Abstract

Advancements in technology have led to increasingly complex structures in high-frequency data, neces

sitating the development of efficient models for accurately forecasting realized measures. This paper introduces

a novel approach known as the multilinear low-rank heterogeneous autoregressive (MLRHAR) model. Distinguishing itself from the conventional heterogeneous autoregressive (HAR) model, our model utilizes a data-driven

method to replace the fixed heterogenous volatility components. To address the calendar effect, we utilize the

fourth-order tensor technique, which simultaneously reduces dimensions in the response, predictor, and short-term

and calendar temporal directions. This not only reduces the parameter space but also enables the automatic selection

of heterogeneous components from both temporal directions. Moreover, we establish the non-asymptotic properties of high-dimensional HAR models, and a projected gradient descent algorithm is proposed with theoretical

justifications for parameter estimation. Through simulation experiments, we evaluate the efficiency of the proposed

model. We apply our method to financial data on the constituent stocks of the S&P 500 Index. The results obtained

from both the simulation and real studies convincingly demonstrate the significant forecasting advantages offered

by our approach.

Key words and phrases: Calendar effect, Heterogeneous autoregressive model, High-dimensional analysis, High- frequency data, Non-asymptotic property, Tensor technique

Information

Preprint No.SS-2024-0308
Manuscript IDSS-2024-0308
Complete AuthorsHuiling Yuan, Guodong Li, Kexin Lu, Alan T.K. Wan, Yong Zhou
Corresponding AuthorsGuodong Li
Emailsgdli@hku.hk

References

  1. Agterberg, J. and A. R. Zhang (2024). Statistical inference for low-rank tensors: Heteroskedasticity, subgaussianity, and applications. arXiv:2410.06381v1.
  2. Andersen, T. G., T. Bollerslev, F. Diebold, and P. Labys (2003). Modeling and forecasting realized volatility. Econometrica 71, 579–625.
  3. Auddy, A. and M. Yuan (2025). Large dimensional independent component analysis: Statistical optimality and computational tractability. Annuals of Statistics 53, 477–505.
  4. Audrino, F. and S. D. Knaus (2016). Lassoing the HAR model: A model selection perspective on realized volatility dynamics. Econometric Reviews 35, 1485–1521.
  5. Basu, S. and G. Michailidis (2015). Regularized estimation in sparse high-dimensional time series models. The Annals of Statistics 43, 1535–1567.
  6. Bauer, G. H. and K. Vorkink (2011). Forecasting multivariate realized stock market volatility. Journal of Econometrics 160, 93–101.
  7. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.
  8. Bollerslev, T., A. J. Patton, and R. Quaedvlieg (2018). Modelling and forecasting (un)reliable realized covariances for more reliable financial decisions. Journal of Econometrics 207, 71–91.
  9. Bubák, V., E. Kočenda, and F. Žikeš (2011). Volatility transmission in emerging European foreign exchange markets. Journal of Banking & Finance 35, 2829–2841.
  10. Chen, H., G. Raskutti, and M. Yuan (2019). Non-convex projected gradient descent for generalized low-rank tensor regression. The Journal of Machine Learning Research 20, 172–208.
  11. Chen, Y., W. K. Härdle, and U. Pigorsch (2010). Localized realized volatility modeling. Journal of the American Statistical Association 105, 1376–1393.
  12. Clements, A. and D. P. A. Preve (2021). A practical guide to harnessing the HAR volatility model. Journal of Banking & Finance 133, 106285.
  13. Cleveland, W. S. and S. J. Devlin (1980). Calendar effects in monthly time series: Detection by spectrum analysis and graphical methods. Journal of the American Statistical Association 75, 487–496.
  14. Cleveland, W. S. and S. J. Devlin (1982). Calendar effects in monthly time series: Modeling and adjustment. Journal of the American Statistical Association 77, 520–528.
  15. Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7, 174–196.
  16. Cubadda, G., B. Guardabascio, and A. Hecq (2017). A vector heterogeneous autoregressive index model for realized volatility measures. International Journal of Forecasting 33, 337–344.
  17. De Lathauwer, L., B. De Moor, and J. Vandewalle (2000). A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications 21, 1253–1278.
  18. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987–1007.
  19. Engle, R. F. and G. M. Gallo (2006). A multiple indicators model for volatility using intra-daily data. Journal of Econometrics 131, 3–27.
  20. Ghysels, E., P. Santa-Clara, and R. Valkanov (2006). Predicting volatility: Getting the most out of return data sampled at different frequencies. Journal of Econometrics 131, 59–95.
  21. Han, R., R. Willett, and A. R. Zhang (2022). An optimal statistical and computational framework for generalized tensor estimation.
  22. Han, Y., R. Chen, D. Yang, and C.-H. Zhang (2024). Tensor factor model estimation by iterative projection. Annuals of Statistics 52, 2541–2667.
  23. Hansen, P. R., Z. Huang, and H. H. Shek (2012). Realized GARCH: A joint model for returns and realized measures of volatility. Journal of Applied Econometrics 27, 877–906.
  24. Hillar, C. J. and L.-H. Lim (2013). Most tensor problems are NP-hard. Journal of the ACM (JACM) 60(6), 1–39.
  25. Hong, W. T., J. Lee, and E. Hwang (2020). A note on the asymptotic normality theory of the least squares estimates in multivariate HAR-RV models. Mathematics 8, doi:10.3390/math8112083.
  26. Levy, T. and Y. Joseph (2012). The week-of-the-year effect: Evidence from around the globe. Journal of Banking & Finance 36, 1963–1974.
  27. Liu, L. Y., A. J. Patton, and K. Sheppard (2015). Does anything beat 5-minute RV? A comparison of realized measures across multiple asset classes. Journal of Econometrics 187, 293–311.
  28. Patton, A. J. and K. Sheppard (2015). Good volatility, bad volatility: Signed jumps and the persistence of volatility. The Reviews of Economics and Statistics 97, 683–697.
  29. Proietti, T. and D. J. Pedregal (2023). Sensonality in high frequency time series. Econometrics and Statistics 27, 62–82.
  30. Shephard, N. and K. Sheppard (2010). Realising the future: Forecasting with high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25, 197–231.
  31. Shin, M., D. Kim, Y. Wang, and J. Fan (2025). Factor and idiosyncratic VAR volatility matrix models for heavy-tailed high-frequency financial observations. Journal of Econometrics 252, doi: 10.1016/j.jeconom.2025.106129.
  32. Souček, M. and N. Todorova (2013). Realized volatility transmission between crude oil and equity futures markets: A multivariate HAR approach. Energy Economics 40, 586–597. American Statistical Association 97, 1167–1179.
  33. Sullivan, R., A. Timmermann, and H. White (2001). Dangers of data mining: The case of calendar effects in stock returns. Journal of Econometrics 105, 249–286.
  34. Taylor, N. (2017). Realised variance forecasting under Box-Cox transformations. International Journal of Forecasting 33, 770–785.
  35. Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society: Series B 58, 267–288.
  36. Wainwright, M. J. (2019). High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge: Cambridge University Press.
  37. Xia, D., A. R. Zhang, and Y. Zhou (2022). Inference for low-rank tensors-no need to debias. Annuals of Statistics 50(2), 1220–1245.
  38. Xu, K., E. Chen, and Y. Han (2025). Statistical inference for low-rank tensor models. arXiv:2501.16223v1.
  39. Yuan, H., K. Lu, and G. Li (2025). Volatility analysis with high-frequency and low-frequency historical data, and options-implied information. Statistica Sinica 35, 2305–2323.

Acknowledgments

We are deeply grateful to the editor, an associate editor and two anonymous referees for their

valuable comments that led to the substantial improvement of the manuscript. Yuan’s research

was partially supported by National Natural Science Foundation of China (No. 72403089), State

Key Program of National Natural Science Foundation of China (No. 72531003), 2025 Shanghai

Bai Yulan Talent Young Program, Shanghai Pujiang Program (No. 23PJ1402400), and China

Postdoctoral Science Foundation (No. 2023M741190). Li’s research was partially supported by

GRF grants 17313722 and 17309625 from the Hong Kong Research Grant Council. Wan’s research

was partially supported by National Natural Science Foundation of China (No. 72273120). Zhou’s

research was partially supported by State Key Program of National Natural Science Foundation of

China (No. 72531003).

Supplementary Materials

The online Supplementary Material contains the tensor notations and Tucker decomposition, the

proofs of the two theorems, Corollary 1, simulation results for the MLR-TT-HAR model, and one

Table for the selected ranks of the MLR-FT-HAR, MLR-TT-HAR and VHARI models in Real data

analysis.

Supplementary materials are available for download.