STVAR {sstvars}R Documentation

Create a class 'stvar' object defining a reduced form or structural smooth transition VAR model

Description

STVAR creates a class 'stvar' object that defines a reduced form or structural smooth transition VAR model

Usage

STVAR(
  data,
  p,
  M,
  d,
  params,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  cond_dist = c("Gaussian", "Student", "ind_Student"),
  parametrization = c("intercept", "mean"),
  identification = c("reduced_form", "recursive", "heteroskedasticity",
    "non-Gaussianity"),
  AR_constraints = NULL,
  mean_constraints = NULL,
  weight_constraints = NULL,
  B_constraints = NULL,
  calc_std_errors = FALSE
)

## S3 method for class 'stvar'
logLik(object, ...)

## S3 method for class 'stvar'
residuals(object, ...)

## S3 method for class 'stvar'
summary(object, ..., digits = 2)

## S3 method for class 'stvar'
plot(x, ..., plot_type = c("trans_weights", "cond_mean"))

## S3 method for class 'stvar'
print(x, ..., digits = 2, summary_print = FALSE, standard_error_print = FALSE)

Arguments

data

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a single times series. NA values are not supported. Ignore if defining a model without data is desired.

p

a positive integer specifying the autoregressive order

M

a positive integer specifying the number of regimes

d

number of times series in the system, i.e. ncol(data). This can be used to define STVAR models without data and can be ignored if data is provided.

params

a real valued vector specifying the parameter values. Should have the form \theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu), where (see exceptions below):

  • \phi_{m,0} = the (d \times 1) intercept (or mean) vector of the mth regime.

  • \varphi_m = (vec(A_{m,1}),...,vec(A_{m,p})) (pd^2 \times 1).

  • if cond_dist="Gaussian" or "Student":

    \sigma = (vech(\Omega_1),...,vech(\Omega_M)) (Md(d + 1)/2 \times 1).

    if cond_dist="ind_Student":

    \sigma = (vec(B_1),...,vec(B_M) (Md^2 \times 1).

  • \alpha = the (a\times 1) vector containing the transition weight parameters (see below).

  • if cond_dist = "Gaussian"):

    Omit \nu from the parameter vector.

    if cond_dist="Student":

    \nu > 2 is the single degrees of freedom parameter.

    if cond_dist="ind_Student":

    \nu = (\nu_1,...,\nu_M) (M \times 1), nu_m > 2.

For models with...

weight_function="relative_dens":

\alpha = (\alpha_1,...,\alpha_{M-1}) (M - 1 \times 1), where \alpha_m (1\times 1), m=1,...,M-1 are the transition weight parameters.

weight_function="logistic":

\alpha = (c,\gamma) (2 \times 1), where c\in\mathbb{R} is the location parameter and \gamma >0 is the scale parameter.

weight_function="mlogit":

\alpha = (\gamma_1,...,\gamma_M) ((M-1)k\times 1), where \gamma_m (k\times 1), m=1,...,M-1 contains the multinomial logit-regression coefficients of the mth 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.

weight_function="exponential":

\alpha = (c,\gamma) (2 \times 1), where c\in\mathbb{R} is the location parameter and \gamma >0 is the scale parameter.

weight_function="threshold":

\alpha = (r_1,...,r_{M-1}) (M-1 \times 1), where r_1,...,r_{M-1} are the threshold values.

weight_function="exogenous":

Omit \alpha from the parameter vector.

AR_constraints:

Replace \varphi_1,...,\varphi_M with \psi as described in the argument AR_constraints.

mean_constraints:

Replace \phi_{1,0},...,\phi_{M,0} with (\mu_{1},...,\mu_{g}) where \mu_i, \ (d\times 1) is the mean parameter for group i and g is the number of groups.

weight_constraints:

If linear constraints are imposed, replace \alpha with \xi as described in the argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop \alpha from the parameter vector.

identification="heteroskedasticity":

\sigma = (vec(W),\lambda_2,...,\lambda_M), where W (d\times d) and \lambda_m (d\times 1), m=2,...,M, satisfy \Omega_1=WW' and \Omega_m=W\Lambda_mW', \Lambda_m=diag(\lambda_{m1},...,\lambda_{md}), \lambda_{mi}>0, m=2,...,M, i=1,...,d.

B_constraints (only for structural models identified by heteroskedasticity):

Replace vec(W) with \tilde{vec}(W) that stacks the columns of the matrix W in to vector so that the elements that are constrained to zero are not included.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth regime, \Omega_{m} denotes the positive definite error term covariance matrix of the mth regime, and B_m is the invertible (d\times d) impact matrix of the mth regime. \nu_m is the degrees of freedom parameter of the mth regime. If parametrization=="mean", just replace each \phi_{m,0} with regimewise mean \mu_{m}. vec() is vectorization operator that stacks columns of a given matrix into a vector. vech() stacks columns of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector. Bvec() is a vectorization operator that stacks the columns of a given impact matrix B_m into a vector so that the elements that are constrained to zero by the argument B_constraints are excluded.

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", or "ind_Student", where the latest is the Student's t distribution with independent components.

parametrization

"intercept" or "mean" determining whether the model is parametrized with intercept parameters \phi_{m,0} or regime means \mu_{m}, m=1,...,M.

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 x 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 x 1),\ m=1,...,M, contains the coefficient matrices and \psi (q x 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 x 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 in order to constrain the the weight parameter 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.

calc_std_errors

should approximate standard errors be calculated?

object

object of class 'stvar'.

...

currently not used.

digits

number of digits to be printed.

x

an object of class 'stvar'.

plot_type

should the series be plotted with the estimated transition weights or conditional means?

summary_print

if set to TRUE then the print will include log-likelihood and information criteria values.

standard_error_print

if set to TRUE, instead of printing the estimates, prints the approximate standard errors using square roots of the diagonal of inverse of the observed information matrix.

Details

If data is provided, then also residuals are computed and included in the returned object.

The plot displays the time series together with estimated transition weights.

Value

Returns an S3 object of class 'stvar' defining a smooth transition VAR model. The returned list contains the following components (some of which may be NULL depending on the use case):

data

The input time series data.

model

A list describing the model structure.

params

The parameters of the model.

std_errors

Approximate standard errors of the parameters, if calculated.

transition_weights

The transition weights of the model.

regime_cmeans

Conditional means of the regimes, if data is provided.

total_cmeans

Total conditional means of the model, if data is provided.

total_ccovs

Total conditional covariances of the model, if data is provided.

uncond_moments

A list of unconditional moments including regime autocovariances, variances, and means.

residuals_raw

Raw residuals, if data is provided.

residuals_std

Standardized residuals, if data is provided.

structural_shocks

Recovered structural shocks, if applicable.

loglik

Log-likelihood of the model, if data is provided.

IC

The values of the information criteria (AIC, HQIC, BIC) for the model, if data is provided.

all_estimates

The parameter estimates from all estimation rounds, if applicable.

all_logliks

The log-likelihood of the estimates from all estimation rounds, if applicable.

which_converged

Indicators of which estimation rounds converged, if applicable.

which_round

Indicators of which round of optimization each estimate belongs to, if applicable.

Functions

About S3 methods

If data is not provided, only the print and simulate methods are available. If data is provided, then in addition to the ones listed above, predict method is also available. See ?simulate.stvar and ?predict.stvar for details about the usage.

References

See Also

fitSTVAR, swap_parametrization, alt_stvar

Examples

# Below examples use the example data "gdpdef", which is a two-variate quarterly data
# of U.S. GDP and GDP implicit price deflator covering the period from 1959Q1 to 2019Q4.

# Gaussian STVAR p=1, M=2, model with the weighted relative stationary densities
# of the regimes as the transition weight function:
theta_122relg <- c(0.734054, 0.225598, 0.705744, 0.187897, 0.259626, -0.000863,
  -0.3124, 0.505251, 0.298483, 0.030096, -0.176925, 0.838898, 0.310863, 0.007512,
  0.018244, 0.949533, -0.016941, 0.121403, 0.573269)
mod122 <- STVAR(data=gdpdef, p=1, M=2, params=theta_122relg)
print(mod122) # Printout of the model
summary(mod122) # Summary printout
plot(mod122) # Plot the transition weights
plot(mod122, plot_type="cond_mean") # Plot one-step conditional means

# Logistic Student's t STVAR with p=1, M=2, and the first lag of the second variable
# as the switching variable:
params12 <- c(0.62906848, 0.14245295, 2.41245785, 0.66719269, 0.3534745, 0.06041779, -0.34909745,
  0.61783824, 0.125769, -0.04094521, -0.99122586, 0.63805416, 0.371575, 0.00314754, 0.03440824,
  1.29072533, -0.06067807, 0.18737385, 1.21813844, 5.00884263, 7.70111672)
fit12 <- STVAR(data=gdpdef, p=1, M=2, params=params12, weight_function="logistic",
  weightfun_pars=c(2, 1), cond_dist="Student")
summary(fit12) # Summary printout
plot(fit12) # Plot the transition weights

# Threshold STVAR with p=1, M=2, the first lag of the second variable as switching variable:
params12thres <- c(0.5231, 0.1015, 1.9471, 0.3253, 0.3476, 0.0649, -0.035, 0.7513, 0.1651,
 -0.029, -0.7947, 0.7925, 0.4233, 5e-04, 0.0439, 1.2332, -0.0402, 0.1481, 1.2036)
mod12thres <- STVAR(data=gdpdef, p=1, M=2, params=params12thres, weight_function="threshold",
  weightfun_pars=c(2, 1))
mod12thres # Printout of the model

# Student's t logistic STVAR with p=2, M=2 with the second lag of the second variable
# as the switching variable and structural shocks identified by heteroskedasticity;
# the model created without data:
params22log <- c(0.357, 0.107, 0.356, 0.086, 0.14, 0.035, -0.165, 0.387, 0.452,
 0.013, 0.228, 0.336, 0.239, 0.024, -0.021, 0.708, 0.063, 0.027, 0.009, 0.197,
  -0.03, 0.24, -0.76, -0.02, 3.36, 0.86, 0.1, 0.2, 7)
mod222logtsh <- STVAR(p=2, M=2, d=2, params=params22log, weight_function="logistic",
 weightfun_pars=c(2, 2), cond_dist="Student", identification="heteroskedasticity")
print(mod222logtsh) # Printout of the model

# STVAR p=2, M=2, model with exogenous transition weights and mutually independent
# Student's t shocks:
set.seed(1); tw1 <- runif(nrow(gdpdef)-2) # Transition weights of Regime 1
params22exoit <- c(0.357, 0.107, 0.356, 0.086, 0.14, 0.035, -0.165, 0.387, 0.452,
 0.013, 0.228, 0.336, 0.239, 0.024, -0.021, 0.708, 0.063, 0.027, 0.009, 0.197,
 -0.1, 0.2, -0.15, 0.13, 0.21, 0.15, 0.11, -0.09, 3, 4)
mod222exoit <- STVAR(p=2, M=2, d=2, params=params22exoit, weight_function="exogenous",
 weightfun_pars=cbind(tw1, 1-tw1), cond_dist="ind_Student")
print(mod222exoit) # Printout of the model

# Linear Gaussian VAR(p=1) model:
theta_112 <- c(0.649526, 0.066507, 0.288526, 0.021767, -0.144024, 0.897103,
  0.601786, -0.002945, 0.067224)
mod112 <- STVAR(data=gdpdef, p=1, M=1, params=theta_112)
summary(mod112) # Summary printout
[Package sstvars version 1.0.1 Index]