# Intuitive meaning of error-in-variables

I understand the explanation of the example of error-in-variables used wikipedia. What I do not understand is how could we explain intuitively the error-in-variables problem? One way would be to say that it happens when a regressor is endogenized when there are errors in measurements of that regressor. But what exactly happens for the variable to become endogenous? How could I recognize it in a daily example / practical problem. The book I'm reading (Hayashi's Econometrics) refers that in data on households the problem is very common, but it doesn't explain why those errors would endogenize a variable of interest.

Any help would be appreciated.

• Can you summarize what exactly does he writes? Not everyone has this book or is interested in reading it just to understand what you are asking. – Tim Feb 10 '15 at 15:18
• @Tim I've edited the question because the fundamental thing I wanted to learn wasn't clear. – An old man in the sea. Feb 10 '15 at 21:18
• Have you ever been concerned about the accuracy of your regressor data? If not, you have been unusually privileged! – whuber Feb 10 '15 at 21:25
• @whuber but does the endogenization occur whenever we have errors in measurements of the regressors? Or do we have to assume a specific type of errors in measurements? How can we measure something without any error whatsoever? I'm thinking mainly in economics where we have to measure the GDP of a country, etc. It's impossible to do it without errors in measurements – An old man in the sea. Feb 10 '15 at 21:32
• That's right! We ignore errors in variables in order to use simple procedures like OLS. Studies show that the errors have to become fairly large--their SDs should be a noticeable percentage of the range--before they really affect the results, so in many cases OLS is just fine. In other circumstances, such as where interest focuses solely on prediction, there is no interest in the true underlying values of the regressors anyway (predictions are based on the measured values). But in every circumstance this kind of judgment needs to be applied when choosing the regression method. – whuber Feb 10 '15 at 21:36

The same can happen with variables if they are misreported as in household surveys, among others. Whenever you have self-reported or non-administrative data this is a worry. Suppose you want to regress $$Y_i = \alpha + \beta X_i + \epsilon_i$$ but you observe $\tilde{X}_i = X_i + \eta_i$ because you were sleepy when you entered the data in your datasheet and every now and then you made a mistake in entering the data. This adds the "noise" we were talking about before. This is represented by $\eta_i$ here. Let's say that you made this sleepiness error at random so it is uncorrelated with $X_i$ and $\epsilon_i$. If you then regress $$Y_i = \alpha + \beta \tilde{X}_i + u_i$$ with $u_i = \epsilon_i - \beta \eta_i$, you know that your estimated coefficient is \begin{align} \widehat{\beta} &= \frac{Cov(Y_i,\tilde{X}_i)}{Var(\tilde{X}_i)} \\ &= \frac{Cov(\alpha + \beta \tilde{X}_i + u_i,\tilde{X}_i)}{Var(\tilde{X}_i)} \\ &= \beta + \frac{Cov(u_i,\tilde{X}_i)}{Var(X_i + \eta_i)} \\ &= \beta + \frac{Cov(\epsilon_i -\beta \eta_i , X_i + \eta_i)}{Var(X_i + \eta_i)} \\ &= \beta \left(1 - \frac{Var(\eta_i)}{{Var(X_i + \eta_i)}} \right) \end{align} The second line expands $Y_i$. The third line splits the covariance into the sum of covariances, the fourth line uses the definitions of $u_i$ and $\tilde{X}_i$. Then use the fact that $\eta_i$ is uncorrelated with $X_i$ and $\epsilon_i$. The last line factors. In the bracket of the last line you have one minus the inverse of the signal-to-noise ratio.
The bigger the noise becomes relative to the signal the worse will be the song in your radio. The signal-to-noise ratio lies between 0 and 1, so if there is only noise you will not hear the song anymore. This is the so-called attenuation bias of your estimated $\widehat{\beta}$ due to the measurement error.
With regards to whuber's comment that you need a very strong noise in order to affect the results: in panel data methods the attenuation bias is propagated (see Griliches and Hausman, 1986). For example, if somebody reports a 9 dollars hourly wage in year 1 when in reality she gets 10 dollars then this is just an error of 10% for OLS. Now if she gets 12 dollars in year 2 (suppose now you have a panel data set) and you want to take advantage of the panel structure by first differencing, your first difference is $12 - 9 = 3$ but in reality it should have been $12 - 10 = 2$. So now the measurment error has increased to a half.
• Sure, ask away. The endogeneity case described in the answer is the typical bias towards 0. If $\eta_i$ is correlated with $\epsilon_i$ then the bias may be up or down depending on whether $Cov(\eta_i,\epsilon_i)$ is positive or negative. – Andy Feb 10 '15 at 22:04