Table 5 Full sample VaR and ES results for long 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

−4.102

-4.376

−4.377

−4.443

−4.086

−4.387

−4.304−

4.404

−4.219

−4.512

−4.452

−4.596

stdVaR

2.038

2.167

2.177

2.253

2.071

2.230

2.182

2.276

2.148

2.263

2.254

2.376

mES

−4.897

−5.599

−5.518

-5.581

−4.871

−5.598

−5.396

−5.518

−5.036

−5.826

−5.666

−5.798

stdES

2.432

2.837

2.847

2.925

2.468

2.914

2.840

2.956

2.564

2.970

2.981

3.082

AE

1.389

1.117

1.107

1.098

1.370

1.126

1.192

1.164

1.389

1.089

1.107

1.051

LRuc

0.000

0.230

0.269

0.313

0.000

0.196

0.051

0.095

0.000

0.361

0.269

0.596

LRcc

0.001

0.479

0.537

0.596

0.000

0.423

0.088

0.158

0.001

0.509

0.537

0.726

DQ

0.000

0.300

0.136

0.178

0.000

0.610

0.052

0.270

0.000

0.016

0.010

0.037

ER

0.000

0.622

0.434

0.315

0.000

0.892

0.527

0.650

0.000

0.898

0.593

0.492

ESR

0.000

0.527

0.376

0.486

0.001

0.706

0.309

0.496

0.003

0.926

0.663

0.862

ADMax

6.918

6.335

6.980

6.451

6.682

5.982

6.677

6.213

6.466

5.857

6.631

5.900

Tick

0.141

0.139

0.140

0.139

0.139

0.137

0.137

0.137

0.141

0.140

0.141

0.140

FZL

1.700

1.669

1.669

1.666

1.688

1.648

1.659

1.657

1.706

1.671

1.680

1.677

Panel B: Confidence level 99%

        

mVaR

−4.873

−5.478

−5.364

v5.561

−4.847

−5.479

−5.240

−5.499

−5.011−

5.686

−5.467

−5.773

stdVaR

2.420

2.755

2.707

2.897

2.456

2.830

2.743

2.926

2.552

2.882

2.829

3.053

mES

−5.586

−6.730

−6.381

−6.669

−5.551

−6.717

−6.211

−6.582

−5.743

−7.051

−6.548

−6.950

stdES

2.774

3.492

3.402

3.594

2.813

3.581

3.375

3.637

2.925

3.661

3.564

3.784

AE

1.924

1.126

1.290

1.103

1.924

1.126

1.361

1.103

1.947

1.056

1.220

1.079

LRuc

0.000

0.417

0.068

0.507

0.000

0.417

0.025

0.507

0.000

0.717

0.163

0.607

LRcc

0.000

0.416

0.084

0.475

0.000

0.613

0.078

0.475

0.000

0.579

0.199

0.530

DQ

0.000

0.243

0.052

0.245

0.000

0.786

0.311

0.764

0.000

0.035

0.066

0.227

ER

0.000

0.093

0.073

0.052

0.000

0.606

0.010

0.002

0.000

0.631

0.034

0.213

ESR

0.088

0.620

0.354

0.573

0.015

0.777

0.278

0.598

0.114

0.894

0.495

0.748

ADMax

5.925

4.801

4.636

4.934

5.899

4.802

4.988

5.087

5.389

4.161

4.387

4.217

Tick

0.069

0.066

0.066

0.067

0.067

0.065

0.066

0.066

0.069

0.067

0.066

0.068

FZL

1.931

1.847

1.850

1.847

1.913

1.818

1.848

1.839

1.936

1.846

1.853

1.856

  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 long 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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