Time series forecasting with the CIR# model: from hectic markets sentiments to regular seasonal tourism
Abstract
This research aims to propose the so-called CIR#, which takes its cue from the well- known Cox-Ingersoll-Ross (CIR) model originally devised for pricing, as a general econometric model. To this end, we present the results on two very different time series such as Polish interest rates (subject to market sentiments) and seasonal tourism (subject to pandemic lock-down measures). For interest rates, as reference models, we consider an improved version of the CIR model (denoted CIRadj), the Hull and White model, the exponentially weighted moving average (EWMA) which is often adopted whenever no structure is assumed in the data and a popular machine learning model such as the short-term memory network (LSTM). For tourism, as a benchmark, we consider seasonal autoregressive integrated moving average (SARIMA) complemented by the generalized autoregressive conditional heteroskedasticity (GARCH) for modelling the variance, the classic Holt-Winters model and the aforementioned LSTM. Results support the claim that the CIR# performs better than the other models in all considered cases being able to deal with erratic behaviour in data.
Keyword : tourism demand prediction, interest rate forecasting, cluster volatility and jumps fitting, SARIMA, CIR model, Hull and White model
How to Cite
Orlando, G., & Bufalo, M. (2023). Time series forecasting with the CIR# model: from hectic markets sentiments to regular seasonal tourism. Technological and Economic Development of Economy, 29(4), 1216–1238. https://doi.org/10.3846/tede.2023.19294
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Akimov, A., Lee, C. L., & Stevenson, S. (2019). Interest rate sensitivity in European public real estate markets. Journal of Real Estate Portfolio Management, 25(2), 138–150. https://doi.org/10.1080/10835547.2020.1803694
Allcock, J. B. (1989). Seasonality. In Witt, S. F. & Moutinho, L. (Eds.), Tourism marketing and management handbook (pp. 387–392). Prentice Hall.
Ampountolas, A. (2021). Modeling and forecasting daily hotel demand: A comparison based on SARIMAX, neural networks, and GARCH models. Forecasting, 3(3), 580–595. https://doi.org/10.3390/forecast3030037
Ascione, G., Mehrdoust, F., Orlando, G., & Samimi, O. (2023). Foreign exchange options on Heston-CIR model under Levy process framework. Applied Mathematics and Computation, 446, 1–31. https://doi.org/10.1016/j.amc.2023.127851
Asteriou, D., & Hall, S. G. (2011). ARIMA models and the Box–Jenkins methodology. Applied Econometrics, 2(2), 265–286.
Bank for International Settlements. (2015). Is the unthinkable becoming routine? Technical report. BIS.
Baum, T., & Lundtorp, S. (2001). Seasonality in tourism: Issues and implications. In Seasonality in Tourism (pp. 13–30). Routledge. https://doi.org/10.4324/9780080516806-6
Bibby, B. M., Jacobsen, M., & Sørensen, M. (2010). Estimating functions for discretely sampled diffusion-type models. In Y. Ait-Sahalia. & L. P. Hansen (Eds.), Handbook of financial econometrics: Tools and techniques (vol. 1, pp. 203–268). North-Holland, Oxford. https://doi.org/10.1016/B978-0-444-50897-3.50007-9
Bjørnland, H. C., & Hungnes, H. (2006). The importance of interest rates for forecasting the exchange rate. Journal of Forecasting, 25(3), 209–221. https://doi.org/10.1002/for.983
Bollerslev, T. (2008). Glossary to ARCH (GARCH). CREATES Research paper 2008-49. SSRN. https://doi.org/10.2139/ssrn.1263250
Brigo, D., & El-Bachir, N. (2006). Credit derivatives pricing with a smile-extended jump stochastic intensity model (ICMA Centre Discussion Papers in Finance DP2006-13). SSRN. https://doi.org/10.2139/ssrn.950208
Brigo, D., & Mercurio, F. (2000). The CIR++ model and other deterministic- shift extensions of short rate models. In Proceedings of the 4th Columbia-JAFEE Conference for Mathematical Finance and Financial Engineering (pp. 563–584). https://doi.org/10.2139/ssrn.292060
Brigo, D., & Mercurio, F. (2001). A deterministic-shift extension of analytically-tractable and time-homogeneous short rate models. Finance and Stochastics, 5, 369–387. https://doi.org/10.1007/PL00013541
Brigo, D., & Mercurio, F. (2006). Interest rate models – Theory and practice: With smile, inflation and credit (2nd ed.). Springer-Verlag.
Butler, R. (1998). Seasonality in tourism: Issues and implications. The Tourist Review, 53(3), 18–24. https://doi.org/10.1108/eb058278
Carmona, R. A., & Tehranchi, M. R. (2006). Interest rate models: An infinite dimensional stochastic analysis perspective. Springer-Verlag.
Chang, Y.-W., & Liao, M.-Y. (2010). A seasonal ARIMA model of tourism forecasting: The case of Taiwan. Asia Pacific Journal of Tourism Research, 15(2), 215–221. https://doi.org/10.1080/10941661003630001
Chen, L. (1996). Stochastic mean and stochastic volatility: A three-factor model of the term structure of interest rates and its applications and its applications in derivatives pricing and risk management. Blackwell Publishers. https://doi.org/10.1007/978-3-642-46825-4_1
Choden, & Unhapipat, S. (2018). ARIMA model to forecast international tourist visit in Bumthang, Bhutan. Journal of Physics: Conference Series, 1039, 012023. https://doi.org/10.1088/1742-6596/1039/1/012023
Claveria, O., Monte, E., & Torra, S. (2017). Data pre-processing for neural network-based forecasting: Does it really matter? Technological and Economic Development of Economy, 23(5), 709–725. https://doi.org/10.3846/20294913.2015.1070772
Corluka, G. (2019). Tourism seasonality – an overview. Journal of Business Paradigms, 4(1), 21–43.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. https://doi.org/10.2307/1911242
Duffie, D. (2005). Credit risk modeling with affine processes. Journal of Banking & Finance, 29(11), 2751–2802. https://doi.org/10.1016/j.jbankfin.2005.02.006
Dutta, A., Mishra, T., Uddin, G. S., & Yang, Y. (2021). Brexit uncertainty and volatility persistence in tourism demand. Current Issues in Tourism, 24(16), 2225–2232. https://doi.org/10.1080/13683500.2020.1822300
Engelen, K. C. (2015). The unthinkable as the new normal. The International Economy, 29(3), 30.
Eurostat. (2022). Database – Tourism. https://ec.europa.eu/eurostat/web/tourism/data/database
Grundey, D. (2008). Managing sustainable tourism in Lithuania: Dream or reality? Technological and Economic Development of Economy, 14(2), 118–129. https://doi.org/10.3846/1392-8619.2008.14.118-129
Gruppe, M., Basse, T., Friedrich, M., & Lange, C. (2017). Interest rate convergence, sovereign credit risk and the European debt crisis: A survey. Journal of Risk Finance, 18(4), 432–442. https://doi.org/10.1108/JRF-01-2017-0013
He, K., Ji, L., Wu, C. W. D., & Tso, K. F. G. (2021). Using SARIMA–CNN–LSTM approach to forecast daily tourism demand. Journal of Hospitality and Tourism Management, 49, 25–33. https://doi.org/10.1016/j.jhtm.2021.08.022
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327–343. https://doi.org/10.1093/rfs/6.2.327
Hochberg, Y., & Tamhane, A. C. (1989). Multiple comparison procedures. John Wiley & Sons.
Holt, C. C. (1957). Forecasting seasonals and trends by exponentially weighted moving averages (Office of Naval Research Memorandum, vol. 52). Carnegie Institute of Technology.
Holt, C. C. (2004). Forecasting seasonals and trends by exponentially weighted moving averages. International Journal of Forecasting, 20(1), 5–10. https://doi.org/10.1016/j.ijforecast.2003.09.015
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. The Review of Financial Studies, 3(4), 573–592. https://doi.org/10.1093/rfs/3.4.573
Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice (2nd ed.). OTexts.
Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149–176. https://doi.org/10.1093/biomet/36.1-2.149
Keller-Ressel, M., & Steiner, T. (2008). Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance and Stochastics, 12(2), 149–172. https://doi.org/10.1007/s00780-007-0059-z
Keynes, J. M. (1936). The general theory of the rate of interest. In The General theory of employment, interest, and money (pp. 145–153). Springer. https://doi.org/10.1007/978-3-319-70344-2_13
Kladıvko, K. (2007). Maximum likelihood estimation of the Cox-Ingersoll- Ross process: The Matlab implementation. https://it.mathworks.com/matlabcentral/fileexchange/37297-maximum-likelihood-estimation\-of-the-cox-ingersoll-ross-process-the-matlab-implementation
Kudo, M., Toyama, J., & Shimbo, M. (1999). Multidimensional curve classification using passing-through regions. Pattern Recognition Letters, 20(11), 1103–1111. https://doi.org/10.1016/S0167-8655(99)00077-X
Li, X., Law, R., Xie, G., & Wang, S. (2021). Review of tourism forecasting research with internet data. Tourism Management, 83, 104245. https://doi.org/10.1016/j.tourman.2020.104245
Li, Y., & Cao, H. (2018). Prediction for tourism flow based on LSTM neural network. Procedia Computer Science, 129, 277–283. https://doi.org/10.1016/j.procs.2018.03.076
Lim, C., & McAleer, M. (2001). Forecasting tourist arrivals. Annals of Tourism Research, 28(4), 965–977. https://doi.org/10.1016/S0160-7383(01)00006-8
Lock, S. (2022). Global tourism industry – statistics & facts. https://www.statista.com/topics/962/global-tourism/#dossierContents__outerWrapper
Lu, W., Li, J., Li, Y., Sun, A., & Wang, J. (2020). A CNN-LSTM-based model to forecast stock prices. Complexity, 2020, 6622927. https://doi.org/10.1155/2020/6622927
MATLAB. (2022). MathWorks. https://www.mathworks.com/products/matlab.html
Mininni, M., Orlando, G., & Taglialatela, G. (2020). Challenges in approximating the black and scholes call formula with hyperbolic tangents. Decisions in Economics and Finance, 1–28. https://doi.org/10.1007/s10203-020-00305-8
Moreno, M., & Platania, F. (2015). A cyclical square-root model for the term structure of interest rates. European Journal of Operational Research, 241(1), 109–121. https://doi.org/10.1016/j.ejor.2014.08.010
Najafi, A. R., & Mehrdoust, F. (2017). Bond pricing under mixed generalized CIR model with mixed Wishart volatility process. Journal of Computational and Applied Mathematics, 319(C), 108–116. https://doi.org/10.1016/j.cam.2016.12.039
Orlando, G., & Bufalo, M. (2021). Interest rates forecasting: Between Hull and White and the CIR#. How to make a single factor model work. Journal of Forecasting, 40(8), 1566–1580. https://doi.org/10.1002/for.2783
Orlando, G., Mininni, R. M., & Bufalo, M. (2018). A new approach to CIR short-term rates modelling. In Mili, M., Samaniego Medina, R., & di Pietro, F. (Eds.), New methods in fixed income modeling – fixed income modeling (pp. 35–44). Springer International. https://doi.org/10.1007/978-3-319-95285-7_2
Orlando, G., Mininni, R. M., & Bufalo, M. (2019a). Interest rates calibration with a CIR model. The Journal of Risk Finance, 20(4), 370–387. https://doi.org/10.1108/JRF-05-2019-0080
Orlando, G., Mininni, R. M., & Bufalo, M. (2019b). A new approach to forecast market interest rates through the CIR model. Studies in Economics and Finance, 37(2), 267–292. https://doi.org/10.1108/SEF-03-2019-0116
Orlando, G., Mininni, R. M., & Bufalo, M. (2020). Forecasting interest rates through Vasicek and CIR models: A partitioning approach. Journal of Forecasting, 39(4), 569–579. https://doi.org/10.1002/for.2642
Perry, M. B. (2010). The exponentially weighted moving average. Wiley Encyclopedia of Operations Research and Management Science. https://doi.org/10.1002/9780470400531.eorms0314
Polyzos, S., Samitas, A., & Spyridou, A. Ef. (2021). Tourism demand and the COVID-19 pandemic: An LSTM approach. Tourism Recreation Research, 46(2), 175–187. https://doi.org/10.1080/02508281.2020.1777053
Qadeer, K., Rehman, W. U., Sheri, A. M., Park, I., Kim, H. K., & Jeon, M. (2020). A Long Short-Term Memory (LSTM) network for hourly estimation of PM2.5 concentration in two cities of South Korea. Applied Sciences, 10(11), 3984. https://doi.org/10.3390/app10113984
Santamaria, D., & Filis, G. (2019). Tourism demand and economic growth in Spain: New insights based on the yield curve. Tourism Management, 75, 447–459. https://doi.org/10.1016/j.tourman.2019.06.008
Schwarzbach, C., Kunze, F., Rudschuck, N., & Windels, T. (2012). Asset management in the German insurance industry: The quality of interest rate forecasts. Zeitschrift für die gesamte Versicherungswissenschaft, 101(5), 693–703. https://doi.org/10.1007/s12297-012-0218-y
Shanika, A., & Jahufer, A. (2021). Volatility analysis of international tourist arrivals to Sri Lanka using GARCH models. Faculty of Applied Sciences, South Eastern University of Sri Lanka, Sammanthurai.
Supriatna, A., Lesmana, E., Aridin, L., Sukono, & Napitupulu, H. (2019). Comparison between multiplicative Holt Winter and decomposition method in predicting the number of incoming international tourists to Indonesia. IOP Conference Series: Materials Science and Engineering, 567(1), 012047. https://doi.org/10.1088/1757-899X/567/1/012047
Thakur, B. P. S., Kannadhasan, M., & Goyal, V. (2018). Determinants of corporate credit spread: Evidence from India. Decision, 45(1), 59–73. https://doi.org/10.1007/s40622-018-0179-7
Trading Economics. (2023). Polish Zloty - 2023 Data – 1993–2022 Historical - Quote. https://tradingeconomics.com/poland/currency
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188. https://doi.org/10.1016/0304-405X(77)90016-2
Winters, P. R. (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6(3), 324–342. https://doi.org/10.1287/mnsc.6.3.324
Yıldırım, D. C., Toroslu, I. H., & Fiore, U. (2021). Forecasting directional movement of Forex data using LSTM with technical and macroeconomic indicators. Financial Innovation, 7(1), 1–36. https://doi.org/10.1186/s40854-020-00220-2
Zhao, Z., Chen, W., Wu, X., Chen, P. C. Y., & Liu, J. (2017). LSTM network: A deep learning approach for short-term traffic forecast. IET Intelligent Transport Systems, 11(2), 68–75. https://doi.org/10.1049/iet-its.2016.0208
Zhu, L. (2014). Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps. Journal of Applied Probability, 51(3), 699–712. https://doi.org/10.1239/jap/1409932668
Zong, C.-L., & Wang, L. (2018). Prediction of urban residents’ travel rate in China based on ARIMA models. Journal of Interdisciplinary Mathematics, 21(5), 1285–1290. https://doi.org/10.1080/09720502.2018.1497999
Allcock, J. B. (1989). Seasonality. In Witt, S. F. & Moutinho, L. (Eds.), Tourism marketing and management handbook (pp. 387–392). Prentice Hall.
Ampountolas, A. (2021). Modeling and forecasting daily hotel demand: A comparison based on SARIMAX, neural networks, and GARCH models. Forecasting, 3(3), 580–595. https://doi.org/10.3390/forecast3030037
Ascione, G., Mehrdoust, F., Orlando, G., & Samimi, O. (2023). Foreign exchange options on Heston-CIR model under Levy process framework. Applied Mathematics and Computation, 446, 1–31. https://doi.org/10.1016/j.amc.2023.127851
Asteriou, D., & Hall, S. G. (2011). ARIMA models and the Box–Jenkins methodology. Applied Econometrics, 2(2), 265–286.
Bank for International Settlements. (2015). Is the unthinkable becoming routine? Technical report. BIS.
Baum, T., & Lundtorp, S. (2001). Seasonality in tourism: Issues and implications. In Seasonality in Tourism (pp. 13–30). Routledge. https://doi.org/10.4324/9780080516806-6
Bibby, B. M., Jacobsen, M., & Sørensen, M. (2010). Estimating functions for discretely sampled diffusion-type models. In Y. Ait-Sahalia. & L. P. Hansen (Eds.), Handbook of financial econometrics: Tools and techniques (vol. 1, pp. 203–268). North-Holland, Oxford. https://doi.org/10.1016/B978-0-444-50897-3.50007-9
Bjørnland, H. C., & Hungnes, H. (2006). The importance of interest rates for forecasting the exchange rate. Journal of Forecasting, 25(3), 209–221. https://doi.org/10.1002/for.983
Bollerslev, T. (2008). Glossary to ARCH (GARCH). CREATES Research paper 2008-49. SSRN. https://doi.org/10.2139/ssrn.1263250
Brigo, D., & El-Bachir, N. (2006). Credit derivatives pricing with a smile-extended jump stochastic intensity model (ICMA Centre Discussion Papers in Finance DP2006-13). SSRN. https://doi.org/10.2139/ssrn.950208
Brigo, D., & Mercurio, F. (2000). The CIR++ model and other deterministic- shift extensions of short rate models. In Proceedings of the 4th Columbia-JAFEE Conference for Mathematical Finance and Financial Engineering (pp. 563–584). https://doi.org/10.2139/ssrn.292060
Brigo, D., & Mercurio, F. (2001). A deterministic-shift extension of analytically-tractable and time-homogeneous short rate models. Finance and Stochastics, 5, 369–387. https://doi.org/10.1007/PL00013541
Brigo, D., & Mercurio, F. (2006). Interest rate models – Theory and practice: With smile, inflation and credit (2nd ed.). Springer-Verlag.
Butler, R. (1998). Seasonality in tourism: Issues and implications. The Tourist Review, 53(3), 18–24. https://doi.org/10.1108/eb058278
Carmona, R. A., & Tehranchi, M. R. (2006). Interest rate models: An infinite dimensional stochastic analysis perspective. Springer-Verlag.
Chang, Y.-W., & Liao, M.-Y. (2010). A seasonal ARIMA model of tourism forecasting: The case of Taiwan. Asia Pacific Journal of Tourism Research, 15(2), 215–221. https://doi.org/10.1080/10941661003630001
Chen, L. (1996). Stochastic mean and stochastic volatility: A three-factor model of the term structure of interest rates and its applications and its applications in derivatives pricing and risk management. Blackwell Publishers. https://doi.org/10.1007/978-3-642-46825-4_1
Choden, & Unhapipat, S. (2018). ARIMA model to forecast international tourist visit in Bumthang, Bhutan. Journal of Physics: Conference Series, 1039, 012023. https://doi.org/10.1088/1742-6596/1039/1/012023
Claveria, O., Monte, E., & Torra, S. (2017). Data pre-processing for neural network-based forecasting: Does it really matter? Technological and Economic Development of Economy, 23(5), 709–725. https://doi.org/10.3846/20294913.2015.1070772
Corluka, G. (2019). Tourism seasonality – an overview. Journal of Business Paradigms, 4(1), 21–43.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. https://doi.org/10.2307/1911242
Duffie, D. (2005). Credit risk modeling with affine processes. Journal of Banking & Finance, 29(11), 2751–2802. https://doi.org/10.1016/j.jbankfin.2005.02.006
Dutta, A., Mishra, T., Uddin, G. S., & Yang, Y. (2021). Brexit uncertainty and volatility persistence in tourism demand. Current Issues in Tourism, 24(16), 2225–2232. https://doi.org/10.1080/13683500.2020.1822300
Engelen, K. C. (2015). The unthinkable as the new normal. The International Economy, 29(3), 30.
Eurostat. (2022). Database – Tourism. https://ec.europa.eu/eurostat/web/tourism/data/database
Grundey, D. (2008). Managing sustainable tourism in Lithuania: Dream or reality? Technological and Economic Development of Economy, 14(2), 118–129. https://doi.org/10.3846/1392-8619.2008.14.118-129
Gruppe, M., Basse, T., Friedrich, M., & Lange, C. (2017). Interest rate convergence, sovereign credit risk and the European debt crisis: A survey. Journal of Risk Finance, 18(4), 432–442. https://doi.org/10.1108/JRF-01-2017-0013
He, K., Ji, L., Wu, C. W. D., & Tso, K. F. G. (2021). Using SARIMA–CNN–LSTM approach to forecast daily tourism demand. Journal of Hospitality and Tourism Management, 49, 25–33. https://doi.org/10.1016/j.jhtm.2021.08.022
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327–343. https://doi.org/10.1093/rfs/6.2.327
Hochberg, Y., & Tamhane, A. C. (1989). Multiple comparison procedures. John Wiley & Sons.
Holt, C. C. (1957). Forecasting seasonals and trends by exponentially weighted moving averages (Office of Naval Research Memorandum, vol. 52). Carnegie Institute of Technology.
Holt, C. C. (2004). Forecasting seasonals and trends by exponentially weighted moving averages. International Journal of Forecasting, 20(1), 5–10. https://doi.org/10.1016/j.ijforecast.2003.09.015
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. The Review of Financial Studies, 3(4), 573–592. https://doi.org/10.1093/rfs/3.4.573
Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice (2nd ed.). OTexts.
Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149–176. https://doi.org/10.1093/biomet/36.1-2.149
Keller-Ressel, M., & Steiner, T. (2008). Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance and Stochastics, 12(2), 149–172. https://doi.org/10.1007/s00780-007-0059-z
Keynes, J. M. (1936). The general theory of the rate of interest. In The General theory of employment, interest, and money (pp. 145–153). Springer. https://doi.org/10.1007/978-3-319-70344-2_13
Kladıvko, K. (2007). Maximum likelihood estimation of the Cox-Ingersoll- Ross process: The Matlab implementation. https://it.mathworks.com/matlabcentral/fileexchange/37297-maximum-likelihood-estimation\-of-the-cox-ingersoll-ross-process-the-matlab-implementation
Kudo, M., Toyama, J., & Shimbo, M. (1999). Multidimensional curve classification using passing-through regions. Pattern Recognition Letters, 20(11), 1103–1111. https://doi.org/10.1016/S0167-8655(99)00077-X
Li, X., Law, R., Xie, G., & Wang, S. (2021). Review of tourism forecasting research with internet data. Tourism Management, 83, 104245. https://doi.org/10.1016/j.tourman.2020.104245
Li, Y., & Cao, H. (2018). Prediction for tourism flow based on LSTM neural network. Procedia Computer Science, 129, 277–283. https://doi.org/10.1016/j.procs.2018.03.076
Lim, C., & McAleer, M. (2001). Forecasting tourist arrivals. Annals of Tourism Research, 28(4), 965–977. https://doi.org/10.1016/S0160-7383(01)00006-8
Lock, S. (2022). Global tourism industry – statistics & facts. https://www.statista.com/topics/962/global-tourism/#dossierContents__outerWrapper
Lu, W., Li, J., Li, Y., Sun, A., & Wang, J. (2020). A CNN-LSTM-based model to forecast stock prices. Complexity, 2020, 6622927. https://doi.org/10.1155/2020/6622927
MATLAB. (2022). MathWorks. https://www.mathworks.com/products/matlab.html
Mininni, M., Orlando, G., & Taglialatela, G. (2020). Challenges in approximating the black and scholes call formula with hyperbolic tangents. Decisions in Economics and Finance, 1–28. https://doi.org/10.1007/s10203-020-00305-8
Moreno, M., & Platania, F. (2015). A cyclical square-root model for the term structure of interest rates. European Journal of Operational Research, 241(1), 109–121. https://doi.org/10.1016/j.ejor.2014.08.010
Najafi, A. R., & Mehrdoust, F. (2017). Bond pricing under mixed generalized CIR model with mixed Wishart volatility process. Journal of Computational and Applied Mathematics, 319(C), 108–116. https://doi.org/10.1016/j.cam.2016.12.039
Orlando, G., & Bufalo, M. (2021). Interest rates forecasting: Between Hull and White and the CIR#. How to make a single factor model work. Journal of Forecasting, 40(8), 1566–1580. https://doi.org/10.1002/for.2783
Orlando, G., Mininni, R. M., & Bufalo, M. (2018). A new approach to CIR short-term rates modelling. In Mili, M., Samaniego Medina, R., & di Pietro, F. (Eds.), New methods in fixed income modeling – fixed income modeling (pp. 35–44). Springer International. https://doi.org/10.1007/978-3-319-95285-7_2
Orlando, G., Mininni, R. M., & Bufalo, M. (2019a). Interest rates calibration with a CIR model. The Journal of Risk Finance, 20(4), 370–387. https://doi.org/10.1108/JRF-05-2019-0080
Orlando, G., Mininni, R. M., & Bufalo, M. (2019b). A new approach to forecast market interest rates through the CIR model. Studies in Economics and Finance, 37(2), 267–292. https://doi.org/10.1108/SEF-03-2019-0116
Orlando, G., Mininni, R. M., & Bufalo, M. (2020). Forecasting interest rates through Vasicek and CIR models: A partitioning approach. Journal of Forecasting, 39(4), 569–579. https://doi.org/10.1002/for.2642
Perry, M. B. (2010). The exponentially weighted moving average. Wiley Encyclopedia of Operations Research and Management Science. https://doi.org/10.1002/9780470400531.eorms0314
Polyzos, S., Samitas, A., & Spyridou, A. Ef. (2021). Tourism demand and the COVID-19 pandemic: An LSTM approach. Tourism Recreation Research, 46(2), 175–187. https://doi.org/10.1080/02508281.2020.1777053
Qadeer, K., Rehman, W. U., Sheri, A. M., Park, I., Kim, H. K., & Jeon, M. (2020). A Long Short-Term Memory (LSTM) network for hourly estimation of PM2.5 concentration in two cities of South Korea. Applied Sciences, 10(11), 3984. https://doi.org/10.3390/app10113984
Santamaria, D., & Filis, G. (2019). Tourism demand and economic growth in Spain: New insights based on the yield curve. Tourism Management, 75, 447–459. https://doi.org/10.1016/j.tourman.2019.06.008
Schwarzbach, C., Kunze, F., Rudschuck, N., & Windels, T. (2012). Asset management in the German insurance industry: The quality of interest rate forecasts. Zeitschrift für die gesamte Versicherungswissenschaft, 101(5), 693–703. https://doi.org/10.1007/s12297-012-0218-y
Shanika, A., & Jahufer, A. (2021). Volatility analysis of international tourist arrivals to Sri Lanka using GARCH models. Faculty of Applied Sciences, South Eastern University of Sri Lanka, Sammanthurai.
Supriatna, A., Lesmana, E., Aridin, L., Sukono, & Napitupulu, H. (2019). Comparison between multiplicative Holt Winter and decomposition method in predicting the number of incoming international tourists to Indonesia. IOP Conference Series: Materials Science and Engineering, 567(1), 012047. https://doi.org/10.1088/1757-899X/567/1/012047
Thakur, B. P. S., Kannadhasan, M., & Goyal, V. (2018). Determinants of corporate credit spread: Evidence from India. Decision, 45(1), 59–73. https://doi.org/10.1007/s40622-018-0179-7
Trading Economics. (2023). Polish Zloty - 2023 Data – 1993–2022 Historical - Quote. https://tradingeconomics.com/poland/currency
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188. https://doi.org/10.1016/0304-405X(77)90016-2
Winters, P. R. (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6(3), 324–342. https://doi.org/10.1287/mnsc.6.3.324
Yıldırım, D. C., Toroslu, I. H., & Fiore, U. (2021). Forecasting directional movement of Forex data using LSTM with technical and macroeconomic indicators. Financial Innovation, 7(1), 1–36. https://doi.org/10.1186/s40854-020-00220-2
Zhao, Z., Chen, W., Wu, X., Chen, P. C. Y., & Liu, J. (2017). LSTM network: A deep learning approach for short-term traffic forecast. IET Intelligent Transport Systems, 11(2), 68–75. https://doi.org/10.1049/iet-its.2016.0208
Zhu, L. (2014). Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps. Journal of Applied Probability, 51(3), 699–712. https://doi.org/10.1239/jap/1409932668
Zong, C.-L., & Wang, L. (2018). Prediction of urban residents’ travel rate in China based on ARIMA models. Journal of Interdisciplinary Mathematics, 21(5), 1285–1290. https://doi.org/10.1080/09720502.2018.1497999