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Note: This FAQ is no longer relevant since the nlogit command was introduced in Stata 7.
Is there a nested logit command in Stata?
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Title
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Comparison of the nested logit and the constrained multinomial models
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Author
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William Gould and William Sribney, StataCorp
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Date
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April 1999
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The short answer is, no. Stata does not presently have a command that does
nested logit. However, Stata does have one feature — the ability to estimate
multinomial models with constraints across the equations — which may help for
some choice models. Consider a choice among {1,2,3} in which you imagine
the choice is made
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Nested Logit model:
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A choice is made first between 1 and (2,3) and then, if (2,3),
a choice is made between 2 and 3.
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Constrained Multinomial model:
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A choice is made among {1,2,3}, but there are some variables that
affect only {1,{2,3}}, and some that affect {1,2,3}.
The Nested Logit and Constrained Multinomial models are somewhat related, but clearly different.
How to estimate the Constrained Multinomial model
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Let W be the outcome 1, 2, or 3.
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Example: 1=do not buy a car, 2=buy a Ford, 3=buy a Mercedes.
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Let X be a variable that affects {1,{2,3}}.
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Example: X=age of current car.
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Let Z be a variable that affects {1,2,3}.
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Example: Z=wealth.
In the unconstrained multinomial logistic model, we write:
r1 = 1
r2 = exp(a2 + X*b2 + Z*c2)
r3 = exp(a3 + X*b3 + Z*c3)
p1 = Pr(W==1) = r1/(r1+r2+r3) = 1/(1+r2+r3)
p2 = Pr(W==2) = r2/(r1+r2+r3) = r2/(1+r2+r3)
p3 = Pr(W==3) = r3/(r1+r2+r3) = r3/(1+r2+r3)
Here we have (arbitrarily) used group 1 as the base group for normalization.
The coefficients to be estimated are a2, b2, c2, a3, b3, and c3.
The Relative Risks (RR) are ratios of the probabilities:
RR(W==1 wrt W==1) = Pr(W==1) / Pr(W==1) = r1 = 1
RR(W==2 wrt W==1) = Pr(W==2) / Pr(W==1) = r2
RR(W==3 wrt W==1) = Pr(W==3) / Pr(W==1) = r3
Relative Risk Ratios are ratios of the Relative Risks (ratios of ratios!). The
Relative Risk Ratio (RRR) for a one-unit change in the corresponding variable,
relative to the (arbitrarily chosen) base group (W==1) is simply the
exponentiated value of a coefficient. For example,
Pr(W==2|age+1) / Pr(W==1|age+1)
------------------------------- = exp(b2)
Pr(W==2|age) / Pr(W==1|age)
or, in words, exp(b2) is the RRR of X (age of car), Ford vs. no new purchase.
Here is the verbal interpretation of all the coefficients:
exp(b2) = RRR of X (age of car), Ford vs. no new purchase
exp(c2) = RRR of Z (wealth), Ford vs. no new purchase
exp(b3) = RRR of X (age of car), Mercedes vs. no new purchase
exp(c3) = RRR of Z (wealth), Mercedes vs. no new purchase
The real trick is learning to think in the Relative Risk metric (meaning
r1, r2, and r3).
We want to constrain variable X — age of current car — so that it's
effect is to leave p2/p3 unchanged. X affects the overall risk
of buying a car (1-p1) but not which car is purchased. Thus, we want to
constrain the change in
p2 r2 exp(a2 + X*b2 + Z*c2)
-- = -- = ---------------------
p3 r3 exp(a3 + X*b3 + Z*c3)
to be 0 for a change in X, meaning b2=b3 (because then X*b2 and X*b3 cancel).
If this is the constraint we want, here's the Stata code to do it.
Let's assume the variables are named
W carpur, 1=no car, 2=Ford, 3=Mercedes
X c_c_age
Z wealth
Then the Stata code is
. constraint define 1 [2]c_c_age=[3]c_c_cage
. mlogit carpur c_cage style wealth, base(1) constrain(1)
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