Transcript Qualitative and Limited Dependent Variable Analysis III
ECON 6002 Econometrics Memorial University of Newfoundland
Qualitative and Limited Dependent Variable Models
Adapted from Vera Tabakova’s notes
16.1 Models with Binary Dependent Variables
16.2 The Logit Model for Binary Choice
16.3 Multinomial Logit
16.4 Conditional Logit
16.5 Ordered Choice Models
16.6 Models for Count Data
16.7 Limited Dependent Variables
Principles of Econometrics, 3rd Edition Slide 16-2
When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is
y
= 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include: The number of trips to a physician a person makes during a year.
The number of fishing trips taken by a person during the previous year.
The number of children in a household.
The number of automobile accidents at a particular intersection during a month.
The number of televisions in a household.
The number of alcoholic drinks a college student takes in a week.
Principles of Econometrics, 3rd Edition Slide16-3
If
Y
is a Poisson random variable, then its probability function is
y
!
y
y
y
e
y
y
!
,
1
y
0,1, 2,
“rate” (16.27)
exp
1 2
x
Also equal To the variance (16.28) This choice defines the
Poisson regression model
for count data.
Principles of Econometrics, 3rd Edition Slide16-4
If we observe 3 individuals: one faces one event, the other two two events each:
L
1
,
2
ln
L
1
,
2
ln ln 0
ln
ln
e
exp
y
!
y
1 2
x
y
ln
y
1
2
2
ln ln 2
2
x
ln
y
ln
L
1
,
2
i N
1
exp
1 2
x
i
y
i
1 2
x
i
ln
y
i
Principles of Econometrics, 3rd Edition Slide16-5
0
0 exp
1
x
2 0
Pr
Y
y
exp
y
!
y
0
,
y
0,1, 2,
So now you can calculate the predicted probability of a certain number y of events
Principles of Econometrics, 3rd Edition Slide16-6
i
x
i
i
2 You may prefer to express this marginal effect as a %:
%
x
i
100
i i
x
i
100
2
%
(16.29)
Principles of Econometrics, 3rd Edition Slide16-7
i
i
exp
1 2
x
i
D
i
i i
0
exp
1 2
x
i
i i
exp
1 2
x
i
100 exp
1 2
x
i
exp
1
exp
2
x
i
1 2
x
i
Principles of Econometrics, 3rd Edition
If there is a dummy Involved, be careful, remember
e
Which would be identical to the effect of a dummy In the log-linear model we saw under OLS
Slide16-8
Extensions: overdispersion Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) This will often not hold Real life data are often overdispersed For example: • a few women will have many affairs and many women will have few • a few travelers will make many trips to a park and many will make few • etc.
Principles of Econometrics, 3rd Edition Slide16-9
Extensions: overdispersion use "C:\bbbECONOMETRICS\Rober\GRAD\GROSMORNE.dta", clear . poisson visits Travelcost educat income Iteration 0: log likelihood = -1321.4696 . poisson persontrip Travelcost educat income, nolog Iteration 2: log likelihood = -1321.4665 LR chi2(3) = 671.71
Prob > chi2 = 0.0000
Log likelihood = -1321.4665 Pseudo R2 = 0.0210
Log likelihood = -2541.5165 Pseudo R2 = 0.1167
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] income -.0019933 .0007191 -2.77 0.006 -.0034027 -.0005839
income -.0014578 .0004404 -3.31 0.001 -.002321 -.0005946
_cons 2.144476 .0688666 31.14 0.000 2.0095 2.279452
Principles of Econometrics, 3rd Edition Slide16-10
Extensions: negative binomial Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) The Poisson will inflate your t-ratios in this case, making you think that your model works better than it actually does Or use a Negative Binomial model instead (
nbreg
) or even a Generalised Negative Binomial (
gnbreg
) , which will allow you to model the overdispersion parameter as a function of covariates of our choice You can also test for overdispersion, to test whether the problem is significant
Principles of Econometrics, 3rd Edition Slide16-11
Extensions: negative binomial sum visits Variable | Obs Mean Std. Dev. Min Max -------------+------------------------------------------------------- visits | 966 1.416149 1.718147 1 26
Principles of Econometrics, 3rd Edition Slide16-12
Extensions: negative binomial . nbreg persontrip Travelcost educat income, nolog Negative binomial regression Number of obs = 919 LR chi2(3) = 236.04
Dispersion = mean Prob > chi2 = 0.0000
Log likelihood = -2038.1155 Pseudo R2 = 0.0547
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -.7135986 .0489137 -14.59 0.000 -.8094676 -.6177295
educat -.0218888 .0248201 -0.88 0.378 -.0705353 .0267578
income -.0014357 .0006578 -2.18 0.029 -.0027249 -.0001465
_cons 1.994577 .1037 19.23 0.000 1.791329 2.197826
/lnalpha -1.190022 .0724583 -1.332038 -1.048006
alpha .3042145 .0220429 .2639388 .3506361
Likelihood-ratio test of alpha=0: chibar2(01) = 1006.80 Prob>=chibar2 = 0.000
Principles of Econometrics, 3rd Edition Slide16-13
Extensions: excess zeros Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model They would underpredict them too There is said to be an “excess zeros” problem You can then use
hurdle models
or
zero inflated
models to accommodate the extra zeros or
zero augmented
Principles of Econometrics, 3rd Edition Slide16-14
Extensions: excess zeros Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model They would underpredict them too
nbvargr
Is a very useful command
Principles of Econometrics, 3rd Edition
0 2 4 6 k mean = 3.296; overdispersion = 5.439 8 observed proportion poisson prob neg binom prob 10
Slide16-15
Extensions: excess zeros You can then use
hurdle models
or
zero inflated
or
zero augmented
models to accommodate the extra zeros They will also allow you to have a different process driving the value of the strictly positive count and whether the value is zero or strictly positive EXAMPLES: •Number of extramarital affairs versus gender •Number of children before marriage versus religiosity In the continuous case, we have similar models (e.g. Cragg’s Model) and an example is that of size of Insurance Claims from fires versus the age of the building
Principles of Econometrics, 3rd Edition Slide16-16
Extensions: excess zeros You can then use
hurdle models
or
zero inflated
or
zero augmented
models to accommodate the extra zeros
Hurdle Models
A hurdle model is a modified count model in which there are two processes, one generating the zeros and one generating the positive values. The two models are not constrained to be the same. In the hurdle model a binomial probability model governs the binary outcome of whether a count variable has a zero or a positive value. If the value is positive, the "hurdle is crossed," and the conditional distribution of the positive values is governed by a zero-truncated count model.
Example: smokers versus non-smokers, if you are a smoker you will smoke!
Principles of Econometrics, 3rd Edition Slide16-17
Extensions: excess zeros
Hurdle Models
In Stata Joseph Hilbe’s downloadable ado HPLOGIT will work, although it does not allow for two different sets of variables, just two different sets of coefficients Example: smokers versus non-smokers, if you are a smoker you will smoke!
Principles of Econometrics, 3rd Edition Slide16-18
Extensions: excess zeros You can then use
hurdle models
or
zero inflated
or
zero augmented
models to accommodate the extra zeros Zero-inflated models (initially suggested by D. Lambert) attempt to account for excess zeros in a subtly different way.
In this model there are two kinds of zeros, "true zeros" and excess zeros. Zero-inflated models estimate also two equations, one for the count model and one for the excess zero's. The key difference is that the count model allows zeros now. It is not a truncated count model, but allows for “corner solutions” Example: meat eaters (who sometime just did not eat meat that week) versus vegetarians who never ever do
Principles of Econometrics, 3rd Edition Slide16-19
Extensions: excess zeros webuse fish We want to model how many fish are being caught by fishermen at a state park. Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish at all, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish (and admitted it ) so there are excess zeros in the data because of the people that did not fish.
Principles of Econometrics, 3rd Edition Slide16-20
Extensions: excess zeros . histogram count, discrete freq Lots of zeros!
0
Principles of Econometrics, 3rd Edition
50 count 100 150
Slide16-21
Extensions: excess zeros . zip naffairs age male relig , inflate( age male relig ) vuong nolog Zero-inflated Poisson regression Number of obs = 601 Nonzero obs = 150 Zero obs = 451 Inflation model = logit LR chi2(3) = 29.67
Log likelihood = -810.055 Prob > chi2 = 0.0000
naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval] naffairs age .015609 .0038029 4.10 0.000 .0081555 .0230625
male -.1598035 .0686006 -2.33 0.020 -.2942583 -.0253487
relig -.0971114 .0292688 -3.32 0.001 -.1544772 -.0397456
_cons 1.581638 .1577305 10.03 0.000 1.272492 1.890784
inflate age -.019041 .0104841 -1.82 0.069 -.0395895 .0015075
male -.1791471 .1948003 -0.92 0.358 -.5609488 .2026546
relig .2884574 .0841492 3.43 0.001 .1235281 .4533867
_cons .9322364 .3901503 2.39 0.017 .1675558 1.696917
Vuong test of zip vs. standard Poisson: z = 11.66 Pr>z = 0.0000
Vuong test
Principles of Econometrics, 3rd Edition Slide16-22
Extensions: excess zeros . zinb naffairs age male relig , inflate( age male relig ) vuong nolog Zero-inflated negative binomial regression Number of obs = 601 Nonzero obs = 150 Zero obs = 451 Inflation model = logit LR chi2(3) = 8.92
Log likelihood = -726.405 Prob > chi2 = 0.0304
naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval] naffairs age .0258188 .0107692 2.40 0.017 .0047115 .046926
male -.2214886 .1660362 -1.33 0.182 -.5469135 .1039364
relig -.1472717 .0749567 -1.96 0.049 -.2941842 -.0003593
_cons 1.273196 .3874106 3.29 0.001 .5138849 2.032506
inflate age -.014892 .0113465 -1.31 0.189 -.0371308 .0073468
male -.2309299 .2091759 -1.10 0.270 -.6409071 .1790474
relig .274744 .0904315 3.04 0.002 .0975014 .4519865
_cons .6673066 .433002 1.54 0.123 -.1813618 1.515975
/lnalpha -.2743069 .2532933 -1.08 0.279 -.7707527 .2221388
alpha .7600988 .1925279 .4626647 1.248745
Vuong test of zinb vs. standard negative binomial: z = 2.82 Pr>z = 0.0024
Vuong test
Principles of Econometrics, 3rd Edition Slide16-23
Extensions: truncation • Count data can be truncated too (usually at zero) • So
ztp
and
ztnb
can accommodate that • Example: you interview visitors at the recreational site, so they all made at least that one trip •In the continuous case we would have to use the truncreg command
Principles of Econometrics, 3rd Edition Slide16-24
Extensions: truncation • This model works much better and showcases the bias in the previous estimates: . ztp persontrip Travelcost educat income, nolog Zero-truncated Poisson regression Number of obs = 919 LR chi2(3) = 885.68
Prob > chi2 = 0.0000
Log likelihood = -2412.6552 Pseudo R2 = 0.1551
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -1.380461 .0571736 -24.15 0.000 -1.492519 -1.268403
educat -.0170332 .0175026 -0.97 0.330 -.0513376 .0172712
income -.0013521 .000473 -2.86 0.004 -.0022791 -.0004251
_cons 2.278878 .0728394 31.29 0.000 2.136116 2.421641
Principles of Econometrics, 3rd Edition
Smaller now estimated Consumer Surplus
Slide16-25
Extensions: truncation This model works much better and showcases the bias in the previous estimates: • Now accounting for overdispersion . ztnb persontrip Travelcost educat income, nolog Zero-truncated negative binomial regression Number of obs = 919 LR chi2(3) = 263.89
Dispersion = mean Prob > chi2 = 0.0000
Log likelihood = -1866.326 Pseudo R2 = 0.0660
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -1.079011 .068793 -15.68 0.000 -1.213843 -.9441795
educat -.0216377 .0322941 -0.67 0.503 -.084933 .0416576
income -.0016369 .0008563 -1.91 0.056 -.0033152 .0000413
_cons 2.015503 .1344308 14.99 0.000 1.752024 2.278983
/lnalpha -.6368613 .101849 -.8364818 -.4372409
alpha .52895 .053873 .433232 .6458158
Likelihood-ratio test of alpha=0: chibar2(01) = 1092.66 Prob>=chibar2 = 0.000
Principles of Econometrics, 3rd Edition Slide16-26
Extensions: truncation and endogenous stratification • Example: you interview visitors at the recreational site, so they all made at least that one trip • You interview patients at the doctors’ office about how often they visit the doctor • You ask people in George St. how often the go to George St… •Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially
Principles of Econometrics, 3rd Edition Slide16-27
Extensions: truncation and endogenous stratification •Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially •It turns out to be supereasy to deal with a Truncated and Endogenously Stratified Poisson Model (as shown by Shaw, 1988): Simply run a plain Poisson on “Count-1” and that will work (In STATA:
poisson
on the corrected count) It is more complex if there is overdispersion though
Principles of Econometrics, 3rd Edition Slide16-28
Extensions: truncation and endogenous stratification •Supereasy to deal with a Truncated and Endogenously Stratified Poisson Model . poisson persontripminusone Travelcost educat income, nolog Poisson regression Number of obs = 919 LR chi2(3) = 1071.95
Prob > chi2 = 0.0000
Log likelihood = -2474.3262 Pseudo R2 = 0.1780
persontrip~e Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -1.657986 .0620722 -26.71 0.000 -1.779646 -1.536327
educat -.0202144 .0191574 -1.06 0.291 -.0577622 .0173333
income -.0016285 .0005184 -3.14 0.002 -.0026446 -.0006124
_cons 2.191885 .0792934 27.64 0.000 2.036473 2.347298
Principles of Econometrics, 3rd Edition
Much smaller now estimated Consumer Surplus
Slide16-29
Extensions: truncation and endogenous stratification •Endogenously Stratified Negative Binomial Model (as shown by Shaw, 1988; Englin and Shonkwiler, 1995): . nbstrat persontrip Travelcost educat income, nolog Negative Binomial with Endogenous Stratification Number of obs = 919 Wald chi2(3) = 283.49
Log likelihood = -1837.3183 Prob > chi2 = 0.0000
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -1.152915 .0695958 -16.57 0.000 -1.289321 -1.01651
educat -.0229483 .0318753 -0.72 0.472 -.0854228 .0395261
income -.0017368 .0008447 -2.06 0.040 -.0033923 -.0000813
_cons 1.189429 .1561017 7.62 0.000 .8834757 1.495383
/lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959
alpha 1.0974 .1626825 .8206915 1.467406
AIC Statistic = 4.007 BIC Statistic = -6243.307
Deviance = 0.000 Dispersion = 0.000
Even after accounting for overdispersion, CS estimate is relatively low
Principles of Econometrics, 3rd Edition Slide16-30
Extensions: truncation and endogenous stratification •How do we calculate the pseudo-R2 for this model???
. nbstrat persontrip Travelcost educat income, nolog Negative Binomial with Endogenous Stratification Number of obs = 919 Wald chi2(3) = 283.49
Log likelihood = -1837.3183 Prob > chi2 = 0.0000
persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval] Travelcost -1.152915 .0695958 -16.57 0.000 -1.289321 -1.01651
educat -.0229483 .0318753 -0.72 0.472 -.0854228 .0395261
income -.0017368 .0008447 -2.06 0.040 -.0033923 -.0000813
_cons 1.189429 .1561017 7.62 0.000 .8834757 1.495383
/lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959
alpha 1.0974 .1626825 .8206915 1.467406
AIC Statistic = 4.007 BIC Statistic = -6243.307
Deviance = 0.000 Dispersion = 0.000
Principles of Econometrics, 3rd Edition Slide16-31
Extensions: truncation and endogenous stratification •GNBSTRAT will also allow you to model the overdispersion parameter in this case, just as
gnbreg
did for the plain case
Principles of Econometrics, 3rd Edition Slide16-32
NOTE: what is the
exposure
• Count models often need to deal with the fact that the counts may be measured over different observation periods, which might be of different length (in terms of time or some other relevant dimension) For example, the number of accidents are recorded for 50 different intersections. However, the number of vehicles that pass through the intersections can vary greatly. Five accidents for 30,000 vehicles is very different from five accidents for 1,500 vehicles. Count models account for these differences by including the log of the exposure variable in model with coefficient constrained to be one.
The use of exposure is often superior to analyzing rates as response variables as such, because it makes use of the correct probability distributions
Principles of Econometrics, 3rd Edition Slide16-33
16.7.1 Censored Data
Figure 16.3 Histogram of Wife’s Hours of Work in 1975
Principles of Econometrics, 3rd Edition Slide16-34
Having
censored data
means that a substantial fraction of the observations on the dependent variable take a limit value. The regression function is no longer given by (16.30).
1 2
x
(16.30) The least squares estimators of the regression parameters obtained by running a regression of
y
on
x
are biased and inconsistent—least squares estimation fails.
Principles of Econometrics, 3rd Edition Slide16-35
Having
censored data
means that a substantial fraction of the observations on the dependent variable take a limit value. The regression function is no longer given by (16.30).
1 2
x
(16.30) The least squares estimators of the regression parameters obtained by running a regression of
y
on
x
are biased and inconsistent—least squares estimation fails.
Principles of Econometrics, 3rd Edition Slide16-36
With truncation, we only observe the value of the regressors when the dependent variable takes a certain value (usually a positive one instead of zero) With censoring we observe in principle the value of the regressors for everyone, but not the value of the dependent variable for those whose dependent variable takes a value beyond the limit
y
i
* Assume
e i
~
N
0,
y i y i
0 if
y i
*
y i
* if
y i
* 0; 0.
1 2
x
i
e
i
9
x
i
e
i Principles of Econometrics, 3rd Edition
(16.31)
Slide16-38
Create
N
= 200 random values of
x i
that are spread evenly (or uniformly) over the interval [0, 20]. These we will keep fixed in further simulations.
Obtain
N
= 200 random values
e i
from a normal distribution with mean 0 and variance 16.
Create
N
= 200 values of the latent variable. Obtain
N
= 200 values of the observed
y i
using
y i
0
y i
* if
y i
* 0 if
y i
* 0
Principles of Econometrics, 3rd Edition Slide16-39
Figure 16.4 Uncensored Sample Data and Regression Function
Principles of Econometrics, 3rd Edition Slide16-40
Figure 16.5 Censored Sample Data, and Latent Regression Function and Least Squares Fitted Line
Principles of Econometrics, 3rd Edition Slide16-41
i
x
i
(se) (.3706) (.0326)
i
x
i
(se) (1.2055) (.0827)
E
MC Principles of Econometrics, 3rd Edition
1
NSAM
NSAM
m
1
b
(16.32a) (16.32b) (16.33)
Slide16-42
The maximum likelihood procedure is called
Tobit
in honor of James Tobin, winner of the 1981 Nobel Prize in Economics, who first studied this model. The probit probability that
y
i = 0 is:
i
i
0] 1 1 2
x i
L
1 , 2 ,
y i
0 1 1 2
x i
y i
0 2 2 1 2 exp 1 2 2
y i
1 2
x i
2
Principles of Econometrics, 3rd Edition Slide16-43
The maximum likelihood estimator is consistent and asymptotically normal, with a known covariance matrix.
Using the artificial data the fitted values are:
y
i
(se) (1.0970) (.0790)
x
i
(16.34)
Principles of Econometrics, 3rd Edition Slide16-44
Principles of Econometrics, 3rd Edition Slide16-45
x
1 2
x
(16.35) Because the cdf values are positive, the sign of the coefficient does tell the direction of the marginal effect, just not its magnitude. If β 2 > 0, as
x
increases the cdf function approaches 1, and the slope of the regression function approaches that of the latent variable model.
Principles of Econometrics, 3rd Edition Slide16-46
Figure 16.6 Censored Sample Data, and Regression Functions for Observed and Positive
y
values
Principles of Econometrics, 3rd Edition Slide16-47
HOURS
1 2
EDUC
3
EXPER
4
AGE
4
KIDSL
6
e
(16.36)
EDUC
2 26.34
Principles of Econometrics, 3rd Edition Slide16-48
Principles of Econometrics, 3rd Edition Slide16-49
Problem: our sample is not a random sample. The data we observe are “selected” by a systematic process for which we do not account.
Solution: a technique called
Heckit
, named after its developer, Nobel Prize winning econometrician James Heckman.
Principles of Econometrics, 3rd Edition Slide16-50
The econometric model describing the situation is composed of two equations. The first, is the
selection equation
that determines whether the variable of interest is observed.
z
i
*
1 2
w
i
u
i
i
1, ,
N
(16.37)
z
i
1 0
z
i
*
0 otherwise
(16.38)
Principles of Econometrics, 3rd Edition Slide16-51
The second equation is the linear model of interest. It is
y i
1 2
x i
e i i
1, ,
n N
n
(16.39)
i i
*
0
1 2
x
i
1
1 2 2
w
i
w
i
i
i
1, ,
n
(16.40) (16.41)
Principles of Econometrics, 3rd Edition Slide16-52
The estimated “Inverse Mills Ratio” is
i
1
1 2 2
w
i
w
i
The estimating equation is
y
i
1 2
x
i
i
v
i
i
1, ,
n
(16.42)
Principles of Econometrics, 3rd Edition Slide16-53
ln
WAGE
EDUC
.0157
EXPER R
2 .1484
(16.43)
AGE
.0838
EDUC
.3139
KIDS
1.3939
MTR
IMR
Principles of Econometrics, 3rd Edition AGE AGE
.0838
EDUC
.0838
EDUC
.3139
KIDS
.3139
KIDS
1.3939
MTR
1.3939
MTR
Slide16-54
ln
WAGE
EDUC
.0163
EXPER
.8664
IMR
(16.44) The maximum likelihood estimated wage equation is ln
WAGE
EDUC
.0118
EXPER
(t-stat) (2.84) (3.96) (2.87) The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method.
Principles of Econometrics, 3rd Edition Slide16-55
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
Principles of Econometrics, 3rd Edition
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
Slide 16-56
Survival analysis (time-to-event data analysis) Multivariate probit (biprobit, triprobit, mvprobit)
Hoffmann, 2004 for all topics Long, S. and J. Freese for all topics Cameron and Trivedi’s book for count data