= Bootstrap inference for a product of regression coefficients
{title2=$\widehat\theta=\widehat\beta_1\widehat\beta_2$}
For $\theta=\beta_1\beta_2$, use $\widehat\theta=\widehat\beta_1\widehat\beta_2$. Under regular regression assumptions, the <delta method> gives <standard error> $\widehat s=(d^\top\widehat Vd)^{1/2}$, where $d=(\widehat\beta_2,\widehat\beta_1)^\top$ and $\widehat V$ estimates the <covariance> of the regression estimate. Resample the observational units, refit, and compute both $\widehat\theta^*$ and $\widehat s^*$. A <percentile bootstrap confidence interval> uses the .025 and .975 <quantiles> of $\widehat\theta^*$. A <bootstrap-t confidence interval> uses <quantiles> of $(\widehat\theta^*-\widehat\theta)/\widehat s^*$ 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.
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