Econometric Tools: Marginal Effects in Stata 1 Introduction Marginal effects tell us how will the outcome variable change when an explanatory variable changes. In many cases the marginal effects are constant, but in some cases they are not. In this lecture we will see a few ways of estimating marginal effects in Stata. 2 Marginal Effects in OLS In OLS, the estimating equation may be given by: Y = β0 + β1 X1 + β2 X2 (1) , where Y is wage, X1 is grade and X2 is tenure. The marginal effect of grade is given by: ∂Y = β1 ∂X1 (2) As we can see, the marginal effect is a constant β1 , and doesn’t depend on anything else. In the regression below, every additional year of schooling will add 70 cents to the hourly wage. The model predicts that for all individuals, irrespective of their grade or any other characteristic. 1 output window . regress wage grade tenure Source | SS df MS -------------+-----------------------------Model | 9291.77652 2 4645.88826 Residual | 64796.1912 2226 29.1088011 -------------+-----------------------------Total | 74087.9678 2228 33.2531274 Number of obs F( 2, 2226) Prob > F R-squared Adj R-squared Root MSE = = = = = = 2229 159.60 0.0000 0.1254 0.1246 5.3953 -----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------grade | .704318 .045626 15.44 0.000 .6148441 .793792 tenure | .1470505 .0209133 7.03 0.000 .1060389 .1880621 _cons | -2.311109 .6063331 -3.81 0.000 -3.500146 -1.122071 ------------------------------------------------------------------------------ To motivate the discussion, say we want to study if the effect of schooling on wages differs between unionized and non-unionized workers. We may start by estimating separate linear regressions for each group. For instance, output window . regress wage grade tenure if union==1 Source | SS df MS -------------+-----------------------------Model | 1472.168 2 736.084001 Residual | 6519.19464 456 14.2964795 -------------+-----------------------------Total | 7991.36264 458 17.44839 Number of obs F( 2, 456) Prob > F R-squared Adj R-squared Root MSE = = = = = = 459 51.49 0.0000 0.1842 0.1806 3.7811 -----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------grade | .5209993 .0641173 8.13 0.000 .3949974 .6470013 tenure | .1455878 .0292043 4.99 0.000 .088196 .2029795 _cons | .4673078 .8892094 0.53 0.599 -1.280149 2.214764 ------------------------------------------------------------------------------ 2 output window . regress wage grade tenure if union==0 Source | SS df MS -------------+-----------------------------Model | 5771.80721 2 2885.9036 Residual | 17930.1479 1404 12.7707606 -------------+-----------------------------Total | 23701.9551 1406 16.8577205 Number of obs F( 2, 1404) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1407 225.98 0.0000 0.2435 0.2424 3.5736 -----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------grade | .7118187 .0388866 18.30 0.000 .6355366 .7881009 tenure | .1567362 .0177014 8.85 0.000 .1220121 .1914604 _cons | -3.012123 .5161758 -5.84 0.000 -4.024682 -1.999564 ------------------------------------------------------------------------------ The first regression shows that the marginal effect of an additional year of schooling on hourly wages is 52 cents for union workers, and 71 cents for non-union workers. The point estimates are different, but in econometrics (and in general, in statistical analysis) we want to know if the difference between those point estimates is statistically significant. One may be tempted to look at their confidence intervals and infer that the coefficients are statistically different from each other if and only if their confidence intervals don’t overlap. However, this is misleading, given that there may be an underlying correlation between both coefficients. Keep in mind that these coefficients are random variables themselves, and there may (or may not) be an underlying correlation between them. To account for any type of correlation between the coefficients, we can estimate a single regression, interacting each explanatory variable with the “union” variable (and adding a “union” dummy). In this case, the estimating equation is given by: Y = β0 + λ1 × union + β1 grade + λ1 grade × union + β2 tenure + λ2 tenure × union (3) Taking derivatives, we find that the marginal effect of schooling is given by: ∂Y = β1 + λ1 union ∂grade (4) Given that union takes two values, either 0 or 1, the marginal effect is given by β1 (=β1 + λ1 ∗ 0) for non-unionized workers and β1 + λ1 (= β1 + λ1 ∗ 1) for unionized workers. We can do this in Stata by creating the interactions directly or by using the xi prefix. To create the interactions directly: g tenure_union = tenure * union g grade_union = grade * union regress wage tenure tenure_union grade grade_union union 3 output window . regress wage tenure tenure_union grade grade_union union Source | SS df MS -------------+-----------------------------Model | 7980.68416 5 1596.13683 Residual | 24449.3425 1860 13.1448078 -------------+-----------------------------Total | 32430.0267 1865 17.3887542 Number of obs F( 5, 1860) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1866 121.43 0.0000 0.2461 0.2441 3.6256 -----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------tenure | .1567362 .0179588 8.73 0.000 .1215147 .1919578 tenure_union | -.0111484 .0332672 -0.34 0.738 -.0763934 .0540965 grade | .7118187 .039452 18.04 0.000 .6344439 .7891936 grade_union | -.1908194 .0730501 -2.61 0.009 -.3340882 -.0475506 union | 3.47943 1.000619 3.48 0.001 1.516975 5.441885 _cons | -3.012123 .5236805 -5.75 0.000 -4.039186 -1.98506 ------------------------------------------------------------------------------ 2.1 The xi prefix A slightly more elegant way of including interactions is by using the “xi:” prefix (see help xi) For illustration purposes lets change the explanatory variables. In this case, we want to see how wages bay by union affiliation and race. Note that race is a categorical variable, so we need to include an indicator variable (dummy variable) for each race, as well as for union affiliation. We can do so easily with the xi prefix: 4 output window . xi: reg wage i.race i.union i.race _Irace_1-3 i.union _Iunion_0-1 (naturally coded; _Irace_1 omitted) (naturally coded; _Iunion_0 omitted) Source | SS df MS -------------+-----------------------------Model | 1300.67748 3 433.559161 Residual | 31312.7649 1874 16.7090528 -------------+-----------------------------Total | 32613.4424 1877 17.3753023 Number of obs F( 3, 1874) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1878 25.95 0.0000 0.0399 0.0383 4.0877 -----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------_Irace_2 | -1.19603 .2144686 -5.58 0.000 -1.616652 -.7754074 _Irace_3 | .7893447 .8421086 0.94 0.349 -.8622244 2.440914 _Iunion_1 | 1.561178 .219941 7.10 0.000 1.129823 1.992533 _cons | 7.491176 .1214895 61.66 0.000 7.252907 7.729445 ------------------------------------------------------------------------------ We may also include interactions between union affiliation and race: output window . xi: reg wage i.race*i.union i.race _Irace_1-3 i.union _Iunion_0-1 i.race*i.union _IracXuni_#_# (naturally coded; _Irace_1 omitted) (naturally coded; _Iunion_0 omitted) (coded as above) Source | SS df MS -------------+-----------------------------Model | 1531.00192 5 306.200385 Residual | 31082.4404 1872 16.6038678 -------------+-----------------------------Total | 32613.4424 1877 17.3753023 Number of obs F( 5, 1872) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1878 18.44 0.0000 0.0469 0.0444 4.0748 ------------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------------+---------------------------------------------------------------_Irace_2 | -1.614053 .2514712 -6.42 0.000 -2.107247 -1.12086 _Irace_3 | 1.881194 1.026421 1.83 0.067 -.1318556 3.894244 _Iunion_1 | 1.153829 .2660411 4.34 0.000 .6320603 1.675597 _IracXuni_2_1 | 1.492629 .4776786 3.12 0.002 .5557899 2.429467 _IracXuni_3_1 | -3.140969 1.784377 -1.76 0.079 -6.640547 .3586095 _cons | 7.5821 .1256907 60.32 0.000 7.335591 7.828608 ------------------------------------------------------------------------------- As you can see, this includes an intercept, dummies for each race (except the comparison category), a dummy for union, and the interactions between each race and union. If we want to include interactions between categorical and continuous variables, we may use 5 output window . xi: reg wage i.union*tenure i.union _Iunion_0-1 i.union*tenure _IuniXtenur_# (naturally coded; _Iunion_0 omitted) (coded as above) Source | SS df MS -------------+-----------------------------Model | 2742.55819 3 914.186063 Residual | 29699.5985 1864 15.933261 -------------+-----------------------------Total | 32442.1567 1867 17.3766238 Number of obs F( 3, 1864) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1868 57.38 0.0000 0.0845 0.0831 3.9916 ------------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------------+---------------------------------------------------------------_Iunion_1 | 1.245279 .344204 3.62 0.000 .5702131 1.920345 tenure | .1897243 .0196568 9.65 0.000 .1511727 .228276 _IuniXtenur_1 | -.0151133 .0363007 -0.42 0.677 -.0863076 .0560809 _cons | 6.06189 .1608928 37.68 0.000 5.746341 6.377439 ------------------------------------------------------------------------------- Note that this includes union, tenure, and the interaction between them. We may also use the vertical line |, as follows. Note that in this syntax the union dummy is not included in the regression (i.e. union and non-union workers are restricted to have the same intercept). output window . xi: reg wage i.union|tenure i.union _Iunion_0-1 i.union|tenure _IuniXtenur_# (naturally coded; _Iunion_0 omitted) (coded as above) Source | SS df MS -------------+-----------------------------Model | 2534.01016 2 1267.00508 Residual | 29908.1465 1865 16.0365397 -------------+-----------------------------Total | 32442.1567 1867 17.3766238 Number of obs F( 2, 1865) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1868 79.01 0.0000 0.0781 0.0771 4.0046 ------------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------------+---------------------------------------------------------------tenure | .1647852 .018468 8.92 0.000 .128565 .2010054 _IuniXtenur_1 | .0870452 .0228859 3.80 0.000 .0421606 .1319299 _cons | 6.333977 .142694 44.39 0.000 6.054121 6.613834 ------------------------------------------------------------------------------- The following regression shows the results. We see that the coefficient on grade is 0.71, and the coefficient on the interaction (“ IuniXgrade 1”) is -.19. Adding them together we get a coefficient of .52, which is the marginal effect for union workers. As you can see, this produces numerically identical marginal effects as estimating the coefficients in each subsample (union vs non-union workers). To get the right result, it is key to interact each explanatory variable with the group 6 variable (in this case union) and to include a dummy for the group variable. This allows unionized workers to have their own intercept (by including the union dummy) and their own slopes (interacting each explanatory variable with the union dummy). Note that I used the vertical line for interactions and added the union dummy by hand. output window . xi: reg wage union i.union|grade i.union|tenure i.union _Iunion_0-1 (naturally coded; _Iunion_0 omitted) i.union|grade _IuniXgrade_# (coded as above) i.union|tenure _IuniXtenur_# (coded as above) Source | SS df MS -------------+-----------------------------Model | 7980.68416 5 1596.13683 Residual | 24449.3425 1860 13.1448078 -------------+-----------------------------Total | 32430.0267 1865 17.3887542 Number of obs F( 5, 1860) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1866 121.43 0.0000 0.2461 0.2441 3.6256 ------------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------------+---------------------------------------------------------------union | 3.47943 1.000619 3.48 0.001 1.516975 5.441885 grade | .7118187 .039452 18.04 0.000 .6344439 .7891936 _IuniXgrade_1 | -.1908194 .0730501 -2.61 0.009 -.3340882 -.0475506 tenure | .1567362 .0179588 8.73 0.000 .1215147 .1919578 _IuniXtenur_1 | -.0111484 .0332672 -0.34 0.738 -.0763934 .0540965 _cons | -3.012123 .5236805 -5.75 0.000 -4.039186 -1.98506 ------------------------------------------------------------------------------- The following commands produce equivalent results . . . . 3 xi: xi: xi: xi: reg reg reg reg wage wage wage wage union i.union|grade i.union|tenure i.union*grade i.union|tenure i.union|grade i.union*tenure i.union*grade i.union*tenure Discrete Choice Models In economics we are interested in understanding how individuals make choices. Most models associate choices with the utility obtained from making those choices, which is unobservable. However, under certain conditions, if we observe people’s choices we could infer something about their preferences. For instance, if we observe that somebody owns a car, we may infer that, for that person, the utility of owning a car is higher than the utility of not owning a car. By observing whether an individual belongs to a union or not we could infer if the utility of belonging to a union is higher than the utility of not belonging to one. Bivariate choice models, like probit and logit, are used to estimate the probability of an event. The outcome is now dichotomous: it takes the value of zero or one. Now the quantity 7 of interest is the probability of having a value of 1. For instance, the probability that a worker belongs to a union. 3.1 Logit Logit models assume this probability is given by the following formula: Pr(y = 1) = exβ = Λ(xβ) 1 + exβ (5) or, equivalently, 1 1 + e−xβ , where xβ = β0 + β1 x1 + β2x2 + ... + βN xN . In this case, the marginal effect of xk is given by Pr(y = 1) = exβ ∂(xβ) ∂ Pr(y = 1) = 2 xβ ∂xk (1 + e ) ∂xk exβ βk = (1 + exβ )2 = Λ(xβ) (1 − Λ(xβ)) βk = Pr(y = 1) × Pr(y = 0) × βk (6) (7) (8) (9) (10) Deriving this formulas is beyond the scope of this lab, but the main take-home point is that the marginal effect depends on the value of all the x variables. For instance, the marginal effect of schooling will be different for a worker with 2 years of tenure than that for a worker with 5 years of tenure. It will also be different for different levels of schooling. To estimate a logit regression we use the logit command, which works very much like the regress command: 8 output window . logit union wage age Iteration Iteration Iteration Iteration 0: 1: 2: 3: log log log log likelihood likelihood likelihood likelihood = = = = -1046.6242 -1026.5473 -1026.2712 -1026.2712 Logistic regression Number of obs LR chi2(2) Prob > chi2 Pseudo R2 Log likelihood = -1026.2712 = = = = 1878 40.71 0.0000 0.0194 -----------------------------------------------------------------------------union | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------wage | .0779941 .0122874 6.35 0.000 .0539111 .102077 age | .008367 .0178993 0.47 0.640 -.0267149 .043449 _cons | -2.06519 .711726 -2.90 0.004 -3.460147 -.6702328 ------------------------------------------------------------------------------ To calculate the marginal effects: output window . predict phat if e(sample), pr . g dydwage = phat * (1 - phat) * _b[wage] (368 missing values generated) . g dydage = phat * (1 - phat) * _b[age] (368 missing values generated) . sum dydwage dydage Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------dydwage | 1878 .0141079 .0019951 .0104284 .0194981 dydage | 1878 .0015135 .000214 .0011187 .0020917 3.2 Probit Probit models assume a normal distribution. Z Pr(y = 1) = x ∞ 9 1 −x2 √ e 2 dx 2π (11) The marginal effect of a continuous variable xk is given by: ∂ Pr(y = 1) = φ(xβ)βk ∂xk (12) −xβ 2 , where φ(xβ) is the standard normal density evaluated at xβ so that φ(xβ) = √12π e 2 . To calculate the value of the normal density at a given value of xb in Stata, we can use normalden. output window . probit union wage age Iteration Iteration Iteration Iteration 0: 1: 2: 3: log log log log likelihood likelihood likelihood likelihood = -1046.6242 = -1025.4675 = -1025.436 = -1025.436 Probit regression Log likelihood = Number of obs LR chi2(2) Prob > chi2 Pseudo R2 -1025.436 = = = = 1878 42.38 0.0000 0.0202 -----------------------------------------------------------------------------union | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------wage | .0483093 .0074246 6.51 0.000 .0337574 .0628611 age | .0044723 .0105121 0.43 0.671 -.016131 .0250755 _cons | -1.243423 .4175672 -2.98 0.003 -2.061839 -.4250059 ------------------------------------------------------------------------------ We can calculate the marginal effects of wages and age on union affiliation for each individual in the sample, as follows: 10 output window . predict phat if e(sample), xb (368 missing values generated) . g index = normalden(phat) (368 missing values generated) . g dydwage_probit = index * _b[wage] (368 missing values generated) . g dydage_probit = index * _b[age] (368 missing values generated) . sum mfx Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------dydwage_pr~t | 1878 .0148534 .0017396 .0114285 .0192726 dydage_pro~t | 1878 .0013751 .000161 .001058 .0017842 As you can see, the marginal effects from the probit model differ slightly from those in the logit model. In practice, there is usually little difference between this models. Note that we should compare the models based on the marginal effects, not the estimated coefficients. The main caveat of this approach is that we don’t have a way to test if these marginal effects are significantly different than zero. We could use bootstrapping techniques, but that is beyond the scope of this lecture. The “mfx” and “margins” commands come to the rescue. 4 mfx As we have seen, the marginal effect of a variable in a discrete choice model depends on the values taken by each of the covariates. The mfx command allows us to estimate quickly the marginal effects evaluated at the mean values of the covariates. As you can see, the marginal effects are relatively close to the values we found with dydage probit and dydwage probit, but these are not the same. The marginal effect evaluated at the average values of the covariates is not the same as the average marginal effect. output window Marginal effects after probit y = Pr(union) (predict) = .24117773 -----------------------------------------------------------------------------variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X ---------+-------------------------------------------------------------------wage | .0150581 .00231 6.53 0.000 .010536 .01958 7.56542 age | .001394 .00328 0.43 0.671 -.005028 .007816 39.2247 ------------------------------------------------------------------------------ 11 The values taken by wage and age are specified in the last column of the output, under the “X”. This means that the marginal effects reported in the table are valid for a 39 year old with an hourly wage of $7.57. The effects will be different for people with different values of age or wage. We can calculate the marginal effects at different values by specifying them with the “at” option. The order of the values matches the order of the covariates in the regression: output window . mfx, at(5,35) Marginal effects after probit y = Pr(union) (predict) = .1989586 -----------------------------------------------------------------------------variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X ---------+-------------------------------------------------------------------wage | .0134823 .00193 6.99 0.000 .009704 .01726 5 age | .0012481 .00289 0.43 0.666 -.004411 .006907 35 ------------------------------------------------------------------------------ 5 margins The margins command estimates the marginal effects for each observation and reports the average marginal effect. As you can see, these numbers match exactly the marginal effects we had calculated earlier by hand for the probit and logit models. The main advantage is that margins also provides standard errors and confidence intervals, so it allows to test if the average marginal effect is statistically significant. output window . logit union wage age [output omitted] . margins, dydx(wage age) Average marginal effects Model VCE : OIM Number of obs = 1878 Expression : Pr(union), predict() dy/dx w.r.t. : wage age -----------------------------------------------------------------------------| Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------wage | .0141079 .0021485 6.57 0.000 .009897 .0183189 age | .0015135 .0032372 0.47 0.640 -.0048313 .0078583 ------------------------------------------------------------------------------ 12 output window . probit union wage age [output omitted] . margins, dydx(wage age) Average marginal effects Model VCE : OIM Number of obs = 1878 Expression : Pr(union), predict() dy/dx w.r.t. : wage age -----------------------------------------------------------------------------| Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------wage | .0148534 .0022088 6.72 0.000 .0105243 .0191825 age | .0013751 .0032318 0.43 0.670 -.0049592 .0077093 ------------------------------------------------------------------------------ 13