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I am having hard time understanding the use of nested and non-nested models.

According to Ben-Akiva's description the nested models are blue bus/red bus problems, where the choice set has multi levels within the logit model estimated.

In some other descriptions, nested models define those which you can get base model by restricting an additional variable. So to my understanding, following is a nested model according to this definition:

  • Model 1

Y = a1 + b1X1 + e

  1. Model 2

Y = a1 + b1X1 + b2X2 + e

Therefore, when comparing these two models, AIC or BIC are used, and Vuong's test in R is irrelevant (to my understanding) as it says it should be used in non-nested models, example ordinary Poisson vs zero-ordinary Poisson model.

On the other hand, I would like to compare two models. In the first one one of the variables are taken as continuous. In the other one, the same continuous variables are divided into different categories and dummied out. Example is:

  1. Model 1

Y = a1 + b1 Income

  1. Model 2

Y = a1 + b1 High Income + b2 Medium Income + b3 Low Income

In these two models, my question is are they considered nested or non-nested? Can I use Vuong non-nested model test to compare these two models?

Any clarification is appreciated.

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