While estimating a regression model you found that the explanatory variable is measured with certain error. Specify the model. What are its consequences on the parameters?

Introduction

In regression analysis, we assume that the explanatory variables are measured accurately. However, in many real-life applications, the independent variables may suffer from measurement errors due to instrument limitations, reporting mistakes, or data entry errors. This issue leads to what is known as the problem of errors-in-variables, which can severely affect the estimation of regression parameters.

Model with Measurement Error

Let the true model be:

Y_i = β₀ + β₁X_i + u_i

Suppose X_i is not directly observed. Instead, we observe X_i* which is measured with error:

X_i* = X_i + v_i

Where:

X_i* is the observed explanatory variable
X_i is the true (unobserved) variable
v_i is the measurement error (assumed to have zero mean and be uncorrelated with X_i and u_i)

Therefore, the regression model using observed data becomes:

Y_i = β₀ + β₁(X_i* − v_i) + u_i

⇒ Y_i = β₀ + β₁X_i* + (u_i − β₁v_i)

The new error term becomes:

ε_i = u_i − β₁v_i

Consequences of Measurement Error

1. Violation of Classical Assumptions

The new error term ε_i is correlated with the explanatory variable X_i* because v_i is a component of both. This violates the assumption that error terms should be uncorrelated with independent variables (Cov(X_i*, ε_i) ≠ 0).

2. Bias and Inconsistency of Estimators

The most serious consequence is that the OLS estimators become biased and inconsistent. This means the estimated coefficients do not converge to the true parameter values even as the sample size increases.

3. Attenuation Bias (or Regression Dilution)

Measurement error usually causes the estimated slope coefficient β̂₁ to be biased toward zero. This phenomenon is known as attenuation bias.

Mathematically, if:

β̂₁ = β₁ × (σ²_X / (σ²_X + σ²_v))

Where σ²_X is the variance of the true variable and σ²_v is the variance of the measurement error.

If σ²_v is large relative to σ²_X, then β̂₁ becomes significantly smaller than β₁.

4. Distorted Hypothesis Testing

Since estimators are biased and inconsistent, the standard errors and t-statistics will also be misleading. This affects hypothesis testing, leading to incorrect conclusions about the significance of explanatory variables.

Possible Remedies

Addressing measurement error in explanatory variables can be challenging, but some of the common solutions include:

1. Instrumental Variables (IV) Method

This method uses an instrument that is correlated with the true explanatory variable but uncorrelated with the measurement error. The IV approach provides consistent estimators.

2. Use of Better Measurement Tools

Improving the accuracy and precision of the data collection methods to reduce measurement error is a practical solution.

3. Errors-in-Variables Models

Special econometric models like Total Least Squares or Structural Equation Modeling (SEM) are designed to handle such issues directly.

4. Repeated Measurements

If repeated measures of the same variable are available, they can be used to estimate and correct the measurement error component.

Conclusion

Measurement error in explanatory variables is a serious problem in regression analysis. It leads to biased and inconsistent OLS estimates, making interpretation and policy implications questionable. The most severe effect is attenuation bias, which systematically underestimates the strength of the relationship between variables. Econometricians must be cautious and apply corrective techniques like Instrumental Variables or use better data sources to ensure reliable and valid regression results.