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Lesson M07.L02: Proxy Variables and Measurement Error

Field Detail
Module 07 — Specification Issues
Level Graduate (ECMT5001 S1 2027)
Duration 30–35 min
Learning Objective Understand how proxy variables can partially address OVB; derive the attenuation bias formula for classical measurement error in a regressor; distinguish measurement error consequences in \(y\) versus \(x\).
Provenance Wooldridge, Introductory Econometrics, Ch 9.2–9.3

Explanation

1. Proxy Variables

In many economic applications the true variable of interest is unobservable. The classic case is ability in a wage equation:

\[\log(\text{wage}) = \beta_0 + \beta_1 \text{educ} + \beta_2 \text{ability} + u\]

We cannot directly measure ability, so we substitute an observable proxy — a variable that is highly correlated with the unobservable concept. Common proxies for ability:

  • IQ test score
  • GPA (grade point average)
  • Armed Forces Qualifying Test (AFQT) score

Formal Requirement for a Valid Proxy

Let \(q\) denote the unobservable variable and \(x_3^*\) the proposed proxy. The proxy is valid if, once we condition on \(x_3^*\), the original unobservable \(q\) provides no additional information about the outcome:

\[E(u \mid x_1, x_2, q) = E(u \mid x_1, x_2, x_3^*)\]

This means: conditioning on the proxy is as good as conditioning on the truth. Equivalently, \(q\) and the outcome are mean-independent given \((x_1, x_2, x_3^*)\).

In the wage example: once we include IQ in the regression, ability itself (beyond IQ) should not directly affect \(\log(\text{wage})\). This is a strong assumption—IQ is an imperfect proxy for ability.

Proxy Variables Reduce but May Not Eliminate OVB

Even with an imperfect proxy:

\[E(\hat{\beta}_1^{\text{proxy}}) = \beta_1 + \text{bias}^{\text{proxy}}\]

The remaining bias satisfies \(|\text{bias}^{\text{proxy}}| \leq |\text{bias}^{\text{no proxy}}|\), so including a proxy almost always moves us closer to the truth. But if the proxy condition fails—if \(q\) affects \(y\) beyond what \(x_3^*\) captures—some OVB remains.

2. Measurement Error in the Dependent Variable (\(y\))

Suppose the true outcome is \(y^*\) but we observe \(y = y^* + e_0\), where \(e_0\) is classical measurement error satisfying:

  • \(E(e_0) = 0\)
  • \(\text{Cov}(e_0, x_j) = 0\) for all regressors \(x_j\)
  • \(\text{Cov}(e_0, u) = 0\)

Substituting \(y^* = y - e_0\) into the true model \(y^* = \beta_0 + \beta_1 x + u\):

\[y = \beta_0 + \beta_1 x + (u + e_0)\]

The composite error \((u + e_0)\) still satisfies zero mean and is uncorrelated with \(x\). Therefore:

OLS remains unbiased and consistent when classical measurement error is in the dependent variable.

The only consequence is reduced efficiency: the error variance \(\text{Var}(u + e_0) = \sigma_u^2 + \sigma_{e_0}^2 > \sigma_u^2\), so standard errors are inflated and estimates are less precise.

3. Measurement Error in an Explanatory Variable (\(x\)): Attenuation Bias

Now suppose the true regressor is \(x^*\) but we observe \(x = x^* + e\), where \(e\) is classical measurement error:

  • \(E(e) = 0\)
  • \(\text{Cov}(e, x^*) = 0\) (error is independent of true value)
  • \(\text{Cov}(e, u) = 0\)

The true model is \(y = \beta_0 + \beta_1 x^* + u\). Substituting \(x^* = x - e\):

\[y = \beta_0 + \beta_1 x + (u - \beta_1 e)\]

The problem: the composite error \((u - \beta_1 e)\) is correlated with \(x = x^* + e\), because \(x\) contains \(e\). Specifically:

\[\text{Cov}(x,\; u - \beta_1 e) = \text{Cov}(x^* + e,\; -\beta_1 e) = -\beta_1 \sigma_e^2 \neq 0\]

OLS is therefore biased and inconsistent. The probability limit of the OLS estimator is:

\[\text{plim}(\tilde{\beta}_1) = \beta_1 \cdot \frac{\sigma_{x^*}^2}{\sigma_{x^*}^2 + \sigma_e^2}\]

where \(\sigma_{x^*}^2 = \text{Var}(x^*)\) and \(\sigma_e^2 = \text{Var}(e)\).

The Attenuation Bias Formula

\[\boxed{E(\tilde{\beta}_1) \approx \beta_1 \cdot \frac{\sigma_{x^*}^2}{\sigma_{x^*}^2 + \sigma_e^2}}\]

Equivalently, writing \(\sigma_x^2\) for the variance of the true regressor:

\[E(\tilde{\beta}_1) = \beta_1 \cdot \frac{\sigma_x^2}{\sigma_x^2 + \sigma_e^2}\]

The multiplier \(\dfrac{\sigma_{x^*}^2}{\sigma_{x^*}^2 + \sigma_e^2}\) is the reliability ratio, often denoted \(\lambda \in (0,1)\).

Since \(\lambda < 1\):

  • If \(\beta_1 > 0\): \(\tilde{\beta}_1 < \beta_1\) — OLS understates the true positive effect.
  • If \(\beta_1 < 0\): \(\tilde{\beta}_1 > \beta_1\) — OLS understates the true negative effect in magnitude.
  • In both cases: \(|\tilde{\beta}_1| < |\beta_1|\) — the estimated effect is attenuated toward zero.

This is called attenuation bias or errors-in-variables bias.

Key Insight

As measurement error grows (\(\sigma_e^2 \to \infty\)), the reliability ratio \(\to 0\) and \(\tilde{\beta}_1 \to 0\): the estimate is completely swamped by noise. As \(\sigma_e^2 \to 0\), \(\lambda \to 1\) and we recover the true \(\beta_1\).

4. Solution: Instrumental Variables

The standard remedy for measurement error in \(x\) is instrumental variables (IV) estimation. An instrument \(z\) must:

  1. Be correlated with the true regressor \(x^*\) (relevance).
  2. Be uncorrelated with the measurement error \(e\) and with \(u\) (exogeneity).

Then IV/2SLS consistently estimates \(\beta_1\) despite the measurement error. This will be covered in M10.

Worked Example

Part A — Ability Proxy in a Wage Equation

True model:

\[\log(\text{wage}) = \beta_0 + \beta_1 \text{educ} + \beta_2 \text{ability} + u\]

Since ability is unobservable, consider two specifications:

Specification OLS estimate of \(\beta_1\)
No proxy for ability \(\hat{\beta}_1 = 0.107\) (likely upward bias if educ and ability are positively correlated)
IQ score as proxy \(\hat{\beta}_1 = 0.094\) (closer to true return to education)

Interpretation: Including IQ as a proxy for ability reduces the estimated return to education from 10.7% to 9.4% per year of schooling. If IQ is a valid proxy, the 9.4% estimate better isolates the causal effect of education itself.

Caveat: If IQ is an imperfect proxy (ability affects wages beyond what IQ captures), residual OVB remains, and the 9.4% estimate is still biased—just less so.

Part B — Measurement Error in Self-Reported Income

Researchers often measure income via survey self-reports. Suppose true income is \(x^* = \text{annual income}\) but the survey records \(x = x^* + e\) where \(e \sim N(0, \sigma_e^2)\).

Suppose \(\sigma_{x^*}^2 = 40{,}000\) (AUD²) and \(\sigma_e^2 = 10{,}000\) (AUD²). The reliability ratio is:

\[\lambda = \frac{40{,}000}{40{,}000 + 10{,}000} = 0.80\]

If the true slope (effect of income on some outcome \(y\)) is \(\beta_1 = 0.50\), OLS on the mismeasured regressor yields approximately:

\[\tilde{\beta}_1 \approx 0.50 \times 0.80 = 0.40\]

Conclusion: A 20% downward bias. The true effect is understated by one fifth. Survey-based income regressions will systematically find smaller effects than the truth.

Common Misconception

"Measurement error in \(y\) is benign, and measurement error in \(x\) is also fine if the error is random."

The first part is true (classical ME in \(y\) does not bias OLS), but the second is false.

Classical (random, zero-mean) measurement error in an explanatory variable \(x\) causes systematic attenuation bias. "Random" does not mean "harmless" here—even pure random noise in \(x\) creates a correlation between \(x\) and the composite error, violating the OLS exogeneity assumption.

The intuition: observed \(x\) contains signal (\(x^*\)) plus noise (\(e\)). When we regress \(y\) on noisy \(x\), part of the variation in \(x\) is pure noise that has no relationship with \(y\), pulling the estimated slope toward zero.

Practice Prompts

  1. Proxy validity. A researcher uses mother's education as a proxy for parental ability in a child-outcome study. Under what conditions is this a valid proxy? What would have to be true about the relationship between mother's education and parental ability beyond what mother's education captures? If the condition fails, in which direction do you expect the remaining bias to go?

  2. Attenuation bias calculation. A true model has \(\beta_1 = 2.5\). The true regressor has variance \(\sigma_{x^*}^2 = 16\) and measurement error variance \(\sigma_e^2 = 4\). (a) Compute the reliability ratio \(\lambda\). (b) Compute the expected OLS estimate \(E(\tilde{\beta}_1)\). (c) By what percentage is the true effect understated?

  3. ME in \(y\) vs ME in \(x\). A model explains test scores using hours studied. Suppose:

  4. Scenario A: Test scores are recorded with random rounding error (ME in \(y\)).
  5. Scenario B: Hours studied are self-reported with random error (ME in \(x\)).

For each scenario: (a) Is OLS biased? (b) Is OLS consistent? (c) What happens to standard errors? Compare the two scenarios and explain why they have different implications for OLS.

Further Resources

  • Wooldridge, Introductory Econometrics (7e), §9.2 (Proxy Variables) and §9.3 (Measurement Error).
  • Griliches, Z. (1977). "Estimating the returns to schooling: Some econometric problems." Econometrica, 45(1), 1–22. — Classic paper on ability bias and proxies.
  • Carroll, R.J., Ruppert, D., Stefanski, L.A. & Crainiceanu, C.M. (2006). Measurement Error in Nonlinear Models. — Comprehensive reference.
  • See also: M03.L04 (OVB) and M10 (Instrumental Variables — remedy for ME in \(x\)).