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Is there a nested logit command in Stata?

Title		Comparison of the nested logit and the constrained multinomial models
Author		William Gould and William Sribney, StataCorp
Date		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

Nested Logit model:

A choice is made first between 1 and (2,3) and then, if (2,3), a choice is made between 2 and 3.

Constrained Multinomial model:

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

Let W be the outcome 1, 2, or 3.

Example: 1=do not buy a car, 2=buy a Ford, 3=buy a Mercedes.

Let X be a variable that affects {1,{2,3}}.

Example: X=age of current car.

Let Z be a variable that affects {1,2,3}.

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)