Chapter 16

Qualitative and Limited Dependent Variable Models

ECON 6002Econometrics Memorial University of Newfoundland

Adapted from Vera Tabakova’s notes

Chapter 16: Qualitative and Limited Dependent Variable Models

Nested Logit

Mixed Logit AKA Random Parameters Logit

Generalized Multinomial Logit

Slide 16-2Principles of Econometrics, 3rd Edition

There is the implicit assumption in logit models that the odds between any pair of alternatives is independent of irrelevant alternatives (IIA)

One way to state the assumption If choice A is preferred to choice B out of the choice set {A,B}, then

introducing a third alternative X, thus expanding that choice set to {A,B,X}, must not make B preferable to A.

which kind of makes sense

Slide16-3Principles of Econometrics, 3rd Edition

IIA assumption

There is the implicit assumption in logit models that the odds between any pair of alternatives is independent of irrelevant alternatives (IIA)

In the case of the multinomial logit model, the IIA implies that adding another alternative or changing the characteristics of a third alternative must not affect the relative odds between the two alternatives considered.

This is not realistic for many real life applications involving similar (substitute) alternatives.

Slide16-4Principles of Econometrics, 3rd Edition

IIA assumption

This is not realistic for many real life applications with similar (substitute) alternatives

Examples: Beethoven/Debussy versus another of Beethoven’s Symphonies

(Debreu 1960; Tversky 1972) Bicycle/Pony (Luce and Suppes 1965) Red Bus/Blue Bus (McFadden 1974). Black slacks, jeans, shorts versus blue slacks (Hoffman, 2004) Etc.

Slide16-5Principles of Econometrics, 3rd Edition

IIA assumption

Red Bus/Blue Bus (McFadden 1974).

Imagine commuters first face a decision between two modes of transportation: car and red bus

Suppose that a consumer chooses between these two options with equal probability, 0.5, so

that the odds ratio equals 1.

Now add a third mode, blue bus. Assuming bus commuters do not care about the color of the bus (they are perfect substitutes), consumers are expected to choose between bus and car still with equal probability, so the probability of car is still 0.5, while the probabilities of each of the two bus types should go down to 0.25

However, this violates IIA: for the odds ratio between car and red bus to be preserved, the new probabilities must be: car 0.33; red bus 0.33; blue bus 0.33

Te IIA axiom does not mix well with perfect substitutes

IIA assumption

We can test this assumption with a Hausman-McFadden test which compares a logistic model with all the choices with one with restricted choices (mlogtest, hausman base in STATA, but check option detail too: mlogtest, hausman detail)

However, see Cheng and Long (2007)

Another test is Small and Hsiao’s (1985)STATA’s command is mlogtest, smhsiao (careful: the sample is randomly split every time, so you must set the seed if you want to replicate your results)

See Long and Freese’s book for details and worked examples

IIA assumption

Extensions have arisen to deal with this issue

The multinomial probit and the mixed logit are alternative models for nominal outcomes that relax IIA, by allowing correlation among the errors (to reflect similarity among options) but these models often have issues and assumptions themselves

IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is called the McFadden’s nested logit model, which allows correlation among some errors, but not all (e.g. Heiss 2002)

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution (Steenburgh 2008), which itself also suggests similarly counterintuitive real-life individual choice behavior

The multinomial probit has serious computational disadvantages too, since it involves calculating multiple (one less than the number of categories) integrals. With integration by simulation this problem is being ameliorated now…

IIA assumption

IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives

The most popular of these is called the McFadden’s nested logit model, which allows correlation among some errors, but not all (e.g. Heiss 2002)

IIA assumption

The nested logit is a partial relaxation of the IID and IIA assumptions of the MNL model

It is relatively straightforward to estimate

It also has a closed-form solution

IIA assumption

Most NL models have only two hierarchical levels, Very few NL models are estimated with three levels, and even fewer with four levels

Note that the “tree structure” does not have an actual “sequential” interpretation of any sort

It is only there to allow for differentials in the degree of correlation within and between “nests”

IIA assumption

Nested Logit By default, nowadays Stata’s nlogit uses a parameterization

that is consistent with RUM

Before Stata 10, a nonnormalized version of the nested logit model was used by Stata (and other packages) and you will see some papers pointing that out

This can still be requested by specifying the nonnormalized option

nonnormalized requests a nonnormalized parameterization of the model that does not scale the inclusive values by the degree of dissimilarity of the alternatives within each nest. Use this option to replicate results from older versions of Stata (Stata help)

Nested Logit By default, NOW Stata’s nlogit uses a parameterization that

is consistent with RUM

Before Stata 10, a nonnormalized version of the nested logit model was used by Stata (and other packages) and you will see some papers pointing that out

Both versions are valid, but only the RUM-consistent version is based on a sound model of consumer behavior

(the normalization is about scaling the coefficients in the second level choice, dividing them by the dissimilarity parameters, so that the utilities can be meaningfully compared, see Heiss (2002) for details)

Nested Logit

Adapted from Stata’s help file, let us consider a model of restaurant choice

use http://www.stata-press.com/data/r13/restaurant

Or look it up (it is one of Stata’s example datasets)

run describe

Nested Logit

“Fake” data on 300 families and their choice of seven local restaurants: Freebirds and Mama’s Pizza sell fast food Cafe Eccell, Los Nortenos, and Wings ’N

More are family restaurants Christopher’s and Mad Cows are fancy

restaurants

Nested Logit

Model the decision of where to eat as a function of: household income Number of kids rating, of the restaurant (coded 0–5) average meal cost per person distance between the household and the

restaurant

Nested Logit

Note that: income and kids are attributes of the family rating is an attribute of the alternative (the

restaurant) cost and distance are attributes of the

alternative as perceived by the families—that is, each family has its own cost and distance for each restaurant.

Nested Logit

Thus:

income and kids are case-specific

rating is alternative-specific

cost and distance are both

Nested Logit

Why not only 300 obs.?

Nested Logit

Why not only 300 obs.?

Nested Logit

You could fit a conditional logit model to this data as arranged

Since income and kids are case-specific, you would use asclogit instead of clogit*

*asclogit is great…in the “old days” you would need to work a bit harder with dummies and interactions to e able to run a mixed model with the old clogit command. This is still a good exercise tough.

Nested Logit

You could fit a conditional logit model to this data as arranged

However, the conditional logit may be inappropriate, since it assumes that the random errors are independent, and as a result it forces the odds ratio of any two alternatives to be independent of the other alternatives, the IIA!!!

Nested Logit

distance -.085346 .0437514 -1.95 0.051 -.1710973 .0004052

rating .8669793 .0981221 8.84 0.000 .6746635 1.059295

cost -.1543089 .0173858 -8.88 0.000 -.1883845 -.1202333

income 0 (omitted)

kids 0 (omitted)

chosen Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -537.06643 Pseudo R2 = 0.0800

Prob > chi2 = 0.0000

LR chi2(3) = 93.41

Conditional (fixed-effects) logistic regression Number of obs = 2100

Iteration 3: log likelihood = -537.06643

Iteration 2: log likelihood = -537.06643

Iteration 1: log likelihood = -537.06762

Iteration 0: log likelihood = -538.52688

note: income omitted because of no within-group variance.

note: kids omitted because of no within-group variance.

. clogit chosen kids income cost rating distance , group(family_id)

This is the pure conditional logit!

Nested Logit

.

_cons .8012166 1.821835 0.44 0.660 -2.769514 4.371948

kids -.2062462 .2713623 -0.76 0.447 -.7381066 .3256142

income .0963617 .0221248 4.36 0.000 .052998 .1397254

MadCows

_cons 1.626207 1.672155 0.97 0.331 -1.651157 4.903571

kids -.0374476 .2535199 -0.15 0.883 -.5343376 .4594424

income .0714976 .021214 3.37 0.001 .029919 .1130763

Christophers

_cons .6935955 .9268626 0.75 0.454 -1.123022 2.510213

kids .2442661 .2268803 1.08 0.282 -.2004111 .6889434

income .0428583 .0194924 2.20 0.028 .0046538 .0810627

WingsNmore

_cons 1.483996 .9318191 1.59 0.111 -.3423353 3.310328

kids .1761988 .2257644 0.78 0.435 -.2662912 .6186889

income .0325719 .0193533 1.68 0.092 -.0053599 .0705037

LosNortenos

_cons .3677203 .9075505 0.41 0.685 -1.411046 2.146486

kids .3486398 .226436 1.54 0.124 -.0951667 .7924463

income .0426504 .0193985 2.20 0.028 .0046301 .0806708

CafeEccell

_cons -1.228673 1.108607 -1.11 0.268 -3.401503 .944157

kids .3248002 .274624 1.18 0.237 -.2134529 .8630533

income .0239198 .0232255 1.03 0.303 -.0216013 .069441

MamasPizza

Freebirds (base alternative)

distance -.2127489 .0482651 -4.41 0.000 -.3073468 -.1181511

cost -.1330786 .0675342 -1.97 0.049 -.2654432 -.0007141

restaurant

chosen Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -482.21731 Prob > chi2 = 0.0000

Wald chi2(14) = 60.28

max = 7

avg = 7.0

Alternative variable: restaurant Alts per case: min = 7

Case variable: family_id Number of cases = 300

Alternative-specific conditional logit Number of obs = 2100

Iteration 4: log likelihood = -482.21731

Iteration 3: log likelihood = -482.21731

Iteration 2: log likelihood = -482.21859

Iteration 1: log likelihood = -483.15644

Iteration 0: log likelihood = -487.34059

. asclogit chosen cost dist, case(family_id) alternatives(restaurant) casevars(income kids)

I could not estimate rating at the same timeIt did not converge

Could you run a Plain MNL With this dataset?

_cons .8012166 1.821835 0.44 0.660 -2.769514 4.371948

kids -.2062462 .2713623 -0.76 0.447 -.7381066 .3256142

income .0963617 .0221248 4.36 0.000 .052998 .1397254

MadCows

_cons 1.626207 1.672155 0.97 0.331 -1.651157 4.903571

kids -.0374476 .2535199 -0.15 0.883 -.5343376 .4594424

income .0714976 .021214 3.37 0.001 .029919 .1130763

Christophers

. asclogit chosen cost dist, case(family_id) alternatives(restaurant) casevars(income kids)

distance -.1922521 .0465579 -4.13 0.000 -.2835039 -.1010003

cost -.1046302 .0656595 -1.59 0.111 -.2333204 .02406

restaurant

chosen Coef. Std. Err. z P>|z| [95% Conf. Interval]

Nested Logit

Here we suspect that restaurants should be grouped by type (fast, family, or fancy)

Why?

Nested Logit

Assuming that “unobserved stuff” affecting a decision about one alternative has no effect on the choice other alternatives may seem innocuous, but often this assumption is too restrictive

Example: when a family was deciding which restaurant to visit, they were pressed for time because of plans to attend a movie later

Nested Logit

The unobserved shock (being in a hurry) would raise the likelihood that of going to either fast food restaurant (Freebirds or Mama’s Pizza)

Another family might be choosing a restaurant to celebrate a birthday and therefore be inclined to attend a fancy restaurant (Christopher’s or Mad Cows)

Nested Logit

With the nested logit,we are not assuming that families first choose whether to attend a fast, family, or fancy restaurant and then choose the particular restaurant

We assume merely that they choose one of the seven restaurants

Nested Logit We now must first create a variable that

defines the structure of our “decision tree”

nlogitgen type = restaurant(fast: Freebirds | MamasPizza, family: CafeEccell | LosNortenos| WingsNmore, fancy: Christophers | MadCows)

Nested Logit We now must first create a variable that

defines the structure of our “decision tree”

Nested Logit

Our new type variable defines the three types of restaurants

We can now see how the alternative-specific attributes (cost, rating, and distance) apply to the bottom alternative set (the seven restaurants)

and how family-specific attributes (income and kid) apply to the alternative set at the first decision level (the three types of restaurants)

Nested Logit

nlogit chosen cost rating distance || type: income kids, base(family) || restaurant:, noconstant case(family_id)

Nested Logit

nlogit chosen cost rating distance || type: income kids, base(family) || restaurant:, noconstant case(family_id)

LR test for IIA (tau = 1): chi2(3) = 6.87 Prob > chi2 = 0.0762

/fancy_tau 4.099844 2.810123 -1.407896 9.607583

/family_tau 2.505113 .9646351 .614463 4.395763

/fast_tau 1.712878 1.48685 -1.201295 4.627051

type

dissimilarity parameters

kids -.3959413 .1220356 -3.24 0.001 -.6351267 -.1567559

income .0461827 .0090936 5.08 0.000 .0283595 .0640059

fancy

kids 0 (base)

income 0 (base)

family

kids -.0872584 .1385026 -0.63 0.529 -.3587184 .1842016

income -.0266038 .0117306 -2.27 0.023 -.0495952 -.0036123

fast

type equations

distance -.3797474 .1003828 -3.78 0.000 -.5764941 -.1830007

rating .463694 .3264935 1.42 0.156 -.1762215 1.10361

cost -.1843847 .0933975 -1.97 0.048 -.3674404 -.0013289

restaurant

chosen Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -485.47331 Prob > chi2 = 0.0000

Wald chi2(7) = 46.71

max = 7

avg = 7.0

Alternative variable: restaurant Alts per case: min = 7

Case variable: family_id Number of cases = 300

RUM-consistent nested logit regression Number of obs = 2100

Nested Logit

nlogit chosen cost rating distance || type: income kids, base(family) || restaurant:, noconstant case(family_id)

Option noconstant suppresses the constant terms for the bottom-level alternatives

Needed for convergence in this example unless you simplify things a little:nlogit chosen distance || type: income , base(family) || restaurant:, case(family_id)

Nested Logit

The error correlation parameters are re-expressed as dissimilarity parameters

In Stata notation nlogit estimates a tau, in this example for each “type” (upper level branch) with subcategories (lower level branches, twigs,…)

In Cameron and Trivedi’s (MMA) notation, dissimilarity parameters are rhos and called scale parameters

Nested Logit

In the normalised version of the nlogit model the dissimilarity parameters are used to scale the logsums or inclusive values:

In Cameron and Trivedi’s (MMA) notation:

Inclusive value or logsum

The inclusive value for the mth nest is the expected value of the maximum utility that Individual i can obtains from choosing an alternative within nest m

Nested Logit Nlogit estimates a tau (dissimilarity parameter, which is the

coefficient of the inclusive value/logsum) for each “type” (upper level branch) with subcategories (lower level branches, twigs,…)

dissimilarity parameters (inversely) measure the degree of correlation (rho =1-tau2 or for the mth nest) of random shocks within each of the three types of restaurants

If greater than one the model is inconsistent with RUM

Nested Logit Dissimilarity parameters must fall between 0 and 1

If one of them (say the one for fast food) were less than zero, something that increased the likelihood of choosing Freebirds would decrease the likelihood of choosing a fast food restaurant, which simply does not make any sense

If the dissimilarity parameter is zero, the changes in restaurant probabilities will not affect the choice of type of restaurant and the correct model is recursive (separated)

Nested Logit

The conditional logit model is a special case of nested logit where all the dissimilarity parameters equal one

Our Likelihood-ratio test of this hypothesis here shows mixed evidence of the null hypothesis that all the dissimilarity parameters are equal to one

IIA holds if and only if all dissimilarity parameters are equal to one

Nested Logit

In LIML (two-step or sequential estimation) it was often assumed for convenience that all of the dissimilarity parameters were equal

This is a restriction you can impose on our stata code too

Nested Logit

You could estimate the NLOGIT in two steps using LIML but you would need some complex corrections of the standard errors in the second step

Nowadays we have toys powerful enough to run NLOGIT all in one step using FIML

The latter is preferable (nlogit in Stata uses FIML), since it is more efficient

The LIML sequential estimation might still help to provide starting values, as the FIML log-likelihood is not globally concave

Nested Logit

Try:

. nlogit chosen rating distance || type: income kids, base(family) || restaurant: cost, noconstant case(family_id)

. nlogit chosen rating distance cost || type: kids, base(family) || restaurant: income, noconstant case(family_id)

Nested Logit

Issues: you can build your tree in different ways, some will work better than others

Those choices in general will yield different results anyway

No test to choose among trees

Mixed Logit AKA Random Parameters Logit

multinomial logit models with unobserved heterogeneity

They allow the parameters to vary randomly across individuals

See mixlogit command (Hole, A. R. Fitting mixed logit models by using maximum simulated likelihood Stata Journal, 2007, 7,

388-401 )

(find C:/… traindata.dta)

Mixed Logit AKA Random Parameters Logit

multinomial logit models with unobserved heterogeneity

The RPL allows for correlation across alternatives through an individual-specific random effect

Keywords

Slide 16-46Principles of Econometrics, 3rd Edition

binary choice models censored data conditional logit count data models feasible generalized least squares Heckit identification problem independence of irrelevant

alternatives (IIA) index models individual and alternative specific

variables individual specific variables latent variables likelihood function limited dependent variables linear probability model

logistic random variable logit log-likelihood function marginal effect maximum likelihood estimation multinomial choice models multinomial logit odds ratio ordered choice models ordered probit ordinal variables Poisson random variable Poisson regression model probit selection bias tobit model truncated data

References

Cameron and Trivedi’s MMA and MUS Hensher, Rose, and Greene’s (2005) Applied

Choice Analysis: A Primer, available (electronically too) at the QEII

Next

Ordered Choice Count data