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 component MXL models – Identification issue – Variability of parameter estimates – Estimation of choice probabilities • Usefulness of MNL models • Conclusions and future research 2

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 3

Error component MXL models • Examined is a trinomial MXL model 2 explanatory variables Standard iid Gumbel = β + β + µ + ε  u X X 1 1 11 2 21 1 1 n n n n n  = β + β + µ + ε  u X X 2 1 12 2 22 2 2 n n n n n  = β + β + µ + ε  u X X 3 1 13 2 23 2 3 n n n n n   π 2 2 error components   + 2 0 0 s 1   6 µ π   2 2 ~ ( 0 , ) N s Σ = + 2 2 s s 1 1   n 2 2 6   µ π 2 2 ~ ( 0 , ) N s   + 2 s 2 2   n 4 2  6 

Error component MXL models Simulated discrete choice data Generated by a probit model = β + β + ξ    u X X 1 0 0 1 1 11 2 21 1 n n n n     ∑ = ρ = β + β + ξ    0 1 u X X ξ 2 1 12 2 22 2 n n n n    ρ  0 1   = β + β + ξ  u X X 3 1 13 2 23 3 n n n n β =  1 . 0 1  β = ρ = 0.00, 0.10, 0.30, 0.50, 0.70, 0.90, 0.95, 0.99  0 . 5 2 Each data set contains 1,000 cases 25 data sets are generated for each value of ρ ~ N ( 0 , 1 ) X jin 5

Error component MXL models Identification issue For trinomial probit models, Dansie (1985) suggests σ σ       0 0 1 0 0 ' ' 0 0 11 11       = σ = σ = Σ Σ Σ 0 1 0 1 ' 0 1 0       23 23 A B C     σ   σ  0 1   0 ' 1   0 0 1  23 23 • 3 matrices are equivalent, and produce the same , thus Σ A is not estimable likelihood value • Model estimation would not be able to indicate which is most likely 6

Error component MXL models Identification issue (cont.) For GEV models, Börsch-Supan (1990) and Munizaga et al. (2000) estimated in the case of 4 alternatives Nested logit model HEV model σ σ       0 0 1 0 0 ' ' 0 0 11 11       = σ = σ = Σ Σ Σ 0 1 0 1 ' 0 1 0       23 23 A B C     σ   σ  0 1   0 ' 1   0 0 1  23 23 • and found that nested logit models have some capacity to accommodate heteroscedasticity 7

Error component MXL models Identification issue (cont.) • In this study, data sets are simulated by Σ B σ   σ     0 0 1 0 0 ' ' 0 0 11 11       = σ = σ = Σ Σ Σ 0 1 0 1 ' 0 1 0       23 A 23 B C     σ σ    0 1  0 ' 1    0 0 1  23 23   π 2   + 2 0 0 s • Error component MXL 1   6 π   model examined in 2 Σ = + 2 2 s s   2 2 this study is consistent 6   π with Σ A 2   + 2 s   2  6  8

Error component MXL models Identification issue (cont.) 1000 1000 2 ˆ 2 s ˆ s 2 1 100 100 Parameter Estimate Parameter Estimate 10 10 1 1 0.1 0.1 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ ρ • 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)) 9

Error component MXL models Identification issue (cont.) = = 2 2 2 • Hereafter, we constrain s s s 1 2 σ   σ     0 0 1 0 0 ' ' 0 0 11 11       = σ = σ = Σ Σ Σ 0 1 0 1 ' 0 1 0       23 A 23 B C     σ σ    0 1  0 ' 1    0 0 1  23 23   1 0 0   π 2     π   2 + 2 0 0 s   + ρ 2   1 1   s   6   π 6     2 Σ = + 2 2 1   s s   2 2 6   π 2 2   + s 2 s  ρ =  2  6  π 2 + 2 s 10 6

Error component MXL models Variability of parameter estimates 10 ˆ 9 β 8 1 Parameter estimate 推定パラメータ値 7 6 5 4 3 2 β 1 = 1 . 0 1 0 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 Error correlation coefficient ρ • Parameter estimates are quite instable especially for the case with higher ρ 11

Error component MXL models Variability of parameter estimates (cont.) • This instability is caused by the dependence of coefficient estimates on error variance Probit model Error component MXL model     1 0 0 1 0 0 2 s       π ρ = 2 ∑ = ρ   Σ = + ρ 2   π   2 0 1 1 s   ξ + 2 s  6      ρ  0 1  6  1  • Error variance is not standardized in MXL model • Needs for normalization ~ 1 β = β ˆ of parameter estimates j j π 2 2 + ˆ s 6 12

Error component MXL models Variability of parameter estimates (cont.) 1.4 ~ β 1.2 1 β 1 = 1 . 0 Paraemter Estimate 1 0.8 0.6 0.4 0.2 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ • After normalization, utility coefficients are unbiased and stable 13

Error component MXL models Variability of parameter estimates (cont.) 1000 2 ˆ s 100 Parameter Estimate 10 True value 1 0.1 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ρ • Estimated variances of the error components tend to be biased upward 14

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, 1.6 2 ) rather than N(0, π 2 /3) fits better to L(0, π 2 /3), though the latter has equal variance to L(0, π 2 /3) (1.6 < π /3 0.5 ≈ 1.8) Cumulative distribution function 0.695 1.2 0.69 1 0.685 0.8 0.68 0.6 0.675 0.4 0.67 0.2 0.665 0 -2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0 3 -3 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 0.66 15 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

Error component MXL models Estimation of choice probabilities • Choice probabilities are calculated by ( ) η = ˆ ˆ  β + β + η or 1 ˆ if i j exp X X s ∫∫∑ ˆ = η η η η = ( ) 1 1 1 2 2  ( ) in in i ( ) ( ) , P i df df η = 1 2 n β ˆ + β ˆ + η i j  or 2 or 3 exp ˆ if i j X X s 2 1 1 jn 2 2 jn j j = = = = = = for the case 1 . 0 X X X X X X 11 21 12 22 13 23 • The effects of biased estimate of s is examined by introducing q, and calculate ( ) β + β + η π 2 exp 6 q X X s ∫∫∑ ( ) = η η 1 1 2 2 i i i ( | ) ( ) ( ) P i q df df 1 2 β + β + η π 2 exp 6 q X X s 1 1 2 2 j j j j • True probability is obtained when q ≈ 1.29 16

Error component MXL models Estimation of choice probabilities (cont.) 0.6 0.6 ρ = 0.1 0.5 0.5 P(1) P(1) 0.4 0.4 Choice Probability Choice Probability P(2) P(2) 0.3 0.3 True value P(3) P(3) 0.2 0.2 0.1 0.1 1.15 1.15 1.73 1.73 0 0 0.1 0.1 1 1 10 10 Range of estimated probability q q • True probabilities are contained in the 17 range of the estimated probability

Error component MXL models Estimation of choice probabilities (cont.) 0.6 0.6 ρ = 0.5 0.5 0.5 P(1) P(1) 0.4 0.4 Choice Probability Choice Probability 0.3 0.3 P(2) P(2) True value P(3) P(3) 0.2 0.2 0.1 0.1 1.57 1.57 2.29 2.29 0 0 0.1 0.1 1 1 10 10 Range of estimated probability q q • True probabilities are NOT contained in 18 the range of the estimated probability

Error component MXL models Estimation of choice probabilities (cont.) 0.6 0.6 ρ = 0.9 0.5 0.5 P(1) P(1) 0.4 0.4 Choice Probability Choice Probability 0.3 0.3 P(2) P(2) True value P(3) P(3) 0.2 0.2 0.1 0.1 3.45 3.45 5.33 5.33 0 0 0.1 0.1 1 1 10 10 Range of estimated probability q q • True probabilities are NOT contained in 19 the range of the estimated probability