Monotone functional regression with hybrid depth weighting and block-conformal prediction bands for forward realized variance paths

Çağlar SÖZEN; Onur ŞEYRANLIOĞLU; Arif ÇİLEK; Abdulmuttalip PİLATİN; Çağlar SÖZEN; Onur ŞEYRANLIOĞLU; Arif ÇİLEK; Abdulmuttalip PİLATİN

doi:10.3934/math.2026753

AIMS Mathematics

2026, Volume 11, Issue 6: 18525-18552. doi: 10.3934/math.2026753

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Research article

Monotone functional regression with hybrid depth weighting and block-conformal prediction bands for forward realized variance paths

1.
Görele School of Applied Sciences, Department of Finance and Banking, Giresun University, Giresun, Turkey
2.
Faculty of Economics and Administrative Sciences, Department of Business Administration, Giresun University, Giresun, Turkey
3.
Bulancak School of Applied Sciences, Department of International Trade and Finance, Giresun University, Giresun, Turkey
4.
Department of Finance and Banking, Recep Tayyip Erdoğan University, Rize, Turkey

Received: 03 April 2026 Revised: 08 June 2026 Accepted: 15 June 2026 Published: 23 June 2026
MSC : 62M10, 62M20, 62P20, 91G70

We forecast forward realized variance (FRV) paths, defined as cumulative future daily variance proxy curves over a finite trading horizon, using a leakage-disciplined functional framework for multiday risk assessment. The framework combines multiresponse ridge regression, hybrid depth weighting, horizon-weighted blocked cross-validation, and isotonic post-projection to preserve the monotone structure of FRV paths. Uncertainty is summarized through upper one-sided block-calibrated conformal bands, interpreted as empirical risk envelopes under temporal dependence rather than exact distribution-free guarantees. In a fixed panel design for four liquid exchange-traded funds, GDX, GDXJ, XLE, and UUP, over the period 2010–2025, the proposed model reduces long-horizon mean squared error relative to rolling historical FRV by approximately 31.8%, 20.4%, 36.5%, and 28.0%, respectively, over $h = 20{:}30$. Comparisons with heterogeneous autoregressive (HAR) ridge and functional principal component autoregressive (FPCA-AR) benchmarks are asset-dependent. The proposed model is most favorable for GDX and remains close to HAR ridge for GDXJ, whereas HAR ridge and FPCA-AR remain competitive for XLE and UUP. Coverage is conservative or close to nominal at $\alpha = 0.05$ but more heterogeneous at $\alpha = 0.10$. Robustness checks support a cautious interpretation of the method as a shape-aware enhancement of rolling FRV forecasting.
- forward realized variance,
- functional regression,
- depth weighting,
- isotonic projection,
- block-conformal prediction,
- liquid exchange-traded funds
Citation: Çağlar SÖZEN, Onur ŞEYRANLIOĞLU, Arif ÇİLEK, Abdulmuttalip PİLATİN. Monotone functional regression with hybrid depth weighting and block-conformal prediction bands for forward realized variance paths[J]. AIMS Mathematics, 2026, 11(6): 18525-18552. doi: 10.3934/math.2026753

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Abstract

We forecast forward realized variance (FRV) paths, defined as cumulative future daily variance proxy curves over a finite trading horizon, using a leakage-disciplined functional framework for multiday risk assessment. The framework combines multiresponse ridge regression, hybrid depth weighting, horizon-weighted blocked cross-validation, and isotonic post-projection to preserve the monotone structure of FRV paths. Uncertainty is summarized through upper one-sided block-calibrated conformal bands, interpreted as empirical risk envelopes under temporal dependence rather than exact distribution-free guarantees. In a fixed panel design for four liquid exchange-traded funds, GDX, GDXJ, XLE, and UUP, over the period 2010–2025, the proposed model reduces long-horizon mean squared error relative to rolling historical FRV by approximately 31.8%, 20.4%, 36.5%, and 28.0%, respectively, over $h = 20{:}30$. Comparisons with heterogeneous autoregressive (HAR) ridge and functional principal component autoregressive (FPCA-AR) benchmarks are asset-dependent. The proposed model is most favorable for GDX and remains close to HAR ridge for GDXJ, whereas HAR ridge and FPCA-AR remain competitive for XLE and UUP. Coverage is conservative or close to nominal at $\alpha = 0.05$ but more heterogeneous at $\alpha = 0.10$. Robustness checks support a cautious interpretation of the method as a shape-aware enhancement of rolling FRV forecasting.

References

[1]	T. G. Andersen, T. Bollerslev, Answering the skeptics: Yes, standard volatility models do provide accurate forecasts, Int. Econ. Rev., 39 (1998), 885–905. https://doi.org/10.2307/2527343 doi: 10.2307/2527343
[2]	O. E. Barndorff-Nielsen, N. Shephard, Econometric analysis of realized volatility and its use in estimating stochastic volatility models, J. R. Stat. Soc. B, 64 (2002), 253–280. https://doi.org/10.1111/1467-9868.00336 doi: 10.1111/1467-9868.00336
[3]	T. G. Andersen, T. Bollerslev, F. X. Diebold, P. Labys, Modeling and forecasting realized volatility, Econometrica, 71 (2003), 579–625. https://doi.org/10.1111/1468-0262.00418 doi: 10.1111/1468-0262.00418
[4]	F. Corsi, A simple approximate long-memory model of realized volatility, J. Financ. Economet., 7 (2009), 174–196. https://doi.org/10.1093/jjfinec/nbp001 doi: 10.1093/jjfinec/nbp001
[5]	A. E. Hoerl, R. W. Kennard, Ridge regression: Biased estimation for nonorthogonal problems, Technometrics, 12 (1970), 55–67. https://doi.org/10.1080/00401706.1970.10488634 doi: 10.1080/00401706.1970.10488634
[6]	R. Fraiman, G. Muniz, Trimmed means for functional data, TEST, 10 (2001), 419–440. https://doi.org/10.1007/BF02595706 doi: 10.1007/BF02595706
[7]	S. López-Pintado, J. Romo, On the concept of depth for functional data, J. Am. Stat. Assoc., 104 (2009), 718–734. https://doi.org/10.1198/jasa.2009.0108 doi: 10.1198/jasa.2009.0108
[8]	S. Nagy, I. Gijbels, D. Hlubinka, Depth-based recognition of shape outlying functions, J. Comput. Graph. Stat., 26 (2017), 883–893. https://doi.org/10.1080/10618600.2017.1336445 doi: 10.1080/10618600.2017.1336445
[9]	C. Bergmeir, J. M. Benítez, On the use of cross-validation for time series predictor evaluation, Inform. Sci., 191 (2012), 192–213. https://doi.org/10.1016/j.ins.2011.12.028 doi: 10.1016/j.ins.2011.12.028
[10]	C. Bergmeir, R. J. Hyndman, B. Koo, A note on the validity of cross-validation for evaluating autoregressive time series prediction, Comput. Stat. Data Anal., 120 (2018), 70–83. https://doi.org/10.1016/j.csda.2017.11.003 doi: 10.1016/j.csda.2017.11.003
[11]	M. Ayer, H. D. Brunk, G. M. Ewing, W. T. Reid, E. Silverman, An empirical distribution function for sampling with incomplete information, Ann. Math. Statis., 26 (1955), 641–647. https://doi.org/10.1214/aoms/1177728423 doi: 10.1214/aoms/1177728423
[12]	R. E. Barlow, Statistical inference under order restrictions: The theory and application of isotonic regression, New York: John Wiley & Sons, 1972.
[13]	M. J. Best, N. Chakravarti, Active set algorithms for isotonic regression: A unifying framework, Math. Program., 47 (1990), 425–439. https://doi.org/10.1007/BF01580873 doi: 10.1007/BF01580873
[14]	V. Vovk, A. Gammerman, G. Shafer, Algorithmic learning in a random world, New York: Springer, 2005. https://doi.org/10.1007/b106715
[15]	J. Lei, L. Wasserman, Distribution-free prediction bands for nonparametric regression, J. R. Stat. Soc. B, 76 (2014), 71–96. https://doi.org/10.1111/rssb.12021 doi: 10.1111/rssb.12021
[16]	I. Gibbs, E. Candès, Adaptive conformal inference under distribution shift, Adv. Neural Inform. Process. Syst., 34 (2021), 1660–1672.
[17]	M. Parkinson, The extreme value method for estimating the variance of the rate of return, J. Bus., 53 (1980), 61–65. https://doi.org/10.1086/296071 doi: 10.1086/296071
[18]	M. B. Garman, M. J. Klass, On the estimation of security price volatilities from historical data, J. Bus., 53 (1980), 67–78. https://doi.org/10.1086/296072 doi: 10.1086/296072
[19]	L. C. G. Rogers, S. E. Satchell, Estimating variance from high, low and closing prices, Ann. Appl. Probab., 1 (1991), 504–512. https://doi.org/10.1214/aoap/1177005835 doi: 10.1214/aoap/1177005835
[20]	J. O. Ramsay, B. W. Silverman, Functional data analysis, 2 Eds., New York: Springer, 2005. https://doi.org/10.1007/b98888
[21]	L. Horváth, P. Kokoszka, Inference for functional data with applications, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-3655-3
[22]	J. S. Morris, Functional regression, Annu. Rev. Stat. Appl., 2 (2015), 321–359. https://doi.org/10.1146/annurev-statistics-010814-020413 doi: 10.1146/annurev-statistics-010814-020413
[23]	F. Ferraty, P. Vieu, Nonparametric functional data analysis: Theory and practice, New York: Springer, 2006. https://doi.org/10.1007/0-387-36620-2
[24]	A. Cuevas, A partial overview of the theory of statistics with functional data, J. Stat. Plan. Infer., 147 (2014), 1–23. https://doi.org/10.1016/j.jspi.2013.04.002 doi: 10.1016/j.jspi.2013.04.002
[25]	Y. Romano, E. Patterson, E. Candès, Conformalized quantile regression, Adv. Neural Inform. Proces. Syst., 2019.
[26]	H. R. Künsch, The jackknife and the bootstrap for general stationary observations, Ann. Statist., 17 (1989), 1217–1241. https://doi.org/10.1214/aos/1176347265 doi: 10.1214/aos/1176347265
[27]	S. N. Lahiri, Resampling methods for dependent data, New York: Springer, 2003. https://doi.org/10.1007/978-1-4757-3803-2
[28]	P. T. Reiss, R. T. Ogden, Functional principal component regression and functional partial least squares, J. Am. Stat. Assoc., 102 (2007), 984–996. https://doi.org/10.1198/016214507000000527 doi: 10.1198/016214507000000527
[29]	T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics, 31 (1986), 307–327. https://doi.org/10.1016/0304-4076(86)90063-1 doi: 10.1016/0304-4076(86)90063-1
[30]	J. Lei, M. G'Sell, A. Rinaldo, R. J. Tibshirani, L. Wasserman, Distribution-free predictive inference for regression, J. Am. Stat. Assoc., 113 (2018), 1094–1111. https://doi.org/10.1080/01621459.2017.1307116 doi: 10.1080/01621459.2017.1307116
[31]	F. X. Diebold, R. S. Mariano, Comparing predictive accuracy, J. Bus. Econ. Stat., 13 (1995), 253–263. https://doi.org/10.1080/07350015.1995.10524599 doi: 10.1080/07350015.1995.10524599
[32]	W. K. Newey, K. D. West, A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55 (1987), 703–708. https://doi.org/10.2307/1913610 doi: 10.2307/1913610
[33]	P. R. Hansen, A. Lunde, J. M. Nason, The model confidence set, Econometrica, 79 (2011), 453–497. https://doi.org/10.3982/ECTA5771 doi: 10.3982/ECTA5771
[34]	Historical price data (Adjusted Close and OHLC) for GDX, GDXJ, XLE, and UUP, Yahoo Finance, 2025. Available from: https://finance.yahoo.com/.

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