I got a copy of the JMP by Jann Spiess in December to look at sometime. I had to look at multiple versions because it kept on being updated, even until the end of flyout season in February. Here are some of the versions.

November 29, 2017 version: https://www.gsb.stanford.edu/sites/gsb/files/jmp_jann-spiess.pdf

December 8, 2017 version: https://economics.yale.edu/sites/default/files/jmp_3.pdf

February 4, 2018 version: https://scholar.harvard.edu/files/spiess/files/spiess_jmp_feb4.pdf

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I was reading the intro to Spiess's JMP (December 8, 2017 version) and noticed in footnote 5 a reference to the "leave-one-out potential outcomes" (LOOP) estimator from Wu and Gagnon-Bartsch (WGB). I was curious to learn more since I am not familiar with these topics and noticed the title of the paper "The LOOP Estimator: Adjusting for Covariates in Randomized Experiments" which sounded highly relevant for Spiess but was only cited in footnotes and was not mentioned anywhere in the literature review section.

I looked through WGB [http://www-personal.umich.edu/~johanngb/working/loop.pdf] and in particular read through section 3.

When I turned back to Spiess's section 1 ("A Simple Example"), I noticed that what I was reading sounded very familiar.

First is notation (WGB --> Spiess):

i's treatment assignment: X_i --> d_i

assume treatment is assigned randomly: p_i = P(X_i = 1) --> p = P(d_i = 1)

potential outcome if i is assigned to treatment: t_i --> y_i(1)

potential outcome if i is assigned to control: c_i --> y_i(0)

individual treatment effect / unit-level causal effect: tau_i --> tau_i

Next is the observation that it is "not possible to observe any single participant's treatment effect, because for each participant we are only able to observe the treatment response or the control response" (WGB) compared to "We do not observe both potential outcomes for one unit simultaneously, but observe only the treatment status and the realized outcome" (Spiess).

Next, "Less well known is the fact that it is also possible to provide an unbiased estimate of an individual participant's treatment effect. For example, Y_i*U_i is one such estimator" (WGB) compared to "we can still obtain an unbiased estimate of the unit treatment effect. Indeed, I will argue below that (d_i-p)/(p(1-p))y is an unbiased estimator for tau_i" (Spiess). Compare the equation at the top of Section 1.3 in Spiess to Equation (9) in WGB. The line following that equation in Spiess shows why it's unbiased, identical to Equation (10) and Equation (11) in WGB.

Next, "Although this does result in an unbiased estimator of tau_i, it is clearly useless for all practical purposes. A more sanguine way of putting this would be that the estimator, despite being unbiased, likely has very high variance" (WGB) compared to "But this estimator can have very high variance" (Spiess).

Next, "As an alternative estimator of tau_i, consider [Equation (12)]" (WGB) compared to "We can modify this estimator by regression adjustments y-hat to obtain [Equation (1)]" (Spiess). The equations are again the same, with the adjustment called m-hat in WGB and y-hat in Spiess.

Next, "if m-hat is independent of the i-th participant's treatment assignment - then tau_i-hat is an unbiased estimator of tau" (WGB) compared to "As long as y_i-hat only uses information from x_i and z_-i and not the outcome y_i or treatment assignment d_i, tau_i-hat will still be unbiased" (Spiess).

Next, "The advantage of this estimator is that it will have a low variance as long as m-hat ~= m" (WGB) compared to "Which regressio...See full post