Question

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).

1. List a few other numerical measures of health that one could use as Xi in this regression.

2. Suppose we use BMI as our measure of Xi . What is the interpretation of β0? What is the interpretation of β1? (Remember, Yi represents the income of person i – let’s assume this is measured in dollars).

Answer 1

1. There are several other measures of health that can be used instead of BMI, such as mBMI (modified body mass index), BMI Prime (another modification of BMI) and SBSI (surface-based body shaped index, which is a function of height, waist circumference, body surface area and vertical trunk circumference). All these would be at least interval scale variable, and can be used instead of the above explanatory variable.

2. The intercept coefficient $\dpi{80} \beta_0$ is basically the average $\dpi{80} Y_i$ (income in this case) when the $\dpi{80} X_i$ (BMI in this case) is equal to zero. The intercept may or may not make sense depending on the model. In this case, neither body mass can be zero, nor the height can be infinite, and hence, the explanatory variable can not be equal to zero, and hence, the intercept for this particular case does not makes sense.

The slope coefficient $\dpi{80} \beta_1$ is the change in the average $\dpi{80} Y_i$ (income in this case) for a unit change in the $\dpi{80} X_i$ (BMI in this case). This means, if $\dpi{80} X_i$ increases by one unit, $\dpi{80} Y_i$ would increase by $\dpi{80} \beta_1$ units (income in dollars).