After data has been transformed it would be interesting to see if there is any autocorrelation left. If it is possible to remove some time dependency in data the modelling process gets simpler.
Figure4.2ais a plot of the autocorrelation in daily log return data. Comparing this with gure3.3it is easy to see that many signicant lags has been removed through the transformation. Some has even switched to being negative. Com-paring gure4.2bwith partial autocorrelation in daily log return data to gure 3.4, it is easy to see that the transformation has removed a lot of signicance at lag= 1. But there is still a lot of autocorrelation left in data after transfor-mation that cannot be ignored, especially CSIYHYI and DK00S.N:Index have many signicant lags of lower order that certainly not can be assumed to be white noise.
4.2 Autocorrelation in log return indices 23
ACF
0 5 15 25 35
−0.060.04
Lag
ACF
KAXGI
0 5 15 25 35
−0.060.04
Lag
ACF
NDDUE15
0 5 15 25 35
−0.060.02
Lag
ACF
NDDUJN
0 5 15 25 35
−0.060.04
Lag
ACF
NDDUNA
0 5 15 25 35
−0.040.04
Lag
ACF
NDUEEGF
0 5 15 25 35
−0.040.02
Lag
ACF
TPXDDVD
0 5 15 25 35
0.00.3
Lag
ACF
CSIYHYI
0 5 15 25 35
0.00.2
Lag
ACF
JPGCCOMP
0 5 15 25 35
−0.040.06
Lag
ACF
NDEAGVT
0 5 15 25 35
−0.050.15
Lag
ACF
NDEAMO
0 5 10 20 30
−0.4−0.1
Lag
ACF
DK00S.N.Index
(a) ACF in daily log return data with 95 % condence interval (red).
PACF
0 5 15 25 35
−0.060.04
Lag
Partial ACF
KAXGI
0 5 15 25 35
−0.060.04
Lag
Partial ACF
NDDUE15
0 5 15 25 35
−0.060.02
Lag
Partial ACF
NDDUJN
0 5 15 25 35
−0.060.04
Lag
Partial ACF
NDDUNA
0 5 15 25 35
−0.040.04
Lag
Partial ACF
NDUEEGF
0 5 15 25 35
−0.040.02
Lag
Partial ACF
TPXDDVD
0 5 15 25 35
0.00.3
Lag
Partial ACF
CSIYHYI
0 5 15 25 35
0.00.2
Lag
Partial ACF
JPGCCOMP
0 5 15 25 35
−0.040.06
Lag
Partial ACF
NDEAGVT
0 5 15 25 35
−0.050.15
Lag
Partial ACF
NDEAMO
0 5 10 20 30
−0.40.0
Lag
Partial ACF
DK00S.N.Index
(b) PACF in daily log return data with 95 % condence interval (red).
Figure 4.2
A way to deal with autocorrelation in data is to use weekly data instead. Using weekly data, we might lose some extreme events, but using e.g. data from Friday every week the variance is kept realistic. If the mean value for the week is used instead the true variance is reduced resulting in a weak model. The few extreme events that are not in weekly data would anyway have vanished on the long run when modelling and generating scenarios. Therefore the use of weekly (Friday) data is acceptable, and is a technique already widely used in statistical nance exactly to get independent data. Using weekly data, the estimate of weekly volatility is more accurate.
If the Shapiro-Wilk test is applied on the weekly log return indices the result is that all p-value>0.1, and thereby the null hypothesis of level-stationarity cannot be rejected. Another way to check if weekly log return indices are stationary is to estimate their mean recursively. The recursive estimation has been done using a forgetting factor λ = 0.9 such that the recursive estimate at time t, becomes a weighting of the previous t−1 observations. The weighting of the i'th observation is given by :
W(i) =λ−(i−t),
where i∈[1;t]. Afterwards, the weighting is scaled such thatPt
i=1W(i) = 1. In practice the eective number of previous values used in the estimation is given by:
nef f = 1
1−λ= 1
1−0.90 = 10.
In gure 4.4 the recursive estimate of the mean for each weekly log return series is plotted. It is clearly seen that the mean has small uctuations around zero (except DK00S.N.Index), therefore the weekly log return indices might be stationary.
This was already expected cf. earlier results and thereby the plots of ACF and PACF show a more exact picture of what is going on and not disturbed by time dependency. Stationarity is a nice property when we want to model the data, because a lot of dierent models require that the input must be stationary. The ACF and PACF for weekly log returns are plotted in gure4.3aand4.3b.
As expected even more signicant autocorrelation have been removed, now to an acceptable level. The bond indices except CSIYHYI have only one or two lags just outside the 95 % condence bands in the ACF, which acceptable. The
4.2 Autocorrelation in log return indices 25
ACF
0 5 10 15 20 25
−0.05
Lag
ACF
KAXGI
0 5 10 15 20 25
−0.050.15
Lag
ACF
NDDUE15
0 5 10 15 20 25
−0.100.05
Lag
ACF
NDDUJN
0 5 10 15 20 25
−0.05
Lag
ACF
NDDUNA
0 5 10 15 20 25
−0.050.10
Lag
ACF
NDUEEGF
0 5 10 15 20 25
−0.100.05
Lag
ACF
TPXDDVD
0 5 10 15 20 25
−0.10.3
Lag
ACF
CSIYHYI
0 5 10 15 20 25
−0.050.15
Lag
ACF
JPGCCOMP
0 5 10 15 20 25
−0.050.10
Lag
ACF
NDEAGVT
0 5 10 15 20 25
−0.05
Lag
ACF
NDEAMO
0 5 10 15 20 25
−0.30.1
Lag
ACF
DK00S.N.Index
(a) ACF in weekly log return data with 95 % condence interval (red).
PACF
0 5 10 15 20 25
−0.100.10
Lag
Partial ACF
KAXGI
0 5 10 15 20 25
−0.050.15
Lag
Partial ACF
NDDUE15
0 5 10 15 20 25
−0.100.05
Lag
Partial ACF
NDDUJN
0 5 10 15 20 25
−0.100.05
Lag
Partial ACF
NDDUNA
0 5 10 15 20 25
−0.050.10
Lag
Partial ACF
NDUEEGF
0 5 10 15 20 25
−0.100.05
Lag
Partial ACF
TPXDDVD
0 5 10 15 20 25
−0.10.3
Lag
Partial ACF
CSIYHYI
0 5 10 15 20 25
−0.100.10
Lag
Partial ACF
JPGCCOMP
0 5 10 15 20 25
−0.050.10
Lag
Partial ACF
NDEAGVT
0 5 10 15 20 25
−0.05
Lag
Partial ACF
NDEAMO
0 5 10 15 20 25
−0.30.0
Lag
Partial ACF
DK00S.N.Index
(b) PACF in weekly log return data with 95 % condence interval (red).
Figure 4.3
stock indices have a few more lags just outside the condence bands but this is still acceptable. CSIYHYI still has some pattern in autocorrelation with lag= 1,2and3 being very signicant and lag 1 signicant in the partial auto-correlation. The other bond and stock indices also have a few signicant lags in PACF, but it is acceptable on a 95 % signicance level even though it is a bit suspiciously that almost all the stock indices have signicance lag around lag
=13. There is no trading or market related explanation for this structure and as long as there only is a few lags of higher order just outside the condence bands then data is accepted as being independent. The ACF and PACF in DK00S.N.Index now behave more like the other indices but there still seems to be too much time dependency left.
Recursiv estimation of mean in weekly log return with
lambda=0.90
2000 2004 2008 2012
−0.040.00
KAXGI
2000 2004 2008 2012
−0.040.00
NDDUE15
2000 2004 2008 2012
−0.030.00
NDDUJN
2000 2004 2008 2012
−0.030.01
NDDUNA
2000 2004 2008 2012
−0.060.00
NDUEEGF
2000 2004 2008 2012
−0.030.01
TPXDDVD
2000 2004 2008 2012
−0.0200.005
CSIYHYI
2000 2004 2008 2012
−0.020.01
JPGCCOMP
2000 2004 2008 2012
−0.0020.006
NDEAGVT
2000 2004 2008 2012
−0.0060.002
NDEAMO
2000 2004 2008 2012
−0.250.00
DK00S.N.Index
Figure 4.4: Recursive estimate of mean of each weekly log return index using forgetting factorλ= 0.90. Time is on the rst axis, and mean on the secondary axis.
The reason for the strange behaviour of CSIYHYI might be that the log trans-formation is too eective. Therefore a square root of simple gross return might be a usable transformation for exactly this index. The ACF and PACF for the square root simple gross CSIYHYI index is plotted in Appendix A. There is
4.2 Autocorrelation in log return indices 27
only a slightly dierence compared to the log return data, and therefore the log return transformation will be used for now.
Now it can be assumed, a little roughly, that weekly log returns are independent, with the exception of CSIYHYI and DK00S.N.Index. This is an important feature that is very useful when the indices are modelled in chapter7.