Nested Logit Model

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Nested Logit Model
Phd Graduate Seminar in advance Statistics

• by
• Asif Khan

Institute of Rural Development (IRE) Georg-August
University Goettingen July 24, 2006
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Contents

• Independence of Irrelevant Alternatives

Nested Logit Model
Random Utility Model
GEV distribution
Seperable Utility
Seperable Probabilities
Inclusive value
Estimation
Shortcoming of Nested Logit Model
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• Independence of Irrelevant Alternatives

• Multinomial logit Conditional logit models
  based on IIA.
• The odds do not depend on other outcomes that are
  available. So alternative outcomes
  are irrelevant.
• What this means is that adding or deleting
  outcomes does not affect the odds among the
  remaining outcomes.
• IIA assume that unobserable or latent
  attributes of all alternatives are perceived as
  equally similar.

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Example IIA
Choices of travel to a city dwellers
20
20
60
Share of 3 alternatives
The ratio b/w bus car 1 3
60 15 75
20 5 25
IIA assumption
The ratio b/w bus car must stay at 1 3
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Real world situation Problem with IIA

• IIA property convenient for estimation but fails
  on consumer behavior.
• Unrealistic assumption why?
• b/coz people will travel by white bus, if grey
  bus is not available without switching to taxi,
  which may be expansive.
• More realistic situation may be
• White bus 40
• Taxi 60
• IIA biggest drawback of MNLM model

• Tests for validity of IIA
• Hausman McFadden test (1984)
• Small and Hsiao test of IIA

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Nested Logit Model

• If MNLM fails then
• Multinomial Probit computation problems
• Nested Logit partial relaxation of IIA

Independence from IIA

• Nested Logit
• also called structured logit, sequential logit,
  GEV model
• Useful when alternatives similar in unoberved
  factors to other alternatives
• Developed by Ben-Akiva (1973) McFadden (1978)
• Widely used in transportation, housing, energy
  etc.

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Nested Logit Model
Travel choices available to a worker to workplace
Probability Probability Probability Probability Probability
With alternatives removed With alternatives removed With alternatives removed With alternatives removed
Alternative Original Car Carpool Bus Train
Car .40 - .45 (12.5) .52(30) .48(20)
Carpool .10 .20 (100) - .13(30) .12(20)
Bus .30 .48 (60) .33(10) - .40(33)
Train .20 .32 (60) .22(10) .35(70) -
IIA does not hold across nest
No Proportional substitution across nest
IIA hold within nest
Proportional substitution within nest
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Random Utility Model
NLM a discrete choice mode In DC situation, a
decision maker is assumed to associate a value
(utlity) to each available alternative. Utility
of an alternative f(alt. Char. decison maker
char.) Decision maker choose alternative with
higgest utility Unj gt ULm Since we cannot
observe all utility so it is modelled as random
variables and group them into following
model Unj Vnj enj Total utility
representative/observed utility unknown
utility
Treat them random with cumulative distribution
and collect them into a vector relating to
alternatives at hand enj(en1...., enj) So
based on this enj we are making good guesses or
probablity statement what the choice will be.
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GEV distribution

• In NLM we assume that unoberved utility has GEV
  distribution
• exp(-?kk1 (?e-enj/?k) ?k)
• Generalization of univerate distribution in logit
  model.
• The unoberved utility is correlated within nests.
• enj uncorrelated across nests
• Parameter ?k is a measure of degree of
  independence in enj in nest k. Higher ?k means
  less correlation higher
  independence vice versa.
• McFadden (1978) used 1-?k as indication for
  correlation
• If ?k 1 means complete independence or no
  correlation
• If ?k 1 nested logit model reduce to standard
  logit model

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Seperable Utility

• Observed Utility (Unj) is
• Unj UT UCT
• UT utility from travel mode e.g., auto or
  transit
• UCT utility from travel choice e.g., car, bus
  etc.
• Random utility Marginal utility Conditional
  utility

Constant for all alts. within a nest. Vars. that
describe a nest. These var. differ over nest but
not for alts.within each nest.
UT Marginal U
UCT Conditional U
Varies over alts. within a nest. Vars. that
describe an alt. These vars. vary over
alts.within each nest.
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Seperable Probabilities

• Probabilities in nested logit is a product of two
  simple logits.
• Pi Prob (nest containing i) x Prob (i, given
  nest containing i)
• e.g., Pi Prob (auto) x Prob (car, given
  auto)
• Pi Pn Pin

Pi Marginal Prob. Conditional Prob.
(Upper model)
(Lower model)
Where Yi are vars. that vary over alternatives
within the nest. Zn are variables that vary over
nests but not within alternatives within each
nest In is the inclusive value of nest n ?
parameter of In
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Inclusive value
In E(max Un) E(max Vjej)

• Also called log-sum for nest n or inclusive
  utility
• It is the expected maximum utility that a
  decision maker recived from a choice within
  the alternatives in a nest.
• Ben-Akiva (1973) considered it a link b/w lower
  upper model.
• Hence it brings information from the conditional
  prob. (lower model) to the marginal prob. (upper
  model) as it is the denominator of lower model.
• ? is the log-sum coefficent showing degree of
  independence in the unobserved part of utility
  for alternatives in a nest. Lower ? means
  less independence more correlation. (remember
  1-? is a measure of correlation)
• ? 1 (non correlation so a standard logit)
• ? 0 (means perfect correlation)

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Estimation
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Shortcoming of Nested Logit Model

• For some choices there is a natural tree
  structure for other there is not.
• This natural tree structure is derived from
  seperable utility function arguement
  (for e.g., choose b/w flying ground transport
  then choose b/w bus, car train).
• Hence the behavioral characteristics of
  separability translates into an estimating
  approach that allows nesting procedure to equate
  behavioral estimating considerations.
• The partitioning of some choices is adhoc
  leads to troubling possibilities that the results
  might be dependent on the branches so defined. So
  there will be different results based on
  different specification of tree structure.
• There is no test for discriminating among tree
  structures, a problematic aspect of these
  models (Greene, 2003)

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References

• Greene, William H. 2003. Econometric Analysis.
  5th ed. Prentice Hall, USA.
• Jeffrey, Wooldridge M. 2001. Econometric Analysis
  of Cross Section and Panel Data. The MIT,
  USA.
• The Nested Logit Regression Model.
• http//www.indiana.edu/statmath/stat/all/cdvm/
  cdvm8.html
• Kenneth, Train. 2003. Discrete Choice Methods
  with Simulation. Cambridge University Press, USA.
• Discrete Dependent Variable Models.
  http//onlinepubs.trb.org/onlinepubs/nchrp/cd-22/v
  2chapter5.html
• Maddala, G. S. 1983. Limited-Dependent and
  Qualitative Variables in
  Econometrics. Cambridge University Press USA.
• McFadden, D. L. 2000. Disaggregate Travel
  Demand's RUM Side A 30-Year Retrospective.
  manuscript, Department of Economics,
  University of California, Berkeley.

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The end

• Thank you