Analyzing RCT reemployment experiment

6. Analyzing RCT reemployment experiment#

6.1. Data#

In this lab, we analyze the Pennsylvania re-employment bonus experiment, which was previously studied in “Sequential testing of duration data: the case of the Pennsylvania ‘reemployment bonus’ experiment” (Bilias, 2000), among others. These experiments were conducted in the 1980s by the U.S. Department of Labor to test the incentive effects of alternative compensation schemes for unemployment insurance (UI). In these experiments, UI claimants were randomly assigned either to a control group or one of five treatment groups. Actually, there are six treatment groups in the experiments. Here we focus on treatment group 4, but feel free to explore other treatment groups. In the control group the current rules of the UI applied. Individuals in the treatment groups were offered a cash bonus if they found a job within some pre-specified period of time (qualification period), provided that the job was retained for a specified duration. The treatments differed in the level of the bonus, the length of the qualification period, and whether the bonus was declining over time in the qualification period; see http://qed.econ.queensu.ca/jae/2000-v15.6/bilias/readme.b.txt for further details on data.

# install.packages("librarian", quiet = T)
librarian::shelf(tidyverse, lmtest, sandwich, broom, hdm, quiet = T)
## loading the data
Penn <- as_tibble(read.table("https://github.com/d2cml-ai/14.388_r/raw/main/Data/penn_jae.dat", header=T ))
n <- dim(Penn)[1]
p_1 <- dim(Penn)[2]
Penn <- filter(Penn, tg == 4 | tg == 0)# subset(Penn, tg==4 | tg==0)
attach(Penn)

T4 <- (Penn$tg==4)
summary(T4)

   Mode   FALSE    TRUE 
logical    3354    1745 

glimpse(Penn)

Rows: 5,099
Columns: 23
$ abdt       <int> 10824, 10824, 10747, 10607, 10831, 10845, 10831, 10859, 105…
$ tg         <int> 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0,…
$ inuidur1   <int> 18, 1, 27, 9, 27, 27, 9, 27, 15, 28, 12, 22, 18, 1, 7, 18, …
$ inuidur2   <int> 18, 1, 27, 9, 27, 27, 9, 27, 15, 11, 12, 22, 18, 1, 7, 18, …
$ female     <int> 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0,…
$ black      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,…
$ hispanic   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,…
$ othrace    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ dep        <int> 2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0,…
$ q1         <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ q2         <int> 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0,…
$ q3         <int> 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0,…
$ q4         <int> 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,…
$ q5         <int> 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0,…
$ q6         <int> 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,…
$ recall     <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ agelt35    <int> 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,…
$ agegt54    <int> 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,…
$ durable    <int> 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,…
$ nondurable <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,…
$ lusd       <int> 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0,…
$ husd       <int> 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ muld       <int> 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1,…

6.1.1. Model#

To evaluate the impact of the treatments on unemployment duration, we consider the linear regression model:

\[ Y = D \beta_1 + W'\beta_2 + \varepsilon, \quad E \varepsilon (D,W')' = 0, \]

where \(Y\) is the log of duration of unemployment, \(D\) is a treatment indicators, and \(W\) is a set of controls including age group dummies, gender, race, number of dependents, quarter of the experiment, location within the state, existence of recall expectations, and type of occupation. Here \(\beta_1\) is the ATE, if the RCT assumptions hold rigorously.

We also consider interactive regression model:

\[ Y = D \alpha_1 + D W' \alpha_2 + W'\beta_2 + \varepsilon, \quad E \varepsilon (D,W', DW')' = 0, \]

where \(W\)’s are demeaned (apart from the intercept), so that \(\alpha_1\) is the ATE, if the RCT assumptions hold rigorously.

Under RCT, the projection coefficient \(\beta_1\) has the interpretation of the causal effect of the treatment on the average outcome. We thus refer to \(\beta_1\) as the average treatment effect (ATE). Note that the covariates, here are independent of the treatment \(D\), so we can identify \(\beta_1\) by just linear regression of \(Y\) on \(D\), without adding covariates. However we do add covariates in an effort to improve the precision of our estimates of the average treatment effect.

6.1.2. Analysis#

We consider

classical 2-sample approach, no adjustment (CL)
classical linear regression adjustment (CRA)
interactive regression adjusment (IRA)

and carry out robust inference using the estimatr R packages.

6.2. Carry out covariate balance check#

This is done using “lm_robust” command which unlike “lm” in the base command automatically does the correct Eicher-Huber-White standard errors, instead othe classical non-robus formula based on the homoscdedasticity command.

m <- lm(T4 ~ (female + black + othrace + factor(dep) + q2 +q3 + q4 +
        q5 + q6 + agelt35 + agegt54 + durable + lusd + husd)^2, data = Penn)
coeftest(m, vcov = vcovHC(m, type="HC1"))

t test of coefficients:

                        Estimate  Std. Error t value  Pr(>|t|)    
(Intercept)           0.32145725  0.16125607  1.9935 0.0462656 *  
female                0.10423328  0.13624779  0.7650 0.4442914    
black                 0.07164803  0.08288534  0.8644 0.3873969    
othrace               0.02801517  0.40496815  0.0692 0.9448502    
factor(dep)1         -0.07363340  0.20094637 -0.3664 0.7140574    
factor(dep)2         -0.10854072  0.15754307 -0.6890 0.4908810    
q2                   -0.02667937  0.16255836 -0.1641 0.8696419    
q3                   -0.00567387  0.16218178 -0.0350 0.9720934    
q4                    0.04334425  0.16233956  0.2670 0.7894821    
q5                    0.09386458  0.16157184  0.5809 0.5613028    
q6                   -0.22156423  0.15984049 -1.3862 0.1657604    
agelt35              -0.10923976  0.13323486 -0.8199 0.4123101    
agegt54              -0.43668630  0.13581268 -3.2154 0.0013111 ** 
durable              -0.12500967  0.17083793 -0.7317 0.4643590    
lusd                  0.11428653  0.06824019  1.6748 0.0940422 .  
husd                  0.09467997  0.07364116  1.2857 0.1986096    
female:black          0.08886596  0.04381635  2.0281 0.0425983 *  
female:othrace       -0.41350947  0.16481280 -2.5090 0.0121400 *  
female:factor(dep)1   0.05547519  0.04647943  1.1935 0.2327135    
female:factor(dep)2   0.04544801  0.04053454  1.1212 0.2622494    
female:q2            -0.18880964  0.13587953 -1.3895 0.1647314    
female:q3            -0.16526367  0.13565839 -1.2182 0.2231926    
female:q4            -0.17632875  0.13530942 -1.3032 0.1925829    
female:q5            -0.20321654  0.13480991 -1.5074 0.1317636    
female:q6            -0.04271603  0.14234389 -0.3001 0.7641207    
female:agelt35        0.07342181  0.03004676  2.4436 0.0145765 *  
female:agegt54        0.02612867  0.05090915  0.5132 0.6078053    
female:durable        0.01975687  0.04372711  0.4518 0.6514168    
female:lusd           0.00206955  0.03395251  0.0610 0.9513982    
female:husd           0.01178851  0.03747209  0.3146 0.7530827    
black:factor(dep)1   -0.11726334  0.06797874 -1.7250 0.0845893 .  
black:factor(dep)2   -0.02222434  0.06342353 -0.3504 0.7260446    
black:q2             -0.03349395  0.08989631 -0.3726 0.7094737    
black:q3             -0.19656696  0.08506268 -2.3108 0.0208818 *  
black:q4             -0.12493576  0.08604499 -1.4520 0.1465694    
black:q5             -0.20988793  0.08398490 -2.4991 0.0124822 *  
black:agelt35         0.06226414  0.04512447  1.3798 0.1677004    
black:agegt54         0.05126323  0.08453009  0.6064 0.5442439    
black:durable         0.10475086  0.06615415  1.5834 0.1133854    
black:lusd           -0.02103874  0.05600033 -0.3757 0.7071637    
black:husd            0.24964245  0.16874707  1.4794 0.1390996    
othrace:factor(dep)1 -0.85807661  0.33077992 -2.5941 0.0095115 ** 
othrace:factor(dep)2  0.24153715  0.13330391  1.8119 0.0700573 .  
othrace:q2            0.43717264  0.34672729  1.2609 0.2074202    
othrace:q3           -0.37343006  0.37701782 -0.9905 0.3219856    
othrace:q4            0.41589086  0.32950089  1.2622 0.2069414    
othrace:q5            0.00605385  0.31112921  0.0195 0.9844768    
othrace:agelt35       0.31647921  0.18343682  1.7253 0.0845395 .  
othrace:agegt54       0.30916352  0.15913988  1.9427 0.0521066 .  
othrace:durable      -0.19077665  0.13757582 -1.3867 0.1655946    
othrace:lusd         -0.09108976  0.20264053 -0.4495 0.6530804    
othrace:husd          0.00516356  0.20312636  0.0254 0.9797206    
factor(dep)1:q2       0.16575553  0.20084375  0.8253 0.4092430    
factor(dep)2:q2       0.08749363  0.15663627  0.5586 0.5764745    
factor(dep)1:q3       0.12098515  0.19967547  0.6059 0.5446028    
factor(dep)2:q3       0.13964196  0.15680429  0.8905 0.3732138    
factor(dep)1:q4       0.08476950  0.19909174  0.4258 0.6702857    
factor(dep)2:q4       0.09099807  0.15685510  0.5801 0.5618457    
factor(dep)1:q5       0.11045395  0.19829410  0.5570 0.5775381    
factor(dep)2:q5       0.09599502  0.15542385  0.6176 0.5368449    
factor(dep)1:q6       0.10069661  0.20903291  0.4817 0.6300216    
factor(dep)2:q6       0.04840278  0.16392190  0.2953 0.7677926    
factor(dep)1:agelt35 -0.07372986  0.04856221 -1.5183 0.1290132    
factor(dep)2:agelt35 -0.02438958  0.03750210 -0.6504 0.5154946    
factor(dep)1:agegt54 -0.08058890  0.06829342 -1.1800 0.2380409    
factor(dep)2:agegt54  0.02804903  0.12779824  0.2195 0.8262859    
factor(dep)1:durable -0.06027564  0.05959803 -1.0114 0.3118885    
factor(dep)2:durable  0.12466452  0.05025495  2.4806 0.0131473 *  
factor(dep)1:lusd     0.05896976  0.05246189  1.1240 0.2610461    
factor(dep)2:lusd     0.05157515  0.04708540  1.0954 0.2734147    
factor(dep)1:husd    -0.04332567  0.05587546 -0.7754 0.4381417    
factor(dep)2:husd    -0.08820323  0.04775494 -1.8470 0.0648067 .  
q2:agelt35            0.12754085  0.13321346  0.9574 0.3384031    
q2:agegt54            0.52955745  0.13686983  3.8691 0.0001106 ***
q2:durable            0.12611989  0.16953217  0.7439 0.4569544    
q2:lusd              -0.07617774  0.07121471 -1.0697 0.2848100    
q2:husd              -0.01882666  0.07542000 -0.2496 0.8028882    
q3:agelt35            0.13059171  0.13296592  0.9821 0.3260764    
q3:agegt54            0.47820240  0.13404885  3.5674 0.0003640 ***
q3:durable            0.05420566  0.16940199  0.3200 0.7489950    
q3:lusd              -0.05379374  0.06990174 -0.7696 0.4415960    
q3:husd              -0.06007985  0.07486872 -0.8025 0.4223197    
q4:agelt35            0.13100899  0.13279101  0.9866 0.3238962    
q4:agegt54            0.54033014  0.13143961  4.1109 4.005e-05 ***
q4:durable            0.09625171  0.16983764  0.5667 0.5709246    
q4:lusd              -0.10342038  0.07052957 -1.4663 0.1426184    
q4:husd              -0.17191877  0.07455451 -2.3059 0.0211543 *  
q5:agelt35            0.07077473  0.13203088  0.5360 0.5919501    
q5:agegt54            0.44377887  0.13056413  3.3989 0.0006818 ***
q5:durable            0.07466571  0.16765628  0.4453 0.6560862    
q5:lusd              -0.07611869  0.07005218 -1.0866 0.2772661    
q5:husd              -0.16162391  0.07301750 -2.2135 0.0269085 *  
q6:agelt35            0.17030213  0.13886226  1.2264 0.2201021    
q6:agegt54            0.56552338  0.15041890  3.7597 0.0001721 ***
q6:durable            0.25578701  0.17834228  1.4342 0.1515642    
agelt35:durable       0.00948986  0.04062983  0.2336 0.8153294    
agelt35:lusd         -0.02586155  0.03541632 -0.7302 0.4652926    
agelt35:husd          0.03548765  0.03973732  0.8931 0.3718702    
agegt54:durable      -0.06231416  0.06390387 -0.9751 0.3295462    
agegt54:lusd         -0.03662895  0.05793021 -0.6323 0.5272235    
agegt54:husd         -0.07421790  0.06125689 -1.2116 0.2257288    
durable:lusd         -0.05517360  0.04342256 -1.2706 0.2039228    
durable:husd         -0.00016918  0.05273588 -0.0032 0.9974404    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We see that that even though this is a randomized experiment, balance conditions are failed.

6.3. Model Specification#

# model specifications


# no adjustment (2-sample approach)
formula_cl <- log(inuidur1) ~ T4

# adding controls
formula_cra <- log(inuidur1) ~ T4 + (female + black + othrace + factor(dep) +
              q2 + q3 + q4 + q5 + q6 + agelt35 +agegt54 + durable +lusd + husd )^2
# Omitted dummies: q1, nondurable, muld


ols.cl <- lm(formula_cl, data = Penn)
ols.cra <- lm(formula_cra, data = Penn)


ols.cl = coeftest(ols.cl, vcov = vcovHC(ols.cl, type="HC1"))
ols.cra = coeftest(ols.cra, vcov = vcovHC(ols.cra, type="HC1"))

tidy(ols.cl)
tidy(ols.cra)

A tibble: 2 × 5
term	estimate	std.error	statistic	p.value
<chr>	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	2.05682971	0.02095476	98.155713	0.00000000
T4TRUE	-0.08545541	0.03585569	-2.383315	0.01719391

A tibble: 104 × 5
term	estimate	std.error	statistic	p.value
<chr>	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	2.63308058	0.36809947	7.1531767	9.706770e-13
T4TRUE	-0.07968012	0.03559092	-2.2387766	2.521432e-02
female	-0.11460926	0.31639042	-0.3622400	7.171880e-01
black	-0.89902649	0.24024626	-3.7421040	1.845347e-04
othrace	-2.62692161	0.45415507	-5.7841953	7.729399e-09
factor(dep)1	-0.71969778	0.58121824	-1.2382574	2.156788e-01
factor(dep)2	-0.04055836	0.35983953	-0.1127124	9.102631e-01
q2	-0.15966999	0.37043166	-0.4310376	6.664596e-01
q3	-0.53989061	0.37007942	-1.4588507	1.446691e-01
q4	-0.43335396	0.37119426	-1.1674587	2.430809e-01
q5	-0.34524113	0.36886953	-0.9359438	3.493474e-01
q6	-0.49367729	0.37119655	-1.3299620	1.835915e-01
agelt35	-0.62586241	0.30753420	-2.0350986	4.189365e-02
agegt54	-0.36137313	0.71234128	-0.5073034	6.119644e-01
durable	-0.27922898	0.49151635	-0.5680970	5.699946e-01
lusd	-0.22287017	0.18725950	-1.1901675	2.340371e-01
husd	-0.17018519	0.20097701	-0.8467894	3.971531e-01
female:black	-0.15495072	0.12190711	-1.2710557	2.037680e-01
female:othrace	0.31020732	0.42359960	0.7323126	4.640122e-01
female:factor(dep)1	-0.02787782	0.11750120	-0.2372556	8.124682e-01
female:factor(dep)2	0.14815022	0.10289219	1.4398587	1.499700e-01
female:q2	-0.08741017	0.31593476	-0.2766716	7.820437e-01
female:q3	0.20487663	0.31575477	0.6488473	5.164669e-01
female:q4	0.27319290	0.31552433	0.8658378	3.866207e-01
female:q5	0.06114805	0.31475173	0.1942739	8.459693e-01
female:q6	0.28721990	0.33836991	0.8488341	3.960143e-01
female:agelt35	0.12638618	0.07712465	1.6387261	1.013333e-01
female:agegt54	0.03996642	0.12730239	0.3139487	7.535731e-01
female:durable	0.06800195	0.10982362	0.6191924	5.358179e-01
female:lusd	0.08374007	0.08494090	0.9858627	3.242482e-01
⋮	⋮	⋮	⋮	⋮
q2:agegt54	0.67121854	0.70879486	0.9469856	0.34369192
q2:durable	-0.10641283	0.48764157	-0.2182194	0.82726711
q2:lusd	-0.04279247	0.18957620	-0.2257270	0.82142298
q2:husd	-0.04166835	0.19687813	-0.2116454	0.83239231
q3:agelt35	0.45449799	0.30738644	1.4785883	0.13931345
q3:agegt54	0.86600818	0.70915532	1.2211827	0.22207451
q3:durable	0.24499851	0.48719509	0.5028756	0.61507392
q3:lusd	0.08586741	0.18721407	0.4586589	0.64649906
q3:husd	0.17030374	0.19683494	0.8652109	0.38696460
q4:agelt35	0.38822344	0.30732383	1.2632390	0.20656227
q4:agegt54	0.55533438	0.70744223	0.7849890	0.43249724
q4:durable	0.21733822	0.49110821	0.4425465	0.65811295
q4:lusd	-0.09500584	0.18830290	-0.5045373	0.61390608
q4:husd	-0.12060827	0.19915192	-0.6056094	0.54480168
q5:agelt35	0.27795870	0.30602492	0.9082878	0.36377001
q5:agegt54	0.45145164	0.70619808	0.6392706	0.52267625
q5:durable	0.28812900	0.48721766	0.5913764	0.55429504
q5:lusd	-0.10402235	0.18908115	-0.5501466	0.58224345
q5:husd	-0.15233600	0.19691991	-0.7735937	0.43920770
q6:agelt35	0.33811077	0.33272643	1.0161825	0.30959171
q6:agegt54	0.94934693	0.72897660	1.3023010	0.19287356
q6:durable	0.40916723	0.50818373	0.8051561	0.42076793
agelt35:durable	0.02527879	0.10150649	0.2490362	0.80334302
agelt35:lusd	-0.06527572	0.08831815	-0.7390975	0.45988254
agelt35:husd	0.05771190	0.10184206	0.5666803	0.57095684
agegt54:durable	0.03198905	0.16583044	0.1929022	0.84704343
agegt54:lusd	-0.14816240	0.13744208	-1.0779988	0.28108634
agegt54:husd	-0.30154665	0.15375360	-1.9612331	0.04990731
durable:lusd	0.11578319	0.11005470	1.0520513	0.29282689
durable:husd	0.23764114	0.12901037	1.8420313	0.06552981

The interactive specificaiton corresponds to the approach introduced in Lin (2013).

#interactive regression model;

X = model.matrix ( ~ (female + black + othrace + factor(dep) + 
                      q2 + q3 + q4 + q5 + q6 + agelt35 + agegt54 + durable + lusd + husd)^2)[,-1]

dim(X)

demean<- function(x){ x - mean(x)}

X = apply(X, 2, demean)

ols_ira = lm(log(inuidur1) ~ T4*X) 
ols_ira= coeftest(ols_ira, vcov = vcovHC(ols_ira, type="HC1"))
tidy(ols_ira)

5099
119

A tibble: 203 × 5
term	estimate	std.error	statistic	p.value
<chr>	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	2.05761308	0.02077237	99.05527109	0.000000e+00
T4TRUE	-0.07550055	0.03560489	-2.12051040	3.401317e-02
Xfemale	-0.66628233	0.40902136	-1.62896709	1.033843e-01
Xblack	-0.86348620	0.29766982	-2.90081876	3.738472e-03
Xothrace	-3.81768810	0.93891012	-4.06608473	4.856015e-05
Xfactor(dep)1	0.03592643	0.64926578	0.05533393	9.558747e-01
Xfactor(dep)2	0.21175557	0.45232670	0.46814740	6.397000e-01
Xq2	-0.25464364	0.45645285	-0.55787502	5.769552e-01
Xq3	-0.62123263	0.45607674	-1.36212303	1.732217e-01
Xq4	-0.47992691	0.45723626	-1.04962565	2.939421e-01
Xq5	-0.37186746	0.45499834	-0.81729410	4.138001e-01
Xq6	-0.67704739	0.45325585	-1.49374218	1.353075e-01
Xagelt35	-0.67770538	0.41036858	-1.65145534	9.870976e-02
Xagegt54	-0.30410918	0.70232279	-0.43300486	6.650303e-01
Xdurable	-0.83800816	0.59780754	-1.40180260	1.610376e-01
Xlusd	-0.09948902	0.22848318	-0.43543258	6.632677e-01
Xhusd	-0.06264193	0.23548601	-0.26601127	7.902417e-01
Xfemale:black	-0.21547992	0.15087950	-1.42815910	1.533099e-01
Xfemale:othrace	0.59943867	0.56394269	1.06294252	2.878604e-01
Xfemale:factor(dep)1	-0.17342448	0.14222988	-1.21932523	2.227795e-01
Xfemale:factor(dep)2	0.21664171	0.12664550	1.71061508	8.721554e-02
Xfemale:q2	0.39188855	0.40767215	0.96128361	3.364571e-01
Xfemale:q3	0.68503908	0.40733779	1.68174697	9.268169e-02
Xfemale:q4	0.72136790	0.40714421	1.77177491	7.649412e-02
Xfemale:q5	0.56577628	0.40577292	1.39431748	1.632850e-01
Xfemale:q6	0.90816973	0.43147278	2.10481349	3.535838e-02
Xfemale:agelt35	0.17007540	0.09529775	1.78467387	7.437613e-02
Xfemale:agegt54	0.23575096	0.16051680	1.46869959	1.419785e-01
Xfemale:durable	0.09741318	0.13795391	0.70612846	4.801419e-01
Xfemale:lusd	0.07200772	0.10678103	0.67434937	5.001211e-01
⋮	⋮	⋮	⋮	⋮
T4TRUE:Xq2:agelt35	-0.301665476	0.6517732	-0.462838087	0.64350100
T4TRUE:Xq2:agegt54	0.114639342	0.5672954	0.202080509	0.83986218
T4TRUE:Xq2:durable	-0.516827505	0.7780445	-0.664264703	0.50655219
T4TRUE:Xq2:lusd	0.165773320	0.4309371	0.384680981	0.70049050
T4TRUE:Xq2:husd	0.532257575	0.4617869	1.152604360	0.24912908
T4TRUE:Xq3:agelt35	-0.225663605	0.6520806	-0.346067031	0.72930721
T4TRUE:Xq3:agegt54	-0.193722861	0.5659725	-0.342283187	0.73215248
T4TRUE:Xq3:durable	-0.849033377	0.7822032	-1.085438330	0.27778100
T4TRUE:Xq3:lusd	0.314719307	0.4225769	0.744762296	0.45645118
T4TRUE:Xq3:husd	0.311301439	0.4634820	0.671658129	0.50183305
T4TRUE:Xq4:agelt35	-0.134792991	0.6512222	-0.206984641	0.83603046
T4TRUE:Xq4:agegt54	0.352913656	0.5502729	0.641342975	0.52132989
T4TRUE:Xq4:durable	-0.628083141	0.7948229	-0.790217764	0.42943888
T4TRUE:Xq4:lusd	0.302349224	0.4223630	0.715851553	0.47411715
T4TRUE:Xq4:husd	0.302876538	0.4690497	0.645723827	0.51848849
T4TRUE:Xq5:agelt35	0.096951649	0.6493474	0.149306284	0.88131811
T4TRUE:Xq5:agegt54	0.482271378	0.5557568	0.867774136	0.38556052
T4TRUE:Xq5:durable	-1.239607497	0.7855332	-1.578045962	0.11461965
T4TRUE:Xq5:lusd	0.615845077	0.4260090	1.445615158	0.14834918
T4TRUE:Xq5:husd	0.294625460	0.4633891	0.635805799	0.52493272
T4TRUE:Xq6:agelt35	-0.001044032	0.7295168	-0.001431128	0.99885818
T4TRUE:Xq6:durable	-1.365249811	0.8496439	-1.606849354	0.10815190
T4TRUE:Xagelt35:durable	0.143948477	0.2229223	0.645733914	0.51848195
T4TRUE:Xagelt35:lusd	-0.013860069	0.1899414	-0.072970247	0.94183277
T4TRUE:Xagelt35:husd	0.029405619	0.2174912	0.135203718	0.89245637
T4TRUE:Xagegt54:durable	0.544931584	0.3430590	1.588448479	0.11224953
T4TRUE:Xagegt54:lusd	-0.579361484	0.2877536	-2.013394456	0.04412777
T4TRUE:Xagegt54:husd	0.191048615	0.3277012	0.582996413	0.55992257
T4TRUE:Xdurable:lusd	-0.374439530	0.2400369	-1.559924845	0.11884227
T4TRUE:Xdurable:husd	-0.338109614	0.2774443	-1.218657845	0.22303280

Next we try out partialling out with lasso

# library(hdm)

T4 = demean(T4)

DX = model.matrix(~T4 * X)[ ,-1]

rlasso.ira = summary(rlassoEffects(DX, log(inuidur1), index = 1))

print(rlasso.ira)

[1] "Estimates and significance testing of the effect of target variables"
   Estimate. Std. Error t value Pr(>|t|)  
T4  -0.07889    0.03555  -2.219   0.0265 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

6.4. Results#

# glimpse(ols_ira)
ols_ira[2,1]

-0.0755005494164947

# library(xtable)
table<- matrix(0, 2, 4)
table[1,1]<-  ols.cl[2,1]
table[1,2]<-  ols.cra[2,1]
table[1,3]<-  ols_ira[2,1]
table[1,4]<-  rlasso.ira[[1]][1]

table[2,1]<-  ols.cl[2,2]
table[2,2]<-  ols.cra[2,2]
table[2,3]<-  ols_ira[2,2]
table[2,4]<-  rlasso.ira[[1]][2]


colnames(table)<- c("CL","CRA","IRA", "IRA w Lasso")
rownames(table)<- c("estimate", "standard error")
# tab<- xtable(table, digits=5)
table

A matrix: 2 × 4 of type dbl
	CL	CRA	IRA	IRA w Lasso
estimate	-0.08545541	-0.07968012	-0.07550055	-0.07888608
standard error	0.03585569	0.03559092	0.03560489	0.03555130

Treatment group 4 experiences an average decrease of about \(7.8\%\) in the length of unemployment spell.

Observe that regression estimators delivers estimates that are slighly more efficient (lower standard errors) than the simple 2 mean estimator, but essentially all methods have very similar standard errors. From IRA results we also see that there is not any statistically detectable heterogeneity. We also see the regression estimators offer slightly lower estimates – these difference occur perhaps to due minor imbalance in the treatment allocation, which the regression estimators try to correct.