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The plug-in estimate isFor a concrete bootstrap construction, suppose the rows $(x_i,y_i)$ are independent and identically distributed, $\mathbb{E}(e_i\mid x_i)=0$, the design moment matrix is nonsingular, and the required moments exist. Use a paired bootstrap: sample $n$ rows with replacement, form $X^{\ast },Y^{\ast }$, refit by ordinary least squares, and compute ${\hat{\theta }}^{\ast }={\hat{\beta }}_1^{\ast }{\hat{\beta }}_2^{\ast }$. Repeat many times. Singular resampled designs require a suitable rank-preserving implementation or replacement replicate; with a regular continuous design their frequency vanishes asymptotically.

The bootstrap-t confidence interval method also needs a standard error in the original sample and every replicate. One asymptotically valid heteroscedasticity-robust covariance estimate isThe delta method for the coefficient product givesRecompute ${\hat{V}}^{\ast },d^{\ast },{\hat{s}}^{\ast }$ in each paired replicate. This is bootstrap inference for a product of regression coefficients.

With fixed rather than random design, resample centered residuals if errors are identically distributed with homoskedasticity, or use a properly specified wild bootstrap for heteroscedastic errors, keeping $X$ fixed and refitting after generating $Y^{\ast }$. The printed assumption $\mathbb{E}{e}_i=0$ alone does not establish independent observations or validity of a bootstrap confidence interval. The formulas below give asymptotic 95% coverage under the stated regularity conditions, not an exact finite-sample guarantee. In particular the ordinary first-order studentization needs $({\beta }_1,{\beta }_2)\ne (0,0)$ and a nonzero asymptotic variance; when both coefficients vanish the product has nonstandard behavior.

For a bootstrap-t confidence interval, estimate the original standard error $\hat{s}$ of $\hat{r}$, for example by the standard deviation of the outer paired bootstrap correlations. For each outer resample $b$, estimate its own standard error $s_b^{\ast }$ by drawing many inner paired resamples from that outer sample and taking the standard deviation of their correlations. The resulting studentized statistics areLet $t_p^{\ast }$ be their empirical quantiles. Approximating the distribution of $(\hat{r}-r)/\hat{s}$ by that of $T^{\ast }$ and solving the two inequalities for $r$ givesThe reversal of quantiles is essential. The pivot requires positive standard errors and nondegenerate correlations; its claimed 95% coverage is an approximation under bootstrap regularity, not an exact finite-sample guarantee. One may intersect the final interval with the parameter space $[-1,1]$.

The same percentile and studentized procedures can be applied to the sample mean of $X$, and then only the $X$ observations need be resampled. They are useful when a normal approximation is doubtful. However the mean already has the simple standard-error estimate $S_X/10$ here. For normal observations, the classical exact interval isand the same form is an asymptotic approximation for independent finite-variance observations under appropriate large-sample conditions. Thus bootstrap intervals are valid alternatives, but are often unnecessary for a mean with a reliable classical pivot; the correlation has no corresponding distribution-free elementary pivot. An infinite-variance distribution would require other theory rather than this ordinary standard-error argument.

Calculate the observed sample correlation coefficient $\hat{r}$. In each paired bootstrap, draw 100 indices independently with replacement from $1,\dots ,100$ and select the corresponding whole pairs. Compute the bootstrap correlation $r_b^{\ast }$ from each resample; do not resample the two margins independently, since that would destroy their empirical dependence.

Let $q_p$ be the empirical $p$-quantile of those bootstrap correlations. The percentile bootstrap confidence interval isThis is the requested algorithmic interval; the actual endpoints require the observed pairs. Its coverage is approximate, based on the sampling law of the correlation being well approximated by the bootstrap law. Both sample variances must be positive. Resamples having a constant margin have undefined correlation and cannot be repaired by arbitrarily assigning correlation zero. One may regenerate these exceptional resamples, explicitly conditioning on well-defined correlations; under the usual nondegenerate large-sample conditions their probability is negligible. Frequent degeneracy instead indicates that this ordinary correlation-bootstrap construction is unsuitable.