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Table 1 Comparison results of using 500 and 250 simulated data

From: A model-free approach to do long-term volatility forecasting and its variants

500 size

GE

GA

P-GA

GARCH

250 size

GE

GA

P-GA

GARCH

M1-1step

0.98241

0.97574

0.91931

1.00000

M1-1step

0.91538

0.91120

0.83034

1.00000

M1-5steps

0.75332

0.75716

0.70509

1.00000

M1-5steps

0.49169

0.48479

0.43247

1.00000

M1-30steps

0.16629

0.16108

0.15950

1.00000

M1-30steps

0.25009

0.24752

0.23035

1.00000

M2-1step

0.97948

0.97391

0.98569

1.00000

M2-1step

0.91369

0.91574

0.87614

1.00000

M2-5steps

0.91726

0.88645

0.91872

1.00000

M2-5steps

0.61001

0.61094

0.51712

1.00000

M2-30steps

0.78096

0.72696

0.80259

1.00000

M2-30steps

0.7725

0.74083

0.75251

1.00000

M3-1step

1.00334

0.99333

0.98694

1.00000

M3-1step

0.97796

0.96632

0.93693

1.00000

M3-5steps

0.98281

0.98312

0.96297

1.00000

M3-5steps

0.98127

0.97897

0.99977

1.00000

M3-30steps

0.74129

0.72216

0.72611

1.00000

M3-30steps

1.38353

0.89001*

0.99818

1.00000

M4-1step

1.00390

0.98736

1.04656

1.00000

M4-1step

0.99183

0.95698

0.92811

1.00000

M4-5steps

0.98563

0.94474

1.16346

1.00000

M4-5steps

0.77088

0.72882

0.67894

1.00000

M4-30steps

0.75828

0.79120

0.77923

1.00000

M4-30steps

0.79672

0.60950*

0.81115

1.00000

M5-1step

0.71259

0.70934

0.72124

1.00000

M5-1step

0.83631

0.84134

0. 79075

1.00000

M5-5steps

0.31050

0.30918

0.31147

1.00000

M5-5steps

0.38296

0.38034

0.35155

1.00000

M5-30steps

0.01088

0.01087

0.01069

1.00000

M5-30steps

0.00199

0.00201

0.00194

1.00000

M6-1step

1.02897

1.02439

0.97537

1.00000

M6-1step

0.95939

0.96499

0.93863

1.00000

M6-5steps

1.02441

1.02850

0.93029

1.00000

M6-5steps

0.93594

0.97101

0.85851

1.00000

M6-30steps

1.03845

1.03545

0.89686

1.00000

M6-30steps

0.84401

0.67272*

0.70420

1.00000

M7-1step

0.98459

0.98179

0.96345

1.00000

M7-1step

0.84813

0.83628

0.83216

1.00000

M7-5steps

0.98910

0.99239

0.96265

1.00000

M7-5steps

0.50849

0.50126

0.48020

1.00000

M7-30steps

0.74522

0.73544

0.64106

1.00000

M7-30steps

0.06832

0.06817

0.06507

1.00000

M8-1step

0.79801

0.80723

0.81895

1.00000

M8-1step

0.79561

0.79994

0.83340

1.00000

M8-5steps

0.34915

0.34223

0.35644

1.00000

M8-5steps

0.48028

0.47244

0.45665

1.00000

M8-30steps

0.00106

0.00087*

0.00094

1.00000

M8-30steps

0.00977

0.00942

0.00983

1.00000

  1. Column names “GA” and “GE” represent GE-NoVaS and GA-NoVaS methods, respectively; “GARCH” represents the GARCH-direct method; and “P-GA” represents the P-GA-NoVaS method. The benchmark is the GARCH-direct method, so numerical values in the table corresponding to the GARCH-direct method are 1. Other numerical values are relative performance compared to the GARCH-direct method. “\(Mi\text {-}j\)”steps denotes using data generated from the Model i to do j steps ahead time-aggregated predictions. The bold value means that the corresponding method is the optimal choice for this data case. Cell with \(*\) means that the GA-NoVaS method is at least 10\(\%\) better than the GE-NoVaS method, or inversely, the GE-NoVaS method is at least 10\(\%\) better
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