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Limited Dependent Variable Models: Binary & Multiple Choice, Ordered Probit & Logit, Assignments of Econometrics and Mathematical Economics

Vassar College (VC)Econometrics and Mathematical Economics

An overview of limited dependent variable models in economics, focusing on binary choice models (linear probability, probit, and logit), multiple choice models (ordered probit and logit, multinomial logit, and conditional logit), and truncated dependent variable models (tobit and sample selection). The estimation of coefficients and cutoff values using maximum likelihood estimation and discusses marginal effects. Examples using the mroz dataset and multinomial logit model are included.

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Pre 2010

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Economics 310
Handout # VI Limited Dependent Variable Models
I. Binary Choice Models
A. Linear Probability Model
B. Probit Model (See: Economics 210 handouts)
C. Logit Model
II. Multiple Choice Models
A. Ordered Probit and Logit Models
B. Multinomial Logit Model
C. Conditional Logit Model
D. Poission Model (Count data)
III. Truncated Dependent Variable Models
A. Tobit Model (See: Tobit model handout)
B. Sample Selection Model
IIA. The Ordered Probit and Logit Models
Frequently a dependent variable takes on several unambiguously ordered values. For example,
suppose we consider bond ratings AAA, AA, A, etc. AAA is unambiguously a better rating than
AA and AA is unambiguously a better rating than A and so forth. Other examples would be the
ratings of various goods from consumer surveys, the ranking of politicians from opinion polls,
results from opinion poll on issues (strongly agree, agree, no opinion, disagree, strongly
disagree), levels of education (elementary school, high school, college, graduate study), levels of
employment (unemployed, part time, full time) and levels of risk for insurance. Note that in
each of these cases the ordering is unambiguous.
The Model
We assume that takes on one of the values 0, 1, 2, ..... J depending on the value of .
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Download Limited Dependent Variable Models: Binary & Multiple Choice, Ordered Probit & Logit and more Assignments Econometrics and Mathematical Economics in PDF only on Docsity!

Economics 310

Handout # VI Limited Dependent Variable Models

I. Binary Choice Models

A. Linear Probability Model

B. Probit Model (See: Economics 210 handouts)

C. Logit Model

II. Multiple Choice Models

A. Ordered Probit and Logit Models

B. Multinomial Logit Model

C. Conditional Logit Model

D. Poission Model (Count data)

III. Truncated Dependent Variable Models

A. Tobit Model (See: Tobit model handout)

B. Sample Selection Model

IIA. The Ordered P robit and Logit Models

Frequently a dependent variable takes on several unambiguously ordered values. For example,

suppose we consider bond ratings AAA, AA, A, etc. AAA is unambiguously a better rating than

AA and AA is unambiguously a better rating than A and so forth. Other examples would be the

ratings of various goods from consumer surveys, the ranking of politicians from opinion polls,

results from opinion poll on issues (strongly agree, agree, no opinion, disagree, strongly

disagree), levels of education (elementary school, high school, college, graduate study), levels of

employment (unemployed, part time, full time) and levels of risk for insurance. Note that in

each of these cases the ordering is unambiguous.

The Model

We assume that takes on one of the values 0, 1, 2, ..... J depending on the value of.

We estimate the coefficients and the cutoff values by maximum likelihood

estimation.

Marginal Effects. The marginal effects can be derived simply as follows:

Note that if the coefficient is positive we can say unambiguously that an increase in the

variable will decrease and increase. For the changes in the other

probabilities we need to calculate the marginal effects.

IIA. The Multinomial Logit Model

Suppose the variable which we are interested in takes on various values but there is no logical

ordering to the values. For example suppose the variable y takes on the values 1, 2, 3 depending

upon the mode of travel which a person chooses. 1 = automobile, 2 = train, and 3 = bus. For

clarity let us call these options A, B, and T. We choose one category, automobile as the base

category. We proceed as in the binary logit model. Let. Then

The coefficients are estimated by maximum likelihood estimation. There is also a multinomial

probit model which is similar in spirit but much more complicated to calculate.

Independence of irrelevant alternatives : The multinomial logit model has one very

unattractive property. The ratio of the probabilities (odds) are independent of one another.

Suppose that in the above example that B refers to green buses which is all that are available.

Suppose that for an individual the estimated probabilities are P(A)=.6, P(B)=.2 , and P(T) = .2.

Then P(B)/P(A) = 1/3 and P(T)/P(A) =1/3. Now assume that another color of bus , blue buses

become available. Designate the buses GB and BB respectively. Suppose that individuals are

indifferent between buses of different colors. The alternatives are now A, GB, BB, and T. Since

the individual in question is indifferent between the two colors of buses we might expect the

following Since persons are indifferent between different color buses we would logically expect

the new probabilities to be P(A)=.6, P(GB)=.1 , P(BB)=.1 and P(T) = .2; the same proportion of

persons choose buses as before and they are divided equally between green and blue buses.

However the multinomial logit model requires that the odds of choosing a green bus P(GB)/P(A)

would not be affected by the addition of alternative, blue buses, and would be equal to 1/3.

Similarly P(BB)/P(A)=1/3. Recalculating the probabilities we get P(A) = ½, P(GB)=1/6,

P(BB)=1/6, and P(T)=1/6. So as result of introducing a new color bus the probability of bus

ridership increased from .2 to .33. The implication of all this is that the multinomial logit model

is not appropriate if the alternatives are close substitutes. If the multinomial logit model is to be

used a test by Huasman should be run to indicate whether this problem exists. If the problem is

found to exist then an alternative model such as the nested logit should be used.

Example: Suppose we want to investigate how various personal characteristics affect choice of occupation. Let the variable JC = job classification (1 = management, 2 = sales, 3 = clerical, 4 = service, 5 = professional, 0 = other) Note that there is no sense that these choices are ordered. The explanatory variables are age, ed =years of schooling, and dummy variables for marital status, gender, and race.

. mlogit jc ed age female nonwhite married

Iteration 0: log likelihood = -908. Iteration 5: log likelihood = -708.

Multinomial regression Number of obs = 534 LR chi2(25) = 398. Prob > chi2 = 0. Log likelihood = -708.69521 Pseudo R2 = 0.

jc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1 | ed | .8671075 .1009836 8.59 0.000 .6691832 1. age | .0487994 .0165825 2.94 0.003 .0162983. female | 1.23219 .385499 3.20 0.001 .4766259 1. nonwhite | -.2238054 .5670162 -0.39 0.693 -1.335137. married | .0200836 .3998228 0.05 0.960 -.7635546. _cons | -14.39609 1.59567 -9.02 0.000 -17.52355 -11. -------------+---------------------------------------------------------------- 2 | ed | .5566231 .1076473 5.17 0.000 .3456382. age | .029292 .0175113 1.67 0.094 -.0050295. female | 1.321641 .4014091 3.29 0.001 .5348937 2. nonwhite | -.4456658 .6740512 -0.66 0.509 -1.766782. married | .441271 .4537499 0.97 0.331 -.4480624 1. _cons | -10.03732 1.655088 -6.06 0.000 -13.28123 -6. -------------+---------------------------------------------------------------- 3 | ed | .5042029 .0897659 5.62 0.000 .3282651. age | .0226438 .0137519 1.65 0.100 -.0043094. female | 2.90923 .3403131 8.55 0.000 2.242229 3. nonwhite | .5286729 .4310798 1.23 0.220 -.3162281 1. married | -.3605507 .327685 -1.10 0.271 -1.002801. _cons | -8.725491 1.356261 -6.43 0.000 -11.38371 -6. -------------+---------------------------------------------------------------- 4 | ed | .0837394 .0782126 1.07 0.284 -.0695544. age | .0198725 .0132546 1.50 0.134 -.006106. female | 1.84253 .3091344 5.96 0.000 1.236638 2. nonwhite | .671802 .3909131 1.72 0.086 -.0943736 1. married | -.5703341 .3174866 -1.80 0.072 -1.192596. _cons | -2.777329 1.178795 -2.36 0.018 -5.087724 -. -------------+---------------------------------------------------------------- 5 | ed | 1.129415 .0998826 11.31 0.000 .9336483 1. age | .0334009 .0161435 2.07 0.039 .0017603. female | 1.866768 .3624401 5.15 0.000 1.156398 2. nonwhite | -.735837 .5748099 -1.28 0.200 -1.862444. married | .2192979 .3747328 0.59 0.558 -.5151649. _cons | -17.50938 1.578416 -11.09 0.000 -20.60302 -14.

(Outcome jc==0 is the comparison group)