Measurement Error Bias in Estimates of Income and Income Growth among the Poor: Analytical Results and a Correction Formula

Abstract

In any population, income growth among the poor may be higher or lower than overall income growth. Estimates of income growth among the poor are almost always based on household surveys, but income and expenditure data in those surveys are almost always measured with error. This article uses the assumption that income follows a lognormal distribution to demonstrate that such measurement error can lead to biased estimates of the mean income of the poor and the growth of that mean income over time. In particular, when both income and the measurement error are lognormally distributed, (i) measurement error leads to underestimation of the mean income among the poor at any point in time, (ii) increases (decreases) in measurement error over time, for a given level of inequality, lead to underestimation (overestimation) of income growth among the poor, and (iii) increases (decreases) in inequality over time, for a given level of measurement error, lead to overestimation (underestimation) of income growth among the poor. This article derives a correction formula that calculates the mean income of the poor as a function of the mean of the observed income of the poor, the variance of observed (log) income, and the variance of the (log of) measurement error. This formula can then be used to calculate consistent estimates of income growth among the poor. This article also presents several simulations that relax the assumptions that measurement errors are lognormally distributed, have a mean of zero, and are uncorrelated with income. Relaxing these assumptions has little effect on the results, which implies that the derivations are robust to many different types of measurement error.