Mixed difference integer-valued GARCH model for $ \mathbb{Z}$-valued time series

Abstract

In this paper, we introduce flexible observation-driven $\mathbb{Z}$-valued time series models constructed from mixtures of negative and non-negative components. Compared to models based on the standard Skellam distribution or on a difference of two integer-valued variables, our specification offers greater versatility. For example, it easily allows for skewness and bimodality. Furthermore, the observation of one component of the mixture makes interpretation and statistical analysis easier. We establish conditions for stationarity and mixing, and develop a mixed Poisson quasi-maximum likelihood estimator with proven asymptotic properties. A portmanteau test is proposed to diagnose residual serial dependence. The finite-sample performance of the methodology is assessed via simulation, and an empirical application on tick prices demonstrates its practical usefulness.

Figures (8)

• Figure 1: Plots of time series generated by the MD-INGARCH model with $n=200$. The left panels correspond to the case where $(a,b)=(0,0)$, indicating random transitions of the sign of the time series. The right panels correspond to the case where $(a,b)\neq(0,0)$, where the conditional transition probability of the sign depends on past values, demonstrating more structured dynamics.
• Figure 2: Probability mass functions for various distributions. The top-left panel shows the Skellam distribution, corresponding to the approach in aao18clz21, while the other panels show mixtures of positive and negative distributions with different mixing ratios, corresponding to our approach.
• Figure 3: Boxplots of the bias (estimator minus true value) for each parameter based on 1000 replications. For each parameter, the six boxplots correspond to $n=1800, 3600, 7200$ under data generated from the mixed Poisson model (Pois) and the mixed negative binomial model (NB). In all cases, estimation is carried out by mixed Poisson (Q)MLE.
• Figure 4: Boxplots of estimated standard deviations of the mixed Poisson QMLE based on the asymptotic variance formulas, with empirical standard deviations computed over replications overlaid as solid lines, for $n=1800$, $3600$, and $7200$, under data generated from the mixed Poisson model (Pois) and the mixed negative binomial model (NB).
• Figure 5: Empirical sizes and powers of the portmanteau tests based on $p_{1,n}$ and $p_{2,n}$ across different time series lengths $n$ at the nominal level of 0.05, where the data are generated from Poisson (Pois) and negative binomial (NB) linear MD-INGARCH (1,1) models under the null hypothesis (left panel) and log-linear MD-INGARCH models (1,1) under the alternative hypothesis (right panel), and are fitted using the same linear MD-INGARCH(1,1) models. The vertical and horizontal axes represent the rejection probability and the time series length, respectively.

Theorems & Definitions (36)

• Example 1: Poisson MD-INGARCH model
• Example 2: mixed Poisson MD-INGARCH
• Example 3: i.i.d. Bernoulli sequences
• Example 4: Bernoulli INGARCH model
• Remark 1: Link with the threshold ARCH model
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5