# Detecting the presence of measurement error

When can I know that I'm in the presence of measurement errors? i.e., when can I know that I should use a measurement error model(error-in-variables model)?

Are there any statistical tests for checking that?

• @denesp is it clearer now? – An old man in the sea. Mar 26 '17 at 13:15
• Do you mean measurement error rather than error measurement? – Richard Hardy Mar 26 '17 at 13:40
• @RichardHardy Thanks. I failed to see that. =D – An old man in the sea. Mar 26 '17 at 15:43

Suppose we observe only $\tilde{x}$ and $\tilde{y}$ which are the true values measured with error: $$\tilde{x} = x + u$$ $$\tilde{y} = y + v.$$ We would like to estimate: $$y = \beta x + \epsilon$$ but all we can really estimate is $$\tilde{y} = \hat{\beta} \tilde{x} + \zeta$$

However, if y is measured without error ($\tilde{y}=y$) we can use reverse regression to bound the measurement error in regressions. The regression is:

$$y = \hat{\beta} \tilde{x} + \epsilon$$

You also run the reverse regression $$\tilde{x} = \hat{\frac{1}{\beta}}y + \hat{\frac{1}{\beta}}\epsilon+ u$$ Define the reverse regression beta as $\hat{\beta}_r = \frac{1}{\hat{\frac{1}{\beta}}}$. It turns out that the results from the standard regression and from the reverse regression will bracket the true coeffcient, i.e. $plim\ \hat{\beta} \leq \beta \leq plim\ \hat{\beta}_r$.

This interval is small when the variance of $u$ and $\epsilon$ are small. If one of the two are zero then one of the inequalities hold exactly. Unfortunately, in practice this interval may be large and therefore this approach not very helpful.

• BKay thanks for your answer. I have a doubt. Would we still have a bracket/interval for the coefficient, even if we we're not in the presence of substantial measurement errors? If so, then your approach cannot not tell me when I should consider the meas. errors to be so severe as to use an error-in-variables model... – An old man in the sea. Mar 27 '17 at 18:02

With observational data, I would say measurement error always exists.

The real issue is "how severe" is the measurement error. Evidently in order to be able to assess the severity, "experimental-like" (or quasi-experimental) data should be available.

For an example, see this paper

Boyd, D., Lankford, H., Loeb, S., & Wyckoff, J. (2013). Measuring test measurement error: A general approach. Journal of Educational and Behavioral Statistics, 38(6), 629-663.