Estimator for the Asymptotic Variance Matrix of the CML Parameter Estimator

Estimator for the Asymptotic Variance Matrix of the CML Parameter Estimator

All notations in this supplementary note are drawn from the main paper entitled “On Accommodating Flexible Spatial Dependence Structures in Unordered Multinomial Choice Models: Formulation and Application to Teenagers’ Activity Participation”. The asymptotic variance matrix of the CML estimator is given by:

(S.1)

where

and

.

The “bread” matrix of Equation (S.1) can be estimated in a straightforward manner using the Hessian of the negative of , evaluated at the CML estimate . This is because the information identity remains valid for each pairwise term forming the composite marginal likelihood. Then, can be estimated as:

(S.2)

and

On the other hand, the “vegetable” matrix is not straightforward to estimate due to the underlying spatial dependence in observational units. However, the non-decaying correlation framework (since the spatial dependence fades with distance) allows the use of Heagerty and Lumley’s (2000) windows resampling procedure to estimate . In particular, one can construct suitable overlapping subgroups of the original data that may be viewed as independent replicated observations. Then, may be estimated empirically as the weighted average of the variance of composite score evaluations (computed at ) across the subgroups within each alternative (the weights correspond to the inverse of the size of each subgroup choosing alternative ). In the current spatial context, we can consider all the observational units k in the data choosing the same alternative i as observation q, and within a distance of observation q, as a subgroup or cluster (note that the dependence is weak beyond a distance , and thus each subgroup as just defined would be only weakly dependent on other subgroups). Then, we propose the following as an estimator of the matrix :

(S.3)

where

and

References

Heagerty, P.J., and T. Lumley (2000) Window Subsampling of Estimating Functions with Application to Regression Models. Journal of the American Statistical Association, 95(449), 197-211.