The Effects of Sanction Intensity on Criminal Conduct - JDAI Helpdesk
The Effects of Sanction Intensity on Criminal Conduct - JDAI Helpdesk
The Effects of Sanction Intensity on Criminal Conduct - JDAI Helpdesk
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equati<strong>on</strong>s. <str<strong>on</strong>g>The</str<strong>on</strong>g> first stage equati<strong>on</strong> uses the instrument variables to predict treatment<br />
take-up (the endogenous variable):<br />
T^<br />
= " 0<br />
+ " TA<br />
+ " REGION<br />
+ " GENDER<br />
+ " RACE<br />
+ " AGE<br />
+ " SES<br />
+ " PRIORS<br />
+ " TAR<br />
!<br />
!<br />
!<br />
where<br />
+" TA<br />
* " REGION<br />
+ " TA<br />
* " GENDER<br />
+ " TA<br />
* " RACE<br />
+ " TA<br />
* " AGE<br />
+ " TA<br />
* " SES<br />
+ " TA<br />
* " PRIORS<br />
T^<br />
is the endogenous variable (predicted treatment take-up), " 0<br />
is the intercept,<br />
" TA<br />
is the instrument for assigned treatment, the interacti<strong>on</strong>s between " TA<br />
and the<br />
!<br />
subgroup variables are the additi<strong>on</strong>al instruments, and " TAR<br />
is a c<strong>on</strong>trol (exogenous)<br />
!<br />
variable for post-RA time at risk. Note that <strong>on</strong>ly the interacti<strong>on</strong>s and not the main effects<br />
variables for our subgroups are used as instruments.<br />
!<br />
We hypothesize that race, gender,<br />
etc. predict treatment take-up through their associati<strong>on</strong> with treatment assignment. In the<br />
sec<strong>on</strong>d stage equati<strong>on</strong>, we replace the instruments with the predicted treatment take-up<br />
from the first stage ( "T^ ) to predict the crime outcome Y:<br />
Y = " 0<br />
+ "T^ + " TAR<br />
+ " REGION<br />
+ " GENDER<br />
+ " RACE<br />
+ " AGE<br />
+ " SES<br />
+ " PRIORS<br />
!<br />
<str<strong>on</strong>g>The</str<strong>on</strong>g> main effects for the subgroups remain in the sec<strong>on</strong>d stage model as c<strong>on</strong>trols for any<br />
direct ! variati<strong>on</strong> in outcomes by subgroup. Another important part <str<strong>on</strong>g>of</str<strong>on</strong>g> the 2SLS approach<br />
is the estimati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the ‘reduced form,’ which is simply the OLS estimate <str<strong>on</strong>g>of</str<strong>on</strong>g> the ITT<br />
effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the instrument and exogenous covariates <strong>on</strong> crime. Angrist (2006) notes that it<br />
is acceptable to use OLS even if the outcome variable is dichotomous.<br />
<str<strong>on</strong>g>The</str<strong>on</strong>g> coefficient for<br />
"T^ tells us the actual effect <str<strong>on</strong>g>of</str<strong>on</strong>g> treatment received, or local<br />
average treatment effect (LATE) <strong>on</strong> the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> re<str<strong>on</strong>g>of</str<strong>on</strong>g>fending. It can be compared<br />
with the coefficient ! for assigned treatment in the reduced form model (or the outcomes<br />
from our logistic participati<strong>on</strong> model, as described above) to assess whether the estimated<br />
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