Recentered Instruments in Econometrics

• Recentered instruments are defined as supply shocks adjusted by subtracting their conditional expectation on endogenous controls, ensuring orthogonality.
• They enable valid causal inference in nonparametric and semi-parametric demand models by isolating exogenous variations such as price counterfactuals under faithfulness.
• Practical estimation employs GMM with recentering procedures that yield unbiased, low-variance estimates even when traditional instruments are weak.

Recentered instruments are a class of supply-side instrumental variables constructed by orthogonalizing exogenous shocks with respect to endogenous controls, enabling identification of causal effects when conventional characteristic-based instruments are invalid or weak. This methodology generalizes standard IV approaches to accommodate endogenous product characteristics, diverse sources of exogenous shocks, and settings where policy or price effects must be isolated nonparametrically.

1. Definition and Mathematical Construction

A recentered instrument is formed by taking a supply-side shock, such as a cost or policy variable $Z$, and subtracting its conditional expectation given all potentially endogenous controls. Formally, for a function $R(X,Z)$, the recentered instrument is

$\widehat{R} = R(X,Z) - \mathbb{E}[R(X,Z)\mid X]$

where $X$ represents endogenous characteristics. The orthogonality property $\mathbb{E}[\widehat{R}\mid X]=0$ ensures that any correlation between $R$ and $X$ is purged, satisfying exclusion for identification of the causal parameter in the presence of endogenous $X$ (Borusyak et al., 29 Dec 2025, Borusyak et al., 5 Apr 2025).

This construction applies to a broad set of functions $R$, including $R(X,Z)=Z$ (yielding $Z-\mathbb{E}[Z\mid X]$), $R(X,Z)=Z\otimes X$, or nonlinear model-predicted responses.

2. Identification in Nonparametric and Semi-parametric Demand Models

In flexible demand estimation, such as random-coefficient or nonparametric models, product characteristics often respond endogenously to market conditions, invalidating standard IVs. Recentered instruments circumvent this by utilizing only the exogeneity of $Z$ with respect to unobserved taste shocks $\xi$. Under an index restriction on demand,

$S = \sigma(\delta(\bar X,\xi),\tilde X, P)$

where $\delta$ is an unknown aggregator over product characteristics and unobserved heterogeneity, price counterfactuals ($\sigma(\delta,p')$ for hypothetical $p'$) are identified if the supply-side shocks $Z$ are recentered and satisfy a technical condition called faithfulness. Faithfulness requires that, conditional on $X$, all variation in price effects is detected by $Z$; this is strictly weaker than completeness and holds under several natural economic conditions, including nonparametric pricing equations or location–scale models for $\delta$ (Borusyak et al., 29 Dec 2025).

The identification theorem states that under invertibility of the demand mapping, exogeneity of $Z$, and faithfulness, all price counterfactuals are nonparametrically identified via recentered instruments, independent of instruments for $X$.

3. Construction Procedure and Practical Estimation

The canonical construction follows these steps (Borusyak et al., 5 Apr 2025):

1. Predict Model-Based Responses: For each endogenous variable $y_{jm}$ (e.g., price or its derivative), form a predicted response $\widehat{y}_{jm}$ to $g_{jm}$, the observed cost shock, based on a fitted or baseline model.
2. Recenter by Conditional Mean: Compute $$Z_{jm} = \widehat{y}_{jm} - \mathbb{E}[\widehat{y}_{jm}\mid x_{\bullet m},q_m]$$, demeaning with respect to all controls.
3. Instrument Validity: Under exogeneity of $g_{jm}$ given $x_{\bullet m},q_m$, and recentering, the moment condition $\mathbb{E}[Z_{jm}\,\xi_{jm}]=0$ holds.
4. GMM Estimation and Rank Test: Estimation proceeds via GMM using recentered IVs, with standard first-stage and rank tests adapted for the new construction.

Monte Carlo studies confirm that recentered IVs deliver unbiased and low-variance estimates of price coefficients and random coefficient dispersion, even when product characteristics are endogenous or invariant across markets, while conventional IVs may exhibit severe bias under these conditions (Borusyak et al., 5 Apr 2025).

4. Applied Settings and Supply-Side Proxy Instruments

Recentered instruments generalize to broader macroeconomic and policy analysis contexts. In fiscal SVAR-IVs for small open economies, the "trading-partner forecast-error" instrument

$\varepsilon^*_t = y^*_t - E_{t-1}[y^*_t]$

measures unanticipated foreign output shocks and is constructed to be orthogonal to domestic policy shocks by recentering with respect to expectations. Under the small open economy assumption, feedback from domestic fiscal actions to foreign surprises is negligible, ensuring exogeneity (Keränen et al., 2024).

Similarly, recentered approaches can be applied in triangular models and settings with strategic policy instruments, as in tradeable import certificates (TIC). While TIC are not always recentered instruments in the strict IV sense, their automatic adjustment and flexible linkage between imports and exports create instruments in trade policy analysis that exploit exogenous supply-side relationships (Kranz, 27 Nov 2025).

5. Theoretical Properties and Identification Conditions

The success of recentered instruments depends fundamentally on three conditions:

Condition	Technical Meaning	Practical Verification
Invertibility	$\sigma(\cdot,\tilde X,P)$ maps indices to shares bijectively	Standard in connected substitutes
Exogeneity	$Z\perp \xi\mid \bar X$	Supply-side shock orthogonality
Faithfulness	Zero average price response must imply true price invariance	Richness of supply-side price variation

Faithfulness is weaker than completeness, and is required only for the subset of effects related to price counterfactuals. Empirical verification involves testing for sufficient variation in $Z$ across $X$ and possibly utilizing high-dimensional cost shifters or instrument permutation.

6. Limitations, Extensions, and Empirical Robustness

Recentered instruments identify price effects but do not recover the full demand function nor all endogenous characteristics. Estimation rates, regularization, and practical implementation (nonparametric or sieve GMM) require additional assumptions on smoothness and dimensionality. Empirical applications should verify first-stage strength, faithfulness richness, and instrument orthogonality via generalized method of moments.

Robustness simulations show that recentered IVs are resilient to various forms of endogeneity in characteristics, maintain strong first stages under low shock variance, and degrade less rapidly than characteristic-based IVs when inter-market variation is limited (Borusyak et al., 5 Apr 2025).

A plausible implication is that as the richness and granularity of supply-side cost shocks increases, recentered instruments will offer superior identification power relative to classical approaches, particularly in differentiated products, nested logit models, and strategic sector analyses.

7. Economic Interpretation and Conceptual Advances

Recentered instruments operationalize the insight that mean-zero, supply-driven variation orthogonalized with respect to endogenous controls can serve as valid IVs in models where direct instrument validity for characteristics fails. This enables robust identification in contexts of endogeneity, strategic policy intervention, and real-time supply-side shocks. The technical innovation—especially faithfulness—broadens the scope of IV methodology in econometrics, with direct consequences for policy analysis, flexible demand estimation, and macroeconomic multiplier evaluation (Borusyak et al., 29 Dec 2025, Keränen et al., 2024, Borusyak et al., 5 Apr 2025).

In summary, recentered instruments represent a rigorous, technically grounded methodology for causal identification in the presence of endogenous controls, justified by exact orthogonality conditions and bolstered by both theoretical and empirical robustness across applications in applied microeconometrics and macroeconomic policy analysis.