  Count time series data are frequently analyzed by modeling their conditional
means and the conditional variance is often considered to be a deterministic
function of the corresponding conditional mean and is not typically modeled
independently. We propose a semiparametric mean and variance joint model,
called random rounded count-valued generalized autoregressive conditional
heteroskedastic (RRC-GARCH) model, to address this limitation. The RRC-GARCH
model and its variations allow for the joint modeling of both the conditional
mean and variance and offer a flexible framework for capturing various
mean-variance structures (MVSs). One main feature of this model is its ability
to accommodate negative values for regression coefficients and autocorrelation
functions. The autocorrelation structure of the RRC-GARCH model using the
proposed Laplace link functions with nonnegative regression coefficients is the
same as that of an autoregressive moving-average (ARMA) process. For the new
model, the stationarity and ergodicity are established and the consistency and
asymptotic normality of the conditional least squares estimator are proved.
Model selection criteria are proposed to evaluate the RRC-GARCH models. The
performance of the RRC-GARCH model is assessed through analyses of both
simulated and real data sets. The results indicate that the model can
effectively capture the MVS of count time series data and generate accurate
forecast means and variances.
