# Marginal effects in a linear model

Stata’s margins command has been a powerful tool for many economists. It can calculate predicted means as well as predicted marginal effects. However, we do need to be careful when we use it when fixed effects are included. In a linear model, everything works out fine. However, in a non-linear model, you may not want to use margins, since it’s not calculating what you have in mind.

In a linear model with fixed effects, we can do it either by “demeaning” every variable, or include dummy variables. They return the same results. Fortunately, marginal effects can be calculated the same way in both models.

For example:

clear
sysuse auto
xtset rep78
xtreg price c.mpg##c.trunk, fe
margins , dydx(mpg)
reg price c.mpg##c.trunk i.rep78
margins , dydx(mpg)

##
## . clear
##
## . sysuse auto
## (1978 Automobile Data)
##
## . xtset rep78
##        panel variable:  rep78 (unbalanced)
##
## . xtreg price c.mpg##c.trunk, fe
##
## Fixed-effects (within) regression               Number of obs     =         69
## Group variable: rep78                           Number of groups  =          5
##
## R-sq:                                           Obs per group:
##      within  = 0.2570                                         min =          2
##      between = 0.0653                                         avg =       13.8
##      overall = 0.2237                                         max =         30
##
##                                                 F(3,61)           =       7.03
## corr(u_i, Xb)  = -0.4133                        Prob > F          =     0.0004
##
## ------------------------------------------------------------------------------
##        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -98.12003   226.8708    -0.43   0.667    -551.7763    355.5362
##        trunk |   295.0544   343.3934     0.86   0.394    -391.6032     981.712
##              |
##        c.mpg#|
##      c.trunk |  -12.23318   15.94713    -0.77   0.446    -44.12143    19.65506
##              |
##        _cons |    7574.85   5321.325     1.42   0.160    -3065.797     18215.5
## -------------+----------------------------------------------------------------
##      sigma_u |   992.2156
##      sigma_e |  2631.2869
##          rho |  .12449059   (fraction of variance due to u_i)
## ------------------------------------------------------------------------------
## F test that all u_i=0: F(4, 61) = 0.86                       Prob > F = 0.4948
##
## . margins , dydx(mpg)
##
## Average marginal effects                        Number of obs     =         69
## Model VCE    : Conventional
##
## Expression   : Linear prediction, predict()
## dy/dx w.r.t. : mpg
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -268.4981   74.12513    -3.62   0.000    -413.7807   -123.2156
## ------------------------------------------------------------------------------
##
## . reg price c.mpg##c.trunk i.rep78
##
##       Source |       SS           df       MS      Number of obs   =        69
## -------------+----------------------------------   F(7, 61)        =      3.19
##        Model |   154453046         7  22064720.8   Prob > F        =    0.0061
##     Residual |   422343913        61  6923670.71   R-squared       =    0.2678
## -------------+----------------------------------   Adj R-squared   =    0.1838
##        Total |   576796959        68  8482308.22   Root MSE        =    2631.3
##
## ------------------------------------------------------------------------------
##        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -98.12003   226.8708    -0.43   0.667    -551.7763    355.5362
##        trunk |   295.0544   343.3934     0.86   0.394    -391.6032     981.712
##              |
##        c.mpg#|
##      c.trunk |  -12.23318   15.94713    -0.77   0.446    -44.12143    19.65506
##              |
##        rep78 |
##           2  |   438.0002   2161.922     0.20   0.840    -3885.031    4761.031
##           3  |   987.1363   2022.606     0.49   0.627    -3057.315    5031.587
##           4  |   1240.944   2046.417     0.61   0.547     -2851.12    5333.008
##           5  |    2605.83   2161.837     1.21   0.233    -1717.031    6928.691
##              |
##        _cons |   6355.731   5209.899     1.22   0.227    -4062.105    16773.57
## ------------------------------------------------------------------------------
##
## . margins , dydx(mpg)
##
## Average marginal effects                        Number of obs     =         69
## Model VCE    : OLS
##
## Expression   : Linear prediction, predict()
## dy/dx w.r.t. : mpg
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -268.4981   74.12513    -3.62   0.001    -416.7205   -120.2758
## ------------------------------------------------------------------------------
##
## .

All is fine.

# Marginal effects in a non-linear model

In a nonlinear model, we need to be more careful:

clear
sysuse auto
xtset rep78
xtpoisson price mpg trunk, fe
margins , dydx(mpg)
margins , dydx(mpg) predict(nu0)
poisson price mpg trunk i.rep78
margins , dydx(mpg)

##
## . clear
##
## . sysuse auto
## (1978 Automobile Data)
##
## . xtset rep78
##        panel variable:  rep78 (unbalanced)
##
## . xtpoisson price mpg trunk, fe
##
## Iteration 0:   log likelihood = -39282.052
## Iteration 1:   log likelihood = -27527.055
## Iteration 2:   log likelihood = -27518.944
## Iteration 3:   log likelihood = -27518.944
##
## Conditional fixed-effects Poisson regression    Number of obs     =         69
## Group variable: rep78                           Number of groups  =          5
##
##                                                 Obs per group:
##                                                               min =          2
##                                                               avg =       13.8
##                                                               max =         30
##
##                                                 Wald chi2(2)      =   22890.68
## Log likelihood  = -27518.944                    Prob > chi2       =     0.0000
##
## ------------------------------------------------------------------------------
##        price |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -.0450221   .0003814  -118.05   0.000    -.0457696   -.0442746
##        trunk |   .0047349   .0004772     9.92   0.000     .0037996    .0056702
## ------------------------------------------------------------------------------
##
## . margins , dydx(mpg)
##
## Average marginal effects                        Number of obs     =         69
## Model VCE    : OIM
##
## Expression   : Linear prediction, predict()
## dy/dx w.r.t. : mpg
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -.0450221   .0003814  -118.05   0.000    -.0457696   -.0442746
## ------------------------------------------------------------------------------
##
## . margins , dydx(mpg) predict(nu0)
##
## Average marginal effects                        Number of obs     =         69
## Model VCE    : OIM
##
## Expression   : Predicted number of events (assuming u_i=0), predict(nu0)
## dy/dx w.r.t. : mpg
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -.0190939   .0001245  -153.35   0.000    -.0193379   -.0188498
## ------------------------------------------------------------------------------
##
## . poisson price mpg trunk i.rep78
##
## Iteration 0:   log likelihood = -27550.942
## Iteration 1:   log likelihood = -27550.912
## Iteration 2:   log likelihood = -27550.912
##
## Poisson regression                              Number of obs     =         69
##                                                 LR chi2(6)        =   24962.86
##                                                 Prob > chi2       =     0.0000
## Log likelihood = -27550.912                     Pseudo R2         =     0.3118
##
## ------------------------------------------------------------------------------
##        price |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -.0450221   .0003814  -118.05   0.000    -.0457696   -.0442746
##        trunk |   .0047349   .0004772     9.92   0.000     .0037996    .0056702
##              |
##        rep78 |
##           2  |   .1476657   .0117935    12.52   0.000     .1245509    .1707805
##           3  |   .2295466   .0111741    20.54   0.000     .2076458    .2514474
##           4  |   .2726354   .0112656    24.20   0.000     .2505552    .2947155
##           5  |   .4682657   .0115137    40.67   0.000     .4456992    .4908321
##              |
##        _cons |   9.323117   .0149274   624.57   0.000      9.29386    9.352374
## ------------------------------------------------------------------------------
##
## . margins , dydx(mpg)
##
## Average marginal effects                        Number of obs     =         69
## Model VCE    : OIM
##
## Expression   : Predicted number of events, predict()
## dy/dx w.r.t. : mpg
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          mpg |  -276.7079   2.382193  -116.16   0.000    -281.3769   -272.0389
## ------------------------------------------------------------------------------
##
## .

In this example, “xtpoisson, fe” and “poisson i.rep78” returns the same results. Fixed effect Poisson model (sometimes called conditional fixed effect Poisson) is the same models as a Poisson model with dummies, just like a linear model (OLS with dummies is the same as fixed effect OLS). Poisson model and OLS are unique in this sense that there is no “incidental paramater” problem.

We see in this example, margins commands do not return the same marginal effects, even though the models are the same. The reason behind this is that in a conditional fixed effect Poisson, the fixed effects are not estimated (they are not in the final likelihood function that gets estimated). Therefore, we’ll have to make a decision what values to use as the values of the fixed effects. “margins, predict(nu0)” simply set all fixed effects to zero. On the other hand, margins after Poisson model with dummies does not do that. The fixed effect in that case gets estimated. Therefore the marginal effects in that case make more sense.

So our advise for a conditioanl Poisson model is that we should not use margins to calculate marginal effects afterwards; instead, we should simply stick with the original coefficient estimates.

The same logic applies to the conditional logit model. Fixed effects are not estimated in that model; simply setting them to zero does not make too much sense. In addition, conditional logit model is not the same model as a logit model with dummies, since there is the “incidental paramater” problem. Again, we should just focus on the coefficient estimates as the effect on the logged odds.

In other words, for fixed effect (conditional) logit model, the situation is worse: you cannot do logit with dummies, unless you have a deep panel. That is, when you have, say, more than 20 observations per group, the “incidental parameter” bias becomes negligible. If you stay with conditional logit model, the fixed effects are not estimated. Unfortunately the predicted probability depends on the fixed effects. Stata’s margins command after clogit (or xtlogit, fe) comes with a few options, but none is reasonable for the fixed effects. For example, the pu0 option is to assume all fixed effects being 0.

In a fixed effect logit model,

$log(P(y=1)/(1-P(y=1))) = \alpha_i + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1*x_2$

Here $$\alpha_i$$ is fixed effect for each firm. Therefore,

$P(y=1) = F(\alpha_i + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1*x_2)$

$$F$$ can be a normal CDF or a logit function. Therefore, without estimating $$\alpha_i$$, there is no way to predict $$P$$ in a reasonable way (assuming $$\alpha=0$$ is not reasonable to me).

However, if we stick with logged odds ($$LO=log(P(y=1)/(1-P(y=1)))$$), then $$LO$$ is a linear function of $$\alpha_i$$ and other covariates. In that case, the marginal effects of $$x_1$$ or $$x_2$$ on $$Y$$ has nothing to do with $$\alpha_i$$.

Therefore, we can use margins command to calcuate effects on the logged odds, which will be “predict(xb)” option. This is in fact, not different from the orginal coefficients; but allow you to make linear extrapolations.

clear
webuse union
clogit union c.age##i.south not_smsa grade, group(idcode)
margins, at( age=(15 20 25 30 35 40) south=(0 1)) predict(xb)
marginsplot
##
## . clear
##
## . webuse union
## (NLS Women 14-24 in 1968)
##
## . clogit union c.age##i.south not_smsa grade, group(idcode)
## note: multiple positive outcomes within groups encountered.
## note: 2,744 groups (14,165 obs) dropped because of all positive or
##       all negative outcomes.
##
## Iteration 0:   log likelihood = -4518.8815
## Iteration 1:   log likelihood = -4512.8224
## Iteration 2:   log likelihood = -4512.8192
## Iteration 3:   log likelihood = -4512.8192
##
## Conditional (fixed-effects) logistic regression
##
##                                                 Number of obs     =     12,035
##                                                 LR chi2(5)        =      74.73
##                                                 Prob > chi2       =     0.0000
## Log likelihood = -4512.8192                     Pseudo R2         =     0.0082
##
## ------------------------------------------------------------------------------
##        union |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          age |   .0096842   .0050265     1.93   0.054    -.0001676     .019536
##      1.south |  -1.382178    .276966    -4.99   0.000    -1.925022   -.8393346
##              |
##  south#c.age |
##           1  |   .0208997   .0081247     2.57   0.010     .0049756    .0368238
##              |
##     not_smsa |   .0195233   .1131292     0.17   0.863    -.2022058    .2412523
##        grade |   .0822276   .0419062     1.96   0.050      .000093    .1643622
## ------------------------------------------------------------------------------
##
## . margins, at( age=(15 20 25 30 35 40) south=(0 1)) predict(xb)
##
## Predictive margins                              Number of obs     =     12,035
## Model VCE    : OIM
##
## Expression   : Linear prediction, predict(xb)
##
## 1._at        : age             =          15
##                south           =           0
##
## 2._at        : age             =          15
##                south           =           1
##
## 3._at        : age             =          20
##                south           =           0
##
## 4._at        : age             =          20
##                south           =           1
##
## 5._at        : age             =          25
##                south           =           0
##
## 6._at        : age             =          25
##                south           =           1
##
## 7._at        : age             =          30
##                south           =           0
##
## 8._at        : age             =          30
##                south           =           1
##
## 9._at        : age             =          35
##                south           =           0
##
## 10._at       : age             =          35
##                south           =           1
##
## 11._at       : age             =          40
##                south           =           0
##
## 12._at       : age             =          40
##                south           =           1
##
## ------------------------------------------------------------------------------
##              |            Delta-method
##              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##          _at |
##           1  |   1.202147   .5190753     2.32   0.021      .184778    2.219516
##           2  |    .133464   .5599015     0.24   0.812    -.9639228    1.230851
##           3  |   1.250568   .5153257     2.43   0.015      .240548    2.260588
##           4  |   .2863834   .5465398     0.52   0.600    -.7848148    1.357582
##           5  |   1.298989   .5127819     2.53   0.011     .2939548    2.304023
##           6  |   .4393029   .5349589     0.82   0.412    -.6091973    1.487803
##           7  |    1.34741   .5114619     2.63   0.008     .3449629    2.349857
##           8  |   .5922223   .5252767     1.13   0.260    -.4373011    1.621746
##           9  |   1.395831   .5113752     2.73   0.006     .3935538    2.398108
##          10  |   .7451418   .5175997     1.44   0.150    -.2693351    1.759619
##          11  |   1.444252   .5125224     2.82   0.005     .4397264    2.448777
##          12  |   .8980612   .5120182     1.75   0.079    -.1054761    1.901598
## ------------------------------------------------------------------------------
##
## . marginsplot
##
##   Variables that uniquely identify margins: age south

In this example, we have a fixed effect logit on union status, with age and south interaction, age as continuous variable. Suppose we’d like to see the predicted logged odds of union status for different age and south/north, then we can still use margins to predict logged odds. But we cannot use margins to predict probability, since the fixed effects are not estimated.