This paper proposes an interval-based methodology to model and forecast the price range or range-based volatility process of financial asset prices. Comparing with the existing volatility models, the proposed model utilizes more information contained in the interval time series than using the range information only or modeling the high and low price processes separately. An empirical study of the U.S. stock market daily data shows that the proposed interval-based model produces more accurate range forecasts than the classic point-based linear models for range process, in terms of both in-sample and out-of-sample forecasts. The statistical tests show that the forecasting advantages of the interval-based model are statistically significant in most cases. In addition, some stability tests have been conducted for ascertaining the advantages of the interval-based model through different sample windows and forecasting periods, which reveals similar results. This study provides a new interval-based perspective for volatility modeling and forecasting of financial time series data.

References

References
1.
Merton
,
R. C.
,
1969
, “
Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case
,”
Rev. Econ. Stat.
,
51
(
3
), pp. 
247
257
.10.2307/1926560
2.
Black
,
F.
, and
Scholes
,
M.
,
1973
, “
The Pricing of Options and Corporate Liabilities
,”
J. Polit. Econ.
,
81
(
3
), pp. 
637
654
.10.1086/jpe.1973.81.issue-3
3.
Engle
,
R. F.
,
1982
, “
Autoregressive Conditional Heteroscedasticity With Estimates of the Variance of United Kingdom Inflation
,”
Econometrica
,
50
(
4
), pp. 
987
1008
.10.2307/1912773
4.
Bollerslev
,
T.
,
1986
, “
Generalized Autoregressive Conditional Heteroskedasticity
,”
J. Econometrics
,
31
(
3
), pp. 
307
327
.10.1016/0304-4076(86)90063-1
5.
Hull
,
J.
, and
White
,
A.
,
1987
, “
The Pricing of Options on Assets with Stochastic Volatilities
,”
J. Financ.
,
42
(
2
), pp. 
281
300
.10.1111/j.1540-6261.1987.tb02568.x
6.
Poon
,
S. H.
, and
Granger
,
C. W. J.
,
2003
, “
Forecasting the Volatility in Financial Market: A Review
,”
J. Econ. Lit.
,
41
(
2
), pp. 
478
539
.10.1257/jel.41.2.478
7.
Francq
,
C.
, and
Zakoian
,
J. M.
,
2010
,
GARCH Models: Structure, Statistical Inference and Financial Applications
,
Wiley
,
Hoboken, NJ
.
8.
Alizadeh
,
S.
,
Brandt
,
M. W.
, and
Diebold
,
F. X.
,
2002
, “
Range-Based Estimation of Stochastic Volatility Models
,”
J. Financ.
,
57
(
3
), pp. 
1047
1092
.10.1111/1540-6261.00454
9.
Brandt
,
M. W.
, and
Diebold
,
F. X.
,
2006
, “
A No-Arbitrage Approach to Range-Based Estimation of Return Covariances and Correlations
,”
J. Bus.
,
79
(
1
), pp. 
61
74
.10.3386/w9664
10.
Parkinson
,
M.
,
1980
, “
The Extreme Value Method for Estimating the Variance of the Rate of Return
,”
J. Bus.
,
53
(
1
), pp. 
61
65
.10.1086/296071
11.
Kunitomo
,
N.
,
1992
, “
Improving the Parkinson Method of Estimating Security Price Volatilities
,”
J. Bus.
,
65
(
2
), pp. 
295
302
.10.1086/296570
12.
Yang
,
D.
, and
Zhang
,
Q.
,
2000
, “
Drift-Independent Volatility Estimation Based on High, Low, Open and Close Prices
,”
J. Bus.
,
73
(
3
), pp. 
477
491
.10.1086/209650
13.
Chou
,
R. Y.
,
2005
, “
Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model
,”
J. Money Credit Bank.
,
37
(
3
), pp. 
561
582
.
14.
Fernandes
,
M.
,
de Sá Mota
,
B.
, and
Rocha
,
G.
,
2005
, “
A Multivariate Conditional Autoregressive Range Model
,”
Econ. Lett.
,
86
(
3
), pp. 
435
440
.10.1016/j.econlet.2004.09.005
15.
Martens
,
M.
, and
van Dijk
,
D.
,
2007
, “
Measuring Volatility with the Realized Range
,”
J. Econometrics
,
138
(
1
), pp. 
181
207
.10.1016/j.jeconom.2006.05.019
16.
Christensen
,
K.
, and
Podolskij
,
M.
,
2007
, “
Realized Range-Based Estimation of Integrated Variance
,”
J. Econometrics
,
141
(
2
), pp. 
323
349
.
17.
Brownlees
,
C. T.
, and
Gallo
,
G. M.
,
2010
, “
Comparison of Volatility Measures: A Risk Management Perspective
,”
J. Financ. Econometrics
,
8
(
1
), pp. 
29
56
.10.1093/jjfinec/nbp009
18.
Cheung
,
Y. W.
,
2007
, “
An Empirical Model of Daily Highs and Lows
,”
Int. J. Financ. Econ.
,
12
(
1
), pp. 
1
20
.10.1002/ijfe.303
19.
Cheung
,
Y. L.
,
Cheung
,
Y. W.
, and
Wan
,
A. T.
,
2009
, “
A High-Low Model of Daily Stock Price Ranges
,”
J. Forecast.
,
28
(
2
), pp. 
103
119
.10.1002/for.1087
20.
Bertrand
,
P.
, and
Goupil
,
F.
,
2000
, “Descriptive Statistic for Symbolic Data,”
Analysis of Symbolic Data
,
H.-H. Bock
, and
E. Diday
, eds.,
Springer
,
Heidelberg, Germany
, pp. 
106
124
.
21.
Billard
,
L.
, and
Diday
,
E.
,
2000
, “
Regression Analysis for Interval-Valued Data
,”
Data Analysis, Classification and Related Methods
,
Proceedings of the 7th Conference of the International Federation of Classification Societies (IFCS00)
,
Springer
,
Namur, Belgium
, pp. 
369
374
.
22.
Billard
,
L.
, and
Diday
,
E.
,
2002
, “
Symbolic Regression Analysis
,”
Classification, Clustering and Data Analysis
,
Proceedings of the 8th Conference of the International Federation of Classification Societies (IFCS02)
,
Springer
,
Cracow, Poland
, pp. 
281
288
.
23.
Billard
,
L.
, and
Diday
,
E.
,
2003
, “
From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis
,”
J. Am. Stat. Assoc.
,
98
(
462
), pp. 
470
487
.10.1198/016214503000242
24.
Lima Neto
,
E. A.
,
de Carvalho
,
F. A. T.
, and
Freire
,
E. S.
,
2008
, “
Centre and Range Method for Fitting a Linear Regression Model to Symbolic Interval Data
,”
Comput. Stat. Data Anal.
,
52
(
3
), pp. 
1500
1515
.10.1016/j.csda.2007.04.014
25.
Lima Neto
,
E. A.
, and
de Carvalho
,
F. A. T.
,
2010
, “
Constrained Linear Regression Models for Symbolic Interval-Valued Variables
,”
Comput. Stat. Data Anal.
,
54
(
2
), pp. 
333
347
.10.1016/j.csda.2009.08.010
26.
Han
,
A.
,
Hong
,
Y. M.
, and
Wang
,
S. Y.
,
2012
, “
Autoregressive Conditional Models for Interval-Valued Time Series Data
.” Available at: http://economics.yale.edu/sites/default/files/hong-120926.pdf.
27.
Yang
,
W.
,
Han
,
A.
,
Cai
,
K.
, and
Wang
,
S. Y.
,
2012
, “
ACIX Model With Interval Dummy Variables and its Application in Forecasting Interval-Valued Crude Oil Prices
,”
Procedia Comput. Sci.
,
9
(
2
), pp. 
1273
1282
.10.1016/j.procs.2012.04.139
28.
Yang
,
W.
,
Han
,
A.
, and
Wang
,
S. Y.
,
2013
, “
Analysis of the Interaction Between Crude Oil Price and US Stock Market Based on Interval Data
,”
Int. J. Energy Stat.
,
1
(
2
), pp. 
85
98
.10.1142/S2335680413500063
29.
Diebold
,
F. X.
, and
Mariano
,
R. S.
,
1995
, “
Comparing Predictive Accuracy
,”
J. Bus. Econ. Stat.
,
13
(
3
), pp. 
253
263
.10.1080/07350015.1995.10524599
30.
Harvey
,
D.
,
Leybourne
,
S.
, and
Newbold
,
P.
,
1997
, “
Testing the Equality of Prediction Mean Squared Errors
,”
Int. J. Forecast.
,
13
(
2
), pp. 
281
291
.10.1016/S0169-2070(96)00719-4
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