Classical Measurement Error in Quadratic Models

Applied economists are often interested in models where a covariate enters in quadratic form. For instance, the Kuznets curve and Environmental Kuznets curve posit inverted-U relationships between inequality and pollution, respectively, and income. The Mincer wage equation includes a quadratic for age or experience. Many other theoretical models give rise to non-linear effects of a covariate on outcomes.

 How classical measurement error impacts the estimates is not given a lot of thought. But ... it should be.

 Suppose one estimates a quadratic model

 y = a + b1*x + b2*x^2 + e

 that satisfies all the assumptions of the CLRM except x suffers from classical measurement error. Griliches & Ringstad (1970) show that (under normality) the OLS estimates of b1 and b2 both suffer from attenuation bias. However, the bias of b2 is more severe; the plim for b2 is b2 times the squared reliability ratio. Since the reliability ratio (the ratio of the variance of the true x to the variance of the observed x) lies strictly in the unit interval, the squared reliability ratio is (much) smaller than the reliability ratio.

 Interestingly, the greater bias on b2 makes the estimated turning point suffer from expansion bias (biased away from zero in absolute value)! Again, this matters, as we often care about the turning point. When does pollution begin to fall with income? When do wages turn down with age?


References:
Zvi Griliches and Vidar Ringstad (1970), "Error-in-the-Variables Bias in Nonlinear Contexts," Econometrica, 38(2), 368-70

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