Quadratic GARCH-in-Mean Models for Volatility Feedback

Fits quadratic generalized autoregressive conditional heteroskedasticity-in-mean (QGARCH-M) models motivated by Campbell and Hentschel (1992). The package supports models with lambda fixed at zero, lambda restricted to a function of the remaining parameters, lambda estimated freely, and a threshold extension with state-dependent lambda. It also provides tools for starting values, estimation, forecasting, likelihood-ratio testing, moment diagnostics, and replication with the included monthly U.S. stock market dataset.

qgarch

qgarch is an R package for estimating and working with quadratic GARCH-in-mean models in the Campbell and Hentschel (1992) volatility-feedback framework.

We replicated and extended the original paper in: Jedrzej Bialkowski, Sanghyun Hong and Moritz Wagner (2025). "Is no news still good news? Volatility feedback revisited." Pacific-Basin Finance Journal, Volume 91. https://doi.org/10.1016/j.pacfin.2025.102708.

In the Campbell and Hentschel model, expected returns vary with conditional variance, and stock returns include a volatility-feedback term that helps capture asymmetric return dynamics. The framework is designed to accommodate features such as predictive asymmetry, negative skewness, and excess kurtosis in stock returns. See, John Y. Campbell, Ludger Hentschel (1992). "No news is good news: An asymmetric model of changing volatility in stock returns." Journal of Financial Economics, Volume 31, Issue 3, 281-318. https://doi.org/10.1016/0304-405X(92)90037-X.

What the package does

qgarch provides tools to:

• fit generalized QGARCH-in-mean models with nonlinear maximum likelihood,
• work with several model specifications through a common interface,
• extract coefficients, fitted values, residuals, variance-covariance matrices, and log-likelihood values,
• generate forecasts of the conditional mean and conditional variance,
• compute moment diagnostics, and
• compare nested specifications with likelihood-ratio tests.

The main estimation function is qgarch_fit(). It supports four model specifications:

• model = "zero": sets lambda = 0,
• model = "restricted": links lambda to the remaining parameters through the restricted mapping,
• model = "free": estimates lambda freely, and
• model = "threshold": allows a state-dependent lambda_t.

Installation

You can install the development version of qgarch directly from GitHub:

install.packages("remotes")
remotes::install_github("sho-125/qgarch")
library(qgarch)

Example

The example below installs the package from GitHub, loads the bundled monthly U.S. dataset, fits a QGARCH(1,1) model with freely estimated volatility-feedback parameter lambda, and then plots the fitted conditional variance and standardized residuals.

# Install from GitHub
install.packages("remotes")
remotes::install_github("sho-125/qgarch")

# Load package
library(qgarch)

# Load bundled example data
data("us_monthly", package = "qgarch")

# Extract excess returns
x <- us_monthly$ER

# Fit a QGARCH(1,1) model
fit11 <- qgarch_fit(
  x,
  model = "free",
  arch_order = 1L,
  garch_order = 1L
)

# Review estimation results
print(fit11)
summary(fit11)
coef(fit11)
logLik(fit11)

# Plot fitted conditional variance
plot(fit11, which = "sigma2")

# Plot standardized residuals
plot(fit11, which = "standardized")