Type:	Package
Title:	Winsorized ARMA Estimation for Higher-Order Stochastic Volatility Models
Version:	0.1.0
Description:	Estimation, simulation, hypothesis testing, and forecasting for univariate higher-order stochastic volatility SV(p) models. Supports Gaussian, Student-t, and Generalized Error Distribution (GED) innovations, with optional leverage effects. Estimation uses closed-form Winsorized ARMA-SV (W-ARMA-SV) moment-based methods that avoid numerical optimization. Hypothesis testing includes Local Monte Carlo (LMC) and Maximized Monte Carlo (MMC) procedures for leverage effects, heavy tails, and autoregressive order selection. Forecasting is based on Kalman filtering and smoothing. See Ahsan and Dufour (2021) <doi:10.1016/j.jeconom.2020.01.018>, Ahsan, Dufour, and Rodriguez Rondon (2025) for details.
License:	GPL (≥ 3)
URL:	https://github.com/roga11/wARMASVp
BugReports:	https://github.com/roga11/wARMASVp/issues
Encoding:	UTF-8
Imports:	Rcpp (≥ 1.0.0), gsignal, stats
Suggests:	pso, GenSA, testthat (≥ 3.0.0), knitr, rmarkdown
LinkingTo:	Rcpp, RcppArmadillo
RoxygenNote:	7.3.3
VignetteBuilder:	knitr
NeedsCompilation:	yes
Packaged:	2026-04-21 20:58:28 UTC; gabrielrodriguez
Author:	Gabriel Rodriguez Rondon [aut, cre], Nazmul Ahsan [aut], Jean-Marie Dufour [aut]
Maintainer:	Gabriel Rodriguez Rondon <gabriel.rodriguezrondon@mail.mcgill.ca>
Repository:	CRAN
Date/Publication:	2026-04-22 08:20:08 UTC

wARMASVp: Winsorized ARMA Estimation for Higher-Order Stochastic Volatility Models

Description

Estimation, simulation, hypothesis testing, and forecasting for univariate higher-order stochastic volatility SV(p) models. Supports Gaussian, Student-t, and GED innovations with optional leverage effects.

Details

The main user-facing functions are:

• svp – Estimate SV(p) model with optional leverage

• svpSE – Simulation-based standard errors

• sim_svp – Simulate SV(p) processes

• filter_svp – Kalman/mixture/particle filtering

• forecast_svp – Multi-step volatility forecasts

• lmc_ar, mmc_ar – AR order tests

• lmc_lev, mmc_lev – Leverage tests

• lmc_t, mmc_t – Student-t tail tests

• lmc_ged, mmc_ged – GED tail tests

Author(s)

Maintainer: Gabriel Rodriguez Rondon gabriel.rodriguezrondon@mail.mcgill.ca (ORCID)

Authors:

• Nazmul Ahsan

• Jean-Marie Dufour

References

Ahsan, N. and Dufour, J.-M. (2021). Simple estimators and inference for higher-order stochastic volatility models. Journal of Econometrics, 224(1), 181-197.

Ahsan, N., Dufour, J.-M., and Rodriguez Rondon, G. (2025). Estimation and inference for higher-order stochastic volatility models with leverage. Journal of Time Series Analysis.

Ahsan, N., Dufour, J.-M., and Rodriguez Rondon, G. (2025). Estimation and inference for stochastic volatility models with heavy-tailed distributions.

See Also

Useful links:

• https://github.com/roga11/wARMASVp

• Report bugs at https://github.com/roga11/wARMASVp/issues

Add leverage estimation to a fitted SV(p) model (post-processing) Works for all error distributions: Gaussian, Student-t, GED. - Gaussian: closed form (C_F = 1) - Student-t: closed form with C_t(nu) correction - GED: exact 1D root-finding via uniroot + Gauss-Hermite quadrature

Description

Add leverage estimation to a fitted SV(p) model (post-processing) Works for all error distributions: Gaussian, Student-t, GED. - Gaussian: closed form (C_F = 1) - Student-t: closed form with C_t(nu) correction - GED: exact 1D root-finding via uniroot + Gauss-Hermite quadrature

Usage

.add_leverage(out, y, p, rho_type, del, trunc_lev, wDecay, errorType)

Arguments

Value

Updated model object with leverage fields added

Compute EH cross-moment for leverage estimation

Description

EH = demeaned sample cross-moment (1/(T-2)) \sum(|y_t| - \bar{|y|})(y_{t-1} - \bar{y}).

Usage

.compute_EH(y, rho_type = "pearson")

Arguments

Value

EH value

Gauss-Hermite quadrature nodes and weights for N(0,1) integration Computes nodes z_i and weights w_i such that sum(w_i * f(z_i)) approximates E[f(Z)] where Z ~ N(0,1). Uses the Golub-Welsch algorithm.

Description

Gauss-Hermite quadrature nodes and weights for N(0,1) integration Computes nodes z_i and weights w_i such that sum(w_i * f(z_i)) approximates E[f(Z)] where Z ~ N(0,1). Uses the Golub-Welsch algorithm.

Usage

.gauss_hermite_normal(n = 200L)

Arguments

Value

List with components nodes and weights

Get cached Gauss-Hermite nodes/weights for N(0,1)

Description

Get cached Gauss-Hermite nodes/weights for N(0,1)

Usage

.get_gh(n = 200L)

Arguments

Value

List with nodes and weights

GMM moments for SVL(p)-GED with fixed A matrix (p+4 conditions) Uses exact GED leverage moment.

Description

GMM moments for SVL(p)-GED with fixed A matrix (p+4 conditions) Uses exact GED leverage moment.

Usage

LRT_moment_lev_ged(y, mdl_out, Amat, rho_type, del = 1e-10)

GMM moments for SVL(p)-GED with HAC weighting (p+4 conditions)

Description

GMM moments for SVL(p)-GED with HAC weighting (p+4 conditions)

Usage

LRT_moment_lev_ged_Amat(y, mdl_out, rho_type, del = 1e-10, Bartlett = TRUE)

GMM moments for SVL(p)-Student-t with fixed A matrix (p+4 conditions)

Description

GMM moments for SVL(p)-Student-t with fixed A matrix (p+4 conditions)

Usage

LRT_moment_lev_t(y, mdl_out, Amat, rho_type, del = 1e-10)

GMM moments for SVL(p)-Student-t with HAC weighting (p+4 conditions)

Description

GMM moments for SVL(p)-Student-t with HAC weighting (p+4 conditions)

Usage

LRT_moment_lev_t_Amat(y, mdl_out, rho_type, del = 1e-10, Bartlett = TRUE)

Construct Companion Matrix for AR(p) Process

Description

Builds the companion matrix representation for an AR(p) process, which is used in the state-space formulation of SV(p) models.

Usage

companionMat(phi, p, q)

Arguments

Value

A (q*p) x (q*p) companion matrix.

Approximate correction factor C_g(nu) for GED leverage (diagnostic use only)

Description

C_g(\nu) = E[|u|] \cdot \textrm{Cov}(z, F_{GED}^{-1}(\Phi(z))) / \sqrt{2/\pi}. This is a first-order approximation; estimation uses the exact implicit equation.

Usage

correction_factor_ged_approx(nu, n_nodes = 200L)

Arguments

Value

Approximate C_g(nu)

Correction factor C_t(nu) for Student-t leverage estimation

Description

Under scale mixture u_t = z_t \lambda_t^{-1/2}, C_t(\nu) = [E[\lambda^{-1/2}]]^2 = (\nu/2) [\Gamma((\nu-1)/2) / \Gamma(\nu/2)]^2. Exact, parameter-free.

Usage

correction_factor_t(nu)

Arguments

Value

C_t(nu), always > 1 for finite nu (approaches 1 as nu -> Inf)

Density of Centered log-GED^2 Measurement Noise

Description

Density of Centered log-GED^2 Measurement Noise

Usage

density_eps_ged(y, nu)

Arguments

Value

Density values.

Density of Centered log-F(1,nu) Measurement Noise (Student-t)

Description

Density of Centered log-F(1,nu) Measurement Noise (Student-t)

Usage

density_eps_t(y, nu)

Arguments

Value

Density values.

Filter Latent Volatility from an Estimated SV(p) Model

Description

Applies Kalman filtering (corrected or Gaussian mixture) and RTS smoothing to extract the latent log-volatility process from an estimated SV(p) model.

Usage

filter_svp(
  object,
  method = c("corrected", "mixture", "particle"),
  K = 7,
  M = 1000,
  seed = 42,
  del = 1e-10
)

Arguments

Value

An object of class "svp_filter", a list containing:

w_filtered

Filtered log-volatility (T-vector).

w_smoothed

Smoothed log-volatility (T-vector).

zt

Filtered standardized residuals.

zt_smoothed

Smoothed standardized residuals.

P_filtered

Filtered MSE of first state component.

P_predicted

Predicted MSE of first state component.

xi_filtered

Full filtered state vectors (p x T matrix).

xi_smoothed

Full smoothed state vectors (p x T matrix).

loglik

Approximate log-likelihood.

method

Filter method used.

model

The input model object.

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)$y
fit <- svp(y, p = 1)
filt <- filter_svp(fit)
plot(filt$w_smoothed, type = "l")

Fit K-Component Gaussian Mixture to Measurement Noise Density

Description

Uses EM algorithm to approximate the measurement noise density with a Gaussian mixture. For Gaussian SV, returns the pre-computed KSC (1998) 7-component table.

Usage

fit_ksc_mixture(
  distribution = c("gaussian", "student_t", "ged"),
  nu = NULL,
  K = 7,
  n_sample = 10000,
  max_iter = 500,
  tol = 1e-08,
  seed = 42
)

Arguments

Value

List with weights, means, vars, KL_div.

Multi-Step Ahead Volatility Forecast

Description

Applies Kalman filtering/smoothing to an estimated SV(p) model and produces multi-step ahead volatility forecasts with uncertainty quantification.

Usage

forecast_svp(
  object,
  H = 1,
  output = c("log-variance", "variance", "volatility"),
  filter_method = "corrected",
  K = 7,
  M = 1000,
  seed = 42,
  del = 1e-10
)

Arguments

Value

An object of class "svp_forecast", a list containing:

w_forecasted

Primary forecast (scale determined by output).

log_var_forecast

Log-volatility forecasts w_{T+h|T}.

var_forecast

Conditional variance forecasts \sigma^2_{T+h|T}.

vol_forecast

Conditional volatility forecasts \sigma_{T+h|T}.

P_forecast

Forecast MSE P_{T+h|T} for each horizon.

w_estimated

Filtered log-volatility.

w_smoothed

Smoothed log-volatility.

zt

Filtered standardized residuals.

zt_smoothed

Smoothed standardized residuals.

ys

Demeaned log-squared returns.

mdl

The estimated model object.

H

The forecast horizon.

output

The chosen output scale.

filter_output

The "svp_filter" object from filtering.

Examples

sim <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2,
               leverage = TRUE, rho = -0.3)
fit <- svp(sim$y, p = 1, leverage = TRUE)
fc <- forecast_svp(fit, H = 10)
plot(fc)

E[|u|] for standardized GED(nu) with Var = 1 Closed form: sqrt(Gamma(1/nu)/Gamma(3/nu)) * Gamma(2/nu) / Gamma(1/nu)

Description

E[|u|] for standardized GED(nu) with Var = 1 Closed form: sqrt(Gamma(1/nu)/Gamma(3/nu)) * Gamma(2/nu) / Gamma(1/nu)

Usage

ged_E_abs_u(nu)

Arguments

Value

E[|u|]

LMC Test for AR Order in SV(p) Models

Description

Performs a Local Monte Carlo (LMC) test of the null hypothesis H_0: \phi_{p_0+1} = \cdots = \phi_p = 0 (i.e., that an SV(p_0) model is sufficient against an SV(p) alternative).

Usage

lmc_ar(
  y,
  p_null,
  p_alt,
  J = 10,
  N = 99,
  burnin = 500,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  sigvMethod = "factored"
)

Arguments

Details

When Bartlett = FALSE (default), the test statistic is T \sum_{j=p_0+1}^{p} \hat\phi_j^2, i.e., the sum of squared extra AR coefficients scaled by sample size.

When Bartlett = TRUE, the test statistic is based on the GMM-LRT approach with a Bartlett kernel HAC weighting matrix: S = T \times (M_{H_0} - M_{H_1}), where M denotes the GMM criterion evaluated at the null and alternative estimates. Simulations yielding negative test statistics are discarded and re-drawn.

Value

An object of class "svp_test", a list containing:

s0

Test statistic from observed data.

sN

Simulated null distribution (vector of length N).

pval

Monte Carlo p-value.

test_type

Character string identifying the test.

null_param

Name of the parameter(s) tested.

null_value

Value(s) under the null hypothesis.

call

The matched call.

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)$y
test <- lmc_ar(y, p_null = 1, p_alt = 2, J = 10, N = 49)
print(test)

LMC Test for GED Shape Parameter

Description

Performs a Local Monte Carlo (LMC) test of the null hypothesis H_0: \nu = \nu_0 for the shape parameter in an SV(p) model with GED errors. Testing \nu_0 = 2 corresponds to testing normality.

Usage

lmc_ged(
  y,
  p = 1,
  J = 10,
  N = 99,
  nu_null,
  burnin = 500,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  direction = c("two-sided", "less", "greater"),
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

An object of class "svp_test".

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, errorType = "GED", nu = 1.5)$y
test <- lmc_ged(y, p = 1, J = 10, N = 49, nu_null = 2)
print(test)

LMC Test for Leverage in SV(p) Models

Description

Performs a Local Monte Carlo (LMC) test of the null hypothesis H_0: \rho = \rho_0 (typically \rho_0 = 0, i.e., no leverage) using a GMM likelihood-ratio type statistic.

Usage

lmc_lev(
  y,
  p = 1,
  J = 10,
  N = 99,
  rho_null = 0,
  burnin = 500,
  rho_type = "pearson",
  del = 1e-10,
  trunc_lev = TRUE,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  errorType = "Gaussian",
  logNu = FALSE,
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

An object of class "svp_test", a list containing:

s0

Test statistic from observed data.

sN

Simulated null distribution (vector of length N).

pval

Monte Carlo p-value.

test_type

Character string identifying the test.

null_param

Name of the parameter tested.

null_value

Value under the null hypothesis.

call

The matched call.

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, leverage = TRUE, rho = -0.3)$y
test <- lmc_lev(y, p = 1, J = 10, N = 99)
print(test)

LMC Test for Student-t Tail Parameter

Description

Performs a Local Monte Carlo (LMC) test of the null hypothesis H_0: \nu = \nu_0 for the degrees of freedom parameter in an SV(p) model with Student-t errors. Testing \nu_0 = \infty (or a large value) corresponds to testing for normality.

Usage

lmc_t(
  y,
  p = 1,
  J = 10,
  N = 99,
  nu_null,
  burnin = 500,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  logNu = TRUE,
  direction = c("two-sided", "less", "greater"),
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

An object of class "svp_test".

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, errorType = "Student-t", nu = 5)$y
test <- lmc_t(y, p = 1, J = 10, N = 49, nu_null = 5)
print(test)

MMC Test for AR Order in SV(p) Models

Description

Performs a Maximized Monte Carlo (MMC) test of H_0: \phi_{p_0+1} = \cdots = \phi_p = 0 by maximizing the MC p-value over nuisance parameters (\phi_1, \ldots, \phi_{p_0}, \sigma_y, \sigma_v).

Usage

mmc_ar(
  y,
  p_null,
  p_alt,
  J = 10,
  N = 99,
  burnin = 500,
  eps = NULL,
  threshold = 1,
  method = "pso",
  maxit = NULL,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  sigvMethod = "factored"
)

Arguments

Value

A list with optimization output including value (maximized p-value) and par (nuisance parameters at the maximum).

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)$y
mmc <- mmc_ar(y, p_null = 1, p_alt = 2, J = 10, N = 19,
              method = "pso", maxit = 10)
mmc$value

MMC Test for GED Shape Parameter

Description

Performs a Maximized Monte Carlo (MMC) test of H_0: \nu = \nu_0 for the GED shape parameter.

Usage

mmc_ged(
  y,
  p = 1,
  J = 10,
  N = 99,
  nu_null,
  burnin = 500,
  eps = NULL,
  threshold = 1,
  method = "pso",
  maxit = NULL,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  direction = c("two-sided", "less", "greater"),
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

A list with optimization output including value (maximized p-value) and par (nuisance parameters at the maximum).

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, errorType = "GED", nu = 1.5)$y
mmc <- mmc_ged(y, p = 1, J = 10, N = 19, nu_null = 2, method = "pso", maxit = 10)
mmc$value

MMC Test for Leverage in SV(p) Models

Description

Performs a Maximized Monte Carlo (MMC) test of the null hypothesis H_0: \rho = \rho_0 by maximizing the MC p-value over nuisance parameters (phi, sigma_y, sigma_v).

Usage

mmc_lev(
  y,
  p = 1,
  J = 10,
  N = 99,
  rho_null = 0,
  burnin = 500,
  eps = NULL,
  threshold = 1,
  method = "pso",
  maxit = NULL,
  rho_type = "pearson",
  del = 1e-10,
  trunc_lev = TRUE,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  errorType = "Gaussian",
  logNu = FALSE,
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

A list with the optimization output including:

value

Maximized p-value.

par

Nuisance parameter values at the maximum.

Additional fields depend on the optimization method used.

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, leverage = TRUE, rho = -0.3)$y
mmc <- mmc_lev(y, p = 1, J = 10, N = 19, method = "pso", maxit = 10)
mmc$value

MMC Test for Student-t Tail Parameter

Description

Performs a Maximized Monte Carlo (MMC) test of H_0: \nu = \nu_0 by maximizing the MC p-value over nuisance parameters (phi, sigma_y, sigma_v).

Usage

mmc_t(
  y,
  p = 1,
  J = 10,
  N = 99,
  nu_null,
  burnin = 500,
  eps = NULL,
  threshold = 1,
  method = "pso",
  maxit = NULL,
  del = 1e-10,
  wDecay = FALSE,
  Bartlett = FALSE,
  Amat = NULL,
  logNu = TRUE,
  direction = c("two-sided", "less", "greater"),
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Value

A list with optimization output including value (maximized p-value) and par (nuisance parameters at the maximum).

Examples

y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, errorType = "Student-t", nu = 5)$y
mmc <- mmc_t(y, p = 1, J = 10, N = 19, nu_null = 5, method = "pso", maxit = 10)
mmc$value

CDF of Standardized GED

Description

Computes the CDF of the standardized GED(\nu) distribution with unit variance.

Usage

pged_std(u, nu)

Arguments

Value

Numeric vector of probabilities.

Quantile function for standardized GED(nu) with Var = 1 Uses the relationship between GED CDF and incomplete gamma function.

Description

Quantile function for standardized GED(nu) with Var = 1 Uses the relationship between GED CDF and incomplete gamma function.

Usage

qged_std(p, nu)

Arguments

Value

Quantile value

Simulate from a Stochastic Volatility Model

Description

Master simulation function for SV(p) models. Supports Gaussian, Student-t, and GED error distributions, with optional leverage effects. This mirrors the interface of svp for estimation.

Usage

sim_svp(
  n,
  phi,
  sigy,
  sigv,
  errorType = "Gaussian",
  leverage = FALSE,
  rho = 0,
  nu = NULL,
  burnin = 500
)

Arguments

Details

The model is:

y_t = \sigma_y \exp(w_t / 2) z_t

w_t = \phi_1 w_{t-1} + \cdots + \phi_p w_{t-p} + \sigma_v v_t

where z_t follows a distribution specified by errorType (Gaussian, Student-t, or GED), and v_t is i.i.d. standard normal. When leverage = TRUE, the correlation between z_t and v_{t+1} is \rho.

For Student-t errors with leverage, the scale-mixture representation z_t = \zeta_t \lambda_t^{-1/2} is used, where leverage operates through the Gaussian component \zeta_t. For GED errors with leverage, a Gaussian copula construction z_t = F_{\mathrm{GED}}^{-1}(\Phi(\zeta_t)) is used. In both cases the returned z is the effective return innovation (not the latent \zeta_t), with marginal distribution matching the errorType.

Value

A named list of four length-n numeric vectors:

y

Observed returns y_t.

h

Log-volatility process w_t (equivalently h_t).

z

Return innovation such that y_t = \sigma_y \exp(h_t/2)\, z_t. Marginal distribution matches errorType: N(0,1) for Gaussian, t(\nu) for Student-t, unit-variance GED(\nu) for GED.

v

Volatility innovation such that h_t - \sum_{j=1}^p \phi_j h_{t-j} = \sigma_v\, v_t. Always N(0,1); under leverage, v_t = \rho\, \zeta_{t-1} + \sqrt{1-\rho^2}\, \epsilon_t.

See Also

svp for estimation.

Examples

# Gaussian SV(1), no leverage
sim <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)
plot(sim$y, type = "l")

# Gaussian SV(1) with leverage
sim_lev <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2,
                   leverage = TRUE, rho = -0.5)
plot(sim_lev$y, type = "l")

# Student-t SV(1)
sim_t <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2,
                 errorType = "Student-t", nu = 5)
plot(sim_t$y, type = "l")

# GED SV(1)
sim_ged <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2,
                   errorType = "GED", nu = 1.5)
plot(sim_ged$y, type = "l")

Solve Discrete Lyapunov Equation

Description

Solves X = F X t(F) + Q for X using the vectorization approach.

Usage

solve_lyapunov_discrete(F_mat, Q)

Arguments

Value

Solution matrix X.

Estimate a Stochastic Volatility Model

Description

Master estimation function for SV(p) models using the Winsorized ARMA-SV (W-ARMA-SV) method. Supports Gaussian, Student-t, and GED error distributions, with optional leverage effects.

Usage

svp(
  y,
  p = 1,
  J = 10,
  leverage = FALSE,
  errorType = "Gaussian",
  rho_type = "pearson",
  del = 1e-10,
  trunc_lev = TRUE,
  wDecay = FALSE,
  logNu = FALSE,
  sigvMethod = "factored",
  winsorize_eps = 0
)

Arguments

Details

The model is:

y_t = \sigma_y \exp(w_t / 2) z_t

w_t = \phi_1 w_{t-1} + \cdots + \phi_p w_{t-p} + \sigma_v v_t

where z_t follows a distribution specified by errorType (Gaussian, Student-t, or GED), and v_t is i.i.d. standard normal. When leverage = TRUE, the correlation between z_t and v_t is estimated as \rho.

For Student-t errors with leverage, the correction factor C_t(\nu) from the scale-mixture representation is applied. For GED errors with leverage, the exact implicit equation is solved via 1D root-finding with Gauss-Hermite quadrature.

Value

Depending on errorType:

• "Gaussian": An object of class "svp" (see below).

• "Student-t": An object of class "svp_t".

• "GED": An object of class "svp_ged".

The "svp" class contains:

mu

Mean of \log(y_t^2 + \delta).

phi

Numeric vector of AR coefficients.

sigv

Standard deviation of volatility innovations.

sigy

Unconditional standard deviation.

rho

Leverage parameter (if estimated, otherwise NA).

y

The original data.

p, J

Model order and winsorizing parameter.

errorType

The error distribution used.

call

The matched call.

References

Ahsan, N. and Dufour, J.-M. (2021). Simple estimators and inference for higher-order stochastic volatility models. Journal of Econometrics, 224(1), 181-197.

Ahsan, N., Dufour, J.-M., and Rodriguez Rondon, G. (2025). Estimation and inference for stochastic volatility models with heavy-tailed distributions.

See Also

svpSE for standard errors.

Examples

# Gaussian SV(1) without leverage (default)
y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)$y
fit <- svp(y)
summary(fit)

# With leverage
y2 <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2, leverage = TRUE, rho = -0.3)$y
fit2 <- svp(y2, leverage = TRUE)
coef(fit2)

# Student-t errors
y3 <- sim_svp(1000, phi = 0.9, sigy = 1, sigv = 0.2, errorType = "Student-t", nu = 5)$y
fit3 <- svp(y3, errorType = "Student-t")
summary(fit3)

Simulation-Based Standard Errors for SV(p) Models

Description

Computes standard errors and confidence intervals for estimated parameters by simulating from the fitted model and re-estimating. Supports all model types returned by svp: Gaussian (with or without leverage), Student-t, and GED.

Usage

svpSE(object, n_sim = 199, alpha = 0.05, burnin = 500, logNu = FALSE)

Arguments

Value

A list with:

CI

2 x k matrix of confidence intervals (lower, upper).

SEsim0

Standard errors relative to true parameter values.

SEsim

Standard errors relative to sample mean.

ISEconservative

Conservative interval-based standard errors.

ISEliberal

Liberal interval-based standard errors.

thetamat

Matrix of parameter estimates from simulations.

Examples

# Gaussian SV(1)
y <- sim_svp(1000, phi = 0.95, sigy = 1, sigv = 0.2)$y
fit <- svp(y)
se <- svpSE(fit, n_sim = 49)
se$CI