structure in that the copula parameterizes the full range of dependence from perfect negative dependence to zero dependence to perfect positive dependence. Also, the Spearman’s values tracks the correlation parameters closely for the Gaussian copula. Now partition the random variable vector into two sub-vectors (of size Y Z K-variate and , so that Let the elements of the vector each have a pre-specified but non-normal W Z E×1) continuous parametric distribution so that (note that the cumulative distribution functions can vary across the elements of Let each element of the vector be normally distributed with Z). W mean and standard deviation , so that . Then, defining , we may write the multivariate distribution in Equation (6) as: (7) The important point to note is that we now have the multivariate distribution of translated to the multivariate normal distribution of where and Next, partition the correlation matrix as Γ follows: . Immediately then, the conditional distribution properties of the multivariate normal distribution, and defining , and a diagonal matrix with the diagonal element being , we L×L lth are able to write the conditional distribution of the vector conditional on as follows: W Z (8) This conditional distribution for while accommodating the dependence between the W Z, two random vectors, plays a central role in the estimation of the proposed Gaussian copula model, as discussed in the next section. 3. THE MODEL Consider a repeated choice situation (or a panel situation), with the index for the individual, , q index for the alternative , and index for the choice occasion. For ease in presentation, we will i t use the same number of choice occasions for every individual. Extension to the case of varying T number of choice occasions per individual is straightforward. Also note that the cross-sectional case corresponds to the case of T=1. Consider the random-coefficients formulation in which the utility that an individual q associates at time period with alternative is written as: t i (t=1, 2, 3,…,T) (9)
structure in that the copula parameterizes the full range of dependence from perfect negative dependence to zero dependence to perfect positive dependence. Also, the Spearman’s values tracks the correlation parameters closely for the Gaussian copula. Now partition the K-variate random variable vector Y into two sub-vectors Z (of size E×1) and W , so that Let the elements of the Z vector each have a pre-specified but non-normal continuous parametric distribution so that (note that the cumulative distribution functions can vary across the elements of Z). Let each element of the W vector be normally distributed with mean and standard deviation , so that . Then, defining , we may write the multivariate distribution in Equation (6) as: (7) The important point to note is that we now have the multivariate distribution of translated to the multivariate normal distribution of where and Next, partition the correlation matrix Γ as follows: . Immediately then, the conditional distribution properties of the multivariate normal distribution, and defining , and a diagonal L×L matrix with the lth diagonal element being , we are able to write the conditional distribution of the vector W conditional on Z as follows: (8) This conditional distribution for W Z, while accommodating the dependence between the two random vectors, plays a central role in the estimation of the proposed Gaussian copula model, as discussed in the next section. 3. THE MODEL Consider a repeated choice situation (or a panel situation), with the index q for the individual, , index i for the alternative , and index t for the choice occasion. For ease in presentation, we will use the same number of choice occasions T for every individual. Extension to the case of varying number of choice occasions per individual is straightforward. Also note that the cross-sectional case corresponds to the case of T=1. Consider the random-coefficients formulation in which the utility that an individual q associates at time period t with alternative i is written as: (t=1, 2, 3,…,T) (9)
non-normal_coefficients.docx