BES 2, Statistica economica

3 - Multiple choice models

In many applications, we may want to analyse phenomena of discrete nature, (phenomena that are not quantitative, e.g. preference on public transports) where the number of possible alternatives that can be chosen is larger than two.

For example:

• An individual can choose between a full-time work, a part-time work or not working;
• A firm can choose Europe, Asia or the United States as its main export destination.

In other cases, some quantitative variables can only be observed to lie within certain thresholds: individuals are often unwilling or unable to give precise answers, so survey questions ask respondents to select an answer among predetermined ranges (i.e. classes).

Examples of this are questions about a household’s disposable income or the value of a house, about a firm’s total assets, or about an individual’s job or income satisfaction.

Multiple choice models (or multiresponse) models have been developed to describe the probability of each of the possible discrete outcomes as a function of individual-specific and/or alternative-specific characteristics.

An important goal of such models is to describe these probabilities with a limited number of unknown parameters and in a logically consistent way. (Choice probabilities should lie between 0 and 1 and add up to 1 over all the alternatives).

Among multiple choice models, an important distinction exists between ordered response models and unordered response models.

Ordered response models

Generally more parsimonious (i.e. it requires a lower number of unknown parameters), but is only appropriate if there exists a logical ordering of the alternatives.

The reason is that it assumes there is one underlying latent variable that drives the choice between the ordered alternatives: results will be sensitive to the ordering of the alternatives, so this ordering should make sense.

Unordered models

Are not sensitive to the ordering of the alternatives and are derived upon the assumption that each alternative has a specific utility level and individuals choose the alternative that yields the highest utility.

Ordered choice models

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3 - Multiple choice models

Let us consider the choice between M mutually exclusive alternatives, numbered from 1 to M (NB: these are simply numerical labels attached to the possible outcomes).

If there is a logical ordering in these alternatives, so that they can be ordered in a non-decreasing way (e.g., “Not working” (1), “Working part-time” (2), “Working full-time” (3)), an ordered response model can be used to model the probability of each outcome.

Those are discrete variables so labels 1,2,3 tells us that these variables are not quantitative.

Like binary choice models, this model is based on one underlying continuous latent variable, but with a different match from the latent variable to the observed y dependent variable, which can assume ordered values between 1 and M.

y_i*

We change the observation rule.

The observed ordered outcome is thus a monotonic increasing (many-to-one) transformation of a single continuous latent outcome that is naturally ordered.

Formally, we assume that the continuous latent variable can be expressed as a linear function of observed characteristics and unobserved factors (latent model):

x ϵ_i i* ′ 2y = x β + ϵ with ϵ |x ∼ IID(0, σ )
i i ii i

and that the latent variable is monotonically related to the observed ordered variable by the following observation rule:

*y = j if γ < y ≤ γ
i j−1 j
i

where are increasing unknown parameters (with for ), defining the

γ γ > γ j = 1, . . . , M
j j j−1

extremes of the classes into which the distribution of the continuous latent variable is M divided (threshold parameters or cut-points).

Consequently, the probability that alternative is chosen, conditional on , , is

j x P (y = j|x )
i i i

equal to the conditional probability that the latent variable is between the two boundaries

*y i

and , .

*γ γ P (γ < y ≤ γ |x )
j−1 j j−1 j i i

In order to identify model parameters and some additional distributional assumptions

β γ

and normalization constraints are required.

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3 - Multiple choice models

First, it is conventionally assumed that and , so that the continuous

γ = −∞ γ = ∞
0 M

latent variable is defined to vary between and .

−∞ ∞

Second, to fix the location of and identify the intercept term of the model, one of the

*y i

threshold parameters must be normalized to zero. Usually, it is assumed that

γ = 0
1

Third, to uniquely identify all the parameters we need to normalize the scale of ; as

*β y i

in binary models, the convention is to fix the variance of the errors at a given value, by specifying the distribution of the error terms.

By assuming that the errors are IID with a standard normal distribution we obtain the ordered probit model. The standard logistic distribution gives the ordered logit model.

The ordered probit model

In what follows, we will focus exclusively on the ordered probit model (OP) (analogous considerations apply for the ordered logit model too).

We thus assume that the errors , conditional on , follow a standard normal distribution

ϵ x
i i

(i.e. ) and impose the normalization .

ϵ |x ∼ N (0, 1) γ = 0
i i 1

Given these assumptions, the structure of the OP model is given by:

* ′Latent model y = x β + ϵ , with ϵ |x ∼ N (0, 1)
i i i i i

*1 if y ≤ 0
⎧ i
*2 if 0 < y ≤ γ 2
i
*3 if γ < y ≤ γ Observation rule y = 2 3
⎨ i i
⋮
⎩*M if y > γ M −1
i

It is worth noticing that when , we obtain again the structure of the binary probit

M = 2

model (with taking on values of 1 or 2, instead of 0 or 1, but this is irrelevant).

y i

Given the standard normal assumption for , it is straightforward to derive the distribution of

ϵ i

conditional on from the implied conditional probabilities of the alternatives:

y x M
i i

* ′ ′P (y = 1|x ) = P (y ≤ 0|x ) = P (x β + ϵ ≤ 0|x ) = P (ϵ ≤ −x β|x )
i i i i i i i i i

′= Φ(−x β)
i

* * *P (y = 2|x ) = P (0 < y ≤ γ |x ) = P (y ≤ γ |x ) − P (y ≤ 0|x )
i i 2 i 2 i i i i

′ ′= P (ϵ ≤ γ − x β|x ) − P (ϵ ≤ 0 − x β|x )
i 2 i i i i i i

′ ′= Φ(γ − x β) − Φ(−x β)
2 i i

⋮

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3 - Multiple choice models

* * *P (y = j|x ) = P (γ < y ≤ γ |x ) = P (y ≤ γ |x ) − P (y ≤ γ |x )
i i j−1 j i j i j−1 i i i

′ ′= P (ϵ ≤ γ − x β|x ) − P (ϵ ≤ γ − x β|x )
i j i i j−1 i i

′ ′= Φ(γ − x β) − Φ(γ − x β)
j j−1 i i

⋮

* * ′P (y = M |x ) = P (y > γ |x ) = 1 − P (y ≤ γ |x ) = 1 − P (ϵ ≤ γ − x β|x )
i i M −1 i M −1 i i M −1 i i i

′= 1 − Φ(γ − x β)
M −1 i

It is easy to show that these probabilities add up to 1, ensuring the logically consistency of the model.

Estimating the ordered probit model

The model can be estimated by maximum likelihood methods and the log-likelihood function can be defined based on the conditional probabilities of the M alternatives.

The contribution to the log-likelihood function of each observation can be written as:

i

log L (β, γ) = 1(y = 1) log[P (y = 1|x )]+. . . +1(y = M ) log[P (y = M |x )]
i i i i i i i

M M ′ ′= ∑ 1(y = j) log[P (y = j|x )] = ∑ 1(y = j) log[Φ(γ − x β) − Φ(γ − x β)]
i i i i j j−1
i i
j=1 j=1

The log-likelihood function of the sample can be then obtained by summing these contributions over the observat