diff --git a/missing_data/missing_data.Rmd b/missing_data/missing_data.Rmd
index b3d2546..f5eaaac 100644
--- a/missing_data/missing_data.Rmd
+++ b/missing_data/missing_data.Rmd
@@ -266,6 +266,8 @@ coef(lower)[2]
Our interval estimate of the ATE using Manski bounds is thus [`r round(coef(lower)[2], 2)`, `r round(coef(upper)[2], 2)`].
+Manski bounds are often relatively wide. See Lee (2009) on how to calculate bounds that may be tighter but require additional assumptions.
+
10. Multiple imputation for missing outcomes allows for point estimation of ATEs, but relies on stronger assumptions than bounding.
==
The methods in #8 and #9 describe methods of single imputation, where a single value is substituted for missing values. In multiple imputation, we impute missing values of the dataset multiple times according to an assumed stochastic data generating process. Different methods for multiple imputation impose different structures and assumptions about the probability distributions governing the data generating processes used to impute missing values. In general, multiple imputation proceeds via three stages:
@@ -282,6 +284,8 @@ References:
==
Gerber, Alan S. and Donald P. Green. (2013). *Field Experiments: Design, Analysis, and Interpretation.* New York: W.W. Norton.
+Lee, David S. (2009). "Training, wages, and sample selection: Estimating sharp bounds on treatment effects." *The Review of Economic Studies* 76 (3): 1071-1102.
+
Lin, Winston, Donald P. Green, and Alexander Coppock. (2016). "Standard Operating Procedures for Don Green's Lab at Columbia." Available at [https://alexandercoppock.com/Green-Lab-SOP/Green_Lab_SOP.html](https://alexandercoppock.com/Green-Lab-SOP/Green_Lab_SOP.html).
Rubin Donald B. (2004). *Multiple Imputation for Nonresponse in Surveys.* New York: John Wiley and Sons.
diff --git a/missing_data/missing_data.html b/missing_data/missing_data.html
index 936f0fb..dc6997c 100644
--- a/missing_data/missing_data.html
+++ b/missing_data/missing_data.html
@@ -572,7 +572,7 @@ 2. Missing treatment or outcome data can limit our ability to
geom_vline(aes(xintercept = med_obs), col = "blue", lty = 2, lwd = 1) +
scale_x_continuous("x") +
theme_minimal()
-
+
Similarly, when we seek to estimate causal effects, some patterns of
missing data can lead to biased estimates of causal effects. In
particular, missingness of the treatment assignment indicator or the
@@ -664,7 +664,7 @@
2. Missing treatment or outcome data can limit our ability to
geom_vline(xintercept = .5, col = "red") +
theme_minimal() +
xlab("ATE Estimates")
-
+
3. The potential for bias increases in the proportion of treatment
@@ -1286,6 +1286,9 @@ 9. We can bound ATEs to account for missing outcome data without
## 0.5
Our interval estimate of the ATE using Manski bounds is thus [0.5,
1.5].
+
Manski bounds are often relatively wide. See Lee (2009) on how to
+calculate bounds that may be tighter but require additional
+assumptions.
10. Multiple imputation for missing outcomes allows for point
@@ -1327,6 +1330,9 @@ 10. Multiple imputation for missing outcomes allows for point
References:
Gerber, Alan S. and Donald P. Green. (2013). Field Experiments:
Design, Analysis, and Interpretation. New York: W.W. Norton.
+
Lee, David S. (2009). “Training, wages, and sample selection:
+Estimating sharp bounds on treatment effects.” The Review of
+Economic Studies 76 (3): 1071-1102.
Lin, Winston, Donald P. Green, and Alexander Coppock. (2016).
“Standard Operating Procedures for Don Green’s Lab at Columbia.”
Available at https://alexandercoppock.com/Green-Lab-SOP/Green_Lab_SOP.html.