Unversitat Pompeu Fabra Dr. Kurt Schmidheiny Multinomial Choice (Basic Models) 2 Lecture Notes in Microeconometrics June 17, 2007
Multinomial Choice (Basic Models) Dependent variables can often only a countable number of values, e.g. yn ∈{1, 2,...J} .
This applies often to a context where an agent (individual, household, firm, decision maker, ...) chooses from a set of alternatives. Contents Sometimes the values/categories of such discrete variables can be nat- 1 Ordered Probit 2 urally ordered, i.e. larger values are assumed to correspond to “higher” 1.1TheEconometricModel...... 2 outcomes. The ordered probit model is a latent variable model that of- 1.2Identification...... 4 fers a data generating process for this kind of dependent variables. Some 1.3InterpretationofParameters...... 4 Examples: 1.4Estimation...... 5 • Likert-scale questions in opinion surveys: 1 = “Strongly Disagree”, 1.5ImplementationinSTATA...... 5 2 = “Somewhat Disagree”, 3 = “Undecided”, 4 = “Somewhat 2 Conditional Logit 7 Agree”, 5 = “Strongly Agree”. 2.1TheEconometricModel...... 7 • Employment status queried as 1 = “unemployed”, 2 = “part time”, 2.2Identification...... 8 3 = “full time”. (Although often used as example one might ques- 2.3InterpretationofParameters...... 9 tion the ‘natural’ order in this case and apply unordered models.) 2.4Estimation...... 10 2.5ImplementationinSTATA...... 11 1.1 The Econometric Model
3 Multinomial Logit 12 Consider a latent random variable yn for individual n =1, ..., N 3.1TheEconometricModel...... 12 ∗ ∼ 2 3.2Identification...... 13 yn = xnβ + εn with εi N(0,σ ) 3.3InterpretationofParameters...... 14 that linearly depends on xn. The error term εn is independently and 3.4Estimation...... 15 normally distributed with mean 0 and variance σ2. The distribution of 3.5ImplementationinSTATA...... 15 ∗ ∗ 2 yn given xn is therefore also normal: yn|xn ∼ N(xnβ,σ ). The expected ∗ 4 See also ... 16 value of the latent variable is Eyn = xnβ. Observed is only whether individual n’s index lies in a category j =
References 16 1, 2, ..., J which is defined through its unknown lower µj−1 and upper 3 Lecture Notes in Microeconometrics Multinomial Choice (Basic Models) 4
0.4 1.2 Identification
The choice probabilities Pnj allow only to identify the ratios β/σ and µ/σ
f(y*|x ) but not β, µ and σ individually. Therefore, one usually assumes σ =1. 0.3 f(y*|x ) 2 1 Suppose that the index function contains a constant, i.e. xnβ = β0 + β1x1 + ... + βK xK .Thenβ0 and µ1,...,µJ−1 are not identified as
0.2 only the differences (µj − β0) appear in the choice probabilities Pnj.The f(y*|x) model is usually identified by either setting µ1 =0orβ0 =0.
0.1 1.3 Interpretation of Parameters y = 1 y = 2 y = 3 [The individual index n is skipped in this section] The sign of the es- 0 y* µ x’ β µ x’ β timated parameters β can directly be interpreted: a positive sign tells 1 1 2 2 whether the answer/choice probabilities shift to higher categories when
the independent variable increases. The null hypothesis βk = 0 means Figure 1: Probabilities in the ordered probit model with 3 alternatives. that the variable xk, x =(x1, ..., xk, ..., xK ), has no influence on the choice probabilities. Note, however, that the absolute magnitude of the para- bound µj, i.e. the observed choice yn is meters is meaningless as it is arbitrarily scaled by the assumption σ =1. ⎧ ∗ One can therefore e.g. not directly compare parameter estimates for the ⎪ 1if y ≤ µ1 ⎪ n ⎪ ∗ same variable in different subgroups. ⎨⎪ 2ifµ1 ∂P(y =1|x) The probability that individual n chooses alternative j is easily derived = −φ(x β)βk with help of Figure 1: ∂xk ⎧ | ⎪ Φ[(µ1 − x β)/σ] for j =1 ∂P(y =2x) ⎪ n =[φ(x β) − φ(µ2 − x β)]βk ⎪ − − − ∂xk ⎨⎪ Φ[(µ2 xnβ)/σ] Φ[(µ1 xnβ)/σ] for j =2 | − − − | Pnj = P (yn = j xn) ⎪ Φ[(µ3 xnβ)/σ] Φ[(µ2 xnβ)/σ] for j =3 ∂P(y =3x) − − − ⎪ =[φ(µ2 x β) φ(µ3 x β)]βk ⎪ . ∂xk ⎪ ⎩ . 1 − Φ[(µJ−1 − xnβ)/σ] for j = J . ∂P(y = J|x) = φ(µJ−1 − x β)βk where Φ(.) is the cumulative standard normal distribution. ∂xk 5 Lecture Notes in Microeconometrics Multinomial Choice (Basic Models) 6 Note that the marginal effects can only be reported for specified types x. predict p1 p2 p3, p When βk is positive, then the probability of choosing the first category stores P (yn =1|xn), P (yn =2|xn)andP (yn =3|xn) in the respective P (y = 1) decreases with xk and the probability of the last category P (y = new variables p1, p2 and p3. J) increases. However, the effect on middle categories is ambiguous and The marginal effects on the probability of choosing the alternative depends on x. with value 1 is computed by 1.4 Estimation mfx compute, predict(outcome(1)) for an individual with mean characteristicsx ¯n.Theat option is used to The ordered probit model can be estimated using maximum likelihood evaluate further types xn. (ML). The log likelihood function is