Specialized IV applications and domain extensions

The paper identifies that the composite error term is correlated with the lagged dependent variable, necessitating the use of instrumental variables.

“The second one is that the composite error U_it in the observed variable equation […] is correlated with the lagged dependent variable Y_it-1 and we may therefore need to use instrumental variables (IVs).”◌ not checked against the paper’s text as it now stands

The paper proposes using deeper lags of the observed variable as instruments for the endogenous lagged regressor, based on the MA(1)-type serial dependence of the error.

“Since U_it has an MA(1) type serial dependence structure, we have E( U_itY_it-1-s) =0 for all s≥ 1. This suggests to choose Z_it=( Z_1,it,...,Z_L,it) ^'=( Y_it-2,...,Y_it-1-L) ^' for the IVs of the endogenous regressor Y_it-1.”◌ not checked against the paper’s text as it now stands

The paper develops a two-step LS-MD estimation procedure that incorporates these instrumental variables via an augmented least squares regression in the presence of interactive fixed effects.

“Step 1: For given 010B, we solve the least squares problem augmented by the instrumental variables Z_it, that is, we run the OLS regression of Y_it- 010B Y_it-1 on Z_it with interactive fixed effects 0115 _if_t and solve”◌ not checked against the paper’s text as it now stands

The paper establishes a relevance condition assumption for the instruments and shows it is equivalent, under certain conditions, to a lower bound on the dynamic coefficient, and notes that the LS-MD estimator reduces to conventional 2SLS when there are no interactive fixed effects.

“When there is no interactive fixed effect one can show that the LS-MD estimator is equivalent to the conventional 2SLS estimator for an appropriate weight matrix W_NT^ 010D.”◌ not checked against the paper’s text as it now stands

The paper addresses endogeneity between regressors and latent factors by proposing to use instrumental variables to recover the original regression coefficient consistently.

“Then we may employ a set of instrument variables w_t in the sense that w_t is correlated with _t but uncorrelated with both _t and _t.”◌ not checked against the paper’s text as it now stands

It constructs an explicit instrumental-variable estimator for the regression coefficient matrix using a normal-equation-like approach with a matching constant matrix R.

“This leads to the following estimator for: D=(1/T∑_t=1^ T_t _t^ T R^ T)(1/T∑_t=1^T z_t w_t^ T R^ T)^-1.”◌ not checked against the paper’s text as it now stands

The paper establishes a condition on the instrumental variables (uniform boundedness away from zero of a related eigenvalue) ensuring identifiability and correlation with regressors.

“The smallest eigenvalue of {E( w_t z_t^ T)}^ T R^ T R{E( w_t z_t^ T)} is uniformly bounded away from zero for all t.”◌ not checked against the paper’s text as it now stands

The paper proves that the instrumental-variable estimator achieves the same optimal convergence rate as the least squares estimator without endogeneity.

The paper develops a linear instrumental variables model with a single endogenous regressor and many instruments and constructs a new minimum distance objective function using invariance arguments.

“I analyze a linear instrumental variables model with a single endogenous regressor and many instruments. I use invariance arguments to construct a new minimum distance objective function.”◌ not checked against the paper’s text as it now stands

It shows the resulting minimum distance estimator with a particular weight matrix coincides with the limited information maximum likelihood estimator for the instrumental variables coefficient, and proposes a more efficient estimator when errors are non-normal.

“With respect to a particular weight matrix, the minimum distance estimator is equivalent to the random effects estimator of […], and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator. This weight matrix is inefficient unless the errors are normal, and I construct a new, more efficient estimator based on the optimal weight matrix.”◌ not checked against the paper’s text as it now stands

The paper extends the instrumental variables minimum distance approach to the case where the proportionality restriction (heterogeneous causal effects) is violated, linking it to bias-corrected two-stage least squares and constructing robust confidence intervals.

The paper proposes an instrumental variable quantile regression (IVQR) estimator to handle the endogenous spatial lag term in the general SAR panel model.

“In this section, we employ the instrumental variable quantile regression (IVQR) method for estimation. Let d_it=∑_j≠ im_ijy_jt denote a scalar endogenous variable, which is related to a vector of instruments ω_it.”◌ not checked against the paper’s text as it now stands

The paper develops a three-step estimation procedure for the IVQR estimator based on grid search over the spatial autoregressive parameter and minimizing a weighted distance function of the instrument coefficients.

“Following Chernozhukov and Hansen (2006, 2008) and Galvao (2011), and assuming the availability of instrumental variables ω_it, we can derive the IVQR estimator via the following three steps:”◌ not checked against the paper’s text as it now stands

The paper establishes asymptotic properties, including consistency and asymptotic normality, of the proposed IVQR estimator under a set of regularity conditions.

“Under conditions A1-A5 and Lemma […], for a given τ∈(0,1), θ̂=(λ̂,α̂^*)=(λ̂,ρ̂,β̂,λρ,λβ) converges to a Gaussian distribution: √(NT)(θ̂(τ)-θ(τ))d→N(0,Λ),”

The paper addresses the endogeneity of the spatial lag variable by adopting the instrumental variable quantile regression (IVQR) approach for the partially linear varying coefficient spatial autoregressive model.

“Due to the presence of endogenous variable, we employ the instrumental variable quantile regression (IVQR) method to attenuate the bias.”◌ not checked against the paper’s text as it now stands

It formally defines an instrumental variable conditional quantile relationship linking the endogenous spatial lag term to a vector of instruments independent of the error.

“The endogenous variable d_i is related to a vector of instruments ω_i which are independent of ε_i. Then we can define the following conditional instrumental quantile relationship: Q_τ(y_i|ℱ_-i,X_i,Z_i,U_i) =ρ(τ)d_i+X^⊤_iβ(τ)+Z^⊤_iγ(τ,U_i)+ω_iζ(τ),”◌ not checked against the paper’s text as it now stands

The paper develops a three-step IVQR estimation procedure combining B-spline approximation of varying coefficients with instrumental variables to estimate the model parameters.

“Following Chernozhukov and Hansen (2006, 2008) and Galvao (2011), and assuming the availability of instrumental variables ω_i, we can derive the IVQR estimator via the following three steps:”◌ not checked against the paper’s text as it now stands

The paper studies the instrumental variables quantile regression model, defined via a moment/estimating equation condition on the instrument vector Z_j.

“We are interested in estimating the instrumental variables quantile regression (IV-QR) model Y_j=X_j^'β _0+U_j where 𝔼[Z_j( 1{U_j<0}-q)] =0 for instrument vector Z_j∈ℝ^d and 1{·} is the indicator function.”◌ not checked against the paper’s text as it now stands

The paper proposes smoothing the instrumental-variable moment condition (rather than the objective function) to obtain smoothed estimating equations that are easier and more reliable to compute, especially with more endogenous regressors.

“Computationally, solving our problem is faster and more reliable than the IV-QR method in […], which requires specification of a grid of endogenous coefficient values to search over, computing a conventional QR estimator for each grid point. This advantage is important particularly when there are more endogenous variables.”◌ not checked against the paper’s text as it now stands

The paper extends its IV-QR framework to the overidentified case by constructing an optimal or feasible transformation matrix to combine excess instruments into an exactly identified set of instruments.

“If the model is overidentified with (Z_j)> (X_j), we can use a (X_j)× (Z_j) matrix 𝕎 to transform the original moment conditions 𝔼[Z_j(q-1{Y_j<X_j^'β})]=0 into 𝔼[ Z̃_j( q-1{Y_j<X_j^'β}) ] =0, for Z̃_j=𝕎Z_j∈ℝ^ (X_j).”◌ not checked against the paper’s text as it now stands

The paper uses the standard instrumental-variable estimation logic from dynamic panel data models to identify the homogeneous autoregressive coefficient ρ by eliminating the individual effect λ_i through a forward-differencing transformation.

“First, the identification of the homogeneous regression coefficient ρ follows from a standard argument used in the instrumental variable (IV) estimation of dynamic panel data models.”◌ not checked against the paper’s text as it now stands

The paper constructs orthogonality (moment) conditions between transformed variables and lagged dependent variables that serve as instruments to identify ρ.

“Then, because 𝔼[U_it|Y_i^0:t-1,λ_i] = 0, the orthogonality conditions 𝔼[ (Y_it^* - ρ X_it-1^*) Y_it-1] = 0 for t=1,…,T-1 in combination with a relevant rank condition can be used to identify ρ (see, e.g., […]).”◌ not checked against the paper’s text as it now stands

The paper notes that beyond the moment conditions, a relevant rank condition is also required for identification via this IV-type approach.

“in combination with a relevant rank condition can be used to identify ρ (see, e.g., […]).”◌ not checked against the paper’s text as it now stands

The paper's proposed CQL estimator avoids the need for instrumental variables because it remains unbiased despite misspecification of individual/group-specific fixed effects.

“The proposed CQL estimator remains unbiased in the presence of misspecification of the unobserved individual/group-specific fixed effects; therefore, neither instrumental variables nor bias corrections/reductions are required.”◌ not checked against the paper’s text as it now stands

The paper argues that since its estimator does not rely on within-group transformation, it avoids the endogeneity bias that typically necessitates instrumental variables.

“Moreover, it is worth noting at this point that, since the withingroup transformation in a linear dynamic panel gives rise to endogeneity, thus a least-squares estimator can be severely biased for small T, the proposed estimator does not rely on within-group transformation, thus it also does not suffer from an endogeneity bias. Therefore, instrumental variables (IV’s) or bias correction are not required to implement our method.”◌ not checked against the paper’s text as it now stands

The paper points out a practical limitation of instrumental variable estimation in dynamic panels with long time horizons, namely the difficulty of choosing optimal instruments among many lagged variables.

“In a dynamic panel with long time horizon the IV estimation strategy may not be feasible as the number of lagged variables that can be used as IV’s is large, thus another issue related to choice of optimal IV’s needs to be dealt with.”

The paper develops a new econometric framework for shift-share (Bartik) instrumental variable regressions that identifies effects through quasi-random assignment of shocks rather than exposure shares.

“We provide a new econometric framework for shift-share instrumental variable (SSIV) regressions in which identification follows from the quasi-random assignment of shocks, while exposure shares are allowed to be endogenous.”◌ not checked against the paper’s text as it now stands

The paper derives an equivalence result showing that the orthogonality condition for a shift-share instrument can be rewritten as an orthogonality condition at the level of the underlying shocks.

“The framework is motivated by an equivalence result: the orthogonality between a shift-share instrument and an unobserved residual can be represented as the orthogonality between the underlying shocks and a shock-level unobservable.”◌ not checked against the paper’s text as it now stands

The paper derives two sufficient conditions for consistency of the shift-share IV estimator based on quasi-random shock assignment and a shock-level law of large numbers.

“We use these equivalence results to derive two conditions sufficient for SSIV consistency. First, we assume shocks are as-good-as-randomly assigned as if arising from a natural experiment. This is enough for the shift-share instrument to be valid: i.e. for the shocks to be uncorrelated with the relevant unobservables in expectation. Second, we assume that a shock-level law of large numbers applies—that the instrument incorporates many sufficiently independent shocks, each with sufficiently small average exposure.”

The paper derives an analogous inference formula for the case where the shift-share variable is used as an instrument in an IV regression.

“We also derive an analogous formula when X_i is used as an instrument in an instrumental variables regression, which follows directly from the fact that the associated first-stage and reduced-form regressions take the form in […].”◌ not checked against the paper’s text as it now stands

The paper establishes the asymptotic properties of an instrumental variables estimator that uses a shift-share variable as an instrument.

“establishes the asymptotic properties of the OLS estimator of β in […], as well as the properties of an instrumental variables estimator that uses a shift-share variable as an instrument.”◌ not checked against the paper’s text as it now stands

The paper relates its results to prior work studying the statistical properties of shift-share instrumental variables, comparing assumptions about randomness of shares versus shifters.

“Our paper is related to two other papers studying the statistical properties of shift-share instrumental variables. First, […] consider using the full vector of shares (w_i1, …, w_iS) as an instrument for endogenous treatment. They conclude that this approach requires the entire vector of shares to be as good as randomly assigned conditional on the shifters. Second, […], focusing on the use of a shift-share regressor as an instrument, show it is a valid instrument if the set of shifters is as good as randomly assigned conditional on the shares, and discuss consistency of the instrumental variables estimator in this context.”

The paper proposes a new risk-premia estimator based on sample-splitting instrumental variables regression to fix the two-pass procedure's inconsistency.

“We propose a new estimation procedure based on sample-splitting instrumental variables regression.”◌ not checked against the paper’s text as it now stands

It designs the estimator to be implementable with standard instrumental-variables and two-stage least squares regression tools.

“are easily implementable using standard regression tools (in particular, instrumental variables regressions and two-stage least squares).”◌ not checked against the paper’s text as it now stands

It uses sample-splitting to construct instruments that correct for first-step beta estimation error via an IV regression.

“Our new estimation approach makes use of the idea of sample-splitting in order to create multiple estimates for loadings β_i and to correct for the first-step estimation error via an instrumental variables regression.”◌ not checked against the paper’s text as it now stands

It draws an analogy between weak observed pricing factors and the classical weak instrument problem to motivate its IV-based fix.

The paper formalizes an instrumental relevance condition for proxy controls that treats latent confounders as endogenous regressors instrumented by the proxies.

“The proxies V and Z must each be sufficiently informative about W. In particular, V and Z must each satisfy an instrumental relevance condition in an IV model in which W is a vector of endogenous regressors, with X and D acting as exogenous regressors.”◌ not checked against the paper’s text as it now stands

It shows that in the classical linear Griliches model, the rank condition for proxy informativeness is exactly the standard IV relevance/rank condition.

“Full row rank of E[W Ṽ'] is precisely the rank condition for identification when V is used as an instrument for W with X acting as exogenous regressors, and similarly for E[W Z̃']. Thus the proxies are sufficiently informative if they are relevant instruments for W.”◌ not checked against the paper’s text as it now stands

The paper generalizes this IV-style relevance condition to a nonparametric setting as Assumption 2 (Informative Proxies).

“For every function δ with the property that 0<E[δ(W)^2|X=x,D=d]<∞: i. E[E[δ(W)|X,D,Z]^2|X=x,D=d]>0 ii. E[E[δ(W)|X,D,V]^2|X=x,D=d]>0”◌ not checked against the paper’s text as it now stands

The paper applies a Generalized Method of Moments robust instrumental variable technique (IVGMM) to estimate the six-factor model, alongside OLS.

“we use OLS and Generalized method of moments based robust instrumental variables technique (IVGMM).”◌ not checked against the paper’s text as it now stands

The study performs a weak instrumental variable test to check the validity of the instruments used in the IVGMM estimation.

“Simultaneously, we perform a weak instrumental variable test (Racicot & Rentz, 2015) to check for the validity of the instruments, tests of overidentifying restrictions (Hansen, 1982; Olea & Pflueger, 2013), and the Hausman test (Hausman, 1978) to check the specification and measurement errors.”◌ not checked against the paper’s text as it now stands

The paper constructs robust instruments as a filtered version of the endogenous variables using Durbin and Pal type instruments combined via GLS, following Racicot and Rentz.

“Z is retrieved by optimally combining the Durbin (1954) and Pal (1980) estimators using GLS. The result is based on the Bayesian approach of Theil and Goldberger (1961).”◌ not checked against the paper’s text as it now stands

The results show that the IVGMM approach using these instruments produces more robust and superior parameter estimates than OLS across all four portfolio sets.

The paper develops a new estimator for causal effects that uses aggregate instruments while addressing unobserved aggregate confounding that invalidates conventional IV assumptions.

“We develop an estimator for applications where the variable of interest is endogenous and researchers have access to aggregate instruments. Our method addresses the critical identification challenge – unobserved confounding, which renders conventional estimators invalid.”◌ not checked against the paper’s text as it now stands

It shows that naively constructing unit-specific instrumental variables estimators using the aggregate instrument fails when the instrument correlates with unobserved aggregate confounders.

“Having access to an aggregate instrument suggests an easy solution: as long as Z_t satisfies conventional assumptions of […], we can establish a causal link between Y_it and W_it by constructing an instrumental variables (IV) estimator separately for each unit i and reporting a summary of these estimators, e.g., the average. In practice, however, this naive approach is unlikely to produce a credible answer. Researchers suspect that Z_t correlates with other unobserved aggregate variables that affect the outcomes, and thus, the assumptions of […] are not satisfied.”◌ not checked against the paper’s text as it now stands

It relates its approach to the literature on invalid instruments by relaxing the requirement that a sufficient number of instruments or a known combination of them be valid, instead exploiting a factor structure for aggregate confounders.

The paper describes the Generalized Instrumental Variables (GIV) approach combined with GMM for estimating the behavioral NK model equations.

“In this case we can demonstrate how a Generalized Instrumental Variables (GIV) approach with a Generalized Method of Moments (GMM) estimator can be valid.”◌ not checked against the paper’s text as it now stands

It specifies that identification in the GIV setup relies on exclusion restrictions using lagged variables as valid instruments satisfying moment conditions.

“Conditions for identification hinge on exclusion restrictions implied by excluding lags of the model and using them as instruments.”◌ not checked against the paper’s text as it now stands

The paper adopts a novel approach of not pretesting or screening instrument sets before estimation, using predetermined variables as instruments based on rational expectations assumptions.

“Thus, we can have unconditional moment restriction for the form of equation […] with Z_t^i = Y_t-1^i, for any vectors of predetermined variables. Any vector of variables Y known at time t-1 can be used as instruments and implementations of GIV will differ in these choices. In this paper we take a novel approach in the sense that we don't pretest or screen for sets of instruments prior to estimation.”◌ not checked against the paper’s text as it now stands

The paper notes that the moment/estimating function underlying its bootstrap framework may depend on instrumental variables in semiparametric models.

“The function g in […] can be the (conditional) likelihood in full parametric models, or it can be obtained using the (conditional) moments and/or may depend on instrumental variables in semiparametric models.”◌ not checked against the paper’s text as it now stands

The paper reports that prior researchers have applied instrumental variable approaches to estimate CBI's effect on inflation but failed to find strong instruments.

“Several authors have thus tried to use instrumental variable approaches to estimate the effect of CBI on inflation within a causal framework, but have been unable to find strong instruments […].”◌ not checked against the paper’s text as it now stands

The paper situates instrumental variable designs as one of several causal inference methods historically used in economics, contrasting them with its own longitudinal doubly robust approach.

“Using causal inference in economics has a long history, starting with path analyses and potential outcome language […] and continuing with regression discontinuity analyses […], instrumental variable designs […], and propensity score approaches in the context of the potential outcome framework […], among many other methods.”◌ not checked against the paper’s text as it now stands

The paper discusses a prior interactive fixed effects estimator that relies on lagged regressors as instruments to estimate common factors and parameters after removing factor loadings.

“They first remove factor loadings from the estimation equation and then estimate the remaining common factors and parameters using lagged regressors as instruments.”◌ not checked against the paper’s text as it now stands

The paper notes that this instrumental-variable-based approach suffers from bias when both the number of instruments and parameters grows with T.

“While this estimator is consistent under asymptotic sequences in which T is fixed, it is well known that for large T, the number of instruments and parameters causes bias (see […]).”◌ not checked against the paper’s text as it now stands

The paper notes that extending the model to include lagged dependent variables would require an instrumental variable estimator using lagged regressors as instruments, but chooses not to pursue this extension.

“A straightforward extension that allows for lagged endogenous variables { y_p,t-ℓ} _ℓ =1^k_1, as in Hidalgo and Schafgans ( 2017), requires the use of the instrumental variable estimator, where { x_p,t-ℓ} _ℓ =1^k_1 provide natural instruments for { y_p,t-ℓ} _ℓ =1^k_1.”◌ not checked against the paper’s text as it now stands

The paper explicitly identifies the covariates at lagged periods as natural instruments for the lagged dependent variable in this potential extension.

“{ x_p,t-ℓ} _ℓ =1^k_1 provide natural instruments for { y_p,t-ℓ} _ℓ =1^k_1.”◌ not checked against the paper’s text as it now stands

The paper justifies omitting the instrumental variable extension by arguing it would only add technical complications well understood in the single cross-section (n=1) case.

“We have avoided this generalization as it would detract from the main contribution of the paper and it will only add some extra technicalities and/or considerations which are well known and understood when n=1.”◌ not checked against the paper’s text as it now stands

The paper applies its new KPS test to instrumental variable regression specifications from fifteen highly cited empirical papers.

“Re-examining fifteen highly cited papers conducting instrumental variable regressions, we find that KPS is not rejected in 56 out of 118 specifications at the 5% nominal size.”◌ not checked against the paper’s text as it now stands

The paper identifies the linear instrumental variables regression model as a prominent example where KPS covariance structure matters for computational tractability of the CUE/LIML estimator.

“Prominent examples of such models are the linear instrumental variables (IV) regression model and the linear factor model in asset pricing.”◌ not checked against the paper’s text as it now stands

The paper empirically tests KPS across various specifications of linear IV models from top-ranked economic journal studies to assess whether KPS covariance assumptions are justified in practice.

“Finally, we apply the new KPS test to various specifications of linear IV models employed in fifteen highly cited empirical studies recently published in top ranked economic journals.”◌ not checked against the paper’s text as it now stands

The paper shows in a companion work that the KPS test can be used as a key step in a two-step testing procedure with correct asymptotic size for subvector inference in the linear IV model.

The paper defines the SVMA-IV model in which external instruments correlate with the shock of interest but not with other shocks, allowing for classical measurement error.

“Each of the n_z IVs z_t = (z_1,t,…,z_n_z,t)' are assumed to correlate with the first shock but not the other shocks, after controlling for lagged variables: For all i=1,…,n_z, E(z̃_i,tε_1,t) ≠ 0, E(z̃_i,tε_j,τ)=0 for all (j,τ) ≠ (i,t),”◌ not checked against the paper’s text as it now stands

The paper shows that without further restrictions, external instruments only interval-identify variance decompositions of the instrumented shock, with sharp bounds derived by treating the model as a dynamic measurement error problem.

“First, without further restrictions, the variance decomposition of the instrumented shock's contribution to macroeconomic fluctuations is interval-identified, with informative lower and upper bounds.”◌ not checked against the paper’s text as it now stands

The paper demonstrates that instrumental variable methods for shock identification do not require the invertibility assumption needed in SVAR analysis, and derives a Granger causality test for invertibility exploiting the IV.

“Unlike SVAR analysis, our methods do not require invertibility.”

The paper studies bias of instrumental variable quantile regression (IVQR) estimators, which use instruments Z that can differ from the covariates W.

“We study the bias of classical quantile regression and instrumental variable quantile regression estimators.”◌ not checked against the paper’s text as it now stands

It derives a higher-order stochastic expansion for exact IVQR estimators obtained via mixed-integer programming, extending classical Bahadur-Kiefer theory to the instrumental variable setting.

“We define exact IVQR estimators as estimators that exactly minimize a norm of the sample moment conditions. Such estimators can be obtained using mixed-integer programming (MIP) methods […].”◌ not checked against the paper’s text as it now stands

The paper derives an explicit second-order bias formula for IVQR estimators and proposes a feasible finite-difference bias correction that eliminates this bias, showing gains are especially large for IVQR under endogeneity of the instrumented regressors.

“The gains from bias correction are particularly prominent in settings with large bias such as for the IVQR estimators under endogeneity.”◌ not checked against the paper’s text as it now stands

The simulation study varies instrument strength and endogeneity across DGPs to evaluate the IVQR bias correction procedure, including designs with weak instruments in the appendix.

The paper applies Instrumental Variable Bayesian Model Averaging (IVBMA) to simultaneously address model uncertainty and endogeneity in the study of democracy determinants.

“By utilizing Instrumental Variable Bayesian Model Averaging (IVBMA), introduced by Karl and Lenkoski (2012) and Koop et al. (2012), we simultaneously account for model uncertainty and endogeneity.”◌ not checked against the paper’s text as it now stands

The authors instrument each potentially endogenous explanatory variable using its own lagged value from an earlier period.

“We instrument each possibly endogenous variable with its corresponding lagged value2 (i.e. its averaged annual value from 1991 to 2000)3.”◌ not checked against the paper’s text as it now stands

The paper notes that finding proper instruments satisfying the exclusion restriction is especially challenging within the IVBMA framework because the method aims to encompass all relevant variables.

“Finding proper instruments is a demanding challenge in the IVBMA context. Since the IVBMA technique is meant to encompass all relevant variables, it is tough to find instruments which satisfy the exclusion restriction (Jetter and Parmeter, 2018).”◌ not checked against the paper’s text as it now stands

The paper conducts a simulation study of estimation procedures under IV identification, examining mean bias and standard deviation of estimators.

“[…] plot the mean bias and standard deviation of the estimation procedures in the case of IV identification. The relative ranking of the various estimation procedures is essentially the same as in the median bias and interquartile range plots presented in our main analysis in […].”◌ not checked against the paper’s text as it now stands

The paper shows that SVAR-IV median bias is particularly elevated when the degree of invertibility of the DGP is small, consistent with theoretical predictions.

“As expected, the median bias for SVAR-IV is particularly elevated relative to other estimation methods if the degree of invertibility is small, as predicted by theory.”◌ not checked against the paper’s text as it now stands

The paper reports median bias results for IV-based estimation separately for DGPs with the smallest and largest degrees of invertibility.

“[…] show the median bias of our estimation procedures, but now for the 10 percent of DGPs with the smallest and largest degrees of invertibility, respectively.”◌ not checked against the paper’s text as it now stands

The paper addresses endogeneity in the use of mobile financial services by employing an instrumental variable approach.

“The study employs an instrumental variable approach to address the endogeneity.”◌ not checked against the paper’s text as it now stands

The study identifies and uses two specific instruments: self-reported ability to adapt to technology and the share of higher education in the town/village.

“The two instruments considered for the analysis are self-reported ability to adapt to technology and the share of higher education in the town/ village.”◌ not checked against the paper’s text as it now stands

The paper implements an instrumental variable two-stage least squares (IV-2SLS) method using a linear probability model in both stages, and also reports an IV-probit model.

“The study adopts an instrumental variable two-stage least squares method (IV-2SLS) to examine the relationship using a linear probability model in both the first and second stages.”◌ not checked against the paper’s text as it now stands

The paper validates the strength and validity of the instruments using the first-stage F-statistic compared to Stock-Yogo critical values and the Hansen J-statistic overidentification test.

The paper extends its general dyadic double/debiased machine learning framework to linear instrumental variable (IV) regression models as an additional worked example.

“In addition, we also present a couple of simpler examples with the linear regression models and the linear IV regression models in Appendix […].”◌ not checked against the paper’s text as it now stands

The paper establishes identification of the number of latent factors and the factor space using an overidentified vector of instrumental variables that are uncorrelated with errors but correlated with betas.

“There exists a K-dimensional vector of instrumental variables z_i, for K > k, such that: (i) n →∞plim 1/n∑_i=1^n z_i ε_i' = E[z_i ε_i'] = 0, (ii) The K × k matrix Γ = n →∞plim 1/n∑_i=1^n z_i β_i' has full column rank.”◌ not checked against the paper’s text as it now stands

It shows that with instrumental variables, the asymptotic portfolio returns formed from the instruments follow a singular rank-k factor structure, so the number of factors equals the rank of a reduced-rank matrix V_ξ.

“Under Assumption […], we get from Equations […] and […]: ξ_t = Γ f_t, t=1,...,T, or equivalently in matrix notation Ξ = F Γ', where Ξ is a T × K matrix, i.e., a rank-k exact matrix factorization.”◌ not checked against the paper’s text as it now stands

The paper develops eigenvalue-based test statistics for the number of latent factors using this instrumental-variables identification approach, in addition to the sphericity-based approach.

“Section 3 develops the eigenvalue test statistics based on instrumental variables and based on eigenvalues of the return variance-covariance.”◌ not checked against the paper’s text as it now stands

The paper introduces a novel instrumental variable that exploits the imperfect overlap between local labour markets and local employment agencies.

“We propose a novel identification strategy to overcome the simultaneity of ALMP and labour market outcomes at the regional level. It exploits the imperfect overlap of local labour markets and local employment agencies that decide on the local implementation of policies.”◌ not checked against the paper’s text as it now stands

Specifically, they instrument for ALMP use in a local labour market using the policy mix implemented in overlapping but external employment agency areas.

“we instrument for the use of ALMP in a local labour market with the mix of ALMP implemented outside this market but in local employment agencies that partially overlap with this market.”◌ not checked against the paper’s text as it now stands

The paper argues its instrument is less prone to endogeneity than instruments used in prior literature, such as lagged policy variables.

“Our instruments rely on the imperfect overlap between local labour markets and administrative regions in charge of policy decisions and are less likely to be endogenously determined compared to lagged policy variables.”◌ not checked against the paper’s text as it now stands

They implement the instrument by using a dynamic panel data model where policy use is instrumented with policy use in municipalities of overlapping but external local employment agencies.

The paper identifies the instrumental variable (IV) method as one of the standard approaches used to correct Nickell bias in general dynamic panel models.

“For general dynamic panel models, the Nickell bias is usually addressed with the instrumental variable (IV) method […] or analytical formulas […].”◌ not checked against the paper’s text as it now stands

The paper critiques the IV method for being inefficient and performing poorly in finite samples when the regressor is persistent.

“The IV method is subject to the lack of efficiency and performs poorly with finite samples when the regressor is persistent […].”◌ not checked against the paper’s text as it now stands

The paper notes that its proposed SPJ estimator avoids the weak instrument problem associated with IV-based bias correction methods.

“[…] tackle the Nickell bias by SPJ, which is an “automated” estimator that spares applied researchers from the potential weak instrument issue and complex analytical derivations.”◌ not checked against the paper’s text as it now stands

The paper contrasts its analytical bias correction with instrumental variable approaches that use lagged outcomes as instruments.

“The exact bias correction approach has some appealing characteristics in comparison to alternative corrections, which are based on instrumental variables […]. The instrumental variable methods are based on using further outcome lags as instruments for outcome lags.”◌ not checked against the paper’s text as it now stands

The paper argues that instrumental variable methods suffer from weak instrument problems and unclear instrument choice.

“The correct choice of instrument is often unclear and can lead to problems caused by weak instruments.[Problems caused by weak instruments are discussed by […].]”◌ not checked against the paper’s text as it now stands

The paper shows via simulation that instrumental variable methods yield larger standard errors than its proposed analytical correction.

“In simulations, I find that my analytical solution keeps standard errors as small as the original linear regressions and maintains proper coverage, as compared to instrumental variable methods which lead to larger standard errors.”◌ not checked against the paper’s text as it now stands

The paper notes that IV estimates are highly sensitive to instrument choice, causing sign flips in an applied replication.