Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
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Inference, Selection, and Simulation

This book connects population arguments to sampling evidence. It covers conditioning, randomized contrasts, partial identification, competing rates, extremum estimators, simulation error, generated objects, spectral uncertainty, dependent-data covariance, bootstrap frontiers, and out-of-time validation.

13 chapters65 focused topics

Conditional Probability, Design, and Partial Identification

II-01

Conditional Probability and a Model's Information Structure

Probability calculations must follow the information generated by the model. Conditioning, marginalization, selection, and transformations can change expectations and variances even when marginal distributions look familiar. Independence, conditional independence, and mean independence support different algebraic steps.

  1. Probability mappings
  2. Conditional means and variances
  3. Forms of independence
  4. Integration order and limits
  5. Changes in statistical experiments
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II-02

Symmetry, Order Statistics, and Endogenous Selection

An odd statistic has zero expectation under central symmetry when its index is fixed. If the index is selected using the same shocks, the composed statistic may lose that symmetry. Order-statistic calculations and joint reflection conditions reveal the missing step.

  1. Symmetry of fixed transformations
  2. Order statistics
  3. Selected indices
  4. Joint reflection conditions
  5. Numbers and sizes of markets
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II-03

Random Assignment and Dynamic Policy Contrasts

Policy effects depend on the assignment unit, timing, history, availability, and interference structure. ITT, factorial effects, sequential contrasts, and loop experiments therefore require separate estimands and assignment probabilities. Selection and nonresponse define further observation rules.

  1. ITT and factorial contrasts
  2. Sequential randomization
  3. Policy-loop experiments
  4. Contrasts across time windows
  5. Selection and nonresponse
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II-04

Moment Inequalities, Partial Identification, and Sharp Bounds

Economic optimality conditions can produce inequalities rather than equations. Validity determines the population identified set, optimization gives target extrema, and constructive witnesses establish sharpness. Sampling uncertainty is projected onto this population geometry as a separate step.

  1. Economic optimality
  2. Valid population moments
  3. Identified sets
  4. Sharp extrema
  5. Projection of uncertainty
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Asymptotics and Simulation

II-05

Stochastic Orders and Competing Sample Sizes

Asymptotic negligibility depends on the normalization used by the estimator. Several sample sizes can grow at different rates, so convergence of a raw error term does not determine its contribution after root-n scaling. Triangular arrays and counterexample sequences make those comparisons explicit.

  1. Convergence notation
  2. Statistical scales
  3. Triangular arrays
  4. Competing rates
  5. Counterexample sequences
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II-06

Extremum Estimators and Uniform Local Expansions

An optimization result, statistical consistency, and an asymptotic linear representation answer different questions. Extremum estimation requires a population criterion, uniform approximation, local curvature, controlled generated objects, and numerical tolerances that vanish on the correct scale.

  1. Population and sample criteria
  2. Consistency
  3. Scores and Hessians
  4. Generated first stages
  5. The role of high-level conditions
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II-07

An Error Ledger for Monte Carlo and Forward Simulation

Simulation enters as numerical integration, forward value approximation, nested averaging, and optimization input. Centered simulation noise, bias, discretization, and common-random-number dependence scale differently. A useful computation reports each component against the estimator's statistical normalization.

  1. Numerical integration
  2. Forward simulators
  3. Nested averages
  4. Dependence from random numbers
  5. Computational budgets
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II-08

Kinks, Binding Inequalities, and Simulation Inference

Squared violations and other kinked criteria behave differently at binding and slack moments. Centered simulation noise changes covariance, while mass at the kink can create a mean shift. The limit depends on the relative simulation rate summarized by kappa.

  1. Nonsmooth criteria
  2. Mean and centered components
  3. Probability mass at the kink
  4. Limits indexed by kappa
  5. Separate bias rates
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Generated Objects and Empirical Inference

II-09

Generated Responses, Estimated Operators, and Noisy Derivatives

Estimated derivatives and operators carry first-stage error into spectra, bridges, and counterfactual maps. Same-sample covariance, local range stability, noise-square bias, and moving frames must enter the expansion. Sample splitting solves only the terms its independence structure addresses.

  1. Same-sample covariance
  2. Stable bridges
  3. Limits of pointwise derivatives
  4. Bias in noisy Gram matrices
  5. Moving coordinates and frames
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II-10

Inference for Rank, Singular Values, and Eigenspaces

Exact rank, numerical rank, and effective rank answer different questions. Perturbation bounds transfer matrix error into singular-value and subspace error only when separation conditions hold. Near ties and lower-rank strata require set-valued or threshold-aware reporting.

  1. Exact and numerical rank
  2. Perturbation bounds
  3. Rank strata
  4. Risk frontiers
  5. Robust reporting
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II-11

Local Projections and Covariance for Dependent Data

A local-projection result is defined by its horizon, outcome unit, shock normalization, controls, sample, and covariance estimator. Serial, spatial, and clustered dependence call for different sandwich constructions. Robustness means stability across declared protocols, not silent changes in scale.

  1. The local-projection contract
  2. Shock units
  3. Serial dependence
  4. Spatial and clustered dependence
  5. Meaning of robustness
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II-12

Dependent Bootstraps and Nonsmooth Frontiers

Block and multiplier resampling must reproduce the relevant dependence and rerun generated stages. Minima, maxima, ties, and boundaries can be only directionally differentiable. Interior regularity, boundary inference, and fallback projection methods therefore belong to separate branches of the procedure.

  1. Blocks and multipliers
  2. Re-estimating generated stages
  3. Minima, maxima, and ties
  4. Regular subproblems
  5. Boundaries and fallbacks
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II-13

Predictive Validation, Coordinate Selection, and Oracle Bounds

Prediction exercises require chronological splits, separation of selection and evaluation, explicit baselines, and finite candidate classes. Oracle inequalities bound selection error under stated conditions, while cross-fitting controls reuse of estimated geometry. Predictive success remains evidence about the declared forecast task.

  1. Chronological validation
  2. Separating selection and evaluation
  3. Finite-candidate guarantees
  4. Cross-fitting estimated geometry
  5. Prediction as a diagnostic
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