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1
2
Outline
- Introduction
- Error component MXL models
– Identification issue – Variability of parameter estimates – Estimation of choice probabilities
- Usefulness of MNL models
- Conclusions and future research
Shall We Mixed Logit? Estimation stability and prediction - - PowerPoint PPT Presentation
Shall We Mixed Logit? Estimation stability and prediction - - PowerPoint PPT Presentation
Shall We Mixed Logit? Estimation stability and prediction reliability of error component mixed logit models Shusaku NAKAI Ryuichi KITAMURA Kyoto University Toshiyuki YAMAMOTO Nagoya University 1 Outline Introduction Error
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SLIDE 3
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Introduction
MXL models
- considered the most promising discrete choice
model
- widespread applications in recent years
However
- properties of parameter estimates are not well
understood Objective
- Estimation stability and prediction reliability of
error component MXL models are examined with simulated data
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Error component MXL models
- Examined is a trinomial MXL model
+ + + = + + + = + + + =
n n n n n n n n n n n n n n n
X X u X X u X X u
3 2 23 2 13 1 3 2 2 22 2 12 1 2 1 1 21 2 11 1 1
ε µ β β ε µ β β ε µ β β
2 explanatory variables Standard iid Gumbel 2 error components
) , ( ~ ) , ( ~
2 2 2 2 1 1
s N s N
n n
µ µ
+ + + = Σ 6 6 6
2 2 2 2 2 2 2 2 2 2 1
π π π s s s s
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Simulated discrete choice data
Generated by a probit model
Error component MXL models
+ + = + + = + + =
n n n n n n n n n n n n
X X u X X u X X u
3 23 2 13 1 3 2 22 2 12 1 2 1 21 2 11 1 1
ξ β β ξ β β ξ β β = ∑ 1 1 1 ρ ρ
ξ
= = 5 . . 1
2 1
β β ) 1 , ( ~ N X jin
ρ = 0.00, 0.10, 0.30, 0.50, 0.70, 0.90, 0.95, 0.99 Each data set contains 1,000 cases 25 data sets are generated for each value of ρ
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Identification issue
For trinomial probit models, Dansie (1985) suggests
- 3 matrices are equivalent, and produce the same
likelihood value
- Model estimation would not be able to indicate
which is most likely
Error component MXL models
= 1 1
23 23 11
σ σ σ
A
Σ = 1 ' ' 1 1
23 23
σ σ
B
Σ = 1 1 ' '
11
σ
C
Σ
, thus ΣA is not estimable
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Identification issue (cont.)
For GEV models, Börsch-Supan (1990) and Munizaga et al. (2000) estimated in the case of 4 alternatives
- and found that nested logit models have some
capacity to accommodate heteroscedasticity
Error component MXL models
= 1 1
23 23 11
σ σ σ
A
Σ = 1 ' ' 1 1
23 23
σ σ
B
Σ = 1 1 ' '
11
σ
C
Σ
Nested logit model HEV model
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Identification issue (cont.)
- In this study, data sets are simulated by ΣB
Error component MXL models
= 1 1
23 23 11
σ σ σ
A
Σ = 1 ' ' 1 1
23 23
σ σ
B
Σ = 1 1 ' '
11
σ
C
Σ
+ + + = Σ 6 6 6
2 2 2 2 2 2 2 2 2 2 1
π π π s s s s
- Error component MXL
model examined in this study is consistent with ΣA
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Identification issue (cont.)
- Standard deviation becomes extremely large, implying
covariance structure is unidentified
- MXL model is subject to the same identification
problem of probit model (consistent with Walker et al. (2007))
Error component MXL models
0.1 1 10 100 1000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ Parameter Estimate 0.1 1 10 100 1000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ Parameter Estimate
2 1
s ˆ
2 2
s ˆ
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Identification issue (cont.)
- Hereafter, we constrain
Error component MXL models
= 1 1
23 23 11
σ σ σ
A
Σ = 1 ' ' 1 1
23 23
σ σ
B
Σ = 1 1 ' '
11
σ
C
Σ
+ + + = Σ 6 6 6
2 2 2 2 2 2 2 2 2 2 1
π π π s s s s
2 2 2 1 2
s s s = =
+ 1 1 1 6
2 2
ρ π s
6
2 2 2
π ρ + = s s
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Variability of parameter estimates
Error component MXL models
- Parameter estimates are quite instable
especially for the case with higher ρ
1 2 3 4 5 6 7 8 9 10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 誤差相関係数ρ 推定パラメータ値
Error correlation coefficient ρ Parameter estimate
1
ˆ β
. 1
1 =
β
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Variability of parameter estimates (cont.)
- This instability is caused by the dependence of
coefficient estimates on error variance
- Error variance is not standardized in MXL model
- Needs for normalization
- f parameter estimates
Error component MXL models
= ∑ 1 1 1 ρ ρ
ξ
Probit model
+ = Σ 1 1 1 6
2 2
ρ π s
6
2 2 2
π ρ + = s s
Error component MXL model
j j
s β π β ˆ 6 ˆ 1 ~
2 2 +
=
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0.2 0.4 0.6 0.8 1 1.2 1.4 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ Paraemter Estimate
Error component MXL models
- After normalization, utility coefficients are
unbiased and stable
. 1
1 =
β
Variability of parameter estimates (cont.)
1
β ~
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0.1 1 10 100 1000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ Parameter Estimate
Error component MXL models
Variability of parameter estimates (cont.)
- Estimated variances of the error components
tend to be biased upward
True value
2
s ˆ
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Error component MXL models
Variability of parameter estimates (cont.)
- Biases in estimated variances might be related to the
difference in shape of Normal and Gumbel distribution
- Amemiya (1981) suggests in binary case
N(0, 1.62) rather than N(0, π2/3) fits better to L(0, π2/3), though the latter has equal variance to L(0, π2/3) (1.6 < π/30.5 ≈ 1.8)
0.2 0.4 0.6 0.8 1 1.2
- 3
- 2.7
- 2.4
- 2.1
- 1.8
- 1.5
- 1.2
- 0.9
- 0.6
- 0.3
0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
Cumulative distribution function
0.66 0.665 0.67 0.675 0.68 0.685 0.69 0.695 0.75 0.75 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.8 0.8
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Estimation of choice probabilities
Error component MXL models
- Choice probabilities are calculated by
for the case
- The effects of biased estimate of s is
examined by introducing q, and calculate
- True probability is obtained when q ≈ 1.29
( ) ( )
) ( ) ( ˆ ˆ ˆ exp ˆ ˆ ˆ exp ) ( ˆ
2 1 2 2 1 1 2 2 1 1
η η η β β η β β df df s X X s X X i P
j j jn jn i in in n
∫∫∑
+ + + + = = = = 3
- r
2
- r
1
- r
,
2 1
j i if j i if
j i
η η η η
. 1
23 13 22 12 21 11
= = = = = = X X X X X X
( ) ( )
) ( ) ( 6 exp 6 exp ) | (
2 1 2 2 2 1 1 2 2 2 1 1
η η π η β β π η β β df df s X X q s X X q q i P
j j j j i i i
∫∫∑
+ + + + =
SLIDE 17
17 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability 1.15 1.73 P(1) P(3) P(2) 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability 1.15 1.73 P(1) P(3) P(2)
Estimation of choice probabilities (cont.)
Error component MXL models
- True probabilities are contained in the
range of the estimated probability
0.1 = ρ
Range of estimated probability
True value
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18 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability P(1) P(3) P(2) 1.57 2.29 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability P(1) P(3) P(2) 1.57 2.29
Estimation of choice probabilities (cont.)
Error component MXL models
- True probabilities are NOT contained in
the range of the estimated probability
Range of estimated probability
True value
0.5 = ρ
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19 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability P(1) P(3) P(2) 3.45 5.33 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 q Choice Probability P(1) P(3) P(2) 3.45 5.33
Estimation of choice probabilities (cont.)
Error component MXL models
- True probabilities are NOT contained in
the range of the estimated probability
Range of estimated probability
True value
0.9 = ρ
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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ Parameter Estimate
Usefulness of MNL models
- MNL models are estimated using the same
data sets
- Utility coefficient estimates are biased upward,
but up to about 30%, smaller than MXL model
. 1
1 =
β
1
ˆ β
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Conclusions and future research
For the error component MXL model 1. Variance structure cannot be uniquely identified through model estimation 2. Parameter estimates are quite instable especially for the case with a high error correlation 3. After proper normalization, utility coefficients are unbiased and stable 4. Estimated variances of the error components tend to be biased upwards 5. Choice probabilities are biased unless the error correlation is very small MNL model can produce relatively unbiased utility coefficient estimates
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Conclusions and future research (cont.)
- One would adopt MXL model in search of
covariance specification
- > The model is incapable of identifying the true
structure, and parameter estimates are instable
- One may opt to develop adequately specified
MNL through careful selection of explanatory variables, utility formulation or definition of alternatives (consistent with suggestion by Pinjari & Bhat (2006))
- Needs for further research on properties of