# Test for “dynamic stability”

According to a few source codes I read, the "dynamically stability" of a VAR can be examined by calculating the roots. A necessary condition for stability of a VAR system is that all characteristic roots lie within the "unit circle".

Following example that you can reproduce:

library(vars)
data(Canada)

# Data
Canada[,2:3]

# Optimum lag length (lag.max = 4 because we have quarterly data)
VARselect(Canada[,2:3], lag.max = 4, type = "const")\$selection

# Test for residual serial correlation lag 1
serial.test(VAR(Canada[,2:3], p = 1, type = "const"), type="PT.adjusted")
# cannot reject null hypothesis (H0: no serial correlation)

# Test for residual serial correlation lag 2
serial.test(VAR(Canada[,2:3], p = 2, type = "const"), type="PT.adjusted")
# cannot reject null hypothesis (H0: no serial correlation)

# Test for residual serial correlation lag 3
serial.test(VAR(Canada[,2:3], p = 3, type = "const"), type="PT.adjusted")
# better lag selection

# Roots lag 1
roots(VAR(Canada[,2:3], p = 1, type = "const"))
# Roots lag 2
roots(VAR(Canada[,2:3], p = 2, type = "const"))
# Roots lag 3
roots(VAR(Canada[,2:3], p = 3, type = "const"))


Questions:

1. How would you explain the "unit circle" in a few words?
2. Am I correct that the roots measure the dynamic stability of the residuals of the VAR?
3. I observed that most authors using the roots to test the dynamic stability of a VAR usually only look that the first two root values: roots(...)[],roots(...)[] and that they are lower than Why only the first two values?
4. Is there another way to examine or visualize "dynamic stability"?
• 2. The roots measure the dynamic stability of the dependent variable (as modelled by VAR), not the residuals. – Richard Hardy Sep 4 '17 at 13:42
• And testing roots(...)[] and roots(...)[] is the way to go? – FenleyK Sep 5 '17 at 6:55
• If the roots are ordered as monotonically decreasing, the first is the largest. If it lies within the unit circle, so will the second one and all the others. If the roots are not ordered, all of them should be inspected. Btw, a unit circle is a circle of radius 1 centered at zero in the complex plane. – Richard Hardy Sep 5 '17 at 7:01

## 1 Answer

1. A unit circle is a circle of radius 1 centered at zero in the complex plane.
2. The roots measure the dynamic stability of the dependent variable (as modelled by VAR), not the residuals.
3. If the roots are ordered as monotonically decreasing in their absolute value (note that we are dealing with complex numbers here), the first is the largest. If it lies within the unit circle, so will the second one and all the others. Thus finding the first one to be inside the unit circle guarantees stability of the process. But if the roots are not ordered, all of them should be inspected.
4. Not that I know. Visualizing the roots on the complex plane an drawing a unit circle is quite convenient, IMHO.