Potential Outcomes Framework Flashcards
What is the fundamental problem in causal inference?
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!
What is the Individual Treatment Effect
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
In an Individual Treatment Effect framework what is Y_i0 and Y_i1?
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
What would the Average Treatment Effect equal if we could observe the Individual Treatment Effect for every individual in our population?
The average of all the individual ITEs
ATE = E[ Y_i1 - Y_i0]
What does Rubin’s causal model state?
That it is possible to estimate the ATE when individuals have been randomly assigned to treatment (because both groups should be equal in expectation)
What is the estimated Average Treatment Effect?
The difference between the average outcomes of the sample treatment and the average outcomes of the sample control groups.
How can the estimated ATE be expressed in a regression framework?
E[Y_i | D = 1] = B_0 + B_1
E[Y_i | D = 0] = B_0
What is regression?
A technique used to estimate the conditional expectation function for Y_i given the values of one or more other variables X_ik
What is the population regression function?
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
What does the Conditional Expectation Function equal the Population Regression Function?
When the CEF happens to be linear (unlikely)
When there is joint normality
In saturated regression models
The Population Regression Function is the best (blank) predictor of Y_i given X_i
linear
What is B_1 in the Population Regression Function?
B_1 =
Cov(Y_i, X_i)
/
v(X_i)
Covariance of Y_i and X_i divided by the variance of X_i
What is B_0 in the Population Regression Function?
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
What is a saturated regression model?
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
When will the Population Regression Function slope coefficient have a causal interpretation?
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
What is the counterfactual?
Outcome for an individual in a different state
What is ceteris paribus?
The change in (expected) outcomes across states, holding all else equal
Does the Population Regression Function always represent the Conditional Expectation Function?
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.
What is the omitted variables bias formula?
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
Why can you not test whether the error term is correlated with an included covariate?
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
Why does a randomized experiment ensure that E[ Epislon_i | X ] = 0
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.
Is it a problem if characteristics are not equally distributed between participants in the treatment and control groups in the sample?
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
What is selection bias/omitted variable bias?
Difference in potential outcomes (under no treatment) between those who get treated and those who dont
An estimator is unbiased if
E[ hat_Beta_1 ] = Beta_1
The estimated parameter equals the population parameter
An estimator is efficient (precise) if
It has the lowest variance of all unbiased estimators
