Potential Outcomes Framework

Question 1

What is the fundamental problem in causal inference?

Answer 1

We want to observe the outcomes of each individual under 2 concurrent experimental conditions 1) the treatment condition 2) the counterfactual condition. However it is impossible to observe this!

Question 2

What is the Individual Treatment Effect

Answer 2

If we could observe Y_i1 and Y_i0 for each individual and subtract the difference between the individual’s two outcomes.

ITE = Y_i1 - Yi0

Question 3

In an Individual Treatment Effect framework what is Y_i0 and Y_i1?

Answer 3

Y_i0 = Value of ith individual’s outcome if the individual where assigned to the treatment condition

Y_i1 = value of the ith individual’s outcome if the individual where assigned to the control condition

Question 4

What would the Average Treatment Effect equal if we could observe the Individual Treatment Effect for every individual in our population?

Answer 4

The average of all the individual ITEs

ATE = E[ Y_i1 - Y_i0]

Question 5

What does Rubin’s causal model state?

Answer 5

That it is possible to estimate the ATE when individuals have been randomly assigned to treatment (because both groups should be equal in expectation)

Question 6

What is the estimated Average Treatment Effect?

Answer 6

The difference between the average outcomes of the sample treatment and the average outcomes of the sample control groups.

Question 7

How can the estimated ATE be expressed in a regression framework?

Answer 7

E[Y_i | D = 1] = B_0 + B_1

E[Y_i | D = 0] = B_0

Question 8

What is regression?

Answer 8

A technique used to estimate the conditional expectation function for Y_i given the values of one or more other variables X_ik

Question 9

What is the population regression function?

Answer 9

The line that best fits the population distribution of (Y_i, X_i) in that it minimizes the sum of squared errors in the population

Question 10

What does the Conditional Expectation Function equal the Population Regression Function?

Answer 10

When the CEF happens to be linear (unlikely)
When there is joint normality
In saturated regression models

Question 11

The Population Regression Function is the best (blank) predictor of Y_i given X_i

Answer 11

linear

Question 12

What is B_1 in the Population Regression Function?

Answer 12

B_1 =

Cov(Y_i, X_i)
/
v(X_i)

Covariance of Y_i and X_i divided by the variance of X_i

Question 13

What is B_0 in the Population Regression Function?

Answer 13

B_0 = E[Y_i] - B_1 * E[X]

Average of Y_i minus the average of X_i multiplied by the population slope coefficient

Question 14

What is a saturated regression model?

Answer 14

A regression with discrete explanatory variables where the model includes a separate parameter for every possible combination of values taken on by the explanatory variables

Basically every dummy variable + all possible interactions between all dummy variables

Question 15

When will the Population Regression Function slope coefficient have a causal interpretation?

Answer 15

When the Conditional Expectation Function that it approximates is causal, this is true when the CEF describes bifferenes in the average potential outcomes between the treatment and control groups

Question 16

What is the counterfactual?

Answer 16

Outcome for an individual in a different state

Question 17

What is ceteris paribus?

Answer 17

The change in (expected) outcomes across states, holding all else equal

Question 18

Does the Population Regression Function always represent the Conditional Expectation Function?

Answer 18

No. The CEF tells us how the mean of y varies with X in the population. This relationship may not be linear, so the PRF estimated may not represent the CEF in all cases.

Question 19

What is the omitted variables bias formula?

Answer 19

B_s = B_1 + Pie_1 * Gamma

Pie_1 = the slope of the coefficient from a regression on the omitted variable on the included variable (the relationship between the omitted variable and the other variable in the model)

Gamma = the slope coefficient of the omitted variable in the “long” regression where you do not have omitted variable bias

Question 20

Why can you not test whether the error term is correlated with an included covariate?

Answer 20

Because by construction, OLS models chose an intercept and a slope such that Xi is uncorrelated with the estimated hat_Epislon_i

C( hat_Epsilon_i, x_i ) = 0 mathematically

Question 21

Why does a randomized experiment ensure that E[ Epislon_i | X ] = 0

Answer 21

Because treatment level is distributed independently of any other determinants of the outcome

Random assignment ensures that other characteristics that might influence an outcome are randomly, and equally distributed between participants in the treatment and control groups.

Question 22

Is it a problem if characteristics are not equally distributed between participants in the treatment and control groups in the sample?

Answer 22

No. What is important is that potential members of the treatment and control group are identical on all observable and unobservable dimensions on average in the population

Due to idiosyncrasies in sampling the actual treatment and control groups may differ slightly on observable and unobservable dimensions but such idiosyncrasies are accounted for in the margin of error built into the statistical analysis

Question 23

What is selection bias/omitted variable bias?

Answer 23

Difference in potential outcomes (under no treatment) between those who get treated and those who dont

Question 24

An estimator is unbiased if

Answer 24

E[ hat_Beta_1 ] = Beta_1

The estimated parameter equals the population parameter

Question 25

An estimator is efficient (precise) if

Answer 25

It has the lowest variance of all unbiased estimators