{smcl} {title:Title} {phang}{cmd:aprlb} {hline 2} Estimate the lower bound on the average persuasion rate {title:Syntax} {p 8 8 2} {cmd:aprlb} {it:depvar} {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} {p 4 4 2} {bf:aprlb} estimates the lower bound on the average persuasion rate (APR). {it:varlist} should include {it:depvar} {it:instrvar} {it:covariates} in order. Here, {it:depvar} is binary outcomes ({it:y}), {it:instrvar} is binary instruments ({it:z}), and {it:covariates} ({it:x}) are optional. {p 4 4 2} There are two cases: (i) {it:covariates} are absent and (ii) {it:covariates} are present. {break} - Without {it:x}, the lower bound ({cmd:theta_L}) on the APR is defined by {cmd:theta_L} = {Pr({it:y}=1|{it:z}=1) - Pr({it:y}=1|{it:z}=0)}/{1 - Pr({it:y}=1|{it:z}=0)}. {p 4 4 2} The estimate and its standard error are obtained by the following procedure: {break} 1. Pr({it:y}=1|{it:z}=1) and Pr({it:y}=1|{it:z}=0) are estimated by regressing {it:y} on {it:z}. {break} 2. {cmd:theta_L} is computed using the estimates obtained above. {break} 3. The standard error is computed via STATA command {bf:nlcom}. {break} - With {it:x}, the lower bound ({cmd:theta_L}) on the APR is defined by {cmd:theta_L} = E[{cmd:theta_L_num}({it:x})]/E[{cmd:theta_L_den}({it:x})], {p 4 4 2} where {cmd:theta_L_num}({it:x}) = Pr({it:y}=1|{it:z}=1,{it:x}) - Pr({it:y}=1|{it:z}=0,{it:x}) {p 4 4 2} and {cmd:theta_L_den}({it:x}) = 1 - Pr({it:y}=1|{it:z}=0,{it:x}). {p 4 4 2} The estimate is obtained by the following procedure. {p 4 4 2} If {cmd:model}("no_interaction") is selected (default choice), {break} 1. Pr({it:y}=1|{it:z},{it:x}) is estimated by regressing {it:y} on {it:z} and {it:x}. {p 4 4 2} Alternatively, if {cmd:model}("interaction") is selected, {break} 1a. Pr({it:y}=1|{it:z}=1,{it:x}) is estimated by regressing {it:y} on {it:x} given {it:z} = 1. {break} 1b. Pr({it:y}=1|{it:z}=0,{it:x}) is estimated by regressing {it:y} on {it:x} given {it:z} = 0. {p 4 4 2} Ater step 1, both options are followed by: {p 4 8 2}2. For each {it:x} in the estimation sample, {cmd:theta_L_num}({it:x}) and {cmd:theta_L_den}({it:x}) are evaluated. {p 4 8 2}3. The estimates of {cmd:theta_L_num}({it:x}) and {cmd:theta_L_den}({it:x}) are averaged to estimate {cmd:theta_L}. {p 4 4 2} 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 of {it:y} on {it:z} and {it:x}. {p 4 4 2} This option is only relevant when {it:x} is present. 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; this is accomplished by estimating Pr({it:y}=1|{it:z}=1,{it:x}) and Pr({it:y}=1|{it:z}=0,{it:x}), separately. {cmd:title}({it:string}) specifies a title. {title:Remarks} {p 4 4 2} It is recommended to use this package{c 39}s command {bf:persuasio} instead of calling {bf:aprlb} directly. {title:Examples} {p 4 4 2} We first call the dataset included in the package. {p 4 4 2} . use GKB_persuasio, clear {p 4 4 2} The first example estimates the lower bound on the APR without covariates. {p 4 4 2} . aprlb voteddem_all post {p 4 4 2} The second example adds a covariate. {p 4 4 2} . aprlb voteddem_all post MZwave2 {p 4 4 2} The third example estimates the lower bound by the covariate. . by MZwave2, sort: aprlb voteddem_all 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(lb_coef)}: estimate of the lower bound on the average persuasion rate {p 8 8 2} {bf:e(lb_se)}: standard error of the lower 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(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