\in \{0.0, 0.2, 0.5, 0.8, 1.0\}$, and various bandwidths $b$. (Notice that $a_2=0$ corresponds to the null hypothesis.)

We first demonstrate the ability of the bootstrap procedure to approximate the distribution of the test statistic under the null. For this, and in order to estimate the exact distribution of the studentized test $t_\mathcal U$ (see (\[eq.stud-test\])), 10,000 replications of the process (\[mod-1\]) and (\[mod-2\]) with $a_2=0$ have been generated, and a kernel density estimate of this exact distribution has been obtained using a Gaussian kernel with bandwidth $h$. The suggested bootstrap procedure is then applied to three randomly selected time series and the bootstrap studentized test $ t^\ast_\mathcal U$ (see (\[tst-fanis\])) has been calculated. Two sample sizes of $ T=50$ and $T=500$ observations have been considered. Figure 5.1 shows the results obtained together with the approximation of the distribution of $t_\mathcal U$ provided by the central limit theorem, i.e., the $\mathcal N(0,1)$ distribution. As it can be seen from this figure, the convergence towards the asymptotic Gaussian distribution is very slow. Even for sample sizes as large as $T=500$, the exact distribution retains its skewness which is not reproduced by the $\mathcal N(0,1)$ distribution. In contrast to this, the bootstrap 