(y)) {
    y[, i] = A^(i-1)%*%y[, i-1]
}
require(ggplot2)
require(reshape2)
y_m <- melt(y)
ggplot() +
  geom_line(data = y_m, aes(x = X2, y = value, color = factor(X1))) + 
  xlab('Step') + ylab('Value')

Edit 2:

Here is the result for the case when all entries of $A$ are $0.9$. In the previous code I changed

A <- matrix(runif(n^2), ncol = n, nrow = n)

to

A <- matrix(0.9, ncol = n, nrow = n)

Still have convergence, though the shape looks strange.

enter image description here

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  • $\begingroup$ VAR is an acronym, so there is no need to format it as a formula. Good question, though. $\endgroup$ – Richard Hardy Oct 29 '17 at 14:09
  • $\begingroup$ That was just good (or maybe bad) luck. It is not a sufficient condition. Try $A$ with all entries equal to .9 and it will not be stationary! $\endgroup$ – Jarle Tufto Oct 29 '17 at 14:47
  • $\begingroup$ @JarleTufto please see my second edit. $\endgroup$ – tosik Oct 29 '17 at 14:56
  • $\begingroup$ I don't understand your simulation code. Inside your for-loop you do 'y[, i] = A^(i-1)%*%y[, i-1]'. Why do you raise all elements of A to the $(i-1)$ power? And you need to add the white noise term $u_t$. $\endgroup$ – Jarle Tufto Oct 29 '17 at 14:59
  • $\begingroup$ @JarleTufto oops, agree. Indeed, you are right, this is not a sufficient condition. Forgot that ^ should be replaced by %^% in R (with loading expm package). $\endgroup$ – tosik Oct 29 '17 at 15:29
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A VAR(1) process is stationary if the largest eigenvalue of $A$ has modulus smaller than 1, see Wei 2006, ch. 16. The condition you refer to is neither a necessary or sufficient condition for stationarity.

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I think your simulations are just luck. Take this example:

$$ \begin{bmatrix} 0.8 & 0.2 \\ 0.4 & 0.6 \end{bmatrix} $$

All entries are less than 1 but the matrix has 1 as an eigenvalue. So it is not stationary. Any starting point other than (0,0) that is less than one in the two entries will eventually converge to (2/3 , 1/3). And if they are higher than 1, it will diverge.

Take a look at the definition of markov matrices.

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