[Limdep Nlogit List] CLogit Likelihood Function 2nd Derivative - Clarification

[Richard Turner]

Greetings Community,

I am trying to implement a *conditional logit* model in Excel to gain a
better understanding of the estimation procedure. To my knowledge, the
procedure for estimating *conditional logits* is the same as
estimating *multinomial
logits*, at least in terms of the probability equation, 1st derivatives for
the Jacobian, and the *2nd partial derivatives for the Hessian*.

After implementing the model in Excel, I still do not fully understand
the *second
partial derivatives that create the Hessian*. The second partial
derivatives are:

Where x_ik is the ith observation of the kth variable, π_ik is the
probability of the ith observation of the jth alternative.

After reading pages 9-13 of this document
<https://stats.stackexchange.com/questions/294118/how-to-calculate-multinomial-conditional-logit-hessian>,
I am not sure when you would need to use the second derivative when j′ is
not equal to j. *Could someone explain, as simply as possible, when this
derivative would be used?*

In my Excel model, I have 3 alternatives and two variables, Var1 and Var2,
which I represent as x1 and x2. I've implemented the newton-raphson
algorithm to maximize the likelihood function using only the second
derivative where j′ equals to j. After several iterations, my Newton and
Jacobian values were sufficiently equal to zero and my parameters equaled
parameters estimated using NLOGIT.



I assume, the model was estimated correctly using only the second
derivative for the Hessian when j′ equals to j, but I am still not sure
when to use the other derivative when j′ is not equal to j. For reference,
I have attached my excel model

Thank you very much for your time,
______________________________________________

*Richard Turner*

*iMarketResearch*
*Easier, Faster, Better Advanced Market Research*
www.imarketresearch.com

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