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Table 1 The empirical 1, 5 and 10 % significance levels for different fitted SAR \((1)_{s}\) models, with different SAR coefficients \(\Phi _{1}= 0.1,0.3, 0.5, 0.7,\) and 0.9, for the seasonal portmanteau test statistics \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\), and \(\mathfrak {D}_{m}(s)\), where \(s=4,12\), \(n=200\) and lags \(m=5,15\)

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

 

\(\Phi _{1}\)

\(\hat{Q}_{5}(s)\)

\(\tilde{Q}_{5}(s)\)

\(\mathfrak {D}_{5}(s)\)

\(\hat{Q}_{15}(s)\)

\(\tilde{Q}_{15}(s)\)

\(\mathfrak {D}_{15}(s)\)

\((s=4)\,\alpha =0.01\)

0.1

0.010

0.013

0.007

0.013

0.012

0.008

0.3

0.007

0.017

0.007

0.009

0.010

0.007

0.5

0.009

0.010

0.008

0.014

0.007

0.009

0.7

0.010

0.017

0.009

0.014

0.015

0.006

0.9

0.016

0.009

0.009

0.019

0.005

0.007

\(\alpha =0.05\)

0.1

0.050

0.041

0.034

0.051

0.038

0.033

0.3

0.043

0.051

0.039

0.035

0.040

0.037

0.5

0.049

0.049

0.042

0.055

0.039

0.041

0.7

0.049

0.070

0.041

0.068

0.050

0.041

0.9

0.060

0.055

0.044

0.064

0.043

0.042

\(\alpha =0.10\)

0.1

0.091

0.081

0.077

0.109

0.070

0.070

0.3

0.092

0.105

0.089

0.084

0.084

0.088

0.5

0.095

0.087

0.082

0.101

0.081

0.081

0.7

0.093

0.134

0.090

0.106

0.102

0.086

0.9

0.122

0.107

0.085

0.113

0.090

0.088

\((s=12)\,\alpha =0.01\)

0.1

0.018

0.018

0.006

0.014

0.011

0.005

0.3

0.015

0.010

0.007

0.018

0.008

0.007

0.5

0.015

0.018

0.010

0.016

0.012

0.008

0.7

0.019

0.014

0.012

0.015

0.010

0.011

0.9

0.023

0.015

0.008

0.022

0.013

0.009

\(\alpha =0.05\)

0.1

0.070

0.061

0.031

0.046

0.041

0.030

0.3

0.068

0.063

0.044

0.067

0.041

0.040

0.5

0.072

0.070

0.040

0.050

0.047

0.041

0.7

0.075

0.076

0.039

0.069

0.054

0.037

0.9

0.073

0.069

0.043

0.072

0.060

0.038

\(\alpha =0.10\)

0.1

0.119

0.126

0.083

0.088

0.091

0.080

0.3

0.129

0.142

0.090

0.118

0.089

0.121

0.5

0.133

0.140

0.102

0.091

0.104

0.099

0.7

0.150

0.144

0.074

0.114

0.104

0.088

0.9

0.141

0.140

0.088

0.130

0.120

0.111