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An Example of a Weak Instrument

A public health specialist names Phoebe is interested in how a person's wealth (their assets minus their debts) might effect their health. Phoebe has longitudinal data on people's wealth and health, but she's having trouble determining whether wealth has a causal effect on health with traditional regression models. Phoebe's concern is that a person's health varies endogenously with their weath; a person's wealth might have a positive causal effect on their health, but a person's health could also have a positive causal effect on their wealth.

To untangle this causal relationship, Phoebe decides to conduct an instrumental variable analysis of wealth's effect on health with inheritance as an exogenous instrument of wealth (that is, inheritance effects a person's wealth but does not have any direct or indirect relationship with a person's health other than through how it might affect wealth). However, Phoebe's analysis yields no statistically significant results. Phoebe also tests the relationship between inheritance and wealth, and finds that there is only a very weak correlation. Which of the following could this imply?

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