On Discrete Location Choice Models

When estimating location choices, Poisson regressions and conditional logit models yield identical coefficient estimates (Guimarães et al., 2003). These econometric models involve polar assumptions as regards the similarity of the different locations. Schmidheiny and Brülhart (2011) reconcile these polar cases by introducing a fixed outside option transforming the conditional logit into a nested logit framework. This gives rise to a dissimilarity parameter (λ ∈ [0; 1]) equalling 1 in Poisson regressions (with completely dissimilar locations) and 0 in conditional logit models (with completely similar locations). The dissimilarity parameter is not identified in Schmidheiny and Brülhart (2011). We show that a choice consistent normalisation identifies λ and that, with panel data, its estimation is facilitated by adopting a Poisson regression approach.


Introduction
This paper extends recent firm location choice models of Schmidheiny and Brülhart (2011)  have been chosen instead, that is A conditional logit model employs (1) as the dependent variable. Thereby, choice-specific variables x sh (reported in logarithms) where δ s absorbs source-specific factors. Furthermore, β are coefficients to be estimated. The stochastic component ϵ i,sh follows a Gumbel distribution with location and scale parameter normalised to, respectively, 0 and 1. The probability that a firm of s chooses h equals The log-likelihood function equals and permits us to estimate β. Guimarães et al. (2003) show that a count regression onto n sh (the number of location choices) provides an alternative to estimate β. To see this, multiply (3) with the denominator, which yields the (panel) Poisson regression where α s = ln(δ s ) and E[n pc sh ] = exp(x ′ sh β). Assuming that  n pc sh is Poisson distributed with the conditional mean function exp(δ s + x ′ sh β) of (5) yields a log-likelihood contribution of s given by Equating the first derivative with respect to α s with 0, and solving for α s yields the maximum likelihood estimator of Hence, α s absorbs the discrepancy between the observed number of location choices n s and the number E[n s ] expected from a Poisson distribution. Thereby, 0 < α s < 1 implies that the observed number of location choices is ''underreported''. Substituting (7) into (6) and summing over S yields the log-likelihood function of the fixed effects Poisson regression which looks like a multinomial logit model (Hausman et al., 1984, p. 919). 1 Specifically, since (8) differs from (4) only by a constant, the corresponding estimates for β are identical! SB observe that the elasticity of the Poisson regression, given by differs from the conditional logit model, given by whereby β k denotes the coefficient pertaining to x sh,k . This reflects that Poisson regressions deem the locations to be completely dissimilar. Hence, a change of x sh,k affects the number of location choices with h, but not with h ′ . SB refer to this as a ''positive sum world''. Conversely, the conditional logit model is a ''zero sum world'' where the locations represent completely similar options.
Hence, when more firms choose h, this triggers an equivalent reduction elsewhere. SB show that the introduction of an outside option transforms the conditional logit into a nested logit model covering the continuum between the zero and positive sum world. The outside option Since the outside option contains only one alternative, this nested logit model, depicted in Fig. 1, involves the partial degeneracy discussed in Hunt (2000). The probability P sh depends now on the probability P øs of not choosing the outside option and the (conditional) probability P sh|ø to locate in h > 0, that is The inclusive value parameter (λ ø s /ς ø s ) ∈ [0, 1] measures the dissimilarity between the locations h > 0. Specifically, where ρ ø s ∈ [0, 1] is the correlation between the stochastic profit components ϵ i,sh|ø of investing in different locations. Consider the outside option o offering only the basic ''choice'' of h = 0. Hunt (2000) observes that the distinction between unconditional and conditional probabilities is here obsolete, as P s0|o = 1 and P s0 = P os × P s0|o . The probability of choosing h = 0 equals The coefficients β can be estimated by maximum likelihood from (13) and (15). However, empirically, only the correlation ρ ø s , but not the scale parameters λ ø s and ς ø s , can be estimated from the data (Hunt, 2000). This over-identification problem necessitates some normalisation. SB (p. 217) set ς ø s = 1, ς o s = 1, and λ o s = 1 wherefore (13) and (15) become The nested logit log-likelihood function coincides with (4). Again, the estimates for β are identical and the normalisation of the scale parameters does not affect β.
The nested logit elasticity equals and maps mathematically into the conditional logit model when λ ø s = 0 and into the Poisson regression when λ ø s = 1 and P s0 = 1. Hence, though the conditional logit model and the Poisson regression may have different econometric underpinnings, they share a functional form that connects them with the same, suitably parameterised nested logit model.

Choice consistent normalisation
The SB approach has two shortcomings. Firstly, the normalisation of λ and ς leads to different nested logit models with different elasticities (Hensher and Greene, 2002). Koppelman and Wen (1998) suggest that any normalisation should at least be invariant to adding a constant ∆ to all profits of (2) and (11) since this leaves the ranking of the final options unchanged. Appendix A shows that (16) and (17) do not fulfil this property. Secondly, δ s and λ ø s cannot be separately identified (SB, 2011, p. 217) since they appear in the same first order condition where n os and n øs denote, respectively, the number of times the outside option h = 0 or location h > 0 has been chosen.
Adopting the following choice-consistent normalisation avoids these caveats. Proof. Appendix A.
With Proposition 1, (13) and (15) become Proposition 1 implies that exp(δ s ς o s ) λ o s = exp(0) λ s = 1 and, hence, normalises to 1 the contribution of the outside option. This is unproblematic since the parameter λ s pertaining to E[N ø ] λ s already weights the importance between the outside option h = 0 and the locations h > 0.
When ς ø s = 1, (14) implies that λ s = √ 1 − ρ s , and the correlation ρ s reflects the degree of dissimilarity between the locations.

Poisson regression with λ s
Estimating λ s from (19) involves n os . However, how many times the outside option has been chosen is often unobservable. Counting location choices provides a possible remedy for this.
Similar to Section 1, transforming the nested logit model into a Poisson regression requires the multiplication of P sh of (20) with This resembles (5) .
Recall that α s can be estimated from (7) and reflects the degree of underreporting. The Poisson regression with λ s = 1 requires that α s = 1. This means that there is no underreporting, which is maybe intuitive since the outside option does not affect the choice of a location that differs completely from the alternatives. The conditional logit model with λ s = 0 requires α s = 1/E [N ø ]. This implies a degree of underreporting increasing with the number E[N ø ] of location choices. Typically, α s is close to 0, even when the sample contains a relatively modest number of location choices N ø .
The identification of (23) requires that E [N ø ] > 0 (we do actually have location choices) and that we have panel data permitting us to calculate the group effects (7). Appendix B shows that the elasticity of E[ n pcu sh ] with respect to This is again consistent with SB. Specifically, λ s = 1 returns the Poisson elasticity of (9) and λ s = 0 the conditional logit elasticity of (10). Also, evaluating (24) yields η pcu sh,k = λ s β k +(1−λ s )(1−P sh|ø )β k , meaning that the elasticity of the Poisson regression with a fixed outside option is a linear average between (9) and (10).

Conclusion
This paper extends SB by a choice consistent framework, which permits us to identify the dissimilarity parameter λ. Furthermore, using a Poisson regression facilitates the estimation of the dissimilarity parameter.

Acknowledgement
Insightful comments provided by the referee helped us to clarify several aspects of this paper.