smart_weightpars function

Create random transition weight parameter values

smart_weightpars generates random transition weight parameter values relatively close to the ones given in weight_pars

smart_weightpars(
  M,
  weight_pars,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weight_constraints = NULL,
  accuracy
)

Arguments

• M: a positive integer specifying the number of regimes

• weight_pars: a vector containing transition weight parameter values.

  • If weight_function == "relative_dens":: a length M-1 vector $(\alpha_1,...,\alpha_{M-1})$.

  • If weight_function == "logistic":: a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

  • If weight_function == "mlogit":: a length $((M-1)k\times 1)$ vector $(\gamma_1,...,\gamma_{M-1})$, where $\gamma_m$ $(k\times 1)$, $m=1,...,M-1$ contains the mlogit-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 $I$.
    

  • If weight_function == "exponential":: a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

  • If weight_function == "threshold":: a length $M-1$ vector $(r_1,...,r_{M-1})$, where $r_1,...,r_{M-1}$ are the threshold values in an increasing order.

  • If weight_function == "exogenous":: of length zero.

• weight_function: What type of transition weights $\alpha_{m,t}$ should be used?

  • "relative_dens":: c("$\\alpha_{m,t}=\n$", "$\\frac{\\alpha_mf_{m,dp}(y_{t-1},...,y_{t-p+1})}{\\sum_{n=1}^M\\alpha_nf_{n,dp}(y_{t-1},...,y_{t-p+1})}$"), where $\alpha_m\in (0,1)$ are weight parameters that satisfy $\sum_{m=1}^M\alpha_m=1$ and $f_{m,dp}(\cdot)$ is the $dp$-dimensional stationary density of the $m$th regime corresponding to $p$

     consecutive observations. Available for Gaussian conditional distribution only.
    

  • "logistic":: $M=2$, $\alpha_{1,t}=1-\alpha_{2,t}$, and $\alpha_{2,t}=[1+\exp\lbrace -\gamma(y_{it-j}-c) \rbrace]^{-1}$, where $y_{it-j}$ is the lag $j$

     observation of the $i$th variable, $c$ is a location parameter, and $\gamma > 0$ is a scale parameter.
    

  • "mlogit":: c("$\\alpha_{m,t}=\\frac{\\exp\\lbrace \\gamma_m'z_{t-1} \\rbrace}\n$", "${\\sum_{n=1}^M\\exp\\lbrace \\gamma_n'z_{t-1} \\rbrace}$"), where $\gamma_m$ are coefficient vectors, $\gamma_M=0$, and $z_{t-1}$ $(k\times 1)$ is the vector containing a constant and the (lagged) switching variables.

  • "exponential":: $M=2$, $\alpha_{1,t}=1-\alpha_{2,t}$, and $\alpha_{2,t}=1-\exp\lbrace -\gamma(y_{it-j}-c) \rbrace$, where $y_{it-j}$ is the lag $j$

     observation of the $i$th variable, $c$ is a location parameter, and $\gamma > 0$ is a scale parameter.
    

  • "threshold":: $\alpha_{m,t} = 1$ if $r_{m-1}<y_{it-j}\leq r_{m}$ and $0$ otherwise, where $-\infty\equiv r_0<r_1<\cdots <r_{M-1}<r_M\equiv\infty$ are thresholds $y_{it-j}$ is the lag $j$

     observation of the $i$th variable.
    

  • "exogenous":: Exogenous nonrandom transition weights, specify the weight series in weightfun_pars.

  See the vignette for more details about the weight functions.

• weight_constraints: a list of two elements, $R$ in the first element and $r$ in the second element, specifying linear constraints on the transition weight parameters $\alpha$. The constraints are of the form $\alpha = R\xi + r$, where $R$ is a known $(a\times l)$

  constraint matrix of full column rank ($a$ is the dimension of $\alpha$), $r$ is a known $(a\times 1)$ constant, and $\xi$ is an unknown $(l\times 1)$ parameter. Alternatively , set $R=0$ to constrain the weight parameters to the constant $r$ (in this case, $\alpha$ is dropped from the constrained parameter vector).

• accuracy: a positive real number adjusting how close to the given parameter vector the returned individual should be. Larger number means larger accuracy. Read the source code for details.

Returns

Returns a numeric vector ...

• If weight_function == "relative_dens":: a length M-1 vector $(\alpha_1,...,\alpha_{M-1})$.

• If weight_function == "logistic":: a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

• If weight_function == "mlogit":: a length $((M-1)k\times 1)$ vector $(\gamma_1,...,\gamma_{M-1})$, where $\gamma_m$ $(k\times 1)$, $m=1,...,M-1$ contains the mlogit-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 $I$.
  

• If weight_function == "exponential":: a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

• If weight_function == "threshold":: a length $M-1$ vector $(r_1,...,r_{M-1})$, where $r_1,...,r_{M-1}$ are the threshold values in an increasing order.

• If weight_function == "exogenous":: of length zero.