{smcl}
{title:Title}
{phang}{cmd:aprub} {hline 2} Estimate the upper bound on the average persuasion rate
{title:Syntax}
{p 8 8 2} {cmd:aprub} {it:depvar} {it:treatrvar} {it:instrvar} [{it:covariates}] [{it:if}] [{it:in}] [, {cmd:model}({it:string}) {cmd:title}({it:string})]
{p 4 4 2}{bf:Options}
{col 5}{it:option}{col 24}{it:Description}
{space 4}{hline 44}
{col 5}{cmd:model}({it:string}){col 24}Regression model when {it:covariates} are present
{col 5}{cmd:title}({it:string}){col 24}Title
{space 4}{hline 44}
{title:Description}
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{bf:aprub} estimates the upper bound on the average persuasion rate (APR).
{it:varlist} should include {it:depvar} {it:treatrvar} {it:instrvar} {it:covariates} in order.
Here, {it:depvar} is binary outcomes ({it:y}), {it:treatrvar} is binary treatment ({it:t}),
{it:instrvar} is binary instruments ({it:z}), and {it:covariates} ({it:x}) are optional.
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There are two cases: (i) {it:covariates} are absent and (ii) {it:covariates} are present.
{break} - Without {it:x}, the upper bound ({cmd:theta_U}) on the APR is defined by
{cmd:theta_U} = {E[{it:A}|{it:z}=1] - E[{it:B}|{it:z}=0]}/{1 - E[{it:B}|{it:z}=0]},
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where {it:A} = 1({it:y}=1,{it:t}=1)+1-1({it:t}=1) and
{it:B} = 1({it:y}=1,{it:t}=0).
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The estimate and its standard error are obtained by the following procedure:
{break} 1. E[{it:A}|{it:z}=1] is estimated by regressing {it:A} on {it:z}.
{break} 2. E[{it:B}|{it:z}=0] is estimated by regressing {it:B} on {it:z}.
{break} 3. {cmd:theta_U} is computed using the estimates obtained above.
{break} 4. The standard error is computed via STATA command {bf:nlcom}.
{break} - With {it:x}, the upper bound ({cmd:theta_U}) on the APR is defined by
{cmd:theta_U} = E[{cmd:theta_U_num}({it:x})]/E[{cmd:theta_U_den}({it:x})],
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where
{cmd:theta_U_num}({it:x}) = E[{it:A}|{it:z}=1,{it:x}] - E[{it:B}|{it:z}=0,{it:x}]
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and
{cmd:theta_U_den}({it:x}) = 1 - E[{it:B}|{it:z}=0,{it:x}].
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The estimate is obtained by the following procedure.
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If {cmd:model}("no_interaction") is selected (default choice),
{break} 1. E[{it:A}|{it:z}=1,{it:x}] is estimated by regressing {it:A} on {it:z} and {it:x}.
{break} 2. E[{it:B}|{it:z}=0,{it:x}] is estimated by regressing {it:B} on {it:z} and {it:x}.
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Alternatively, if {cmd:model}("interaction") is selected,
{break} 1. E[{it:A}|{it:z}=1,{it:x}] is estimated by regressing {it:A} on {it:x} given {it:z} = 1.
{break} 2. E[{it:B}|{it:z}=0,{it:x}] is estimated by regressing {it:B} on {it:x} given {it:z} = 0.
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Ater step 1, both options are followed by:
{p 4 8 2}3. For each {it:x} in the estimation sample, {cmd:theta_U_num}({it:x}) and {cmd:theta_U_den}({it:x}) are evaluated.
{p 4 8 2}4. The estimates of {cmd:theta_U_num}({it:x}) and {cmd:theta_U_den}({it:x}) are averaged to estimate {cmd:theta_U}.
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When {it:covariates} are present, the standard error is missing because an analytic formula for the standard error is complex.
Bootstrap inference is implemented when this package{c 39}s command {bf:persuasio} is called to conduct inference.
{title:Options}
{cmd:model}({it:string}) specifies a regression model.
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This option is only relevant when {it:x} is present.
The dependent variable is
either {it:A} or {it:B}.
The default option is "no_interaction" between {it:z} and {it:x}.
When "interaction" is selected, full interactions between {it:z} and {it:x} are allowed.
{cmd:title}({it:string}) specifies a title.
{title:Remarks}
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It is recommended to use this package{c 39}s command {bf:persuasio} instead of calling {bf:aprub} directly.
{title:Examples}
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We first call the dataset included in the package.
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. use GKB_persuasio, clear
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The first example estimates the upper bound on the APR without covariates.
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. aprub voteddem_all readsome post
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The second example adds a covariate.
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. aprub voteddem_all readsome post MZwave2
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The third example estimates the upper bound by the covariate.
. by MZwave2,sort: aprub voteddem_all readsome post
{title:Stored results}
{p 4 4 2}{bf:Scalars}
{p 8 8 2} {bf:e(N)}: sample size
{p 8 8 2} {bf:e(ub_coef)}: estimate of the upper bound on the average persuasion rate
{p 8 8 2} {bf:e(ub_se)}: standard error of the upper bound on the average persuasion rate
{p 4 4 2}{bf:Macros}
{p 8 8 2} {bf:e(outcome)}: variable name of the binary outcome variable
{p 8 8 2} {bf:e(treatment)}: variable name of the binary treatment variable
{p 8 8 2} {bf:e(instrument)}: variable name of the binary instrumental variable
{p 8 8 2} {bf:e(covariates)}: variable name(s) of the covariates if they exist
{p 8 8 2} {bf:e(model)}: regression model specification ("no_interaction" or "interaction")
{p 4 4 2}{bf:Functions:}
{p 8 8 2} {bf:e(sample)}: 1 if the observations are used for estimation, and 0 otherwise.
{title:Authors}
{p 4 4 2}
Sung Jae Jun, Penn State University,
{p 4 4 2}
Sokbae Lee, Columbia University,
{title:License}
{p 4 4 2}
GPL-3
{title:References}
{p 4 4 2}
Sung Jae Jun and Sokbae Lee (2022),
Identifying the Effect of Persuasion,
{browse "https://arxiv.org/abs/1812.02276":arXiv:1812.02276 [econ.EM]}
{title:Version}
{p 4 4 2}
0.2.1 20 November 2022