# Session 5 Estimands and Estimators

## 5.1 Core content

Review: A causal effect, \(\tau_i\), is a comparison of unobserved potential outcomes for each unit \(i\): examples \(\tau_{i} = Y_{i}(Z_{i}=1) - Y_{i}(Z_{i}=0)\) or \(\tau_{i} = \frac{Y_{i}(Z_{i}=1)}{ Y_{i}(Z_{i}=0)}\).

To learn about \(\tau_{i}\), we can treat \(\tau_{i}\) as an

**estimand**or target quantity to be estimated (today) or as a target quantity to be hypothesized about (recall session on hypothesis testing).Many focus on the Average Treatment Effect (ATE), \(\bar{\tau}=\sum_{i=1}^n \tau_{i}\), in part, because it allows easy

**estimation**An

**estimator**is a recipe for calculating a guess about the value of an estimand. For example, the difference of observed means for \(m\) treated units is one estimator of \(\bar{\tau}\): \(\hat{\bar{\tau}} = \sum_{i=1}^n (Z_i Y_i)/m - \sum_{i=1}^n ( ( 1 - Z_i) Y_i)/(n-m)\).Different randomizations will produce different values of the same estimator targeting the same estimand. A

**standard error**summarizes this variability in an estimator.A \(100(1-\alpha)\)%

**confidence interval**is a collection of hypotheses that cannot be rejected at the \(\alpha\) level. We tend to report confidence intervals containing hypotheses about values of our estimand and use our estimator as a test statistic.**Estimators should**(1) avoid systematic error in their guessing of the estimand (be unbiased); (2) vary little in their guesses from experiment to experiment (be precise or efficient); and perhaps ideally (3) converge to the estimand as they use more and more information (be consistent).**Analyze as you randomize**in the context of estimation means that (1) our standard errors should measure variability from randomization and (2) our estimators should target estimands defined in terms of potential outcomes.We do not

**control for**background covariates when we analyze data from randomized experiments. But covariates can make our estimation more**precise**. This is called**covariance adjustment**.**Covariance adjustment**in randomized experiments differs from controlling for in observational studis.

## 5.2 Slides

Below are slides with the core content that we cover in this session.

## 5.3 Resources

- EGAP Methods Guide 10 types treatment effect you should know about
- Chapter 3 in Gerber and Green (2012)

## 5.4 Quizzes and Exercises

## 5.5 Examples

### References

Gerber, Alan S., and Donald P. Green. 2012. *Field Experiments: Design, Analysis, and Interpretation*. New York, NY: W. W. Norton & Company.