Applied Microeconometrics Chapter 3 Multinomial and ordered models

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Applied MicroeconometricsChapter 3Multinomial
and ordered models
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• The Multinomial Logit Model (MNL)
• Estimation
• The IIA Assumption
• Applications
• Extensions to MNL
• Ordered Probit

Train, K. (2003), Discrete Choice Methods with
Simulation (downloadable from http//elsa.berkeley
.edu/books/choice2.html) Wooldridge, J.M. (2002),
Econometric Analysis of Cross Section and Panel
Data, Ch. 15
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Making the right decision between alternative
models
yes
mlogit
IIA valid ?
unordered
no
mprobit nested logit
ordered logit/ probit
ordered
IIAindependence of irrelevant alternatives
(assumption)
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Multinomial models
Multiple alternatives without obvious ordering
? Choice of a single alternative out of a number
of distinct alternatives
e.g. which means of transportation do you use to
get to work?
bus, car, bicycle etc.
? Example for ordered structure how do you feel
today very well, fairly well, not too well,
miserably
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Derivation of the Multinomial Logit model

• Choice between M alternatives
• Decision is determined by the utility level Uij,
  an individual i derives from choosing alternative
  j
• Let
• where i1,,N individuals j0,,J alternatives
• The alternative providing the highest level of
  utility will be chosen.

(1)
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Derivation of the Multinomial Logit model

• Probability that alternative j will be chosen
• In order to calculate this probability, the
  maximum of a number of random variables has to
  be determined.
• In general, this requires solving
  multidimensional integrals ? analytical
  solutions do not exist

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Derivation of the Multinomial Logit model

• Exception If the error terms eij in (1) are
  assumed to be independently identically
  standard extreme value distributed, then an
  analytical solution exists.
• In this case, similar to binary logit, it can
  be shown that the choice probabilities are

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Derivation of the Multinomial Logit model

• Standardization ß00

• The special case where J1 yields the binary
  Logit model.

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Independent variables

• Different kinds of independent variables
• Characteristics that do not vary over
  alternatives (e.g., socio-demographic
  characteristics, time effects)
• Characteristics that vary over alternatives
  (e.g., prices, travel distances etc.)

In the latter case, the multinomial logit is
often called conditional logit (CLOGIT in
Stata) It requires a different arrangement of the
data (one line per alternative for each i)
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Estimation of the MNL

• Estimation is by Maximum Likelihood
• The log likelihood function
• is globally concave and easy to maximize
  (McFadden, 1974) ? big computational advantage
  over multinomial probit or nested logit

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Interpretation of coefficients

• The coefficients themselves cannot be
  interpreted easily but the exponentiated
  coefficients have an interpretation as the
  relative risk ratios (RRR)

• Let
• then

risk ratio
(for simplicity, only one regressor considered)
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Interpretation of coefficients

• The relative risk ratio tells us how the
  probability of choosing j relative to 0 changes
  if we increase x by one unit

such that
relative risk ratio RRR
Note some people also use the term odds ratio
for the relative risk
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Interpretation of coefficients
Interpretation
Variable x increases (decreases) the probability
that alternative j is chosen instead of the
baseline alternative if RRR gt (lt) 1.
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Marginal effects in the MNL

• Marginal Effects

• Elasticities

• relative change of pij if x increases by 1 per
  cent

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Independence of Irrelevant Alternatives (IIA)
Important assumption of the multinomial
Logit-Model ? it implies that the decision
between two alternatives is independent from the
existence of more alternatives
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Independence of Irrelevant Alternatives (IIA)

• Ratio of the choice probabilities between two
  alternatives j and k is independent from any
  other alternative

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Independence of Irrelevant Alternatives (IIA)

• Problem This assumption is invalid in many
  situations.

Example red bus - blue bus - problem

• initial situation only red buses
• an individual chooses to walk with probability
  2/3
• - probability of taking a red bus is 1/3
• probability ratio 21

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Independence of Irrelevant Alternatives (IIA)
Introduction of blue buses

• It is rational to believe that that the
  probability of walking will not change.
• If the number of red buses number of blue
  buses Person walks with P4/6
• Person takes a red bus with P1/6
• Person takes a blue bus with P1/6

New probability ratio for walking vs. red bus
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Not possible according to IIA!
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Independence of Irrelevant Alternatives (IIA)

• The following probabilities result from the
  IIA- assumption
• P(by foot)2/4
• P(red bus)1/4
• P(blue bus)1/4, such that
• Problem probability of walking decreases from
  2/3 to 2/4 due to the introduction of blue buses
  ? not plausible!

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Independence of Irrelevant Alternatives (IIA)

• Reason of IIA property assumption that error
  termns are independently distributed over all
  alternatives.
• The IIA property causes no problems if all
  alternatives considered differ in almost the
  same way.

e.g., probability of taking a red bus is highly
correlated with the probability of taking a blue
bus substitution patterns
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Hausman Test for validity of IIA

• H0 IIA is valid (odds ratios are independent
  of additional alternatives)
• Procedure omit a category
  ? Do the estimated coefficients change
  significantly?
• If they do reject H0
• cannot apply multinomial logit
• choose nested logit or multinomial probit
  instead

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Cramer-Ridder Test

• Often one would like to know whether certain
  alternatives can be merged into one
• e.g., do employment states such as
  unemployment and nonemployment need to be
  distinguished?
• The Cramer-Ridder tests the null hypothesis
  that the alternatives can be merged. It has the
  form of a LR test
• 2(logLU-logLR)?²

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Cramer-Ridder Test

• Derive the log likelihood value of the
  restricted model where two alternatives (here, A
  and N) have been merged

where log
is the log likelihood of the
restricted model, log
is the log likelihood
of the pooled model, and nA and nN are the
number of times A and N have been chosen
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Application
Data
616 observations of choice of a particular health
insurance
3 alternatives

• indemnity plan deductible has to be paid
  before the benefits of the policy can apply
• prepaid plan prepayment and unlimited usage
  of benefits
• uninsured no health insurance

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Application
Observation group nonwhite
0 white
1 black
1 black
Is the choice of health care insurance determined
by the variable nonwhite?
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Application
Stata estimation output for the MNL
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Application

• If one does not choose a category as baseline,
  Stata uses the alternative with the highest
  frequency.

here indemnity is used as the baseline category
used for comparison
customized choice of basic category in
Stata mlogit depvar indepvars, base ()
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Interpreting the output

• The estimated coefficients are difficult to
  interpret quantitatively

• The coefficient indicates how the logarithmized
  probability of choosing the alternative
  prepaid instead of indemnity changes if
  nonwhite changes from 0 to 1. More intuitive
  to exponentiate coeffs and form RRRs

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Calculation of RRR
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Calculation of RRR

• Probability of choosing
• prepaid over indemnity is 1.9 times higher
  for black individuals
• uninsure over indemnity is 1.5 times
  higher for black individuals

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Marginal effects

• Stata computes the marginal effect of
  nonwhite for each alternative separately.

(AKA margeff)
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Marginal effects

• Interpretation If the variable nonwhite
  changes from 0 to 1
• the probability of choosing alternative
  indemnity decreases by 15.2 per cent.
• the probability of choosing alternative
  prepaid increases by 15.0 per cent.
• the probability of choosing alternative
  uninsure rises by 0.2 per cent
• (However, none of the coefficients is
  significant)