Bayesian Estimation for a Novel Geometric Integer-Valued Generalized Autoregressive Conditional Heteroscedastic (NoGe-INGARCH) Model

Główne pojęcia

The authors propose a novel geometric integer-valued generalized autoregressive conditional heteroscedastic (NoGe-INGARCH) model and develop Bayesian estimation techniques using the Hamiltonian Monte Carlo (HMC) algorithm to estimate the model parameters.

Streszczenie

The authors introduce a novel geometric integer-valued generalized autoregressive conditional heteroscedastic (NoGe-INGARCH) model and discuss its properties. The model is defined as:
Xt | Ft-1 ~ NoGe(θt, φ), where 1-φ/θt = λt = α0 + Σ αi Xt-i + Σ βj λt-j
The authors derive the necessary and sufficient conditions for stationarity of the NoGe-INGARCH model and provide the expressions for the unconditional mean and variance.
For parameter estimation, the authors use two methods: conditional maximum likelihood estimation (CMLE) and Bayesian estimation using the Hamiltonian Monte Carlo (HMC) algorithm. The HMC algorithm is employed to draw samples from the posterior distribution of the model parameters.
The simulation study demonstrates that the Bayesian estimates obtained using the HMC algorithm have lower mean absolute bias and root mean squared error compared to the CMLE, especially for smaller sample sizes. The authors also apply the proposed model and estimation techniques to two real-world datasets: weekly Hepatitis-B cases and stock transaction counts. The results show that the NoGe-INGARCH model outperforms other INGARCH models in terms of model selection criteria.

Statystyki

The number of Hepatitis-B cases reported weekly in the states of Bremen, Hamburg and Saxony-Anhalt from January 2017 to February 2019.
The number of transactions of the Ericsson-B stock per minute between 9:35 and 17:14 on 2 July 2002.

Cytaty

"The novel geometric distribution is defined by the pmf: Pr[X = x] = δφ + (1-δ)(1-φ) (1-θ)^(x-1) θ, x ∈ N0, where 0 < φ < 1 and 0 < θ ≤ 1."
"A necessary and sufficient condition for the NoGe-INGARCH(p,q) model (assuming p > q) to be stationary in the mean is that the roots of the equation 1 - Σ (αi + βi) b^-i - Σ αi b^-i = 0 all lie inside the unit circle."

Kluczowe wnioski z

Bayesian estimation for novel geometric INGARCH model

by Divya Kutten... o arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01283.pdf

Głębsze pytania

What are the potential applications of the NoGe-INGARCH model beyond the examples provided in the article?

The NoGe-INGARCH model, as an integer-valued generalized autoregressive conditional heteroskedasticity model, has a wide range of potential applications beyond the specific examples of Hepatitis-B cases and stock transactions mentioned in the article. Some notable applications include:

Epidemiology: The model can be utilized to analyze the spread of infectious diseases, where the count of new cases over time can exhibit overdispersion and autocorrelation, making it suitable for modeling outbreaks and understanding transmission dynamics.

Finance and Insurance: In finance, the NoGe-INGARCH model can be applied to model the frequency of claims in insurance, where the number of claims can vary significantly over time due to external factors. It can also be used to analyze transaction counts in high-frequency trading data.

Telecommunications: The model can be employed to analyze call counts or message transmissions over time, where the data often exhibit count characteristics and temporal dependencies.

Manufacturing and Quality Control: In manufacturing processes, the NoGe-INGARCH model can be used to monitor the count of defects or failures over time, allowing for better quality control and process optimization.

Environmental Studies: The model can be applied to count data related to environmental events, such as the number of extreme weather events or pollution incidents, helping researchers understand trends and patterns in environmental data.

Sports Analytics: In sports, the model can be used to analyze the count of goals, points, or other events over time, providing insights into team performance and game dynamics.

By leveraging the NoGe-INGARCH model in these diverse fields, researchers and practitioners can gain valuable insights into count time series data, enhancing decision-making and predictive capabilities.

How can the proposed Bayesian estimation approach be extended to handle more complex INGARCH model structures or alternative count data distributions?

The proposed Bayesian estimation approach can be extended to accommodate more complex INGARCH model structures and alternative count data distributions through several strategies:

Hierarchical Modeling: By incorporating hierarchical structures, the Bayesian framework can handle multiple levels of variability, allowing for the modeling of nested data or data with varying group effects. This is particularly useful in applications where data is collected from different sources or populations.

Flexible Priors: The use of informative priors can be tailored to reflect prior knowledge about the parameters, especially in complex models. For instance, hierarchical priors can be employed to share information across parameters, improving estimation in sparse data scenarios.

Alternative Count Distributions: The Bayesian framework can be adapted to include various count data distributions, such as the negative binomial, generalized Poisson, or zero-inflated models. This can be achieved by modifying the likelihood function to reflect the chosen distribution while maintaining the Bayesian estimation approach.

Dynamic Models: The Bayesian estimation can be extended to dynamic models that incorporate time-varying parameters or covariates. This allows for the modeling of more complex temporal dependencies and external influences on the count data.

Model Selection and Comparison: Bayesian model comparison techniques, such as the use of Bayes factors or the deviance information criterion (DIC), can be employed to evaluate and select among competing INGARCH models, facilitating the identification of the best-fitting model for the data.

Computational Techniques: Advanced computational techniques, such as variational inference or parallel computing, can be utilized to improve the efficiency of the Bayesian estimation process, especially in high-dimensional parameter spaces or complex models.

By implementing these strategies, the Bayesian estimation approach can effectively handle more intricate INGARCH model structures and diverse count data distributions, enhancing its applicability across various fields.

What are the implications of the stationarity and second-order stationarity conditions derived for the NoGe-INGARCH model in terms of practical model building and interpretation?

The stationarity and second-order stationarity conditions derived for the NoGe-INGARCH model have significant implications for practical model building and interpretation:

Model Validity: The conditions for stationarity ensure that the NoGe-INGARCH model produces stable and meaningful estimates over time. Practitioners must verify that the parameters satisfy the stationarity conditions before applying the model to real data, as non-stationary processes can lead to misleading conclusions.

Interpretation of Parameters: The stationarity conditions provide insights into the long-term behavior of the process. For instance, the unconditional mean derived from the model allows practitioners to interpret the average count over time, which is crucial for understanding the underlying process being modeled.

Forecasting: Stationarity is essential for reliable forecasting. If the model is stationary, forecasts can be made with confidence that they will reflect the underlying data-generating process. Conversely, if the model is non-stationary, forecasts may diverge significantly from actual future counts.

Model Selection: The implications of stationarity conditions guide model selection. When building models, practitioners should consider whether the chosen structure meets the necessary conditions for stationarity, which can influence the choice of model complexity and the inclusion of lagged terms.

Robustness to Shocks: The second-order stationarity conditions indicate how the model responds to shocks or changes in the underlying process. Understanding the autocovariance structure helps in assessing the model's robustness and its ability to capture the persistence of shocks over time.

Policy Implications: In applied settings, such as public health or finance, the implications of stationarity can inform policy decisions. For example, if a model indicates that a process is stationary, interventions can be designed with the expectation that their effects will be consistent over time.

In summary, the stationarity and second-order stationarity conditions are critical for ensuring the validity, interpretability, and applicability of the NoGe-INGARCH model in real-world scenarios, guiding practitioners in their modeling efforts and decision-making processes.