All the (variables) we cannot see
Stella wants to run the following regression to estimate the effects of schooling on wages:
wages_i = B0 + B1 educ_i + e_i
What is a reason why B1 could be biased?
Explains: (i) omitted variable (ii) the relationship to educ and wages and (iii) the resulting direction of bias.
What is a_i in the following regression model?
behavioral_health_i = B0 + B1 fed_housing_i + a_i + e_i
... where i indexes an individual between the ages of 12-16.
Individual fixed effect.
What transformation does the following equation implement?
dot(y)_it = y_it - bar(y)_i
(Write this on the board, Maddy!)
Within-transformation.
Stella wants to study the effect of snow on the number of bunnies in a given park:
bunnies_i = B0 + B1 snow_i + e_i
What are reasons why B1 might be biased?
You get the drill, folks.
What is the within-transformation?
Demeaning data on a certain dimension, in order to estimate a fixed effect model.
What transformation is the following equation implementing?
dot(y)_it = y_{i, t+1} - y_{i, t}
(Maddy, board again!)
First differencing.
Stella wants to study the effect of a solar subsidy on the number of people who purchase solar panels:
adopters_i = B0 + B1 subsidy_i + a_i + e_i
... where i indexes a state. What is a reason why B1 might be biased?
Same as before :)
Frannie (dog) estimates the following regression,
y_it = B0 + B1 X_it + a_it + e_it
What would you need to assume in order to interpret B1 as the causal effect of X on y?
No remaining omitted variables that are not captured by a_i.
What concept are the following equations related to?
Long regression: Y = B0 + B1 X + B2 Z + e
Short regression: Y = H0 + H1 X + nu
(Also write on the board as needed!)
Omitted variable bias.
Billie wants to run the following regression to estimate the effect of a policy that weakens labor protections on the number of employees a firm hires:
employees_it = B0 + B1 policy_t + a_i + e_it
... where i indexes a firm and t indexes a year. What is a source of omitted variable bias?
Same three criteria!
Stella wants to study the effect of a solar subsidy on the number of people who purchase solar panels:
adopters_i = B0 + B1 subsidy_i + a_i + e_i
... where i indexes a state.
Name at least one factor that could be captured by the fixed effect, a_i.
State-specific factors that affect solar adoption that are constant in time. For example, the amount of sunlight.
What research design or method are the following two equations related to?
First stage: D = B0 + B1 Z + e
Second stage: Y = B0 + B1 hat(D) + eTwo-stage least squares or instrumental variables.
Groot is interested in how a chemical pollutant affects children's health outcomes,
health_it = B0 + B1 pollutant_exposure_it + a_i + b_t + e_it
where i indexes a child and t indexes a year. What is a source of omitted variable bias in this model?
Two-way fixed effects! Tricky! But you can do it!
bunnies_i = B0 + B1 snow_i + a_i + e_i
1. First differences
2. Within transformation
3. Dummy variables
What is this equation?
(X^T X)^(-1) (X^T Y)
(Board as needed!)OLS coefficient estimates!