A2.3 Calculating marginal effects
- The example that follows shows how to calculate a ME using mtable and mlincom
- This is to show you exactly what is happening behind the scenes
- I recommend letting mchange do this for you in most cases
The examples use the model with a quadratic term for age from 2.2:
use "https://tdmize.github.io/data/data/cda_gss", clear(cda_gss.dta | GSS 1972-2021 CDA - Categorical Data Analysis | date created 202
> 3)regress occprest c.age##c.age i.woman i.college Source | SS df MS Number of obs = 58,304
-------------+---------------------------------- F(4, 58299) = 4746.11
Model | 2797112.11 4 699278.027 Prob > F = 0.0000
Residual | 8589596.79 58,299 147.336949 R-squared = 0.2456
-------------+---------------------------------- Adj R-squared = 0.2456
Total | 11386708.9 58,303 195.302281 Root MSE = 12.138
-------------------------------------------------------------------------------
occprest | Coefficient Std. err. t P>|t| [95% conf. interval]
--------------+----------------------------------------------------------------
age | 0.284 0.017 16.988 0.000 0.251 0.317
|
c.age#c.age | -0.002 0.000 -14.751 0.000 -0.003 -0.002
|
woman |
Women | -0.159 0.101 -1.576 0.115 -0.357 0.039
|
college |
College De.. | 16.238 0.121 134.149 0.000 16.001 16.475
_cons | 31.359 0.389 80.511 0.000 30.595 32.122
-------------------------------------------------------------------------------est store opmod- First, find the mean of age and calculate what mean + SD is
sum age if e(sample) // Ensure same sample as regression model Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
age | 58,304 46.165 17.15691 18 89di 46.165 + 17.1569163.32191Calculating marginal effects with mtable and mlincom
- Can use mtable to make predictions at the mean and one standard deviation above the mean
- Two options are important here: “atmeans” tells Stata to hold the control variables constant at their means; “post” allows you to test the predictions with subsequent commands
mtable, at(age=(46.165 63.32191)) atmeans postExpression: Linear prediction, predict()
| age xb
----------+-------------------
1 | 46.2 42.850
2 | 63.3 43.143
Specified values of covariates
| 1. 1.
| woman college
----------+--------------------
Current | .542 .226- mlincom allows you test whether the predictions that mtable gave you are significantly different (i.e. whether the marginal effect for that independent variable is statistically significant)
- mlincom’s syntax refers to the row number of the prediction from mtable. See the example above: the first column of mtable tells you the row number for that prediction
mlincom 2 - 1 | lincom pvalue ll ul
-------------+---------------------------------------
1 | 0.293 0.000 0.180 0.406 Calculating marginal effects with mchange
mchange is a command that automatically calculates marginal effects for you. By default, for continuous independent variables mchange calculates discrete changes of mean + 1, mean + SD, and the instantaneous change (labeled as “marginal”). For binary variables it reports 1 vs. 0 (note this is one of the reasons it is vital to use the c. and i. prefixes in your regression command so you get the correct MEs)
est restore opmod(results opmod are active now)mchange age collegeregress: Changes in xb | Number of obs = 58304
Expression: Linear prediction, predict()
| Change p-value
---------------------------------+---------------------
age |
+1 | 0.056 0.000
+SD | 0.293 0.000
Marginal | 0.059 0.000
college |
College Degree vs No Col Degree | 16.238 0.000
Average prediction
42.133 On average, a standard deviation increase in age (about 17.2 years) is predicted to increase occupational prestige by 0.293 points
* AMEs
est restore opmod(results opmod are active now)mtable, at(age=(46.165 63.32191)) postExpression: Linear prediction, predict()
| age xb
----------+-------------------
1 | 46.2 42.850
2 | 63.3 43.143
Specified values where .n indicates no values specified with at()
| No at()
----------+---------
Current | .nmlincom 2 - 1 | lincom pvalue ll ul
-------------+---------------------------------------
1 | 0.293 0.000 0.180 0.406 Quick note: Predictions vs MEs and significance tests
- Note that a marginal effect (ME) is an effect: the result of an increase in a focal IV from a starting value to an ending value
- The two individual pieces of the ME (prediction at starting value and prediction at ending values) are predictions
- We can make predictions at all values of an IV, e.g. when we plot a continuous IV (e.g. graph age predictions from age 18 to 90)
- Often want to construct a statistical significance test on a ME: almost always whether the ME is not zero (this is what mchange is doing)
- Statistical significance tests for predictions are less useful as whether the prediction is zero or not is rarely of interest (e.g. what margins, at(age=(20(10)80)) gives you)
- However, the confidence intervals on predictions can be useful (especially for a plot)
- I always recommend adding CIs to plots of predictions
- Statistical significance tests for predictions are less useful as whether the prediction is zero or not is rarely of interest (e.g. what margins, at(age=(20(10)80)) gives you)