The error terms of similar alternatives are correlated. The stochastic part of the utility function is additively composed of an i. Gumbel-distributed term corresponding to the standard logit approach and a stochastic scaling parameter to account for heterogeneity. Not every single scaling parameter is estimated, but the parameters of their distribution instead. The nested logit approach is predominantly used in the field of transportation research and logistics Train, ; Bhat, ; Knapp et al, , but can also be appropriate for marketing issues Kannan and Wright, ; Chin- tagunta, ; Chintagunta and Vilcassim, ; Guadagni and Little, ; Chib et al, In the field of marketing the nested logit model is mainly applied in brand choice modelling Kamakura et al, ; Ailawadi and Neslin, ; Guadagni and Little, ; Sun et al, ; Chib et al, , where brands are nested, for example, regarding manu- facturer Anderson and de Palma, ; in a purchase incidence decision Chintagunta, ; Chintagunta and Vilcassim, ; or regarding brand type Baltas et al, One important point to make is that the nested logit model is a combina- tion of standard logit models.
Marginal and conditional choice decisions are combined by a nesting structure Hensher et al, The only goal of this process is to accommodate the violation of the IIA-assumption. The nested logit model differs from the standard logit model in that the er- ror components of the choice alternatives do not necessarily need to have the same distribution.
Thus the nested logit model accounts for the fact that each alternative may have specific information in its unobservable utility component, which plays a role in the decision process. Subsets of alterna- tives may have similar information content, such that correlations between pairs of alternatives may exist Hensher et al, The classification of alternatives regarding their similarities into nests and the thus resulting tree structure does not have anything in common with a stochastic valuation of alternatives within the scope of a decision tree.
Nested logit models do not define the process of decision-finding, but account for differences in variances in the unobservable utility components Hensher et al, In this case, the choice probability Pim of an alternative i within nest m results from the product of the marginal choice probability Pm for nest m Level 2 and the conditional choice probability Pi m for alternative i within nest m Level 1. Both the marginal and the conditional probability are standard logit models.
The inclusive value IVm as the expected utility of nest m connects the two decision levels. Unconsidered utility components can variously impact the random components. This leads to different variances, which are explicitly accounted for by the introduction of these scale parameters.
Thus the variances on the upper level cannot be smaller than those on the lower level. The difference between these nested logit model specifications lies in the ex- plicit scaling of the deterministic utility component in the UMNL form. Table 1 compares the two specifications Hunt, ; Koppelman and Wen, a.
The set of all elemental alternatives within nest m is called Cm. Due to identification problems, one of the scale parameters in the util- ity maximization nested logit UMNL specification needs to be normal- ized to 1 Daly, ; Hunt, Table 2 summarizes the results. Without imposing restrictions, only the RU2 UMNL specification satisfies the demand of consistency with utility theory. Because of the generally not theory-consistent results on the level of the marginal choice probabilities in the non-normalized nested logit NNNL and the Level 1 normalized utility maximization nested logit RU1 UMNL specification, only the Level 2 normalized utility maximization nested logit RU2 UMNL specification satisfies condition 6.
When the sample size is large, the estimated parameters should be very close to the true parameters Cameron and Trivedi, When simulating a utility function, mainly the stochastic part needs to be taken into account.
Whereas the observable exoge- nous variables are relatively straightforward to simulate by imposing specific assumptions on the distribution and correlation patterns, the simulation of the unobservable influences requires falling back on the assumptions 2 , 15 , 19 , and In case only NNNL software is available, there are several particularities in model estimation to take into consideration.
The crucial point is whether there are only alternative-specific coefficients in the model, or also at least one generic coefficient. Generic coefficients are constant for all alternatives. A variation on the utility contribution could be reached via alternative-specific values of the corresponding variables.
Moreover, Hunt points to the peculiarities of partially degenerate model structures. Nests with only one elemental alternative are called de- generate nests. For further and detailed information regarding the estimation procedure when degenerate nests enter the model, the reader is referred to the literature Heiss, ; Hensher et al, ; Hunt, Only then is a correct interpretation possible. It must be taken into account which alternative belongs to which nest.
Table 3 refers to this with an example of the conditional deterministic utility component. And this is a violation of the definition of generic coefficients. By imposing restrictions it can be guaranteed that, even when using NNNL software, parameters consistent with random utility can be estimated model C in Figure 2.
It has to be assured that the coefficients in each nest are scaled equally. The IV-parameters are thus to be made equal for all nests. But, of course, each restriction on the parameter estimates means a loss of information in the data. In this simulation study the coffee market is simulated in a very simplistic manner. The simulated market consists of only two brands A and B, where both offer variants containing caffeine and decaffeinated. Figure 3 shows the nest structure of this discrete choice situation.
In this study, the deterministic marginal utility component Vm is neglected. It is often hard to find any vari- ables that are nest- rather than alternative-specific. But even if a nest-specific variable does exist, specifying this variable for the nest or for all alternatives within this nest does not make a difference Heiss, Alternative-specific constants ASC are neglected in this simulation study, but must be integrated in the model when estimating with real data.
As was described in detail in Section 3. Table 4 gives an overview of the simulated data sets for models A and B. The data sets 1. When generating the simulated data, a higher correlation in nest B was assumed. The Level 2 normalized utility maximization nested logit UMNL RU2 model is consistent with random utility theory see Table 2 even without imposing restrictions and should be able to reproduce the input-parameters with a high reliability.
This can only partially be confirmed. At this point, one can not speak of a satisfying and reliable reproduction of the data. The reasons for this are unknown for the present but should be part of fur- ther investigations. In the utility maximization nested logit UMNL model, the IV-parameters only capture the dis- similarity of the alternatives within the nest. The IV-parameters in the non-normalized nested logit NNNL model capture another effect: the relative importance of the variables with generic coeffi- cients for the alternatives within the corresponding nest see Heiss , p.
Only if it is a priori assumed that the IV-parameters are the same in all nests, the scaling problem of the NNNL model can be avoided. The presence of generic coefficients then does not bias the estimates of the NNNL model, because the coefficients are equally scaled in each nest. Table 9 gives an overview over the simu- lated data sets for the models C and D. The data sets 2. The coefficients estimated in the NNNL models do not have any meaning before their re-scaling, i.
In the data sets 2. The reasons for a suboptimal parameter reproduction in the data sets 2. Effectively then, we show it is possible to obtain maximum likelihood estimates of the logit model without requiring the use of a package specifically designed for this purpose and without compromising on estimator efficiency.
The model can be viewed as a process in which individuals with char- acteristics defined by the x matrix, make a choice between two possibili- ties A and B. The characteristics, in conjunction with the p vector. A question important for practising econometricians is how to esti- mate logistic models of this type where access to a ML routine specifi- cally designed for this purpose is not available. The standard response to this problem is to use non-linear least squares in the hope that the estimates so obtained will approximate the ML estimates.
McFadden , p. It turns out to be extremely simple to obtain exact ML estimates of the coefficients of the above model and all of the associate statistics using non-linear least squares. O, is a stochastic error normally distributed.
It is the log of the likelihood function defined in eq. We thus propose the estimation of eq. The estimation of goodness of fit measures in this framework poses no problems. The only estimates for which there exists a computational difference between least squares and ML are the standard errors.
However, this arises simply because of the approximation to the elements of the information matrix used in the least squares optimisation. A simple reformula- tion of the traditionally utilised non-linear specification yields the exact ML estimates. The advantages of this method lie in its simplicity and accessibility. As the computing costs are higher than with ML it is likely to be of greatest use to researchers whose data base is not excessively large.
We have found that some experimentation with a reasonably large data base has resulted in quite moderate costs. Details can be obtained from the authors. Breshw, I. This can be rewritten [using eqs.
It follows immediately from A. Domencich, T. Greene, W. Heckman, J. Tobin, J. Econometrica 26, Related Papers TSP 5. Baltagi econometric analysis of panel data himmy By Manali Mnl.
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