Chapter 2 Jackknife Instrument Variable Estimator
We can use the Jackknife Instrument Variable Estimator (JIVE) for Judget Fixed Effects.
First we will start by installing the Jackknife IV estimator.
Next we will load our data from Stevenson (2018) on guilt verdicts and cash bail/pretrail detention.
Next, we will set up some global macros for sets of covariates.
*Set up global macros
global judge_pre judge_pre_1 judge_pre_2 judge_pre_3 judge_pre_4 judge_pre_5 judge_pre_6 judge_pre_7 judge_pre_8
global demo black age male white
global off fel mis sum F1 F2 F3 F M1 M2 M3 M
global prior priorCases priorWI5 prior_felChar prior_guilt onePrior threePriors
global control2 day day2 day3 bailDate t1 t2 t3 t4 t5 t62.1 Naive OLS
We will estimate a couple specifications using an OLS estimator. Our first model will have minimal controls for time variables, but our second model will have additional controls for demographics and prior justice involvement.
\[ guilt_i = \delta_0 +\delta_1 PretrailDentention_i + \gamma'Dates_i + \beta'Demo_i + \alpha'Crime_i + \varepsilon_i \]
eststo minimal: quietly reg guilt jail3 $control2, robust
eststo maximum: quietly reg guilt jail3 possess robbery DUI1st drugSell aggAss $demo $prior $off $control2 , robust
esttab minimal maximum, mtitle("Minimal" "Max Controls") keep(jail3) (1) (2)
Minimal Max Controls
--------------------------------------------
jail3 -0.000735 0.0292***
(-0.42) (15.23)
--------------------------------------------
N 331971 331971
--------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001The minimal OLS estimate of pretrail detention on guilty plea is -0.0007 increase in percentage points of a guily plea and not statistically significant at the 5 percent level, while maximum controls yields an estimate of 2.9 percentage points.
2.2 Instrument Variables
We will estimate the reduced form equation for the first stage of the instrument variable estimator. We will estimate the first stage with two reduced form equations. One with date variables \(t_i\) and one with data variables plus demographics \(d_i\) and crime and/or prior justice involement \(c_i\).
\[ PreTrailDetention_i = \pi_0 + \pi_1 JFE_i + t_i + d_i+c_i + \varepsilon_i \]
est clear
eststo rf1: quietly reg jail3 $judge_pre $control2, robust
eststo rf2: quietly reg jail3 possess robbery DUI1st drugSell aggAss $demo $prior $off $control2 $judge_pre, robust
esttab rf1 rf2, mtitle("Min Controls" "Max Controls") keep(judge_pre_1 judge_pre_2 judge_pre_3 judge_pre_4 judge_pre_5 judge_pre_6 judge_pre_7 judge_pre_8) (1) (2)
Min Controls Max Controls
--------------------------------------------
judge_pre_1 -0.0325*** -0.0306***
(-5.80) (-6.11)
judge_pre_2 0 0
(.) (.)
judge_pre_3 -0.0237*** -0.0192***
(-4.63) (-4.20)
judge_pre_4 -0.0458*** -0.0442***
(-8.97) (-9.71)
judge_pre_5 -0.0338*** -0.0301***
(-5.99) (-6.02)
judge_pre_6 -0.00910 -0.00740
(-1.78) (-1.62)
judge_pre_7 -0.0307*** -0.0253***
(-5.63) (-5.22)
judge_pre_8 -0.0427*** -0.0361***
(-8.36) (-7.89)
--------------------------------------------
N 331971 331971
--------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001Next, we will estimate the \(F\)-statistics for our instrument relevance tests for both models.
( 1) judge_pre_1 = 0
( 2) o.judge_pre_2 = 0
( 3) judge_pre_3 = 0
( 4) judge_pre_4 = 0
( 5) judge_pre_5 = 0
( 6) judge_pre_6 = 0
( 7) judge_pre_7 = 0
( 8) judge_pre_8 = 0
Constraint 2 dropped
F( 7,331954) = 35.25
Prob > F = 0.0000Our judge fixed effects appear to be relevant instruments since \(F-stat=35.25\) for our minimal controls model.
quietly {
reg jail3 $judge_pre possess robbery DUI1st drugSell aggAss $demo $prior $off $control2 , robust
}
test $judge_pre ( 1) judge_pre_1 = 0
( 2) o.judge_pre_2 = 0
( 3) judge_pre_3 = 0
( 4) judge_pre_4 = 0
( 5) judge_pre_5 = 0
( 6) judge_pre_6 = 0
( 7) judge_pre_7 = 0
( 8) judge_pre_8 = 0
Constraint 2 dropped
F( 7,331928) = 42.67
Prob > F = 0.0000Our judge fixed effects appear to be relevant instruments since \(F-stat=42.67\) for our maximum controls model, as well.
Now let’s estimate our two models with an IV estimator with the ivregress 2sls command.
est clear
eststo iv1: quietly ivregress 2sls guilt (jail3= $judge_pre) $control2, robust first
eststo iv2: quietly ivregress 2sls guilt (jail3= $judge_pre) possess robbery DUI1st drugSell aggAss $demo $prior $off $control2 , robust first
esttab iv1 iv2, mtitle("Min IV Model" "Max IV Model") keep(jail3) (1) (2)
Min IV Model Max IV Model
--------------------------------------------
jail3 0.151* 0.186**
(2.32) (2.88)
--------------------------------------------
N 331971 331971
--------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001Our \(LATE\) estimate is an increase of guilty plea by 15.1 percentage points for individuals with a guilty plea with the minimal-controls model.
Our \(LATE\) estimate is an increase of guilty plea by 18.6 percentage points for individuals with a guilty plea with the maximum-controls model.
2.3 Jackknife Instrument Variable Estimator
Now we will implement our Jackknife IV estimator and estimate miminal-control and maximum-control models. We use the jive command after doing our install of package st0108.
note: t6 dropped due to collinearity
note: judge_pre_8 dropped due to collinearity
Jackknife instrumental variables regression (UJIVE1)
First-stage summary Number of obs = 331971
------------------------- F( 10,331960) = 270.73
F( 7,331954) = 35.36 Prob > F = 0.0000
Prob > F = 0.0000 R-squared = -0.0183
R-squared = 0.0026 Adj R-squared = -0.0183
Root MSE = 0.5045
------------------------------------------------------------------------------
guilt | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
jail3 | .1624352 .0701409 2.32 0.021 .024961 .2999094
day | -.0000271 .0000664 -0.41 0.683 -.0001574 .0001031
day2 | -1.18e-07 3.25e-07 -0.36 0.716 -7.56e-07 5.19e-07
day3 | 2.65e-10 4.23e-10 0.63 0.531 -5.64e-10 1.09e-09
bailDate | .0000606 9.00e-06 6.74 0.000 .000043 .0000783
t1 | .0094375 .0174909 0.54 0.589 -.0248442 .0437192
t2 | -.0118243 .0136833 -0.86 0.388 -.0386431 .0149946
t3 | .0101888 .0123284 0.83 0.409 -.0139746 .0343521
t4 | .0195249 .0080497 2.43 0.015 .0037478 .035302
t5 | .0017162 .0055469 0.31 0.757 -.0091555 .0125879
_cons | -.675075 .1748001 -3.86 0.000 -1.017678 -.332472
------------------------------------------------------------------------------
Instrumented: jail3
Instruments: day day2 day3 bailDate t1 t2 t3 t4 t5 judge_pre_1 judge_pre_2
judge_pre_3 judge_pre_4 judge_pre_5 judge_pre_6 judge_pre_7
------------------------------------------------------------------------------Our estimated \(LATE\) with JIVE is a 16.2 percentage point increase in a guilty plea for individuals with a pre-trail detention.
jive guilt (jail3= $judge_pre) possess robbery DUI1st drugSell aggAss $demo $prior $off $control2 , robustnote: t6 dropped due to collinearity
note: judge_pre_8 dropped due to collinearity
Jackknife instrumental variables regression (UJIVE1)
First-stage summary Number of obs = 331971
------------------------- F( 36,331934) = 1375.47
F( 7,331928) = 42.83 Prob > F = 0.0000
Prob > F = 0.0000 R-squared = 0.0925
R-squared = 0.2486 Adj R-squared = 0.0924
Root MSE = 0.4763
-------------------------------------------------------------------------------
guilt | Coef. Std. Err. t P>|t| [95% Conf. Interval]
--------------+----------------------------------------------------------------
jail3 | .2120002 .0755804 2.80 0.005 .0638648 .3601355
possess | -.0609555 .0038514 -15.83 0.000 -.0685041 -.0534068
robbery | -.102243 .0091054 -11.23 0.000 -.1200894 -.0843966
DUI1st | .0583614 .0066367 8.79 0.000 .0453536 .0713692
drugSell | .1154289 .0096137 12.01 0.000 .0965863 .1342715
aggAss | .004039 .0040886 0.99 0.323 -.0039746 .0120526
black | .0570339 .0061167 9.32 0.000 .0450452 .0690225
age | .001521 .0001193 12.75 0.000 .0012871 .0017548
male | -.0544413 .008064 -6.75 0.000 -.0702465 -.0386362
white | .1002602 .0038903 25.77 0.000 .0926353 .1078851
priorCases | -.0057862 .0002551 -22.68 0.000 -.0062861 -.0052862
priorWI5 | .0245624 .0060695 4.05 0.000 .0126663 .0364585
prior_felChar | -.0076116 .0008469 -8.99 0.000 -.0092714 -.0059518
prior_guilt | .023201 .001188 19.53 0.000 .0208726 .0255293
onePrior | .0473753 .0035872 13.21 0.000 .0403445 .0544062
threePriors | -.0015179 .0048536 -0.31 0.754 -.0110307 .0079949
fel | -.0220026 .0107113 -2.05 0.040 -.0429965 -.0010087
mis | .1393341 .0115767 12.04 0.000 .1166441 .1620242
sum | .0657736 .0036656 17.94 0.000 .0585891 .072958
F1 | -.0156777 .012377 -1.27 0.205 -.0399363 .0085808
F2 | .027801 .0111437 2.49 0.013 .0059597 .0496423
F3 | .0923061 .0031874 28.96 0.000 .086059 .0985532
F | -.0929481 .0119417 -7.78 0.000 -.1163534 -.0695427
M1 | .0094806 .0076985 1.23 0.218 -.0056083 .0245695
M2 | -.0772227 .0045121 -17.11 0.000 -.0860662 -.0683791
M3 | .1242172 .0051198 24.26 0.000 .1141825 .1342518
M | .2706689 .0053935 50.18 0.000 .2600978 .28124
day | -.00003 .0000608 -0.49 0.622 -.0001492 .0000892
day2 | -7.03e-09 2.95e-07 -0.02 0.981 -5.85e-07 5.71e-07
day3 | 2.42e-10 3.87e-10 0.63 0.532 -5.16e-10 9.99e-10
bailDate | .0000418 8.57e-06 4.88 0.000 .000025 .0000586
t1 | -.0186899 .016402 -1.14 0.255 -.0508374 .0134576
t2 | -.0243829 .0127911 -1.91 0.057 -.049453 .0006872
t3 | .0033173 .0110964 0.30 0.765 -.0184314 .025066
t4 | .0142448 .0072415 1.97 0.049 .0000517 .028438
t5 | .009051 .0052937 1.71 0.087 -.0013245 .0194264
_cons | -.6960786 .164429 -4.23 0.000 -1.018355 -.3738024
-------------------------------------------------------------------------------
Instrumented: jail3
Instruments: possess robbery DUI1st drugSell aggAss black age male white
priorCases priorWI5 prior_felChar prior_guilt onePrior
threePriors fel mis sum F1 F2 F3 F M1 M2 M3 M day day2 day3
bailDate t1 t2 t3 t4 t5 judge_pre_1 judge_pre_2 judge_pre_3
judge_pre_4 judge_pre_5 judge_pre_6 judge_pre_7
------------------------------------------------------------------------------Our estimated \(LATE\) with JIVE is a 21.2 percentage point increase in a guilty plea for individuals with a pre-trail detention
Also, don’t forget to clear your global macros! Local macros expire at the end of a run, but global macros will remain, and you need to tell Stata to remove them.