L8 - Multinomial Choice Models Flashcards
What is different about Multinomial choice models?
The left hand side variable (The choice set) consists of more than two alternatives
- instead of just 0 and 1, it could be the set {1,2,3,4,5} etc
What data do we use in Multinomial choice models?
- Both of these models have more than two outcomes in the dependent variables but their are slight nuances that differentiate them
- Multinomial (Conditional) conditional choice models
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Nominal scale variable:
- Unique values
- Classification
- No indication of order (e.g. hair colour)
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Nominal scale variable:
- Ordered Choice models
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Ordinal scaled data:
- Unique values
- Provide indication of order
- Cannot determine distance between the objects (this makes them different form linear or cardinal variables)
- e.g. in this case cant say 2 is twice a big as 4, just that 4 is greater than 2
- like a satisfaction rating about a product or service
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Ordinal scaled data:
What is the distribution of the errors term in the Multinomial Logit (MNL) Model?
- we now have more than 1 alternative j
- what do we mean by independently and identically distribution extreme values
- independent –> the error term for alternatives does not give away any information for the error term of another alternative
- identically distributed –> variance of the error term of one alternative is the same of the of another alternatives –> kind of like Homoscedasticity
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In the MNL model, what is the probability that the decision makers chooses alternative i over all other options?
- So the Utility of alternative i > Utility of alternative j
- assuming that εni is known, then this probability is a cumulative distribution function εnj evaluated at the second formula on the slide
- can multiple across alternatives because we can assume they are independent to one another
- if it isn’t know then we have to integrate the formula see at the end of the slide with respect to each value of εni
- assuming that εni is known, then this probability is a cumulative distribution function εnj evaluated at the second formula on the slide
What does the MNL model probability formula collapse to?
WILL NEED THIS EXPRESS FOR THE EXAM
- This happens because we assume the variance of the error term is equal to π2/6
Denominator is just the numerator added up across all the alternatives
What are the properties of logit probabilities?
What is the maximum likelihood estimator for the MNL model?
- assuming alternatives are independent we can multiple probabilities across alternatives
- However, Non chose alternatives we are raised to the power of 1, so this entire product function collapses to the probability of person n choosing alternative i
- assuming each individual decision making is independent we can further multiple these probabilities across each person in the data set
- log the previous function to make it easier to compute
- set the first-order condition equal to zero to find the maximum likelihood estimator
What can we decompose the probabilities down into?
- Marginal Effects:
- Direct effects
- Cross-effects
- Elasticities:
- Direct elasticities
- Cross-elasticities
What are the marginal effects of probability in Multinomial models?
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Direct Marginal Effect represents the change in the choice probability of an alternative given a unit change in a variable related to that same alternative
- how does the probability of taking the bus change if a bus ticket increases by £1
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Cross-Marginal Effect represents the change in the choice probability of an alternative given a unit change in a variable related to a competing alternative.
- how does the probability of taking the bus change if a rail ticket increases by £1
MUST MULTIPLE RESULT BY 100
What are the elasticity effects of probability in Multinomial models?
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Direct elasticity measures the change in probability of choosing a particular alternative in the choice set with respect to a given percentage change in an attribute of that same alternative.
- how does the probability of taking the bus change if a bus ticket increases by 1%
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Cross-elasticity measures the change in probability of choosing a particular alternative in the choice set with respect to a given percentage change in an attribute of a competing alternative.
- how does the probability of taking the bus change if a train ticket increases by 1%
TAKE RESULT AS IT COMES (DON’T MULTIPLE BY 100)
What is something special about MNL model with respect to the constants included?
With J alternatives, at most J-1 alternative-specific constants can enter the model, with one of the constants normalised to zero.
- so if you have 4 alternatives, you would include 3 constants in the model –> with one normalised to zero otherwise the model wouldn’t be identified
How is data arranged from MNL models?
- Much include the dependent variables for each alternative for Each individual (separated by colour)
- Hence there are 4 lines for each individual with their own respective explanatory variables
What are the 4 ways in which you can test the ‘fit’ of the model given you have alternative specifications?
- Likelihood Ratio Index
- Can compute this with other specifications as the base to create a comparison
- Pseudo R-squared
- Log-likelihood function (MLE) –> LLmodel
- Log-likelihood function (constant only) –> LLbase
- LL function where you only include the constant(s) of your model
What do the changes in probabilities of the MNL models sum to?
- 0
- This should be for elasticities and marginal effects
How do we deal with the individual decision-makers characteristics in a MNL model?
- e.g. income may be different across participants (cannot estimate coefficient for these variable as differences in these are based across the decision makers)
- How do we deal with them in Multinomial models
- Interact them with the alternative specific constant –> multiple the individual charactistics with alternative specific constants?/ –>multiple income with the constant variable of the different alternatives
- as those with lower incomes would be more sensitive to changes in the costs?