Stability in Threshold VAR Models
-
Pu Chen
and
Willi Semmler
Abstract
This paper investigates the stability of threshold autoregressive models. We review recent research on stability issues from both a theoretical and empirical standpoint. We provide a sufficient condition for the stationarity and ergodicity of threshold autoregressive models by applying the concept of joint spectral radius to the switching system. The joint spectral radius criterion offers a generally applicable criterion to determine the stability in a threshold autoregressive model.
Acknowledgment
We want to acknowledge comments on a previous version of the paper, presented at the 6TH international FMND Workshop, organized in Paris, June 2022.
R codes for the calculation of upper and lower bounds of a joint spectra radius using Julia.
getwd()
library(JuliaCall)
julia_setup(installJulia = TRUE)
## https://blegat.github.io/SwitchOnSafety.jl/latest/generated/AJPR14e54/
julia_command(“using SwitchOnSafety”)
julia_command(“A1 = [−1 −1 ; −4 0]”)
julia_command(“A2 = [ 3 3 ; −2 1]”)
julia_command(“s = discreteswitchedsystem([A1, A2])”)
julia_command(“import CSDP”)
julia_command(“optimizer_constructor = optimizer_with_attributes(CSDP.Optimizer,
MOI.Silent() => true);”)
julia_command(“lb, ub = soslyapb(s, 1, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
## SNDE codes
##
## Hansen Example 1.1 US GNP data
julia_command(“A1 = [0.51 −0.93 0 0 −0.38 ; 1 0 0 0 0 ; 0 1 0 0 0 ; 0 0 1 0 0; 0 0
0 1 0]”)
julia_command(“A2 = [0.30 0.18 0 0 −0.16 ; 1 0 0 0 0 ; 0 1 0 0 0 ; 0 0 1 0 0; 0 0
0 1 0]”)
julia_command(“s = discreteswitchedsystem([A1, A2])”)
julia_command(“import CSDP”)
julia_command(“optimizer_constructor = optimizer_with_attributes(CSDP.Optimizer,
MOI.Silent() => true);”)
julia_command(“lb, ub = soslyapb(s, 1, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
julia_command(“seq = sosbuildsequence(s, 1, p_0=:Primal)”)
julia_command(“psw = findsmp(seq)”)
julia_command(“lb, ub = soslyapb(s, 2, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
## (0.8188989463625622, 0.9738413822096623)
## Hansen Exampl does not meet the condition of Brockwell, Liu, and Tweedie (1992)
G = abs(c(0.51, −0.93, 0, 0, −0.38 ))
dim(G) = c(1,1,5)
STAT(G)
G = abs(c(0.3, 0.18, 0, 0, −0.16 ))
dim(G) = c(1,1,5)
STAT(G)
sum(G)
## Tsay89 Exampl 2.1 attic temperature data
julia_command(“A1 = [2.51 −1.86 0.02 0.18 0.06 0.17 −0.17 ; 1 0 0 0 0 0 0 ;
0 1 0 0 0 0 0 ; 0 0 1 0 0 0 0 ; 0 0 0 1 0 0 0 ; 0 0 0 0 1 0 0 ; 0 0 0 0 0 1 0 ;]”)
julia_command(“A2 = [1.50 -0.68 0.11 0.43 -0.84 0.34 0 ; 1 0 0 0 0 0 0 ;
0 1 0 0 0 0 0 ; 0 0 1 0 0 0 0 ; 0 0 0 1 0 0 0 ; 0 0 0 0 1 0 0 ; 0 0 0 0 0 1 0 ;]”)
julia_command(“s = discreteswitchedsystem([A1, A2])”)
julia_command(“import CSDP”)
julia_command(“optimizer_constructor = optimizer_with_attributes(CSDP.Optimizer,
MOI.Silent() => true);”)
julia_command(“lb, ub = soslyapb(s, 1, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
julia_command(“seq = sosbuildsequence(s, 1, p_0=:Primal)”)
julia_command(“psw = findsmp(seq)”)
julia_command(“lb, ub = soslyapb(s, 2, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
## Chen21 Exampl 2.2 Canadian lxyn data
julia_command(“A1 = [−0.4 1.02 0.29 −0.593 ; 1 0 0 0 ; 0 1 0 0 ; 0 0 1 0;]”)
julia_command(“A2 = [ 1.5 −0.979 0 0 ; 1 0 0 0 ; 0 1 0 0 ; 0 0 1 0;]”)
julia_command(“s = discreteswitchedsystem([A1, A2])”)
julia_command(“import CSDP”)
julia_command(“optimizer_constructor = optimizer_with_attributes(CSDP.Optimizer,
MOI.Silent() => true);”)
julia_command(“lb, ub = soslyapb(s, 1, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
julia_command(“seq = sosbuildsequence(s, 1, p_0=:Primal)”)
julia_command(“psw = findsmp(seq)”)
julia_command(“lb, ub = soslyapb(s, 2, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
### Amendola09
julia_command(“A1 = [1.5 −0.51 ; 1 0]”)
julia_command(“A2 = [ 1 −0.5 ; 1 0]”)
julia_command(“s = discreteswitchedsystem([A1, A2])”)
julia_command(“import CSDP”)
julia_command(“optimizer_constructor = optimizer_with_attributes(CSDP.Optimizer,
MOI.Silent() => true);”)
julia_command(“lb, ub = soslyapb(s, 1, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
julia_command(“seq = sosbuildsequence(s, 1, p_0=:Primal)”)
julia_command(“psw = findsmp(seq)”)
julia_command(“lb, ub = soslyapb(s, 2, optimizer_constructor=optimizer_constructor,
tol=1e-6, verbose=1)”)
A1 = matrix(c(1.5, 1, −0.51, 0),2,2)
A2 = matrix(c(1, 1, −0.5, 0),2,2)
eigen(abs(A1))
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Articles in the same Issue
- Frontmatter
- Research Articles
- Welfare cost of inflation, when credit card transaction services are included among monetary services
- Co-Jumping of Treasury Yield Curve Rates
- Have European natural gas prices decoupled from crude oil prices? Evidence from TVP-VAR analysis
- Stability in Threshold VAR Models
- Examining the Impact of Energy Policies on CO2 Emissions with Information and Communication Technologies and Renewable Energy
- Interfuel Substitution and Inflation Dynamics in India
Articles in the same Issue
- Frontmatter
- Research Articles
- Welfare cost of inflation, when credit card transaction services are included among monetary services
- Co-Jumping of Treasury Yield Curve Rates
- Have European natural gas prices decoupled from crude oil prices? Evidence from TVP-VAR analysis
- Stability in Threshold VAR Models
- Examining the Impact of Energy Policies on CO2 Emissions with Information and Communication Technologies and Renewable Energy
- Interfuel Substitution and Inflation Dynamics in India
