n_params: Calculate the number of (freely estimaed) parameters in the...

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

Calculate the number of (freely estimaed) parameters in the model

Description

n_params calculates the number of (freely estimaed) parameters in the model.

Usage

n_params(
  p,
  M,
  d,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t"),
  identification = c("reduced_form", "recursive", "heteroskedasticity",
    "non-Gaussianity"),
  AR_constraints = NULL,
  mean_constraints = NULL,
  weight_constraints = NULL,
  B_constraints = NULL
)

Arguments

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.
cond_dist	specifies the conditional distribution of the model as "Gaussian", "Student", "ind_Student", or "ind_skewed_t", where "ind_Student" the Student's t distribution with independent components, and "ind_skewed_t" is the skewed t distribution with independent components (see Hansen, 1994).
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".
AR_constraints	a size (Mpd^2 \times q) constraint matrix C specifying linear constraints to the autoregressive parameters. The constraints are of the form (\varphi_{1},...,\varphi_{M}) = C\psi, where \varphi_{m} = (vec(A_{m,1}),...,vec(A_{m,p})) \ (pd^2 \times 1),\ m=1,...,M, contains the coefficient matrices and \psi (q \times 1) contains the related parameters. For example, to restrict the AR-parameters to be the identical across the regimes, set C = [I:...:I]' (Mpd^2 \times pd^2) where I = diag(p*d^2).
mean_constraints	Restrict the mean parameters of some regimes to be identical? Provide a list of numeric vectors such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if M=3, the argument list(1, 2:3) restricts the mean parameters of the second and third regime to be identical but the first regime has freely estimated (unconditional) mean. Ignore or set to NULL if mean parameters should not be restricted to be the same among any regimes. This constraint is available only for mean parametrized models; that is, when parametrization="mean".
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).
B_constraints	a (d \times d) matrix with its entries imposing constraints on the impact matrix B_t: NA indicating that the element is unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict negative sign constraint, and zero indicating that the element is constrained to zero. Currently only available for models with identification="heteroskedasticity" or "non-Gaussianity" due to the (in)availability of appropriate parametrizations that allow such constraints to be imposed.

Value

Returns the number of parameters in the parameter vector of the specified model.

Warning

No argument checks!

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.