## Threshold autoregressive models for interval-valued time series data.(English)Zbl 1452.62681

Summary: Modeling and forecasting symbolic data, especially interval-valued time series (ITS) data, has received considerable attention in statistics and related fields. The core of available methods on ITS analysis is based on various applications of conventional linear modeling. However, few works have considered possible nonlinearities in ITS data. In this paper, we propose a new class of threshold autoregressive interval (TARI) models for ITS data. By matching the interval model with interval observations, we develop a minimum-distance estimation method for TARI models, and establish the asymptotic theory for the proposed estimators. We show that the threshold parameter estimator is $$T$$-consistent and follows an asymptotic compound Poisson process as the sample size $$T\rightarrow\infty$$. And the estimators for other TARI model parameters are root-$$T$$ consistent and asymptotically normal. Simulation studies show that the proposed TARI model provides more accurate out-of-sample forecasts than the existing center-radius self-exciting threshold (CR-SETAR) model for ITS data in the literature. Empirical applications to the S&P 500 Price Index document significant asymmetric reactions of the stock markets in Japan, U.K. and France to shocks from the U.S. stock market and that incorporating this asymmetric effect yield better out-of-sample forecasts than a variety of popular models available in the literature.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62J05 Linear regression; mixed models 62P05 Applications of statistics to actuarial sciences and financial mathematics 62P20 Applications of statistics to economics

SODAS
Full Text:

### References:

 [1] Andrews, D. W.K., Generic uniform covergence, Econometric Theory, 8, 02, 241-257, (1992) [2] Arroyo, J.; Espínola, R.; Maté, C., Different approaches to forecast interval time series: a comparison in finance, Comput. Econ., 37, 2, 169-191, (2011) · Zbl 1206.91087 [3] Bertoluzza, C.; Corral Blanco, N.; Salas, A., On a new class of distances between fuzzy numbers, Mathware Soft Comput., 2, 2, 71-84, (1995) · Zbl 0887.04003 [4] Billard, L., 2006. Symbolic data analysis: what is it? Compstat 2006-Proceedings in Computational Statistics, pp. 261-269. [5] Billard, L.; Diday, E., Regression analysis for interval-valued data, (Data Analysis, Classification, and Related Methods, (2000), Springer Berlin Heidelberg), 369-374 · Zbl 1026.62073 [6] Billard, L.; Diday, E., From the statistics of data to the statistics of knowledge: symbolic data analysis, J. Amer. Statist. Assoc., 98, 462, 470-487, (2003) [7] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201 [8] Brito, P., Modelling and analysing interval data, (Advances in Data Analysis, (2007), Springer Berlin Heidelberg), 197-208 [9] Chan, K. S., Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. technical report, 245, (1988), Statistics, University of Chicago [10] Chan, K. S., Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, Ann. Statist., 21, 1, 520-533, (1993) · Zbl 0786.62089 [11] Chan, K. S.; Tong, H., On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference eequation, Adv. Appl. Probab., 17, 3, 666-678, (1985) · Zbl 0573.60056 [12] Chan, K. S.; Tong, H., On estimating thresholds in autoregressive models, J. Time Series Anal., 7, 3, 179-190, (1986) · Zbl 0596.62085 [13] Chan, K. S.; Tsay, R. S., Limiting properties of the least squares estimator of a continuous threshold autoregressive model, Biometrika, 85, 2, 413-426, (1998) · Zbl 0938.62089 [14] Chen, C. W.; Chiang, T. C.; So, M. K., Asymmetric reaction to US stock-return news: evidence from major stock markets based on a double-threshold model, J. Econ. Bus., 55, 5, 487-502, (2003) [15] Chen, C. W.; So, M. K.; Gerlach, R. H., Asymmetric response and interaction of US and local news in financial markets, Appl. Stoch. Models Bus. Ind., 21, 3, 273-288, (2005) · Zbl 1087.62114 [16] De Carvalho, F.D.A., Neto, E.D.A.L., Tenorio, C.P., 2004. A new method to fit a linear regression model for interval-valued data. In: KI 2004: Advances in Artificial Intelligence, pp. 295-306. · Zbl 1132.68617 [17] (Diday, E.; Noirhomme-Fraiture, M., Symbolic Data Analysis and the SODAS Software, (2008), J. Wiley & Sons) · Zbl 1275.62029 [18] Fan, Y.; Xu, J. H., What has driven oil prices Since 2000? A structural change perspective, Energy Econ., 33, 6, 1082-1094, (2011) [19] Golan, A.; Ullah, A., Interval estimation: an information theoretic approach, Econometric Rev., 36, 1-15, (2017) [20] Goldfeld, S. M.; Quandt, R. E., A Markov model for switching regressions, J. Econometrics, 1, 1, 3-15, (1973) · Zbl 0294.62087 [21] González-Rodríguez, G.; Blanco, Á.; Colubi, A.; Lubiano, M. A., Estimation of a simple linear regression model for fuzzy random variables, Fuzzy Sets and Systems, 160, 3, 357-370, (2009) · Zbl 1175.62073 [22] González-Rodríguez, G.; Blanco, Á.; Corral, N.; Colubi, A., Least squares estimation of linear regression models for convex compact random sets, Adv. Data Anal. Classif., 1, 1, 67-81, (2007) · Zbl 1131.62058 [23] González-Rivera, G.; Lin, W., Constrained regression for interval-valued data, Journal of Businiess Economic Statistics, 31, 4, 473-490, (2013) [24] Hamilton, J. D., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 2, 357-384, (1989) · Zbl 0685.62092 [25] Han, A., Hong, Y., Wang, S., 2017. Autoregressive conditional models for interval-valued time series data. Manuscript, Department of Economics, Cornell University. [26] Han, A.; Hong, Y.; Wang, S.; Yun, X., A vector autoregressive moving average model for interval-valued time series data, (Essays in Honor of Aman Ullah, (2016), Emerald Group Publishing Limited), 417-460 [27] Hansen, B. E., Inference when a nuisance parameter is not identified under the null hypothesis, Econometrica, 64, 2, 413-430, (1996) · Zbl 0862.62090 [28] Hansen, B. E., Threshold effects in non-dynamic panels: estimation, testing, and inference, J. Econometrics, 93, 2, 345-368, (1999) · Zbl 0941.62127 [29] Hansen, B. E., Sample splitting and threshold estimation, Econometreica, 68, 3, 575-603, (2000) · Zbl 1056.62528 [30] Henriques, I.; Sadorsky, P., Oil prices and the stock prices of alternative energy companies, Energy Econ., 30, 3, 998-1010, (2008) [31] Huang, B.; Yang, C. W.; Hwang, M. J., The dynamics of a nonlinear relationship between crude oil spot and futures prices: a multivariate threshold regression approach, Energy Econ., 31, 1, 91-98, (2009) [32] Hukuhara, M., Integration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac., 10, 3, 205-223, (1967) · Zbl 0161.24701 [33] Jung, H. Y.; Lee, W. J.; Yoon, J. H., A unified approach to asymptotic behaviors for the autoregressive model with fuzzy data, Inform. Sci., 257, 127-137, (2014) · Zbl 1321.62109 [34] Kaucher, E., Interval analysis in the extended interval space IR, (Fundamentals of Numerical Computation, Computer-Oriented Numerical Analysis, vol. 2, (1980), Springer Vienna), 33-49 · Zbl 0419.65031 [35] Kim, C. J.; Nelson, C. R.; Piger, J., The less-volatile US economy: a Bayesian investigation of timing, breadth, and potential explanations, J. Bus. Econom. Statist., 22, 1, 80-93, (2004) [36] Körner, R., On the variance of fuzzy random variables, Fuzzy Sets and Systems, 92, 1, 83-93, (1997) · Zbl 0936.60017 [37] Körner, R.; Näther, W., On the variance of random fuzzy variables, Stat. Model. Anal. Manag. Fuzzy Data, 87, 25-42, (2002) [38] Kumar, S.; Managi, S.; Matsuda, A., Stock prices of Clean energy firms, oil and carbon markets: A vector autoregressive analysis, Energy Econ., 34, 1, 215-226, (2012) [39] Lee, S. W.; Hansen, B. E., Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator, Econometric Theory, 10, 01, 29-52, (1994) [40] Li, D.; Ling, S., On the least squares estimation of multiple-regine threshold autoregressive models, J. Econometrics, 167, 1, 240-253, (2012) · Zbl 1441.62795 [41] Lima Neto, E. D.A. L.; de Carvalho, F. D.A., Constrained linear regression models for symbolic interval-valued variables, Comput. Statist. Data Anal., 54, 2, 333-347, (2010) · Zbl 1464.62055 [42] Lima Neto, E. D.A. L.; de Cavalho, F. D.A., Centre and range method for Fitting a linear regression model to symbolic interval data, Comput. Statist. Data Anal., 52, 3, 1500-1515, (2008) · Zbl 1452.62493 [43] Lima Neto, E. D.A. L.; Cordeiro, G. M.; de Carvalho, F. D.A., Bivariate symbolic regression models for interval-valued variables, J. Stat. Comput. Simul., 81, 11, 1727-1744, (2011) · Zbl 1431.62328 [44] Lin, J. G.; Zhuang, Q. Y.; Huang, C., Fuzzy statistical analysis of multiple regression with crisp and fuzzy covariates and applications in analyzing economic data of China, Comput. Econ., 39, 1, 29-49, (2012) · Zbl 1241.91143 [45] Lin, W., González-Rivera, G., 2017. Extreme returns and intensity of trading. Working Paper. [46] Lin, W.; González-Rivera, G., Interval-valued time series models: estimation based on order statistics exploring the agriculture marketing service data, Comput. Statist. Data Anal., 100, 694-711, (2016) · Zbl 1466.62139 [47] Lindgren, G., Markov regime models for mixed distributions and switching regressions, Scand. J. Stat., 5, 2, 81-91, (1978) · Zbl 0382.62073 [48] Maia, A. L.S.; de Carvalho, F. D.A.; Ludermir, T. B., Forecasting models for interval-valued time series, Neurocomputing, 71, 16, 3344-3352, (2008) [49] McMillan, D. G., Nonlinear predictability of stock market returns: evidence from nonparametric and threshold models, Int. Rev. Econ. Finance, 10, 4, 353-368, (2001) [50] Meyn, S. P.; Tweedie, R. L., Markov chains and stochastic stability, (2012), Springer Science & Business Media · Zbl 0925.60001 [51] Moore, R. E.; Kearfott, R. B.; Cloud, M. J., Introduction to interval analysis, (2009), Society for Industrial and Applied Mathematics · Zbl 1168.65002 [52] Narayan, P. K., The behaviour of US stock prices: evidence from a threshold autoregressive model, Math. Comput. Simulation, 71, 2, 103-108, (2006) · Zbl 1134.91566 [53] Peel, D. A.; Speight, A. E., Modelling business cycle nonlinearity in conditional mean and conditional variance: some international and sectoral evidence, Economica, 65, 258, 211-229, (1998) [54] Peel, D. A.; Speight, A., The nonlinear time seris properties of unemployment rates: some further evidence, Appl. Econ., 30, 2, 287-294, (1998) [55] Petruccelli, J. D., On the consistency of least squares estimator of a threshold AR(1) model, J. Time Series Anal., 7, 4, 269-278, (1986) · Zbl 0601.62110 [56] Qian, L., On maximum likelihood estimators for a threshold autoregression, J. Statist. Plann. Inference, 75, 1, 21-46, (1998) · Zbl 0953.62098 [57] Rodrigues, P.M.M., Salish, N., 2011. Modeling and forecasting interval time series with threshold models: an application to S&P 500 index returns. Economics and Research Department, Banco de Portugal. [58] Rodrigues, P. M.; Salish, N., Modeling and forecasting interval time series with threshold models, Adv. Data Anal. Classif., 9, 1, 41-57, (2015) [59] Seo, M. H.; Linton, O., A smoothed least squares estimator for threshold regression models, J. Econometrics, 141, 2, 704-735, (2007) · Zbl 1418.62355 [60] Stock, J. H.; Watson, M. W., Understanding changes in international business cycle dynamics, J. Eur. Econ. Assoc., 3, 5, 968-1006, (2005) [61] Teles, P.; Brito, P., Modeling interval time series with space-time processes, Comm. Statist. Theory Methods, 44, 17, 3599-3627, (2015) · Zbl 1342.37076 [62] Teräsvirta, T., Specification, estimation, and evaluation of smooth transition autoregressive models, J. Amer. Statist. Assoc., 89, 425, 208-218, (1994) · Zbl 1254.91686 [63] Tong, H., 1978. On a threshold model. Sijthoff & Noordhoff, vol. 29, pp. 575-586. [64] Yang, W.; Han, A., A new approach for forecasting the price range with financial interval-valued time series data, ASCE-ASME J. Risk Uncertain. Eng. Syst. B, 1, 2, (2015) [65] Yang, W.; Han, A.; Cai, K.; Wang, S., ACIX model with interval dummy variables and its application in forecasting interval-valued crude oil prices, Proc. Comput. Sci., 9, 1273-1282, (2012) [66] Yang, W.; Han, A.; Hong, Y.; Wang, S., Analysis of crisis impact on crude oil prices: a new approach with interval time series modelling, Quant. Finance, 16, 12, 1917-1928, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.