^{1}

^{2}

In this paper, we explore the properties of a positive-part Stein-like estimator which is a stochastically weighted convex combination of a fully correlated parameter model estimator and uncorrelated parameter model estimator in the Random Parameters Logit (RPL) model. The results of our Monte Carlo experiments show that the positive-part Stein-like estimator provides smaller MSE than the pretest estimator in the fully correlated RPL model. Both of them outperform the fully correlated RPL model estimator and provide more accurate information on the share of population putting a positive or negative value on the alternative attributes than the fully correlated RPL model estimates. The Monte Carlo mean estimates of direct elasticity with pretest and positive-part Stein-like estimators are closer to the true value and have smaller standard errors than those with fully correlated RPL model estimator.

The random parameters logit (RPL) model is a generalization of the conditional logit model for multinomial choices. The conditional logit model is derived from an assumption that the errors in the underlying random utility functions for each choice alternative are statistically independent and identically distributed (iid) extreme value type I. This leads to the property known as the Independence of Irrelevant Alternatives (IIA): The ratio of the probability of two alternatives remains constant no matter how many choices there are. This is widely regarded to be a very restrictive assumption.

The key feature of the RPL model is that response parameters can vary randomly, following a chosen distribution, across the population from which samples are drawn. The random coefficients capture individual heterogeneity and the model does not suffer from the independence of irrelevant alternatives assumption. The random coefficients can be correlated in the RPL model as generally expected in reality, because the unobservable preference of each individual is used to evaluate the attributes of all alternatives in each choice situation. Estimation is by maximum simulated likelihood (MSL), which is described by [

In this paper we explore a problem that can exist in any correlated random parameters model. Let

Most applied researchers will test the significance of the covariance parameters before deciding to rely on the fully correlated random parameter model instead the model in which the parameters are random but uncorrelated, so that

The RPL model is described in [

bability that individual n chooses alternative i is of the usual logistic form,

that ^{1} with mean vector

For estimation purposes we use Cholesky’s decomposition and write

where

where

Stein-rule estimators follow the work of [

Following is the Stein-rule estimator which dominates the maximum likelihood estimator (MLE) in linear regression under weighted quadratic loss with weight matrix W,

where

strictions,

Sufficient conditions for minimaxity, meaning that the estimator minimizes the maximum risk over the entire parameter space, are

where

where u is the test statistic for the hypothesis

The positive-part Stein-like estimator

where

pothesis that the coefficient covariance matrix is diagonal, or equivalently that the Cholesky elements in A below the diagonal are zero. The scalar a controls the amount of shrinkage towards the UCRPLM estimates. The shrinkage estimator

With test statistic u, the pretest estimator

where

In our experiments the number of choice alternatives is

Our simulation and RPL model estimation were carried out in NLOGIT 5.0. Based on our Monte Carlo experiment results, [

To study how the pretest and shrinkage estimators reduce the estimation risk of the FCRPLM estimators, we calculate the MSEs of the estimated parameters mean, variance, covariance with the pretest, shrinkage and FCRPLM estimators respectively. First, we compare the MSE of the fully correlated estimators and those of

UCRPLM estimators, where MSE is the Monte Carlo average of the squared error loss

0.0 | 0.454 | 0.217 | 0.000 |

0.2 | 0.450 | 0.251 | 0.010 |

0.4 | 0.385 | 0.221 | 0.025 |

0.6 | 0.327 | 0.141 | 0.044 |

0.8 | 0.329 | 0.231 | 0.091 |

In

The covariance elements reveal important information about the joint effect of alternative attributes on people' decisions. If two random coefficients are highly positively correlated with each other, it means people are attracted and motivated by both of the related attributes. In our Monte Carlo experiments, the shrinkage estimators with higher shrinkage constant a outperform estimators with less shrinkage and most of the pretest estimators.

Since one of the advantages of RPL model is providing the information on the share of population that places a positive or negative value on the alternative attributes, we also calculate the joint probability of the first two estimated parameters are less than zero.

To analyze the sensitivity of the RPL model in response to a change in the level of alternative attribute, we calculate the mean estimates of direct elasticity with the true parameters

Pretest Estimator | Shrinkage Estimator | ||||
---|---|---|---|---|---|

LR_25% | LR_5% | LR_1% | |||

0.0 | 0.80 | 0.55 | 0.47 | 0.66 | 0.45 |

0.2 | 0.70 | 0.51 | 0.45 | 0.60 | 0.42 |

0.4 | 0.74 | 0.48 | 0.43 | 0.58 | 0.44 |

0.6 | 0.80 | 0.52 | 0.41 | 0.58 | 0.41 |

0.8 | 0.91 | 0.73 | 0.52 | 0.60 | 0.43 |

Pretest Estimator | Shrinkage Estimator | ||||

LR_25% | LR_5% | LR_1% | |||

0.0 | 0.76 | 0.43 | 0.26 | 0.52 | 0.25 |

0.2 | 0.60 | 0.34 | 0.25 | 0.45 | 0.23 |

0.4 | 0.77 | 0.45 | 0.35 | 0.52 | 0.30 |

0.6 | 0.81 | 0.49 | 0.26 | 0.52 | 0.24 |

0.8 | 0.81 | 0.63 | 0.40 | 0.59 | 0.37 |

Pretest Estimator | Shrinkage Estimator | ||||

LR_25% | LR_5% | LR_1% | |||

0.0 | 0.71 | 0.32 | 0.17 | 0.36 | 0.10 |

0.2 | 0.56 | 0.19 | 0.08 | 0.27 | 0.06 |

0.4 | 0.75 | 0.39 | 0.30 | 0.39 | 0.15 |

0.6 | 0.83 | 0.49 | 0.23 | 0.44 | 0.18 |

0.8 | 0.87 | 0.68 | 0.37 | 0.45 | 0.22 |

True Prob. | UCRPLM | FCRPLM | Pretest | Shrinkage | Shrinkage | |
---|---|---|---|---|---|---|

0.0 | 0.025 | 0.047 | 0.120 | 0.014 | 0.027 | 0.015 |

[0.003] | [0.049] | [0.000] | [0.008] | [0.001] | ||

{0.022} | {0.095} | {−0.011} | {0.002} | {−0.010} | ||

0.2 | 0.100 | 0.060 | 0.110 | 0.021 | 0.037 | 0.024 |

[0.006] | [0.077] | [0.007] | [0.016] | [0.009] | ||

{−0.040} | {0.010} | {−0.079} | {−0.063} | {−0.076} | ||

0.4 | 0.213 | 0.071 | 0.137 | 0.034 | 0.062 | 0.038 |

[0.026] | [0.094] | [0.033] | [0.042] | [0.034] | ||

{−0.142} | {−0.076} | {−0.179} | {−0.151} | {−0.175} | ||

0.6 | 0.334 | 0.084 | 0.210 | 0.052 | 0.123 | 0.066 |

[0.069] | [0.133] | [0.082] | [0.087] | [0.080] | ||

{−0.250} | {−0.124} | {−0.282} | {−0.211} | {−0.268} | ||

0.8 | 0.406 | 0.115 | 0.292 | 0.090 | 0.259 | 0.171 |

[0.094] | [0.153] | [0.117] | [0.113] | [0.104] | ||

{−0.291} | {−0.114} | {−0.316} | {−0.147} | {−0.235} |

Note: [ ] provides the MSE results, {} provides bias results.

True RPL Model Parameters | ||||
---|---|---|---|---|

0.0 | 2.009 | 1.960 | 2.053 | 2.042 |

0.2 | 2.014 | 1.957 | 2.052 | 2.049 |

0.4 | 2.020 | 1.954 | 2.051 | 2.057 |

0.6 | 2.025 | 1.951 | 2.051 | 2.065 |

0.8 | 2.031 | 1.947 | 2.051 | 2.075 |

Comparing the results in

According to our Monte Carlo experiment results, the UCRPLM estimators have smaller estimation risk than the

FCRPLM Estimator | Pretest Estimator (with | |||||||
---|---|---|---|---|---|---|---|---|

0.0 | 2.058 | 1.995 | 2.108 | 2.108 | 1.779 | 1.720 | 1.805 | 1.799 |

(0.026) | (0.026) | (0.028) | (0.028) | (0.019) | (0.019) | (0.020) | (0.020) | |

0.2 | 2.169 | 2.097 | 2.216 | 2.219 | 1.842 | 1.785 | 1.872 | 1.864 |

(0.027) | (0.026) | (0.028) | (0.028) | (0.019) | (0.019) | (0.020) | (0.021) | |

0.4 | 2.297 | 2.219 | 2.347 | 2.356 | 1.885 | 1.828 | 1.915 | 1.907 |

(0.031) | (0.030) | (0.032) | (0.033) | (0.021) | (0.021) | (0.022) | (0.022) | |

0.6 | 2.408 | 2.324 | 2.463 | 2.486 | 1.873 | 1.819 | 1.904 | 1.896 |

(0.031) | (0.030) | (0.032) | (0.033) | (0.021) | (0.021) | (0.022) | (0.022) | |

0.8 | 2.568 | 2.475 | 2.629 | 2.673 | 1.879 | 1.828 | 1.914 | 1.911 |

(0.031) | (0.030) | (0.032) | (0.033) | (0.026) | (0.025) | (0.027) | (0.028) | |

Shrinkage Estimator (with | Shrinkage Estimator (with | |||||||

0.0 | 1.882 | 1.823 | 1.918 | 1.915 | 1.801 | 1.742 | 1.829 | 1.824 |

(0.022) | (0.021) | (0.023) | (0.023) | (0.019) | (0.019) | (0.021) | (0.021) | |

0.2 | 1.956 | 1.894 | 1.992 | 1.989 | 1.866 | 1.807 | 1.896 | 1.890 |

(0.021) | (0.021) | (0.023) | (0.023) | (0.020) | (0.019) | (0.021) | (0.021) | |

0.4 | 2.032 | 1.969 | 2.070 | 2.068 | 1.914 | 1.856 | 1.946 | 1.939 |

(0.025) | (0.024) | (0.026) | (0.026) | (0.021) | (0.021) | (0.022) | (0.022) | |

0.6 | 2.082 | 2.018 | 2.122 | 2.127 | 1.922 | 1.866 | 1.956 | 1.950 |

(0.025) | (0.025) | (0.027) | (0.027) | (0.021) | (0.021) | (0.022) | (0.022) | |

0.8 | 2.206 | 2.137 | 2.254 | 2.273 | 1.961 | 1.906 | 1.999 | 2.001 |

(0.027) | (0.026) | (0.028) | (0.029) | (0.024) | (0.023) | (0.025) | (0.025) |

Note: ( ) provides the standard error results.

FCRPLM estimators. The pretest and positive-part Stein-like estimators both perform better than the FCRPLM estimators. The positive-part Stein-like estimators with higher shrinkage constant a outperform those with a smaller one and the pretest estimators. Shrinkage estimation reduces the risk of the FCRPLM estimators by shrinking the FCRPLM estimates towards the UCRPLM estimates. Providing the information on the share of population putting a negative or positive value on the alternative attributes is one of the advantages of the RPL model. When the random coefficients are correlated to each other, the FCRPLM estimator of this quantity has a smaller bias and slightly larger MSE than the UCRPLM estimator. Based on our Monte Carlo experiments, the pretest and shrinkage estimates can reduce the MSEs of the estimated results of share of the population putting a positive or negative value on alternative attributes as well. The Monte Carlo mean estimates of direct elasticity based on the pretest and shrinkage estimators with a larger shrinkage constant are closer to the true value with smaller standard errors than those based on the FCRPLM estimators.

Tong Zeng,R. Carter Hill, (2016) Shrinkage Estimation in the Random Parameters Logit Model. Open Journal of Statistics,06,667-674. doi: 10.4236/ojs.2016.64056