Treatment effects in observational studies
Despite the popularity of randomized experiements in economics nowadays, most situations we have observational data in economic studies. One reason is experiemnts are expensive; the other reason is that sometimes it is simply not feasible to have experiments. If we have observational data, and we’d like to draw causal conclusions, then we have a few different situations. The worse situation is that we have an endogenous treatement. Or to say, we have unobserved confounders. In that case, we need instrumental variables. However instruments are hard to find, and even harder to justify. If we assume we don’t have unobserved counfounders. Or to say, we have conditional independence of the treatment variable. That is, conditional on other variables in the model, there is no unobserved confounders. If that assumption holds, then we can make causal inference.
Stata has a set of eteffects and teffects commands are designed for treatment effects. eteffects is for treatment effects when you have endogenous treatment. In that case, you’ll need instruments to model treatment at first stage. Then eteffects use control function approach for the second stage of modeling treatment effects on outcome. In this blog, we are trying to understand how teffects works. That is, when we don’t need an instruments, or say we assume no unobserved confounders.
Regression adjustement (RA)
The RA method is to allow all coveriates’ effects differ between treatment and control. That is, it is the same model as an outcome model with treatment interacting with all other coveriates.
Example:
clear
webuse bweightex
teffects ra (bweight prenatal1 mmarried mage fbaby) (mbsmoke)
reg bweight i.mbsmoke##c.(prenatal1 mmarried mage fbaby)
margins r.mbsmoke##
## . clear
##
## . webuse bweightex
## (Hypothetical birthweight data)
##
## . teffects ra (bweight prenatal1 mmarried mage fbaby) (mbsmoke)
##
## Iteration 0: EE criterion = 2.967e-25
## Iteration 1: EE criterion = 1.674e-26
##
## Treatment-effects estimation Number of obs = 60
## Estimator : regression adjustment
## Outcome model : linear
## Treatment model: none
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATE |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -389.3099 37.00882 -10.52 0.000 -461.8458 -316.7739
## -------------+----------------------------------------------------------------
## POmean |
## mbsmoke |
## nonsmoker | 3613.769 29.97438 120.56 0.000 3555.021 3672.518
## ------------------------------------------------------------------------------
##
## . reg bweight i.mbsmoke##c.(prenatal1 mmarried mage fbaby)
##
## Source | SS df MS Number of obs = 60
## -------------+---------------------------------- F(9, 50) = 18.11
## Model | 1376306.34 9 152922.927 Prob > F = 0.0000
## Residual | 422241.592 50 8444.83184 R-squared = 0.7652
## -------------+---------------------------------- Adj R-squared = 0.7230
## Total | 1798547.93 59 30483.8633 Root MSE = 91.896
##
## ------------------------------------------------------------------------------
## bweight | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mbsmoke |
## smoker | -32.87702 193.2827 -0.17 0.866 -421.0967 355.3426
## prenatal1 | -23.8431 38.31014 -0.62 0.537 -100.7913 53.10507
## mmarried | -40.39753 50.21942 -0.80 0.425 -141.2662 60.47114
## mage | 27.9537 7.671423 3.64 0.001 12.54519 43.36221
## fbaby | -2.030497 40.76029 -0.05 0.960 -83.89995 79.83896
## |
## mbsmoke#|
## c.prenatal1 |
## smoker | 8.436558 56.38855 0.15 0.882 -104.8232 121.6963
## |
## mbsmoke#|
## c.mmarried |
## smoker | 33.92362 61.34015 0.55 0.583 -89.2817 157.1289
## |
## mbsmoke#|
## c.mage |
## smoker | -15.98608 9.006494 -1.77 0.082 -34.07616 2.103995
## |
## mbsmoke#|
## c.fbaby |
## smoker | 14.86782 57.58771 0.26 0.797 -100.8005 130.5362
## |
## _cons | 2976.807 151.0117 19.71 0.000 2673.491 3280.123
## ------------------------------------------------------------------------------
##
## . margins r.mbsmoke
##
## Contrasts of predictive margins
## Model VCE : OLS
##
## Expression : Linear prediction, predict()
##
## ------------------------------------------------
## | df F P>F
## -------------+----------------------------------
## mbsmoke | 1 116.06 0.0000
## |
## Denominator | 50
## ------------------------------------------------
##
## ------------------------------------------------------------------------
## | Delta-method
## | Contrast Std. Err. [95% Conf. Interval]
## -----------------------+------------------------------------------------
## mbsmoke |
## (smoker vs nonsmoker) | -389.3099 36.13675 -461.8927 -316.7271
## ------------------------------------------------------------------------We can see the teffects return the same ATE(Average Treatment Effect) as the margins command after a regression with treatment interacting with all other covariates.
To estimate ATET (or ATT, Average Treatment effected on the Treated),
clear
webuse bweightex
teffects ra (bweight prenatal1 mmarried mage fbaby) (mbsmoke), atet
reg bweight i.mbsmoke##c.(prenatal1 mmarried mage fbaby)
margins r.mbsmoke, subpop(mbsmoke)##
## . clear
##
## . webuse bweightex
## (Hypothetical birthweight data)
##
## . teffects ra (bweight prenatal1 mmarried mage fbaby) (mbsmoke), atet
##
## Iteration 0: EE criterion = 2.792e-25
## Iteration 1: EE criterion = 1.692e-26
##
## Treatment-effects estimation Number of obs = 60
## Estimator : regression adjustment
## Outcome model : linear
## Treatment model: none
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATET |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -437.4721 53.43513 -8.19 0.000 -542.203 -332.7412
## -------------+----------------------------------------------------------------
## POmean |
## mbsmoke |
## nonsmoker | 3693.639 53.80537 68.65 0.000 3588.182 3799.095
## ------------------------------------------------------------------------------
##
## . reg bweight i.mbsmoke##c.(prenatal1 mmarried mage fbaby)
##
## Source | SS df MS Number of obs = 60
## -------------+---------------------------------- F(9, 50) = 18.11
## Model | 1376306.34 9 152922.927 Prob > F = 0.0000
## Residual | 422241.592 50 8444.83184 R-squared = 0.7652
## -------------+---------------------------------- Adj R-squared = 0.7230
## Total | 1798547.93 59 30483.8633 Root MSE = 91.896
##
## ------------------------------------------------------------------------------
## bweight | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mbsmoke |
## smoker | -32.87702 193.2827 -0.17 0.866 -421.0967 355.3426
## prenatal1 | -23.8431 38.31014 -0.62 0.537 -100.7913 53.10507
## mmarried | -40.39753 50.21942 -0.80 0.425 -141.2662 60.47114
## mage | 27.9537 7.671423 3.64 0.001 12.54519 43.36221
## fbaby | -2.030497 40.76029 -0.05 0.960 -83.89995 79.83896
## |
## mbsmoke#|
## c.prenatal1 |
## smoker | 8.436558 56.38855 0.15 0.882 -104.8232 121.6963
## |
## mbsmoke#|
## c.mmarried |
## smoker | 33.92362 61.34015 0.55 0.583 -89.2817 157.1289
## |
## mbsmoke#|
## c.mage |
## smoker | -15.98608 9.006494 -1.77 0.082 -34.07616 2.103995
## |
## mbsmoke#|
## c.fbaby |
## smoker | 14.86782 57.58771 0.26 0.797 -100.8005 130.5362
## |
## _cons | 2976.807 151.0117 19.71 0.000 2673.491 3280.123
## ------------------------------------------------------------------------------
##
## . margins r.mbsmoke, subpop(mbsmoke)
##
## Contrasts of predictive margins
## Model VCE : OLS
##
## Expression : Linear prediction, predict()
##
## ------------------------------------------------
## | df F P>F
## -------------+----------------------------------
## mbsmoke | 1 73.30 0.0000
## |
## Denominator | 50
## ------------------------------------------------
##
## ------------------------------------------------------------------------
## | Delta-method
## | Contrast Std. Err. [95% Conf. Interval]
## -----------------------+------------------------------------------------
## mbsmoke |
## (smoker vs nonsmoker) | -437.4721 51.09606 -540.1015 -334.8426
## ------------------------------------------------------------------------It is the comparison of treatment’s effect on the potential outcomes, for the treated. That’s why in margins, we have subpop(mbsmoke) option.
Inverse Probability Weighting (IPW)
IPW estimators have two steps. The first step is to estimate the treatment model, that is, treatment as a function of some covariates. Usually a logit model is used. Then the probability of treatment is estimated. In the second stage, the inverse probability is used as weights to compute the outcome difference between treatment versus control units.
These steps produce consistent estimates of the effect parameters because the treatment is assumed to be independent of the potential outcomes after conditioning on the covariates.
We can manually conduct the two steps, but the nice thing about using Stata’s teffects is that it takes account of the noise of estimating probability in the first step when calculating standard errors in the second step.
example
clear
webuse cattaneo2
teffects ipw (bweight) (mbsmoke mmarried c.mage##c.mage fbaby medu, probit)
probit mbsmoke mmarried c.mage##c.mage fbaby medu
predict ps
replace ps = 1/ps if mbsmoke==1
replace ps = 1/(1-ps) if mbsmoke==0
reg bweight mbsmoke [pweight=ps]##
## . clear
##
## . webuse cattaneo2
## (Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)
##
## . teffects ipw (bweight) (mbsmoke mmarried c.mage##c.mage fbaby medu, probit)
##
## Iteration 0: EE criterion = 4.621e-21
## Iteration 1: EE criterion = 9.606e-26
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : inverse-probability weights
## Outcome model : weighted mean
## Treatment model: probit
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATE |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -230.6886 25.81524 -8.94 0.000 -281.2856 -180.0917
## -------------+----------------------------------------------------------------
## POmean |
## mbsmoke |
## nonsmoker | 3403.463 9.571369 355.59 0.000 3384.703 3422.222
## ------------------------------------------------------------------------------
##
## . probit mbsmoke mmarried c.mage##c.mage fbaby medu
##
## Iteration 0: log likelihood = -2230.7484
## Iteration 1: log likelihood = -2042.6734
## Iteration 2: log likelihood = -2040.5088
## Iteration 3: log likelihood = -2040.5061
## Iteration 4: log likelihood = -2040.5061
##
## Probit regression Number of obs = 4,642
## LR chi2(5) = 380.48
## Prob > chi2 = 0.0000
## Log likelihood = -2040.5061 Pseudo R2 = 0.0853
##
## ------------------------------------------------------------------------------
## mbsmoke | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mmarried | -.6484821 .0526991 -12.31 0.000 -.7517705 -.5451938
## mage | .1744327 .0352437 4.95 0.000 .1053562 .2435092
## |
## c.mage#|
## c.mage | -.0032559 .0006462 -5.04 0.000 -.0045224 -.0019894
## |
## fbaby | -.2175962 .0491066 -4.43 0.000 -.3138433 -.121349
## medu | -.0863631 .0098692 -8.75 0.000 -.1057064 -.0670198
## _cons | -1.558255 .4511589 -3.45 0.001 -2.44251 -.674
## ------------------------------------------------------------------------------
##
## . predict ps
## (option pr assumed; Pr(mbsmoke))
##
## . replace ps = 1/ps if mbsmoke==1
## (864 real changes made)
##
## . replace ps = 1/(1-ps) if mbsmoke==0
## (3,778 real changes made)
##
## . reg bweight mbsmoke [pweight=ps]
## (sum of wgt is 9,193.96990537643)
##
## Linear regression Number of obs = 4,642
## F(1, 4640) = 79.08
## Prob > F = 0.0000
## R-squared = 0.0389
## Root MSE = 573.36
##
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mbsmoke | -230.6886 25.94182 -8.89 0.000 -281.5469 -179.8303
## _cons | 3403.463 9.616992 353.90 0.000 3384.609 3422.317
## ------------------------------------------------------------------------------In the above example, we run teffects ipw and then manually replicate the two steps. The only thing different is the standard error for the treatment effect.
Doubly robust estimators
RA estimator estimates the outcome directly, IPW esitmates the treatment assignment. There is also a group of estimators called doubly-robust estimators. They estimate both stages, and only require one of these stages to be correctly specified.
Stata implemented the augmented-IPW (AIPW) combination proposed by Robins and Rotnitzky (1995) and the IPW-regression-adjust ment (IPWRA) combination proposed by Wooldridge (2010).
The AIPW estimator augments the IPW estimator with a correction term. The term removes the bias if the treatment model is wrong and the outcome model is correct, and the term goes to 0 if the treatment model is correct and the outcome model is wrong.
The IPWRA estimator uses IPW probability weights when performing RA. The weights do not affect the accuracy of the RA estimator if the treatment model is wrong and the outcome model is correct. The weights correct the RA estimator if the treatment model is correct and the outcome model is wrong.
I have not figured out how to do these two commands manually. So we’ll just run two examples of how they are used.
clear
webuse cattaneo2
teffects ipwra (bweight mmarried mage prenatal1 fbaby) (mbsmoke mmarried mage prenatal1 fbaby)
teffects aipw (bweight mmarried mage prenatal1 fbaby) (mbsmoke mmarried mage prenatal1 fbaby)##
## . clear
##
## . webuse cattaneo2
## (Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)
##
## . teffects ipwra (bweight mmarried mage prenatal1 fbaby) (mbsmoke mmarried mag
## > e prenatal1 fbaby)
##
## Iteration 0: EE criterion = 1.034e-21
## Iteration 1: EE criterion = 6.631e-26
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : IPW regression adjustment
## Outcome model : linear
## Treatment model: logit
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATE |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -238.7679 24.38353 -9.79 0.000 -286.5587 -190.977
## -------------+----------------------------------------------------------------
## POmean |
## mbsmoke |
## nonsmoker | 3402.851 9.538741 356.74 0.000 3384.155 3421.546
## ------------------------------------------------------------------------------
##
## . teffects aipw (bweight mmarried mage prenatal1 fbaby) (mbsmoke mmarried mage
## > prenatal1 fbaby)
##
## Iteration 0: EE criterion = 1.917e-22
## Iteration 1: EE criterion = 3.005e-27
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : augmented IPW
## Outcome model : linear by ML
## Treatment model: logit
## ------------------------------------------------------------------------------
## | Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATE |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -239.0294 24.2524 -9.86 0.000 -286.5632 -191.4955
## -------------+----------------------------------------------------------------
## POmean |
## mbsmoke |
## nonsmoker | 3402.839 9.538926 356.73 0.000 3384.143 3421.535
## ------------------------------------------------------------------------------Matching
First let’s emphasize that matching usually does not deal with endogenous treatment problem. Some people have the wrong impression that matching resolves the endogeneity problem, but in fact it only helps for selection on observables. If you have selection on unobservables, or unobserved confounders, matching does not help.
Matching aims to blance the distribution of covariates in the treatment and control groups. That’s all it does, to make comparisons between apples, not apples to oranges. In that sense, regression does that too. We use regression to adjust for differences between treatment and control. However, regression sometimes rely on extrapolation too much, when data do not overlap between treatment and control. It’s not that matching can do magic on non-overlapping data, but it can make it clear that how bad the non-overlapping problem is. Simply running regression blindly will not have the researchers realize the non-overlapping problem. Combining matching with regression is probably a better idea. That is, running regression on matched sample is recommended.
Here we introduced a few popular matching matheds implemented in Stata, namely nnmatch, psmatch, and cem.
Propensity score matching
The ideal situation of matching is exact matching; that is, treatment and control units can match to each other with exactly the same value of all covariates. This is often not possible, as the number of covariates increase and if they are not discrete. In high dimention (many covariates to match), it is very hard or even impossible to find exact matches. Therefore, how to reduce from high dimension to low dimenstion is key. Rosenbaum and Rubin 1983 proved that propensity score provides the one-dimension representation of high-dimension of covariates. Therefore, we only need to find matches based on propensity scores.
In Stata, teffects psmatch can do estimation after matching on propensity scores. Stata’s psmatch2 command has been popular for propensity score matching too. The nice thing of these commands is that it does two steps in one command: first it estimate the logit or probit model for propensity score, then match the treatment and control groups, then estimate the outcome equation on matched sample. The standard errors are correct based on all these steps. If we do this manually, standard errors will not be correct.
example
clear
webuse cattaneo2
teffects psmatch (bweight) (mbsmoke mmarried c.mage##c.mage fbaby medu), atet
psmatch2 mbsmoke mmarried c.mage##c.mage fbaby medu, logit ties
reg bweight mbsmoke [aweight=_weight]
##
## . clear
##
## . webuse cattaneo2
## (Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)
##
## . teffects psmatch (bweight) (mbsmoke mmarried c.mage##c.mage fbaby medu), atet
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : propensity-score matching Matches: requested = 1
## Outcome model : matching min = 1
## Treatment model: logit max = 74
## ------------------------------------------------------------------------------
## | AI Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATET |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -236.7848 26.57789 -8.91 0.000 -288.8765 -184.693
## ------------------------------------------------------------------------------
##
## . psmatch2 mbsmoke mmarried c.mage##c.mage fbaby medu, logit ties
##
## Logistic regression Number of obs = 4,642
## LR chi2(5) = 375.00
## Prob > chi2 = 0.0000
## Log likelihood = -2043.2504 Pseudo R2 = 0.0841
##
## ------------------------------------------------------------------------------
## mbsmoke | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mmarried | -1.145706 .0918962 -12.47 0.000 -1.32582 -.965593
## mage | .321518 .0638472 5.04 0.000 .1963798 .4466563
## |
## c.mage#|
## c.mage | -.0060368 .0011849 -5.09 0.000 -.0083592 -.0037144
## |
## fbaby | -.3864258 .0880445 -4.39 0.000 -.5589898 -.2138618
## medu | -.1420833 .0173215 -8.20 0.000 -.1760328 -.1081338
## _cons | -2.950915 .8102504 -3.64 0.000 -4.538976 -1.362853
## ------------------------------------------------------------------------------
##
## . reg bweight mbsmoke [aweight=_weight]
## (sum of wgt is 1,728)
##
## Source | SS df MS Number of obs = 3,671
## -------------+---------------------------------- F(1, 3669) = 151.87
## Model | 51455506 1 51455506 Prob > F = 0.0000
## Residual | 1.2431e+09 3,669 338808.237 R-squared = 0.0397
## -------------+---------------------------------- Adj R-squared = 0.0395
## Total | 1.2945e+09 3,670 352736.492 Root MSE = 582.07
##
## ------------------------------------------------------------------------------
## bweight | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mbsmoke | -236.7848 19.21387 -12.32 0.000 -274.4557 -199.1138
## _cons | 3374.444 13.58626 248.37 0.000 3347.807 3401.082
## ------------------------------------------------------------------------------
##
## .In the above example, suppose we are interesting in mbsmoke’s effect on bweight. If we match smoking mothers with non-smoking mothers, by age, first babe, and education, the the teffects psmatch gives us the treatment effect on the treated. We can do this in two steps. First, by psmatch2 we generate _weight, which is a number for how many times a control unit is used to match to a treatment unit. Notice that to match teffects result, we use logit, and ties option. By default, teffects psmatch includes all ties (control units that have the same propensity scores that are close enough), but psmatch2 by default only include one. Then in the second step, we run a regression with _weight as the weight. Notice the standard errors will differ. We should use the standard errors reported in teffects.
Nearest neighbor matching
Stata has the nnmatch option for teffects. nnmatch by default uses Mahalanobis distance, it can also specify to have exact matching. The advantage is this is nonparametric.
clear
webuse cattaneo2
teffects nnmatch (bweight mmarried mage fbaby medu) (mbsmoke) , atet
teffects psmatch (bweight) (mbsmoke mmarried mage fbaby medu), atet
##
## . clear
##
## . webuse cattaneo2
## (Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)
##
## . teffects nnmatch (bweight mmarried mage fbaby medu) (mbsmoke) , atet
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : nearest-neighbor matching Matches: requested = 1
## Outcome model : matching min = 1
## Distance metric: Mahalanobis max = 74
## ------------------------------------------------------------------------------
## | AI Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATET |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -239.2433 25.68524 -9.31 0.000 -289.5854 -188.9011
## ------------------------------------------------------------------------------
##
## . teffects psmatch (bweight) (mbsmoke mmarried mage fbaby medu), atet
##
## Treatment-effects estimation Number of obs = 4,642
## Estimator : propensity-score matching Matches: requested = 1
## Outcome model : matching min = 1
## Treatment model: logit max = 74
## ------------------------------------------------------------------------------
## | AI Robust
## bweight | Coef. Std. Err. z P>|z| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## ATET |
## mbsmoke |
## (smoker |
## vs |
## nonsmoker) | -245.711 26.38675 -9.31 0.000 -297.4281 -193.9939
## ------------------------------------------------------------------------------
##
## .Notice the specification is different in these two models.
Coarsened Exact Matching (CEM)
CEM is not part of teffects. But Stata does have cem package implemented.
clear
webuse cattaneo2
cem mmarried mage fbaby medu, treatment(mbsmoke)
reg bweight mbsmoke [aweight=cem_weights]
##
## . clear
##
## . webuse cattaneo2
## (Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)
##
## . cem mmarried mage fbaby medu, treatment(mbsmoke)
##
## Matching Summary:
## -----------------
## Number of strata: 274
## Number of matched strata: 135
##
## 0 1
## All 3778 864
## Matched 3421 827
## Unmatched 357 Multivariate L1 distance: .25424705
##
## Univariate imbalance:
##
## L1 mean min 25% 50% 75% max
## mmarried 6.4e-15 8.4e-15 0 0 0 0 0
## mage .04955 -.00887 1 0 1 0 -2
## fbaby 6.1e-15 6.1e-15 0 0 0 0 0
## medu .03809 -.03316 0 0 0 0 0
##
## . reg bweight mbsmoke [aweight=cem_weights]
## (sum of wgt is 4,248.00000000005)
##
## Source | SS df MS Number of obs = 4,248
## -------------+---------------------------------- F(1, 4246) = 121.12
## Model | 40566772.9 1 40566772.9 Prob > F = 0.0000
## Residual | 1.4221e+09 4,246 334928.257 R-squared = 0.0277
## -------------+---------------------------------- Adj R-squared = 0.0275
## Total | 1.4627e+09 4,247 344401.26 Root MSE = 578.73
##
## ------------------------------------------------------------------------------
## bweight | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## mbsmoke | -246.8017 22.42533 -11.01 0.000 -290.7671 -202.8364
## _cons | 3383.394 9.894625 341.94 0.000 3363.996 3402.793
## ------------------------------------------------------------------------------
##
## .In the above example, we use CEM on the same covariates, then apply weights in the final regression. The disandvantage is that we have not accounted for the noise in calculating those weights.