get_alpha_mt: Get the transition weights alpha_mt

Description

get_alpha_mt computes the transition weights.

Usage

get_alpha_mt(
  data,
  Y2,
  p,
  M,
  d,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  all_A,
  all_boldA,
  all_Omegas,
  weightpars,
  all_mu,
  epsilon,
  log_mvdvalues = NULL
)

Value

Returns the mixing weights a $(T x M)$ matrix, so that the $t$th row is for the time period $t$

and $m$:th column is for the regime $m$.

Arguments

data

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. Missing values are not supported.

Y2

the data arranged as obtained from reform_data(data, p) but excluding the last row

p

a positive integer specifying the autoregressive order

M

a positive integer specifying the number of regimes

d

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

weight_function

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

"relative_dens":: $\alpha_{m,t}= \frac{\alpha_mf_{m,dp}(y_{t-1},...,y_{t-p+1})}{\sum_{n=1}^M\alpha_nf_{n,dp}(y_{t-1},...,y_{t-p+1})}$, 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":

$\alpha_{m,t}=\frac{\exp\lbrace \gamma_m'z_{t-1} \rbrace} {\sum_{n=1}^M\exp\lbrace \gamma_n'z_{t-1} \rbrace}$, 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.

weightfun_pars

If weight_function == "relative_dens":: Not used.

If weight_function %in% c("logistic", "exponential", "threshold"):

a numeric vector with the switching variable $i\in\lbrace 1,...,d \rbrace$ in the first and the lag $j\in\lbrace 1,...,p \rbrace$ in the second element.

If weight_function == "mlogit":

a list of two elements:

The first element $vars:: a numeric vector containing the variables that should used as switching variables in the weight function in an increasing order, i.e., a vector with unique elements in $\lbrace 1,...,d \rbrace$.

The second element $lags:

an integer in $\lbrace 1,...,p \rbrace$ specifying the number of lags to be used in the weight function.

If weight_function == "exogenous":

a size (nrow(data) - p x M) matrix containing the exogenous transition weights as [t, m] for time $t$ and regime $m$. Each row needs to sum to one and only weakly positive values are allowed.

all_A

4D array containing all coefficient matrices $A_{m,i}$, obtained from pick_allA.

all_boldA

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.

all_Omegas

A 3D array containing the covariance matrix parameters obtain from pick_Omegas...

If cond_dist %in% c("Gaussian", "Student"):: all covariance matrices $\Omega_{m}$ in [, , m].

If cond_dist=="ind_Student":

all impact matrices $B_m$ of the regimes in [, , m].

weightpars

numerical vector containing the transition weight parameters, obtained from pick_weightpars.

all_mu

an $(d \times M)$ matrix containing the unconditional regime-specific means

epsilon

the smallest number such that its exponent is wont classified as numerically zero (around -698 is used).

log_mvdvalues

a $T x M$ matrix containing log multivariate normal densities (can be used with relative dens weight function only)

Details

Note that we index the time series as $-p+1,...,0,1,...,T$.

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.

@keywords internal