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For $\theta ={\beta }_1{\beta }_2$, use $\hat{\theta }={\hat{\beta }}_1{\hat{\beta }}_2$. Under regular regression assumptions, the delta method gives standard error $\hat{s}=(d^{\top }\hat{V}d{)}^{1/2}$, where $d=({\hat{\beta }}_2,{\hat{\beta }}_1{)}^{\top }$ and $\hat{V}$ estimates the covariance of the regression estimate. Resample the observational units, refit, and compute both ${\hat{\theta }}^{\ast }$ and ${\hat{s}}^{\ast }$. A percentile bootstrap confidence interval uses the .025 and .975 quantiles of ${\hat{\theta }}^{\ast }$. A bootstrap-t confidence interval uses quantiles of $({\hat{\theta }}^{\ast }-\hat{\theta })/{\hat{s}}^{\ast }$ and reverses them when solving for $\theta$. If both true coefficients vanish, the first derivative is zero and the standard first-order justification is unavailable; product-specific nonregular inference is then needed.

For each bootstrap replicate form the studentized pivotLet $t_{.025}^{\ast },t_{.975}^{\ast }$ be its empirical quantiles. The bootstrap-t confidence interval approximatesSolving both inequalities for $\theta$ reverses the quantile order, givingUsing the same fixed standard error in every bootstrap numerator would not implement this studentized procedure; each replicate needs its own ${\hat{s}}^{\ast }$.

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.