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Mathematics, 10.02.2020 17:01 lindsey0456

I have the following dilemma:

Calculate Marginal effect by hand with logit function and dummy binary variables

I understand-ish what marginal effects are, also the calculation of it, derivation of the sigmoid function and how to interpret it (as a the change in probability by increasing your variable of interest by "a little bit", this little bit being 1 for discrete vars or by a std(x)/1000 for continuous ). Now, the part I find tricky is to corroborate the results of the marginal effects by hand and recalculating the probabilities for x=0 and then x=1 (for example) with and then get a difference in probability equal to the marginal effect I got earlier, I am particularly stuck dummy variables since If I increase one, I have to decrease the other one, so I am not so sure how to work around it and interpret it. (this question also applies for highly correlated variables)

To make it more clear, let's say I have the following dataset:

x0, x1, x2, x3, x4, x5, x6
[1 , 0. , 0. , 4.6, 3.1, 1.5, 0.2],
[1 , 0. , 1. , 5. , 3.6, 1.4, 0.2],
[1 , 1. , 0. , 5.4, 3.9, 1.7, 0.4],
[1 , 0. , 1. , 4.6, 3.4, 1.4, 0.3],
[1 , 1. , 0. , 5. , 3.4, 1.5, 0.2],
[1 , 0. , 0. , 4.4, 2.9, 1.4, 0.2],
[1 , 0. , 1. , 4.9, 3.1, 1.5, 0.1],
[1 , 1. , 0. , 5.4, 3.7, 1.5, 0.2],
...

x0 = What will eventually the intercept.
x1, x2 = Dummy variables, I left 2/3, one dropped to avoid co linearity.
x3,4,5,6 = regular continuous variables

Coefficients:
[ 7.56986405, 0.75703164, 0.27158741, -0.37447474, -2.79926022, 1.43890492, -2.95286947]

logit
[-3.34739217,
-2.27001103,
-1.49517926,
-0.77178644,
-0.808111,
-2.48474722,
-1.76183804,
-0.90621541
...]

Probabilities
[0.03398066,
0.09363728,
0.18314562,
0.31609279,
0.30829318,
0.0769344 ,
0.14656029,
0.28777491,
...]

Marginal effect = p*(1-p) * B_j

Now let's say that I am interested in the marginal effect of x1 (one of the dummies), I will simply do: p*(1-p) * 0.7570 (its respective coefficient), which will result in an array of length n (# of obs) with different marginal effects (which is fine because I understand that the effects are non constant and non-linear). Let's say that this array goes from [0.0008 to 0.0495]

Now the problem is, how can you verify this results? How can I measure the marginal effect when the dummy goes from values 0 to 1?
You could argue that I could do two things MEM and AME methods:

MEM: Leave all the values at its mean and then calculate all over again for x1 = 0 and then for x1 = 1 (MEM method)
(you can't really do this because that you will be assuming that you can have some observations where x1 and x2 will be equal to 1 at the same time, which incorrect since the mean for a dummy is like a proportion of how many "1s" there are for that column)

AME: Leave as observed, but changing all the values of x1 to 0 (making all the values of x2 = 1) and then do the opposite (x1 = 1, x2 =0, you have to do this since it can't belong to two categories at the same time), and then take the average of the results (AME method) (Side comment: One thing I am not sure if it is the average between the difference in marginal effects when x1 = 0 and then 1, or if it is an average between the probabilities when x1 =0 and then 1, I used both, but probability I think it makes more sense to me)
Now, if I try the 2nd approach I get very different results to what I originally got ( which were values between [0.0008 to 0.0495]), it gives me values between [0.0022 to 0.1207], which is a massive difference.

To summarise:
How can do a mathematical corroboration to get the same values I got initially ([0.0008 to 0.0495])?

How can I interpret these original values in the first place? Because if I take 0.0495, I am basically saying "if I increase x1 by 1-unit (from 0 to 1), I will have a 4.95% increase in probability of my event happening", the problems is that it doesn't consider that to make the 1-unit increase I need to, by default, decrease the other dummy variable (x2), so I will be doing something of a double-change in the variables or like a double marginal effect at the same time.

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Calculate Marginal effect by hand with logit function and...
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