GMM Estimation and Instruments for Price Endogeneity

Summary

After share inversion, BLP is a GMM / IV problem: the structural error ${\xi }_{jt}$ (unobserved product quality) is correlated with price $p_{jt}$ because firms set higher prices for higher-quality goods, so OLS on the demand index is biased. Valid instruments $Z_{jt}^D$ — cost shifters, the BLP instruments (functions of rival product characteristics), the Gandhi-Houde differentiation IV, and the approximation to the optimal instruments — give moment conditions $E[{\xi }_{jt}{Z}_{jt}^D]=0$. Adding a supply side yields additional moments $E[{\omega }_{jt}{Z}_{jt}^S]=0$ and cross-equation restrictions. The objective is minimized over $\theta$ with a weighting matrix $W$.

Overview

Price is endogenous: $p_{jt}$ depends on the unobserved quality ${\xi }_{jt}$ through the firm’s pricing decision, so $E[{\xi }_{jt}{p}_{jt}]\ne 0$. Identification of the price coefficient $\alpha$ (and the random-coefficient parameters ${\theta }_2$) requires instruments that shift prices/markups but are uncorrelated with ${\xi }_{jt}$. Because ${\theta }_2$ governs the nonlinear substitution, each nonlinear parameter needs its own excluded instrument; Berry & Haile (2014) show $D_t^{-1}({\mathcal{S}}_t,{\tilde{\theta }}_2)$ depends on the endogenous shares of all products in the market, so each ${\tilde{\theta }}_2$ parameter requires an additional instrument.

Main Content

Demand moment conditions

From the inverted linear index ${\delta }_{jt}=[x_{jt},v_{jt}]\beta -\alpha p_{jt}+{\xi }_{jt}$, with instruments $Z_{jt}^D$ (which include the exogenous regressors $x_{jt},v_{jt}$):

$${\xi }_{jt}={\delta }_{jt}({\mathcal{S}}_t,{\tilde{\theta }}_2)-[x_{jt},v_{jt}]\beta +\alpha p_{jt},E[{\xi }_{jt}{Z}_{jt}^D]=0.$$

The demand-only GMM program is ${\min }_{\theta }{q}_D(\theta )=g_D(\theta {)}^{\prime }W{g}_D(\theta )$ with $g_D(\theta )=\frac{1}{N}{\sum }_{j,t}{\xi }_{jt}{Z}_{jt}^D$.

The full GMM estimator (supply + demand)

Stack demand and supply sample moments and minimize over $\theta =[\beta ,\alpha ,{\tilde{\theta }}_2,\gamma ]$:

$$\underset{\theta }{\min }q(\theta )\equiv g(\theta {)}^{\prime }Wg(\theta ),g(\theta )=\left[\begin{array}{c}\frac{1}{N}{\sum }_{j,t}{\xi }_{jt}{Z}_{jt}^D\\ \frac{1}{N}{\sum }_{j,t}{\omega }_{jt}{Z}_{jt}^S\end{array}\right].$$

where ${\omega }_{jt}=f_{MC}(p_{jt}-{\eta }_{jt})-[x_{jt},w_{jt}]\gamma$ is the supply-side (marginal cost) error and ${\eta }_t={\Delta }_t({\theta }_2{)}^{-1}{\mathbf{s}}_t$ the markup (see Supply Side and Markups). The program is solved twice: once to obtain a consistent estimate of the efficient weighting matrix $W$, and again for the efficient two-step GMM estimator.

Why instruments — and what valid ones look like

Three families of instruments, in increasing sophistication:

• Cost shifters $w_{jt}$. Exogenous variables that shift marginal cost (hence price) but not demand — the classic exclusion restriction. These provide $K_3-K_x$ overidentifying restrictions for demand.
• BLP instruments. Functions of the exogenous characteristics of rival (and own) products, e.g. sums or averages of competitors’ characteristics. They are relevant because markups ${\eta }_{jt}$ and the inverse-share term $D_t^{-1}({\mathcal{S}}_t,{\tilde{\theta }}_2)$ both depend on the characteristics of all products in the market. Armstrong (2016): plain BLP instruments become weak as the number of products grows absent strong cost shifters.
• Differentiation IV (Gandhi & Houde 2019). A second-order polynomial basis in the differences of product characteristics $d_{jkt}=x_{kt}-x_{jt}$. Two flavors: the Local measure counts the number of rival products within one standard deviation of product $j$; the Quadratic measure sums the aggregate distance between $j$ and other products. These outperform the sums-of-characteristics BLP instruments because differentiation more directly captures local competition.

Approximation to the optimal instruments (Amemiya 1977 / Chamberlain 1987)

The asymptotic GMM variance depends on $(D^{\prime }{\Omega }^{-1}D)$ with $D=E[(\partial {\xi }_{jt}/\partial \theta ,\partial {\omega }_{jt}/\partial \theta )\mid Z_t]$ and $\Omega =E[({\xi }_{jt},{\omega }_{jt}{)}^{\prime }({\xi }_{jt},{\omega }_{jt})\mid Z_t]$. Chamberlain (1987): the optimal instruments are the expected Jacobian $Z_{jt}^{Opt}=E[D_{jt}(Z_t){\Omega }_{jt}^{-1}\mid Z_t]$. Conlon & Gortmaker partition these into demand and supply instruments,

$$Z_{jt}^{Opt,D}\equiv E[(D_{jt}{\Omega }_t^{-1}\odot \Theta {)}_{\cdot 1}\mid Z_t],Z_{jt}^{Opt,S}\equiv E[(D_{jt}{\Omega }_t^{-1}\odot \Theta {)}_{\cdot 2}\mid Z_t].$$

The optimal instruments from the linear parts are exogenous regressors rescaled by covariances; those for ${\theta }_2$ are nonlinear functions of the data (“quantity/markup shifters”). This formulation makes the exclusion restrictions explicit: $w_{jt}$ (cost shifters excluded from demand) give $K_3-K_x$ restrictions; $v_{jt}$ (demand shifters excluded from supply) give $K_1-K_x$ restrictions; and joint estimation adds $K_2$ cross-equation restrictions. The true optimal IV are infeasible (they require knowing the equilibrium pricing function); the feasible approximation (Berry et al. 1999, Algorithm 2) draws structural errors $({\xi }_t^{\ast },{\omega }_t^{\ast })$, re-solves for equilibrium $({\hat{p}}_t,{\hat{s}}_t)$ via the $\zeta$-markup fixed point, and averages the analytic Jacobian. The “approximate” variant (replacing errors by their expectation 0) performs as well as the costlier “asymptotic” and “empirical” variants.

Testing supply-side validity (overidentification)

Including a (possibly misspecified) supply side adds moments that can be tested. A Hausman-style likelihood-ratio test compares the full-model objective with the demand-only objective:

$$\mathrm{LR}=N[g(\hat{\theta }{)}^{\prime }Wg(\hat{\theta })-g_D({\hat{\theta }}_D{)}^{\prime }{W}_D{g}_D({\hat{\theta }}_D)]\sim {\chi }_{K-K_x}^2.$$

In Monte Carlo, the authors reject misspecified conduct assumptions but not correctly specified ones.

Examples

A merger-evaluation pipeline (e.g. automobiles): instrument price with (i) cost shifters $w_{jt}$ such as wages/steel prices in the assembly region, (ii) Gandhi-Houde differentiation IV built from horsepower/size differences to rivals, then (iii) in a second stage compute feasible optimal instruments assuming Bertrand conduct. The recommended workflow (Gandhi & Houde 2019): start with differentiation IV plus an “expected price” instrument in a first stage; if conduct is known, compute feasible optimal instruments in a second stage. Small-sample gains from optimal IV are largest with multiple random coefficients.

Connections

• Instrumental Variables — BLP’s identification rests on IV/exclusion restrictions for endogenous price.
• The BLP Contraction Mapping — supplies the inverted ${\delta }_{jt}$ that defines the residual ${\xi }_{jt}$ (the inner loop).
• Random Coefficients Logit Model — the nonlinear parameters ${\theta }_2$ that each require an instrument.
• Supply Side and Markups — source of the supply moments $E[{\omega }_{jt}{Z}_{jt}^S]=0$ and cross-equation restrictions.
• Method of Simulated Moments — the simulated moments analogue.

See Also

• BLP Demand Estimation - Overview
• Numerical Integration and Optimization in PyBLP
• Parameter Estimation in Market Response
• _Index