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 \(n \to \infty\):

Instrument Exogeneity

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

Instruments must be uncorrelated with the structural error term.
Guarantees instrument validity.

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 \(\beta\).

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.

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:

Homoskedasticity (Optional but Simplifying)

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

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

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

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