Missing data can pose major problems when estimating econometric models since it is generally unlikely that missing values are Missing Completely At Random. A strategy to address this issue without using more complex econometric approaches is represented by multiple imputation, that is the process of replacing missing values by multiple sets of plausible values. This post provides a simple example in which xsmle
is used together with mi
, a Stata's suite of commands that deals with multiple data imputation. Consider the following data in which the only regressor (x1
) has 14 missing values
. sum y x1
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
y | 940 4.144785 1.674932 -.7510805 9.170195
x1 | 926 1.375773 1.202928 -2.039139 4.966914
Since these missing values make the panel unbalanced, xsmle
will not be able to work. Nonetheless, we can overcome the obstacle by exploiting mi
and xsmle
jointly.
The first step is to declare the dataset as a mi
dataset. The command mi set wide
is the appropriate one. Indeed, data must be mi set
ted before other mi
commands can be used. It does not matter which mi
style you choose since you can always change it using mi convert
. In this example, I choose the wide
style.
. mi set wide
. mi register imputed x1
. set seed 1712
. mi impute regress x1 = i.cat, add(10)
Univariate imputation Imputations = 10
Linear regression added = 10
Imputed: m=1 through m=10 updated = 0
| Observations per m
|----------------------------------------------
Variable | complete incomplete imputed | total
---------------+-----------------------------------+----------
x1 | 926 14 14 | 940
--------------------------------------------------------------
(complete + incomplete = total; imputed is the minimum across m
of the number of filled in observations.)
Once the dataset has been mi set
ted, the second step is to register the variables with missing values. In this example, x1
is a variable that has missing values and mi register imputed x1
declares this variable as a variable to be imputed. A good practice for reproducible results is to set the seed of the Stata's pseudo random number generator using the command set seed #
, where #
is any number between between 0 and 2^31-1.
Then, the command mi impute regress x1 = i.cat, add(10)
can be used to fill in missing values of x1
using the set of dummy variables from the categorical variable cat
through the regress
method (see help mi impute
for detail on the available methods). The option add(10)
specifies the number of imputations to add to the mi
data (currently, the total number of imputations cannot exceed 1,000). After mi impute regress x1 = i.cat, add(10)
has been executed, ten new variables _#_x1
(with # = 1,...,10) will be created in the dataset, each representing an imputed version of x1
.
. mi estimate, dots: xsmle y x1, wmat(W) model(sdm) fe type(ind)
Imputations (10):
.........10 done
Multiple-imputation estimates Imputations = 10
SDM with spatial fixed-effects Number of obs = 940
Average RVI = 0.0409
Complete DF = 748
DF adjustment: Small sample DF: min = 421.51
avg = 624.79
max = 732.54
Model F test: Equal FMI F( 7, 730.5) = 214.64
Within VCE type: OIM Prob > F = 0.0000
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Main |
x1 | .3927311 .0358267 10.96 0.000 .3223164 .4631458
-------------+----------------------------------------------------------------
Wx |
x1 | .7172546 .0759222 9.45 0.000 .5682036 .8663056
-------------+----------------------------------------------------------------
Spatial |
rho | .3342481 .0404076 8.27 0.000 .2548957 .4136005
-------------+----------------------------------------------------------------
Variance |
sigma2_e | .8201919 .0385477 21.28 0.000 .7445121 .8958717
-------------+----------------------------------------------------------------
Direct |
x1 | .4641954 .0315439 14.72 0.000 .4021925 .5261983
-------------+----------------------------------------------------------------
Indirect |
x1 | 1.215779 .1155787 10.52 0.000 .9888711 1.442687
-------------+----------------------------------------------------------------
Total |
x1 | 1.679974 .127172 13.21 0.000 1.430302 1.929647
------------------------------------------------------------------------------
Finally, as documented in help mi estimate
, the prefix command mi estimate: estimation_command
can be used to execute the estimation_command
on the imputed _#_x1
variables. This command will adjust coefficients and standard errors for the variability between imputations according to the combination rules by Rubin (1987). In this example, the command
mi estimate, dots: xsmle y x1, wmat(W) model(sdm) fe type(ind)
estimates a spatial fixed effects Durbin model on the ten imputed versions of the x1
variable.
References
Official Stata manuals and help files.
Rubin, D. B. 1987. Multiple Imputation for Nonresponse in Surveys. New York: Wiley.