Attenuation bias refers to the systematic underestimation of the magnitude of an estimated regression coefficient, typically causing it to be biased towards zero. This type of bias primarily occurs when there is measurement error in the independent variable(s) of a regression analysis.
The measurement errors in the independent variables (or variable of interest) dilutes the true relationship between independent variable and the dependent variable. It is most problematic in observational data where measurement errors are common.
Mathematical intuition
Suppose that we want to investigate the relationship between per capita expenditure of a household (`Y`) and distance to the nearest market (`D`). Distance to the nearest market is self-reported by the household, so it is likely to suffer from measurement error (E). This error (E) is random and independent of distance. It has zero mean and some finite variance `\sigma_E^2`.
`Y = \alpha+\beta D+\varepsilon---(1)`
We want to estimate Equation (1) and `\hat{\beta}` is our coefficient of interest. But, we cannot observe the true `\hat{\beta}` due to measurement error. The estimated `\hat{\beta}` is smaller than the `\hat{\beta_{True}}` (given that there is no omitted variable bias and reverse causality bias).
`\hat{\beta} = \frac{cov(Y,D)}{var(D)}`
`\hat{\beta}=\frac{cov(Y,D+E)}{var(D+E)}`
`\hat{\beta}=\frac{cov(Y,D)+cov(Y,E)}{var(D)+var(E)}`
`\hat{\beta}=\frac{cov(Y,D)+0}{var(D)+var(E)}`
`\hat{\beta}=\frac{cov(Y,D)}{var(D)+var(E)}---(2)`
The observed `\hat{\beta}` from Equation (2) is smaller than the `\hat{\beta_{True}}` because of `\sigma_E^2` or `var(E)`.
How to Address Attenuation Bias
Use Better Measurements: Reduce measurement errors by improving the quality of data collection.
Instrumental Variables (IV): If a reliable instrument for the mismeasured variable exists, it can be used to correct the bias.
Reliability Ratios: If the extent of measurement error is known, the bias can be adjusted using reliability ratios.
Structural Equation Modeling: Can be used to explicitly model measurement error.
Attenuation bias is a fundamental problem in econometrics and statistics because it can lead to incorrect conclusions about the importance or existence of relationships between variables.
Understanding Practically
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