2.2 Bootstrap as a Data Analytical Tool

In a data analysis the statistician wants to get a basic understanding of the stochastic nature of the data. For this purpose he/she applies several data analytic tools and interprets the results. A basic problem of a data analysis is over-interpretation of the results after a battery of statistical methods has been applied. A similar situation occurs in multiple testing but there exist approaches to capture the joint stochastics of several test procedures. The situation becomes more involved in modern graphical data analysis. The outcomes of a data analytic tool are plots and the interpretation of the data analysis relies on the interpretation of these (random) plots. There is no easy way to have an understanding of the joint distribution of the inspected graphs. The situation is already complicated if only one graph is checked. Typically it is not clearly specified for which characteristics the plot is checked. We will illustrate this by a simple example. We will argue that the bootstrap and other resampling methods offer a simple way to get a basic understanding for the stochastic nature of plots that depend on random data. In the next section we will discuss how this more intuitive approach can be translated into the formal setting of mathematical decision theoretical statistics. Our example is based on the study of a financial time series. Figure 2.1 shows the daily values of the German DAX index from end of 1993 until November 2003. In Fig. 2.2 mean-corrected log returns are shown. Logreturns for a series $x_t$ are defined as $\log x_t - \log x_{t-1}$. Mean-corrected logreturns $r_t$ are defined as this difference minus its sample average. Under the Black-Sholes model the logreturns $r_t$ are i.i.d. It belongs to folklore in finance that this does not hold. We now illustrate how this could be seen by application of resampling methods.

Figure 2.1: Plot of DAX data

Figure 2.2: 1-lag mean-corrected logreturn of the DAX data

Figure 2.3: Random permutation of the data in Fig. 2.2

Figure 2.4: Nonparametric bootstrap sample of the data in Fig. 2.2

Figure 2.3 shows a plot of the same logreturns as in Fig. 2.2 but with changed order. The logreturns are plotted against a random permutation of the days. The clusters appearing in Fig. 2.2 dissappear. Figure 2.3 shows that these clusters could not be explained by stochastic fluctuations. The same story is told in Fig. 2.4. Here a bootstrap sample of the logreturns is shown. Logreturns are drawn with replacement from the set of all logreturns (i.i.d. resampling) and they are plotted in the order as they were drawn. Again the clusters disappear and the same happens for typical repetitions of random permutation or bootstrap plots. The clusters in Fig. 2.2 can be interpreted as volatility clusters. The volatility of a logreturn for a day is defined as the conditional variance of the logreturn given the logreturns of all past days. The volatilities of neighbored days are positively correlated. This results in volatility clusters. A popular approach to model the clusters are GARCH (Generalized Autoregressive Conditionally Heteroscedastic) models. In the GARCH(1,1) specification one assumes that $r_t=\sigma_t \varepsilon_t$ where $\varepsilon_t$ are i.i.d. errors with mean zero and variance 1 and where $\sigma_t^2$ is a random conditional variance process fulfilling $\sigma_t^2= a_0 + a_1 \sigma^2_{t-1} +b_1 r_{t-1}^2$. Here $a_0$, $a_1$ and $b_1$ are unknown parameters. Figure 2.5 shows a bootstrap realization of a fitted GARCH(1,1) model. Fitted parameters $\widehat{a}_0$, $\widehat{a}_1$ and $\widehat{b}_1$ are calculated by a quasi-likelihood method (i.e. likelihood method for normal $\varepsilon_t$). In the bootstrap resampling the errors $\varepsilon_t$ are generated by i.i.d. resampling from the residuals $r_t / \widehat{\sigma}_t$ where $\widehat{\sigma}_t^2$ is the fitted volatility process $\widehat{\sigma}_t^2= \widehat{a}_0 + \widehat{a}_1 \widehat{\sigma}^2_{t-1} +\widehat{b}_1 r_{t-1}^2$. An alternative resampling would to generate normal i.i.d. errors in the resampling. This type of resampling is also called parametric bootstrap. At first sight the volatility clusters in the parametric bootstrap have similar shape as in the plot of the observed logreturns. Figure 2.6 shows local averages $\widehat{m}(t)$ over squared logreturns. We have chosen $\widehat{m}(t)$ as Nadaraya-Watson estimate $\widehat{m}_h(t) = [\sum_{s=1}^N K\{(s-t)/h\}r_t^2] / [\sum_{s=1}^N K\{(s-t)/h\}]$. We used a Gaussian kernel $K$ and bandwidth $h= 5$ days. Figures 2.7-2.9 show the corresponding plots for the three resampling methods. Again the plots for random permutation resampling and nonparametric i.i.d. bootstrap qualitatively differ from the plot for the observed time series (Figs. 2.7 and 2.8). In Fig. 2.9 the GARCH bootstrap shows a qualitatively similar picture as the original logreturns ruling again not out the GARCH(1,1) model.

Figure 2.5: GARCH(1,1) bootstrap sample of the data in Fig. 2.2

Figure 2.6: Plot of kernel smooth of squared logreturns from the data in Fig. 2.2

Figure 2.7: Plot of kernel smooth of squared logreturns from the data in Fig. 2.3

Figure 2.8: Plot of kernel smooth of squared logreturns from the data in Fig. 2.4

Figure 2.9: Plot of kernel smooth of squared logreturns from the data in Fig. 2.5

As a last example we consider plots that measure local and global shape characteristics of the time series. We consider the number of local maxima of the kernel smoother $\widehat {m}_h$ as a function of the bandwidth $h$. We compare this function with the number of local maxima for resamples. Figures 2.10-2.12 show the corresponding plots for the permutation resampling, the nonparametric bootstrap and the GARCH(1,1) bootstrap. The plot of the original data set is always compared with the plots for $10$ resamples. Again i.i.d. structures are not supported by the resampling methods. GARCH(1,1) bootstrap produces plots that are comparable to the original plot.

Figure 2.10: Number of local maxima of kernel smoother $\widehat {m}_h$ of squared mean-corrected logreturns of DAX data from Fig. 2.1 (black line) compared with number of local maxima for 10 random permutation resamples (dashed lines)

Figure 2.11: Number of local maxima of kernel smoother $\widehat {m}_h$ of squared mean-corrected logreturns of DAX data from Fig. 2.1 (black line) compared with number of local maxima for 10 nonparametric bootstrap resamples (dashed lines)

The last approach could be formalized to a test procedure. This could e.g. be done by constructing uniform resampling confidence bands for the expected number of local maxima. We will discuss resampling tests in the next section. For our last example we would like to mention that there seems to be no simple alternative to resampling. An asymptotic theory for the number of maxima that could be used for asymptotic confidence bands is not available (to our knowledge) and it would be rather complicated. Thus, resampling offers an attractive way out. It could be used for a more data analytic implementation as we have used it here. But it could also be used for getting a formal test procedure.

The first two problems, discussed in Figs. 2.1-2.9, are too complex to be formalized as a testing approach. It is impossible to describe for what differences the human eye is looking in the plots and to summarize the differences in one simple quantity that can be used as a test statistic. The eye is using a battery of ''tests'' and it is applying the same or similar checks for the resamples. Thus, resampling is a good way to judge statistical findings based on the original plots.

Figure 2.12: Number of local maxima of kernel smoother $\widehat {m}_h$ of squared mean-corrected logreturns of DAX data from Fig. 2.1 (black line) compared with number of local maxima for GARCH(1,1) bootstrap resamples (dashed lines)