34.4 Asymptotic Properties of the IV Estimator

IV estimation provides consistent and asymptotically normal estimates of structural parameters under a specific set of assumptions. Understanding the asymptotic properties of the IV estimator requires clarity on the identification conditions and the large-sample behavior of the estimator.

Consider the linear structural model:

\[ Y = X \beta + u \]

Where:

  • \(Y\) is the dependent variable (\(n \times 1\))

  • \(X\) is a matrix of endogenous regressors (\(n \times k\))

  • \(u\) is the error term

  • \(\beta\) is the parameter vector of interest (\(k \times 1\))

Suppose we have a matrix of instruments \(Z\) (\(n \times m\)), with \(m \ge k\).

The IV estimator of \(\beta\) is:

\[ \hat{\beta}_{IV} = (Z'X)^{-1} Z'Y \]

Alternatively, when using 2SLS, this is equivalent to:

\[ \hat{\beta}_{2SLS} = (X'P_ZX)^{-1} X'P_ZY \]

Where:

  • \(P_Z = Z (Z'Z)^{-1} Z'\) is the projection matrix onto the column space of \(Z\).

34.4.1 Consistency

For \(\hat{\beta}_{IV}\) to be consistent, the following conditions must hold as \(n \to \infty\):

  1. Instrument Exogeneity

\[ \mathbb{E}[Z'u] = 0 \]

  • Instruments must be uncorrelated with the structural error term.

  • Guarantees instrument validity.

  1. Instrument Relevance

\[ \mathrm{rank}(\mathbb{E}[Z'X]) = k \]

  • Instruments must be correlated with the endogenous regressors.

  • Ensures identification of \(\beta\).

  • If this fails, the model is underidentified, and \(\hat{\beta}_{IV}\) does not converge to the true \(\beta\).

  1. Random Sampling (IID Observations)
  • \(\{(Y_i, X_i, Z_i)\}_{i=1}^n\) are independent and identically distributed (i.i.d.).
  • In more general settings, stationarity and mixing conditions can relax this.
  1. Finite Moments
  • \(\mathbb{E}[||Z||^2] < \infty\) and \(\mathbb{E}[||u||^2] < \infty\)
  • Ensures Law of Large Numbers applies to sample moments.

If these conditions are satisfied: \[ \hat{\beta}_{IV} \overset{p}{\to} \beta \] This means the IV estimator is consistent.

34.4.2 Asymptotic Normality

In addition to consistency conditions, we require:

  1. Homoskedasticity (Optional but Simplifying)

\[ \mathbb{E}[u u' | Z] = \sigma^2 I \]

  • Simplifies variance estimation.

  • If violated, heteroskedasticity-robust variance estimators must be used.

  1. Central Limit Theorem Conditions
  • Sample moments must satisfy a CLT: \[ \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n Z_i u_i \right) \overset{d}{\to} N(0, \Omega) \] Where \(\Omega = \mathbb{E}[Z_i Z_i' u_i^2]\).

Under the above conditions: \[ \sqrt{n}(\hat{\beta}_{IV} - \beta) \overset{d}{\to} N(0, V) \]

Where the asymptotic variance-covariance matrix \(V\) is: \[ V = (Q_{ZX})^{-1} Q_{Zuu} (Q_{ZX}')^{-1} \] With:

  • \(Q_{ZX} = \mathbb{E}[Z_i X_i']\)

  • \(Q_{Zuu} = \mathbb{E}[Z_i Z_i' u_i^2]\)

34.4.3 Asymptotic Efficiency

  1. Optimal Instrument Choice
  • Among all IV estimators, 2SLS is efficient when the instrument matrix \(Z\) contains all relevant information.
  • Generalized Method of Moments (GMM) can deliver efficiency gains in the presence of heteroskedasticity, by optimally weighting the moment conditions.

GMM Estimator

\[ \hat{\beta}_{GMM} = \arg \min_{\beta} \left( \frac{1}{n} \sum_{i=1}^n Z_i (Y_i - X_i' \beta) \right)' W \left( \frac{1}{n} \sum_{i=1}^n Z_i (Y_i - X_i' \beta) \right) \]

Where \(W\) is an optimal weighting matrix, typically:

\[ W = \Omega^{-1} \]

Result

  • If \(Z\) is overidentified (\(m > k\)), GMM can be more efficient than 2SLS.
  • When instruments are exactly identified (\(m = k\)), IV, 2SLS, and GMM coincide.

Summary Table of Conditions

Condition Requirement Purpose
Instrument Exogeneity \(\mathbb{E}[Z'u] = 0\) Instrument validity
Instrument Relevance \(\mathrm{rank}(\mathbb{E}[Z'X]) = k\) Model identification
Random Sampling IID (or stationary and mixing) LLN and CLT applicability
Finite Second Moments \(\mathbb{E}[||Z||^2] < \infty\), etc. LLN and CLT applicability
Homoskedasticity (optional) \(\mathbb{E}[u u' | Z] = \sigma^2 I\) Simplifies variance formulas
Optimal Weighting \(W = \Omega^{-1}\) in GMM Asymptotic efficiency