sstvars (version 1.1.6)

bound_jsr_G: Calculate upper bound for the joint spectral radius of a set of matrices

Description

bound_jsr_G calculates lower and upper bounds for the joint spectral radious of a set of square matrices, typically the "bold A" matrices, using the algorithm by Gripenberg (1996).

Usage

bound_jsr_G(
  S,
  epsilon = 0.01,
  adaptive_eps = FALSE,
  ncores = 2,
  print_progress = TRUE
)

Value

Returns an upper bound for the joint spectral radius of the "companion form AR matrices" of the regimes.

Arguments

S

the set of matrices the bounds should be calculated for in an array, in STVAR applications, all \(((dp)x(dp))\) "bold A" (companion form) matrices in a 3D array, so that [, , m] gives the matrix the regime m.

epsilon

a strictly positive real number that approximately defines the goal of length of the interval between the lower and upper bounds. A smaller epsilon value results in a narrower interval, thus providing better accuracy for the bounds, but at the cost of increased computational effort. Note that the bounds are always wider than epsilon and it is not obvious what epsilon should be chosen obtain bounds of specific tightness.

adaptive_eps

logical: if TRUE, starts with a large epsilon and then decreases it gradually whenever the progress of the algorithm requires, until the value given in the argument epsilon is reached. Usually speeds up the algorithm substantially but is an unconventional approach, and there is no guarantee that the algorithm converges appropriately towards bounds with the tightness given by the argument epsilon.

ncores

the number of cores to be used in parallel computing.

print_progress

logical: should the progress of the algorithm be printed?

Details

The upper and lower bounds are calculated using the Gripenberg's (1996) branch-and-bound method, which is also discussed in Chang and Blondel (2013). This function can be generally used for approximating the JSR of a set of square matrices, but the main intention is STVAR applications (for models created with sstvars, the function bound_JSR should be preferred). Specifically, Kheifets and Saikkonen (2020) show that if the joint spectral radius of the companion form AR matrices of the regimes is smaller than one, the STVAR process is ergodic stationary. Virolainen (2025) shows the same result for his parametrization of of threshold and smooth transition vector autoregressive models. Therefore, if the upper bound is smaller than one, the process is stationary ergodic. However, as the condition is not necessary but sufficient and also because the bound might be too conservative, upper bound larger than one does not imply that the process is not ergodic stationary. You can try higher accuracy, and if the bound is still larger than one, the result does not tell whether the process is ergodic stationary or not.

Note that with high precision (small epsilon), the computational effort required are substantial and the estimation may take long, even though the function takes use of parallel computing. This is because with small epsilon the the number of candidate solutions in each iteration may grow exponentially and a large number of iterations may be required. For this reason, adaptive_eps=TRUE can be considered for large matrices, in which case the algorithm starts with a large epsilon, and then decreases it when new candidate solutions are not found, until the epsilon given by the argument epsilon is reached.

References

• C-T Chang and V.D. Blondel. 2013 . An experimental study of approximation algorithms for the joint spectral radius. Numerical algorithms, 64, 181-202.

• Gripenberg, G. 1996. Computing the joint spectral radius. Linear Algebra and its Applications, 234, 43–60.

• I.L. Kheifets, P.J. Saikkonen. 2020. Stationarity and ergodicity of Vector STAR models. Econometric Reviews, 39:4, 407-414.

• Virolainen S. 2025. Identification by non-Gaussianity in structural threshold and smooth transition vector autoregressive models. Unpublished working paper, available in ArXiv.

See Also

bound_JSR

Examples

# \donttest{
## Below examples take approximately 5 seconds to run.

# A set of two (5x5) square matrices:
set.seed(1); S1 <- array(rnorm(20*20*2), dim=c(5, 5, 2))

# Bound the joint spectral radius of the set of matrices S1, with the
# approximate tightness epsilon=0.01:
bound_jsr_G(S1, epsilon=0.01, adaptive_eps=FALSE)

# Obtain bounds faster with adaptive_eps=TRUE:
bound_jsr_G(S1, epsilon=0.01, adaptive_eps=TRUE)
# Note that the upper bound is not the same as with adaptive_eps=FALSE.

# A set of three (3x3) square matrices:
set.seed(2); S2 <- array(rnorm(3*3*3), dim=c(3, 3, 3))

# Bound the joint spectral radius of the set of matrices S2:
bound_jsr_G(S2, epsilon=0.01, adaptive_eps=FALSE)

# Larger epsilon terminates the iteration earlier and results in wider bounds:
bound_jsr_G(S2, epsilon=0.05, adaptive_eps=FALSE)

# A set of eight (2x2) square matrices:
set.seed(3); S3 <- array(rnorm(2*2*8), dim=c(2, 2, 8))

# Bound the joint spectral radius of the set of matrices S3:
bound_jsr_G(S3, epsilon=0.01, adaptive_eps=FALSE)
# }