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.

2. 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 $\beta$.

3. 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.

4. 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:

5. Homoskedasticity (Optional but Simplifying)

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

• Simplifies variance estimation.

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

6. 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

7. 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

---