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Table 2 The empirical power for a nominal 5 % level test comparing the approximation distributions of the seasonal portmanteau test statistics \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\), and \(\mathfrak {D}_{m}(s)\), at lag \(m=10\) based on \(10^{4}\) simulations. In each simulation, the SAR \((1)_{s}\) and SMA \((1)_{s}\) are fitted to data of series length \(n=200\) generated from SARMA \((2,2)\times (2,2)_{s}\) models where asterisk (*) refers to NULL and \(s=4,12\)

From: Portmanteau test statistics for seasonal serial correlation in time series models

Model

\(\phi _{1}\)

\(\phi _{2}\)

\(\theta _{1}\)

\(\theta _{2}\)

\(\Phi _{1}\)

\(\Phi _{2}\)

\(\Theta _{1}\)

\(\Theta _{2}\)

\(\hat{Q}_{m}(4)\)

\(\tilde{Q}_{m}(4)\)

\(\mathfrak {D}_{m}(4)\)

\(\hat{Q}_{m}(12)\)

\(\tilde{Q}_{m}(12)\)

\(\mathfrak {D}_{m}(12)\)

Fitted by SAR (1)

 1

*

*

*

*

*

*

−0.5

*

0.111

0.133

0.129

0.100

0.113

0.119

 2

*

*

*

*

*

*

−0.6

0.3

0.333

0.301

0.389

0.301

0.301

0.356

 3

*

*

*

*

0.7

*

−0.4

*

0.087

0.091

0.063

0.084

0.090

0.062

 4

*

*

*

*

0.1

0.3

*

*

0.102

0.119

0.120

0.092

0.095

0.089

 5

0.3

*

*

*

−0.35

*

*

*

0.786

0.663

0.652

0.763

0.641

0.661

 6

0.4

*

*

*

*

*

−0.8

*

0.961

0.998

0.971

0.951

0.996

0.965

 7

*

*

*

*

0.4

−0.6

0.3

*

0.371

0.401

0.442

0.363

0.396

0.406

 8

0.7

0.2

*

*

−0.5

*

*

*

1.000

1.000

1.000

1.000

1.000

1.000

 9

0.7

*

0.7

*

−0.8

*

*

*

0.124

0.096

0.075

0.119

0.094

0.066

 10

0.1

0.3

*

*

*

*

−0.8

*

0.867

0.918

0.889

0.844

0.907

0.887

Fitted by SMA (1)

 11

*

*

*

*

0.5

*

*

*

0.126

0.148

0.150

0.111

0.144

0.143

 12

*

*

*

*

*

*

−0.6

0.3

0.175

0.172

0.264

0.161

0.165

0.244

 13

*

*

*

*

0.7

*

−0.4

*

0.382

0.701

0.665

0.378

0.700

0.662

 14

*

*

*

*

0.1

0.3

*

*

0.113

0.100

0.211

0.109

0.098

0.203

 15

0.3

*

*

*

−0.35

*

*

*

0.760

0.689

0.662

0.755

0.679

0.654

 16

0.4

*

*

*

*

*

−0.8

*

0.958

0.961

0.913

0.953

0.958

0.908

 17

*

*

*

*

0.4

*

0.3

*

0.204

0.367

0.265

0.201

0.295

0.257

 18

0.7

0.2

*

*

−0.5

*

*

*

1.000

1.000

1.000

1.000

1.000

1.000

 19

0.7

*

0.7

*

−0.8

*

*

*

0.632

0.761

0.935

0.628

0.757

0.931

 20

0.1

0.3

*

*

*

*

−0.8

*

0.779

0.705

0.700

0.770

0.698

0.698