stab_conds_satisfied checks whether the stability condition is satisfied
for each of the regimes.
stab_conds_satisfied(p, M, d, params, all_boldA = NULL, tolerance = 0.001)Returns TRUE if the stability condition is satisfied for all regimes and FALSE if not.
According to the argument tolerance, stab_conds_satisfied may return FALSE when the parameter
vector satisfies the stability conditions but is very close to the boundary (this is used to ensure numerical stability
in the estimation of the model parameters).
the autoregressive order of the model
the number of regimes
the number of time series in the system, i.e., the dimension
a real valued vector specifying the parameter values. Should have the form \(\theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)\), where (see exceptions below):
\(\phi_{m,0} = \) the \((d \times 1)\) intercept (or mean) vector of the \(m\)th regime.
\(\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p}))\) \((pd^2 \times 1)\).
cond_dist="Gaussian" or "Student":\(\sigma = (vech(\Omega_1),...,vech(\Omega_M))\) \((Md(d + 1)/2 \times 1)\).
cond_dist="ind_Student" or "ind_skewed_t":\(\sigma = (vec(B_1),...,vec(B_M)\) \((Md^2 \times 1)\).
\(\alpha = \) the \((a\times 1)\) vector containing the transition weight parameters (see below).
cond_dist = "Gaussian"):Omit \(\nu\) from the parameter vector.
cond_dist="Student":\(\nu > 2\) is the single degrees of freedom parameter.
cond_dist="ind_Student":\(\nu = (\nu_1,...,\nu_d)\) \((d \times 1)\), \(\nu_i > 2\).
cond_dist="ind_skewed_t":\(\nu = (\nu_1,...,\nu_d,\lambda_1,...,\lambda_d)\) \((2d \times 1)\), \(\nu_i > 2\) and \(\lambda_i \in (0, 1)\).
For models with...
weight_function="relative_dens":\(\alpha = (\alpha_1,...,\alpha_{M-1})\) \((M - 1 \times 1)\), where \(\alpha_m\) \((1\times 1), m=1,...,M-1\) are the transition weight parameters.
weight_function="logistic":\(\alpha = (c,\gamma)\) \((2 \times 1)\), where \(c\in\mathbb{R}\) is the location parameter and \(\gamma >0\) is the scale parameter.
weight_function="mlogit":\(\alpha = (\gamma_1,...,\gamma_M)\) \(((M-1)k\times 1)\), where \(\gamma_m\) \((k\times 1)\), \(m=1,...,M-1\) contains the multinomial logit-regression coefficients of the \(m\)th regime. Specifically, for switching variables with indices in \(I\subset\lbrace 1,...,d\rbrace\), and with \(\tilde{p}\in\lbrace 1,...,p\rbrace\) lags included, \(\gamma_m\) contains the coefficients for the vector \(z_{t-1} = (1,\tilde{z}_{\min\lbrace I\rbrace},...,\tilde{z}_{\max\lbrace I\rbrace})\), where \(\tilde{z}_{i} =(y_{it-1},...,y_{it-\tilde{p}})\), \(i\in I\). So \(k=1+|I|\tilde{p}\) where \(|I|\) denotes the number of elements in \(I\).
weight_function="exponential":\(\alpha = (c,\gamma)\) \((2 \times 1)\), where \(c\in\mathbb{R}\) is the location parameter and \(\gamma >0\) is the scale parameter.
weight_function="threshold":\(\alpha = (r_1,...,r_{M-1})\) \((M-1 \times 1)\), where \(r_1,...,r_{M-1}\) are the threshold values.
weight_function="exogenous":Omit \(\alpha\) from the parameter vector.
identification="heteroskedasticity":\(\sigma = (vec(W),\lambda_2,...,\lambda_M)\), where \(W\) \((d\times d)\) and \(\lambda_m\) \((d\times 1)\), \(m=2,...,M\), satisfy \(\Omega_1=WW'\) and \(\Omega_m=W\Lambda_mW'\), \(\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})\), \(\lambda_{mi}>0\), \(m=2,...,M\), \(i=1,...,d\).
Above, \(\phi_{m,0}\) is the intercept parameter, \(A_{m,i}\) denotes the \(i\)th coefficient matrix of the \(m\)th
regime, \(\Omega_{m}\) denotes the positive definite error term covariance matrix of the \(m\)th regime, and \(B_m\)
is the invertible \((d\times d)\) impact matrix of the \(m\)th regime. \(\nu_m\) is the degrees of freedom parameter
of the \(m\)th regime. If parametrization=="mean", just replace each \(\phi_{m,0}\) with regimewise mean
\(\mu_{m}\).
3D array containing the \(((dp)x(dp))\) "bold A" (companion form) matrices of each regime,
obtained from form_boldA. Will be computed if not given.
Returns FALSE if modulus of any eigenvalue of "bold A" is larger or equal to 1-tolerance.
No argument checks!
Does not support constrained parameter vectors.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.