# Instrumental variables and noisy measurement

I am interested in the effect of the unemployment rate at the time of labor market entry ($u^{LME}$) on wages later in life (this is an old question, but I have a new data set). I'd like to run something like the following regression:

$w_{i,t} = \alpha + \beta_0 u_i^{LME} + \text{other controls}$

$u_i^{LME}$ is endogenous (because to some extent time of labor market entry is a choice), so I need an instrument for $u_i^{LME}$. Past authors have used the birth year as an instrument, which I seek to do.

Problem: I only observe individuals later in life (a 2-year window at different ages for different individuals), so I must somehow identify the time of labor market entry (which is not explicitly reported) to find the unemployment rate at that time. I construct this variable as:

$$\text{Year of LME} = \text{Current Year} - (\text{Age} - \text{Years of Education})$$

This is an imperfect proxy for Year of LME, primarily because Years of Education is an approximation based on level of education attained, for which there are only 6 reported options.

Question: Assuming Year of Birth is a valid instrument for $u^{LME}$ if we had data on the year of labor market entry, is it still a reasonable instrument for the noisy measure of $u^{LME}$ that I have constructed?

Any other thoughts would be helpful.

The question is: if you had data on year of labor market entry, why would you still use your constructed measure? Typically instrumental variables are a way around measurement error. In your case you want to regress $$w_{it} = \alpha + \beta_0 u^{TE}_{it} + X'_{it} \gamma + \epsilon_{it}$$ but the noisy measure of labor market entry $u^{LME}_{it} = u^{TE}_{it} + \nu$, i.e. the true entry year $u^{TE}$ plus some measurement error $\nu$, generates a spurious negative correlation with the error. When you substitute this back into the regression \begin{align} w_{it} &= \alpha + \beta_0 \left(u^{LME}_{it} - \nu \right) + X'_{it} \gamma + \epsilon_{it} \newline &= \alpha + \beta_0 u^{LME}_{it} + X'_{it} \gamma + \epsilon_{it} - \beta_0 \nu \newline &= \alpha + \beta_0 u^{LME}_{it} + X'_{it} \gamma + u_{it} \end{align} where $u_{it} = \epsilon_{it} - \beta_0 \nu$ is the error term (since the measurement error is unobserved) such that $Cov(u^{LME}_{it},u_{it})\neq 0)$. This gives the typical attenuation bias towards zero, however, there is another source of bias from selection. Suppose there is a positive wage shock $\epsilon_{it}\uparrow$ and because individuals see this shock they enter the labor market earlier than they would have in the absence of the shock. Then selection will make it look again like entering earlier increases your wage.
Also omitted variable bias is a problem here since $u^{LME}_{it}$ is a function of education which depends on unobserved ability. So you could also imagine that high-ability workers enter the labor market later because they want to signal their ability with a college degree. Later wages are higher but not necessarily because of later entry per se but because these individuals are high ability, i.e. they would have had higher wages also in the case that they had entered the market earlier. The problem is that you don't observe ability.
Long story short: using an instrumental variable for your constructed measure is a good idea. However, even if you were to observe the true time of labor market entry it is still a good idea to use the instrument because of the selection problem. One more thing about your constructed measure is that you should add a constant equal to the school entry age. Otherwise you are assuming that people enter school straight from birth, so a better measure would be $$u^{LME}_{it} = \text{year} - \text{age} + \text{years of education} + 6$$ If you have U.S. data, for instance, you can get the school entry age by state.