{smcl} {title:Title} {phang}{cmd:persuasio4yz} {hline 2} Conduct causal inference on persuasive effects for binary outcomes {it:y} and binary instruments {it:z} {title:Syntax} {p 8 8 2} {cmd:persuasio4yz} {it:depvar} {it:instrvar} [{it:covariates}] [{it:if}] [{it:in}] [, {cmd:level}(#) {cmd:model}({it:string}) {cmd:method}({it:string}) {cmd:nboot}(#) {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:level}(#){col 24}Set confidence level; default is {cmd:level}(95) {col 5}{cmd:model}({it:string}){col 24}Regression model when {it:covariates} are present {col 5}{cmd:method}({it:string}){col 24}Inference method; default is {cmd:method}("normal") {col 5}{cmd:nboot}(#){col 24}Perform # bootstrap replications {col 5}{cmd:title}({it:string}){col 24}Title {space 4}{hline 44} {title:Description} {cmd:persuasio4yz} conducts causal inference on persuasive effects. {p 4 4 2} It is assumed that binary outcomes {it:y} and binary instruments {it:z} are observed. This command is for the case when persuasive treatment ({it:t}) is unobserved, using an estimate of the lower bound on the average persuasion rate (APR) via this package{c 39}s command {cmd:aprlb}. {p 4 4 2} {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} When treatment {it:t} is unobserved, the upper bound on the APR is simply 1. {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 confidence interval 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} 4. Then, a confidence interval for the APR is set by {p 8 8 2} [ {it:est} - {it:cv} * {it:se} , 1 ], {p 4 4 2} where {it:est} is the estimate, {it:se} is the standard error, and {it:cv} is the one-sided standard normal critical value (e.g., {it:cv} = 1.645 for {cmd:level}(95)). {break} - With {it:x}, the lower bound ({cmd:theta_L}) on the APR is defined by {cmd:theta_L} = E[{cmd:theta_L}(x)], {p 4 4 2} where {cmd:theta_L}(x) = {Pr({it:y}=1|{it:z}=1,{it:x}) - Pr({it:y}=1|{it:z}=0,{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}. {break} 4. A bootstrap confidence interval for the APR is set by {p 8 8 2} [ bs_est({it:alpha}) , 1 ], {p 4 4 2} where bs_est({it:alpha}) is the {it:alpha} quantile of the bootstrap estimates of {cmd:theta_L} and 1 - {it:alpha} is the confidence level. {break} {p 4 4 2} The bootstrap procedure is implemented via STATA command {cmd:bootstrap}. {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:level}(#) sets confidence level; default is {cmd:level}(95). {cmd:method}({it:string}) refers the method for inference. {p 4 4 2} The default option is {cmd:method}("normal"). By the naure of identification, one-sided confidence intervals are produced. {p 4 8 2}1. When {it:x} is present, it needs to be set as {cmd:method}("bootstrap"); otherwise, the confidence interval will be missing. {p 4 8 2}2. When {it:x} is absent, both options yield non-missing confidence intervals. {cmd:nboot}(#) chooses the number of bootstrap replications. {p 4 4 2} The default option is {cmd:nboot}(50). It is only relevant when {cmd:method}("bootstrap") is selected. {cmd:title}({it:string}) specifies a title. {title:Remarks} {p 4 4 2} It is recommended to use {cmd:nboot}(#) with # at least 1000. A default choice of 50 is meant to check the code initially because it may take a long time to run the bootstrap part. The bootstrap confidence interval is based on percentile bootstrap. Normality-based bootstrap confidence interval is not recommended because bootstrap standard errors can be unreasonably large in applications. {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 conducts inference on the APR without covariates, using normal approximation. {p 4 4 2} . persuasio4yz voteddem_all post, level(80) method("normal") {p 4 4 2} The second example conducts bootstrap inference on the APR. {p 4 4 2} . persuasio4yz voteddem_all post, level(80) method("bootstrap") nboot(1000) {p 4 4 2} The third example conducts bootstrap inference on the APR with a covariate, MZwave2, interacting with the instrument, post. {p 4 4 2} . persuasio4yz voteddem_all post MZwave2, level(80) model("interaction") method("bootstrap") nboot(1000) {title:Stored results} {p 4 4 2}{bf:Matrices} {p 8 8 2} {bf:e(apr_est)}: (1*2 matrix) bounds on the average persuasion rate in the form of [lb, 1] {p 8 8 2} {bf:e(apr_ci)}: (1*2 matrix) confidence interval for the average persuasion rate in the form of [lb_ci, 1] {p 4 4 2}{bf:Macros} {p 8 8 2} {bf:e(cilevel)}: confidence level {p 8 8 2} {bf:e(inference_method)}: inference method: "normal" or "bootstrap" {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