Wavelet-\(L_2 E\) stochastic volatility models: an application to the water-energy nexus. (English) Zbl 07705112

Summary: Forecasting commodity markets are difficult due to the time-varying nature and complexity of the financial return series representing these markets. Over the past few decades, statistical tools have come about to remedy these challenges. These methods focus on identifying the time-varying behavior and incorporating the characteristics within proposed models. This paper augments a series of well-known stochastic volatility models by first applying the \(WaveL_2E\) thresholding method. The \(WaveL_2E\) denoises non-stationary time series by dynamic multivariate complex wavelet thresholding combined with multivariate minimum distance mixture density estimation. Volatility clustering within the signal is estimated through the models’ ability to identify abrupt changes in the mean behavior as well as accurately threshold groups of outliers. Through optimization and recovery of the mixture density parameters as time series, the \(WaveL_2E\) simultaneously identifies the signal and the time-dependent variance components. After an overview of the accuracy of the proposed models, we develop dynamic forecasts of the recovered signal. Our forecasted results show that the new \(WaveL_2E\) stochastic volatility models are promising forecasting tools for highly persistent returns. Finally, we demonstrate the utility of our methods by studying the exchange-traded funds (ETF) for water and energy, identifying a quarterly lead-lag relationship.


62-XX Statistics


astsa; brms; NUTS; BUGS; Stan
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[1] Akintunde, MO; Kgosi, PM; Shangodoyin, DK, Evaluation of GARCH model adequacy in forecasting non-linear economic time series data, J. Comput. Model., 3, 1792-8850 (2013)
[2] Athanasopoulos, G.; Hyndman, RJ; Kourentzes, N.; Petropoulos, F., Forecasting with temporal hierarchies, Eur. J. Oper. Res., 262, 60-74 (2017) · Zbl 1403.62154
[3] Bandi, FM; Russell, JR; Yang, C., Realized volatility forecasting in the presence of Time-Varying noise, J. Bus. Econ. Stat., 31, 331-345 (2013)
[4] Barunik, J.; Vacha, L., Realized wavelet-based estimation of integrated variance and jumps in the presence of noise, Quant. Finance, 15, 1347-1364 (2015) · Zbl 1406.91432
[5] Bera, AK; Higgins, ML, ARCH models: properties, estimation and testing, J. Econ. Surv., 7, 305-366 (1993)
[6] Berger, T., Forecasting based on decomposed financial return series: a wavelet analysis, J. Forecast., 35, 419-433 (2016)
[7] Berument, H.; Yalcin, Y.; Yildirim, J., The effect of inflation uncertainty on inflation: Stochastic volatility in mean model within a dynamic framework, Econ. Model., 26, 1201-1207 (2009)
[8] Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, J. Econ., 31, 307-327 (1986) · Zbl 0616.62119
[9] Bollerslev, T.; Chou, RY; Kroner, KF, ARCH modeling in finance: a review of the theory and empirical evidence, J. Econ., 52, 5-59 (1992) · Zbl 0825.90057
[10] Bürkner, P-C, brms: an R package for bayesian multilevel models using stan, J. Stat. Softw., 80, 1-28 (2017)
[11] Canale, A.; Ruggiero, M., Bayesian nonparametric forecasting of monotonic functional time series, Electron. J. Stat., 10, 3265-3286 (2016) · Zbl 1357.62278
[12] Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M. A., Guo, J., Li, P. and Riddell, A. (2017). Stan: a probabilistic programming language. J. Stat. Softw. 76(1). doi:10.18637/jss.v076.i01.
[13] Carriero, A.; Clark, TE; Marcellino, M., Measuring uncertainty and its impact on the economy, Rev. Econ. Stat., 100, 799-815 (2018)
[14] Carriero, A.; Galvão, AB; Kapetanios, G., A comprehensive evaluation of macroeconomic forecasting methods, Int. J. Forecast., 35, 1226-1239 (2019)
[15] Chan, JCC, The stochastic volatility in mean model with time-varying parameters: an application to inflation modeling, J. Bus. Econ. Stat., 35, 17-28 (2017)
[16] Chiann, C.; Morettin, PA, A wavelet analysis for time series, J.Nonparametric Stat., 10, 1-46 (1998) · Zbl 0922.62094
[17] Chib, S.; Nardari, F.; Shephard, N., Markov chain monte carlo methods for stochastic volatility models, J. Econ., 108, 281-316 (2002) · Zbl 1099.62539
[18] Daubechies, I. (1992). Ten lectures on wavelets. SIAM. · Zbl 0776.42018
[19] Deyoreo, M.; Kottas, A.; Deyoreo, BM, A Bayesian nonparametric Markovian model for non-stationary time series, Stat. Comput., 27, 1525-1538 (2017) · Zbl 1384.62287
[20] Engle, RF, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007 (1982) · Zbl 0491.62099
[21] Ensor, KB; Koev, GM, Computational finance: correlation, volatility, and markets, WIREs Comput. Stat., 6, 326-340 (2014)
[22] Faria, G.; Verona, F., Forecasting stock market returns by summing the frequency-decomposed parts, J. Empir. Finance, 45, 228-242 (2018)
[23] Frühwirth-Schnatter, S.; Wagner, H., Stochastic model specification search for Gaussian and partial non-Gaussian state space models, J. Econ., 154, 85-100 (2010) · Zbl 1431.62373
[24] Fouladi, SH; Hajiramezanali, M.; Amindavar, H.; Ritcey, JA; Arabshahi, P., Denoising based on multivariate stochastic volatility modeling of multiwavelet coefficients, IEEE Trans. Signal Process., 61, 5578-5589 (2013) · Zbl 1393.94226
[25] Gallant, AR; Rossi, PE; Tauchen, G., Stock prices and volume, Rev. Financ. Stud., 5, 199-242 (1992)
[26] Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Technical Report 3. · Zbl 1331.62139
[27] Ghosh, H.; Kumar, R.; Prajneshu, PA, The GARCH and EGARCH nonlinear time-series models for volatile data: an application, J. Stat. Appl., 5, 161-177 (2010)
[28] Ghysels, E., Harvey, A. C., Renault, E. M., Ghysels, E., Harvey, A. and Renault, E. (1996). Stochastic Volatility. In Statistical methods in finance. Centre Interuniversitaire De Recherche En Économie Quantitative, CIREQ.
[29] Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994. · Zbl 0725.62083
[30] Harvey, A.; Ruiz, E.; Shephard, N., Multivariate stochastic variance models, Rev. Econ. Stud., 61, 247-264 (1994) · Zbl 0805.90026
[31] Hoffman, MD; Gelman, A., The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, J. Mach. Learn. Res., 15, 1593-1623 (2014) · Zbl 1319.60150
[32] Hosszejni, D. and Kastner, G. (2019). Modeling univariate and multivariate stochastic volatility in R with stochvol and factorstochvol. · Zbl 1435.62109
[33] In, F. and Kim, S. (2012). An introduction to wavelet theory in finance : a wavelet multiscale approach, p. 204. World Scientific. · Zbl 1276.91003
[34] Joo, TW; Kim, SB, Time series forecasting based on wavelet filtering, Expert Syst. Appl., 42, 3868-3874 (2015)
[35] Kastner, G. and Frühwirth-Schnatter, S. (2017). Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models. Technical report. · Zbl 1506.62094
[36] Kastner, G.; Frühwirth-Schnatter, S.; Freitas Lopes Insper, H.; Lopes, HF, Efficient bayesian inference for multivariate factor stochastic volatility models, J. Comput. Graph. Stat., 26, 905-917 (2017)
[37] Kim, S.; Shephard, N.; Chib, S., Stochastic volatility : likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65, 361-393 (1998) · Zbl 0910.90067
[38] Koopman, SJ; Hol Uspensky, E., The stochastic volatility in mean model: empirical evidence from international stock markets, J. Appl. Econ., 17, 667-689 (2002)
[39] Leão, WL; Abanto-Valle, CA; Chen, MH, Bayesian analysis of stochastic volatility-in-mean model with leverage and asymmetrically heavy-tailed error using generalized hyperbolic skew Student’s t-distribution, Stat. Interface, 10, 529-541 (2017) · Zbl 1390.62212
[40] McGee, M.; Ensor, KB, Tests for harmonic components in the spectra of categorical time series, J. Time Ser. Anal., 19, 309-323 (1998) · Zbl 0906.62099
[41] Meyer Jun, RY; Meyer, R.; Yu, J., BUGS for a bayesian analysis of stochastic volatility models, Econom. J., 3, 1-17 (2000) · Zbl 0970.91060
[42] Moelya Artha, S. E., Yasin, H., Warsito, B. and Santoso, R. (2018). Suparti: application of wavelet neuro-fuzzy system (WNFS) method for stock forecasting. J. Phys.: Conf. Ser. 1-12. doi:10.1088/1742-6596/1025/1/012101.
[43] Mumtaz, H.; Zanetti, F., The impact of the volatility of monetary policy shocks, J. Money Credit Bank., 45, 535-558 (2013)
[44] Neal, RM, An improved acceptance procedure for the hybrid Monte Carlo algorithm, J. Comput. Phys., 111, 194-203 (1994) · Zbl 0797.65115
[45] Paul, RK, ARIMAX-GARCH-WAVELET model for forecasting volatile data, Model. Assist. Stat. Appl., 10, 243-252 (2015)
[46] Power, GJ; Eaves, J.; Turvey, C.; Vedenov, D., Catching the curl: wavelet thresholding improves forward curve modelling, Econ. Model., 64, 312-321 (2017)
[47] Raath, KC; Ensor, KB, Time-varying wavelet-based applications for evaluating the water-energy nexus, Front. Energy Res., 8, 118 (2020)
[48] Raath, K. C., Ensor, K. B., Scott, D. W. and Crivello, A. (2020). Denoising non-stationary signals by dynamic multivariate complex wavelet thresholding. doi:10.2139/ssrn.3528714.
[49] Risse, M., Combining wavelet decomposition with machine learning to forecast gold returns, Int. J. Forecast., 35, 601-615 (2019)
[50] Rostan, P.; Rostan, A., The versatility of spectrum analysis for forecasting financial time series, J. Forecast., 37, 327-339 (2018)
[51] Schlüter, S. and Deuschle, C. (2010). Using wavelets for time series forecasting-does it pay off? Technical report, Friedrich-Alexander University Erlangen-Nuremberg Institute for Economics.
[52] Scott, DW, Parametric statistical modeling by minimum integrated square error, Technometrics, 43, 274-285 (2001)
[53] Shumway, R. H. and Stoffer, D. S. (2005). Time series analysis and its applications (Springer texts in statistics). Springer.
[54] Sudheer, G.; Suseelatha, A., Short term load forecasting using wavelet transform combined with Holt-Winters and weighted nearest neighbor models, Int. J. Electr. Power Energy Syst., 64, 340-346 (2015)
[55] Taylor, SJ, Modeling stochastiv volatility: a review and comparative study, Math. Financ., 4, 183-204 (1994) · Zbl 0884.90054
[56] Taylor, JW; Yu, K., Using auto-regressive logit models to forecast the exceedance probability for financial risk management, J. R. Stat. Soc.: Ser. A (Stat. Soc.), 179, 1069-1092 (2016)
[57] Thomson, D.; Van Vuuren, G., Forecasting the south african business cycle using fourier analysis, Int. Bus. Econ. Res. J. (IBER), 15, 175 (2016)
[58] Tripathy, N.; Garg, A., Forecasting stock market volatility: evidence from six emerging markets, J. Int. Bus. Econ., 14, 25 (2013)
[59] Vankov, ER; Guindani, M.; Ensor, KB, Filtering and estimation for a class of stochastic volatility models with intractable likelihoods, Bayesian Anal., 14, 29-52 (2019) · Zbl 1409.62184
[60] Wilhelmsson, A., Garch forecasting performance under different distribution assumptions, J. Forecast., 25, 561-578 (2006)
[61] Wold, H., A study in analysis of stationary time series. By Herman Wold, J. R. Stat. Soc., 102, 295-298 (1939)
[62] Yule, GU, On a method of investigating periodicities in disturbed series, with special reference to wolfer’s sunspot numbers, Philos. Trans. R. Soc. Lond. Ser. A, 226, 267-298 (1927) · JFM 53.0509.02
[63] Zhang, Y-J; Zhang, J-L, Volatility forecasting of crude oil market: a new hybrid method, J. Forecast., 37, 781-789 (2018) · Zbl 1414.62507
[64] Zhang, K.; Gençay, R.; Yazgan, ME, Application of wavelet decomposition in time-series forecasting, Econ. Lett., 158, 41-46 (2017) · Zbl 1396.62221
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