Dependent Bootstraps and Nonsmooth Frontiers

Conceptual map

II-12.01Blocks and multipliers
II-12.02Re-estimating generated stages
II-12.03Minima, maxima, and ties
II-12.04Regular subproblems
II-12.05Boundaries and fallbacks

1. A minimum is directionally differentiable at a tie

Smooth components can produce a nonsmooth frontier. At a tie, the first-order map is a minimum over active directional changes rather than a single gradient.

Let $\phi(\theta)=\min\{\theta _{1},\theta _{2}\}$ . If $\theta _{1}<\theta _{2}$ , the derivative in direction $$h$$ is $h_{1}$ . At $\theta _{1}=\theta _{2}$ ,

Proposition 1 · Directional derivative of a two-way minimum

\[\phi ^{\prime}_{\theta }(h)=\operatorname{min}\{h_{1},h_{2}\}.\](1)

Proof. At a tie $\theta _{1}=\theta _{2}=c$ , $[\phi(\theta+th)-\phi(\theta)]/t=[c+t\min(h_{1},h_{2})-c]/t$ for $$t>0$$ . ∎

The map in (1) is continuous and positively homogeneous, but it is nonlinear. For direction (1, $$-2$$ ), the derivative is $$-2$$ ; for ( $$-2$$ ,1), it is also $$-2$$ .

2. The ordinary centered bootstrap can fail at the tie

Suppose $\sqrt{n}(\hat{\theta}-\theta)\Rightarrow Z$ with a continuous bivariate limit and $\theta=(0,0)$ . Then $\sqrt{n}\,\phi(\hat{\theta})\Rightarrow\min(Z_{1},Z_{2})$ . If a bootstrap perturbation has limit $Z^{*}$ , the ordinary centered plug-in statistic behaves conditionally like

\[\operatorname{min}(Z_{1}+Z_{1}^{*},Z_{2}+Z_{2}^{*})-\operatorname{min}(Z_{1},Z_{2}),\](2)

whose conditional law depends on realized $$Z$$ and generally differs from the target law of $\operatorname{min}(Z_{1},Z_{2})$ . A directional bootstrap estimates the active set and applies an estimate of (1) to the bootstrap perturbation under conditions governing tie detection.

3. Resampling must reproduce the dependence unit

For a stationary weakly dependent time series, moving-block procedures resample consecutive blocks. Typical smooth-statistic conditions send block length $\ell _{T}$ to infinity while $\ell _{T}/T\to0$ , alongside mixing and moment restrictions. Cluster multipliers attach one weight to every observation in the same cluster. Time multipliers require a dependence structure in the weights or in preaggregated score blocks.

Every generated response, nuisance estimate, covariance, active set, and optimization frontier that contributes first-order randomness must be recomputed within the draw. Holding a first stage fixed changes the resampled statistic unless a separate influence adjustment accounts for it.

4. Interior, tie, and support boundaries are separate branches

Failure case · Smooth bootstrap applied after an estimated tie

Selecting one active component by an arbitrarily small sample difference treats the frontier as differentiable. When the population components tie, that selection remains random at first order.

A parameter-support boundary adds tangent-cone projection beyond the minimum map. A block length of one reduces to an iid bootstrap; a block as long as the sample supplies essentially one resampled series. If the directional-bootstrap regularity or tie threshold cannot be justified, projection of a joint confidence set for the smooth components gives a more conservative fallback.

5. Implementation, exercises, and sources

Record resampling unit, block or multiplier construction, block length, number of draws, tie threshold, active-set rule, parameter support, random seed, and all rerun stages. Compare interior and tie data-generating processes in a coverage simulation and report Monte Carlo error for the coverage estimate.

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Exercises

Verify (1) for four directions, including a coordinate tie.
Simulate the two Gaussian expressions in the target limit and (2) conditional on fixed $$Z$$ .
List every first-stage object that must be rerun for a frontier built from estimated local projections.

Partial solutions

1. Substitute $\theta+t h$ and divide by positive $$t$$ . 2. The target redraws a fresh minimum; the naive increment shifts the active comparison by the realized first-stage vector. 3. Include shock construction, projection coefficients, covariance, model refit, response map, and frontier optimization when each is estimated.

Zheng Fang and Andres Santos (2019), “Inference on Directionally Differentiable Functions,” Review of Economic Studies 86, 377–412.Bootstrap limits for directional maps.
S. N. Lahiri, Resampling Methods for Dependent Data.Block bootstrap constructions and rates.
Donald W. K. Andrews (2000), “Inconsistency of the Bootstrap When a Parameter Is on the Boundary of the Parameter Space,” Econometrica 68, 399–405.Boundary bootstrap failure.

6. Audit checkpoint

State the smooth inputs, active set, interior or tie status, directional derivative, parameter support, dependence unit, block or multiplier rule, block-length limits, rerun stages, tie threshold, bootstrap draw count, fallback projection, and simulated coverage error.

7. Scope boundary

The chapter gives one directional-minimum calculation and resampling design requirements. Each frontier still needs a theorem matched to its dependence and active-set geometry.