Recently, we proposed a new paradigm for choice set generation in the context of route choice model estimation. As detailed in Frejinger and Bierlaire (2007), we assume that choice sets contain all paths connecting each origin-destination pair. Although this is behaviorally questionable, this assumption is made in order to avoid bias in the econometric model. These sets are in general impossible to generate explicitly. Therefore, we propose an importance sampling approach to generate subsets of paths suitable for model estimation. Using only a subset of alternatives requires the path utilities to be corrected according to the sampling protocol in order to obtain unbiased parameter estimates. In Frejinger and Bierlaire (2007) we derive such a sampling correction for the Multinomial Logit (MNL) model.

The Path Size Logit model is a MNL model where a Path Size (PS) attribute is included in the deterministic utilities. The PS attribute should capture the correlation among routes. It is generally computed based on sampled paths only but we argue that it should capture the correlation among all routes (universal choice set). This becomes problematic since the universal choice set is unknown in practice. In this paper we present a generalization of the PS attribute called Expanded PS (EPS). It is computed based on sampled paths only but involves an expansion factor that corrects for the sampling. 

We have estimated close to 300 models based on synthetic data in order perform a sensitivity analysis of the estimation results with respect to the sampling correction, the PS and EPS attributes, the parameters of the path generation algorithm as well as the parameter values in the postulated model. In this context, synthetic data is important because the universal choice set as well as the true parameter values are known and we can hence evaluate the bias in the parameter estimates. 

Unbiased parameter estimates are only obtained for models including a sampling correction. Moreover, the results clearly show that EPS is superior to the original PS attribute. Few draws for the choice set generation is needed if the PS attribute is computed based on the universal choice set. More draws is required in order to obtain unbiased parameter estimates with the EPS attribute but we show that the estimates converge rapidly to the true values when the number of draws increase. Parameter estimates in models with the original PS attribute (based on sampled paths only) are biased even for very high number of draws.