## Modeling and forecasting interval time series with threshold models.(English)Zbl 1414.62076

Summary: This paper proposes threshold models to analyze and forecast interval-valued time series. A relatively simple algorithm is proposed to obtain least square estimates of the threshold and slope parameters. The construction of forecasts based on the proposed model and methods for the analysis of their forecast performance are also introduced and discussed, as well as forecasting procedures based on the combination of different models. To illustrate the usefulness of the proposed methods, an empirical application on a weekly sample of S&P500 index returns is provided. The results obtained are encouraging and compare very favorably to available procedures.

### MSC:

 62F10 Point estimation 62P20 Applications of statistics to economics
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### References:

 [1] Arroyo J, Maté C (2006) Introducing interval time series: accuracy measures. COMPSTAT 2006. Proceeding in Computational statistics, Heidelberg, pp 1139-1146 [2] Arroyo, J.; Espínola, R.; Maté, C., Different approaches to forecast interval time series: a comparison in finance, Comput Econ, 37, 169-191, (2011) · Zbl 1206.91087 [3] Arroyo J, Gonzalez-Rivera G, Maté C (2010) Forecasting with Interval and Histogram Data. Some Financial Applications. In: Ullah A, Giles D (eds.) Handbook of Empirical Economics and Finance, Chapman and Hall, pp 247-280 [4] Bai, J., Estimating multiple breaks one at a time, Econ Theory, 13, 315-352, (1997) [5] Beckers, S., Variance of security price return based on high, low and closing prices, J Bus, 56, 97-112, (1983) [6] Cheung, YW, An empirical model of daily highs and lows, Int J Financ Econ, 12, 1-20, (2007) [7] Chou, RY, Forecasting financial volatilities with extreme values: the conditional autoregressive range (CARR) model, J Money Credit Bak, 37, 561-582, (2005) [8] Clements, MP; Smith, J., A monte carlo study of the forecasting performance of empirical SETAR models, J Appl Econ, 14, 123-141, (1999) [9] Diebold, F.; Mariano, R., Comparing predictive accuracy, J Bus Econ Statistics, 13, 253-263, (1995) [10] Dueker, M.; Martin, S.; Spangnolo, F., Contemporaneous threshold autoregressive models: estimation, testing and forecasting, J Econ, 141, 517-547, (2007) · Zbl 1418.62315 [11] Eitrheim, Ã; Teräsvirta, T., Testing the adequacy of smooth transition autoregressive models, J Econ, 74, 59-75, (1996) · Zbl 0864.62058 [12] Gonzalo, J.; Pitarakis, J-Y, Estimation and model selection based inference in single and multiple threshold models, J Econ, 110, 319-352, (2002) · Zbl 1043.62068 [13] Granger CWJ, Terasvirta T (1993) Modelling non-linear economic relationships. OUP Catalogue. Oxford University Press. ISBN 9780198773207 [14] Guidolin, M.; Hyde, S.; McMillan, D.; Ono, S., Non-linear predictability in stock and bond returns: when and where is it exploitable?, Int J Forecast, 25, 373-399, (2009) [15] Hansen, B., Inference when a nuisance parameter is not identified under the null hypothesis, Econometrica, 64, 413-430, (1996) · Zbl 0862.62090 [16] Hansen B (1997) Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics 2 · Zbl 1078.91558 [17] Henry, Ó; Olekaln, N.; Summers, PM, Exchange rate instability: a threshold autoregressive approach, Econ Record, 77, 160-166, (2001) [18] Hu, C.; He, LT, An application of interval methods to stock market forecasting, Reliab Comput, 13, 423-434, (2007) · Zbl 1125.91348 [19] Hsu, HL; Wu, B., Evaluating forecasting performance for interval data, Comput Math Appl, 56, 2155-2163, (2008) · Zbl 1165.62341 [20] Ichino, M.; Yaguchi, H., Generalized Minkowski metrics for mixed and feature-type data analysis, IEEE Trans Systems Man Cybern, 24, 698-708, (1994) · Zbl 1371.68235 [21] Maia, ALS; de Carvalho, FdAT, Holt’s exponential smoothing and neural network models for forecasting interval-valued time series, Int J Forecast, 27, 740-759, (2011) [22] Muñoz SR, Maté AC, Arroyo J, Sarabia Á (2007) iMLP: Applying multi-layer perceptrons to interval-valued data. Neural Process Lett 25:157-169 [23] Nieto, FH, Modelling bivariate threshold autoregressive processes in the presence of missing data, Commun Stat Theory Methods, 34, 905-930, (2005) · Zbl 1073.62079 [24] Pitarakis J-Y (2006) Model selection uncertainty and detection of threshold effects. Studies in Nonlinear Dynamics and Econometrics 10 · Zbl 1225.62160 [25] Timmermann A (2006) Forecast Combinations. Handbook of Economic Forecasting, Elsevier, Amsterdam [26] Tong H (2011) Threshold models in time series analysis â 30 years on (with discussions by P. Whittle, M. Rosenblatt, B. E. Hansen, P. Brockwell, N. I. Samia and F. Battaglia). Stat Interface 4:107-136 [27] Tsay, RS, Testing and modelling multivariate threshold models, J Am Stat Assoc, 93, 1188-1202, (1998) · Zbl 1063.62578 [28] Zhang, G., Time series forecasting using a hybrid ARIMA and neural network model, Neurocomputing, 50, 159-175, (2003) · Zbl 1006.68828 [29] Zou, H.; Yang, Y., Combining time series models for forecasting, Int J Forecast, 20, 69-84, (2004)
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