/*** Title ----- {phang}{cmd:aprlb} {hline 2} Estimate the lower bound on the average persuasion rate Syntax ------ > {cmd:aprlb} _depvar_ _instrvar_ [_covariates_] [_if_] [_in_] [, {cmd:model}(_string_) {cmd:title}(_string_)] ### Options | _option_ | _Description_ | |-------------------|-------------------------| | {cmd:model}(_string_) | Regression model when _covariates_ are present | | {cmd:title}(_string_) | Title | Description ----------- __aprlb__ estimates the lower bound on the average persuasion rate (APR). _varlist_ should include _depvar_ _instrvar_ _covariates_ in order. Here, _depvar_ is binary outcomes (_y_), _instrvar_ is binary instruments (_z_), and _covariates_ (_x_) are optional. There are two cases: (i) _covariates_ are absent and (ii) _covariates_ are present. - Without _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)}. The estimate and its standard error are obtained by the following procedure: 1. Pr({it:y}=1|{it:z}=1) and Pr({it:y}=1|{it:z}=0) are estimated by regressing _y_ on _z_. 2. {cmd:theta_L} is computed using the estimates obtained above. 3. The standard error is computed via STATA command __nlcom__. - With _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})], 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}) and {cmd:theta_L_den}({it:x}) = 1 - Pr({it:y}=1|{it:z}=0,{it:x}). The estimate is obtained by the following procedure. If {cmd:model}("no_interaction") is selected (default choice), 1. Pr({it:y}=1|{it:z},{it:x}) is estimated by regressing _y_ on _z_ and _x_. Alternatively, if {cmd:model}("interaction") is selected, 1a. Pr({it:y}=1|{it:z}=1,{it:x}) is estimated by regressing _y_ on _x_ given _z_ = 1. 1b. Pr({it:y}=1|{it:z}=0,{it:x}) is estimated by regressing _y_ on _x_ given _z_ = 0. Ater step 1, both options are followed by: {p 4 8 2}2. For each _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}. When _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's command __persuasio__ is called to conduct inference. Options ------- {cmd:model}(_string_) specifies a regression model of _y_ on _z_ and _x_. This option is only relevant when _x_ is present. The default option is "no_interaction" between _z_ and _x_. When "interaction" is selected, full interactions between _z_ and _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}(_string_) specifies a title. Remarks ------- It is recommended to use this package's command __persuasio__ instead of calling __aprlb__ directly. Examples -------- We first call the dataset included in the package. . use GKB_persuasio, clear The first example estimates the lower bound on the APR without covariates. . aprlb voteddem_all post The second example adds a covariate. . aprlb voteddem_all post MZwave2 The third example estimates the lower bound by the covariate. . by MZwave2, sort: aprlb voteddem_all post Stored results -------------- ### Scalars > __e(N)__: sample size > __e(lb_coef)__: estimate of the lower bound on the average persuasion rate > __e(lb_se)__: standard error of the lower bound on the average persuasion rate ### Macros > __e(outcome)__: variable name of the binary outcome variable > __e(instrument)__: variable name of the binary instrumental variable > __e(covariates)__: variable name(s) of the covariates if they exist > __e(model)__: regression model specification ("no_interaction" or "interaction") ### Functions: > __e(sample)__: 1 if the observations are used for estimation, and 0 otherwise. Authors ------- Sung Jae Jun, Penn State University, Sokbae Lee, Columbia University, License ------- GPL-3 References ---------- Sung Jae Jun and Sokbae Lee (2022), Identifying the Effect of Persuasion, [arXiv:1812.02276 [econ.EM]](https://arxiv.org/abs/1812.02276) Version ------- 0.2.1 20 November 2022 ***/ capture program drop aprlb program aprlb, eclass sortpreserve byable(recall) version 16.1 syntax varlist (min=2) [if] [in] [, model(string) title(string)] marksample touse gettoken Y varlist_without_Y : varlist gettoken Z X : varlist_without_Y quietly levelsof `Y' if "`r(levels)'" != "0 1" { display "`Y' is not a 0/1 variable" error 450 } quietly levelsof `Z' if "`r(levels)'" != "0 1" { display "`Z' is not a 0/1 variable" error 450 } display " " display as text "{hline 65}" display "{bf:aprlb:} Estimating the Lower Bound on the Average Persuasion Rate" display as text "{hline 65}" display " " display " - Binary outcome: `Y'" display " - Binary instrument: `Z'" display " - Covariates (if exist): `X'" display " " * if there are no covariates (X) if "`X'" == "" { quietly reg `Y' `Z' if `touse', robust local nobs = e(N) quietly nlcom lower_bound:_b[`Z']/(1-_b[_cons]) tempname b V lb se matrix `b' = r(b) matrix `V' = r(V) ereturn post `b' `V', obs(`nobs') esample(`touse') ereturn display, nopv scalar `lb' = r(table)[1,1] scalar `se' = r(table)[2,1] display " " display "Note: It is recommended to use {bf:persuasio} for causal inference." display " " ereturn scalar lb_coef = `lb' ereturn scalar lb_se = `se' ereturn local outcome `Y' ereturn local instrument `Z' } * if there are covariates (X) if "`X'" != "" { tempvar yhat yhat1 yhat0 thetahat_num thetahat_den thetahat if "`model'" == "" | "`model'" == "no_interaction" { quietly reg `Y' `Z' `X' if `touse', robust tempname bhat b_coef matrix `bhat' = e(b) scalar `b_coef' = `bhat'[1,1] quietly predict `yhat' if `touse' gen `yhat1' = `yhat' + `b_coef' - `b_coef'*`Z' gen `yhat0' = `yhat' - `b_coef'*`Z' } if "`model'" == "interaction" { quietly { reg `Y' `X' if `Z'==1 & `touse', robust predict `yhat1' if `touse' reg `Y' `X' if `Z'==0 & `touse', robust predict `yhat0' if `touse' } } quietly replace `yhat1' = min(max(`yhat1',0),1) quietly replace `yhat0' = min(max(`yhat0',0),1) gen `thetahat_num' = `yhat1' - `yhat0' gen `thetahat_den' = 1 - `yhat0' tempname lb_num lb_den quietly sum `thetahat_num' if `touse' scalar `lb_num' = r(mean) quietly sum `thetahat_den' if `touse' scalar `lb_den' = r(mean) local nobs = r(N) tempname b lb se scalar `lb' = `lb_num'/`lb_den' scalar `se' = . matrix `b' = `lb_num'/`lb_den' matrix colnames `b' = lower_bound ereturn post `b', obs(`nobs') esample(`touse') ereturn display, nopv display " " display "Notes: It is recommended to use {bf:persuasio} for causal inference." display " Standard errors are missing if covariates are present." display " " ereturn scalar lb_coef = `lb' ereturn scalar lb_se = `se' ereturn local outcome `Y' ereturn local instrument `Z' ereturn local covariates `X' ereturn local model `model' } display "Reference: Jun and Lee (2022), arXiv:1812.02276 [econ.EM]" end