<p>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
$$\tilde{y} = \hat{\beta} \tilde{x} + \zeta$$</p>

<p>However, if  y is measured without error ($\tilde{y}=y$) we can use <a href="http://davegiles.blogspot.ca/2014/11/reverse-regression-follow-up.html" rel="nofollow noreferrer">reverse regression</a> to bound the measurement error in regressions. The regression is:</p>

<p>$$ y = \hat{\beta} \tilde{x} + \epsilon$$</p>

<p>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 <a href="http://econ.lse.ac.uk/staff/spischke/ec524/Merr_new.pdf" rel="nofollow noreferrer">the reverse regression will bracket the true
coeffcient, i.e.</a> $plim\  \hat{\beta} \leq \beta \leq plim\  \hat{\beta}_r$.</p>

<p>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. </p>

CC BY-SA 3.0
2017-03-27T18:02:44.193Z
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.
CC BY-SA 3.0
2017-03-27T12:39:06.467Z
2017-03-27T12:39:06.467Z
BKay
<p>With observational data, I would say measurement error always exists. </p>

<p>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.  </p>

<p>For an example, see this paper</p>

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

<p><a href="http://www.nber.org/papers/w18010" rel="nofollow noreferrer">here downloadable freely</a> from its initial NBER version.</p>

<p>They treat test-scores essentially as a proxy for "student achievement" (knowledge, skills and abilities),  so we are not talking about errors in grading the tests. As the authors write</p>

<blockquote>
  <p><em>"Reliability coefficients based on the test-retest approach using parallel test forms is recognized in the psychometric literature as
  the gold standard for quantifying measurement error from all sources.
  Students take alternative, but parallel (i.e., interchangeable), tests
  on two or more occurrences sufficiently separated in time so as to
  allow for the “random variation within each individual in health,
  motivation, mental efficiency, concentration, forgetfulness,
  carelessness, subjectivity or impulsiveness in response and luck in
  random guessing” but sufficiently close in time that the knowledge,
  skills and abilities of individuals taking the tests are unchanged."</em></p>
</blockquote>

CC BY-SA 3.0
2017-03-26T20:17:51.207Z
ALecos thanks for your answer.  So, should I always use a error-in-variables model? If not, then how would we quantify that severity and at what level would we resort to measurement error models? Your answer seems to be a bit too specific. I was looking for something more general, statistically based. ;)
An old man in the sea.
CC BY-SA 3.0
2017-03-26T18:36:00.940Z
2017-03-27T12:18:38.387Z
2017-03-27T12:18:38.387Z
Alecos Papadopoulos
<p>Testing for measurement error in linear regressions: Hausman (1978, Econometrica, www.jstor.org/stable/1913827)</p>

<p>Testing for measurement error in general nonlinear, nonparametric models: Wilhelm (2018, CeMMAP Working Paper, <a href="http://www.ucl.ac.uk/~uctpdwi/papers/cwp451818.pdf" rel="nofollow noreferrer">http://www.ucl.ac.uk/~uctpdwi/papers/cwp451818.pdf</a>)</p>

CC BY-SA 4.0
2018-08-22T09:50:55.770Z
These seem like helpful and relevant links. You could make your answer more useful by bringing the lessons from these papers into your answer.
BKay
CC BY-SA 4.0
2018-08-22T08:51:47.203Z
2018-08-22T08:51:47.203Z
Daniel
<p>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)?</p>

<p>Are there any statistical tests for checking that?</p>

CC BY-SA 3.0
2017-03-26T13:15:18.890Z
@denesp is it clearer now?
An old man in the sea.
CC BY-SA 3.0
2017-03-26T13:40:16.987Z
Do you mean *measurement error* rather than *error measurement*?
Richard Hardy
CC BY-SA 3.0
2017-03-26T15:43:50.117Z
@RichardHardy Thanks. I failed to see that. =D
An old man in the sea.
<econometrics><applied-econometrics>
Detecting the presence of measurement error
An old man in the sea.
# Detecting the presence of measurement error

(Asked by: username_0 on 2017-03-26)

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?

**Tags:** econometrics, applied-econometrics

> **Comments:**
> **username_0**: @denesp is it clearer now?  
> **username_1**: Do you mean *measurement error* rather than *error measurement*?  
> **username_0**: @RichardHardy Thanks. I failed to see that. =D  

---

## 3 answers

### Answer by username_2 on 2017-03-27. Score: 3

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](http://davegiles.blogspot.ca/2014/11/reverse-regression-follow-up.html) 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.](http://econ.lse.ac.uk/staff/spischke/ec524/Merr_new.pdf) $plim\ \hat{\beta} \leq \beta \leq plim\ \hat{\beta}\_r$.

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.

> **Comments:**
> **username_0**: username_2 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...  

---

### Answer by username_3 on 2017-03-26. Score: 1

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.**

[here downloadable freely](http://www.nber.org/papers/w18010) from its initial NBER version.

They treat test-scores essentially as a proxy for "student achievement" (knowledge, skills and abilities), so we are not talking about errors in grading the tests. As the authors write

> *"Reliability coefficients based on the test-retest approach using parallel test forms is recognized in the psychometric literature as
> the gold standard for quantifying measurement error from all sources.
> Students take alternative, but parallel (i.e., interchangeable), tests
> on two or more occurrences sufficiently separated in time so as to
> allow for the “random variation within each individual in health,
> motivation, mental efficiency, concentration, forgetfulness,
> carelessness, subjectivity or impulsiveness in response and luck in
> random guessing” but sufficiently close in time that the knowledge,
> skills and abilities of individuals taking the tests are unchanged."*

> **Comments:**
> **username_0**: ALecos thanks for your answer.  So, should I always use a error-in-variables model? If not, then how would we quantify that severity and at what level would we resort to measurement error models? Your answer seems to be a bit too specific. I was looking for something more general, statistically based. ;)  

---

### Answer by username_4 on 2018-08-22. Score: 0

Testing for measurement error in linear regressions: Hausman (1978, Econometrica, www.jstor.org/stable/1913827)

Testing for measurement error in general nonlinear, nonparametric models: Wilhelm (2018, CeMMAP Working Paper, <http://www.ucl.ac.uk/~uctpdwi/papers/cwp451818.pdf>)

> **Comments:**
> **username_2**: These seem like helpful and relevant links. You could make your answer more useful by bringing the lessons from these papers into your answer.  

---

