Summary Statistics - Multi-Factor Dynamic Modelling and Forecasting of Interest Rates and Equit

The statistical analysis commences with the computation of central tendency measures such as the mean, median and mode, followed by variability measures like the standard deviation, minimum and maximum values. Further characteristics of the data such as skewness, kurtosis and the results for the Jarque-Bera (1980) normality test are also presented. The sample estimates for all these statistical measures are reported in Table 3.1 to Table 3.5 for the interbank markets and and in Table 3.6 the UK bond market data, respectively.

In addition, the correlogram results are presented individually for each interbank market rates in tables 3.7 to 3.11 and for the UK bond market in Table 3.12. The Sample Autocorrelation Function (ACF) with six lags is computed for both level and first difference of the time series, followed by the results of the Liung-Box (1979) Q-statistic test for autocorrelation. Two lag orders, ten and twenty lags respectively, have been used to compute the sample test statistic, in order to avoid the misinterpretation associated with this test in practice. If the lag order is too small the test may not detect serial correlation at higher-order lags, whereas if it is too large the power of the test may be diminished due to potential dilution of the significant correlation coefficient by insignificant correlation at other lags. For all investigated time-series the sample autocorrelation coefficients decay

0 1 2 3 4 5 6 7

00 01 02 03 04 05 06 07 08 09 10 11 12 13

SPOT1Y SPOT7Y SPOT10Y SPOT15Y SPOT20Y

101 very slowly in a linear manner, a characteristic of nonstationary time series. For the differenced series, the autocorrelation coefficients seem to cut off straight from the first lag without any discernable pattern (although they are mostly positive), suggesting that the series do not need a higher order differencing. Regarding the Liung-Box tests, all the sample values of the Q-statistic were in excess of the critical value of the corresponding 99% quantile from the



(10)

_and



(20)

distributions, respectively. Therefore, the null hypothesis of no autocorrelation has been rejected in all cases for all series, both levels and first-differences.

102

Table 3.1 Standard Statistics for LIBOR–GBP Interest Rates: 2000-2013.

LIBOR-

GBP LEVEL FIRST DIFFERENCES

Interest Rate 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M

Observations 3,455 3,455 3,455 3,455 3,455 3,454 3,454 3,454 3,454 3,454

Mean 3.438 3.5237 3.6636 3.7855 3.9983 -0.0009 -0.0014 -0.0016 -0.0016 -0.0017

Median 4.1369 4.0992 4.1891 4.39 4.5663 0 0 0 0 0

Maximum 6.9409 6.75 6.9038 6.7988 6.8877 1.25 0.6238 0.2125 0.1981 0.2056

Minimum 0.48 0.4913 0.5069 0.6013 0.9081 -1.0313 -1.1975 -1.065 -1.0763 -1.0875 Std. Dev. 2.0715 2.0955 2.0537 1.9732 1.8462 0.1132 0.0342 0.0259 0.0272 0.0335 Skewness -0.4511 -0.4429 -0.4214 -0.3926 -0.3388 0.1083 -11.2208 -21.1702 -18.3976 -9.8028

Kurtosis 1.5768 1.5958 1.6462 1.6562 1.6776 22.6081 483.2458 843.1311 713.5563 327.4826 Jarque-Bera 408.76 396.8128 366.0793 348.702 317.8565 5.53E+04 3.33E+07 1.02E+08 7.29E+07 1.52E+07

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both level and first difference of the LIBOR–GBP rates. The statistics comprise measures of central tendency – the mean and the median, measures of variability – maximum, minimum, standard deviation, and measures of normality – skewness, kurtosis and the JB normality test.

The sample estimates of these statistics indicate that the LIBOR-GBP level rates are increasing in the mean as the maturity increases. The opposite is true regarding the volatility, for longer maturity the rates are less volatile. The distributions implied by the data are all asymmetrical and platykurtic(k 3), therefore they are not normal distributions. This is also confirmed by the JB test - the null of normality is rejected for all the time-series. For the first difference series, the distributions are closely centred around a mean of almost zero, skewed to the left except for the one-week series, leptokurtic(k 3) and not normal according to the JB test.

103 Table 3.2 Standard Statistics for LIBOR–USD Interest Rates: 2000-2013.

LIBOR-USD LEVEL FIRST DIFFERENCES

Interest Rate 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M

Observations 3,455 3,455 3,455 3,455 3,455 3,454 3,454 3,454 3,454 3,454

Mean 2.3997 2.4367 2.5498 2.6849 2.9042 -0.0017 -0.0016 -0.0017 -0.0016 -0.0017

Median 1.8213 1.8388 1.8794 1.9888 2.385 0 0 0 0 0

Maximum 6.9275 6.8213 6.8688 7.1088 7.5013 1.3813 0.4688 0.2925 0.2738 0.3544 Minimum 0.1585 0.1851 0.245 0.3825 0.7203 -0.6413 -0.5288 -0.42 -0.415 -0.4175 Std. Dev. 2.1309 2.1243 2.0943 2.0328 1.9291 0.0528 0.0326 0.0281 0.0317 0.0425 Skewness 0.5617 0.5591 0.5621 0.5792 0.6362 3.356 -3.4891 -4.7984 -2.415 -0.4388 Kurtosis 1.8875 1.8931 1.9185 1.9766 2.151 186.5144 94.0157 79.4077 38.4838 15.3207 Jarque-Bera 359.8349 356.4129 350.3285 343.9954 336.798 4.85E+06 1.20E+06 8.53E+05 1.85E+05 2.20E+04

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both level and first difference of the LIBOR –USD rates. The statistics comprise certain measures of central tendency – the mean and the median, measures of variability – maximum, minimum, standard deviation, and measures of relative normality – skewness, kurtosis and the JB normality test.

The sample estimates of these statistics indicate that the LIBOR-GBP level rates are increasing in the mean as the maturity increases.

The opposite is true regarding the volatility, for longer maturity the rates are less volatile. The distributions implied by the data are all asymmetrical and platykurtic(k 3) , therefore they are not normal distributions. This is also confirmed by the JB test - the null of normality is rejected for all the time-series. For the first difference series, the distributions are closely spread around a mean of almost zero; they are skewed to the left and leptokurtic(k 3) and not normal according to the JB test.

104 Table 3.3 Standard Statistics for LIBOR–EUR Interest Rates: 2000-2013.

LIBOR-EUR LEVEL FIRST DIFFERENCES

Interest Rate 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M

Observations 3,455 3,455 3,455 3,455 3,455 3,454 3,454 3,454 3,454 3,454

Mean 2.3864 2.4561 2.5974 2.7121 2.8761 -0.0009 -0.0009 -0.0009 -0.001 -0.001

Median 2.1559 2.173 2.2754 2.2925 2.4129 0 0 0 0 -0.0007

Maximum 5.1719 5.1863 5.3913 5.4375 5.5138 0.5288 0.6431 0.1569 0.1731 0.4263 Minimum 0.0233 0.0514 0.1207 0.2143 0.4264 -0.445 -0.4166 -0.3539 -0.2568 -0.4194 Std. Dev. 1.5059 1.4964 1.475 1.4088 1.3514 0.0371 0.0228 0.0167 0.0183 0.0269 Skewness -0.0533 -0.0252 0.0728 0.1459 0.201 0.2174 4.0166 -3.1464 -0.7601 0.6049 Kurtosis 1.756 1.779 1.8369 1.8809 1.9175 47.1072 243.0815 79.9075 28.4649 47.6 Jarque-Bera 224.4304 214.9826 197.8105 192.5621 191.9396 2.80E+05 8.30E+06 8.57E+05 9.37E+04 2.86E+05

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both level and first difference of the LIBOR–EUR rates. The statistics comprise certain measures of central tendency – the mean and the median, measures of variability – maximum, minimum, standard deviation and measures of relative normality – skewness, kurtosis and the JB normality test.

The sample estimates of these statistics indicate that the LIBOR-EUR level rates are increasing in the mean as the maturity increases, suggesting an upward shape of the yield curve. The opposite is true regarding the volatility, for longer maturity the rates are less volatile. The distributions implied by the data are slightly asymmetrical and platykurtic (k 3),therefore they are not normal distributions. This is also confirmed by the JB test - the null of normality is rejected for all the time-series. For the first difference series, the distributions are closely centred around a mean of almost zero, skewed to the left except for the one-week series, leptokurtic(k 3) and not normal according to the JB normality test.

105 Table 3.4 Standard Statistics for LIBOR–JPY Interest Rates: 2000-2013.

LIBOR-JPY LEVEL FIRST DIFFERENCES

Interest Rate 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M

Observations 3,455 3,455 3,455 3,455 3,455 3,454 3,454 3,454 3,454 3,454

Mean 0.1921 0.2241 0.2887 0.3643 0.4714 -7.19E-06 -8.17E-06 -9.36E-06 7.86E-06 5.16E-05

Median 0.1121 0.138 0.1935 0.3344 0.535 0 0 0 0 0

Maximum 2.0725 1.06 1.0938 1.185 1.3325 0.8488 0.3688 0.16 0.0688 0.1

Minimum 0.0313 0.0363 0.0455 0.0573 0.0831 -1.335 -0.455 -0.1338 -0.0513 -0.0625 Std. Dev. 0.2188 0.2415 0.2851 0.3097 0.3526 0.0389 0.0164 0.0071 0.0054 0.0053 Skewness 1.8253 1.5222 1.2735 0.8986 0.4411 -5.7647 0.8541 2.1448 1.9071 1.5414 Kurtosis 6.5736 4.1784 3.3632 2.7178 2.0335 533.5677 328.6445 143.1974 39.781 65.9803 Jarque-Bera 3757.035 1534.209 952.9288 476.4578 246.5006 4.05E+07 1.53E+07 2.83E+06 1.97E+05 5.72E+05

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both level and first difference of the LIBOR–JPY rates. The statistics comprise certain measures of central tendency – the mean and the median, measures of variability – maximum, minimum, standard deviation and measures of relative normality – skewness, kurtosis and the JB normality test. Both mean and standard deviation are increasing with the maturity, reflecting the uncertainty within the Japanese financial system as the result of many policies failing to have any impact on Japanese financial markets. For all the different maturity time series considered, the skewness and the kurtosis estimates indicate non-normality as confirmed by the JB test statistics and its p-values. For the first difference series, the distributions are closely centred around a mean of almost zero, skewed to the left except for the one-week series, leptokurtic(k 3) and not normal according to the JB normality test.

106 Table 3.5 Standard Statistics for LIBOR–CAD Interest Rates: 2000-2013.

LIBOR-CAD LEVEL FIRST DIFFERENCES

Interest Rates 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M

Observations 3,455 3,455 3,455 3,455 3,455 3,454 3,454 3,454 3,454 3,454

Mean 2.6829 2.7196 2.8085 2.9349 3.193 -0.0011 -0.0011 -0.0011 -0.0011 -0.0011

Median 2.625 2.6867 2.7517 2.8225 3 0 0 0 0 0

Maximum 5.8567 5.8767 6.025 6.1892 6.4933 0.695 0.6866 0.1917 0.28 0.325

Minimum 0.22 0.2917 0.3983 0.6933 1.2333 -0.4733 -0.6987 -0.33 -0.3433 -0.365 Std. Dev. 1.5949 1.5763 1.5272 1.4469 1.3111 0.0352 0.0307 0.0246 0.0293 0.0366 Skewness 0.217 0.2254 0.2777 0.3797 0.5671 1.387 -1.2261 -3.2007 -1.6188 -0.4882

Kurtosis 2.0326 2.0455 2.0907 2.1828 2.4719 98.142 186.6345 41.5035 27.0744 18.4641 Jarque-Bera 161.8163 160.4362 163.4585 179.1474 225.3419 1.30E+06 4.85E+06 2.19E+05 8.49E+04 3.46E+04

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both level and first difference of the LIBOR–CAD rates. The statistics comprise certain measures of central tendency – the mean and the median, measures of variability – maximum, minimum, standard deviation and measures of relative normality – skewness, kurtosis and the JB normality test. The mean of the series increases with its maturity suggesting higher on average, interbank rates for longer maturities, therefore an upward interbank yield curve. For all the different maturity time series considered, the skewness and the kurtosis estimates indicate non-normality as confirmed by the JB test statistics and its p-values. For the first difference series, the distributions are closely centred around a mean of almost zero, skewed to the left except for the one-week series, leptokurtic(k 3) and not normal according to the JB normality test.

107 Table 3.6 Standard Statistics for UK Spot Rates: 2000-2013.

UK Spot LEVEL FIRST DIFFERENCES

Interest Rates 1Y 7Y 10Y 15Y 25Y 1Y 7Y 10Y 15Y 25Y

Observations 3,346 3,346 3,346 3,346 3,346 3,345 3,345 3,345 3,345 3,345

Mean 3.3274 4.0256 4.208 4.3472 4.3156 -0.0018 -0.0014 -0.0011 -0.0007 -0.0003 Median 4.2342 4.4636 4.5004 4.5239 4.3997 -0.0011 -0.0016 -0.0012 -0.0007 -0.0005 Maximum 6.3652 6.1509 5.7299 5.2352 5.0466 0.4633 0.2278 0.2476 0.2373 0.2347 Minimum 0.1346 0.9909 1.5889 2.2856 3.0762 -0.328 -0.2869 -0.3654 -0.4462 -0.4257 Std. Dev. 2.0136 1.2198 0.9389 0.6596 0.4055 0.037 0.049 0.0492 0.0463 0.0439 Skewness -0.4447 -0.9565 -1.2547 -1.6127 -1.1542 0.0517 0.0297 -0.0704 -0.2742 -0.2481 Kurtosis 1.5894 2.9242 3.7589 4.8883 3.9158 17.728 4.6897 5.4403 7.7347 7.8254 Jarque-Bera 387.6715 510.9648 958.1979 1947.49 859.8786 30233.9 398.4083 832.7736 3166.269 3279.579

Probability 0 0 0 0 0 0 0 0 0 0

Notes: This table reports the standard statistics for both, level and first difference of the UK nominal interest rates. The statistics comprise certain measures of central tendency the – mean and the median, certain measures of variability – maximum, minimum, standard deviation and measures of relative normality – skewness, kurtosis and the JB normality test. The mean of the series increases with its maturity suggesting higher on average, interbank rates for longer maturities, therefore an upward interbank yield curve. The skewness and the kurtosis estimates indicate non-normality as confirmed by the JB test statistics and its p-values. For the first difference series, the distributions are closely centred around a mean of almost zero, skewed to the left except for the one- and seven-year series, leptokurtic

(k 3) and not normal according to the JB normality test.

108 Table 3.7 Coefficients of Autocorrelation LIBOR-GBP Interest Rates, 2000-2013.

LIBOR-GBP LEVEL FIRST DIFFERENCE

RATES 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M



0.9980 0.9990 0.9990 0.9990 0.9990 -0.0930 0.2120 0.2850 0.2370 0.1800



₂ 0.9970 0.9990 0.9990 0.9990 0.9980 -0.1090 0.1180 0.1850 0.1610 0.1000



0.9950 0.9980 0.9980 0.9980 0.9970 -0.0310 0.1050 0.1770 0.1460 0.0880



4 _0.9940 _0.9970 _0.9970 _0.9970 _0.9960 _-0.0630 _0.1010 _0.2190 _0.1700 _0.1110



5 _0.9930 _0.9970 _0.9960 _0.9960 _0.9950 _-0.0090 _0.0960 _0.1390 _0.1040 _0.0530



6 _0.9930 _0.9960 _0.9960 _0.9950 _0.9940 _0.0030 _0.0820 _0.1130 _0.1020 _0.0810

LB1 Q -stat. 34,161.00* 34,362.00* 34,348.00* 34,314.00* 34,236.00* 140.77* 395.63* 925.43* 631.49* 280.12*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 67,827.00* 68,233.00* 68,195.00* 68,061.00* 67,751.00* 180.26* 581.14* 1,336.40* 939.08* 410.45*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table section we report the values of the first six autocorrelation coefficients for the level and first difference of all LIBOR-GBP time series.

2. In the lower section, the modified Q statistic suggested by Liung-Box (1979) for ten lags (LB1) and twenty lags (LB2) is presented together with its p-values; * indicates 1% level of statistical significance.

109 Table 3.8 Coefficients of Autocorrelation for LIBOR-USD Interest Rates, 2000-2013.

LIBOR-USD LEVEL FIRST DIFFERENCES

RATES 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M



1 _0.999 _0.999 _0.999 _0.999 _0.999 _0.356 _0.456 _0.262 _0.156 _0.106



₂ 0.998 0.999 0.998 0.998 0.998 0.102 0.304 0.107 0.042 0.01



3 _0.997 _0.998 _0.998 _0.997 _0.997 _-0.045 _0.228 _0.066 _0.031 _0.02



4 _0.996 _0.997 _0.997 _0.996 _0.996 _-0.016 _0.198 _0.115 _0.06 _0.038



5 _0.995 _0.996 _0.996 _0.996 _0.995 _-0.07 _0.202 _0.095 _0.053 _0.016



6 _0.994 _0.995 _0.995 _0.995 _0.994 _0.069 _0.204 _0.157 _0.112 _0.082

LB1 Q -stat. 34,243.00* 34,303.00* 34,309.00* 34,278.00* 34,237.00* 527.20* 1,788.80* 553.47* 203.68* 93.12*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 67,833.00* 67,957.00* 64,689.00* 67,926.00* 67,781.00* 579.20* 1,897.90* 652.83* 244.01* 116.38*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table section we report the values of the first six autocorrelation coefficients for the level and first difference of all LIBOR-USD time series.

2. In the lower section, the modified Q statistic suggested by Liung-Box (1979) for ten lags (LB1) and twenty lags (LB2) is presented together with its p-values. * indicates 1% level of statistical significance.

110 Table 3.9 Coefficients of Autocorrelation for LIBOR-EUR Interest Rates, 2000-2013.

LIBOR-EUR LEVEL FIRST DIFFERENCE

RATES 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M



1 ^0.999 ^0.999 ^0.999 ^0.999 ^0.999 ^0.189 ^0.267 ^0.404 ^0.305 ^0.065



2 ^0.999 ^0.999 ^0.999 ^0.999 ^0.998 ^0.065 ^0.167 ^0.325 ^0.25 ^0.094



3 ^0.998 ^0.998 ^0.998 ^0.998 ^0.998 ^0.029 ^0.178 ^0.273 ^0.213 ^0.092



4 ^0.997 ^0.998 ^0.998 ^0.997 ^0.997 ^0.02 ^0.115 ^0.253 ^0.188 ^0.083



5 ^0.996 ^0.997 ^0.997 ^0.997 ^0.996 ^-0.154 ^0.113 ^0.181 ^0.126 ^0.026



6 ^0.995 ^0.996 ^0.996 ^0.996 ^0.995 ^0.005 ^0.126 ^0.224 ^0.176 ^0.099

LB1 Q -stat. 34,323.00* 34,381.00* 34,389.00* 34,358.00* 34,301.00* 227.78* 817.75* 2,258.50* 1,370.70* 223.07*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 68,201.00* 68,285.00* 68,316.00* 68,190.00* 67,966.00* 288.88* 991.33* 3,188.00* 2,080.20* 393.22*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table section we report the values of the first six autocorrelation coefficients for the level and first difference of all LIBOR-EUR time series.

111 Table 3.10 Coefficients of Autocorrelation LIBOR-JPY Interest Rates, 2000-2013.

LIBOR-JPY LEVEL FIRST DIFFERENCE

RATES 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M



1 ^0.984 ^0.998 ¹ ^0.179 ¹ ^-0.138 ^0.033 ^0.179 ^0.353 ^0.34



2 ^0.973 ^0.995 ^0.999 ^0.127 ¹ ^-0.016 ^-0.021 ^0.127 ^0.244 ^0.214



3 ^0.962 ^0.993 ^0.999 ^0.081 ^0.999 ^-0.011 ^0.049 ^0.081 ^0.169 ^0.139



4 ^0.951 ^0.99 ^0.998 ^0.076 ^0.999 ^-0.002 ^-0.057 ^0.076 ^0.161 ^0.144



5 ^0.941 ^0.988 ^0.997 ^0.099 ^0.999 ^-0.358 ^0.037 ^0.099 ^0.144 ^0.114



6 ^0.941 ^0.985 ^0.997 ^0.103 ^0.998 ^0.147 ^0.115 ^0.103 ^0.153 ^0.116

LB1 Q -stat. 31,135.00* 33,622.00* 34,404.00* 34,469.00* 34,510.00* 636.24* 222.39* 336.64* 1,107.10* 843.94*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 59,618.00* 64,876.00* 68,299.00* 65,189.00* 68,808.00* 672.43* 348.82* 382.83* 1,148.60* 872.91*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table section we report the values of the first six autocorrelation coefficients for the level and first difference of all LIBOR-JPY time series.

2. In the lower section, the modified Q statistic suggested by Liung-Box (1979) for ten lags (LB1) and twenty lags (LB2) are presented together with their p-values; *indicates 1% level of statistical significance.

112 Table 3.11 Coefficients of Autocorrelation for LIBOR-CAD Interest Rates, 2000-2013.

LIBOR-CAD LEVEL FIRST

DIFFERENCE

RATES 1W 1M 3M 6M 12M 1W 1M 3M 6M 12M



1 ^0.999 ^0.999 ^0.999 ^0.999 ^0.999 ^0.194 ^-0.008 ^0.219 ^0.222 ^0.218



₂ 0.999 0.999 0.999 0.998 0.998 0.059 0.101 0.156 0.076 0.065



3 ^0.998 ^0.998 ^0.998 ^0.997 ^0.996 ^0.014 ^0.074 ^0.082 ^0.052 ^0.041



4 ^0.997 ^0.997 ^0.997 ^0.996 ^0.995 ^-0.021 ^0.077 ^0.129 ^0.093 ^0.038



5 ^0.996 ^0.997 ^0.996 ^0.996 ^0.993 ^0.023 ^0.074 ^0.11 ^0.071 ^0.039



6 ^0.995 ^0.996 ^0.996 ^0.994 ^0.992 ^0.034 ^0.059 ^0.064 ^0.043 ^0.037

LB1 Q -stat. 34,334.00* 34,358.00* 34,339.00* 34,272.00* 34,110.00* 166.84* 157.57* 520.30* 295.26* 219.19*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 68,190.00* 68,219.00* 68,130.00* 67,878.00* 67,231.00* 187.23* 209.12* 653.00* 339.04* 236.26*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table section we report the values of the first six autocorrelation coefficients for the level and first difference of all LIBOR-CAD time series.

113 Table 3.12 Coefficients of Autocorrelation for UK Spot Interest Rates, 2000-2013.

UK SPOT LEVEL FIRST DIFFERENCE

RATES 1Y 7Y 10Y 15Y 25Y 1Y 7Y 10Y 15Y 25Y



₁ _0.999 _0.998 _0.997 _0.996 _0.993 _0.119 _0.038 _0.039 _0.051 _0.071



₂ 0.998 0.996 0.995 0.993 0.986 0.028 -0.025 -0.04 -0.055 -0.074



₃

0.997 0.994 0.992 0.989 0.979 -0.003 -0.039 -0.052 -0.065 -0.085



₄ 0.996 0.992 0.99 0.986 0.973 0.007 0.038 0.043 0.04 0.028



₅ 0.995 0.99 0.987 0.983 0.968 -0.006 -0.022 -0.013 -0.004 -0.007



₆ 0.995 0.988 0.985 0.979 0.962 0.029 -0.031 -0.033 -0.035 -0.041

LB1 Q -stat. 33,198.00* 32,820.00* 32,616.00* 32,304.00* 31,301.00* 82.44* 26.25* 38.33* 54.43* 83.88*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

LB2 Q-stat. 65,832.00* 64,520.00* 63,830.00* 59,858.00* 59,773.00* 180.56* 50.35* 61.38* 71.60* 93.85*

p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1. In the upper section of this table we report the values of the first six autocorrelation coefficients for the level and first difference of all U.K. spot rates.

2. In the lower section, the modified Q statistic suggested by Liung-Box (1979) for ten lags (LB1) and twenty lags (LB2) is presented together with its p-values.

114 Table 3.13 The Correlations between the first-difference time-series

GBP-LIBOR 1W 1M 3M 6M 12M

USD-LIBOR 1W 1M 3M 6M 12M

1W 1 1W 1

1M 0.51 1 1M 0.60 1

3M 0.28 0.85 1 3M 0.49 0.85 1

6M 0.21 0.75 0.93 1 6M 0.35 0.70 0.88 1

12M 0.16 0.61 0.79 0.93 1 12M 0.21 0.49 0.67 0.91 1

EUR-LIBOR 1W 1M 3M 6M 12M

JPY-LIBOR 1W 1M 3M 6M 12M

1W 1 1W 1

1M 0.50 1 1M 0.47 1

3M 0.43 0.72 1 3M 0.32 0.59 1

6M 0.33 0.59 0.88 1 6M 0.19 0.45 0.76 1

12M 0.19 0.39 0.64 0.85 1 12M 0.10 0.33 0.61 0.84 1

CAD-LIBOR 1W 1M 3M 6M 12M UK-SPOT 1Y 7Y 10Y 15Y 25Y

1W 1 1Y 1

1M 0.50 1 7Y 0.64 1

3M 0.46 0.63 1 10Y 0.54 0.98 1

6M 0.30 0.49 0.84 1 15Y 0.47 0.93 0.97 1

12M 0.21 0.38 0.70 0.89 1 25Y 0.38 0.81 0.86 0.94 1

Unit Root Testing

As nonstationarity is a dominant characteristic of all the time series under study, a more formal assessment is required. It is assumed that the possibility of the nonstationarity feature is implied by the presence of a single unit root with the time series being I(1). This particular kind of nonstationarity will be tested for consistency with the data. The widely used Augmented Dickey-Fuller (1979) (ADF) test for a single unit root is known to have a low statistical power especially if a structural break is potentially present, see Patterson (2000). In the light of the credit and liquidity crisis within the interbank market during September 2008 this is highly plausible, as indicated by the sharp fall of the level of interest rates at that point in time. Additionally, the ADF can be unreliable if the time series contains a moving average disturbance term. To overcome these problems, the decision regarding the existence or non-existence of a unit root has to be assessed in conjunction with other unit root test statistics. Consequently, another two

115 unit-root tests¹⁹ are considered. They are the Phillips-Perron (PP) (1988) test and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) (1992) test. While the first two tests, ADF and PP, are unit root tests with the null hypothesis

H

₀: the series has a unit root, the KPSS test is a stationarity test with an opposite null in contrast with former tests.

The ADF test is an extension of the Dickey-Fuller test for higher-order serial correlation in the series. The ADF test tests for the existence of a single unit root in an autoregressive AR p( ), p 1specification. The testing procedure is adjusted by adding more lagged difference terms in the test regression - a parametric correction. While there are three version of the test, we consider the most general augmented specification including two exogeneous variables in the regression model, a constant



and a linear trend t:

1 1 1

( ) : _t _t _t ... _p _{t p} _t

AR p    r

  

t r_     



r_



r_ 



(3.33) Where the disturbance terms are white noise,



_t iid(0,



²)independent and identically distributed. The null hypothesis

H

₀

:   0

is tested against the one-sided alternative hypothesis

H

₁

:   0

The test statistic under the null hypothesis follows a nonstandard distribution and the critical values are extracted automatically by EViews from the MacKinnon (1996) table which is a larger set of simulations than the original Dickey and Fuller table of critical values. The ADF statistic is always negative. The more negative the sample critical value is, the higher the probability of rejecting the null will be. The number of lagged first difference terms, p, is determined using the Schwarz (1978) Information Criterion (SIC).

An alternative to the ADF test for unit root is the PP test that like the ADF test controls for the higher-order correlation but in a non-parametric way. To account for any serial correlation and heteroskedasticity in the residuals of the regression, therefore allowing for processes



,

that are not iid

(0, 

)

distributed, the PP test constructs a modified ADF

t

- statistic using a correction factor. The asymptotic distribution of the PP unit root

t

-statistic is the same as the ADF

t

- statistic and the same MacKinnon critical or p-values are used for decision criteria.

19 Despite the fact that some of the series may indicate structural breaks, in this study we don’t consider more general unit root tests that account for parameter instability such as Bai-Perron (2003) and Zivot-Andrews (1992).

116 The classic ADF and PP unit root tests treat asymmetrically the null hypothesis of

“the series is I(1)” against the alternative “the series is I(0)” giving a dominant role to the null. To overcome this shortcoming another class of stationarity tests have been developed such as Leybourne and McCabe (LMc) (1994) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) (1992). These tests invert the hypotheses, with the null of stationarity or ARIMA(p,0,0) process against the alternative hypothesis of nonstationarity or an ARIMA(p,1,1) process. In addition, they are more powerful than ADF and PP unit root test when the ARMA processes have a large moving average component²⁰. For this study, the KPSS test is employed including both deterministic regressors, the constant and the time trend. The critical values have been simulated and tabulated in Kwiatkowski et al. (1992) and they are identical to the Leybourne and Macabe (1994) test statistics.

The results for the unit root testing obtained using these three tests are very consistent as can be inferred from Tables 3.14 to 3.19.

Table 3.14 LIBOR-GBP Rates: The Unit Root ADF, PP and KPPS Tests.

Unit Root Tests ADF PP KPSS

LIBOR-GBP t-Stat. Prob.* Adj. t-stat Prob.* LM - Stat. Crit. Val.**

1-week Level -1.41538 0.8567 -2.54069 0.3082 0.954851 0.216

LIBOR - GBP First Diff. -28.2672 0.0000 -68.8717 0.0000 0.074986 0.216 1-month Level -1.1874 0.912 -1.37301 0.8687 0.945881 0.216 LIBOR - GBP First Diff. -20.5227 0.0000 -54.6061 0.0000 0.128029 0.216

2-month Level -1.83251 0.6888 -1.23469 0.9024 0.961849 0.216

LIBOR - GBP First Diff. -7.98846 0.0000 -55.9544 0.0000 0.146846 0.216 3-month Level -1.02112 0.9395 -1.1936 0.9108 0.954575 0.216 LIBOR - GBP First Diff. -19.7532 0.0000 -56.0292 0.0000 0.152881 0.216

6-month Level -0.9729 0.9459 -1.18539 0.9124 0.930066 0.216

LIBOR - GBP First Diff. -21.1581 0.0000 -57.1767 0.0000 0.152708 0.216 12-Month Level -1.1219 0.9239 -1.32233 0.8821 0.854629 0.216 LIBOR - GBP First Diff. -23.4627 0.0000 -55.6623 0.0000 0.135482 0.216 This table presents the sample test statistics and the probabilities for ADF and PP unit root tests; the sample test statistic and the critical values of the KPSS test, computed using EViews.

*MacKinnon (1996) one-sided p-values

**Kwiatkowski-Phillips-Schmidt-Shin (1992, Table1)

20 Schwert (1989) showed that the ADF and the PP tests could suffer from severe size distortion, results being biased towards rejecting the null when it is true (type I error).

117 Table 3.15 LIBOR-USD Rates: The Unit Root ADF, PP and KPPS Tests.

This table presents the sample test statistics and the probabilities for ADF and PP unit root tests; the sample test statistic and the critical values of the KPSS test, computed using EViews.

*MacKinnon (1996) one-sided p-values

**Kwiatkowski-Phillips-Schmidt-Shin (1992, Table1)

Table 3.16 LIBOR-EUR Rates: The Unit Root ADF, PP and KPPS Tests.

Unit Root Tests ADF PP KPSS

LIBOR - EUR t- stat Prob.* Adj. t-stat Prob.* LM - Stat. Crit.val**

1-week Level -1.4634 0.8419 -1.4471 0.8470 0.5756 0.216

LIBOR - EUR First Diff. -24.2384 0.0000 -48.2664 0.0000 0.1705 0.216

1-month Level -1.5206 0.8227 -1.4682 0.8403 0.5868 0.216

LIBOR -EUR First Diff. -12.5899 0.0000 -55.5019 0.0000 0.1411 0.216

2-month Level -1.3981 0.8617 -1.3659 0.8707 0.6103 0.216

LIBOR -EUR First Diff. -12.7543 0.0000 -58.0599 0.0000 0.1492 0.216

3-month Level -1.7327 0.7366 -1.2872 0.8906 0.6215 0.216

LIBOR - EUR First Diff. -8.0070 0.0000 -56.0441 0.0000 0.1534 0.216

6-month Level -1.1539 0.9182 -1.1922 0.9111 0.6326 0.216

LIBOR - EUR First Diff. -13.0708 0.0000 -60.9614 0.0000 0.1538 0.216

12-Month Level -0.9613 0.9474 -1.1491 0.9191 0.6140 0.216

LIBOR - EUR First Diff. -16.8580 0.0000 -64.5011 0.0000 0.1497 0.216 This table presents the sample test-statistics and the probabilities for ADF and PP unit root tests; the sample test statistic and the critical values of the KPSS test, computed using EViews.

*MacKinnon (1996) one-sided p-values

**Kwiatkowski-Phillips-Schmidt-Shin (1992, Table1)

Unit Root Tests ADF PP KPSS LIBOR-USD t-Stat. Prob.* Adj. t-stat Prob.* LM - Stat Crit. Val.**

1-week Level -1.3479 0.8755 -1.3396 0.8776 0.6667 0.216 LIBOR - USD First Diff. -21.3138 0.0000 -40.0117 0.0000 0.1896 0.216

1-month Level -1.4538 0.8449 -1.296183 0.8885 0.6784 0.216

LIBOR -USD First Diff. -15.6319 0.0000 -41.72224 0.0000 0.1956 0.216

2-month Level -1.4613 0.8426 -1.2412 0.901 0.6984 0.216

LIBOR - USD First Diff. -12.1556 0.0000 -42.2856 0.0000 0.2249 0.216

3-month Level -1.4748 0.8382 -1.2460 0.8999 0.7055 0.216

LIBOR - USD First Diff. -12.1453 0.0000 -42.9282 0.0000 0.2336 0.216

6-month Level -1.3243 0.8816 -1.1986 0.9098 0.7152 0.216

LIBOR - USD First Diff. -13.3258 0.0000 -51.3386 0.0000 0.2810 0.216

12-Month Level -1.2388 0.9015 -1.3050 0.8864 0.6954 0.216

LIBOR - USD First Diff. -52.8442 0.0000 -54.5147 0.0000 0.2831 0.216

118 Table 3.17 LIBOR –JPY Rates: The Unit Root ADF, PP and KPPS Tests.

Unit Root Tests ADF PP KPSS LIBOR - JPY t-Stat. Prob.* Adj. t-stat Prob.* LM - Stat. C Crit.Val.**

1-week Level -3.1112 0.1036 -3.8276 0.0152 0.7309 0.216

LIBOR - JPY First Diff. -24.9361 0.0000 -91.0419 0.0000 0.0447 0.216

1-month Level -1.9296 0.6386 -2.2471 0.4625 0.7404 0.216

LIBOR - JPY First Diff. -15.7659 0.0000 -57.1410 0.0000 0.0477 0.216

2-month Level -1.6243 0.7835 -1.3951 0.8626 0.8018 0.216

LIBOR - JPY First Diff. -13.5408 0.0000 -52.5393 0.0000 0.1081 0.216

3-month Level -1.1410 0.9206 -1.0501 0.9354 0.8286 0.216

LIBOR - JPY First Diff. -16.7229 0.0000 -54.0182 0.0000 0.1685 0.216

6-month Level -0.8975 0.9548 -0.8429 0.9604 0.8589 0.216

LIBOR - JPY First Diff. -16.8506 0.0000 -47.6673 0.0000 0.2177 0.216

12-Month Level -0.5921 0.9790 -0.6885 0.973 0.8598 0.216

LIBOR - JPY First Diff. -21.8245 0.0000 -46.7694 0.0000 0.2899 0.216 This table presents the sample test-statistics and the probabilities for ADF and PP unit root tests; the sample test statistic and the critical values of the KPSS test, computed using EViews.

*MacKinnon (1996) one-sided p-values

**Kwiatkowski-Phillips-Schmidt-Shin (1992, Table1)

Table 3.18 LIBOR – CAD Rates: The Unit Root ADF, PP and KPPS Tests.

Unit Root Tests ADF PP KPSS LIBOR - CAD t-Stat. Prob.* Adj. t-stat Prob.* LM - Stat. Crit.

Val.**

1-week Level -1.1041 0.9269 -1.1886 0.9117 0.5094 0.216

LIBOR - CAD First Diff. -48.3407 0.0000 -48.9854 0.0000 0.1809 0.216

1-month Level -1.0835 0.9302 -1.2755 0.8933 0.5148 0.216

LIBOR - CAD First Diff. -21.2242 0.0000 -65.6230 0.0000 0.1626 0.216

2-month Level -1.3380 0.8781 -1.2708 0.8944 0.5214 0.216

LIBOR - CAD First Diff. -13.6088 0.0000 -71.4236 0.0001 0.1679 0.216

3-month Level -1.3426 0.8769 -1.2999 0.8876 0.5263 0.216

LIBOR - CAD First Diff. -14.0961 0.0000 -55.6726 0.0000 0.1643 0.216

6-month Level -1.2805 0.8922 -1.3831 0.8659 0.5397 0.216

LIBOR - CAD First Diff. -24.4136 0.0000 -50.8584 0.0000 0.1593 0.216

12-Month Level -1.6112 0.7888 -1.7290 0.7384 0.5268 0.216

LIBOR - CAD First Diff. -47.0961 0.0000 -48.7181 0.0000 0.1187 0.216 This table presents the sample test-statistics and the probabilities for ADF and PP unit root tests; the sample

In document Multi-Factor Dynamic Modelling and Forecasting of Interest Rates and Equity Markets (Page 120-139)