A note on the nested Logit model

Transcription

1 Erik Biørn Version of September ECON ECONOMETRICS: MICROECONOMETRICS AND DISCRETE CHOICE AUTUMN 2008 A note on the nested Logit model In this note we present the basic idea of the nested multiple Logit model and how it relates to and generalizes the standard multiple Logit model. At the start we restate some important implications of the latter model. Multinomial Logit: Conditional probabilities Essentially the standard multinomial Logit model can be viewed as a modelling of relative response probabilities by functions of the form P ji G(x i β j ) = P ji +P ki G(x i β j )+G(x i β k ) P ji = G(x iβ j ) P ki G(x i β k ) j k =1... J k j where G( ) is a suitable positive function with one argument. Combining this with the condition J j=1 P ji = 1 it follows that (1) P 1i = F 1 (x i β) =. P Ji = F J (x i β) = G(x i β 1 ) J k=1 G(x iβ k ) G(x i β J ) J k=1 G(x iβ k ). [1]: In the Logit model we let the common function G( ) be a simple exponential function: G(x i β j )=e x iβ j (j = 1... J). [2]: A property of this function on which much of the algebra for Logit models relies is: G(a 1 )G(a 2 ) G(a P ) G(a 1 + a a P ). We further let β 1 = 0 = G(x i β 1 ) = G(0) = 1 i. This indicates that alternative 1 is a kind of base alternative. We then obtain the system of Logit probabilities (2) P 1i = P (y 1i = 1) = J k=2 ex iβ k P ji = P (y ji = 1) = e x iβ j 1 + J k=2 ex iβ k 1 j = 2... J i = 1... n.

2 An important property of the standard Logit model is that all conditional probabilities also have Logit-form i.e. the same form as (2). The following examples illustrate this: The probability of choosing (i) alternative j given that either j or k is chosen and (ii) alternative j given that one of (j k l) is chosen are respectively (i) P (j jk)i = P (y ji =1 y ji =1 y ki =1) = P ji P ji +P ki = e x iβ j e x iβ j +e x iβ k ex i(β j β k) 1+e x i(β j β k ) (ii) P (j jkl)i = P (y ji =1 y ji =1 y ki =1 y li =1) = P ji P ji +P ki +P li = e x iβ j e x iβ j +e x iβ k +e x i β l e x i(β j β l) 1+e x i(β j β l ) +e x i(β k β l ) j l k = 2... J; j k l letting symbolize or. We see that the level of the coefficients (β j β k ) in example (i) (β j β k β l ) in example (ii) will be indeterminate even if the conditional response probabilities are known. These conditional probabilities are uniquely characterized by coefficient differences: (β k β j ) in example (i) (β k β j β l β j ) in example (ii). Hence it follows from (2) and examples (i) and (ii) that [ ] [ ] [ ] Pji P(j jkl)i P(j jk)i (3) ln = ln = ln = x i (β j β k ) j k =2... J. P ki P (k jkl)i P (k jk)i Such ratios are as in the binomial model denoted as log-odds ratios. The remarkable thing is that the three log-odds-ratios one marginal and two conditional are equal and described by only the variable vector and the coefficient vector for alternatives j and k. Thus it is immaterial whether the choice set we condition on includes all J alternatives only (j k l) or only (j k). The general structure of the probabilities The standard multinomial Logit model has the limitation that all alternatives are considered as being on the same level co-ordinated. In the model class now to be considered we assume that there is two levels in the hierarchy M groups (often denoted as nests) indexed by m (m = 1... M) at the upper level and K m in group m. Let alternative k = 1... K m in group m be indexed by (mk). Examples will be given in the lectures. Two types of variable vectors occur one being specific for the group and common to all alternatives in the group for group m denoted as z m and one being specific for the alternative within a group for alternative (mk) denoted as x mk Further let x m be the variable vector which contains all alternative specific variables in group m that 2

3 is x m = (x m1 x m2... x mkm ). We assume that no coefficients are alternative specific some are group specific and some are common to all groups: For group m the vector β m is attached to the alternative specific vector x mk while the vector α is attached to the group specific vectors z m. Let in the following k l be shorthand notation for K j k=1 K j l=1 and m j r be shorthand notation for M m=1 M j=1 M r=1. Within each group (nest) it is assumed that den conditional probability to choose (mk) given that m is chosen is of the form (1): (4) p k m = G(x mkβm) l G(x mlβm) k p k m = 1 m m=1... M; k =1... K m where G( ) is a so far unspecified function which in nested Logit models like in the standard ones is parameterized as an exponential function. A simple specification of the probability of choosing an alternative in group m regardless of which might be that it has has the standard Logit form: p m = G(z mα) j G(z m = 1... M. jα) Then however no variables in x m would have affected whether group m is chosen or not: The probability that alternative (mk) is chosen would be the product of two multinomial choice probabilities: = p m p k m = G(z mα) j G(z jα) G(x mk β m) l G(x mlβ m) m=1... M; k =1... K m. Why would this be inconvenient when the alternatives form a hierarchic structure as indicated? Nested Logit as a special case In the nested multinomial Logit model a more er general parametrization of the choice probabilities is chosen: For group m the group specific choice probability is assumed to be of the form (5) p m = G(z mα)h(x m βm ρ m ) r G(z m = 1... M rα)h(x r βr ρ r ) where H(x m β m ρ m ) is a so far unspecified positive function which in addition to x m and β m also contains the scalar parameter ρ m. The latter serves to weight the different G(z m α)s. From (4) and (5) it follows that (6) = p m p k m = G(z m α)h(x m β m ρ m )G(x mk β m) [ r G(z rα)h(x r β r ρ r )][ l G(x mlβ m)] m = 1... M; k = 1... K m. 3

4 Will now the very restrictive IIA-axiom hold? [See Dagsvik Section 3.2 (in syllabus) for a precise definition of IIA and discussion of its implications.] First from (5) it follows that the odds-ratio for groups m and j is (7) p m = G(z mα)h(x m βm ρ m ) p j G(z j α)h(x j βj ρ m j = 1... M. j) Since no attributes for groups outside m and j enter this expression we can conclude that IIA still holds between all pairs of groups. Second from (6) it follows that the odds-ration for alternatives (mk) and (jl) is = G(z mα)h(x m βm ρ m )G(x mk βm) r p jl G(z j α)h(x j βj ρ j)g(x jl βj ) G(x jrβj (8) ) r G(x mrβm) m j =1... M; k =1... K m ; l =1... K j. Because attributes for alternatives outside (mk) and (jl) enter this expression we know that IIA does not in general hold between all pairs of alternatives. However in two special cases will IIA be satisfied: Case 1: The alternatives belong to the same group (nest) (m=j). Then (8) is simplified to p ml = p k m p l m = G(x mkβ m) G(x ml β m) m = 1... M; k l = 1... K m. Case 2: The restrictions H(x m β m ρ m )= r G(x mrβ m) (m = 1... M) apply. Then (8) is simplified to = G(z mα)g(x mk β m j =1... M; m) p jl G(z j α)g(x jl βj ) k =1... K m ; l =1... K j. So far the functions G( ) and H( ) have been specified as arbitrary functions. In the Nested Logit model we in particular do the following: [1]: We parameterize G( ) as G(wθ) = e wθ = G(w 1 θ 1 )G(w 2 θ 2 ) G(w 1 θ 1 + w 2 θ 2 ). [2]: We rewrite β m as β m = β m /ρ m and at the same time assume (9) H(x m β m ρ m ) = [ k G(x mkβ m)] ρ m = [ k ex mk(β m /ρ m ) ] ρ m. The last assumption can be rewritten as (10) H(x m β m ρ m ) = e wmρm 4

5 where (11) w m = ln[ l ex ml(β m/ρ m) ] e wm = l ex ml(β m/ρ m) If ρ 1... ρ m are free parameters it follows from (4) (6) and (10) that (12) (13) (14) p k m = ex mk(β m/ρ m) l ex ml(β m/ρ m) p m = ez mα [ l ex ml(β m /ρ m ) ] ρ m j ez jα [ l ex jl(β j /ρ j) ] ρ j = ez mα+w m ρ m j ez jα+w j ρ j = ez mα [ l ex ml(β m /ρ m ) ] ρ m r ez rα [ l ex rl(β r /ρ r) ] ρ r = ez mα+w m ρ m exmk(βm/ρm) r ez rα+w r ρ r e w m = ezmα+wm(ρm 1)+x mk(β m/ρ m) r e x mk(β m /ρ m ) l ex ml(β m /ρ m ) l ezrα+wr(ρr 1)+x rl(β r/ρ r) m = 1... M; k = 1... K m. Note that in (13) w m enters as a regressor with coefficient ρ m. From the last equality it follows that the odds-ratios become: (15) p jl = ezmα+wm(ρm 1)+xmk(βm/ρm) e z jα+w j (ρ j 1)+x jl (β j /ρ j ) m j =1... M; k =1... K m ; l =1... K j. Here the IIA-axiom is violated since variables and parameters outside alternatives (mk) and (jk) occur in the second term of the exponents of the numerator and the denominator via the transformed variables w m and w j. Such a model can be utilized to take account of that fact that substitution possibilities or correlation pattern or association within groups (nests) differ from those between groups (nests). These can be represented by κ m = 1 ρ m (m = 1... M) such that ρ m = 1 κ m = 0 (m = 1... M) express that there is no correlation. 1 κ m provides a measure of correlation within group m [see Train (2003 pp )]. Certain authors have denoted κ 1... κ M as dissimilarity parameters. In the special case ρ m = 1 = κ m = 0 βm = β m (m = 1... M) which corresponds to case 2 above since H(x m β m 1)= k G(x mkβ m )= k ex mkβ m 5

6 it follows from (12) (14) that (16) (17) (18) p k m = p m = = ex mkβ m l ex mlβ m l ezmα+x mlβ m j l ez jα+x jl β j ez mα+x mk β m r l ez rα+x rl β r m = 1... M; k = 1... K m while w m and w j drop out of (15) The last equation (18) is a standard multinomial Logit probability which confirms that the IIA-axiom holds. Then all J = M m=1 K m choice alternatives are in a certain sense co-ordinated being on the same level since the substitution and correlation pattern is the same between all pairs of alternatives. The group dimension is uninteresting Nested Logit as a special case A remark on estimation In the nested logit case with free ρ m s estimation can proceed stepwise as follows: (i) Estimate β m = β m /ρ m by Maximum Likelihood as in a standard multinomial Logit model for group by using the conditional probabilities (12). (ii) Compute values of w m (m = 1... M) from (11) by replacing β m = β m /ρ m by their estimates from (i). (iii) Estimate α and ρ m by Maximum Likelihood as in a standard multinomial Logit analysis from the last equality of (13) while replacing w m (m = 1... M) by their values computed from (ii). Here ρ m appear as a standard Logit parameter co-ordinated with the parameter vector α (iv) Finally estimate β m by multiplying the estimate of β m = β m /ρ m from (i) by the estimate of ρ m from (iii). [This is a feasible procedure but from an efficiency point of view not beyond critique; see Train (2003 Section 4.2.4).] REFERENCE Train K.E. (2003): Discrete Choice Methods with Simulation Cambridge University Press. 6