Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 47 / 4 / b

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.

Let $q_{.025}^{\ast }$ and $q_{.975}^{\ast }$ be the empirical quantiles of the paired-bootstrap values ${\hat{\theta }}^{\ast }$. The percentile bootstrap confidence interval isIts endpoints use the bootstrap distribution of the estimate itself. The regularity assumptions and resampling scheme are those in the parent solution.

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 ${\hat{s}}^{\ast }$.