pick_lambdas function

Pick the structural parameter eigenvalues 'lambdas'

pick_lambdas picks the structural parameters eigenvalue 'lambdas' from the parameter vector of a structural model identified by heteroskedasticity.

pick_lambdas(
  p,
  M,
  d,
  params,
  identification = c("reduced_form", "recursive", "heteroskedasticity",
    "non-Gaussianity")
)

Arguments

• p: a positive integer specifying the autoregressive order

• M: a positive integer specifying the number of regimes

• 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 thresholds.
    

  • weight_function="exogenous":: Omit $\alpha$ from the parameter vector.

  • AR_constraints:: Replace $\varphi_1,...,\varphi_M$ with $\psi$ as described in the argument AR_constraints.

  • mean_constraints:: Replace $\phi_{1,0},...,\phi_{M,0}$ with $(\mu_{1},...,\mu_{g})$ where $\mu_i, \ (d\times 1)$ is the mean parameter for group $i$ and $g$ is the number of groups.

  • weight_constraints:: If linear constraints are imposed, replace $\alpha$ with $\xi$ as described in the argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop $\alpha$

     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 $\Omega_m=W\Lambda_mW'$, $\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})$, $\lambda_{mi}>0$, $m=2,...,M$, $i=1,...,d$.

  • B_constraints:: For models identified by heteroskedasticity, replace $vec(W)$ with $\tilde{vec}(W)$

     that stacks the columns of the matrix $W$ in to vector so that the elements that are constrained to zero are not included. For models identified by non-Gaussianity, replace $vec(B_1),...,vec(B_M)$ with similarly with vectorized versions $B_m$ so that the elements that are constrained to zero are not included.
    

  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 $\mu_{m}$. $vec()$ is vectorization operator that stacks columns of a given matrix into a vector. $vech()$ stacks columns of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector. $Bvec()$

  is a vectorization operator that stacks the columns of a given impact matrix $B_m$ into a vector so that the elements that are constrained to zero by the argument B_constraints are excluded.

• identification: is it reduced form model or an identified structural model; if the latter, how is it identified (see the vignette or the references for details)?

  • "reduced_form":: Reduced form model.
  • "recursive":: The usual lower-triangular recursive identification of the shocks via their impact responses.
  • "heteroskedasticity":: Identification by conditional heteroskedasticity, which imposes constant relative impact responses for each shock.
  • "non-Gaussianity":: Identification by non-Gaussianity; requires mutually independent non-Gaussian shocks, thus, currently available only with the conditional distribution "ind_Student".

Returns

Returns the length (d*(M - 1)) vector $(\lambda_{2},...,\lambda_{M})$

(see the argument params) for structural models identified by heteroskedasticity, numeric(0) if $M=1$, and NULL for other models.

Details

Constrained parameter vectors are not supported. Not even constraints in $W$!

References

• Lütkepohl H., Netšunajev A. 2017. Structural vector autoregressions with smooth transition in variances. Journal of Economic Dynamics & Control, 84 , 43-57.