APPLYING THE KNOWLEDGE GENERATOR 95

0 10 20 30 40 50 60 70 80

Intervals returns equal to 0 (in percentage) metal: Gold, interval=60 min

Hours

Percentage

(a) XAU

0 10 20 30 40 50 60 70 80 90

Intervals returns equal to 0 (in percentage) metal: Silver, interval=60 min

Hours

Percentage

(b) XAG

0 10 20 30 40 50 60 70 80 90

Intervals returns equal to 0 (in percentage) metal: Palladium, interval=60 min

Hours

Percentage

(c) XPD

0 10 20 30 40 50 60 70 80 90

Intervals returns equal to 0 (in percentage) metal: Platino, interval=60 min

Hours

Percentage

(d) XPT

Figure A.1: Occurrence of zeros during the trading day (hourly series)

18Additional tables and figures not included in the paper.

0 20 40 60 80 100 120

ACF for the squared returns, metal: Gold, interval=60 min

Lags

ACF for the squared returns, metal: Silver, interval=60 min

Lags

ACF for the squared returns, metal: Palladium, interval=60 min

Lags

ACF for the squared returns, metal: Platino, interval=60 min

Lags

Sample Autocorrelation

(d) XPT

Figure A.2: ACF squared returns

1 6 12 18 24

x 10⁻⁵ Average squared returns within a day, metal: Gold, interval=60 min

Hours

x 10⁻⁵ Average squared returns within a day, metal: Silver, interval=60 min

Hours

x 10⁻⁵ Average squared returns within a day, metal: Palladium, interval=60 min

Hours

x 10⁻⁵ Average squared returns within a day, metal: Platino, interval=60 min

Hours

(d) XPT

Figure A.3: Average hourly squared return

0 20 40 60 80 100 120

ACF for the volume (in 1000 oz.), metal: Gold, interval=60 min

Lags

ACF for the volume (in 1000 oz.), metal: Silver, interval=60 min

Lags

ACF for the volume (in 1000 oz.), metal: Palladium, interval=60 min

Lags

ACF for the volume (in 1000 oz.), metal: Platino, interval=60 min

Lags

Sample Autocorrelation

(d) XPT

Figure A.4: ACF volume

1 6 12 18 24

Average volume (in 1000 oz.) within a day, metal: Gold, interval=60 min

Hours

Average volume (in 1000 oz.) within a day, metal: Silver, interval=60 min

Hours

Average volume (in 1000 oz.) within a day, metal: Palladium, interval=60 min

Hours

Average volume (in 1000 oz.) within a day, metal: Platino, interval=60 min

Hours

(d) XPT

Figure A.5: Average hourly volume

0 20 40 60 80 100 120

ACF for the QS (end), metal: Gold, interval=60 min

Lags

ACF for the QS (Average), metal: Silver, interval=60 min

Lags

ACF for the QS (Average), metal: Palladium, interval=60 min

Lags

ACF for the QS (Average), metal: Platino, interval=60 min

Lags

Sample Autocorrelation

(d) XPT

Figure A.6: ACF PQS

1 6 12 18 24

x 10⁻³ Average QS (Average) within a day, metal: Gold, interval=60 min

Hours

x 10⁻³ Average QS (Average) within a day, metal: Silver, interval=60 min

Hours

Average QS (Average) within a day, metal: Palladium, interval=60 min

Hours

x 10⁻³ Average QS (Average) within a day, metal: Platino, interval=60 min

Hours

(d) XPT

Figure A.7: Average hourly PQS

0 20 40 60 80 100 120

ACF for the OF squared, metal: Gold, interval=60 min

Lags

ACF for the OF squared, metal: Silver, interval=60 min

Lags

ACF for the OF squared, metal: Palladium, interval=60 min

Lags

ACF for the OF squared, metal: Platino, interval=60 min

Lags

Sample Autocorrelation

(d) XPT

Figure A.8: ACF OF squared

1 6 12 18 24

Average OF squared within a day, metal: Gold, interval=60 min

Hours

Average OF squared within a day, metal: Silver, interval=60 min

Hours

Average OF squared within a day, metal: Palladium, interval=60 min

Hours

Average OF squared within a day, metal: Platino, interval=60 min

Hours

(d) XPT

Figure A.9: Average hourly OF squared

0 20 40 60 80 100 120

−0.5 0 0.5 1

ACF hourly volume − Gold

0 20 40 60 80 100 120

−0.5 0 0.5 1

ACF st hourly volume − 5 harm

(a) ACF

Estimated harmonic (5 comp.) for volume − Gold

0 5 10 15 20 250

Figure A.10: ACF of the volume and ACF of volume residuals after a regression on harmonics (left panel) and estimated periodic component (right panel)

0 20 40 60 80 100 120

ACF for the QS (Average), metal: Gold, interval=60 min

Lags

x 10⁻³ Average QS (Average) within a day, metal: Gold, interval=60 min

Hours

ACF for the OF squared, metal: Gold, interval=60 min

Lags

Average OF squared within a day, metal: Gold, interval=60 min

Hours

(d) Avegare OF

Figure A.11: ACF and average hourly QS and OF squared - XAU

Table A.1: Estimation Return Egarch and Egarch-Har models with harmonics

XAU XAU XAG XAG XPD XPD XPT XPT

Egarch Egarch HAR Egarch Egarch HAR Egarch Egarch HAR Egarch Egarch HAR µ −1.4e − 09 −1.4e − 09 3.00e − 09 3.00e − 09 4.28e − 10 4.28e − 10 1.06e − 10 1.06e − 10

(2.2e − 05) (2.2e − 05) (3.9e − 05) (3.9e − 05) (4.6e − 05) (4.6e − 05) (2.9e − 05) (2.9e − 05) φ1 −0.038^b −0.038^b −0.023 −0.023 −0.044^a −0.044^a −0.025^b −0.025^b

(0.019) (0.019) (0.015) (0.015) (0.013) (0.013) (0.012) (0.012)

ω −0.241 −0.107 −0.241 −0.656 −0.053^c −0.115 −0.018 −0.193

(0.383) (0.067) (0.262) (8.834) (0.030) (0.197) (0.014) (0.274)

θ1 0.123^a 0.139^a 0.096^a −0.143 −0.009 −0.169 −0.006 −0.085

(0.025) (0.034) (0.030) (1.193) (0.007) (0.111) (0.013) (0.089)

θ2 0.138^b - 0.106^a - 0.173^a - 0.123^a

-(0.055) (0.039) (0.042) (0.026)

θ3 0.078 0.097^c 0.053^c 0.370 −0.083^b 0.267 −0.031 0.375^a

(0.053) (0.056) (0.028) (2.262) (0.038) (0.526) (0.030) (0.140)

θ6 −0.009 −0.107^b −0.031 0.031 0.032 0.560 −0.085^b 0.140^c

(0.022) (0.045) (0.046) (3.319) (0.022) (0.631) (0.033) (0.084)

θ12 −0.016^c 0.054 0.001 0.914 −0.041^c 0.466 0.043^a −0.251

(0.008) (0.121) (0.008) (3.858) (0.022) (0.306) (0.008) (0.289)

θ24 0.154^a 0.051 0.108^a −0.153 −0.030^b −1.024^c −0.022 0.126

(0.042) (0.193) (0.025) (2.019) (0.015) (0.566) (0.021) (0.495)

β1 0.007 0.625^a 0.285^a −0.644 0.911^a −0.145 1.177^a −1.630^a

(0.034) (0.061) (0.021) (6.078) (0.056) (0.569) (0.013) (0.413)

β2 −0.012^c - −0.453^a - −2.5e − 05 - 3.9e − 05

-(0.007) (0.039) (0.006) (0.012)

β3 0.233^a −0.243^b 0.304^a −1.050 −0.235^b −1.752^b 0.001 3.143^a

(0.058) (0.099) (0.024) (8.508) (0.092) (0.845) (0.011) (0.665)

β6 0.190 0.658^a 0.153^a 0.577 0.336^a −0.886 −0.296^a −0.436

(0.126) (0.232) (0.031) (8.509) (0.046) (1.512) (0.055) (1.030)

β12 0.166^a −0.779^c 0.200^b −3.748 0.001 5.708^a 0.120^b −1.138^b

(0.051) (0.416) (0.080) (2.542) (0.006) (2.171) (0.053) (0.568)

β24 0.390^a 0.728 0.480^a 5.784 −0.021 −1.937 −0.005 1.038

(0.091) (0.526) (0.086) (24.91) (0.014) (4.814) (0.008) (1.560)

γ1 −0.516^a −0.071^b −0.534^a 0.623 −0.483^a −4.222^a 0.011 −0.012

(0.130) (0.034) (0.112) (12.10) (0.051) (0.758) (0.008) (0.017)

φ1 −0.372^a −0.138^c −0.254^b −2.957^b 0.158^a −0.187 0.001 0.087

(0.087) (0.079) (0.102) (1.208) (0.058) (0.462) (0.006) (0.071)

γ2 −0.123 −0.048^a −0.240^a −1.505 −0.181^a −1.364^a 0.448^a −0.238

(0.105) (0.014) (0.090) (6.471) (0.046) (0.327) (0.053) (0.437)

φ2 0.253^a 0.350^a 0.197^a 0.239 0.255^a −0.309 0.266^a 1.185^a

(0.049) (0.049) (0.036) (5.689) (0.059) (0.207) (0.080) (0.274)

γ3 −0.177^b −0.040 −0.342^a −1.155 −0.026^c −0.698 0.254^a −0.626

(0.090) (0.029) (0.091) (10.10) (0.013) (0.429) (0.085) (0.525)

φ3 0.350^a 0.292^a 0.197^b −0.221 0.131^b 0.302 0.645^a 1.199^a

(0.048) (0.033) (0.080) (7.866) (0.065) (0.340) (0.070) (0.323)

γ4 −0.199^a −0.257^a −0.040^b −0.280 −0.113^a −0.044 −0.401^a −1.305^a

(0.054) (0.051) (0.019) (4.217) (0.038) (0.119) (0.089) (0.203)

φ4 −0.014 0.150^a −0.057^a −0.490 0.300^a −0.493 0.329^a −0.295

(0.024) (0.036) (0.013) (3.658) (0.064) (0.318) (0.069) (0.331)

γ5 −0.251^a −0.073 −0.108^c −0.168 −0.174^b −0.163 −0.323^a −0.627^a

(0.046) (0.048) (0.063) (1.890) (0.084) (0.111) (0.083) (0.169)

φ5 0.140^a 0.257^a 0.043 −0.202 0.308^a −0.044 0.107^a −0.657^a

(0.039) (0.036) (0.050) (3.097) (0.111) (0.055) (0.031) (0.254)

LLF 56787.14 56745.73 50203.66 50150.49 48111.29 48167.20 53140.53 53155.10

Note: Estimation results for the return series for the precious metals: Gold (XAU), Silver (XAG), Palladium (XPD) and Platinum (XPT). Egarch and Egarch-HAR models. Period 27th of December 2008 to the 30th of November 2010. Hourly series, 11880 observations. LLF is the Log-likelihood function. Standard errors in bracket. ”a”, ”b” and ”c” indicate significance at the 1%, 5% and 10%.

24 48 72 96 120

ACF std. residuals EGARCH(1 2 3 6 12 24,1 2 3 6 12 24) AR(1) 5 harm, metal:Gold

24 48 72 96 120

ACF std. residuals EGARCH HAR (1 3 6 12 24,1 3 6 12 24) AR(1) 5 harm, metal:Gold

(a) XAU

0 5 10 15 20 25

0 2

4x 10⁻⁵ Mean Estimated series EGARCH and EGARCH HAR, metal:Gold

0 5 10 15 20 250

ACF std. residuals EGARCH(1 2 3 6 12 24,1 2 3 6 12 24) AR(1) 5 harm, metal:Silver

24 48 72 96 120

ACF std. residuals EGARCH HAR (1 3 6 12 24,1 3 6 12 24) AR(1) 5 harm, metal:Silver

(c) XAG

0 5 10 15 20 25

0 0.5

1x 10⁻⁴ Mean Estimated series EGARCH and EGARCH HAR, metal:Silver

0 5 10 15 20 250

ACF std. residuals EGARCH(1 2 3 6 12 24,1 2 3 6 12 24) AR(1) 5 harm, metal:Palladium

24 48 72 96 120

ACF std. residuals EGARCH HAR (1 3 6 12 24,1 3 6 12 24) AR(1) 5 harm, metal:Palladium

(e) XPD

0 5 10 15 20 25

0 0.5

1x 10⁻⁴ Mean Estimated series EGARCH and EGARCH HAR, metal:Palladium

0 5 10 15 20 250

ACF std. residuals EGARCH(1 2 3 6 12 24,1 2 3 6 12 24) AR(1) 5 harm, metal:Platino

24 48 72 96 120

ACF std. residuals EGARCH HAR (1 3 6 12 24,1 3 6 12 24) AR(1) 5 harm, metal:Platino

(g) XPT

3x 10⁻⁵ Mean Estimated series EGARCH and EGARCH HAR, metal:Platino

0 5 10 15 20 250

Figure A.12: ACF standardized squared residuals Return and mean estimated series

Table A.2: Estimation Volume GARMA model

XAU XAU XAG XAG XPD XPD XPT XPT

d1 0.151^a 0.292^a 0.118^a 0.254^a 0.103^a 0.237^a 0.134^a 0.258^a (0.022) (0.008) (0.013) (0.014) (0.006) (0.010) (0.017) (0.008) d2 0.257^a 0.468^a 0.220^a 0.390^a 0.199^a 0.374^a 0.200^a 0.359^a

(0.015) (0.031) (0.006) (0.014) (0.010) (0.013) (0.016) (0.008) d3 0.176^a 0.382^a 0.131^a 0.263^a 0.112^a 0.270^a 0.123^a 0.269^a

(0.016) (0.035) (0.010) (0.014) (0.009) (0.008) (0.012) (0.009) d4 0.180^a 0.362^a 0.142^a 0.248^a 0.132^a 0.259^a 0.182^a 0.319^a

(0.046) (0.081) (0.011) (0.015) (0.007) (0.011) (0.023) (0.009) d5 0.211^a 0.368^a 0.153^a 0.231^a 0.125^a 0.237^a 0.159^a 0.271^a

(0.009) (0.053) (0.011) (0.014) (0.013) (0.011) (0.013) (0.011) d6 0.153^a 0.317^a 0.131^a 0.196^a 0.142^a 0.243^a 0.130^a 0.234^a

(0.038) (0.046) (0.011) (0.016) (0.010) (0.017) (0.021) (0.008) d7 0.179^a 0.323^a 0.137^a 0.191^a 0.119^a 0.209^a 0.133^a 0.232^a

(0.013) (0.055) (0.016) (0.015) (0.010) (0.013) (0.019) (0.012) d8 0.175^a 0.320^a 0.137^a 0.189^a 0.120^a 0.208^a 0.133^a 0.221^a

(0.047) (0.059) (0.010) (0.015) (0.012) (0.011) (0.017) (0.011) d9 0.182^a 0.314^a 0.153^a 0.191^a 0.116^a 0.197^a 0.141^a 0.23^a

(0.045) (0.061) (0.014) (0.018) (0.010) (0.010) (0.023) (0.013) d10 0.176^a 0.316^a 0.140^a 0.172^a 0.126^a 0.204^a 0.143^a 0.224^a

(0.040) (0.108) (0.010) (0.017) (0.012) (0.015) (0.012) (0.010) d11 0.181^a 0.313^a 0.138^a 0.164^a 0.138^a 0.211^a 0.143^a 0.228^a

(0.023) (0.057) (0.017) (0.017) (0.012) (0.010) (0.015) (0.010) d12 0.156^a 0.293^a 0.135^a 0.152^a 0.109^a 0.189^a 0.132^a 0.213^a

(0.018) (0.107) (0.018) (0.019) (0.014) (0.009) (0.025) (0.014) d13 0.077^c 0.143^a 0.075^a 0.083^a 0.063^a 0.101^a 0.077^a 0.116^a

(0.042) (0.010) (0.008) (0.011) (0.008) (0.008) (0.009) (0.010)

φ1 - −0.018 - −0.234^a - −0.055^b - −0.028

(0.159) (0.044) (0.021) (0.026)

φ24 - −0.257^a - −0.157^a - −0.170^a - −0.182^a

(0.069) (0.015) (0.009) (0.008)

φ48 - −0.150^a - −0.068^a - −0.132^a - −0.103^a

(0.025) (0.011) (0.009) (0.010)

φ72 - −0.097^c - −0.083^a - −0.084^a - −0.056^a

(0.055) (0.010) (0.008) (0.009)

φ96 - −0.070^a - −0.033^a - −0.052^a - −0.058^a

(0.013) (0.009) (0.007) (0.012)

φ120 - −0.002 - −0.002 - −0.026^a - −0.006

(0.061) (0.009) (0.009) (0.009)

θ1 - −0.517^a - −0.667^a - −0.587^a - −0.519^a

(0.104) (0.033) (0.037) (0.031)

µ −22.01^a −26.62^a −99.93^a −99.21^a −13.31^a −11.38^a −13.39^a −10.88^a (1.742) (6.688) (6.929) (12.12) (1.090) (0.892) (1.161) (0.540) σ 367.0^a 356.8^a 20908^a 20274^a 706.6^a 690.7^a 441.3^a 433.3^a

(20.12) (35.77) (361.8) (292.9) (8.443) (8.043) (5.404) (3.859) LLF -51936.01 -51768.53 -75947.61 -75764.66 -55826.96 -55691.76 -53030.35 -52921.69

Note: Estimation results for the volume series for the precious metals: Gold (XAU), Silver (XAG), Palladium (XPD) and Platinum (XPT). GARMA models. Period 27th of December 2008 to the 30th of November 2010. Hourly series, 11880 observations. LLF is the Log-likelihood function. Standard errors in bracket. ”a”, ”b” and ”c” indicate significance at the 1%, 5% and 10%.

24 48 72 96 120

ACF std. residuals GARMA 24 ARMA(0,0), metal:Gold

24 48 72 96 120

ACF std. residuals GARMA 24 ARMA(1 24 48 72 96 120,1), metal:Gold

(a) XAU Mean Estimated series, metal:Gold

0 5 10 15 20 250 GARMA 24 ARMA(1 24 48 72 96 120,1) mean Vol

ACF std. residuals GARMA 24 ARMA(0,0), metal:Silver

24 48 72 96 120

ACF std. residuals GARMA 24 ARMA(1 24 48 72 96 120,1), metal:Silver

(c) XAG Mean Estimated series, metal:Silver

0 5 10 15 20 250 GARMA 24 ARMA(1 24 48 72 96 120,1) mean Vol

ACF std. residuals GARMA 24 ARMA(0,0), metal:Palladium

24 48 72 96 120

ACF std. residuals GARMA 24 ARMA(1 24 48 72 96 120,1), metal:Palladium

(e) XPD

0 5 10 15 20 25

0 50

Hours Mean Estimated series, metal:Palladium

0 5 10 15 20 250

50

GARMA 24 ARMA(0,0) GARMA 24 ARMA(1 24 48 72 96 120,1) mean Vol

ACF std. residuals GARMA 24 ARMA(0,0), metal:Platino

24 48 72 96 120

ACF std. residuals GARMA 24 ARMA(1 24 48 72 96 120,1), metal:Platino

(g) XPT Mean Estimated series, metal:Platino

0 5 10 15 20 250 GARMA 24 ARMA(1 24 48 72 96 120,1) mean Vol

(h) XPT

Figure A.13: ACF standardized squared residuals Volume and mean estimated series

Table A.3: Estimation OF²/100 ARFIMA with harmonics

XAU XAU XAG XAG XPD XPD XPT XPT

d 0.051^a 0.046^a 0.168^a 0.163^a 0.080^a 0.077^a 0.070^a 0.071^a (0.014) (0.014) (0.025) (0.025) (0.009) (0.009) (0.009) (0.009) φ1 0.294^a 0.294^a 0.734^a 0.724^a −0.261^c −0.165 −0.547^b −0.616^a (0.068) (0.062) (0.046) (0.045) (0.137) (0.109) (0.255) (0.219) φ24 0.031^a 0.029^a 0.008 0.007 0.055^a 0.054^a 0.021^b 0.020^b (0.008) (0.008) (0.006) (0.006) (0.009) (0.009) (0.008) (0.008)

φ48 - 0.027^a - 0.005 - 0.038^a - −0.001

(0.008) (0.006) (0.009) (0.007)

φ72 - 0.021^b - 0.002 - −0.015^c - −0.001

(0.008) (0.006) (0.009) (0.007)

φ96 - 0.020^b - 0.014^b - 0.016^c - −0.007

(0.008) (0.006) (0.009) (0.007)

φ120 0.051^a 0.048^a 0.011^c 0.010 −0.007 −0.007 0.004 0.005 (0.008) (0.008) (0.006) (0.006) (0.009) (0.009) (0.009) (0.008) θ1 −0.195^a −0.191^a −0.806^a −0.794^a 0.291^b 0.198^c 0.558^b 0.624^a

(0.064) (0.058) (0.033) (0.035) (0.133) (0.107) (0.245) (0.210) µ −0.827^a −0.826^a −0.056^a −0.056^a −0.051^a −0.051^a −0.065^a −0.065^a (0.056) (0.061) (0.007) (0.007) (0.004) (0.004) (0.005) (0.005) γ1 0.587^a 0.587^a 0.035^a 0.035^a 0.037^a 0.037^a 0.044^a 0.044^a

(0.053) (0.059) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) φ1 0.453^a 0.452^a 0.038^a 0.038^a 0.044^a 0.044^a 0.030^a 0.030^a

(0.053) (0.059) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) γ2 0.309^a 0.309^a 0.019^a 0.019^a 0.018^a 0.018^a 0.020^a 0.020^a

(0.050) (0.055) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) φ2 −0.268^a −0.268^a −0.012^a −0.012^a −0.019^a −0.019^a −0.019^a −0.019^a (0.050) (0.055) (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) γ3 −0.151^a −0.150^a −0.002 −0.002 −0.002 −0.002 0.007^c 0.007^c

(0.047) (0.051) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) φ3 −0.245^a −0.245^a −0.016^a −0.016^a −0.018^a −0.018^a −0.029^a −0.029^a (0.047) (0.051) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) γ4 0.093^b 0.093^c 0.002 0.002 −0.005^c −0.005^c −0.004 −0.004 (0.044) (0.048) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) φ4 0.237^a 0.237^a 0.007^b 0.007^b 0.008^a 0.008^a 0.013^a 0.013^a

(0.044) (0.048) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) γ5 0.017 0.017 −0.001 −0.001 −0.000 −0.000 0.006^c 0.006^c

(0.042) (0.045) (0.002) (0.002) (0.002) (0.003) (0.003) (0.003) φ5 −0.013 −0.013 −0.003 −0.003 0.005^c 0.005^c 0.011^a 0.011^a

(0.042) (0.045) (0.002) (0.002) (0.002) (0.003) (0.003) (0.003) σ 9.238^a 9.220^a 0.050^a 0.050^a 0.049^a 0.049^a 0.085^a 0.085^a

(0.118) (0.119) (0.000) (0.000) (0.000) (0.000) (0.001) (0.001) LLF -30063.71 -30052.31 858.8025 861.9133 1032.726 1044.071 -2263.594 -2263.024

Note: Estimation results for the OF²/100 series for the precious metals: Gold (XAU), Silver (XAG), Palladium (XPD) and Platinum (XPT). ARFIMA models.Period 27th of December 2008 to the 30th of November 2010. Hourly series, 11880 observations. LLF is the Log-likelihood function. Standard errors in bracket. ”a”, ”b” and ”c” indicate significance at the 1%, 5% and 10%.

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Gold

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Gold

(a) XAU Mean Estimated series, metal:Gold

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean OF²

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Silver

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Silver

(c) XAG Mean Estimated series, metal:Silver

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean OF²

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Palladium

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Palladium

(e) XPD Mean Estimated series, metal:Palladium

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean OF²

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Platino

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Platino

(g) XPT Mean Estimated series, metal:Platino

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean OF²

(h) XPT

Figure A.14: ACF standardized squared residuals OF 2 and mean estimated series

Table A.4: Estimation QS ∗ 100 ARFIMA with harmonics

XAU XAU XAG XAG XPD XPD XPT XPT

dS 0.083^a 0.076^a 0.281^a 0.274^a 0.283^a 0.267^a 0.513^a 0.493^a (0.010) (0.010) (0.011) (0.011) (0.010) (0.011) (0.023) (0.017) φ1 −0.084^a −0.072^a 0.033 0.056^c −0.031 0.006 0.079 0.080^c

(0.023) (0.022) (0.035) (0.032) (0.032) (0.027) (0.057) (0.045) φ24 0.104^a 0.090^a 0.153^a 0.130^a 0.163^a 0.120^a 0.123^a 0.106^a

(0.008) (0.008) (0.009) (0.009) (0.009) (0.009) (0.009) (0.009)

φ48 - 0.036^a - 0.041^a - 0.100^a - 0.070^a

(0.008) (0.009) (0.009) (0.009)

φ72 - 0.049^a - 0.086^a - 0.102^a - 0.075^a

(0.008) (0.009) (0.009) (0.009)

φ96 - 0.070^a - 0.084^a - 0.077^a - 0.046^a

(0.008) (0.009) (0.009) (0.009)

φ120 0.308^a 0.292^a 0.101^a 0.076^a 0.168^a 0.125^a 0.105^a 0.088^a (0.008) (0.008) (0.009) (0.009) (0.009) (0.009) (0.009) (0.009) θ1 0.234^a 0.227^a 0.149^a 0.125^a 0.196^a 0.177^a −0.071 −0.053 (0.023) (0.022) (0.032) (0.029) (0.029) (0.025) (0.066) (0.046) µ −0.132^a −0.134^a −0.431^a −0.464^a −1.172^a −1.192^a −4.736^a −4.891^a (0.012) (0.014) (0.047) (0.057) (0.091) (0.109) (0.655) (0.433) γ1 −0.118^a −0.120^a −0.152^a −0.154^a −0.325^a −0.324^a −0.227^a −0.232^a (0.009) (0.011) (0.008) (0.010) (0.016) (0.022) (0.012) (0.016) φ1 0.014 0.014 −0.060^a −0.062^a −0.165^a −0.171^a −0.064^a −0.064^a (0.009) (0.011) (0.008) (0.010) (0.016) (0.022) (0.012) (0.016) γ2 −0.100^a −0.101^a −0.113^a −0.114^a −0.253^a −0.250^a −0.169^a −0.172^a (0.008) (0.011) (0.006) (0.008) (0.013) (0.018) (0.009) (0.011) φ2 −0.002 −0.002 −0.030^a −0.032^a −0.085^a −0.085^a −0.063^a −0.065^a (0.008) (0.011) (0.006) (0.008) (0.013) (0.018) (0.009) (0.011) γ3 −0.084^a −0.085^a −0.076^a −0.076^a −0.159^a −0.157^a −0.097^a −0.097^a (0.008) (0.010) (0.005) (0.007) (0.011) (0.016) (0.007) (0.009) φ3 −0.005 −0.005 −0.022^a −0.023^a −0.087^a −0.087^a −0.075^a −0.076^a (0.008) (0.010) (0.005) (0.007) (0.011) (0.016) (0.007) (0.009) γ4 −0.070^a −0.071^a −0.058^a −0.058^a −0.096^a −0.095^a −0.051^a −0.051^a (0.008) (0.010) (0.005) (0.006) (0.010) (0.014) (0.006) (0.008) φ4 −0.009 −0.009 −0.020^a −0.021^a −0.080^a −0.079^a −0.070^a −0.071^a (0.008) (0.010) (0.005) (0.006) (0.010) (0.014) (0.006) (0.008) γ5 −0.056^a −0.057^a −0.039^a −0.039^a −0.061^a −0.059^a −0.030^a −0.029^a (0.008) (0.010) (0.004) (0.006) (0.009) (0.013) (0.005) (0.007) φ5 −0.010 −0.011 −0.013^a −0.014^b −0.083^a −0.083^a −0.060^a −0.060^a (0.008) (0.010) (0.004) (0.006) (0.009) (0.013) (0.005) (0.007) σ 0.130^a 0.128^a 0.075^a 0.074^a 0.253^a 0.245^a 0.140^a 0.138^a

(0.001) (0.001) (0.000) (0.000) (0.003) (0.003) (0.001) (0.001) LLF -4742.710 -4674.295 -1520.893 -1400.621 -8712.461 -8508.625 -5210.185 -5122.007

Note: Estimation results for the QS ∗ 100 series for the precious metals: Gold (XAU), Silver (XAG), Palladium (XPD) and Platinum (XPT). ARFIMA models. Period 27th of December 2008 to the 30th of November 2010. Hourly series, 11880 observations. LLF is the Log-likelihood function. Standard errors in bracket. ”a”, ”b” and ”c” indicate significance at the 1%, 5% and 10%.

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Gold

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Gold

(a) XAU Mean Estimated series, metal:Gold

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean QS

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Silver

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Silver

(c) XAG Mean Estimated series, metal:Silver

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean QS

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Palladium

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Palladium

(e) XPD Mean Estimated series, metal:Palladium

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean QS

ACF std. residuals ARFIMA(1 24 120,1,1) 5 harm, metal:Platino

24 48 72 96 120

ACF std. residuals ARFIMA(1 24 48 72 96 120,1,1) 5 harm, metal:Platino

(g) XPT Mean Estimated series, metal:Platino

ARFIMA(1 24 120,1,1) 5 harm ARFIMA(1 24 48 72 96 120,1,1) 5 harm mean QS

(h) XPT

Figure A.15: ACF standardized squared residuals QS and mean estimated series

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