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 $α_{m, t}$ should be used?

"relative_dens":: $α_{m, t} = \frac{α_{m} f_{m, d p} (y_{t - 1}, . . ., y_{t - p + 1})}{\sum_{n = 1}^{M} α_{n} f_{n, d p} (y_{t - 1}, . . ., y_{t - p + 1})}$ , where $α_{m} \in (0, 1)$ are weight parameters that satisfy $\sum_{m = 1}^{M} α_{m} = 1$ and $f_{m, d p} (\cdot)$ is the $d p$ -dimensional stationary density of the $m$ th regime corresponding to $p$ consecutive observations. Available for Gaussian conditional distribution only.

"logistic":

$M = 2$ , $α_{1, t} = 1 - α_{2, t}$ , and $α_{2, t} = [1 + \exp {- γ (y_{i t - j} - c)}]^{- 1}$ , where $y_{i t - j}$ is the lag $j$ observation of the $i$ th variable, $c$ is a location parameter, and $γ > 0$ is a scale parameter.

"mlogit":

$α_{m, t} = \frac{\exp {γ_{m}^{'} z_{t - 1}}}{\sum_{n = 1}^{M} \exp {γ_{n}^{'} z_{t - 1}}}$ , where $γ_{m}$ are coefficient vectors, $γ_{M} = 0$ , and $z_{t - 1}$ $(k \times 1)$ is the vector containing a constant and the (lagged) switching variables.

"exponential":

$M = 2$ , $α_{1, t} = 1 - α_{2, t}$ , and $α_{2, t} = 1 - \exp {- γ (y_{i t - j} - c)}$ , where $y_{i t - j}$ is the lag $j$ observation of the $i$ th variable, $c$ is a location parameter, and $γ > 0$ is a scale parameter.

"threshold":

$α_{m, t} = 1$ if $r_{m - 1} < y_{i t - j} \leq r_{m}$ and $0$ otherwise, where $- \infty \equiv r_{0} < r_{1} < \dots < r_{M - 1} < r_{M} \equiv \infty$ are thresholds $y_{i t - j}$ 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 $i \in {1, . . ., d}$ in the first and the lag $j \in {1, . . ., p}$ 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 ${1, . . ., d}$ .

The second element $lags:

an integer in ${1, . . ., p}$ 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 $((d p) x (d p))$ "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 $Ω_{m}$ 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 $(T \times M)$ 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.