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Note: This FAQ is for Stata 10 and older versions of Stata. In Stata 11,
the margins command replaced mfx.
I am using a model with interactions. How can I obtain marginal effects and
their standard errors?
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Title
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Obtaining marginal effects and their standard errors after estimations
with interactions
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Author
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May Boggess, StataCorp
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Date
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May 2004; revised August 2004
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The marginal effect of an independent variable is the derivative (that is,
the slope) of a given function of the covariates and coefficients of the
preceding estimation. The derivative is evaluated at a point that is
usually, and by default, the means of the covariates.
The mfx command
assumes that the variables in the estimation are independent. Let’s
start with an example. Say that we have the quadratic model
mpg = b0 + b1*weight + b2*weight^2
which we fit with
regress.
The marginal effect of the linear predictor is the derivative of this
function with respect to weight, evaluated at the average weight. Calling
the mean weight meanwei, we have
dydx = b1 + 2*b2*meanwei
If we ran mfx after
regress with this model, Stata would not know that
the second covariate was a function of the first, so it would calculate the
derivative to be just b1.
. sysuse auto, clear
(1978 Automobile Data)
. generate wei2=weight*weight
. regress mpg weight wei2
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 2, 71) = 72.80
Model | 1642.52197 2 821.260986 Prob > F = 0.0000
Residual | 800.937487 71 11.2808097 R-squared = 0.6722
-------------+------------------------------ Adj R-squared = 0.6630
Total | 2443.45946 73 33.4720474 Root MSE = 3.3587
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | -.0141581 .0038835 -3.65 0.001 -.0219016 -.0064145
wei2 | 1.32e-06 6.26e-07 2.12 0.038 7.67e-08 2.57e-06
_cons | 51.18308 5.767884 8.87 0.000 39.68225 62.68392
------------------------------------------------------------------------------
. mfx
Marginal effects after regress
y = Fitted values (predict)
= 21.297297
------------------------------------------------------------------------------
variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X
---------+--------------------------------------------------------------------
weight | -.0141581 .00388 -3.65 0.000 -.02177 -.006546 3019.46
wei2 | 1.32e-06 .00000 2.12 0.034 9.8e-08 2.6e-06 9.7e+06
------------------------------------------------------------------------------
To obtain the marginal effects of such a function of the covariates we can
use nlcom.
This will not be as easy as using mfx because
nlcom can't do the differentiation for us, as
mfx does. The technique we are about to use relies
on our knowing the derivative of the function beforehand, as we did in our
example above.
The reason we want to use nlcom is that it
calculates the standard error of the marginal effect by the delta method.
We can also use
predictnl
in the same way since it is also designed to use the delta method to obtain
standard errors.
Here are some examples. Let’s start with out first example from above:
. sysuse auto, clear
(1978 Automobile Data)
. generate wei2=weight*weight
. regress mpg weight wei2
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 2, 71) = 72.80
Model | 1642.52197 2 821.260986 Prob > F = 0.0000
Residual | 800.937487 71 11.2808097 R-squared = 0.6722
-------------+------------------------------ Adj R-squared = 0.6630
Total | 2443.45946 73 33.4720474 Root MSE = 3.3587
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | -.0141581 .0038835 -3.65 0.001 -.0219016 -.0064145
wei2 | 1.32e-06 6.26e-07 2.12 0.038 7.67e-08 2.57e-06
_cons | 51.18308 5.767884 8.87 0.000 39.68225 62.68392
------------------------------------------------------------------------------
. quietly summarize weight if e(sample)
. local meanwei = r(mean)
. display " y = " _b[weight]* `meanwei' + _b[wei2]*`meanwei'^2 + _b[_cons]
y = 20.50813
. display "dydx = " _b[weight]+ 2*_b[wei2]* `meanwei'
dydx = -.00616011
. nlcom _b[weight]+ 2*_b[wei2]* `meanwei'
_nl_1: _b[weight]+ 2*_b[wei2]* 3019.45945945946
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_nl_1 | -.0061601 .0005108 -12.06 0.000 -.0071787 -.0051415
------------------------------------------------------------------------------
. predictnl dydx = _b[weight]+ 2*_b[wei2]* `meanwei' in 1, se(se)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydx se in 1
+----------------------+
| dydx se |
|----------------------|
. | -.0061601 .0005108 |
+----------------------+
Here is an example using
probit
with two covariates and their interaction. There are three derivatives we
can obtain with this model. We can obtain the two first derivatives, the
marginal effect of each variable, and the second derivative (the probability
of success differentiated with respect to both variables), which is called
the interaction effect.
. sysuse auto, clear
(1978 Automobile Data)
. replace weight=weight/1000
weight was int now float
(74 real changes made)
. replace length=length/10
length was int now float
(74 real changes made)
. generate wl=weight*length
. probit foreign weight length wl, nolog
Probit estimates Number of obs = 74
LR chi2(3) = 33.47
Prob > chi2 = 0.0000
Log likelihood = -28.29759 Pseudo R2 = 0.3716
------------------------------------------------------------------------------
foreign | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 2.662482 4.160075 0.64 0.522 -5.491116 10.81608
length | .6118874 .6413161 0.95 0.340 -.645069 1.868844
wl | -.2324718 .2263927 -1.03 0.304 -.6761934 .2112498
_cons | -7.115107 10.56744 -0.67 0.501 -27.82691 13.59669
------------------------------------------------------------------------------
Note: 1 failure and 0 successes completely determined.
. quietly summarize weight if e(sample)
. local meanwei = r(mean)
. quietly summarize length if e(sample)
. local meanlen = r(mean)
. local xb _b[weight]*`meanwei' + /*
> */ _b[len]*`meanlen' + _b[wl]*`meanwei'*`meanlen' + _b[_cons]
. predictnl dydw = normalden(`xb')*(_b[weight]+ _b[wl]*`meanlen') in 1, se(sew)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydw sew in 1
+----------------------+
| dydw sew |
|----------------------|
1. | -.5068093 .2714958 |
+----------------------+
. predictnl dydl = normalden(`xb')*(_b[len]+ _b[wl]*`meanwei') in 1, se(sel)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydl sel in 1
+----------------------+
| dydl sel |
|----------------------|
1. | -.0267456 .0940725 |
+----------------------+
. predictnl dydlw =normalden(`xb')*(-(`xb'))*(_b[weight]+ _b[wl]*`meanlen') /*
> */ *(_b[len]+ _b[wl]*`meanwei') /*
> */ + normalden(`xb')*( _b[wl]) in 1, se(selw)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydlw selw in 1
+----------------------+
| dydlw selw |
|----------------------|
1. | -.0339839 .1040486 |
+----------------------+
How do we handle a dummy variable? The marginal effect for a dummy variable
is not obtained by differentiation but as a difference of the predicted
value at 1 and the predicted value at 0. Here is an example of a logit model
with an interaction, where one variable is a dummy.
. sysuse auto, clear
(1978 Automobile Data)
. set seed 12345
. generate dum=uniform()>0.5
. table dum
----------------------
dum | Freq.
----------+-----------
0 | 37
1 | 37
----------------------
. generate td=turn*dum
. probit foreign turn dum td, nolog
Probit estimates Number of obs = 74
LR chi2(3) = 40.64
Prob > chi2 = 0.0000
Log likelihood = -24.712342 Pseudo R2 = 0.4512
------------------------------------------------------------------------------
foreign | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
turn | -.4162473 .1256398 -3.31 0.001 -.6624968 -.1699978
dum | -6.545885 5.52614 -1.18 0.236 -17.37692 4.285149
td | .1644299 .1498474 1.10 0.273 -.1292656 .4581253
_cons | 15.41459 4.641172 3.32 0.001 6.318063 24.51112
------------------------------------------------------------------------------
. quietly summarize turn if e(sample)
. local meantur= r(mean)
. quietly summarize dum if e(sample)
. local meandum = r(mean)
. local xb _b[turn]*`meantur' + /*
> */ _b[dum]*`meandum' + _b[td]*`meantur'*`meandum' + _b[_cons]
. predictnl dydt = normalden(`xb')*(_b[turn] + _b[td]*`meandum') in 1, se(set)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydt set in 1
+----------------------+
| dydt set |
|----------------------|
1. | -.0725871 .0163552 |
+----------------------+
. local xb1 _b[turn]*`meantur' + /*
> */ _b[dum]*1 + _b[td]*`meantur'*1 + _b[_cons]
. local xb0 _b[turn]*`meantur' + /*
> */ _b[dum]*0+ _b[td]*`meantur'*0 + _b[_cons]
. predictnl dydd = normal(`xb1')-normal(`xb0') in 1, se(sed)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dydd sed in 1
+----------------------+
| dydd sed |
|----------------------|
1. | -.0057505 .1292351 |
+----------------------+
. predictnl dyddt =normalden(`xb1')*(_b[turn] + _b[td]) /*
> */ - normalden(`xb0')*(_b[turn]) in 1, se(sedt)
(73 missing values generated)
Warning: prediction constant over observations; perhaps you meant to run nlcom.
. list dyddt sedt in 1
+---------------------+
| dyddt sedt |
|---------------------|
1. | .0378494 .0348098 |
+---------------------+
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