Measurement Error Models¶
Fuller, W. A. (1987). Measurement Error Models. John Wiley & Sons.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Measurement Uncertainty and Observational Noise
- An incomplete map (uncertainty) can be completed; an imprecise instrument (noise) cannot be made infinitely precise, a distinction Fuller (1987) formalizes by treating measurement-error models as a structural class separate from ordinary regression with unknown parameters.
This sourceCanonical statistical reference: distinguishes ordinary regression (uncertainty in unknown parameters) from errors-in-variables structures (noise intrinsic to the measurement apparatus that biases inference if ignored).
- An incomplete map (uncertainty) can be completed; an imprecise instrument (noise) cannot be made infinitely precise, a distinction Fuller (1987) formalizes by treating measurement-error models as a structural class separate from ordinary regression with unknown parameters.
Domain-specific¶
- Attenuation Bias
- When the true model is y = βx + ε but x is observed only as x* = x + u, where u is random additive noise independent of the true x and the outcome y, ordinary least squares applied to the observed data recovers not β but β multiplied by the reliability ratio σ²_x / (σ²_x + σ²_u)
This sourceDerives that OLS on a regressor with classical additive error recovers beta times the reliability ratio sigma2_x/(sigma2_x+sigma2_u).
- When the true model is y = βx + ε but x is observed only as x* = x + u, where u is random additive noise independent of the true x and the outcome y, ordinary least squares applied to the observed data recovers not β but β multiplied by the reliability ratio σ²_x / (σ²_x + σ²_u)
- Endogeneity
- The condition arises through three primary structural mechanisms. Omitted-variable confounding (the most common form): an unobserved variable causes both the regressor and the outcome, so variation in the regressor is partly variation in the confound, and the estimated coefficient picks up the confound's effect in addition to any causal effect of the regressor itself. Simultaneity (reverse causation or mutual determination): the outcome and the regressor jointly determine each other — price and quantity in a demand-supply system, police presence and crime rates, health and income — so the regressor is not causally upstream of the outcome but co-determined with it, and no single regression coefficient captures either causal direction cleanly. Measurement error in the regressor: when a regressor is measured with error, the estimated coefficient is attenuated toward zero (attenuation bias) because the mismeasured variable has noise that is by definition uncorrelated with the true value but correlated with the regression error
This sourceThat classical (truth-independent) measurement error in a regressor attenuates its coefficient toward zero.
- The condition arises through three primary structural mechanisms. Omitted-variable confounding (the most common form): an unobserved variable causes both the regressor and the outcome, so variation in the regressor is partly variation in the confound, and the estimated coefficient picks up the confound's effect in addition to any causal effect of the regressor itself. Simultaneity (reverse causation or mutual determination): the outcome and the regressor jointly determine each other — price and quantity in a demand-supply system, police presence and crime rates, health and income — so the regressor is not causally upstream of the outcome but co-determined with it, and no single regression coefficient captures either causal direction cleanly. Measurement error in the regressor: when a regressor is measured with error, the estimated coefficient is attenuated toward zero (attenuation bias) because the mismeasured variable has noise that is by definition uncorrelated with the true value but correlated with the regression error
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