4  Specification of the error distribution

This section derives the probability distribution of hours worked for a special case of the distribution of error terms. This special distribution results in a multinomial logit model for utility. The multinomial logit model has been used extensively in discrete choice modelling. The discrete error distribution in the previous section was used only for convenience, and it is first necessary to state the problem where is considered to be a continuous random variable. Hence, and are now the density and distribution functions respectively of . It is possible to convert the result in equation (5) into the following form for continuous , remembering that hours continue to be discrete:

(9)    

Essentially, the expression in (9) takes all the possible conditional probabilities, represented by , and integrates out to obtain the required marginal distribution . Given that the conditional probabilities require the product of distribution functions, , it cannot be expected that an arbitrary choice of will be tractable. This section considers a special case generating a highly convenient form for the hours distribution.

4.1  A special case: The extreme value distribution

Suppose the distribution of is described by the following density function:

(10)    

for which the distribution function is:

(11)    

The choice of this ‘thin-tailed’ distribution has the obvious advantage that no further parameters need to be estimated.[13] This is known as an Extreme (Maximum) Value Type I distribution, which is often abbreviated to ‘extreme value’ distribution.[14] It is highly tractable in the present context. These qualities have generally been (implicitly) taken as sufficient justification for its use, though section 7 briefly discusses some alternatives.

The arithmetic mean of this distribution is non-zero, being equal to 0.5772 (Euler’s number); the mode is zero and the median is . The shape of the distribution is illustrated in Figures 2 and 3, showing the density and distribution functions respectively. In Figure 3, the distribution function used in the numerical example of section 3 is shown for comparison: this is obviously a step function for the discrete distribution.

Figure 2 – Extreme value probability density function

Figure 3 – Extreme value cumulative distribution function and the discrete distribution from Table 1

Substitution into gives:

(12)    

Noting that the logarithm of can be expressed as , and using , the expression in (12) becomes:

(13)    

Furthermore, , say. Hence (13) can be rewritten more succinctly as Thus:

(14)    

Further simplification is achieved using the variate transformation, so that and , whereby:

(15)    
    

In this special case, the probability distribution of hours of work for an individual depends in a very simple way on the measured utility levels associated with each hours level.[15] The discrete choice model flowing from the assumption of an extreme value distribution is called a multinomial logit model.[16]

For the numerical example considered earlier, the hypothetical measured utility levels for the four hours points are 5, 7.5, 10 and 9. Substitution into (15) gives the probabilities 0.005, 0.056, 0.686, and 0.253.