Applied Econometrics

1
Applied Econometrics

• William Greene
• Department of Economics
• Stern School of Business

2
Applied Econometrics

• 21. Discrete Choice Modeling

3
A Microeconomics Platform

• Consumers Maximize Utility (!!!)
• Fundamental Choice Problem Maximize U(x1,x2,)
  subject to prices and budget constraints
• A Crucial Result for the Classical Problem
• Indirect Utility Function V V(p,I)
• Demand System of Continuous Choices
• The Integrability Problem Utility is not
  revealed by demands

4
Theory for Discrete Choice

• Theory is silent about discrete choices
• Translation to discrete choice
• Existence of well defined utility indexes
  Completeness of rankings
• Rationality Utility maximization
• Axioms of revealed preferences
• Choice sets and consideration sets consumers
  simplify choice situations
• Implication for choice among a set of discrete
  alternatives
• Commonalities and uniqueness
• Does this allow us to build models?
• What common elements can be assumed?
• How can we account for heterogeneity?
• Revealed choices do not reveal utility, only
  rankings which are scale invariant

5

Choosing Between Two Alternatives

• Modeling the Binary Choice
• Ui,suv ?suv ?Psuv ?suvIncome ?i,suv
• Ui,sed ?sed ?Psed ?sedIncome ?i,sed
• Chooses SUV Ui,suv gt Ui,sed
• Ui,suv - Ui,sed gt 0
• (?SUV-?SED) ?(PSUV-PSED) (?SUV-?sed)Income
• ?i,suv - ?i,sed gt 0
• ?i gt -? ?(PSUV-PSED) ?Income

6
What Can Be Learned from the Data? (A Sample of
Consumers, i 1,,N)

• Are the attributes relevant?
• Predicting behavior
• Individual
• Aggregate
• Analyze changes in behavior when
• attributes change

7
Application

• 210 Commuters Between Sydney and Melbourne
• Available modes Air, Train, Bus, Car
• Observed
• Choice
• Attributes Cost, terminal time, other
• Characteristics Household income
• First application Fly or Other

8
Binary Choice Data
Choose Air Gen.Cost Term Time
Income 1.0000 86.000 25.000
70.000 .00000 67.000 69.000
60.000 .00000 77.000 64.000
20.000 .00000 69.000 69.000
15.000 .00000 77.000 64.000
30.000 .00000 71.000 64.000
26.000 .00000 58.000 64.000
35.000 .00000 71.000 69.000
12.000 .00000 100.00 64.000
70.000 1.0000 158.00 30.000
50.000 1.0000 136.00 45.000
40.000 1.0000 103.00 30.000
70.000 .00000 77.000 69.000
10.000 1.0000 197.00 45.000
26.000 .00000 129.00 64.000
50.000 .00000 123.00 64.000 70.000
9
An Econometric Model

• Choose to fly iff UFLY gt 0
• Ufly ??1Cost ?2Time ?Income ?
• Ufly gt 0 ? ? gt -(??1Cost ?2Time ?Income)
• Probability model For any person observed by
  the analyst, Prob(fly) Prob? gt -(??1Cost
  ?2Time ?Income)
• Note the relationship between the unobserved ?
  and the outcome

10
??1Cost ?2TTime ?Income
11
Modeling Approaches

• Nonparametric relationship
• Minimal Assumptions
• Minimal Conclusions
• Semiparametric index function
• Stronger assumptions
• Robust to model misspecification
  (heteroscedasticity)
• Still weak conclusions
• Parametric Probability function and index
• Strongest assumptions complete specification
• Strongest conclusions
• Possibly less robust. (Not necessarily)

12
Nonparametric
P(Air)f(Income)
13
Semiparametric

• MSCORE Find bx so that
• sign(bx) sign(y) is maximized.
• Klein and Spady Find b to maximize a
  semiparametric likelihood of G(bx)

14
MSCORE
--------------------------------------------------
-------------------- Maximum Score Estimates of
Linear Quantile Regression Model from Binary
Response Data Quantile .500
Number of Parameters 3 Observations input
210 Maximum Iterations 500 End
Game Iterations 100 Bootstrap Estimates
20 Check Ties? No Save
bootstraps? No Start values from MSCORE
(normalized) Normal exit after 100
iterations. Score functions Naive At
theta(0) Maximum Raw .44762
.44762 .45714 Normalized
.72381 .72381 .72857 Estimated MSEs
from 20 bootstrap samples (Nonconvergence in 0
cases) Angular deviation (radians) of bootstraps
from estimate Mean square .873715
Mean absolute .844307 Standard errors below
are based on bootstrap mean squared deviations.
These and the t-ratios are only
approximations. ---------------------------------
------------------------------------ Variable
Coefficient Standard Error b/St.Er. PZgtz
Mean of X --------------------------------------
------------------------------- Constant
-.65410 .55222 -1.184 .2362
HINC -.75595 .67266 -1.124
.2611 34.5476 GC .02652
.13643 .194 .8459
102.648 -----------------------------------------
----------------------------
15
Klein and Spady Semiparametric
--------------------------------------------------
-------------------- Semiparametric Binary Choice
Model Dependent variable CHAIR Log
likelihood function -123.27371 Restricted
log likelihood -123.75705 Chi squared 1
d.f. .96667 Significance level
.32551 McFadden Pseudo R-squared
.0039055 Estimation based on N 210, K
1 Information Criteria Normalization1/N
Normalized Unnormalized AIC
1.18356 248.54743 Fin.Smpl.AIC 1.18365
248.56666 Bayes IC 1.19950
251.89453 Hannan Quinn 1.19000
249.90054 Logistic kernel fn. Bandwidth
.41017 ------------------------------------------
--------------------------- Variable Coefficient
Standard Error b/St.Er. PZgtz Odds
Ratio -------------------------------------------
--------------------------
Characteristics in numerator of ProbY 1
HINC .15671 .09521 1.646
.0998 1.16965 GC 1.00000
......(Fixed Parameter)....... Constant
.000 ......(Fixed Parameter)...... --------
-------------------------------------------------
------------
Note necessary normalizations. Coefficients are
not very meaningful.
16
Parametric Logit Model
--------------------------------------------------
-------------------- Binary Logit Model for
Binary Choice Dependent variable
CHAIR Log likelihood function
-113.18621 Restricted log likelihood
-123.75705 Chi squared 2 d.f.
21.14168 Significance level
.00003 McFadden Pseudo R-squared
.0854161 Estimation based on N 210, K
3 Information Criteria Normalization1/N
Normalized Unnormalized AIC
1.10654 232.37241 Fin.Smpl.AIC 1.10709
232.48892 Bayes IC 1.15435
242.41373 Hannan Quinn 1.12587
236.43175 ---------------------------------------
------------------------------ Variable
Coefficient Standard Error b/St.Er. PZgtz
Mean of X --------------------------------------
-------------------------------
Characteristics in numerator of ProbY
1 Constant -3.75847 .71712
-5.241 .0000 HINC .02722
.00842 3.232 .0012 34.5476
GC .01698 .00542 3.135
.0017 102.648 ------------------------------
---------------------------------------
17
Logit vs. MScore
-------------------------------------------------
-------- Predictions for Binary Choice Model.
Predicted value is 1 when betax is greater
than one, zero otherwise. Note, column
or row total percentages may not sum to
100 because of rounding. Percentages are of
full sample. ----------------------------------
--------------------- Actual
Predicted Value Value
0 1 Total Actual
-------------------------------------------
----------- 0 152 ( 72.4) 0 (
.0) 152 ( 72.4) 1 57 ( 27.1)
1 ( .5) 58 ( 27.6) ---------------
--------------------------------------- Total
209 ( 99.5) 1 ( .5) 210
(100.0) -------------------------------------
----------------- Logit Model --------------
----------------------------------------- Actua
l Predicted Value
Value 0 1
Total Actual ------------------------------
------------------------ 0 145 (
69.0) 7 ( 3.3) 152 ( 72.4) 1
48 ( 22.9) 10 ( 4.8) 58 (
27.6) --------------------------------------
---------------- Total 193 ( 91.9)
17 ( 8.1) 210 (100.0) ------------------
------------------------------------
18
Parametric Model Estimation

• How to estimate ?, ?1, ?2, ??
• Its not regression
• The technique of maximum likelihood
• Proby1
• Prob? gt -(??1Cost ?2Time ?Income)
• Proby0 1 - Proby1
• Requires a model for the probability

19
Completing the Model F(?)

• The distribution
• Normal PROBIT, natural for behavior
• Logistic LOGIT, allows thicker tails
• Gompertz EXTREME VALUE, asymmetric, underlies
  the basic logit model for multiple choice
• Does it matter?
• Yes, large difference in estimates
• Not much, quantities of interest are more stable.

20
Underlying Probability Distributions for Binary
Choice
21
Estimated Binary Choice (Probit) Model
---------------------------------------------
Binomial Probit Model
Maximum Likelihood Estimates
Dependent variable MODE
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -84.09172
Restricted log likelihood -123.7570
Chi squared 79.33066
Degrees of freedom 3
ProbChiSqd gt value .0000000
Hosmer-Lemeshow chi-squared 46.96547
P-value .00000 with deg.fr. 8
---------------------------------------------
----------------------------------------------
-------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
----------------------- Index
function for probability Constant
.43877183 .62467004 .702 .4824 GC
.01256304 .00368079 3.413
.0006 102.647619 TTME -.04778261
.00718440 -6.651 .0000 61.0095238 HINC
.01442242 .00573994 2.513
.0120 34.5476190
22
Estimated Binary Choice Models
LOGIT PROBIT
EXTREME VALUE Variable Estimate t-ratio
Estimate t-ratio Estimate t-ratio Constant
1.78458 1.40591 0.438772 0.702406 1.45189
1.34775 GC 0.0214688 3.15342
0.012563 3.41314 0.0177719 3.14153 TTME
-0.098467 -5.9612 -0.0477826 -6.65089
-0.0868632 -5.91658 HINC 0.0223234 2.16781
0.0144224 2.51264 0.0176815 2.02876 Log-L
-80.9658 -84.0917
-76.5422 Log-L(0) -123.757
-123.757 -123.757
23
Effect on Predicted Probability of an Increase in
Income
??1Cost ?2Time ?(Income1)
(? is positive)
24
Marginal Effects in Probability Models

• ProbOutcome some F(??1Cost)
• Partial effect ? F(??1Cost) / ?x
• (derivative)
• Partial effects are derivatives
• Result varies with model
• Logit ? F(??1Cost) / ?x Prob (1-Prob)
  ?
• Probit ? F(??1Cost) / ?x Normal density ?
• Scaling usually erases model differences

25
The Delta Method

26
Marginal Effects for Binary Choice

• Logit
• Probit

27
Estimated Marginal Effects
Logit Probit
Extreme Value
28
Marginal Effect for a Dummy Variable

• Probyi 1xi,di F(?xi?di)
• conditional mean
• Marginal effect of d
• Probyi 1xi,di1Probyi 1xi,di0
• Logit

29
(Marginal) Effect Dummy Variable

• HighIncm 1(Income gt 50)

-------------------------------------------
Partial derivatives of probabilities with
respect to the vector of characteristics.
They are computed at the means of the Xs.
Observations used are All Obs.
------------------------------------------- -
----------------------------------------------
------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
-----------------------
Characteristics in numerator of ProbY 1 GC
.003598132 .00113543 3.169
.0015 102.64762 TTME -.01759234
.00348663 -5.046 .0000 61.009524
Marginal effect for dummy variable is P1
- P0. HIGHINCM .085653672 .0993467
.862 .3886 .1857143
30
Computing Effects

• Compute at the data means?
• Simple
• Inference is well defined
• Average the individual effects
• More appropriate?
• Asymptotic standard errors. (Not done correctly
  in the literature terms are correlated!)
• Is testing about marginal effects meaningful?

31
Average Partial Effects
32
Elasticities

• Elasticity
• How to compute standard errors?
• Delta method
• Bootstrap
• Bootstrap the individual elasticities? (Will
  neglect variation in parameter estimates.)
• Bootstrap model estimation?

33
Estimated Income Elasticity for Air Choice Model

------------------------------------------
Results of bootstrap estimation of model.
Model has been reestimated 25 times.
Statistics shown below are centered
around the original estimate based on the
original full sample of observations. Result
is ETA .71183 bootstrap
samples have 840 observations. Estimate
RtMnSqDev Skewness Kurtosis .712
.266 -.779 2.258 Minimum .125
Maximum 1.135 --------------------------
----------------
Mean Income 34.55, Mean P .2716, Estimated ME
.004539, Estimated Elasticity0.5774.
34
Odds Ratio Logit Model Only

• Effect Measure? Effect of a unit change in
  the odds ratio.

35
Ordered Outcomes

• E.g. Taste test, credit rating, course grade
• Underlying random preferences Mapping to
  observed choices
• Strength of preferences
• Censoring and discrete measurement
• The nature of ordered data

36
Modeling Ordered Choices

• Random Utility
• Uit ? ?xit ?izit ?it
• ait ?it
• Observe outcome j if utility is in region j
• Probability of outcome probability of cell
• PrYitj F(?j ait) - F(?j-1 ait)

37
Movie Madness
38
Health Care Satisfaction (HSAT)
Self administered survey Health Care
Satisfaction? (0 10)
Continuous Preference Scale
39
Ordered Probability Model
40
Ordered Probabilities
41
Five Ordered Probabilities
42
Coefficients
43
Effects in the Ordered Probability Model
Assume the ßk is positive. Assume that xk
increases. ßx increases. µj- ßx shifts to the
left for all 5 cells. Proby0
decreases Proby1 decreases the mass shifted
out is larger than the mass shifted in. Proby2
decreases same reason. Proby3
increases. Proby4 increases
When ßk gt 0, increase in xk decreases Proby0
and increases ProbyJ. Intermediate cells are
ambiguous, but there is only one sign change in
the marginal effects from 0 to 1 to to J
44
Ordered Probability Model for Health Satisfaction
---------------------------------------------
Ordered Probability Model
Dependent variable HSAT
Number of observations 27326
Underlying probabilities based on Normal
Cell frequencies for outcomes Y
Count Freq Y Count Freq Y Count Freq 0
447 .016 1 255 .009 2 642 .023 3
1173 .042 4 1390 .050 5 4233 .154 6
2530 .092 7 4231 .154 8 6172 .225 9
3061 .112 10 3192 .116
---------------------------------------------
----------------------------------------------
-------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
----------------------- Index
function for probability Constant
2.61335825 .04658496 56.099 .0000
FEMALE -.05840486 .01259442 -4.637
.0000 .47877479 EDUC .03390552
.00284332 11.925 .0000 11.3206310 AGE
-.01997327 .00059487 -33.576
.0000 43.5256898 HHNINC .25914964
.03631951 7.135 .0000 .35208362
HHKIDS .06314906 .01350176 4.677
.0000 .40273000 Threshold
parameters for index Mu(1) .19352076
.01002714 19.300 .0000 Mu(2)
.49955053 .01087525 45.935 .0000 Mu(3)
.83593441 .00990420 84.402
.0000 Mu(4) 1.10524187 .00908506
121.655 .0000 Mu(5) 1.66256620
.00801113 207.532 .0000 Mu(6)
1.92729096 .00774122 248.965 .0000
Mu(7) 2.33879408 .00777041 300.987
.0000 Mu(8) 2.99432165 .00851090
351.822 .0000 Mu(9) 3.45366015
.01017554 339.408 .0000
45
Ordered Probability Effects
-------------------------------------------------
--- Marginal effects for ordered probability
model M.E.s for dummy variables are
Pryx1-Pryx0 Names for dummy
variables are marked by .
-----------------------------------------------
----- ---------------------------------------
--------------------------- Variable
Coefficient Standard Error b/St.Er.PZgtz
Mean of X ----------------------------------
--------------------------------
These are the effects on ProbY00 at means.
FEMALE .00200414 .00043473 4.610
.0000 .47877479 EDUC -.00115962
.986135D-04 -11.759 .0000 11.3206310 AGE
.00068311 .224205D-04 30.468
.0000 43.5256898 HHNINC -.00886328
.00124869 -7.098 .0000 .35208362
HHKIDS -.00213193 .00045119 -4.725
.0000 .40273000 These are the
effects on ProbY01 at means. FEMALE
.00101533 .00021973 4.621 .0000
.47877479 EDUC -.00058810
.496973D-04 -11.834 .0000 11.3206310 AGE
.00034644 .108937D-04 31.802
.0000 43.5256898 HHNINC -.00449505
.00063180 -7.115 .0000 .35208362
HHKIDS -.00108460 .00022994 -4.717
.0000 .40273000 ... repeated for all 11
outcomes These are the effects on
ProbY10 at means. FEMALE -.01082419
.00233746 -4.631 .0000 .47877479 EDUC
.00629289 .00053706 11.717
.0000 11.3206310 AGE -.00370705
.00012547 -29.545 .0000 43.5256898
HHNINC .04809836 .00678434 7.090
.0000 .35208362 HHKIDS .01181070
.00255177 4.628 .0000 .40273000
46
Ordered Probit Marginal Effects
47
Multinomial Choice Among J Alternatives

• Random Utility Basis
• Uitj ?ij ?i xitj ?izit ?ijt
• i 1,,N j 1,,J(i) t 1,,T(i)
• Maximum Utility Assumption
• Individual i will Choose alternative j in
  choice setting t iff Uitj gt Uitk for all k ? j.
• Underlying assumptions
• Smoothness of utilities
• Axioms Transitive, Complete, Monotonic

48
Utility Functions

• The linearity assumption and curvature
• The choice set
• Deterministic and random components The model
• Generic vs. alternative specific components
• Attributes and characteristics
• Coefficients
• Part worths preference weights coefficients
• Alternative specific constants
• Scaling

49
The Multinomial Logit (MNL) Model

• Independent extreme value (Gumbel)
• F(?itj) 1 Exp(-Exp(?itj)) (random part of
  each utility)
• Independence across utility functions
• Identical variances (means absorbed in constants)
• Same parameters for all individuals (temporary)
• Implied probabilities for observed outcomes

50
Specifying Probabilities

• Choice specific attributes (X) vary by choices,
  multiply by generic
• coefficients. E.g., TTME, GC
• Generic characteristics (Income, constants) must
  be interacted with
• choice specific constants. (Else they fall out
  of the probability)
• Estimation by maximum likelihood dij 1 if
  person i chooses j

51
Observed Data

• Types of Data
• Individual choice
• Market shares
• Frequencies
• Ranks
• Attributes and Characteristics
• Choice Settings
• Cross section
• Repeated measurement (panel data)

52
Data on Discrete Choices

• Line MODE TRAVEL INVC INVT
  TTME GC HINC
• 1 AIR .00000 59.000 100.00
  69.000 70.000 35.000
• 2 TRAIN .00000 31.000 372.00
  34.000 71.000 35.000
• 3 BUS .00000 25.000 417.00
  35.000 70.000 35.000
• 4 CAR 1.0000 10.000 180.00
  .00000 30.000 35.000
• 5 AIR .00000 58.000 68.000
  64.000 68.000 30.000
• 6 TRAIN .00000 31.000 354.00
  44.000 84.000 30.000
• 7 BUS .00000 25.000 399.00
  53.000 85.000 30.000
• 8 CAR 1.0000 11.000 255.00
  .00000 50.000 30.000
• 321 AIR .00000 127.00 193.00
  69.000 148.00 60.000
• 322 TRAIN .00000 109.00 888.00
  34.000 205.00 60.000
• 323 BUS 1.0000 52.000 1025.0
  60.000 163.00 60.000
• 324 CAR .00000 50.000 892.00
  .00000 147.00 60.000
• 325 AIR .00000 44.000 100.00
  64.000 59.000 70.000
• 326 TRAIN .00000 25.000 351.00
  44.000 78.000 70.000
• 327 BUS .00000 20.000 361.00
  53.000 75.000 70.000
• 328 CAR 1.0000 5.0000 180.00
  .00000 32.000 70.000

53
Estimated MNL Model
--------------------------------------------------
--------- Discrete choice (multinomial logit)
model Dependent variable Choice Log
likelihood function -256.76133 Estimation
based on N 210, K 7 Information
Criteria Normalization1/N
Normalized Unnormalized AIC
2.51201 527.52265 Fin.Smpl.AIC 2.51465
528.07711 Bayes IC 2.62358
550.95240 Hannan Quinn 2.55712
536.99443 R21-LogL/LogL Log-L fncn R-sqrd
R2Adj Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------
54
Estimated MNL Model
--------------------------------------------------
--------- Discrete choice (multinomial logit)
model Dependent variable Choice Log
likelihood function -256.76133 Estimation
based on N 210, K 7 Information
Criteria Normalization1/N
Normalized Unnormalized AIC
2.51201 527.52265 Fin.Smpl.AIC 2.51465
528.07711 Bayes IC 2.62358
550.95240 Hannan Quinn 2.55712
536.99443 R21-LogL/LogL Log-L fncn R-sqrd
R2Adj Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------
55
Estimated MNL Model
--------------------------------------------------
--------- Discrete choice (multinomial logit)
model Dependent variable Choice Log
likelihood function -256.76133 Estimation
based on N 210, K 7 Information
Criteria Normalization1/N
Normalized Unnormalized AIC
2.51201 527.52265 Fin.Smpl.AIC 2.51465
528.07711 Bayes IC 2.62358
550.95240 Hannan Quinn 2.55712
536.99443 R21-LogL/LogL Log-L fncn R-sqrd
R2Adj Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------
56
Estimated MNL Model
--------------------------------------------------
--------- Discrete choice (multinomial logit)
model Dependent variable Choice Log
likelihood function -256.76133 Estimation
based on N 210, K 7 Information
Criteria Normalization1/N
Normalized Unnormalized AIC
2.51201 527.52265 Fin.Smpl.AIC 2.51465
528.07711 Bayes IC 2.62358
550.95240 Hannan Quinn 2.55712
536.99443 R21-LogL/LogL Log-L fncn R-sqrd
R2Adj Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------
57
Estimated MNL Model
--------------------------------------------------
--------- Discrete choice (multinomial logit)
model Dependent variable Choice Log
likelihood function -256.76133 Estimation
based on N 210, K 7 Information
Criteria Normalization1/N
Normalized Unnormalized AIC
2.51201 527.52265 Fin.Smpl.AIC 2.51465
528.07711 Bayes IC 2.62358
550.95240 Hannan Quinn 2.55712
536.99443 R21-LogL/LogL Log-L fncn R-sqrd
R2Adj Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------
58
Model Fit Based on Log Likelihood

• Three sets of predicted probabilities
• No model Pij 1/J (.25)
• Constants only Pij (1/N)?i dij
• (58,63,30,59)/210.286,.300,.143,.281)
• Estimated model Logit probabilities
• Compute log likelihood
• Measure improvement in log likelihood with
  R-squared 1 LogL/LogL0 (Adjusted for number
  of parameters in the model.)
• NOT A MEASURE OF FIT!

59
Fit the Model with Only ASCs
--------------------------------------------------
------------------------------- Discrete choice
(multinomial logit) model Dependent variable
Choice Log likelihood function
-283.75877 Constants only -283.7588
.0000-.0048 Response data are given as ind.
choices Number of obs. 210, skipped 0
obs ---------------------------------------------
------------- Variable Coefficient Standard
Error b/St.Er. PZgtz ------------------------
---------------------------------- A_AIR
-.01709 .18491 -.092 .9263
A_TRAIN .06560 .18117 .362
.7173 A_BUS -.67634 .22424
-3.016 .0026 ----------------------------------
------------------------ Log likelihood function
-256.76133 Constants only -283.7588 .0951
.0850 Chi-squared 4
53.99489 Prob chi squared gt value
.00000 ------------------------------------------
---------------- Variable Coefficient
Standard Error b/St.Er. PZgtz ---------------
-------------------------------------------
GC .03711 .01484 2.500
.0124 INVC -.05480 .01668
-3.285 .0010 INVT -.00896
.00215 -4.162 .0000 HINCA
.02922 .00931 3.138 .0017
A_AIR -1.88740 .69281 -2.724
.0064 A_TRAIN .69364 .25010
2.773 .0055 A_BUS -.20307
.24817 -.818 .4132 ---------------------
-------------------------------------
60
CLOGIT Fit Measures

• Based on the log likelihood

Values in parentheses below show the number of
correct predictions by a model with only choice
specific constants.
Log likelihood function -256.76133 Constants
only -283.7588 .0951 .0850 Chi-squared 4
53.99489
Based on the model predictions
------------------------------------------------
------ Cross tabulation of actual vs.
predicted choices. Row indicator is
actual, column is predicted.
Predicted total is F(k,j,i)Sum(i1,...,N)
P(k,j,i). Column totals may be subject to
rounding error. -------------------------
----------------------------- Matrix Crosstab
has 5 rows and 5 columns. AIR
TRAIN BUS CAR Total
------------------------------------------
---------------------------- AIR 35.0000
(16) 7.0000 4.0000 13.0000
58.0000 TRAIN 7.0000 41.0000 (19)
4.0000 11.0000 63.0000 BUS
5.0000 4.0000 16.0000 (4) 4.0000
30.0000 CAR 11.0000 11.0000
6.0000 31.0000 (17) 59.0000 Total
58.0000 63.0000 30.0000
59.0000 210.0000
61
Effects of Changes in Attributes on Probabilities

• Partial Effects Effect of a change in attribute
  k of alternative m on the probability that
  choice j will be made is
• Proportional changes Elasticities

Note the elasticity is the same for all choices
j. (IIA)
62
Elasticities for CLOGIT

• Request Effects attribute (choices where
  changes occur )
• Effects GC() (INVT changes in all choices)

-------------------------------------------------
-- Elasticity averaged over
observations. Attribute is GC in choice
AIR Effects on probabilities of
all choices in model Direct Elasticity
effect of the attribute.
Mean St.Dev
ChoiceAIR 2.6625 .8999
ChoiceTRAIN -1.1465 .9546
ChoiceBUS -1.1465
.9546 ChoiceCAR -1.1465
.9546 Attribute is GC in choice
TRAIN ChoiceAIR
-1.2224 .5288 ChoiceTRAIN
3.6089 2.1318 ChoiceBUS
-1.2224 .5288
ChoiceCAR -1.2224 .5288
Attribute is GC in choice BUS
ChoiceAIR -.5847
.2773 ChoiceTRAIN -.5847
.2773 ChoiceBUS
3.6921 1.5194 ChoiceCAR
-.5847 .2773 Attribute is GC
in choice CAR ChoiceAIR
-.8990 .4630
ChoiceTRAIN -.8990 .4630
ChoiceBUS -.8990 .4630
ChoiceCAR 2.6416
1.5498 --------------------------------------
-------------
Own effect Cross effects
Note the effect of IIA on the cross effects.
63
Choice Based Sampling

• Over/Underrepresenting alternatives in the data
  set
• Biases in parameter estimates? (Constants only?)
• Biases in estimated variances
• Weighted log likelihood, weight ?j / Fj for all
  i.
• Fixup of covariance matrix
• Choices list of names / list of true
  proportions

64
Choice Based Sampling Estimators
-------------------------------------------------
--------- Variable Coefficient Standard Error
b/St.Er. PZgtz ------------------------------
---------------------------- GC
.03711 .01484 2.500 .0124
INVC -.05480 .01668 -3.285
.0010 INVT -.00896 .00215
-4.162 .0000 HINCA .02922
.00931 3.138 .0017 A_AIR
-1.88740 .69281 -2.724 .0064
A_TRAIN .69364 .25010 2.773
.0055 A_BUS -.20307 .24817
-.818 .4132 ----------------------------------
------------------------ Weighted ---------------
-------------------------------------------
GC .06656 .02169 3.068
.0022 INVC -.09444 .02586
-3.652 .0003 INVT -.01328
.00330 -4.027 .0001 HINCA .02135
.01646 1.297 .1947 A_AIR
-2.84078 1.20646 -2.355 .0185
A_TRAIN -.40944 .37854 -1.082
.2794 A_BUS -1.20969 .31968
-3.784 .0002 ----------------------------------
------------------------
65
Changes in Estimated Elasticities
-------------------------------------------------
---------------- Elasticity
averaged over observations. Attribute is GC
in choice CAR Effects on
probabilities of all choices in model
Unweighted
Attribute is GC in choice CAR
Mean
St.Dev ChoiceAIR
-.8990 .4630 ChoiceTRAIN
-.8990 .4630 ChoiceBUS
-.8990 .4630 ChoiceCAR
2.6416 1.5498 ----------------
----------------------------------- Weighted

ChoiceAIR -3.5148 1.8585
ChoiceTRAIN -3.5148 1.8585
ChoiceBUS -3.5148
1.8585 ChoiceCAR
2.8358 2.3350 ----------------------------
-----------------------
66
The I.I.D Assumption

• Uitj ?ij ?i xitj ?izit ?ijt
• F(?itj) 1 Exp(-Exp(?itj)) (random part of
  each utility)
• Independence across utility functions
• Identical variances (means absorbed in constants)
• Restriction on scaling
• Correlation across alternatives?
• Implication for cross elasticities (we saw
  earlier)
• Behavioral assumption, independence from
  irrelevant alternatives (IIA)