BLP Demand Estimation - Index

Routing Summary

• Need the big picture / what BLP is and the estimation pipeline? → BLP Demand Estimation - Overview
• Need the demand utility spec, shares as integrals, relaxing IIA, RCNL? → Random Coefficients Logit Model
• Need how observed shares are inverted to mean utilities ${\mathbf{\delta }}_t$ (the contraction, SQUAREM/LM, Lipschitz constant)? → The BLP Contraction Mapping
• Need why price is endogenous and which instruments are valid (cost shifters, BLP, Gandhi-Houde differentiation IV, optimal IV) + the GMM objective? → GMM Estimation and Instruments for Price Endogeneity
• Need quadrature vs Monte Carlo, the nested fixed point, optimization, and concrete best practices? → Numerical Integration and Optimization in PyBLP
• Need the supply side, Bertrand markups, marginal-cost recovery, merger/equilibrium simulation? → Supply Side and Markups

Concept Map

Concept	Note	Type	Depends On	Key Result
BLP model & best-practices overview	BLP Demand Estimation - Overview	overview	RCL, Contraction, GMM, Integration, Supply	BLP = nonlinear change of vars then (two-eq) linear IV; well-behaved with best practices
Random-coefficients (mixed) logit	Random Coefficients Logit Model	definition	Discrete Choice Models	$U_{ijt}={\delta }_{jt}+{\mu }_{ijt}+{ϵ}_{ijt}$; shares are integrals over types; relaxes IIA
Share inversion	The BLP Contraction Mapping	theorem	RCL	${\mathbf{\delta }}^{h+1}={\mathbf{\delta }}^h+\log \mathcal{S}-\log \mathbf{s}$ is a contraction; SQUAREM/LM accelerate
GMM & instruments	GMM Estimation and Instruments for Price Endogeneity	concept	RCL, Contraction, Instrumental Variables	$E[{\xi }_{jt}{Z}_{jt}^D]=0$; differentiation IV > BLP sums; feasible optimal IV
Integration & optimization	Numerical Integration and Optimization in PyBLP	concept	RCL, Contraction, GMM	Quadrature/sparse-grid/Halton; NFXP + analytic gradients; >99% converge
Supply side & markups	Supply Side and Markups	concept	RCL, GMM	${\eta }_t={\Delta }_t^{-1}{\mathbf{s}}_t$, $c_{jt}=p_{jt}-{\eta }_{jt}$; $\zeta$-markup fixed point for equilibria

Notes

• BLP Demand Estimation - Overview — CONTAINS: the BLP problem pipeline; parameter partition $\theta =[{\theta }_1=\beta ,{\theta }_3=\gamma ,{\theta }_2=(\alpha ,{\tilde{\theta }}_2)]$; what is novel in Conlon-Gortmaker (reformulation, optimal IV, best practices); canonical auto/cereal example.
• Random Coefficients Logit Model — CONTAINS: indirect utility (mean + random-coefficient deviation + EV1 error); outside good; shares as integrals over the mixing distribution; the linear $\delta$-index with structural error ${\xi }_{jt}$; RCNL extension with nesting parameter $\rho$ and inclusive value.
• The BLP Contraction Mapping — CONTAINS: the $J_t$-equation share system; tolerance in log-share difference; the contraction and its Lipschitz constant / convergence rate; outside-share effect; dampened RCNL contraction; Newton/Levenberg-Marquardt and SQUAREM acceleration.
• GMM Estimation and Instruments for Price Endogeneity — CONTAINS: why price is endogenous; demand & full GMM moment conditions and objective; cost shifters, BLP instruments, Gandhi-Houde Local/Quadratic differentiation IV; Chamberlain optimal-IV approximation with explicit exclusion/cross-equation restrictions; overidentification LR test.
• Numerical Integration and Optimization in PyBLP — CONTAINS: pMC vs qMC (scrambled Halton) vs Gaussian quadrature, sparse grids, curse of dimensionality, variance reduction; the NFXP algorithm; analytic gradients, gradient-based optimizers, box constraints, tight tolerances; log-sum-exp numerical stability.
• Supply Side and Markups — CONTAINS: Bertrand-Nash FOCs; ownership matrix ${\mathcal{H}}_t$ and intra-firm derivative matrix ${\Delta }_t$; markup and marginal-cost recovery; supply moments; Morrow-Skerlos $\zeta$-markup fixed point for counterfactual equilibria; merger simulation.

Sources

• Conlon Gortmaker 2020 - Best Practices BLP Demand Estimation (PyBLP).pdf — Conlon, C. & Gortmaker, J. (2020), “Best Practices for Differentiated Products Demand Estimation with PyBLP,” RAND Journal of Economics.