Daisuke Nagakura University of Washington. Abstract

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1 A Note on the Relationship of the Ordered and Sequential Probit Models to the Multinomial Probit Model Daisue Nagaura University of Washington Abstract In this note, I show that the ordered and sequential probit models are special cases of the multinomial probit model where the disturbance terms in the latent variables degenerate or those variances converge to zero at a certain rate. Citation: Nagaura, Daisue, (004) "A Note on the Relationship of the Ordered and Sequential Probit Models to the Multinomial Probit Model." Economics Bulletin, Vol., No. 40 pp. 7 Submitted: September 6, 004. Accepted: October 7, 004. URL: 04C40007A.pdf

2 . Introduction The ordered, sequential, and multinomial probit models are widely used in analyzing discrete choice problems (See Davidson and MacKinnon, 004, and Amemiya, 985, for details of these models). These three models have been considered as different types of models, i.e., it is not believed that one of them nests other two as special cases. In this note, I show that the ordered and sequential probit models are obtained as special cases of the multinomial probit model where the disturbance terms in the latent variables are degenerated or those variances converge to zero at a certain rate.. Definitions of the three models First, let us state the definitions of these three models. Let the number of alternatives be J. Define the unobserved latent variable as y * t = x t γ + η t, η t~ NID(0, η ), where x t is a observable explanatory vector without a constant term, and γ is a coefficient vector. Let y t be a discrete random variable whose value ranges from to J. Then, the ordered probit model with J alternatives is defined as follows: Definition (ordered probit model) if y* t < d if d * yt < d yt = J if d * J yt < dj J if d * J yt where d < d < < dj. The parameter d is called threshold parameter. It is customary to set η = for identification of parameters. The estimation can be easily done by the maximum lielihood estimation. Second, I shall define the sequential probit model. The J latent variables for the model + are given by yt = φ + x t δ + ζ t, ζ t ~NID(0,) for =,, J, where φ is a scalar constant term, and δ is a coefficient vector. Then, the sequential probit model is defined as follows: Definition (sequential probit model) yt + if yt if yt < 0, and yt 0 = J if yt < 0, yt < 0,..., ytj < 0, and ytj J if yt < 0, yt < 0,..., ytj < 0, and ytj < 0

3 In the sequential probit model, each decision is made sequentially according to a binary probit model. Whether or not an alternative is selected is determined before an alternative ( > ) is considered. See Amemiya (985, p0) for details of this model. Third, I shall give the definition of the multinomial probit model. The J latent variables for the model is defined as y t = α + x t β + ε t, ε t ~N(0, Ω ) for =,, J, where α is a scalar constant term, β is a coefficient vector, and ε t is a J normal random vector with typical element ε t and the covariance matrix ρ ρj J ρj Ω = J, J where between ε and ρ ( = ρ ) is the correlation coefficient ε. The definition of the multinomial probit model is given as follows: is the standard deviation of ε and Definition (multinomial probit model) yt = when y t > y ti i, i. We usually set α r = 0, β r = 0, r = 0, s = for some r and s, for parameter identification (Note that ε t becomes J normal random vector by this normalization).. Main results Hereafter, the subscript t is suppressed for the sae of notational simplicity. Proposition : The multinomial probit model reduces to the ordered probit mode with ( α + α ) d = ρ ρ + + for =,, J, where ρ =, if the following conditions are satisfied for some : (A) ρ = or, (B) ρ γ = β, (C) ρ < ρ < < < < ρjj, αr α αs α and (D) <, for r < s, r, s, =,, J, r, s ρ ρrr ρ ρss Proof. It can be easily shown that the conditional distribution of ε conditioned on ε is N ( ρ, ( ε ρ )). Under the condition (A), the conditional variance becomes zero and ε is perfectly correlated with ε, i.e., When this is the case, other correlation coefficients also become positive or negative one. α r α αs α It can be shown that this condition is equivalent to < for r < < s. ρ ρ ρ ρ r r s s

4 = () ε ρ ε Additionally, if the condition (B) is satisfied, the condition for y = in the multinomial probit model becomes as follows: y > yi i, i α + x β + ε > αi + x βi + εi i, i x ( β βi) + ε εi > αi α i, i ( ρ ρii )( x γ + ε ) > αi α i, i where the notation means that the conditions on the both sides are equivalent. Furthermore, from the condition (C), the above is equivalent to x γ + ε > c ( ) i for i < and ( ) i (From () and (B)), x γ + ε < c for i >, () ( ) ( αi α ) ( ) ( ) where c i. Note that ci = ci clearly by its definition. From the ρ ρii condition (D), when is fixed, the maximum of ( ) ( ) c i for i < is c, while the ( ) minimum of c i for i > is ( c ) +. Thus, () is rewritten as : ( ) ( ) ( ) < γ + ε < + = + c x c ( c ). () Note that the inequalities in () reduce to only one side for = and J. Setting ε = η, it have been shown that when the conditions (A)~(D) are satisfied, the ( α+ α) multinomial probit model reduces to the ordered probit model with d = ρ ρ + + for =,, J. 4 The conditions for =,, and are illustrated in figure (a), (b), and (c), respectively. The shaded area is the area which satisfies the all required conditions, i.e., y = when x γ + η falls into this area. Note that ρ can be negative one for <, but it must be positive one for <. For example if we set =, ρ = for all. Example: Set J = 4, ρ = for all, = 0 =, =, 4 =, α = 0, α =, α =.6, α 4=, β = 0, β =, β =, and β 4 = in the multinomial probit model 5. Notice that the conditions (A), (C) and (D) in Proposition are satisfied. Then this reduces to the ordered probit model with γ =, =, d =, d = 0.6, and d = 0.4. η ( ) ( ) c < c + is obvious from the condition (D). 4 The equality in the definition (and ) is not essentially important because the latent variable is a continuous random variable, hence the equality holds with probability zero. 5 This multinomial probit model is a normalized one by the previously mentioned way.

5 The next proposition states the relationship between the sequential and the multinomial probit models. Proposition : The sequential probit model can be obtained as a limiting case of the multinomial probit model if the following conditions are satisfied: (E) =, α = φ, δ = β, α 0, β 0, 0 so that lim 0 α 0 lim β l 0 βl 0 α φ and (F) ρ i = 0 i,, i. Proof. Multiplying the both sides by probit model can be rewritten as =, lim β =δ, 0 0 β 0 l 0 = 0, lim l 0 0 l 0 or lim αl = 0, α = for =,, J, and l >, y > yi i, i i α + β i + i ε > i αi + β i i + i εi α + β + ε > αi + β i + εi i, the condition for y = in the multinomial x x, for i < and x x for i >. Under the condition (E), the above reduces to because i i i i 0 > φ + x δ + ε for i <, and φ + x δ + ε > 0 for i > α l, l β, and the variance of (l > ) converge to zero. Set ε l ζ = ε, then the condition (F) ensures that ζ ~ NID(0,). This completes the proof. Intuitively, this is because the terms which converge to zero can be neglected compared with other terms which also converge to zero but at less rapid rate so that the condition (E) holds. To confirm the analytical proof of Proposition, I conduct a small Monte Carlo experiment. The number of choices is set at three. Samples for a multinomial probit model, yt ( m) t =,,0, are generated following the definition with y t= ε t, y t = 0.5/ s + ε t / s, and y t =/s + ε t / s, where ε t =,, are random draws from NID(0,) and s is some () s constant. Samples for a sequential probit model, yt t =,,0, are generated following the + + definition with y t = y t, and yt = syt. According to Proposition, ( m) y t should () s ( m) converge to yt as s becomes large. Figure (a), (b), (c) shows the actual values of y t () s for s =, 0, and 000, respectively, and those of yt are shown in Figure (d). The same ε t for t =,,0, =,, are used for each value of s to mae the result clearer. Observe ( m) () s that y t actually converges to y t as s becomes large, as insisted in Proposition. 4

6 . Conclusion I have shown that the ordered and sequential probit models are special cases of the multinomial probit model where the disturbance terms in the latent variables degenerate or those variances converge to zero at a certain rate. Which probit model should be used is an important problem in an empirical analysis. The results obtained in this note are useful for comparing these three models; specification tests for these models will be developed based on the results. For example, Kobayashi (00) have proposed a test for ordered probit models against multinomial probit models in the case of three alternatives. The results of this note can be used for extending the test to the case of more than three alternatives. Also, a test for sequential probit models in this context has not been proposed as far as I now. These are the subects of further research. Finally, I note that the conditions I have shown in this note are sufficient conditions, so the same results may be obtained under weaer conditions. Reference Amemiya, T. (985) Advanced Econometrics, Cambridege, Mass., Harvard University Press. Kobayashi M. (00) A Test for Ordered Probit Models against Multinomial Probit Models, mimeo. Davidson, R., and J. G.. MacKinnon (004) Econometric Theory and Methods, New Yor, Oxford University press. 5

7 Figure. Conditions of the degenerated multinomial probit model y = c or c c or c c 4 or c 4 (a) conditions for y = y = c or c c or c c 4 or c 4 (b) conditions for y = y = c or c c or c c 4 or c 4 (c) conditions for y = 6

8 Figure. Actual values of the samples from the multinomial and sequential probit models (a) samples from the multinomial probit model when s = (b) samples from the multinomial probit model when s = (c) samples from the multinomial probit model when s = (d) samples from the sequential probit model when s =, 0,000 Note: x axis indicates the no. of a sample, and y axis is for an actual value of a sample. 7