A practically tractable expression of the covariances of the Cross-Nested Logit model

Abstract

This paper proposes a practically tractable mathematical procedure for the calculation of the covariances underlying whatever given Cross-Nested Logit (CNL) model, based on the variance of a one-dimensional random variable, whose cumulative distribution function and density probability function are given in closed form. This allows expressing the CNL covariances as a function of just a one-dimensional integral, which can be evaluated easily and effectively by means of standard numerical techniques, implementable also in basic computer spreadsheets. Firstly, a formal theoretical proof of the procedure is illustrated. Then, a comparison with the calculations performed by Marzano and Papola [Marzano, V., Papola, ., 2008. On the covariance structure of the Cross-Nested Logit model. Transportation Research B 42(2), 83–98] is proposed, and details about the practical implementation of the procedure are discussed. Finally, estimation of the CNL model in contexts with prior expectations on covariances/correlations is addressed practically, thanks to the simplification achieved in the calculation of the CNL covariances.