Measurement error as an endogeneity problem

· by Brian S. Anderson · Read in about 2 min · (319 words) ·

A substantial portion of entrepreneurship research employs latent constructs—unobserved variables measured with psychometric or other indicators meant to capture an underlying, but unobservable, latent phenomenon. Because we do not observe the latent construct directly, we represent the variance in the latent construct (\(\xi\)) as a function of the assumed true score (\(\lambda\)), and random and systematic error (\(\delta\)):

\(\xi=\lambda_{t}+\delta_{r,s}\)

Measurement error will always introduce bias into effect size estimates. Substituting this equation for \(x\) in a regression equation yields:

\(y=\alpha+\beta(\lambda_{t}+\delta_{r,s})+\epsilon\)

Multiplying out and rearranging yields:

\(y=\alpha+\beta(\lambda_{t})+(\beta\delta_{r,s})+\epsilon\)

As is clear from the last equsion, the estimate of \(\beta\) will always be inconsistent so long as r,s does not equal zero. We can think of measurement error then as another form of endogeneity.

Researchers often assess the impact of measurement error by evaluating the reliability of the construct’s indicators; Cronbach’s alpha for example. The assumption being that we infer validity—that the latent construct captures the phenomenon of interest—through reliability. But higher reliability only implies lower measurement error and hence validity, because validity itself is not observable in a latent construct. Further, even highly reliable constructs still retain some measurement error, leading to inconsistent parameter estimates.

If the researcher collapses the indicators into a single summed or mean-scaled score, he or she assumes that measurement error equals zero (r,s= 0). The reality for these models, however, is that r,s can never equal zero, and so measurement error in the latent variable manifests as endogeneity.

The most effective approach for dealing with measurement error in latent constructs is to model the error and adjust the effect size of accordingly. This is what structural equation modeling accomplishes, and it is by far the most robust and effective method for latent constructs. Errors-in-variable regression is another technique, although assumes constant error variance across all indicators. The 2SLS approach, however, is another effective technique to address endogeneity stemming from measurement error, but that’s for another post!