Testing if one regressor is a proxy for another

I'm reading Guiso, Sapienza and Zingales: "Trusting the Stock Market" The Journal of Finance, Vol. 63, No. 6, 2008 and have a question about how they test whether trust is a proxy for risk aversion.

I'm looking at pg. 2584 where they talk about the Poisson count model estimate where the dependent variable is the number of stocks invested in (measure for diversification). Their mathematical model implies (in section I) that both risk aversion and trust increase diversification--I think I understand that part. My problem is with the regression they run (results shown in Panel A of Table VIII).

The authors claim that because the coefficient on trust is positive and significant it tells us that trust is not a proxy for risk aversion. I understand that this result is consistent with the model, but since (i) risk aversion should (theoretically, based on their model) also predict diversification, but (ii) the regression doesn't give a significant coefficient on risk aversion, and (iii) their measure of risk aversion is "noisy", i.e. has measurement error, and finally (iv) risk aversion and trust are correlated--albeit weakly as shown in Table III Panel C, couldn't they trust still be a proxy for risk aversion and with the coefficient estimate for trust being biased??

I might be missing something basic here...

1 Answer

For the benefit of readers I note that, as the authors write at the bottom of p.2557

"We define trust as the subjective probability individuals attribute to the possibility of being cheated."

So they actually measure "lack of trust" (the higher the value of the variable, the lower the trust), in order to see whether it co-varies positively with risk aversion, and so whether it can be considered a proxy for it.

But the correlation shown between "Trust" and Absolute Risk Aversion in the table you mention is $0.017$ which is not just "weak" -it is essentially non-existent, since with a finite number of (observational) data points, correlation even between random variables that are in reality uncorrelated, will not be exactly equal to zero (even with a sample size of $n=1,444$ as theirs).

This alone suffices for the argument that "Trust" cannot be considered a proxy for Risk Aversion, since in order for this to be a valid argument, we need the correlation between the two to be high.

The fact that Risk Aversion appears "statistically insignificant" means that Risk Aversion has no explanatory power on Diversification, but this cannot be attributed to the presence of Trust, since their correlation is so small.