Home Misspecified Discrete Choice Models and Huber-White Standard Errors
Article
Licensed
Unlicensed Requires Authentication

Misspecified Discrete Choice Models and Huber-White Standard Errors

  • Michael Guggisberg EMAIL logo
Published/Copyright: February 1, 2018
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Journal of Econometric Methods
From the journal Journal of Econometric Methods Volume 8 Issue 1

Abstract

I analyze properties of misspecified discrete choice models and the efficacy of Huber-White (sometimes called ‘robust’) standard errors. The Huber-White correction provides asymptotically correct standard errors for a consistent estimator from a misspecified model. There is little justification for using Huber-White standard errors in discrete choice models since misspecification usually leads to inconsistent estimators. I derive necessary and sufficient conditions for consistency of the maximum likelihood estimator of any potentially misspecified random utility model (e.g. conditional logit). I also derive (easily satisfied) sufficient conditions for consistent estimation of the sign of the data generating parameter. It follows the researcher can consistently test the sign (or nullity) of the parameter from the data generating process using the (possibly) misspecified conditional logit. I investigate small sample properties of the Huber-White estimator via a simulation study and find the correction provides little to no improvement for inferences.

Keywords: discrete choice; Huber-White standard errors; misspecified models

Acknowledgement

I am a Research Staff Member in the Strategy, Forces and Resources Division at the Institute for Defense Analyses. I would like to thank David Brownstone, Daniel Gillen, Dale Poirier and Ivan Jeliazkov for their invaluable comments and support. I also thank the University of California, Irvine School of Social Sciences for funding this research.

A Appendix

A.1 Type One Error Rate

Corollary 2 states type one error rates for null coefficients will be at least asymptotically conservative. Table 2 and Table 3 illustrate this result. The coefficients for the observable utility in the DGP were (β10,β20)=(0,1). Then a mixed logit DGP (Table 2) and a heteroskedastic logit DGP (Table 3) were simulated but a conditional logit was estimated. The alpha level was set to α = 0.20. The top row in Table 2 shows the different variances in the random effect. The top row in Table 3 shows the different levels of heteroskedasticity. The ‘J’ and ‘N’ columns represent the number of presented alternatives and individuals respectively. The values in the table are the simulated type one error rates for β1.

Table 2:

Type One Error Rate for α = 0.20 (Mixed Logit DGP).

σ 2 = 0.52
 σ 2 = 12
 σ 2 = 22
J N H R H R H R
2 100 0.18 0.18 0.18 0.15 0.15 0.10
2 500 0.19 0.20 0.18 0.19 0.15 0.16
2 1000 0.18 0.18 0.15 0.16 0.10 0.10
3 100 0.20 0.20 0.22 0.22 0.24 0.24
3 500 0.18 0.18 0.18 0.18 0.13 0.13
3 1000 0.19 0.19 0.19 0.20 0.16 0.16
5 100 0.18 0.20 0.19 0.21 0.20 0.22
5 500 0.19 0.19 0.18 0.19 0.14 0.14
5 1000 0.19 0.20 0.19 0.20 0.17 0.19

  1. H, Hessian standard error; R, Huber-White robust standard error.

Table 3:

Type One Error Rate for α = 0.20 (Heteroskedastic Logit DGP).

γ 1 = 0.5
 γ 1 = 1
 γ 1 = 1.5
J N H R H R H R
2 100 0.18 0.21 0.18 0.20 0.18 0.20
2 500 0.19 0.18 0.20 0.19 0.19 0.18
2 1000 0.20 0.19 0.20 0.20 0.21 0.21
3 100 0.18 0.19 0.19 0.21 0.20 0.23
3 500 0.18 0.18 0.19 0.18 0.22 0.21
3 1000 0.18 0.18 0.18 0.18 0.19 0.19
5 100 0.22 0.24 0.18 0.19 0.19 0.21
5 500 0.21 0.20 0.21 0.21 0.20 0.20
5 1000 0.20 0.20 0.20 0.19 0.20 0.20

  1. H, Hessian standard error; R, Huber-White robust standard.

A.2 Kullback-Leibler Minimizer Estimand

When the model is misspecified the QML estimator is estimating the parameter minimizing the KL divergence from the assumed model to the true model. Table 4 and Table 5 show the KL minimizer from the simulation failing to specificy a random effect (Table 4) and heteroskedastic effect (Table 5). The top row in Table 4 shows the different variances in the random effect. The top row in Table 5 shows the different levels of heteroskedasticity. The ‘J’ and ‘N’ columns represent the number of presented alternatives and individuals respectively. The β column identifies the parameter of interest. The coefficients for the observable utility in the DGP were (β10,β20)=(−2,1). The ‘β⋆’ column shows the analytic KL minimizer found by using numerical optimization. The ‘SE’ column shows the estimated standard error of the QML estimates around the KL minimizer over the simulation. This standard error is calculated by taking the square root of the sample variance of the estimated beta coefficients around the analytic KL minimizer, S−1∑s=1S(β^is−βi⋆)2 for i∈{1,2} where S is the number of simulations. The ‘KL’ column shows the KL divergence evaluated at the KL minimizing parameter.

Table 4:

KL Minimizer (Mixed Logit DGP).

σ 2 = 0.5 2
σ 2 = 1 2
σ 2 = 2 2
J N β β ⋆ SE KL β ⋆ SE KL β ⋆ SE KL
2 100 −2 −1.80 0.46 2.2 −1.48 0.35 7.8 −0.93 0.20 21.4
2 500 −2 −1.91 0.21 8.9 −1.67 0.17 31.7 −1.20 0.10 88.6
2 1000 −2 −1.88 0.14 16.4 −1.63 0.11 59.2 −1.17 0.07 171.7
3 100 −2 −1.93 0.40 2.6 −1.75 0.36 9.2 −1.30 0.25 25.5
3 500 −2 −1.97 0.17 10.3 −1.84 0.16 36.1 −1.50 0.12 100.9
3 1000 −2 −1.92 0.11 19.0 −1.77 0.11 68.0 −1.40 0.08 197.2
5 100 −2 −1.98 0.36 2.3 −1.90 0.37 8.1 −1.58 0.29 24.4
5 500 −2 −2.00 0.16 9.3 −1.96 0.15 34.0 −1.73 0.14 104.0
5 1000 −2 −1.97 0.11 16.7 −1.90 0.11 62.6 −1.65 0.09 197.7
2 100 1 0.82 0.50 2.2 0.64 0.41 7.8 0.44 0.31 21.4
2 500 1 0.89 0.22 8.9 0.76 0.19 31.7 0.54 0.14 88.6
2 1000 1 0.95 0.17 16.4 0.85 0.14 59.2 0.67 0.11 171.7
3 100 1 0.92 0.43 2.6 0.85 0.36 9.2 0.74 0.30 25.5
3 500 1 0.93 0.19 10.3 0.84 0.17 36.1 0.68 0.14 100.9
3 1000 1 0.97 0.13 19.0 0.92 0.12 68.0 0.80 0.11 197.2
5 100 1 1.01 0.38 2.3 1.02 0.38 8.1 0.99 0.34 24.4
5 500 1 0.98 0.16 9.3 0.93 0.16 34.0 0.81 0.14 104.0
5 1000 1 0.99 0.11 16.7 0.95 0.11 62.6 0.86 0.10 197.7

  1. β ⋆, KL minimizer; SE, Standard error of β^ around β⋆; KL, minimized KL distance of assumed model from DGP.

Table 5:

KL Minimizer (Heteroskedastic Logit DGP).

γ 1 = 0.5
 γ 1 = 1
 γ 1 = 1.5
J N β β ⋆ SE KL β ⋆ SE KL β ⋆ SE KL
2 100 −2 −1.67 0.49 4.2 −1.24 0.40 10.5 −1.03 0.33 14.9
2 500 −2 −1.60 0.20 21.7 −1.07 0.14 58.1 −0.80 0.11 84.9
2 1000 −2 −1.64 0.14 41.4 −1.13 0.11 112.9 −0.86 0.09 168.1
3 100 −2 −1.76 0.41 6.4 −1.35 0.35 16.7 −1.10 0.31 24.3
3 500 −2 −1.66 0.17 31.1 −1.15 0.14 84.9 −0.87 0.11 127.5
3 1000 −2 −1.70 0.12 61.6 −1.21 0.10 169.7 −0.92 0.08 254.8
5 100 −2 −1.81 0.35 7.0 −1.47 0.34 19.7 −1.23 0.31 31.2
5 500 −2 −1.73 0.16 39.2 −1.27 0.14 113.7 −0.98 0.13 177.9
5 1000 −2 −1.73 0.11 80.3 −1.27 0.10 230.2 −0.97 0.09 353.9
2 100 1 0.71 0.55 4.2 0.51 0.48 10.5 0.41 0.44 14.9
2 500 1 0.96 0.21 21.7 0.84 0.18 58.1 0.75 0.16 84.9
2 1000 1 0.90 0.15 41.4 0.74 0.13 112.9 0.63 0.12 168.1
3 100 1 0.94 0.38 6.4 0.76 0.35 16.7 0.65 0.30 24.3
3 500 1 1.01 0.18 31.1 0.93 0.16 84.9 0.87 0.15 127.5
3 1000 1 1.01 0.13 61.6 0.91 0.12 169.7 0.83 0.10 254.8
5 100 1 0.89 0.34 7.0 0.74 0.31 19.7 0.64 0.28 31.2
5 500 1 0.99 0.16 39.2 0.92 0.15 113.7 0.87 0.14 177.9
5 1000 1 1.02 0.11 80.3 0.97 0.11 230.2 0.92 0.10 353.9

  1. β ⋆, KL minimizer; SE, Standard error of β^ around β⋆; KL, minimized KL distance of assumed model from DGP.

A.3 Coverage Probabilities

Coverage probabilities of the QML estimators for the data generating parameters, β0, are presented in Table 6, Table 7, and Table 8. Coverage probabilities of the QML estimators for the KL minimizer parameters, β⋆, are presented in Table 9 and Table 10. The column J represents the number of alternatives and N represents the number of individuals. The coefficients for the observable utility in the data generating process were (β10,β20)=(−2,1). Three types of standard errors are used: Hessian (denoted by H), Huber-White robust (denoted by R), and simulation (denoted by S). The simulation standard error is calculated by taking the square root of the variance of the estimated β coefficients around the data generating coefficient, S−1∑s=1S(β^is−βi0)2 for i∈{1,2} where S is the number of simulations. The simulation based standard errors help show if normality of the estimator is being achieved. The confidence intervals calculated have level 80%, thus a better performing estimator will have a coverage probability closer to 0.80.

Table 6 shows the coverage probabilities of the correctly specified conditional logit. Table 7 and Table 9 shows the coverage probabilities of the misspecified conditional logit failing to account for individual level heteroskedasticity. Coverage probabilities for the misspecified conditional logit (denoted M) and the correctly specified heteroskedastic logit (denoted C) are given. In the correctly specified heteroskedastic logit the heteroskedastic parameter θn is estimated (but omitted from the table). Table 8 and Table 10 shows the coverage probabilities for the misspecified conditional logit failing to account for a random alternative specific effect. Coverage probabilities for the misspecified conditional logit (denoted M) and the correctly specified mixed logit (denoted C) are given.

Table 6:

Coverage Probabilities of DGP Parameters (Conditional Logit Correctly Specified).

J I β H R S
2 100 β 1 0.80 0.79 0.86
2 500 β 1 0.80 0.79 0.81
2 1000 β 1 0.82 0.82 0.81
3 100 β 1 0.80 0.79 0.84
3 500 β 1 0.77 0.77 0.80
3 1000 β 1 0.79 0.78 0.80
5 100 β 1 0.83 0.81 0.82
5 500 β 1 0.78 0.78 0.81
5 1000 β 1 0.79 0.79 0.80
2 100 β 2 0.82 0.80 0.82
2 500 β 2 0.82 0.82 0.82
2 1000 β 2 0.80 0.81 0.82
3 100 β 2 0.80 0.79 0.82
3 500 β 2 0.79 0.80 0.80
3 1000 β 2 0.80 0.80 0.80
5 100 β 2 0.80 0.80 0.82
5 500 β 2 0.82 0.81 0.80
5 1000 β 2 0.80 0.80 0.80

  1. H, Hessian standard error; R, Huber-White robust standard error; S, Simulation standard error.

Table 7:

Coverage Probabilities of DGP Parameters (Mixed Logit DGP).

σ = 0.52
 σ = 12
 σ = 22
J I β HM RM HC RC SC HM RM HC RC SC HM RM HC RC SC
2 100 β 1 0.76 0.74 0.98 0.98 0.99 0.48 0.46 0.96 0.96 0.99 0.00 0.00 0.90 0.90 0.98
2 500 β 1 0.77 0.76 0.96 0.97 0.79 0.33 0.32 0.91 0.91 0.82 0.00 0.00 0.65 0.65 0.87
2 1000 β 1 0.66 0.65 0.96 0.96 0.86 0.04 0.04 0.83 0.83 0.88 0.00 0.00 0.63 0.63 0.86
3 100 β 1 0.80 0.79 0.61 0.61 0.89 0.62 0.63 0.62 0.62 0.90 0.14 0.16 0.62 0.62 0.90
3 500 β 1 0.81 0.81 0.39 0.39 0.68 0.59 0.58 0.39 0.39 0.68 0.01 0.01 0.44 0.44 0.73
3 1000 β 1 0.74 0.73 0.42 0.42 0.78 0.23 0.22 0.40 0.40 0.78 0.00 0.00 0.39 0.39 0.75
5 100 β 1 0.80 0.80 0.55 0.55 0.68 0.77 0.77 0.60 0.60 0.73 0.44 0.47 0.66 0.67 0.79
5 500 β 1 0.79 0.79 0.63 0.63 0.81 0.79 0.80 0.66 0.66 0.86 0.29 0.31 0.61 0.61 0.85
5 1000 β 1 0.78 0.79 0.47 0.47 0.61 0.62 0.62 0.43 0.43 0.61 0.01 0.01 0.34 0.34 0.67
2 100 β 2 0.81 0.79 0.96 0.96 0.98 0.75 0.72 0.96 0.96 0.99 0.51 0.44 0.92 0.93 0.98
2 500 β 2 0.77 0.76 0.96 0.96 0.83 0.58 0.56 0.92 0.92 0.87 0.09 0.08 0.79 0.79 0.82
2 1000 β 2 0.78 0.77 0.97 0.97 0.98 0.66 0.65 0.96 0.96 0.95 0.10 0.09 0.88 0.89 0.90
3 100 β 2 0.80 0.79 0.73 0.74 0.88 0.81 0.79 0.75 0.76 0.90 0.74 0.73 0.75 0.76 0.86
3 500 β 2 0.76 0.76 0.48 0.48 0.89 0.63 0.62 0.50 0.51 0.89 0.25 0.24 0.51 0.51 0.87
3 1000 β 2 0.79 0.79 0.53 0.53 0.86 0.75 0.75 0.50 0.50 0.86 0.37 0.36 0.47 0.47 0.87
5 100 β 2 0.80 0.81 0.71 0.71 0.83 0.80 0.80 0.73 0.74 0.84 0.84 0.84 0.73 0.74 0.83
5 500 β 2 0.82 0.82 0.64 0.64 0.79 0.75 0.75 0.65 0.65 0.83 0.54 0.54 0.59 0.59 0.87
5 1000 β 2 0.80 0.80 0.47 0.47 0.73 0.78 0.78 0.43 0.43 0.71 0.52 0.52 0.41 0.41 0.86

  1. First Letter H, Hessian standard error; R, Huber-White robust standard error; S, Simulation standard error. Second Letter M, Misspecified model; C, Correct model.

Table 8:

Coverage Probabilities of DGP Parameters (Heteroskedastic Logit DGP).

σ = 0.52
 σ = 12
 σ = 22
J I β HM RM HC RC SC HM RM HC RC SC HM RM HC RC SC
2 100 β 1 0.64 0.69 0.79 0.74 0.88 0.24 0.30 0.80 0.75 1.00 0.08 0.12 0.81 0.77 0.99
2 500 β 1 0.22 0.28 0.78 0.78 0.82 0.00 0.00 0.80 0.79 0.81 0.00 0.00 0.81 0.80 0.82
2 1000 β 1 0.09 0.12 0.80 0.79 0.81 0.00 0.00 0.80 0.79 0.80 0.00 0.00 0.78 0.78 0.81
3 100 β 1 0.67 0.73 0.79 0.77 0.85 0.26 0.33 0.84 0.81 0.86 0.09 0.13 0.80 0.77 0.87
3 500 β 1 0.22 0.28 0.81 0.80 0.80 0.00 0.00 0.82 0.81 0.82 0.00 0.00 0.78 0.78 0.81
3 1000 β 1 0.10 0.14 0.80 0.80 0.81 0.00 0.00 0.81 0.81 0.81 0.00 0.00 0.78 0.78 0.80
5 100 β 1 0.72 0.76 0.81 0.79 0.81 0.34 0.44 0.77 0.76 0.82 0.12 0.17 0.84 0.81 0.83
5 500 β 1 0.31 0.39 0.81 0.80 0.81 0.00 0.00 0.80 0.79 0.80 0.00 0.00 0.82 0.80 0.82
5 1000 β 1 0.12 0.17 0.79 0.79 0.81 0.00 0.00 0.79 0.79 0.80 0.00 0.00 0.79 0.79 0.79
2 100 β 2 0.74 0.73 0.80 0.74 0.85 0.59 0.60 0.80 0.67 0.95 0.51 0.51 0.85 0.64 0.96
2 500 β 2 0.82 0.82 0.80 0.80 0.80 0.68 0.68 0.82 0.79 0.82 0.47 0.46 0.83 0.79 0.82
2 1000 β 2 0.73 0.73 0.79 0.78 0.80 0.28 0.28 0.81 0.80 0.80 0.06 0.06 0.80 0.76 0.81
3 100 β 2 0.83 0.82 0.78 0.74 0.84 0.74 0.73 0.82 0.71 0.86 0.66 0.64 0.85 0.75 0.89
3 500 β 2 0.83 0.82 0.79 0.78 0.81 0.80 0.78 0.80 0.77 0.82 0.70 0.70 0.81 0.78 0.80
3 1000 β 2 0.81 0.81 0.79 0.79 0.80 0.71 0.70 0.79 0.77 0.81 0.46 0.45 0.80 0.78 0.78
5 100 β 2 0.81 0.81 0.79 0.77 0.83 0.71 0.69 0.79 0.74 0.82 0.57 0.54 0.88 0.78 0.83
5 500 β 2 0.82 0.82 0.81 0.81 0.80 0.76 0.76 0.80 0.78 0.81 0.66 0.66 0.82 0.77 0.81
5 1000 β 2 0.82 0.83 0.81 0.80 0.79 0.80 0.80 0.77 0.76 0.79 0.70 0.71 0.80 0.77 0.81

  1. First Letter H, Hessian standard error; R, Huber-White robust standard error; S, Simulation standard error. Second Letter M, Misspecified model; C, Correct model.

Table 9:

Coverage Probabilities of KL Minimizer (Mixed Logit DGP).

σ = 0.52
 σ = 12
 σ = 22
J N β H R S H R S H R S
2 100 β 1 0.82 0.81 0.84 0.84 0.82 0.82 0.90 0.89 0.83
2 500 β 1 0.81 0.80 0.83 0.84 0.83 0.81 0.91 0.88 0.80
2 1000 β 1 0.82 0.80 0.82 0.85 0.83 0.81 0.92 0.89 0.83
3 100 β 1 0.81 0.80 0.82 0.80 0.81 0.84 0.85 0.85 0.81
3 500 β 1 0.82 0.82 0.80 0.81 0.82 0.81 0.86 0.85 0.79
3 1000 β 1 0.82 0.81 0.79 0.80 0.80 0.81 0.86 0.85 0.80
5 100 β 1 0.80 0.80 0.81 0.79 0.78 0.84 0.82 0.84 0.82
5 500 β 1 0.79 0.79 0.79 0.82 0.82 0.80 0.82 0.83 0.82
5 1000 β 1 0.80 0.80 0.80 0.81 0.81 0.82 0.83 0.83 0.81
2 100 β 2 0.84 0.80 0.78 0.91 0.87 0.79 0.94 0.90 0.84
2 500 β 2 0.82 0.82 0.80 0.87 0.86 0.79 0.93 0.92 0.78
2 1000 β 2 0.81 0.80 0.81 0.86 0.85 0.79 0.91 0.90 0.81
3 100 β 2 0.81 0.80 0.83 0.85 0.84 0.81 0.89 0.88 0.82
3 500 β 2 0.80 0.80 0.81 0.81 0.81 0.79 0.88 0.87 0.78
3 1000 β 2 0.80 0.79 0.79 0.82 0.82 0.78 0.87 0.86 0.81
5 100 β 2 0.80 0.80 0.81 0.80 0.80 0.82 0.83 0.84 0.80
5 500 β 2 0.82 0.82 0.81 0.81 0.81 0.80 0.85 0.85 0.80
5 1000 β 2 0.80 0.80 0.77 0.81 0.81 0.78 0.86 0.86 0.80

  1. H, Hessian standard error; R, Huber White robust standard error; S, Simulation standard error.

Table 10:

Coverage Probabilities of KL Minimizer (Heteroskedastic Logit DGP).

γ 1 = 0.5
 γ 1 = 1
 γ 1 = 1.5
J N β H R S H R S H R S
2 100 β 1 0.77 0.80 0.84 0.75 0.82 0.87 0.79 0.88 0.86
2 500 β 1 0.74 0.81 0.82 0.73 0.85 0.80 0.74 0.88 0.81
2 1000 β 1 0.76 0.82 0.81 0.72 0.84 0.83 0.72 0.87 0.83
3 100 β 1 0.77 0.81 0.83 0.75 0.85 0.83 0.74 0.84 0.84
3 500 β 1 0.75 0.82 0.80 0.72 0.85 0.82 0.70 0.86 0.81
3 1000 β 1 0.75 0.82 0.81 0.68 0.84 0.79 0.71 0.88 0.79
5 100 β 1 0.78 0.82 0.82 0.72 0.82 0.82 0.73 0.85 0.83
5 500 β 1 0.75 0.83 0.81 0.70 0.85 0.81 0.66 0.84 0.81
5 1000 β 1 0.75 0.84 0.80 0.69 0.85 0.79 0.68 0.86 0.84
2 100 β 2 0.82 0.82 0.82 0.79 0.82 0.78 0.85 0.84 0.82
2 500 β 2 0.83 0.83 0.80 0.86 0.86 0.80 0.87 0.87 0.80
2 1000 β 2 0.84 0.83 0.80 0.86 0.86 0.80 0.87 0.88 0.82
3 100 β 2 0.83 0.82 0.80 0.85 0.83 0.80 0.90 0.89 0.81
3 500 β 2 0.82 0.82 0.82 0.86 0.85 0.81 0.88 0.87 0.82
3 1000 β 2 0.81 0.81 0.80 0.85 0.85 0.83 0.88 0.87 0.79
5 100 β 2 0.83 0.82 0.81 0.85 0.84 0.81 0.87 0.86 0.81
5 500 β 2 0.82 0.82 0.80 0.85 0.84 0.82 0.85 0.85 0.81
5 1000 β 2 0.83 0.83 0.80 0.82 0.83 0.80 0.84 0.85 0.79

  1. H, Hessian standard error; R, Huber White robust standard error; S, Simulation standard error.

A.4 MSE of Choice Probabilities

Square root of MSE of choice probabilities is presented in Table 11 and Table 12. In Table 11 the data generating process is mixed logit; a conditional logit and a mixed logit are estimated. In Table 12 the data generating process is heteroskedastic logit; a conditional logit and a heteroskedastic logit are estimated. The square root MSE is calculated as S−1∑s=1SN−1∑n=1N∑j=1J(P^njs−Pnjs0)2, where P^njs is the estimated choice probability and Pnjs0 is the true choice probability for individual n with alternative j in simulation s. The choice probabilities for the mixed logit are with the individual effects marginalized out. The choice probabilities have been scaled by 10 for presentation purposes.

Table 11:

Square Root MSE of Choice Probabilities (Mixed Logit DGP).

σ = 0.52
 σ = 12
 σ = 22
J N M C M C M C
2 100 0.46 1.08 0.46 1.01 0.56 1.00
2 500 0.22 0.62 0.22 0.61 0.23 0.55
2 1000 0.16 0.54 0.17 0.49 0.21 0.55
3 100 0.38 1.24 0.39 1.21 0.52 1.07
3 500 0.18 1.46 0.22 1.37 0.39 1.17
3 1000 0.13 1.54 0.19 1.51 0.38 1.24
5 100 0.25 0.59 0.29 0.56 0.45 0.53
5 500 0.12 0.36 0.18 0.33 0.38 0.48
5 1000 0.09 0.57 0.16 0.57 0.37 0.60

  1. M, Misspecified conditional logit; C, Correctly specified mixed logit.

Table 12:

Square Root MSE of Choice Probabilities (Heteroskedastic Logit DGP).

γ 1 = 0.5
 γ 1 = 1
 γ 1 = 1.5
J N M C M C M C
2 100 1.09 1.55 1.73 1.81 2.12 2.18
2 500 0.99 1.28 1.73 1.65 2.20 2.12
2 1000 0.97 1.28 1.70 1.63 2.17 2.08
3 100 1.04 1.38 1.65 1.61 2.05 2.03
3 500 0.96 1.25 1.64 1.56 2.09 2.01
3 1000 0.94 1.25 1.62 1.54 2.07 1.98
5 100 0.74 1.07 1.26 1.24 1.65 1.64
5 500 0.76 1.02 1.35 1.27 1.76 1.71
5 1000 0.77 1.03 1.35 1.28 1.74 1.69

  1. M, Misspecified conditional logit; C, Correctly specified heteroskedastic logit.

References

Bhat, C. R. 1995. “A Heteroscedastic Extreme Value Model of Intercity Travel Mode Choice.” Transportation Research Part B: Methodological 29: 471–483.10.1016/0191-2615(95)00015-6Search in Google Scholar

Cramer, J. 2005. “Omitted Variables and Misspecified Disturbances in the Logit Model,” Tinbergen Institute Discussion Papers 05-084/4, Tinbergen Institute.Search in Google Scholar

Davidson, R., and J. G. MacKinnon. 1984. “Convenient Specification Tests for Logit and Probit Models.” Journal of Econometrics 25: 241–262.10.1016/0304-4076(84)90001-0Search in Google Scholar

Dubin, J. A., and L. Zeng. 1991. “The Heterogeneous Logit Model.” California Institute of Technology, Social Science Working Paper 759.Search in Google Scholar

Dumont, J., and J. Keller. 2015. RSGHB: Functions for Hierarchical Bayesian Estimation: A Flexible Approach. r package version 1.1.2.Search in Google Scholar

Eicker, F. 1967. “Limit Theorems for Regressions with Unequal and Dependent Errors.” Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability 1: 59–82.Search in Google Scholar

Freedman, D. A. 2006. “On the So-Called “Huber Sandwich Estimator” and “Robust Standard Errors”.” The American Statistician 60: 299–302.10.1017/CBO9780511815874.019Search in Google Scholar

Gartner, S. S., and G. M. Segura. 2000. “Race, Casualties, and Opinion in the Vietnam War.” The Journal of Politics 62: 115–146.10.1111/0022-3816.00006Search in Google Scholar

Gould, E. D., V. Lavy, and M. D. Paserman. 2003. “Immigrating to Opportunity: Estimating the Effect of School Quality Using a Natural Experiment on Ethiopians in Israel.” CEPR Discussion Papers, C.E.P.R. Discussion Papers 4052, C.E.P.R. Discussion Papers.10.2139/ssrn.433261Search in Google Scholar

Gourieroux, C., A. Monfort, and A. Trognon. 1984. “Pseudo Maximum Likelihood Methods: Theory.” Econometrica 52: 681–700.10.2307/1913471Search in Google Scholar

Haan, P. 2006. “Estimation of Multinomial Logit Models with Unobserved Heterogeneity using Maximum Simulated Likelihood.” Stata Journal 6: 229–245.10.1177/1536867X0600600205Search in Google Scholar

Hole, A. R. 2006. “Small-Sample Properties of Tests for Heteroscedasticity in the Conditional Logit Model.” Economics Bulletin 3: 1–14.Search in Google Scholar

Hole, A. R. 2007. “Fitting Mixed Logit Models by using Maximum Simulated Likelihood.” Stata Journal 7: 388–401.10.1177/1536867X0700700306Search in Google Scholar

Huber, P. J. 1967. “The Behavior of Maximum Likelihood Estimates Under Nonstandard Conditions.” Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability 1: 221–233.Search in Google Scholar

Jacobs, D., and J. T. Carmichael. 2002. “The Political Sociology of the Death Penalty: A Pooled Time-Series Analysis.” American Sociological Review 67: 109–131.10.2307/3088936Search in Google Scholar

Jeliazkov, I., and E. H. Lee. 2010. “MCMC Perspectives on Simulated Likelihood Estimation.” In Maximum Simulated Likelihood Methods and Applications (Advances in Econometrics, Vol. 26), edited by William Greene, R. Carter Hill, 3–39. Bingley, UK: Emerald Group Publishing Limited.10.1108/S0731-9053(2010)0000026005Search in Google Scholar

Keane, M. 1992. “A Note on Identification in the Multinomial Probit Model.” Journal of Business and Economic Statistics 10: 193–200.10.1080/07350015.1992.10509898Search in Google Scholar

Koop, G., and D. J. Poirier. 1993. “Bayesian Analysis of Logit Models using Natural Conjugate Priors.” Journal of Econometrics 56: 323–340.10.1016/0304-4076(93)90124-NSearch in Google Scholar

Kullback, S., and R. A. Leibler. 1951. “On Information and Sufficiency.” Annals of Mathematical Statistics 22: 49–86.10.1214/aoms/1177729694Search in Google Scholar

Lange, K. 1999. Numerical Analysis for Statisticians. Statistics and computing, Springer.Search in Google Scholar

Lassen, D. D. 2005. “The Effect of Information on Voter Turnout: Evidence from a Natural Experiment.” American Journal of Political Science 49: 103–118.10.1111/j.0092-5853.2005.00113.xSearch in Google Scholar

Luce, R. D. 1959. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley .Search in Google Scholar

MacKinnon, J. G., and H. White. 1985. “Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties.” Journal of Econometrics 29: 305–325.10.1016/0304-4076(85)90158-7Search in Google Scholar

McFadden, D., and K. Train. 2000. “Mixed MNL Models for Discrete Response.” Journal of Applied Econometrics 15: 447–470.10.1002/1099-1255(200009/10)15:5<447::AID-JAE570>3.0.CO;2-1Search in Google Scholar

Ramalho, E. A., and J. J. Ramalho. 2010. “Is Neglected Heterogeneity Really an Issue in Binary and Fractional Regression Models? A Simulation Exercise for Logit, Probit and Loglog Models.” Computational Statistics and Data Analysis 54: 987–1001.10.1016/j.csda.2009.10.012Search in Google Scholar

Ruud, P. A. 1983. “Sufficient Conditions for the Consistency of Maximum Likelihood Estimation Despite Misspecification of Distribution in Multinomial Discrete Choice Models.” Econometrica 51: 225–228.10.2307/1912257Search in Google Scholar

Ruud, P. A. 1996. “Simulation of the Multinomial Probit Model: An Analysis of Covariance Matrix Estimation.” Working paper.Search in Google Scholar

Train, K. E. 2009. Discrete Choice Methods with Simulation, no. 9780521766555 in Cambridge Books, Cambridge University Press.Search in Google Scholar

White, H. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica 48: 817–838.10.2307/1912934Search in Google Scholar

White, H. 1982. “Maximum Likelihood Estimation of Misspecified Models.” Econometrica 50: 1–25.10.2307/1912526Search in Google Scholar

White, H. 1983. “Corrigendum [Maximum Likelihood Estimation of Misspecified Models].” Econometrica 51: 513.10.2307/1912004Search in Google Scholar

Yatchew, A., and Z. Griliches. 1985. “Specification Error in Probit Models.” The Review of Economics and Statistics 67: 134–39.10.2307/1928444Search in Google Scholar

Supplemental Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/jem-2016-0002).

Published Online: 2018-02-01

©2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Research Articles
  2. Testing Spatial Dependence in Spatial Models with Endogenous Weights Matrices
  3. Uniformity and the Delta Method
  4. On the Size Distortion of a Test for Equality between the ATE and FE Estimands
  5. Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting
  6. Regression Discontinuity and Heteroskedasticity Robust Standard Errors: Evidence from a Fixed-Bandwidth Approximation
  7. Testing for a Functional Form of Mean Regression in a Fully Parametric Environment
  8. Dif-in-Dif Estimators of Multiplicative Treatment Effects
  9. Broken or Fixed Effects?
  10. Misspecified Discrete Choice Models and Huber-White Standard Errors
  11. Review Article
  12. Local Average and Quantile Treatment Effects Under Endogeneity: A Review
https://doi.org/10.1515/jem-2016-0002
Keywords for this article
discrete choice; Huber-White standard errors; misspecified models

Articles in the same Issue

  1. Research Articles
  2. Testing Spatial Dependence in Spatial Models with Endogenous Weights Matrices
  3. Uniformity and the Delta Method
  4. On the Size Distortion of a Test for Equality between the ATE and FE Estimands
  5. Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting
  6. Regression Discontinuity and Heteroskedasticity Robust Standard Errors: Evidence from a Fixed-Bandwidth Approximation
  7. Testing for a Functional Form of Mean Regression in a Fully Parametric Environment
  8. Dif-in-Dif Estimators of Multiplicative Treatment Effects
  9. Broken or Fixed Effects?
  10. Misspecified Discrete Choice Models and Huber-White Standard Errors
  11. Review Article
  12. Local Average and Quantile Treatment Effects Under Endogeneity: A Review
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 28.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jem-2016-0002/html
Scroll to top button