Localized least squares estimation for STGARCH(p,q) model

Abstract

In this paper, we introduce a spatio-temporal GARCH model that incorporates spatial dependencies to extend traditional GARCH frameworks. The model is defined by a spatiotemporal process with non-stationary volatility, allowing for parameter variation across space. Two estimation methods localized and local linear least squares are proposed to address spatially non-stationary data. We examine the asymptotic properties of these estimators and evaluate their performance through simulations, finding the local linear estimator generally more effective for smooth parameters. Finally, we apply the model to real-world data, demonstrating its ability to capture both spatial and temporal dependencies in volatility.

Keywords

• Estimation
• linear local least squares
• local least squares
• properties of estimators
• spatio-temporal GARCH model
• spatial local stationarity

References

1. [1] A. Aouri, M. Boukeloua and S. Kharfouchi, Probabilistic insights into spatio-temporal GARCH model with location-dependent parameters, TWMS J. Appl. Eng. Math. 15 (6), 1473–1486, 2024.
2. [2] A. Aouri and P. Otto, A Heterogeneous Spatiotemporal GARCH Model: A Predictive Framework for Volatility in Financial Networks, arXiv preprint arXiv:2508.20101, 2025.
3. [3] N. A. Awang, A. F. A. Rashid, N. M. Napi and I. A. J. Sopaheluwakan, Ozone prediction using artificial neural network (ANN), J. Phys.: Conf. Ser. 914 (1), 012020, 2017.
4. [4] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (3), 307–327, 1986.
5. [5] G. E. Box, G. M. Jenkins, G. C. Reinsel and G. M. Ljung, Time Series Analysis: Forecasting and Control, John Wiley & Sons, 2015.
6. [6] N. A. Cressie, Statistics for Spatial Data, John Wiley & Sons, 1993.
7. [7] R. Dahlhaus and S. Subba Rao, Statistical inference for time-varying ARCH processes, Ann. Statist. 34, 1075–1114, 2006.
8. [8] F. X. Diebold and R. S. Mariano, Comparing predictive accuracy, J. Bus. Econ. Stat. 13 (3), 253–263, 1995.

9. [9] F. Djebari and P. Otto, Forecasting Oil and Carbon Volatility through Network Models with GARCH-Informed Correlation Weights, J. Econometrics (Preprint), 2026.
10. [10] R. F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50 (4), 987–1007, 1982.
11. [11] J. Fan, Local linear regression smoothers and their minimax efficiencies, Ann. Statist. 21 (1), 196–216, 1993.
12. [12] C. Francq and J. M. Zakoian, GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons, 2019.
13. [13] P. Fryzlewicz, T. Sapatinas and S. Subba Rao, Normalized least-squares estimation in time-varying ARCH models, Ann. Statist. 36 (2), 742–786, 2008.
14. [14] A. E. Gelfand, H. J. Kim, C. F. Sirmans and S. Banerjee, Spatial modeling with spatially varying coefficient processes, J. Amer. Statist. Assoc. 98 (462), 387–396, 2003.
15. [15] R. Giacometti, G. Torri, K. Rujirarangsan and M. Cameletti, Spatial Multivariate GARCH Models and Financial Spillovers, J. Risk Financial Management 16 (9), 397, 2023.
16. [16] E. Gilleland and D. Nychka, Statistical models for monitoring and regulating groundlevel ozone, Environmetrics 16 (5), 535–546, 2005.
17. [17] P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, 1980.
18. [18] S. Holleland and H. A. Karlsen, A stationary spatio-temporal GARCH model, J. Time Ser. Anal. 41 (2), 177–209, 2020.
19. [19] D. Kapadia and N. Jariwala, Prediction of tropospheric ozone using artificial neural network (ANN) and feature selection techniques, Model. Earth Syst. Environ. 8, 2183– 2192, 2022.
20. [20] S. Kharfouchi and H. Mehri, Conditions for the existence of stationary and causal space-time bilinear model and Central Limit Theorem, Int. J. Math. Oper. Res. 4 (6), 638–650, 2012.
21. [21] S. Kharfouchi, Inference for 2-D GARCH models, Statist. Probab. Lett. 92, 99–108, 2014.
22. [22] S. N. Lahiri and K. Mukherjee, Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs, Ann. Inst. Statist. Math. 56, 225–250, 2004.
23. [23] S. Das, T. A. Kuffner, S. N. Lahiri and D. J. Nordman, Second-Order Properties of the Convolved Subsampling Method for Time Series, J. Time Series Anal. 47 (1), 77–89, 2026.
24. [24] A. N. Meli Chrisko, P. Otto and W. Schmid, Exponential Spatiotemporal GARCH Model with Asymmetric Volatility Spillovers, arXiv preprint arXiv:2511.05126, 2025.
25. [25] P. Otto, W. Schmid and R. Garthoff, Generalized spatial and spatiotemporal autoregressive conditional heteroskedasticity, J. Appl. Statist. 43 (8), 1528–1544, 2016.
26. [26] P. Otto and W. Schmid, Spatial and spatiotemporal GARCH models – A unified approach, arXiv:1908.08320, 2019.
27. [27] J. Pan, K. He, K. Wang, Q. Mu and C. Ling, Spatiotemporal joint analysis of PM2.5 and Ozone in California with INLA approach, J. Environ. Manage. 363, 121294, 2024.
28. [28] S. Subba Rao, Statistical analysis of a spatio-temporal model with location-dependent parameters and a test for spatial stationarity, J. Time Ser. Anal. 29 (4), 673–694, 2008.
29. [29] T. Sato and Y. Matsuda, Spatial autoregressive conditional heteroskedasticity models, J. Japan Statist. Soc. 47 (2), 221–236, 2017.
30. [30] B. Seifert and M. Gächter, Finite-sample variance of local polynomials: Analysis and solutions, J. Amer. Statist. Assoc. 91 (434), 267-275, 1996.
31. [31] R. Tibshirani and T. Hastie, Local likelihood estimation, J. Amer. Statist. Assoc. 82 (398), 559–567, 1987.
32. [32] D. Xiang and Z. Hong, Local-Linear Two-Stage Estimation of Local Autoregressive Geographically and Temporally Weighted Regression Model, ISPRS Int. J. Geo-Inf. 14 (7), 276, 2025.
33. [33] D. Zhu, C. Cai, T. Yang and X. Zhou, A machine learning approach for air quality prediction: Model regularization and optimization, Big Data Cogn. Comput. 2 (1), 5, 2018.
34. [34] W. K. Newey and D. McFadden, Large sample estimation and hypothesis testing, in: R. F. Engle and D. McFadden (eds.), Handbook of Econometrics, Vol. 4, Elsevier, Amsterdam, 2111–2245, 1994.