get_alpha_mt: Get the transition weights alpha_mt

In sstvars: Toolkit for Reduced Form and Structural Smooth Transition Vector Autoregressive Models

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
)

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 mth 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 ith 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 ith 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 ith 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.

Value

Returns the mixing weights a (T x M) matrix, so that the tth row is for the time period t and m:th column is for the regime m.

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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sstvars documentation built on April 11, 2025, 5:47 p.m.