12.8 Application

Figure: The S&P 500 Data. The upper plot shows the S&P 500 together with the exponential trend. The lower plot shows the residual process $ X$.
\includegraphics[width=1.4\defpicwidth]{sp500pic1.ps}

Figure 12.1 shows the daily closing value of the S&P 500 share index from the 31st December 1976 to the 31st December 1997, which covers 5479 trading days. In the upper panel, the index series shows a trend of exponential form which is estimated using the method given in Härdle et al. (2001). The lower panel is the residual series after removing the exponential trend. In mathematical finance one assumes often a specific dynamic form of this residual series, Platen (2000). More precisely, Härdle et al. (2001) assume the following model for an index process $ S(t)$

$\displaystyle S(t) = S(0) X(t) \exp \left( \int_0^t \eta(s) ds \right)$ (12.31)

with a diffusion component $ X(t)$ solving the stochastic differential equation

$\displaystyle d X(t) = a \{1-X(t)\} dt + \sigma X^{1/2}(t) d W(t)$ (12.32)

where $ W(t)$ is a Brownian motion and $ \alpha$ and $ \sigma$ are parameters. Discretizing this series with a sampling interval $ \Delta$ leads to the observations $ (X_i, Y_i)$ with $ Y_i= X_{(i+1)\Delta} - X_{i\Delta}$ and $ X_i = X_{i\Delta}$, which will be $ \alpha$-mixing and fulfill all the other conditions assumed in Section 12.3.

We now apply the empirical likelihood test procedure on the S&P 500 data presented in Figure 12.1 to test the parametric mean function $ m(x) = a (1-x)$ given in the Cox-Ingersoll-Ross diffusion model (12.33). The process $ X$ is restored from the observed residuals by the approach introduced in Härdle et al. (2001). The parametric estimate for $ a$ is $ \hat{a} = 0.00968$ by using methods based on the marginal distribution and the autocorrelation structure of $ X$. For details about the procedure see Härdle et al. (2001). The cross validation is used to find the bandwidth $ h$. However, the score function is monotonic decreasing for $ h < 0.15$ and then become a flat line for $ h \in [0.15, 0.8]$. This may be caused by the different intensity level of the design points. Further investigation shows that a $ h$-value larger (smaller) than 0.06 (0.02) produces an oversmoothed (undersmoothed) curve estimate. Therefore, the test is carried out for a set of $ h$ values ranging from 0.02 to 0.06. The P-values of the test as a function of $ h$ are plotted in Figure 12.2.

Figure: The p-values of the S&P 500 Data
\includegraphics[width=1.4\defpicwidth]{ELsp500pvalues.ps}

The P-values indicate that there is insufficient evidence to reject the diffusion model.