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 |