20.2 Principal Component Analysis of the VDAX's Dynamics

We will first check the data with the help of the ``Augmented Dickey-Fuller`` Tests (ADF-Test - see (10.46)) for stationarity. The null hypothesis of a unit root for the individual VDAX sub-indices $ \hat{\sigma}_I(\tau^*_j)$ cannot be rejected at the 90% significance level. Obviously due to this result the first differences $ x_{jt}=\Delta[\hat{\sigma}_{I,t}(\tau^*_j)] = \hat{\sigma}_{I,t+1}(\tau^*_j) - \hat{\sigma}_{I,t}(\tau^*_j), t=1, ...., n-1,$ of the implied volatility indices will be used for further analysis. Additional ADF tests support the assumption of stationarity for the first differences. 34969 SFEAdfKpss.xpl


Table: Empirical covariance matrix $ \hat{\Omega}$ of the first differences (all values have been multiplied by $ 10^5$) 34972 SFEVolaCov.xpl
Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Sub 6 Sub 7 Sub 8
20.8 9.06 6.66 6.84 4.29 2.48 2.11 1.38
9.06 9.86 6.67 4.44 3.21 1.72 1.11 0.92
6.66 6.67 6.43 3.87 2.63 1.49 1.01 0.53
6.84 4.44 3.87 4.23 2.66 1.39 1.38 0.68
4.29 3.21 2.63 2.66 2.62 1.03 1.02 0.51
2.48 1.72 1.49 1.39 1.03 2.19 0.63 0.33
2.11 1.11 1.01 1.38 1.02 0.63 1.76 0.43
1.38 0.92 0.53 0.68 0.51 0.33 0.43 1.52


Let $ \bar{x}_j$ be the respective sample mean of the first differences $ x_{jt}$. Table 19.1 contains the empirical covariance matrix $ \hat{\Omega}$ used as an estimator for the $ 8 \times 8$ matrix $ \Omega$ of the covariance $ {\mathop{\text{\rm Cov}}(x_{it},x_{jt})}, i,j = 1, ..., 8$. With help of the Jordan decomposition we obtain $ \hat{\Omega}=\hat{\Gamma}\hat{\Lambda}\hat{\Gamma}^\top $. The diagonal matrix $ \hat{\Lambda}$ contains the eigenvalues $ \hat{\lambda}_k, k = 1, ..., 8$ of $ \hat{\Omega}$, $ \hat{\Gamma}$ are the eigenvectors. Time series of the principal components can be obtained with the help of $ Y=X_C\hat{\Gamma}$, where $ X_C$ represents the $ 440 \times 8$ matrix of the centered first differences $ x_{jt}^c = x_{jt} - \bar{x}_j, j=1, ..., 8, t= 1,
..., 440,$. The $ 440 \times 8$ matrix $ Y=(Y_1,...,Y_8),
Y_j=(y_{1j},y_{2j},...,y_{440,j})^\top $ contains the principal components.

How accurately the first $ l$ principal components have already determined the process of the centered first differences can be measured using the proportion of variance $ \varphi_l$ with respect to the total variance of the data. The proportion of explained variance corresponds to the relative proportion of the corresponding eigenvalue, i.e.,


$\displaystyle \varphi_l=\frac {\sum^l_{k=1} \lambda_k}
{\sum^8_{k=1} \lambda_k}...
...op{\text{\rm Var}}(y_{tk})}}
{\sum^8_{k=1} {\mathop{\text{\rm Var}}(y_{tk})} },$     (20.2)

where $ \lambda_k, k = 1, ..., 8$ are the eigenvalues of the true covariance matrix $ \Omega$. An estimator for $ \varphi_l$ is

$\displaystyle \hat{ \varphi}_l=\frac {\sum^l_{k=1} \hat{\lambda}_k}
{\sum^8_{k=1} \hat{\lambda}_k}.
$

In Table 19.2 the individual proportions of the variance $ \hat{\lambda}_l / \sum^8_{k=1} \hat{\lambda}_k $ as well as the cumulative variance from the $ l$ decomposed proportions from the principal components, $ \hat{ \varphi}_l$, are displayed. It is obvious that the first principal component already describes 70% of the total variance of the underlying data. With the second principal component an additional 13% of the total variance within the observed time period can be explained. Together 83% of the variance of the analyzed first differences of our VDAX sub-indices can be explained with the help of the first and second principal components. Obviously the explaining power of the principal components significantly declines from the third principal component on.

Table: Explained sample variance using principal components in percentage 34981 SFEVolaPCA.xpl
Principal Explaining proportion cumulative
Component of variance proportion
1 70.05 70.05
2 13.06 83.12
3 5.57 88.69
4 3.11 91.80
5 3.06 94.86
6 2.12 96.97
7 1.93 98.90
8 1.10 100.00


By displaying the eigenvalues in a graph, a form with a strong curvature at the second principal component is shown. In accordance with the well known ``elbow'' criterion, using the first two principal components with an explanation power of over 80% of the total variance is considered to be sufficient in describing the data set. The remaining variance can be interpreted for analytical purposes as the effect of an unsystematic error term. Figure 19.3 contains the factor loading of the first two principal components. Based on the orthogonality of the components the loading factors can be estimated using the least squares regression of the individual equations

$\displaystyle x_{jt}^c = \sum^2_{l=1}b_{jl}y_{lt}+\varepsilon_t,$     (20.3)

Here $ \varepsilon_t$ is an independent error term.

Fig.: Factor loadings of the first and second principal components 34985 SFEPCA.xpl
\includegraphics[width=1\defpicwidth]{sfmpca.ps}

Based on the factor loadings it is clear that a shock to the first factor would affect the implied volatility of all times to maturity considered in a similar way, or would cause a non-parallel shift in the maturities' structure. A shock to the second principal component, on the other hand, causes a tilt of the structure curve: while at short times to maturity it causes a positive change, the longer time to maturities are influenced negatively. The absolute size of the effect of a shock decreases in both factors with the time to maturity.