## Robust Inference Via Heteroskedasticity in Linear Models — Working Paper No. WP/2026/100

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### Notation and useful definitions
- 0_{m×n} denotes the m × n matrix of zeros.
- For any a ∈ R, ⌊a⌋ is the largest b ∈ N such that b ≤ a.
- For any sequence of random vectors {U_t}_{t=1}^T, ̄E[U_t] ≡ E[1/T ∑_{t=1}^T U_t].
- C denotes universal positive constants that can differ across instances.
- ||·||_2 denotes the Euclidean norm, and ||·||_∞ denotes the maximum norm.
- For a sequence {U_t}, B_U(n,n+m) denotes the Borel σ-field generated by {U_t, t = n,...,n+m}.
- Strong mixing coefficient: α_mixing(b) = sup_a sup_{G∈B_U(−∞,a), H∈B_U(a+b,∞)} |P(G∩H) − P(G)P(H)|.
- Convention: call {U_t} “strong mixing of size −d” if α_mixing(b) = O(b^{−d−ω}) for some ω > 0.

### Model setup (Section 2)
- Linear model:
  - y_t = β′Y_t + δ′X_t + ε_t, for t = 1,...,T.
  - y_t: scalar outcome variable.
  - Y_t: g × 1 vector of endogenous variables.
  - X_t: p × 1 vector of exogenous variables.
  - β and δ: coefficient vectors.
  - ε_t: error term with ̄E[X_t ε_t] = 0_{p×1}.
- Auxiliary projection:
  - Y_t = ξ′Q_t + v_t, where Q_t is k_Q × 1 (may include elements of X_t), ξ is coefficient matrix, ζ = [ξ′_1,...,ξ′_g]′, and v_t has components v_{t,j} with ̄E[Q_t v_{t,j}] = 0_{k_Q×1}.
- IV-style validity and strength:
  - Validity (Lewbel [2012] type): ̄E[(Z_t − ̄μ) v_{t,j} ε_t] = 0_{k_Z×1} for all j = 1,...,g. (Equation (2.4))
  - Strength (identification): ̄E[v_t v′_t ⊗ (Z_t − ̄μ)] has full rank. (Equation (2.5))
  - Note: “strength” denotes rank/relevance requirement distinct from validity (exogeneity).

### Discussion of identifying restrictions (Section 3)
- Comparisons and mapping:
  - Standard regime-based identification schemes (Rigobon [2003], Lewis [2022], Prono [2014]) imply the paper’s conditions for particular choices of Z_t and Q_t.
  - Different choices of Z_t and Q_t can relax some restrictions and strengthen others; approach generalisable to multiple regimes.
- Feasibility remarks:
  - In some examples Q_t and Z_t are set equal to functions of ε_t; feasible because ε_t is identified under the null (2.2).
  - Simulations (Section 5) confirm that using functions of the error term under the null yields non-trivial power.
- Single endogenous variable special case:
  - Strength (2.5) reduces to ̄E[v_t^2 (Z_t − ̄μ)] ≠ 0_{k_Z×1}.

### Regime-based identification via heteroskedasticity (Section 3.1)
- Rigobon-style model (3.6) summary:
  - y_t = β Y_t + ε_t; Y_t = θ y_t + ̃v_t; S_t ∈ {0,1}; βθ ≠ 1.
  - Error/comovement and variance notation: c_{t,0}, c_{t,1}; σ^2_{ε,0}, σ^2_{ε,1}, σ^2_{̃v,0}, σ^2_{̃v,1}.
  - Regime probabilities: P(S_t = 0) = ρ_t and P(S_t = 1) = 1 − ρ_t for 0 < ρ_t < 1.
  - Rigobon identification condition: σ^2_{ε,0} σ^2_{̃v,1} ≠ σ^2_{ε,1} σ^2_{̃v,0}. (3.9)
- Two mappings into the framework:
  1. Q_t = ε_t, Z_t = S_t, with c_{t,0} = c_{t,1} = 0:
     - ξ = θ/(1 − βθ), v_t = 1/(1 − βθ) ̃v_t.
     - Validity (2.4) becomes ̄E[(S_t − ̄μ) ̃v_t ε_t]/(1 − βθ) = 0 using (3.8). (3.10)
     - Strength reduces to ̄E[(Z_t − ̄μ) v_t^2] = ̄ρ(1 − ̄ρ)/(1 − βθ)^2 (σ^2_{̃v,1} − σ^2_{̃v,0}). (3.11)
     - Hence ̄E[(Z_t − ̄μ) v_t^2] ≠ 0 ⇐⇒ σ^2_{̃v,0} ≠ σ^2_{̃v,1}. (3.12)
     - Rigobon/Lewis restrictions imply validity and strength under this mapping and allow extension to multiple regimes.
  2. Q_t = Z_t = S_t, allowing c_{t,0} = c_{t,1} = c_t ≠ 0:
     - ξ = 0, v_t = θ/(1 − βθ) ε_t + 1/(1 − βθ) ̃v_t.
     - Validity (2.4) becomes (θ/(1 − βθ)) ̄E[(S_t − ̄μ) ε_t^2]. (3.13)
     - Strength becomes ̄E[(Z_t − ̄μ) v_t^2] = ̄ρ(1 − ̄ρ)/(1 − βθ)^2 [(θ^2 σ^2_{ε,1} + σ^2_{̃v,1}) − (θ^2 σ^2_{ε,0} + σ^2_{̃v,0})]. (3.14)
     - Joint validity and strength can hold in several ways:
       - If θ = 0, validity holds even with error comovement; strength requires σ^2_{̃v,1} ≠ σ^2_{̃v,0}.
       - If σ^2_{ε,0} = σ^2_{ε,1}, validity holds; strength requires σ^2_{̃v,1} ≠ σ^2_{̃v,0}.
     - These restrictions are not nested in the standard regime-based approach.
- Example with common/idiosyncratic decomposition (3.15):
  - ε_t = ε_{id,t} + a_t u_t, ̃v_t = ̃v_{id,t} + b_t u_t with E[·|S_t] conditions and non-random a_t, b_t.
  - c_{t,0} = c_{t,1} = c_t if E[u_t^2 | S_t] = E[u_t^2].
  - Under θ = 0 or σ^2_{ε,0} = σ^2_{ε,1}, strength reduces to E[(̃v_{id,t})^2 | S_t = 0] ≠ E[(̃v_{id,t})^2 | S_t = 1].

### Non-regime-based identification via heteroskedasticity (Section 3.2)
- Two settings without pre-labelled regimes:
  1. Continuous-regime analogue:
     - Existence of mean-zero Z_t such that ̄E[Z_t ̃v_t ε_t] = 0.
     - With Q_t = ε_t, validity becomes ̄E[Z_t v_t ε_t] = 0; strength holds if ̄E[Z_t ̃v_t^2] ≠ 0.
     - Interpretation: Z_t observed, uninformative about error comovement but informative of time-varying variance (tracks regimes continuously).
     - Alternative: impose E[̃v_t | ε_t] = 0 and set Z_t = ε_t^2. Validity satisfied since ̄E[(ε_t^2 − ̄E[ε_t^2]) ̃v_t] = 0; strength if ̄E[ε_t^2 ̃v_t^2] ≠ ̄E[ε_t^2] ̄E[̃v_t^2].
  2. Measurement-error setting (3.17):
     - True model: y_t = Y_t β + X_t δ + ̃ε_t, observed Y*_t = Y_t + ̃v_t.
     - Projection: Y_t = ̆ξ X_t + ̆v_t with ̄E[X_t ̆v_t] = 0.
     - Reduced system: y_t = Y*_t β + X_t δ + ε_t, Y*_t = ̆ξ X_t + v_t where ε_t = ̃ε_t − β ̃v_t, v_t = ̆v_t + ̃v_t.
     - Setting Q_t = Z_t = X_t yields validity because E[X_t ε_t v_t] = 0 under independence and mean-zero assumptions.
     - Strength holds if ̄E[X_t v_t^2] = ̄E[X_t ̆v_t^2] ≠ 0, interpretable as conditional-heteroskedasticity requirement: ̄E[̃v_t^2 | X_t] varying with X_t so that ̄E[X_t E[̃v_t^2 | X_t]] ≠ 0.

### The HetAR test (Section 4)
- Test logic (Anderson–Rubin/AR style):
  - Define y_{t,0} = y_t − β′_0 Y_t. Under H_0: β = β_0, y_{t,0} = δ′ X_t + ε_t.
  - IV validity implies ̄E[(Z_t − ̄μ) v_{t,j} ε_t] = 0 for j = 1,...,g.
  - Population linear projection of y_{t,0} onto [X′_t, (Z_t − ̄μ)′ v_{t,1},..., (Z_t − ̄μ)′ v_{t,g}]′ has zero coefficients on (Z_t − ̄μ) v_{t,j}.
  - Auxiliary regression:
    - y_{t,0} = δ′ X_t + ∑_{j=1}^g α^{(j)′} [(Z_t − ̄μ) v_{t,j}] + ε_t; under H_0, α^{(j)} = 0 for all j.
  - Joint test of α^{(j)} = 0 for j = 1,...,g provides a test of H_0: β = β_0.
  - Strength condition (2.5) not imposed for size control; test is only powerful when (2.5) holds.
- Sample implementation:
  - Define W_t([ˆμ′, ˆζ′]′) = [X′_t, (Z_t − ˆμ)′ v_{t,1}(ˆξ),..., (Z_t − ˆμ)′ v_{t,g}(ˆξ)]′ and γ = [δ′, α′]′.
  - Auxiliary regression (4.18): y_{t,0} = γ′ W_t([ˆμ′, ˆζ′]′) + ε_t, with OLS estimator ˆγ = (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1} (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) y_{t,0}) = [ˆδ′, ˆα′]′.
- Test statistic and critical region:
  - Let R = [0_{k_Z g×p} I_{k_Z g}].
  - HetAR = T ˆγ′ R′ (R ˆΨ R′)^{−1} R ˆγ.
  - Reject H_0 at q significance level if HetAR > the 1 − q quantile of χ^2_{k_Z g}.
  - ˆΨ is an estimator of the covariance matrix of √T(ˆγ − γ) and must be recalculated for each null since it depends on y_{t,0}.
- Variance estimation and generated regressors correction:
  - Use modified Newey and West [1987] estimator:
    - ˆΨ = (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1} ˆV (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1},
    - where ˆV is given in (A.1) in the Appendix and accounts for generated regressors.
  - A sufficient condition for ignoring generated-regressor corrections is ̄E[∇_ζ (Z_t − ̄μ) v_{t,j}(ζ) ε_t] = 0_{k_Z×k_Q}, but this condition may not plausibly hold in several examples; the paper provides discussion and a counterexample.

### Asymptotic properties of the HetAR test (Section 4.2)
- Assumption 4.1 (key parts preserved verbatim):
  - (a) {[X′t, εt, Q′t, Z′t, vec(vt)′]′} is a strong-mixing sequence of size −2r/(r− 2), r > 2.
  - (b) ̄E[Qt vt′] = 0_{k_Q×g}, ̄E[(Zt − ̄μ) vt,j εt] = 0_{k_Z×1} for all j = 1,...,g, ̄E[X′t εt] = 0_{p×1}.
  - (c) max1≤t≤T max1≤j≤kZ E[|Z8t,j|r] <∞,   max1≤t≤T max1≤j≤kQ E[|Q8t,j|r] <∞,  max1≤t≤T max1≤l≤g E[|v8t,l|r] <∞, max1≤t≤T max1≤j≤p E[|X4t,j|r] <∞, and   max1≤t≤T E[|ε4t|r] <∞.
  - (d) MQ,T ≡ ̄E[Qt Qt′], MW,T ≡ ̄E[Wt Wt′], VZ,Z ≡ var(1/√T ∑T t=1 Zt) are uniformly positive definite.
  - (e) The following exist: limT→∞ MQ,T, limT→∞ VZ,T, limT→∞ ̄μ, limT→∞ ̄E[εt vt], limT→∞ ̄E[εt (Zt − ̄μ) Zt].
  - (f) m→∞ as T →∞ and m = o(T1/4). In practice, m =⌊T1/5⌋.
- Remarks on Assumption 4.1:
  - (a) allows weak dependence and nonstationary time-series (covers ARMA and linear/non-linear GARCH under standard conditions).
  - (b) contains the validity condition.
  - (c) provides sufficient moment conditions (weaker conditions are possible).
  - (d) non-degeneracy conditions.
  - (f) standard HAC bandwidth choice; in practice authors set m =⌊T1/5⌋.
  - (e) imposed for simplicity; can be generalised.
- Edge-case note:
  - A particular condition reduces to ̄E[(ε2t − E[ε2t]) ε2t] = 0, which generally does not hold.
  - The condition holds in Section 3.1.2 setup (reduces to ̄E[(St − ̄ρ) St εt] = 0) and in the mismeasured regressor setup (X2t independent of ̃εt and ̃vt so that ̄E[X2t εt] = 0).
  - Authors use the adjusted variance estimator throughout to provide a test usable across all examples in Section 3.
- Main asymptotic result (Theorem 4.1):
  - Suppose Assumption 4.1 holds. Then, under the null hypothesis in (2.2),
    - HetAR d → χ^2_{k_Z g}.
  - Extensions emphasized:
    - No identification assumption is imposed for validity.
    - Result remains valid under heteroskedasticity of arbitrary form and weak mixing conditions (contrast to Lewbel [2012] which relies on i.i.d. or strictly stationary data).
  - Methodological note: validity achieved by adapting two-step estimation results in Newey and McFadden [1994].

### Simulation findings (Section 5 summary of empirical performance)
- Monte Carlo design:
  - 1,000 replications.
  - Sample sizes: T ∈{200, 500}.
  - DGPs considered: DGP-1 to DGP-6 covering regime-based identification, weak identification, the two new regime-based restrictions, a non-regime case, and a mismeasured regressors case.
- Preserved parameter settings (selected examples verbatim):
  - β = 0.5, σ2u = 0.01, ρ = 0.5.
  - DGP-1: σ2ε,0 = 0.05, σ2ε,1 = 0.04, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 0, θ = 0.5, Qt = εt and Zt = St.
  - DGP-2: σ2ε,0 = σ2ε,1 = 0.05, σ2 ̃vid,0 = σ2 ̃vid,1 = 0.05, at = bt = 0, θ = 0.5, Qt = εt and Zt = St.
  - DGP-3: σ2ε,0 = 0.05, σ2ε,1 = 0.04, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 1, θ = 0, Qt = Zt = St.
  - DGP-4: σ2ε,0 = σ2ε,1 = 0.05, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 1, θ = 0.5, Qt = Zt = St.
  - DGP-5 (non-regime): β = θ = 0.5, σ2ε = 0.05, ρ ̃v = 0.01, with log(ε2t)ηt structure; Q t = yt − Yt β0, Zt = (yt − Yt β0)2.
  - DGP-6 (mismeasured regressors): β = 0.5, δ = ̆ξ = 1, σ2X = 0.1, σ2 ̃ε = 0.05, σ2 ̃v = 0.4, ρX,̆v = 1; Qt = Zt = Xt.
- Comparator:
  - Power curves from a Wald test of the null from OLS with Newey and West [1987] variance (labeled Wald-OLS), using the same number of lags as HetAR.
- Simulation conclusions:
  - HetAR has good size properties in all DGPs, including when identification fails in DGP-2.
  - HetAR has good power properties in DGPs where β is identified, though some excessive conservativeness is noted in DGP-4.
  - Size properties are reported at the point on the curve where β = β0; power curves focus on alternatives.

### Empirical implications and applications (summary)
- Three policy-relevant applications:
  - Passthrough of fuel prices to headline inflation in Sierra Leone:
    - Regime-based identification using St ∈{0,1} indexing 'low' vs 'high' global oil-price volatility regimes.
    - Institutional features (PRA pricing formula, predetermined St, and θ = 0) support identification; OLS may underestimate passthrough.
  - Effect of remittances on consumption growth in the Philippines:
    - Mismeasured regressor setup. Using X_t = quarterly US unemployment rate of foreign-born workers as a validity instrument, HetAR finds a significantly positive effect where OLS does not, consistent with attenuation bias from mismeasurement.
  - Exchange-rate passthrough to inflation in a multi-country setup (continuous regime case):
    - y_t: quarterly y-o-y CPI change; Y_t: y-o-y percentage depreciation of the US$/local currency.
    - Controls X_t include lagged GDP growth and lagged inflation change up to second lags.
    - Validity requires ̄E[Zt ̃vt εt | Zt] = 0; strength requires ̄E[Zt ̃v2t]̸= 0.
    - Results: highest passthroughs in EMDEs; many advanced economies and pegged regimes show no rejection at 68% level.

### Concluding methodological claims
- The paper develops a practical framework for inference via heteroskedasticity in linear models that:
  - Enables identification without external instruments, without manual regime labelling, and with mismeasured regressors.
  - Proposes an Anderson–Rubin type test (HetAR) with derived conditions for asymptotic validity in non-i.i.d. settings.
  - Provides simulation evidence documenting good size and power.
  - Demonstrates versatility and scalability through three empirical applications.

### Appendix A — Proofs and technical expressions (high-level items)
- Identification restatement:
  - Structural/reduced-form representation: y_t = β′(ζ′Q_t) + δ′X_t + W_{1,t}, Y_t = ζ′Q_t + W_{2,t} with W_{1,t} = ε_t + β′v_t and W_{2,t} = v_t.
  - Instrument validity (2.4) yields a linear moment condition leading to identification when ̄E[W_{2,t} W′_{2,t} ⊗ (Z_t − ̄μ)] has full rank.
- Expression for ˆV (HAC with generated-regressor corrections):
  - ˆV = 1/T Σ_{t=1}^T ˆZ_t ˆZ′_t + 1/T Σ_{τ=1}^m w_τ Σ_{t=τ+1}^T (ˆZ_t ˆZ′_{t−τ} + ˆZ_{t−τ} ˆZ′_t), where w_τ = 1 − τ/(m + 1). (A.1)
  - ˆZ_t is a composite vector involving X_t ˆε_t([ˆμ′, ˆζ′]′), ˆε_t([ˆμ′, ˆζ′]′) ˆv_t(ˆζ) ⊗ (Z_t − ˆμ), plus additional terms that correct for parameter estimation and generated regressors (definitions provided).
  - Auxiliary estimators: ˆε_t([ˆμ′, ˆζ′]′), ˆρ([ˆμ′, ˆζ′]′, ˆζ), ˆθ([ˆμ′, ˆζ′]′, ˆμ), ˆM_Q.
- Proof sketch for Theorem 4.1:
  - Re-scaled OLS estimator expansion: √T(ˆγ − γ) expressed as inverse of sample W moment times 1/√T Σ W_t times an expansion including ε_t and generated-regressor corrections.
  - Mean-value expansion and Jacobian ∇_{̃λ} W_t(̃λ) simplify substantially under the null because α = 0, yielding leading-term representation in terms of 1/√T Σ Z_t.
  - Limiting normality of 1/√T Σ Z_t (Lemma 1.6) and consistent estimation of V_{Z,T} by ˆV (Lemma 1.8) lead to HetAR d → χ^2_{k_Z g}.

*Source: wpiea2026100-source-pdf - Section 5 provides simulation evidence. Section 6 presents the empirical results.*

### Section  5  provides  simulation  evidence.   Section  6  presents  the  empirical  results.

### Section 5 provides simulation evidence. Section 6 presents the empirical results.

### Notation and useful definitions
- 0_{m×n} denotes the m × n matrix of zeros.
- For any a ∈ R, ⌊a⌋ is the largest b ∈ N such that b ≤ a.
- For any sequence of random vectors {U_t}_{t=1}^T, ̄E[U_t] ≡ E[1/T ∑_{t=1}^T U_t].
- C denotes universal positive constants that can differ across instances.
- ||·||_2 denotes the Euclidean norm, and ||·||_∞ denotes the maximum norm.
- For a sequence {U_t}, B_U(n,n+m) denotes the Borel σ-field generated by {U_t, t = n,...,n+m}.
- Strong mixing coefficient: α_mixing(b) = sup_a sup_{G∈B_U(−∞,a), H∈B_U(a+b,∞)} |P(G∩H) − P(G)P(H)|.
- Convention: call {U_t} “strong mixing of size −d” if α_mixing(b) = O(b^{−d−ω}) for some ω > 0.

### Model (Section 2)
- Linear model: y_t = β′Y_t + δ′X_t + ε_t, for t = 1,...,T.
  - y_t: scalar outcome variable.
  - Y_t: g × 1 vector of endogenous variables.
  - X_t: p × 1 vector of exogenous variables.
  - β and δ: coefficient vectors.
  - ε_t: error term with ̄E[X_t ε_t] = 0_{p×1}.
- Inference goal: invert test H_0: β = β_0 vs. H_1: β ≠ β_0 to construct (1−α) confidence set.
- Auxiliary projection: Y_t = ξ′Q_t + v_t, where:
  - Q_t is k_Q × 1 and includes elements of X_t (possibly only X_t).
  - ξ is coefficient matrix; ξ_j denotes column j.
  - v_t has components v_{t,j} with ̄E[Q_t v_{t,j}] = 0_{k_Q×1} for j = 1,...,g.
  - ζ = [ξ′_1,...,ξ′_g]′.
- IV validity condition (from Lewbel [2012]): letting Z_t be k_Z × 1 and ̄μ = ̄E[Z_t],
  - ̄E[(Z_t − ̄μ) v_{t,j} ε_t] = 0_{k_Z×1} for all j = 1,...,g. (Equation (2.4))
- IV strength condition (identification, Appendix generalisation): 
  - ̄E[v_t v′_t ⊗ (Z_t − ̄μ)] has full rank. (Equation (2.5))
  - Term “strength” denotes rank/relevance requirement distinct from validity (exogeneity).

### Discussion of identifying restrictions (Section 3)
- Paper compares setup and restrictions to other approaches (Rigobon [2003], Lewis [2022], Prono [2014]).
- Highlights:
  - Standard regime-based identification schemes imply the paper’s conditions for particular choices of Z_t and Q_t.
  - Different choices of Z_t and Q_t can relax some restrictions at the expense of strengthening others.
  - Approach is immediately generalisable to multiple regimes.
- Remarks on feasibility:
  - In some examples Q_t and Z_t are set equal to functions of ε_t. This is feasible because ε_t is identified under the null hypothesis (2.2).
  - Simulations in Section 5 confirm that using functions of the error term under the null yields non-trivial power.
- Single endogenous variable case: strength condition (2.5) reduces to ̄E[v_t^2 (Z_t − ̄μ)] ≠ 0_{k_Z×1}.

### Regime-based identification via heteroskedasticity (Section 3.1)
- Consider Rigobon-style model (3.6):
  - y_t = β Y_t + ε_t
  - Y_t = θ y_t + ̃v_t
  - S_t ∈ {0,1}, β and θ scalar with βθ ≠ 1.
  - Error assumptions (3.7): E[ε_t | S_t = 0] = E[ε_t | S_t = 1] = E[̃v_t | S_t = 0] = E[̃v_t | S_t = 1] = 0.
  - Comovements: E[ε_t ̃v_t | S_t = 0] = c_{t,0}, E[ε_t ̃v_t | S_t = 1] = c_{t,1}. (3.8)
  - Variances: E[ε_t^2 | S_t = 0] = σ^2_{ε,0}, E[ε_t^2 | S_t = 1] = σ^2_{ε,1}, E[̃v_t^2 | S_t = 0] = σ^2_{̃v,0}, E[̃v_t^2 | S_t = 1] = σ^2_{̃v,1}.
  - Regime probabilities: P(S_t = 0) = ρ_t and P(S_t = 1) = 1 − ρ_t for 0 < ρ_t < 1.
  - Identification condition in Rigobon: σ^2_{ε,0} σ^2_{̃v,1} ≠ σ^2_{ε,1} σ^2_{̃v,0}. (3.9)
- Two mappings to the paper’s framework:
  1. Q_t = ε_t, Z_t = S_t, with c_{t,0} = c_{t,1} = 0.
     - Solve for Y_t to obtain ξ = θ/(1 − βθ) and v_t = 1/(1 − βθ) ̃v_t.
     - Validity (2.4) becomes ̄E[(S_t − ̄μ) ̃v_t ε_t]/(1 − βθ) = 0, using (3.8). (3.10)
     - Strength reduces to ̄E[(Z_t − ̄μ) v_t^2] = ̄ρ(1 − ̄ρ)/(1 − βθ)^2 (σ^2_{̃v,1} − σ^2_{̃v,0}). (3.11)
     - Therefore ̄E[(Z_t − ̄μ) v_t^2] ≠ 0 ⇐⇒ σ^2_{̃v,0} ≠ σ^2_{̃v,1}. (3.12)
     - Thus, Rigobon/Lewis restrictions imply the paper’s validity and strength conditions under this mapping and allow extension to multiple regimes (e.g., S_t ∈ {0,1,2} via Z_t = e_{2,S_t}).
  2. Q_t = Z_t = S_t, allowing c_{t,0} = c_{t,1} = c_t ≠ 0.
     - Then ξ = 0 and v_t = θ/(1 − βθ) ε_t + 1/(1 − βθ) ̃v_t.
     - Validity (2.4) becomes (θ/(1 − βθ)) ̄E[(S_t − ̄μ) ε_t^2]. (3.13)
     - Strength becomes ̄E[(Z_t − ̄μ) v_t^2] = ̄ρ(1 − ̄ρ)/(1 − βθ)^2 [(θ^2 σ^2_{ε,1} + σ^2_{̃v,1}) − (θ^2 σ^2_{ε,0} + σ^2_{̃v,0})]. (3.14)
     - Joint validity and strength can hold in several ways:
       - If θ = 0, validity holds even if errors comove; strength requires σ^2_{̃v,1} ≠ σ^2_{̃v,0}.
       - If σ^2_{ε,0} = σ^2_{ε,1}, validity holds; strength requires σ^2_{̃v,1} ≠ σ^2_{̃v,0}.
     - These restrictions are not nested in the standard regime-based approach.
- Example with common and idiosyncratic decomposition (3.15):
  - ε_t = ε_{id,t} + a_t u_t, ̃v_t = ̃v_{id,t} + b_t u_t with E[·|S_t] conditions and non-random a_t, b_t.
  - c_{t,0} = c_{t,1} = c_t if E[u_t^2 | S_t] = E[u_t^2].
  - Under θ = 0 or σ^2_{ε,0} = σ^2_{ε,1}, strength reduces to E[(̃v_{id,t})^2 | S_t = 0] ≠ E[(̃v_{id,t})^2 | S_t = 1].

### Non-regime-based identification via heteroskedasticity (Section 3.2)
- Two settings where identifying assumptions hold without pre-labelled regimes:
  1. Continuous-regime analogue:
     - Structural model as in (3.6) but assume existence of mean-zero Z_t such that ̄E[Z_t ̃v_t ε_t] = 0.
     - With Q_t = ε_t, validity (2.4) becomes ̄E[Z_t v_t ε_t] = 0; strength holds if ̄E[Z_t ̃v_t^2] ≠ 0.
     - Interpretation: Z_t is observed, uninformative about error comovement but informative of time-varying variance (tracks regimes continuously).
     - Alternative: impose E[̃v_t | ε_t] = 0 and set Z_t = ε_t^2. Validity satisfied since ̄E[(ε_t^2 − ̄E[ε_t^2]) ̃v_t] = 0, strength satisfied if ̄E[ε_t^2 ̃v_t^2] ≠ ̄E[ε_t^2] ̄E[̃v_t^2].
     - This recovers restrictions similar to Lewis [2021].
  2. Measurement-error setting (3.17):
     - True model: y_t = Y_t β + X_t δ + ̃ε_t, observed Y*_t = Y_t + ̃v_t.
     - Projection: Y_t = ̆ξ X_t + ̆v_t with ̄E[X_t ̆v_t] = 0.
     - Observed Y*_t = ̆ξ X_t + ̆v_t + ̃v_t.
     - Reduced system: y_t = Y*_t β + X_t δ + ε_t, Y*_t = ̆ξ X_t + v_t where ε_t = ̃ε_t − β ̃v_t, v_t = ̆v_t + ̃v_t.
     - Setting Q_t = Z_t = X_t yields validity (2.4) because for all t, E[X_t ε_t v_t] = 0 (independence and mean-zero assumptions).
     - Strength holds if ̄E[X_t v_t^2] = ̄E[X_t ̆v_t^2] ≠ 0, interpretable as conditional-heteroskedasticity requirement: ̄E[̃v_t^2 | X_t] varying with X_t so that ̄E[X_t E[̃v_t^2 | X_t]] ≠ 0.

### The HetAR test (Section 4)
- Test construction (following AR logic, Chernozhukov and Hansen [2008, pp. 69]):
  - Define y_{t,0} = y_t − β′_0 Y_t. Under H_0: β = β_0, y_{t,0} = δ′ X_t + ε_t.
  - IV validity implies ̄E[(Z_t − ̄μ) v_{t,j} ε_t] = 0 for j = 1,...,g.
  - Population linear projection of y_{t,0} onto [X′_t, (Z_t − ̄μ)′ v_{t,1},..., (Z_t − ̄μ)′ v_{t,g}]′ has zero coefficients on (Z_t − ̄μ) v_{t,j}.
  - Auxiliary regression: y_{t,0} = δ′ X_t + ∑_{j=1}^g α^{(j)′} [(Z_t − ̄μ) v_{t,j}] + ε_t, so under H_0, α^{(j)} = 0 for all j.
  - Joint test of α^{(j)} = 0 for j = 1,...,g provides a test of H_0: β = β_0.
  - Note: strength condition (2.5) not imposed for size control; test is only powerful when (2.5) holds.
- Sample analogues:
  - Let W_t([ˆμ′, ˆζ′]′) = [X′_t, (Z_t − ˆμ)′ v_{t,1}(ˆξ),..., (Z_t − ˆμ)′ v_{t,g}(ˆξ)]′ and γ = [δ′, α′]′.
  - Auxiliary regression (4.18): y_{t,0} = γ′ W_t([ˆμ′, ˆζ′]′) + ε_t, where ε_t = γ′(W_t − W_t([ˆμ′, ˆζ′]′)) + ε_t and W_t = [X′_t, (Z_t − ̄μ)′ v_{t,1},..., (Z_t − ̄μ)′ v_{t,g}]′.
  - OLS estimator: ˆγ = (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1} (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) y_{t,0}) = [ˆδ′, ˆα′]′.
- Test statistic:
  - Let R = [0_{k_Z g×p} I_{k_Z g}].
  - HetAR = T ˆγ′ R′ (R ˆΨ R′)^{−1} R ˆγ.
  - Reject H_0 at q significance level if HetAR exceeds the 1 − q quantile of χ^2_{k_Z g}.
  - ˆΨ is estimator of covariance matrix of √T(ˆγ − γ) and must be recalculated for each null since it depends on y_{t,0}.
- Variance estimation and generated regressors:
  - To account for generated-regressor issues, use modified Newey and West [1987] estimator:
    - ˆΨ = (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1} ˆV (1/T ∑_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′))^{−1},
    - where ˆV is the term accounting for generated regressors given in (A.1) in the Appendix.
  - Note on generated regressors: a sufficient condition for ignoring generated-regressor corrections is ̄E[∇_ζ (Z_t − ̄μ) v_{t,j}(ζ) ε_t] = 0_{k_Z×k_Q}, but this condition may not plausibly hold in several examples (discussion and counterexample provided).

*Italicized final source attribution:* _Source: wpiea2026100-source-pdf - Section 5 provides simulation evidence. Section 6 presents the empirical results._

### 4.2    Asymptotic properties of the HetAR test

### 4.2    Asymptotic properties of the HetAR test

### Assumptions used to derive the limiting distribution under the null
- Assumption 4.1 defines the variables Wt and Zt as in the source and imposes:
  - (a) {[X′t, εt, Q′t, Z′t, vec(vt)′]′} is a strong-mixing sequence of size −2r/(r− 2), r > 2.
  - (b) ̄E[Qt vt′] = 0kQ×g, ̄E[(Zt − ̄μ) vt,j εt] = 0kZ×1 for all j = 1,...,g, ̄E[X′t εt] = 0p×1.
  - (c) max1≤t≤T max1≤j≤kZ E[|Z8t,j|r] <∞,   max1≤t≤T max1≤j≤kQ E[|Q8t,j|r] <∞   max1≤t≤T max1≤l≤g E[|v8t,l|r] <∞, max1≤t≤T max1≤j≤p E[|X4t,j|r] <∞, and   max1≤t≤T E[|ε4t|r] <∞.
  - (d) The following matrices are uniformly positive definite: MQ,T ≡ ̄E[Qt Qt′], MW,T ≡ ̄E[Wt Wt′], VZ,Z ≡ var(1/√T ∑T t=1 Zt).
  - (e) The following exist: limT→∞ MQ,T, limT→∞ VZ,T, limT→∞ ̄μ, limT→∞ ̄E[εt vt], limT→∞ ̄E[εt (Zt − ̄μ) Zt].
  - (f) m→∞ as T →∞ and m = o(T1/4). In practice, m =⌊T1/5⌋.

- Additional remarks on Assumption 4.1:
  - (a) is a weak dependence condition allowing nonstationary time-series (covers ARMA under standard regularity conditions and linear/non-linear GARCH models).
  - (b) contains the validity condition.
  - (c) contains sufficient moment conditions (weaker conditions possible with more complex dependence assumptions).
  - (d) contains standard non-degeneracy conditions.
  - (f) is a standard assumption in HAC variance estimation; in practice the authors set m =⌊T1/5⌋.
  - (e) is imposed for simplicity and can be generalized at the expense of additional notation.
  - Alternative assumption sets and alternative NW bandwidths (e.g., as in Lazarus et al. [2018]) are possible.

### Edge cases and adjustments
- A particular condition discussed reduces to ̄E[(ε2t − E[ε2t]) ε2t] = 0, which generally does not hold. The authors note:
  - The condition holds in the setup in Section 3.1.2 where it reduces to ̄E[(St − ̄ρ) St εt] = 0.
  - The condition holds in the mismeasured regressor setup in Section 3.2 where X2t is independent of ̃εt and ̃vt so that ̄E[X2t εt] = 0.
- To provide a test usable across all examples in Section 3, the authors use the adjusted variance estimator throughout.

### Main asymptotic result
- Theorem 4.1. Suppose Assumption 4.1 holds. Then, under the null hypothesis in (2.2),
  - HetAR d → χ2kZg.
- The authors emphasize two extensions relative to the generated-regressor-corrected TSLS estimator of Lewbel [2012]:
  - No identification assumption is imposed.
  - The result remains valid under heteroskedasticity of arbitrary form and weak mixing conditions (Lewbel [2012] relies on i.i.d. or strictly stationary data).
- Methodological note: validity is achieved by adapting two-step estimation results in Newey and McFadden [1994].

### Simulation findings (summary of empirical performance)
- Design details:
  - Monte Carlo: 1,000 replications.
  - Sample sizes: T ∈{200, 500}.
  - The DGPs considered include six cases (DGP-1 to DGP-6) covering regime-based identification, weak identification, the two new regime-based restrictions, a non-regime case, and a mismeasured regressors case. Parameter settings (selected examples preserved verbatim):
    - β = 0.5, σ2u = 0.01, ρ = 0.5.
    - DGP-1: σ2ε,0 = 0.05, σ2ε,1 = 0.04, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 0, θ = 0.5, Qt = εt and Zt = St.
    - DGP-2: σ2ε,0 = σ2ε,1 = 0.05, σ2 ̃vid,0 = σ2 ̃vid,1 = 0.05, at = bt = 0, θ = 0.5, Qt = εt and Zt = St.
    - DGP-3: σ2ε,0 = 0.05, σ2ε,1 = 0.04, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 1, θ = 0, Qt = Zt = St.
    - DGP-4: σ2ε,0 = σ2ε,1 = 0.05, σ2 ̃vid,0 = 0.05, σ2 ̃vid,1 = 0.01, at = bt = 1, θ = 0.5, Qt = Zt = St.
    - DGP-5 (non-regime): β = θ = 0.5, σ2ε = 0.05, ρ ̃v = 0.01, with log(ε2t)ηt structure; Q t = yt − Yt β0, Zt = (yt − Yt β0)2.
    - DGP-6 (mismeasured regressors): β = 0.5, δ = ̆ξ = 1, σ2X = 0.1, σ2 ̃ε = 0.05, σ2 ̃v = 0.4, ρX,̆v = 1; Qt = Zt = Xt.
  - For comparison the authors report power curves from a Wald test of the null from OLS with Newey and West [1987] variance (labeled Wald-OLS), using the same number of lags as for HetAR.
- Simulation conclusions:
  - The proposed HetAR test has good size properties in all DGPs, including when identification fails in DGP-2.
  - HetAR has good power properties in DGPs where β is identified, though some excessive conservativeness is noted in DGP-4.
  - Size properties are reported at the point on the curve where β = β0; power curves focus on alternatives.

### Empirical implications and applications (summary points)
- The methodology is applied in three policy-relevant contexts:
  - Passthrough of fuel prices to headline inflation in Sierra Leone (regime-based identification using St ∈{0,1} indexing 'low' vs 'high' global oil-price volatility regimes). Institutional features (PRA pricing formula, predetermined St, and θ = 0) support identification; OLS may underestimate passthrough.
  - Effect of remittances on consumption growth in the Philippines (mismeasured regressor setup). Using X t = quarterly US unemployment rate of foreign-born workers as a validity instrument, the HetAR approach finds a significantly positive effect where OLS does not, consistent with attenuation bias from mismeasurement.
  - Exchange-rate passthrough to inflation in a multi-country setup (continuous regime case): y t is quarterly y-o-y CPI change, Y t is y-o-y percentage depreciation of the US$/local currency, controls X t include lagged GDP growth and lagged inflation change up to second lags. Validity requires ̄E[Zt ̃vt εt | Zt] = 0; strength requires ̄E[Zt ̃v2t]̸= 0. Results: highest passthroughs in EMDEs; many advanced economies and pegged regimes show no rejection at 68% level.

### Concluding methodological claims
- The paper develops a practical framework for inference via heteroskedasticity in linear models:
  - Identification without external instruments, without manual regime labelling, and with mismeasured regressors.
  - An Anderson–Rubin type test is proposed, with derived conditions for asymptotic validity in non-i.i.d. settings.
  - Simulation evidence documents good size and power.
  - Three empirical applications illustrate versatility and scalability.

*Italic source: wpiea2026100-source-pdf - 4.2    Asymptotic properties of the HetAR test*

### Appendix A: Proofs of Results

### Appendix A: Proofs of Results

### Identification
- Structural and reduced-form representations re-written as
  - y_t = β′(ζ′Q_t) + δ′X_t + W_{1,t}
  - Y_t = ζ′Q_t + W_{2,t}
  - where W_{1,t} = ε_t + β′v_t and W_{2,t} = v_t.
- Instrument validity condition (2.4) yields
  - ̄E[v_t ⊗ (Z_t − ̄μ) ε_t] = 0 ⇐⇒ ̄E[W_{2,t} ⊗ (Z_t − ̄μ)(W_{1,t} − β′W_{2,t})] = 0
  - ⇐⇒ ̄E[W_{2,t} ⊗ (Z_t − ̄μ) W_{1,t}] − ̄E[W_{2,t} W′_{2,t} ⊗ (Z_t − ̄μ)] β = 0.
- Identification condition:
  - β is identified whenever ̄E[W_{2,t} W′_{2,t} ⊗ (Z_t − ̄μ)] has full rank.

### Expression for ˆV
- ˆV is expressed as
  - ˆV = 1/T Σ_{t=1}^T ˆZ_t ˆZ′_t + 1/T Σ_{τ=1}^m w_τ Σ_{t=τ+1}^T (ˆZ_t ˆZ′_{t−τ} + ˆZ_{t−τ} ˆZ′_t), where w_τ = 1 − τ/(m + 1). (A.1)
- Definition of ˆZ_t (component vector) used in ˆV:
  - ˆZ_t =
    [
      X_t ˆε_t([ˆμ′, ˆζ′]′)
      ˆε_t([ˆμ′, ˆζ′]′) ˆv_t(ˆζ) ⊗ (Z_t − ˆμ)
      + ˆρ([ˆμ′, ˆζ′]′, ˆζ) ⊗ I_{k_Z} (Z_t − ˆμ)
      − I_g ⊗ ˆθ([ˆμ′, ˆζ′]′, ˆμ) (ˆv_t(ˆζ) ⊗ ˆM_Q^{-1} Q_t)
    ].
- Auxiliary estimators and definitions:
  - ˆε_t([ˆμ′, ˆζ′]′) = y_{t,0} − ˆγ′ W_t([ˆμ′, ˆζ′]′)
  - ˆρ([ˆμ′, ˆζ′]′, ˆζ) = 1/T Σ_{t=1}^T v_t(ˆζ) ˆε_t([ˆμ′, ˆζ′]′)
  - ˆθ([ˆμ′, ˆζ′]′, ˆμ) = 1/T Σ_{t=1}^T ˆε_t([ˆμ′, ˆζ′]′) (Z_t − ˆμ) Q′_t
  - ˆM_Q = 1/T Σ_{t=1}^T Q_t Q′_t

### Proof of Theorem 4.1 — asymptotic expansion and limiting distribution
- Re-scaled and centred OLS estimator representation:
  - √T(ˆγ − γ) = [1/T Σ_{t=1}^T W_t([ˆμ′, ˆζ′]′) W′_t([ˆμ′, ˆζ′]′)]^{-1}
    × [I: 1/√T Σ_{t=1}^T W_t([ˆμ′, ˆζ′]′) ( (W_t − W_t([ˆμ′, ˆζ′]′))′ γ + ε_t ) ].
- Consistency and preliminary lemmas invoked:
  - By Lemma 1.1, I − M^{-1}_{W,T} = o_p(1).
  - By Lemma 1.2, [ˆμ′, ˆζ′]′ − [̄μ′, ζ′] = o_p(1).
- Mean-value expansion using ̃λ = [ ̃μ′, ̃ξ′_1, ..., ̃ξ′_g ]′ and λ = [̄μ′, ξ′_1, ..., ξ′_g]′:
  - II = 1/√T Σ_{t=1}^T W_t ε_t
    + (1/T)[ Σ_{t=1}^T ε_t ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) − W_t γ′ ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) ] √T([ˆμ′, ˆζ′]′ − λ) + o_p(1)
    = 1/√T Σ_{t=1}^T W_t ε_t + II_a + II_b + o_p(1).
- Structure and simplifications of the Jacobian ∇_{̃λ} W_t(̃λ):
  - Notation: Z_{t,j} = (Z_{t,j} − ̃μ_j).
  - Many partial derivatives are zero:
    - For all j = 1,..., k_Z, l = 1,..., p, m = 1,..., g, ∂X_{t,l}/∂̃μ_j = ∂X_{t,l}/∂̃ξ_{m,s} = 0.
    - For all j,l = 1,..., k_Z and m = 1,..., g,
      ∂Z_{t,j} v_{t,m}(̃ζ) / ∂̃μ_l = −v_{t,m}(̃ζ) if j = l, and 0 otherwise.
    - For j = 1,..., k_Z, l = 1,..., k_Q, m = 1,..., g, q = 1,..., g,
      ∂Z_{t,j} v_{t,m}(̃ζ) / ∂̃ξ_{q,l} = −Q_{t,l} (Z_{t,j} − ̃μ_j) if m = q, and 0 otherwise.
  - Evaluated at ̃λ = λ,
    - ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) =
      [ 0_{p×k_Z} 0_{p×gk_Q} ; v_t ⊗ I_{k_Z} − I_g ⊗ (Z_t − ̄μ) Q′_t ] . (Equation 33)
- Under the null hypothesis (γ = [δ′, α′]′ = [δ′, 0′_{gk_Z×1}]′):
  - γ′ ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) = 0_{1×(k_Z + gk_Q)}.
  - Hence W_t γ′ ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) = 0_{(p + gk_Z)×(k_Z + gk_Q)}.
- Simplified expressions for II_a,t and averaging:
  - II_{a,t} = ε_t ∇_{̃λ}|_{̃λ=λ} W_t(̃λ) =
    [ 0_{p×k_Z} 0_{p×gk_Q} ; ε_t v_t ⊗ I_{k_Z} − ε_t I_g ⊗ (Z_t − ̄μ) Q′_t ].
  - By Lemma 1.4 and Lemma 1.5, 1/T Σ_{t=1}^T II_{a,t} − E[1/T Σ_{t=1}^T II_{a,t}] = o_p(1) (elementwise).
- Behaviour of II_b:
  - By Lemma 1.7, 1/√T Σ_{t=1}^T Q_t v_{t,j} = O_p(1) for all j = 1,..., g.
  - Together with Lemma 1.2,
    - II_b = 1/√T Σ_{t=1}^T [ Z_t − ̄μ ; M^{-1}_{Q,T} Q_t v_{t,1} ; ... ; M^{-1}_{Q,T} Q_t v_{t,g} ] + o_p(1). (A.2)
    - Here M_{Q,T} ≡ E[1/T Σ_{t=1}^T Q_t Q′_t].
- Define G_T ≡ E[1/T Σ_{t=1}^T II_{a,t}]. Combining terms:
  - II = 1/√T Σ_{t=1}^T W_t ε_t + 1/√T Σ_{t=1}^T G_T [ Z_t − ̄μ ; M^{-1}_{Q,T} Q_t v_{t,1} ; ... ; M^{-1}_{Q,T} Q_t v_{t,g} ] + o_p(1)
  - = 1/√T Σ_{t=1}^T [ X_t ε_t ; ε_t v_t ⊗ (Z_t − ̄μ) + (̄E[ε_t v_t] ⊗ I_{k_Z})(Z_t − ̄μ) − (I_g ⊗ ̄E[ε_t (Z_t − ̄μ) Q′_t])(v_t ⊗ M^{-1}_{Q,T} Q_t) ] + o_p(1).
- Final asymptotic representation:
  - √T(ˆγ − γ) = M^{-1}_{W,T} (1/√T Σ_{t=1}^T Z_t) + o_p(1).
- Limiting distribution:
  - By Lemma 1.6, for V_{Z,T} ≡ var(1/√T Σ_{t=1}^T Z_t),
    - V^{-1/2}_{Z,T} (1/√T Σ_{t=1}^T Z_t) d→ N[0, I_{(p + k_Z g)×(p + k_Z g)}].
  - Therefore,
    - √T(ˆγ − γ) = M^{-1}_{W,T} V^{1/2}_{Z,T} V^{-1/2}_{Z,T} (1/√T Σ_{t=1}^T Z_t) + o_p(1)
    - M_{W,T} V^{-1/2}_{Z,T} √T(ˆγ − γ) d→ N[0, I_{(p + k_Z g)×(p + k_Z g)}].
- Consistent estimation of variance:
  - By Lemma 1.8, V_{Z,T} can be consistently estimated by ˆV, yielding the required result.

*Robust Inference Via Heteroskedasticity in Linear Models — Working Paper No. WP/2026/100*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026100-source-pdf.pdf_