Home > Statistics > Exact matching on discrete covariates is the same as regression adjustment

## Exact matching on discrete covariates is the same as regression adjustment

I illustrate that exact matching on discrete covariates and regression adjustment (RA) with fully interacted discrete covariates perform the same nonparametric estimation.

Comparing exact matching with RA

A well-known example from the causal inference literature estimates the average treatment effect (ATE) of pregnant women smoking on the babies’ birth weights. Cattaneo (2010) discusses this example and I use an extract of his data. (My extract is not representative, and the results below only illustrate the methods I discuss.) See Wooldridge (2010, chap. 21) for an introduction to estimating an ATE.

The birth weight of the baby born to a mother is recorded in bweight. mbsmoke is the binary treatment indicating whether each woman smoked while she was pregnant. I also control for the women’s education (medu), a binary indicator for whether this was her first baby (fbaby), and a binary indicator for whether she was married (mmarried).

As is frequently the case, one of my control variables has too many categories for exact matching or to include as a categorical variable in fully interacted regression. In example 1, I impose a priori knowledge that allows me to combine 0–8 years of schooling into the “Before HS” category, 9–11 years into “In HS”, 12 into “HS”, and more than 12 into “HS+”, where HS stands for high school.

Example 1: Cutting medu into four categories

```. generate medu2 = irecode(medu, 8, 11, 12)

. label define  ed2l 0 "before HS"  1 "in HS" 2 "HS" 3 "HS+"

. label values medu2 ed2l
```

Exact matching requires that none of the cells formed by the treatment variable and the values for the discrete variables be empty. In example 2, I create case, which enumerates the set of possible covariate values, and then tabulate case over the treatment levels.

Example 2: Tabulating covariate patterns by treatment level

```. egen case = group(medu2 fbaby mmarried) , label

. tab case mbsmoke

group(medu2 fbaby |  1 if mother smoked
mmarried) | nonsmoker     smoker |     Total
----------------------+----------------------+----------
before HS No notmarri |        29         18 |        47
before HS No married |        63          4 |        67
before HS Yes notmarr |        29         12 |        41
before HS Yes married |        17          3 |        20
in HS No notmarried |       106        103 |       209
in HS No married |        76         53 |       129
in HS Yes notmarried |       173         62 |       235
in HS Yes married |        28         18 |        46
HS No notmarried |       197        119 |       316
HS No married |       706        163 |       869
HS Yes notmarried |       233         90 |       323
HS Yes married |       502         69 |       571
HS+ No notmarried |        77         25 |       102
HS+ No married |       812         58 |       870
HS+ Yes notmarried |        95         26 |       121
HS+ Yes married |       635         41 |       676
----------------------+----------------------+----------
Total |     3,778        864 |     4,642
```

Some further consolidation might be required, because so few smokers with “before HS” education were married. There are only 4 treated cases with “before HS” education, not first baby, and married; there are only 3 treated cases with “before HS” education, first baby, and married. As I discuss in Done and undone, how I combine the categories is critical to obtaining consistent estimates. For this example, I leave the categories as previously defined and proceed to estimate the ATE by matching exactly on the covariates.

Example 3: ATE estimated by exact matching on discrete covariates

```. teffects nnmatch (bweight ) (mbsmoke), ematch(medu2 fbaby mmarried)

Treatment-effects estimation                   Number of obs      =      4,642
Estimator      : nearest-neighbor matching     Matches: requested =          1
Outcome model  : matching                                     min =          3
Distance metric: Mahalanobis                                  max =        812
------------------------------------------------------------------------------
|              AI Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -227.3809   26.99005    -8.42   0.000    -280.2804   -174.4813
------------------------------------------------------------------------------
```

Exact matching with replacement compares each treated case with the mean of the not-treated cases with the same covariate pattern, and it compares each not-treated case with the mean of the treated cases with the same covariate pattern. The mean of the case-level comparisons estimates the ATE.

RA estimates the ATE by the difference between the averages of the predicted values for the treated and not-treated cases. With fully interacted discrete covariates, the predicted values are the outcome averages within each covariate pattern.

Example 4 illustrates that exact matching with replacement produces the same point estimates as RA with fully interacted discrete covariates.

Example 4: ATE estimated by RA on discrete covariates

```. regress bweight ibn.mbsmoke#ibn.case,
>         noconstant vce(robust) vsquish

Linear regression                               Number of obs     =      4,642
F(32, 4610)       =    5472.14
Prob > F          =     0.0000
R-squared         =     0.9731
Root MSE          =     561.89

-------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
mbsmoke#case |
nonsmoker #|
before HS ..  |   3412.345   85.26789    40.02   0.000     3245.179    3579.511
nonsmoker #|
before HS ..  |   3382.048   64.77681    52.21   0.000     3255.054    3509.041
nonsmoker #|
before HS ..  |   3095.897   121.4719    25.49   0.000     2857.753     3334.04
nonsmoker #|
before HS ..  |   3213.588   108.5406    29.61   0.000     3000.797     3426.38
nonsmoker #|
in HS No n..  |   3219.255    66.9732    48.07   0.000     3087.955    3350.554
nonsmoker #|
in HS No m..  |   3454.434   57.21777    60.37   0.000      3342.26    3566.608
nonsmoker #|
in HS Yes ..  |   3227.977   49.20252    65.61   0.000     3131.516    3324.437
nonsmoker #|
in HS Yes ..  |   3467.286   95.52026    36.30   0.000      3280.02    3654.551
nonsmoker #|
HS No notm..  |   3327.249   45.20513    73.60   0.000     3238.625    3415.872
nonsmoker #|
HS No marr..  |   3498.307   20.41325   171.37   0.000     3458.288    3538.327
nonsmoker #|
HS Yes not..  |   3258.069   38.79208    83.99   0.000     3182.018     3334.12
nonsmoker #|
HS Yes mar..  |   3382.054   24.69261   136.97   0.000     3333.644    3430.463
nonsmoker #|
HS+ No not..  |   3227.597   80.73945    39.98   0.000     3069.309    3385.885
nonsmoker #|
HS+ No mar..  |   3514.036   18.78391   187.08   0.000      3477.21    3550.861
nonsmoker #|
HS+ Yes no..  |   3248.295   64.86602    50.08   0.000     3121.126    3375.463
nonsmoker #|
HS+ Yes ma..  |   3441.787   21.05667   163.45   0.000     3400.506    3483.069
smoker #|
before HS ..  |   3181.111   105.5454    30.14   0.000     2974.192    3388.031
smoker #|
before HS ..  |    3373.75   229.6108    14.69   0.000     2923.603    3823.897
smoker #|
before HS ..  |   2924.333   139.0673    21.03   0.000     2651.695    3196.972
smoker #|
before HS ..  |   2863.333   93.69532    30.56   0.000     2679.646    3047.021
smoker #|
in HS No n..  |    3038.68   59.37928    51.17   0.000     2922.268    3155.091
smoker #|
in HS No m..  |   3115.698   58.70879    53.07   0.000     3000.601    3230.795
smoker #|
in HS Yes ..  |   3147.097   62.21084    50.59   0.000     3025.134     3269.06
smoker #|
in HS Yes ..  |   3353.889   111.5621    30.06   0.000     3135.174    3572.604
smoker #|
HS No notm..  |   3061.437   60.37705    50.71   0.000     2943.069    3179.805
smoker #|
HS No marr..  |   3184.221   47.77988    66.64   0.000     3090.549    3277.892
smoker #|
HS Yes not..  |   3131.533   44.98026    69.62   0.000     3043.351    3219.716
smoker #|
HS Yes mar..  |   3199.174   63.82476    50.12   0.000     3074.047    3324.301
smoker #|
HS+ No not..  |    3002.36   89.60639    33.51   0.000     2826.689    3178.031
smoker #|
HS+ No mar..  |   3199.707   82.92361    38.59   0.000     3037.137    3362.277
smoker #|
HS+ Yes no..  |   3161.923   79.54319    39.75   0.000      3005.98    3317.866
smoker #|
HS+ Yes ma..  |   3271.293   90.92146    35.98   0.000     3093.043    3449.542
-------------------------------------------------------------------------------

. margins r.mbsmoke , vce(unconditional) contrast(nowald)

Contrasts of predictive margins

Expression   : Linear prediction, predict()

------------------------------------------------------------------------
|            Unconditional
|   Contrast   Std. Err.     [95% Conf. Interval]
-----------------------+------------------------------------------------
mbsmoke |
(smoker vs nonsmoker)  |  -227.3809   26.82888     -279.9783   -174.7834
------------------------------------------------------------------------
```

The 32 parameters estimated by regress are the means of the outcome for the 32 cases in the table in example 1. The standard errors reported by exact matching and RA are asymptotically equivalent but differ in finite samples.

The regression underlying RA with fully interacted discrete covariates is an interaction between the treatment factor with an interaction between all the discrete covariates. Example 5 illustrates that this regression produces the same results as example 4.

Example 5: RA estimated with interactions

```. regress bweight ibn.mbsmoke#ibn.medu2#ibn.fbaby#ibn.mmarried,
>         noconstant vce(robust) vsquish

Linear regression                               Number of obs     =      4,642
F(32, 4610)       =    5472.14
Prob > F          =     0.0000
R-squared         =     0.9731
Root MSE          =     561.89

------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mbsmoke#|
medu2#fbaby#|
mmarried |
nonsmoker #|
before HS #|
No #|
notmarried  |   3412.345   85.26789    40.02   0.000     3245.179    3579.511
nonsmoker #|
before HS #|
No #|
married  |   3382.048   64.77681    52.21   0.000     3255.054    3509.041
nonsmoker #|
before HS #|
Yes #|
notmarried  |   3095.897   121.4719    25.49   0.000     2857.753     3334.04
nonsmoker #|
before HS #|
Yes #|
married  |   3213.588   108.5406    29.61   0.000     3000.797     3426.38
nonsmoker #|
in HS #|
No #|
notmarried  |   3219.255    66.9732    48.07   0.000     3087.955    3350.554
nonsmoker #|
in HS #|
No #|
married  |   3454.434   57.21777    60.37   0.000      3342.26    3566.608
nonsmoker #|
in HS #|
Yes #|
notmarried  |   3227.977   49.20252    65.61   0.000     3131.516    3324.437
nonsmoker #|
in HS #|
Yes #|
married  |   3467.286   95.52026    36.30   0.000      3280.02    3654.551
nonsmoker #|
HS #|
No #|
notmarried  |   3327.249   45.20513    73.60   0.000     3238.625    3415.872
nonsmoker #|
HS #|
No #|
married  |   3498.307   20.41325   171.37   0.000     3458.288    3538.327
nonsmoker #|
HS #|
Yes #|
notmarried  |   3258.069   38.79208    83.99   0.000     3182.018     3334.12
nonsmoker #|
HS #|
Yes #|
married  |   3382.054   24.69261   136.97   0.000     3333.644    3430.463
nonsmoker #|
HS+ #|
No #|
notmarried  |   3227.597   80.73945    39.98   0.000     3069.309    3385.885
nonsmoker #|
HS+ #|
No #|
married  |   3514.036   18.78391   187.08   0.000      3477.21    3550.861
nonsmoker #|
HS+ #|
Yes #|
notmarried  |   3248.295   64.86602    50.08   0.000     3121.126    3375.463
nonsmoker #|
HS+ #|
Yes #|
married  |   3441.787   21.05667   163.45   0.000     3400.506    3483.069
smoker #|
before HS #|
No #|
notmarried  |   3181.111   105.5454    30.14   0.000     2974.192    3388.031
smoker #|
before HS #|
No #|
married  |    3373.75   229.6108    14.69   0.000     2923.603    3823.897
smoker #|
before HS #|
Yes #|
notmarried  |   2924.333   139.0673    21.03   0.000     2651.695    3196.972
smoker #|
before HS #|
Yes #|
married  |   2863.333   93.69532    30.56   0.000     2679.646    3047.021
smoker #|
in HS #|
No #|
notmarried  |    3038.68   59.37928    51.17   0.000     2922.268    3155.091
smoker #|
in HS #|
No #|
married  |   3115.698   58.70879    53.07   0.000     3000.601    3230.795
smoker #|
in HS #|
Yes #|
notmarried  |   3147.097   62.21084    50.59   0.000     3025.134     3269.06
smoker #|
in HS #|
Yes #|
married  |   3353.889   111.5621    30.06   0.000     3135.174    3572.604
smoker #|
HS #|
No #|
notmarried  |   3061.437   60.37705    50.71   0.000     2943.069    3179.805
smoker #|
HS #|
No #|
married  |   3184.221   47.77988    66.64   0.000     3090.549    3277.892
smoker #|
HS #|
Yes #|
notmarried  |   3131.533   44.98026    69.62   0.000     3043.351    3219.716
smoker #|
HS #|
Yes #|
married  |   3199.174   63.82476    50.12   0.000     3074.047    3324.301
smoker #|
HS+ #|
No #|
notmarried  |    3002.36   89.60639    33.51   0.000     2826.689    3178.031
smoker #|
HS+ #|
No #|
married  |   3199.707   82.92361    38.59   0.000     3037.137    3362.277
smoker #|
HS+ #|
Yes #|
notmarried  |   3161.923   79.54319    39.75   0.000      3005.98    3317.866
smoker #|
HS+ #|
Yes #|
married  |   3271.293   90.92146    35.98   0.000     3093.043    3449.542
------------------------------------------------------------------------------

. margins r.mbsmoke , vce(unconditional) contrast(nowald)

Contrasts of predictive margins

Expression   : Linear prediction, predict()

------------------------------------------------------------------------
|            Unconditional
|   Contrast   Std. Err.     [95% Conf. Interval]
-----------------------+------------------------------------------------
mbsmoke |
(smoker vs nonsmoker)  |  -227.3809   26.82888     -279.9783   -174.7834
------------------------------------------------------------------------
```

Finally, I illustrate that teffects ra produces the same point estimates.

Example 6: RA estimated by teffects

```. teffects ra (bweight bn.medu2#ibn.fbaby#ibn.mmarried, noconstant) (mbsmoke)

Iteration 0:   EE criterion =  2.010e-25
Iteration 1:   EE criterion =  5.818e-26

Treatment-effects estimation                    Number of obs     =      4,642
Outcome model  : linear
Treatment model: none
------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -227.3809   26.73625    -8.50   0.000     -279.783   -174.9788
-------------+----------------------------------------------------------------
POmean       |
mbsmoke |
nonsmoker  |   3402.793    9.59059   354.81   0.000     3383.995     3421.59
------------------------------------------------------------------------------
```

The standard errors are asymptotically equivalent but differ in finite samples because teffects does adjust for the number of parameters estimated in the regression, as regress does.

Done and undone

I illustrated that exact matching on discrete covariates is the same as RA with fully interacted discrete covariates. Key to both methods is that the covariates are in fact discrete. If some collapsing of categories is performed as above, or if a discrete covariate is formed by cutting up a continuous covariate, all the results require that this combining step be performed correctly.

Exact matching on discrete covariates and RA with fully interacted discrete covariates perform the same nonparametric estimation. Collapsing categories or cutting up discrete covariates performs the same function as a bandwidth in nonparametric kernel regression; it determines which observations are comparable with each other. Just as with kernel regression, the bandwidth must be properly chosen to obtain consistent estimates.

References

Cattaneo, M. 2010. Effcient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics 155: 138–154.

Wooldridge, J. M. 2010. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, Massachusetts: MIT Press.

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