Using another person's guess as an IV

In Estimates of the Economic Return to Schooling from a New Sample of Twins by Orley Ashenfelter and Alan Krueger, they correct sampling error with an IV. They claim that their results imply a larger effect of education on earnings than what was found before.

Each respondent has a twin. Each respondent is asked how much money they make, which has inherent sampling error. Then, the respondent's twin is asked how much money the first respondent makes. That estimate is used as an IV, and together they reveal the true income of the first respondent.

Can this work for revealing how many pennies are in a jar? Or how much a person likes a political candidate?

Or how does this work? I see this as magic. If I want to get the truth about something, I just have to get a second opinion?

• I thought this instrument has to do with with insincere revelation as much as noise. You might lie about how much you make but are less likely to lie about how much your brother makes. But you likely don't know exactly how much your brother makes, so the optimal forecast is a combination of your guess and their self report. – BKay Mar 17 '16 at 17:06
• @BKay yeah exactly. So what counts as an 'insincere revelation'? If you ask me how many pennies are in a jar, and I tell you 342, and there aren't. Then you ask a smarter person how many pennies are in the jar, and they say 300, isn't it the same thing? Wouldn't my lack of knowledge be the same as an 'insincere revelation'? And how come they can even do the twin experiment in the first place? – user4207 Mar 17 '16 at 23:39

Instruments are used as a replacement for an independent variable if we think that independent variable is endogenous. That means, we think it may be correlated with our error term. So in the case of estimating money made by a twin, we have a model:

$$\text{salary} = \beta_0 + \beta_1 \cdot \text{guess} + u$$

Where $u$ has standard properties mean zero and normal standard deviation. Here the problem is that the person's "guess" might be correlated with other things that affect a person's salary that isn't measured here, such as truthfulness. We also violate a normal Gauss-Markov assumption of random sampling. So we can use an instrument in place of the guess.

We wish for our instruments to be relevant and valid. Which means we want the instrument to be correlated with the guess, and also uncorrelated with the error term. The other twin's guess would be a good fit because it is probably correlated with the twin's guess, but also their guess probably does not correlate as much with external factors that might affect their sibling's salary.

In your hypothetical for measuring pennies in a jar, taking the other twin's guess itself won't be more accurate, and there might not even be an endogeneity problem. But if you were sampling groups of people instead of individuals, then you could probably expect that result to be more accurate, if groups are clustered together randomly.

In your case with liking a political candidate, you might struggle to argue that a twin's guess of their siblings affinity to a politician could be a relevant instrument. People change their political opinion noticeably when under observation by others, even close family members. So at least you might get some bias there.

• I would suggest to reconsider the existence of $\beta_0$ in this instance. – Alecos Papadopoulos Apr 9 '17 at 17:20

There's no magic. What you have to realize is that the result is conditional on the validity of the assumptions:

A) Under the assumption that there is measurement error, then yes, the average of two measurements will be on average closer to the truth than a single opinion. This is very believable. We all do this kind of thing all the time. For example, when we actually get 'a second opinion' from a doctor, we think that adding another opinion gets us closer to the truth.

B) Under the assumption that there is a smaller incentive to lie to the interviewer about your twin's education than about yours, then one way to get closer to the truth is to use the twin's information about his sibling's education rather than the self-reported one. The authors show that the variances and covariances among the different variables are consistent with this hypothesis. The intuition is that you are less likely to try to hide your sibling's lack of education than you are to hide your own lack of education.

Of course that doesn't make the preferred interpretation by the authors necessarily true. It could still be the case that there is a particular bias in the reporting of your twin's education which they did not consider. Say for example that you exaggerate the difference between your education and his or hers exactly when there is a large wage difference. That could break their result I think.

Lastly, the precise idea of using it as an instrument is not that it is a second opinion. Instead, the assumption is that the 'error' in this second measurement is uncorrelated with income in contrast to the error of the self reported income which is assumed to be correlated with income. This means that the income effects of education estimated using the sibling report is unbiased, while the one using self-reported income is biased. The authors claim the self-reported one leads to a coefficient that is downward biased, resulting in an estimate of a smaller return to education than the real one.