Question: suppose you are interested in estimating how health affects productivity...

Suppose you are interested in estimating how health affects productivity. Specifically, you are interested in the following regression:

where Yi is a variable representing the income earned by person i, and Xi is a variable representing the health of person i.

Income is a fairly straightforward concept to capture. Health, however, is a more complex concept that can be measured in many different ways. For example, body mass index (BMI - weight in kilograms, divided by height in meters squared) is one indicator of health. People with very low BMI are considered underweight (which is typically not healthy) and people with very high BMI are considered overweight (which is typically not healthy).