Table 9 Full sample VaR and ES Results for Short PBW Positions.

From: Tailoring tail risk models for clean energy investments: a dual approach to long and short position forecasting

 

GARCH

GJRGARCH

IGARCH

 

N

ST

HS

CF

N

ST

HS

CF

N

ST

HS

CF

Panel A: Confidence level 97.5%

        

mVaR

−4.146

−3.963

−3.934

−3.945

−4.056

−3.877

−3.872

−3.880

−4.253

−4.054

−3.962

−4.031

stdVaR

2.075

2.025

2.007

1.982

2.082

2.055

2.044

2.008

2.190

2.123

2.109

2.092

mES

−4.941

−4.927

−4.814

−4.764−

4.841

−4.811

−4.758

−4.703

−5.069

−5.078

−4.803

−4.881

stdES

2.470

2.559

2.423

2.442

2.480

2.613

2.494

2.498

2.606

2.687

2.518

2.579

AE

0.985

1.136

1.201

1.154

0.985

1.183

1.229

1.173

0.901

1.089

1.258

1.107

LRuc

0.879

0.165

0.041

0.115

0.879

0.064

0.020

0.078

0.293

0.361

0.010

0.269

LRcc

0.956

0.274

0.105

0.281

0.172

0.095

0.026

0.109

0.494

0.405

0.024

0.501

DQ

0.825

0.238

0.144

0.232

0.235

0.170

0.043

0.149

0.822

0.553

0.094

0.761

ER

0.005

0.294

0.062

0.004

0.000

0.010

0.032

0.000

0.024

0.702

0.122

0.068

ESR

0.162

0.196

0.042

0.022

0.051

0.075

0.024

0.013

0.431

0.499

0.073

0.210

ADMax

4.739

5.212

5.376

5.545

4.992

5.387

5.392

5.263

4.210

4.815

4.922

5.064

Tick

0.129

0.129

0.130

0.129

0.129

0.129

0.130

0.129

0.129

0.128

0.130

0.129

FZL

1.555

1.553

1.569

1.561

1.568

1.571

1.581

1.575

1.554

1.548

1.566

1.554

Panel B: Confidence level 99%

        

mVaR

−4.917

-4.838

−4.735

−4.763

−4.817

−-4.734

−4.665

−4.700

−5.045

−4.975

−4.752−

4.877

stdVaR

2.458

2.500

2.428

2.431

2.468

2.541

2.485

2.482

2.594

2.624

2.524

2.567

mES

−5.630

−5.814

−5.455

−5.546

−5.521

−5.677−

5.421−

5.488

−5.777

−6.028

−5.447

−5.695

stdES

2.812

3.067

2.730

2.897

2.825

3.138

2.852

2.985

2.967

3.225

2.840

3.060

AE

1.126

1.173

1.337

1.314

1.361

1.572

1.619

1.455

1.079

1.079

1.220

1.197

LRuc

0.417

0.269

0.035

0.049

0.025

0.001

0.000

0.005

0.607

0.607

0.163

0.211

LRcc

0.613

0.480

0.105

0.139

0.012

0.000

0.000

0.001

0.716

0.716

0.344

0.411

DQ

0.741

0.641

0.259

0.320

0.004

0.000

0.000

0.000

0.804

0.804

0.513

0.580

ER

0.000

0.097

0.001

0.000

0.001

0.258

0.002

0.000

0.004

0.258

0.000

0.009

ESR

0.030

0.176

0.022

0.027

0.016

0.070

0.006

0.027

0.147

0.467

0.007

0.077

ADMax

3.429

3.339

3.690

3.482

3.969

4.444

4.457

4.365

3.537

3.391

3.495

3.468

Tick

0.061

0.061

0.062

0.062

0.062

0.062

0.063

0.062

0.061

0.061

0.061

0.061

FZL

1.744

1.739

1.753

1.754

1.770

1.779

1.798

1.787

1.737

1.734

1.742

1.742

  1. The table presents the mean values and standard deviations of VaR and ES forecasts at 97.5% (Panel A) and 99% (Panel B) confidence levels for short positions in PBW. Additionally, it includes backtesting outcomes for three GARCH variants paired with different distributions: normal (N), skewed t (ST), historical simulation (HS), or Cornish-Fisher expansion (CF), throughout the entire forecasting horizon. The table details the AE ratio, the p-values for the LRuc, LRcc, DQ, one-sided ER, one-sided Intercept ESR tests, alongside the absolute maximum deviation (ADMax) and mean values of the tick and FZL metrics. Models highlighted in bold denote those included in the superior set of models, which have statistically comparable VaR forecasts based on either the tick loss function or joint VaR and ES forecasts based on the FZL function at a 90% confidence level.
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