change_regime function

Change the parameters of a specific regime of the given parameter vector

change_regime changes the regime parameters $(\phi_{m,0},vec(A_{m,1}),...,\vec(A_{m,p}),vech(\Omega_m))$ (replace $vech(\Omega_m)$

by $vec(B_m)$ for cond_dist="ind_Student") of the given regime to the new given parameters.

change_regime(
  p,
  M,
  d,
  params,
  m,
  regime_pars,
  cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t")
)

Arguments

• p: the autoregressive order of the model

• M: the number of regimes

• d: the number of time series in the system, i.e., the dimension

• params: 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.
  • $\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p}))$ $(pd^2 \times 1)$.

  • • if cond_dist="Gaussian" or "Student":: $\sigma = (vech(\Omega_1),...,vech(\Omega_M))$

        $(Md(d + 1)/2 \times 1)$.
      

    • if 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).
  • • if cond_dist = "Gaussian"):: Omit $\nu$ from the parameter vector.
    • if cond_dist="Student":: $\nu \> 2$ is the single degrees of freedom parameter.
    • if cond_dist="ind_Student":: $\nu = (\nu_1,...,\nu_d)$ $(d \times 1)$, $\nu_i \> 2$.
    • if 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 $\mu_{m}$.

• m: which regime?

• regime_pars: - If cond_dist="Gaussian" or cond_dist="Student":: rhe $(dp + pd^2 + d(d+1)/2)$ vector $(\phi_{m,0},vec(A_{m,1}),...,\vec(A_{m,p}),vech(\Omega_m))$.

  • If cond_dist="ind_Student" or "ind_skewed_t":: the $(dp + pd^2 + d^2 + 1)$ vector $(\phi_{m,0},vec(A_{m,1}),...,\vec(A_{m,p}),vec(B_m))$.

Returns

Returns parameter vector with m:th regime changed to regime_pars.

Details

Does not support constrained models or structural models. Weight parameters and distribution parameters are not changed.

References

• Kheifets I.L., Saikkonen P.J. 2020. Stationarity and ergodicity of Vector STAR models. Econometric Reviews, 39 :4, 407-414.
• Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
• Lanne M., Virolainen S. 2025. A Gaussian smooth transition vector autoregressive model: An application to the macroeconomic effects of severe weather shocks. Unpublished working paper, available as arXiv:2403.14216.
• Virolainen S. 2025. Identification by non-Gaussianity in structural threshold and smooth transition vector autoregressive models. Unpublished working paper, available as arXiv:2404.19707.

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