High-dimensional and sparse IV methods

The paper models instrumental variables as vectors x_i satisfying the moment condition that the structural disturbance has mean zero conditional on them, setting up the IV framework for their analysis.

“E[ϵ_i|x_i] = 0, where x_i is a k_x-vector of instrumental variables.”◌ not checked against the paper’s text as it now stands

The paper develops Lasso and Post-Lasso methods to estimate optimal instruments (the conditional expectation of endogenous variables given instruments) in settings with many instruments, even when the number of instruments exceeds the sample size.

“We develop results for the use of Lasso and Post-Lasso methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p. Our results apply even when p is much larger than the sample size, n.”◌ not checked against the paper’s text as it now stands

The paper shows that the resulting IV estimator using Lasso-based first-stage instrument selection is root-n consistent, asymptotically normal, and semi-parametrically efficient under homoscedasticity, without requiring restrictive beta-min conditions.

“We show that the IV estimator based on using Lasso or Post-Lasso in the first stage is root-n consistent and asymptotically normal when the first-stage is approximately sparse; i.e. when the conditional expectation of the endogenous variables given the instruments can be well-approximated by a relatively small set of variables whose identities may be unknown. We also show the estimator is semi-parametrically efficient when the structural error is homoscedastic. Notably our results allow for imperfect model selection, and do not rely upon the unrealistic "beta-min" conditions that are widely used to establish validity of inference following model selection.”◌ not checked against the paper’s text as it now stands

The paper applies its Lasso-based instrumental variables procedure to an empirical eminent domain example, using demographic characteristics of judges as many potential instruments, and demonstrates improved first-stage strength and precision compared to a baseline instrument set.

“We look at the effect of judicial decisions at the federal circuit court level regarding the government's exercise of eminent domain on house prices and state-level GDP as in […]. We follow the identification strategy of […] who use the random assignment of judges to three judge panels that are then assigned to eminent domain cases to justify using the demographic characteristics of the judges on the realized panels as instruments for their decision.”◌ not checked against the paper’s text as it now stands

The paper proposes using LASSO and related sparse estimators to construct instrumental variables when the number of available instruments is very large, potentially exceeding the sample size.

“In this note, we propose the use of sparse methods (e.g. LASSO, Post-LASSO, √(LASSO), and Post-√(LASSO)) to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments in the canonical Gaussian case.”◌ not checked against the paper’s text as it now stands

It derives the asymptotic distribution of the resulting IV estimator and shows that it can be as efficient as the infeasible optimal instrumental variables estimator.

“That is, the IV estimator based on estimating the first-stage with appropriate sparse methods is asymptotically as efficient as the infeasible optimal IV estimator thus uses D(x_i) and thus achieves the semi-parametric efficiency bound.”◌ not checked against the paper’s text as it now stands

The paper introduces a sample-splitting IV estimator that weakens the growth condition needed on the number of instruments relative to sample size while still achieving oracle efficiency.

“we can replace this condition with the weaker condition that s log p = o(n) by employing a sample splitting method.”◌ not checked against the paper’s text as it now stands

The paper proposes LOLA, a novel auto-driven variable selection algorithm designed specifically to select relevant instrumental variables from a very large candidate pool.

“We propose a selection procedure with no optimization step called LOLA, for Learning Out of Leaders with Adaptation. LOLA is an auto-driven algorithm with two thresholding steps.”◌ not checked against the paper’s text as it now stands

It reframes the problem of choosing instrumental variables as a high-dimensional selection problem to avoid subjective ad hoc choices of instruments.

“To avoid unnecessary discussion about the choice and the pertinence of instrumental variables, we embed the model in a very high dimensional setting.”◌ not checked against the paper’s text as it now stands

The paper proves theoretical consistency of the selection procedure for instrumental variables under sparsity conditions, showing false positive and false negative rates converge to zero.

“The consistency of the procedure is proved under sparsity conditions and simulations are conducted to illustrate the practical good performances of LOLA.”◌ not checked against the paper’s text as it now stands

The paper develops HDS estimation and inference methods specifically for the instrumental variables model with many instruments in a dedicated section.

“We then develop HDS models and methods in instrumental variables models with many instruments in Section […] and a partially linear model with many series terms in Section […], with the main emphasis given to inference.”◌ not checked against the paper’s text as it now stands

The paper states that it focuses the main part of its analysis on applying HDS models and methods to the instrumental variables model.

“We focus the main part of the article on the use of HDS models and methods in the instrumental variables model and the partially linear model.”◌ not checked against the paper’s text as it now stands

The paper presents novel inference results for the instrumental variables model and illustrates them with a returns-to-schooling application, a classic instrumental variables setting.

“We present a set of novel inference results for these models and illustrate their use with applications to returns to schooling and growth regression.”◌ not checked against the paper’s text as it now stands

The paper lists instrumental regression as one of its key methodological contributions alongside partially linear regression.

The paper develops a novel predictor-dependent shrinkage prior for regression coefficients on many instruments in a Bayesian instrumental variables model.

“First, a novel predictor-dependent shrinkage prior is developed for the many instruments setting. The prior is constructed based on a factor model decomposition of the matrix of observed instruments, allowing many instruments to be incorporated into the analysis in a robust way.”◌ not checked against the paper’s text as it now stands

It formalizes the Bayesian linear instrumental variables model with structural and reduced-form equations relating the endogenous regressor to the instrument vector.

“y_i = β x_i + ϵ_y x_i = z_i^t δ + ϵ_x.”◌ not checked against the paper’s text as it now stands

It analyzes how weak factor structure in the instruments matrix relates to the weak instruments problem and can bias first-stage estimates toward zero.

It applies the new prior to an empirical instrumental variables problem, using a battery of macroeconomic instruments to estimate the elasticity of inter-temporal substitution.

“The instrument vector z_i consists of a battery of macroeconomic indicators, twice-lagged. This formulation of the economic problem follows from a linearization of an Euler equation; see […] section II (and references therein) for details.”

The paper shows that better first-stage prediction can reduce the known bias of the standard 2SLS instrumental variable estimator.

“The two-stage least-squares (2SLS) estimator is known to be biased when its first-stage fit is poor. I show that better first-stage prediction can alleviate this bias.”◌ not checked against the paper’s text as it now stands

It proposes a new instrumental variable estimator combining first-stage James–Stein shrinkage with a second-stage control-function approach that dominates 2SLS in bias for at least four instruments.

“For at least four instrumental variables and a single endogenous regressor, I establish that the standard 2SLS estimator is dominated with respect to bias. The dominating IV estimator applies James–Stein type shrinkage in a first-stage high-dimensional Normal-means problem followed by a control-function approach in the second stage.”◌ not checked against the paper’s text as it now stands

The paper establishes that its shrinkage-based instrumental variable estimator, unlike other James-Stein-based IV estimators, uniformly reduces bias relative to 2SLS.

“Unlike other IV estimators based on James--Stein shrinkage, my estimator reduces bias uniformly relative to 2SLS.”◌ not checked against the paper’s text as it now stands

The paper develops a uniform inference method for high-dimensional systems based on a Bahadur representation of de-biased instrumental variable estimators.

“This method is generated from a uniform Bahadur representation of de-biased instrumental variable estimators.”◌ not checked against the paper’s text as it now stands

The paper constructs orthogonalized moment (instrumental variable type) equations using auxiliary regression residuals to enable semiparametrically efficient estimation of target coefficients.

“the construction relies on the semi-parametrically efficient point estimators obtained from the empirical analog of the following orthogonalized moment equation: [(U^0_mk,t - X_k,tγ_mk^0) ν_k,t]=0, k=1,…,K, m=1, …, M,”◌ not checked against the paper’s text as it now stands

The paper frames the auxiliary regressions, whose residuals serve as instruments, as satisfying an orthogonality condition with the covariates.

“X_k,t = X_-k,t^⊤δ_k^0 + ν_k,t, ν_k,t X_-k,t = 0, k=1,…,K,”◌ not checked against the paper’s text as it now stands

The paper introduces a novel high-dimensional mixed-frequency IV regression that identifies a high-dimensional slope parameter using only a low-frequency instrumental variable, without requiring the number of instruments to match the number of regressors.

“First, we show that it is possible to identify and to estimate accurately the high-dimensional slope parameter leveraging on a low-frequency instrumental variable. In contrast, the point identification in the (high-dimensional) linear IV regression typically relies on the order condition postulating that the number of instrumental variables should be at least as large as the number of endogenous regressors.”◌ not checked against the paper’s text as it now stands

It proposes a linear completeness condition on the joint distribution of the regressor process and the instrument as the identification device, generalizing the rank condition from finite-dimensional IV regression.

“The linear completeness is a generalization of the rank condition imposed in the finite-dimensional linear IV regression and requires that the operator L is injective.”◌ not checked against the paper’s text as it now stands

The paper applies the mixed-frequency IV framework empirically by using daily temperature as an instrumental variable to identify the real-time price elasticity of electricity supply on the Australian spot market.

“To that end, we leverage on the daily temperature as an instrumental variable that shifts the demand curve and is exogenous for supply shocks. The temperature is a valid instrumental variable since the electricity demand increases in hot and cold times due to cooling and heating needs.”

The paper identifies that valid instrumental variables for the NKPC arise naturally from the assumption that predetermined variables are uncorrelated with the error term under Rational Expectations, making the set of valid IVs very large.

“In virtually all applications, the relation is assumed to contain an additive error term that is shown (e.g., by the assumption of Rational Expectations (RE)) or primitively assumed to be uncorrelated with predetermined variables excluded from the specified relation. This makes any predetermined variable a valid IV.”◌ not checked against the paper’s text as it now stands

The paper shows through simulation that ad-hoc selection procedures for choosing instrumental variables from a high-dimensional set can invalidate inference due to endogeneity bias.

“This is due to what […] call the `endogeneity bias', which arises when variables are selected on the basis of their in-sample correlation with a model's error terms.”◌ not checked against the paper’s text as it now stands

The paper proposes a novel Sup Score test statistic for conducting instrumental-variable-based inference that is robust to arbitrarily weak identification and allows the number of IVs to grow exponentially with the sample size in a time-series (dependent data) setting.

“I propose a Sup Score test that remains valid under dependent data, arbitrarily weak identification, and a number of IVs that increases exponentially with the sample size.”

The paper proposes an estimator that relaxes the conventional relevance condition in instrumental variable analyses, allowing instruments to be weakly correlated, uncorrelated, or mean-independent of endogenous covariates.

“This paper proposes an estimator that relaxes the conventional relevance condition in instrumental variable (IV) analyses. The method allows endogenous covariates to be weakly correlated, uncorrelated, or even mean-independent—though not independent—of the instruments, enabling the use of the maximal set of relevant instruments in a given application.”◌ not checked against the paper’s text as it now stands

The paper shows that identification of the IV model can be achieved without exclusion restrictions or finite-moment assumptions on the disturbance term.

“Identification is attainable without exclusion restrictions and without finite-moment assumptions on the disturbance term.”◌ not checked against the paper’s text as it now stands

The paper formally derives an MDep analogue of the classical IV relevance condition, showing it is weaker than the standard requirement that instruments be correlated with covariates.

“[…] is the condition of non-independence between non-trivial linear combinations of X and Z; it is the MDep analogue of the relevance condition in the IV setting, e.g., […], and an MDep analogue of the linear completeness condition in ICM estimators, e.g., […]. In the IV setting, the relevance condition requires that no non-zero linear combination of X be uncorrelated with Z.”

The paper defines instrumental variable validity in terms of whether an instrument has a direct effect on the outcome and formalizes valid and invalid IV sets in a linear IV model.

“For j = 1,…,p, the j-th instrument is valid if α^*_j = 0.”◌ not checked against the paper’s text as it now stands

The paper proves an identification theorem for the IV model with unknown validity, showing that alternative parametrizations correspond to distinct instrument groups and that the sparsest solution is equivalent to the plurality rule applied to the whole instrument set.

“Suppose Assumptions 1-3 hold, given 𝒫_0 and {D,Z,γ^*,η}, it can only produce additional G = |{c≠ 0: α^*_j/γ_j^* = c, j ∈𝒱^c*}| groups of different 𝒫_c such that 𝒱^* ∪{∪_c≠ 0ℐ_c} = {1,2…,p}, 𝒱^* ∩ℐ_c = ∅ for any c≠ 0 and ℐ_c ∩ℐ_c̃ = ∅ for c c̃, and E(Z^⊤ϵ̃^c) = 0.”◌ not checked against the paper’s text as it now stands

The paper proposes a new WIT estimator that uses a non-convex MCP penalty as a surrogate for the sparsest rule to select valid instruments in a one-step procedure, improving on two-step selection methods.

“We propose a Weak and Invalid IV robust Treatment effect (WIT) estimator. The sparsest rule is sufficient for identification and is operational in numerical optimization. The proposed procedure has a selection stage (regarding IV validity) and a post-selection estimation stage.”