Table 2 Estimates of ARFIMA-HYAPARCH model for each series

\(\left( {p,\,d_{m} ,\,q} \right)\)

\(\left( {1,\,d,\,1} \right)\)

\(\left( {1,\,d,\,1} \right)\)

\(\left( {0,\,d,\,0} \right)\)

\(\left( {0,\,d,\,1} \right)\)

\(\left( {0,\,d,\,0} \right)\)

\(\left( {1,\,d,\,2} \right)\)

\(\left( {P,d_{v} ,\,Q} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\left( {1,\,\delta ,\,1} \right)\)

\(\mu\)

0.000

(0.389)

0.000

(0.744)

0.000

(0.572)

\(d_{m}\)

0.104

(3.974)***

0.137

(3.085)***

0.117

(4.783)***

0.096

(3.160)***

0.086

(4.670)***

0.129

(3.784)***

\(\theta_{1}\)

 − 0.208

(− 1.742)*

 − 0.234

(− 2.162)**

 − 0.216

(− 1.901)**

\(\theta_{2}\)

\(\phi_{1}\)

 − 0.208

(− 1.742)*

 − 0.234

(− 2.162)**

 − 0.216

(− 2.127)**

\(\phi_{2}\)

 − 

 − 

\(\omega\)

0.000

(0.043)

0.001

(0.032)

0.000

(0.033)

0.000

(0.022)

0.001

(0.016)

0.003

(0.641)

\(d_{v}\)

0.425

(7.886)***

0.347

(4.551)***

0.348

(6.070)***

0.377

(5.547)***

0.435

(6.872)***

0.369

(8.342)***

\(Log\left( \alpha \right)\)

0.217

(3.763)***

0.197

(3.205)***

0.174

(3.879)***

0.238

(4.615)***

0.245

(3.982)***

0.245

(4.704)***

\(\gamma\)

 − 0.182

(− 3.363)***

 − 0.197

(− 3.678)*** − 

 − 0.234

(− 4. 674)***

 − 0.228

(− 3.233)***

 − 0.249

(− 3.861)***

 − 0.145

(− 4.389)***

\(\delta\)

1.972

(11.340)*** − 

2.628

(10.467)***

2.274

(9.892)***

3.157

(12.093)***

2.728

(10.873)***

2.312

(12.520)***

\(\beta_{1}\)

0.677

(5.421)***

0.408

(4.521)***

0.546

(4.764)***

0.479

(4.983)***

0.578

(6.093)***

0.648

(6.169)***

\(\varpi_{1}\)

0.461

(3.125)***

0.293

(3.938)***

0.293

(3.887)***

0.369

(4.673)***

0.398

(3.659)***

0.301

(4.670)***

\(\hat{\upsilon }\)

5.131

(12.811)***

4.178

(13.901)***

5.675

(11.674)***

5.816

(10.632)***

4.755

(10.471)***

4.535

(11.150)***

Skw

0.254

(3.635)***

0.123

(1.768)*

0.124

(3.564)***

0.109

(1.757)*

0.234

(3.698)***

0.084

(1.212)

Ex. Kurt

3.999

(25.646)***

3.628

(25.984)***

3.857

(26.674)***

3.512

(23.982)***

3.853

(22.438)***

3.066

(23.404)***

\(Q\left( {20} \right)\)

22.559

22.601

22.082

21.672

20.874

18.359

\(Q^{2} \left( {20} \right)\)

10.365

11.113

10.874

10.359

10.743

9.105

\(BDS\left( 5 \right)\)

5.143

4.587

5.327

4.875

3.174

3.987

\(\begin{gathered} Log - \hfill \\ Likelihood \hfill \\ \end{gathered}\)

1039.039

1048.083

1046.675

1043.855

1047.972

1049.567

\(Akaike\)

 − 0.072

 − 0.085

 − 0.075

 − 0.088

 − 0.082

 − 0.095