In This Set of Notes We’re Going to

  1. Define the Parameter of Interest
  2. Show why it’s important to define the Parameter of Interest in a Paper using Chetty’s MTO paper as an example

Introduction

One way to think about a Principles of Microeconomic class is that you learn how to articulate an idea, express it mathematically, and visualize it. The thinking is that if you can express an idea in all three ways, you have a pretty good understanding of the it.

Example: Comparative Advantage

  1. Country A has a comparative advantage in a good if its opportunity cost of producing that good is lower than Country B
$$
\\frac{\\text{Good \\#2 per Good \\#1 for Country A}}{\\text{Good \\#2 per Good \\#1 for Country B}} < 1
$$
  1. Visual (To Do)

Given its importance in a paper, the Parameter of Interest really ought to be expressed in all three ways as well. To Do: Add a part about how while the “Population” may be observed like for instance if we’re interested in the Average Effect of a minimum wage on state unemployment — we observe all of the states, how the distribution, $\mathbb{P} \circ \tau{-1}$ is never observed!

As a working example, let’s consider Raj Chetty’s work on Moving to Opportunity. The authors “hypothesize that the gains from moving to a lower-poverty area decline with a child’s age at move.” The language is admittedly a bit ambiguous. Is it that moving families with younger children out of public housing will have a larger effect than moving families with older children?

$$ \frac{d}{da}\mathbb{E}[\tau_i \vert \text{Age}_i = a] \leq 0 $$

Or, is it that moving a child out of public housing at a younger age is more beneficial than moving that same child out of public housing at an older age?

$$ \frac{d}{da}\mathbb{E}[\tau_i(a)] \leq 0 $$

A key way to differentiate between the two is to consider whether age should be thought of a covariate, as in the former, or a mechanism as in the latter. That is, are we interested in how the treatment effect varies by age where age is “fixed” for each person, or are we interested in understanding how the treatment effect varies by age where we allow for the age to vary for each person.

Covariate

Covariate

Mechanism

Mechanism

These address different policy questions. The covariate problem relates to the question of targeting. If policy makers could move certain families, who would benefit the most? The mechanism problem is more appropriate to the question of when should a family move. It is better to move when they’re younger, or older?

Covariate

To understand how the effect varies by age, we can estimate the average difference in outcome across individuals with the same age, and then compare these differences across ages.

$$ \mathbb{E}[Y_i \vert D_i=1, A_i=a] - \mathbb{E}[Y_i \vert D_i=0, A_i=a] = \mathbb{E}[\tau_i \vert A_i=a] $$