Qualitative and Limited Dependent Variable Models Adapted from Vera Tabakova’s notes ECON 4551...
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Transcript of Qualitative and Limited Dependent Variable Models Adapted from Vera Tabakova’s notes ECON 4551...
Chapter 16
Qualitative and Limited Dependent Variable Models
Adapted from Vera Tabakova’s notes
ECON 4551 Econometrics IIMemorial University of Newfoundland
Chapter 16: Qualitative and Limited Dependent Variable Models
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
Slide 16-2Principles of Econometrics, 3rd Edition
16.6 Models for Count Data
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.
Slide16-3Principles of Econometrics, 3rd Edition
16.6 Models for Count Data
If Y is a Poisson random variable, then its probability function is
This choice defines the Poisson regression model for count data.
Slide16-4Principles of Econometrics, 3rd Edition
(16.27) , 0,1,2,!
yef y P Y y y
y
! 1 2 1y y y y
(16.28) 1 2expE Y x
“rate”Also equalTo the variance
16.6.1 Maximum Likelihood Estimation
Slide16-5Principles of Econometrics, 3rd Edition
1 2
1 2
, 0 2 2
ln , ln 0 ln 2 ln 2
L P Y P Y P Y
L P Y P Y P Y
1 2 1 2
ln ln ln ln !!
exp ln !
yeP Y y y y
y
x y x y
1 2 1 2 1 21
ln , exp ln !N
i i i ii
L x y x y
If we observe 3 individuals: one faces no event, the other two two events each:
16.6.2 Interpretation in the Poisson Regression Model
Slide16-6Principles of Econometrics, 3rd Edition
0 0 1 2 0expE y x
0 0expPr , 0,1,2,
!
y
Y y yy
Which is the predicted probabilityof a certain number y of events For someone with characteristics X0
Which is the expected number of occurrences observed
16.6.2 Interpretation in the Poisson Regression Model
Slide16-7Principles of Econometrics, 3rd Edition
(16.29)
2i
ii
E y
x
2
%100 100 %i i
i i
E y E y E y
x x
You may prefer to express this marginal effect as a %:
16.6.2 Interpretation in the Poisson Regression Model
Slide16-8Principles of Econometrics, 3rd Edition
1 2
1 2
1 2
1 2 1 2
1 2
exp
| 0 exp
| 1 exp
exp exp100 % 100 1 %
exp
i i i i
i i i
i i i
i i
i
E y x D
E y D x
E y D x
x xe
x
If there is a dummyInvolved, be careful,remember
Which would be identical to the effect of a dummyIn the log-linear modelwe saw under OLS
Slide16-9Principles of Econometrics, 3rd Edition
Example on Olympic Medals
# Poisson Regressionopen "c:\Program Files\gretl\data\poe\olympics.gdt"smpl year = 88 --restrictgenr lpop = log(pop)genr lgdp = log(gdp)poisson medaltot const lpop lgdpgenr mft = exp($coeff(const)+$coeff(lpop)*median(lpop) \ +$coeff(lgdp)*median(lgdp))*$coeff(lgdp)
Which would give you the marginal effect of GDP for the median countrygenr predicted medals = exp($coeff(const)+$coeff(lpop)*median(lpop) \ +$coeff(lgdp)*median(lgdp))
0.863 medals for those with median GDP and pop
Slide16-10Principles of Econometrics, 3rd Edition
Extensions: overdispersion
Under a plain Poisson the mean of the count is assumed to be equal to the variance (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.
Slide16-11Principles of Econometrics, 3rd Edition
Extensions: overdispersion
_cons .8765791 .1125493 7.79 0.000 .6559865 1.097172 income -.0019933 .0007191 -2.77 0.006 -.0034027 -.0005839 educat -.0307667 .026493 -1.16 0.246 -.0826921 .0211587 Travelcost -.3299655 .0529402 -6.23 0.000 -.4337264 -.2262045 visits Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1321.4665 Pseudo R2 = 0.0210 Prob > chi2 = 0.0000 LR chi2(3) = 56.61Poisson regression Number of obs = 919
Iteration 2: log likelihood = -1321.4665 Iteration 1: log likelihood = -1321.4665 Iteration 0: log likelihood = -1321.4696
. poisson visits Travelcost educat income
_cons 2.144476 .0688666 31.14 0.000 2.0095 2.279452 income -.0014578 .0004404 -3.31 0.001 -.002321 -.0005946 educat -.0206209 .0163568 -1.26 0.207 -.0526797 .0114379 Travelcost -.9570718 .0435943 -21.95 0.000 -1.042515 -.8716285 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -2541.5165 Pseudo R2 = 0.1167 Prob > chi2 = 0.0000 LR chi2(3) = 671.71Poisson regression Number of obs = 919
. poisson persontrip Travelcost educat income, nolog
open C:\Users\rmartinezesp\aaa\bbbECONOMETRICS\Rober\4551\GROSMORNE.dta
Slide16-12Principles of Econometrics, 3rd Edition
Extensions: overdispersion
_cons .8765791 .1125493 7.79 0.000 .6559865 1.097172 income -.0019933 .0007191 -2.77 0.006 -.0034027 -.0005839 educat -.0307667 .026493 -1.16 0.246 -.0826921 .0211587 Travelcost -.3299655 .0529402 -6.23 0.000 -.4337264 -.2262045 visits Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1321.4665 Pseudo R2 = 0.0210 Prob > chi2 = 0.0000 LR chi2(3) = 56.61Poisson regression Number of obs = 919
Iteration 2: log likelihood = -1321.4665 Iteration 1: log likelihood = -1321.4665 Iteration 0: log likelihood = -1321.4696
. poisson visits Travelcost educat income
open C:\Users\rmartinezesp\aaa\bbbECONOMETRICS\Rober\4551\GROSMORNE.dta
educat 938 4.144989 1.120433 1 6 income 966 88.83793 41.94486 20 160 Travelcost 947 .7748112 .6820585 .0036767 7.8652 persontrip 966 3.824017 6.264637 1 91 Variable Obs Mean Std. Dev. Min Max
Slide16-13Principles of Econometrics, 3rd Edition
Extensions: overdispersionopen C:\Users\rmartinezesp\aaa\bbbECONOMETRICS\Rober\4551\GROSMORNE.dta
Slide16-14Principles of Econometrics, 3rd Edition
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
Slide16-15Principles of Econometrics, 3rd Edition
Extensions: negative binomial
Likelihood-ratio test of alpha=0: chibar2(01) = 1006.80 Prob>=chibar2 = 0.000 alpha .3042145 .0220429 .2639388 .3506361 /lnalpha -1.190022 .0724583 -1.332038 -1.048006 _cons 1.994577 .1037 19.23 0.000 1.791329 2.197826 income -.0014357 .0006578 -2.18 0.029 -.0027249 -.0001465 educat -.0218888 .0248201 -0.88 0.378 -.0705353 .0267578 Travelcost -.7135986 .0489137 -14.59 0.000 -.8094676 -.6177295 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -2038.1155 Pseudo R2 = 0.0547Dispersion = mean Prob > chi2 = 0.0000 LR chi2(3) = 236.04Negative binomial regression Number of obs = 919
. nbreg persontrip Travelcost educat income, nolog
Slide16-16Principles of Econometrics, 3rd Edition
Extensions: negative binomial
Slide16-17Principles of Econometrics, 3rd Edition
Extensions: excess zeros
Often the numbers of zeros in the sample cannot be accommodatedproperly 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 or zero augmented models to accommodate the extra zeros
Slide16-18Principles of Econometrics, 3rd Edition
Extensions: excess zeros
Often the numbers of zeros in the sample cannot be accommodatedproperly by a Poisson or Negative Binomial model
They would underpredict them too
nbvargr Is a very useful command
0.2
.4.6
Pro
port
ion
0 2 4 6 8 10k
observed proportion neg binom probpoisson prob
mean = 3.296; overdispersion = 5.439
Slide16-19Principles of Econometrics, 3rd Edition
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
Slide16-20Principles of Econometrics, 3rd Edition
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!
Slide16-21Principles of Econometrics, 3rd Edition
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!
Slide16-22Principles of Econometrics, 3rd Edition
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 sometimes just did not eat meat that week) versus vegetarians who never ever do
Slide16-23Principles of Econometrics, 3rd Edition
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.
Slide16-24Principles of Econometrics, 3rd Edition
Extensions: excess zeros
05
01
001
50F
req
uenc
y
0 50 100 150count
. histogram count, discrete freq
Lots of zeros!
Slide16-25Principles of Econometrics, 3rd Edition
Extensions: excess zeros (sample restricted to count<29)
. histogram count, discrete freq
Lots of zeros!
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25
Den
sity
count
countgamma(0.21421,8.5873)
Test statistic for gamma:
z = -1.384 pvalue = 0.16642
Slide16-26Principles of Econometrics, 3rd Edition
Extensions: excess zeros number of affairs (Fair 1978)
. histogram count, discrete freq
Lots of zeros!
We sill showcase zero-inflated models using STATA now…
LIMDEP has an extra option to run this from Poisson or Negative Binomial dialogs
You would need to program it in GRETL using its maximum likelihood routines (there is a ZIP example on the pdf user’s guide) LIMDEP has an extra option to run this from Poisson or Negative Binomial dialogs
You would need to program it in GRETL using its maximum likelihood routines (there is a ZIP example on the pdf user’s guide)
Slide16-27Principles of Econometrics, 3rd Edition
Extensions: excess zeros (greene22_2.gdt)
Vuong test
Vuong test of zip vs. standard Poisson: z = 11.66 Pr>z = 0.0000 _cons .9322364 .3901503 2.39 0.017 .1675558 1.696917 relig .2884574 .0841492 3.43 0.001 .1235281 .4533867 male -.1791471 .1948003 -0.92 0.358 -.5609488 .2026546 age -.019041 .0104841 -1.82 0.069 -.0395895 .0015075inflate _cons 1.581638 .1577305 10.03 0.000 1.272492 1.890784 relig -.0971114 .0292688 -3.32 0.001 -.1544772 -.0397456 male -.1598035 .0686006 -2.33 0.020 -.2942583 -.0253487 age .015609 .0038029 4.10 0.000 .0081555 .0230625naffairs naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -810.055 Prob > chi2 = 0.0000Inflation model = logit LR chi2(3) = 29.67
Zero obs = 451 Nonzero obs = 150Zero-inflated Poisson regression Number of obs = 601
. zip naffairs age male relig , inflate( age male relig ) vuong nolog
genr ANYAFFAIRS = ( Y>0)
Slide16-28Principles of Econometrics, 3rd Edition
Extensions: excess zeros
Vuong test
Vuong test of zinb vs. standard negative binomial: z = 2.82 Pr>z = 0.0024 alpha .7600988 .1925279 .4626647 1.248745 /lnalpha -.2743069 .2532933 -1.08 0.279 -.7707527 .2221388 _cons .6673066 .433002 1.54 0.123 -.1813618 1.515975 relig .274744 .0904315 3.04 0.002 .0975014 .4519865 male -.2309299 .2091759 -1.10 0.270 -.6409071 .1790474 age -.014892 .0113465 -1.31 0.189 -.0371308 .0073468inflate _cons 1.273196 .3874106 3.29 0.001 .5138849 2.032506 relig -.1472717 .0749567 -1.96 0.049 -.2941842 -.0003593 male -.2214886 .1660362 -1.33 0.182 -.5469135 .1039364 age .0258188 .0107692 2.40 0.017 .0047115 .046926naffairs naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -726.405 Prob > chi2 = 0.0304Inflation model = logit LR chi2(3) = 8.92
Zero obs = 451 Nonzero obs = 150Zero-inflated negative binomial regression Number of obs = 601
. zinb naffairs age male relig , inflate( age male relig ) vuong nolog
Slide16-29Principles of Econometrics, 3rd Edition
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
Slide16-30Principles of Econometrics, 3rd Edition
Extensions: truncation
•
This model works much better and showcases the bias in the previous estimates:
_cons 2.278878 .0728394 31.29 0.000 2.136116 2.421641 income -.0013521 .000473 -2.86 0.004 -.0022791 -.0004251 educat -.0170332 .0175026 -0.97 0.330 -.0513376 .0172712 Travelcost -1.380461 .0571736 -24.15 0.000 -1.492519 -1.268403 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -2412.6552 Pseudo R2 = 0.1551 Prob > chi2 = 0.0000 LR chi2(3) = 885.68Zero-truncated Poisson regression Number of obs = 919
. ztp persontrip Travelcost educat income, nolog
Smaller now estimated Consumer Surplus
Slide16-31Principles of Econometrics, 3rd Edition
Extensions: truncation
• Now accounting for overdispersion
This model works much better and showcases the bias in the previous estimates:
Likelihood-ratio test of alpha=0: chibar2(01) = 1092.66 Prob>=chibar2 = 0.000 alpha .52895 .053873 .433232 .6458158 /lnalpha -.6368613 .101849 -.8364818 -.4372409 _cons 2.015503 .1344308 14.99 0.000 1.752024 2.278983 income -.0016369 .0008563 -1.91 0.056 -.0033152 .0000413 educat -.0216377 .0322941 -0.67 0.503 -.084933 .0416576 Travelcost -1.079011 .068793 -15.68 0.000 -1.213843 -.9441795 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1866.326 Pseudo R2 = 0.0660Dispersion = mean Prob > chi2 = 0.0000 LR chi2(3) = 263.89Zero-truncated negative binomial regression Number of obs = 919
. ztnb persontrip Travelcost educat income, nolog
Slide16-32Principles of Econometrics, 3rd Edition
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
Slide16-33Principles of Econometrics, 3rd Edition
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
Slide16-34Principles of Econometrics, 3rd Edition
Extensions: truncation and endogenous stratification
• Supereasy to deal with a Truncated and Endogenously Stratified Poisson Model
_cons 2.191885 .0792934 27.64 0.000 2.036473 2.347298 income -.0016285 .0005184 -3.14 0.002 -.0026446 -.0006124 educat -.0202144 .0191574 -1.06 0.291 -.0577622 .0173333 Travelcost -1.657986 .0620722 -26.71 0.000 -1.779646 -1.536327 persontrip~e Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -2474.3262 Pseudo R2 = 0.1780 Prob > chi2 = 0.0000 LR chi2(3) = 1071.95Poisson regression Number of obs = 919
. poisson persontripminusone Travelcost educat income, nolog
Much smaller now estimated Consumer Surplus
Slide16-35Principles of Econometrics, 3rd Edition
Extensions: truncation and endogenous stratification
• Endogenously Stratified Negative Binomial Model (as shown by Shaw, 1988; Englin and Shonkwiler, 1995):
Deviance = 0.000 Dispersion = 0.000AIC Statistic = 4.007 BIC Statistic = -6243.307 alpha 1.0974 .1626825 .8206915 1.467406 /lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959 _cons 1.189429 .1561017 7.62 0.000 .8834757 1.495383 income -.0017368 .0008447 -2.06 0.040 -.0033923 -.0000813 educat -.0229483 .0318753 -0.72 0.472 -.0854228 .0395261 Travelcost -1.152915 .0695958 -16.57 0.000 -1.289321 -1.01651 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1837.3183 Prob > chi2 = 0.0000 Wald chi2(3) = 283.49Negative Binomial with Endogenous Stratification Number of obs = 919
. nbstrat persontrip Travelcost educat income, nolog
Even after accounting for overdispersion, CS estimate is relatively low
Slide16-36Principles of Econometrics, 3rd Edition
Extensions: truncation and endogenous stratification
• How do we calculate the pseudo-R2 for this model???
Deviance = 0.000 Dispersion = 0.000AIC Statistic = 4.007 BIC Statistic = -6243.307 alpha 1.0974 .1626825 .8206915 1.467406 /lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959 _cons 1.189429 .1561017 7.62 0.000 .8834757 1.495383 income -.0017368 .0008447 -2.06 0.040 -.0033923 -.0000813 educat -.0229483 .0318753 -0.72 0.472 -.0854228 .0395261 Travelcost -1.152915 .0695958 -16.57 0.000 -1.289321 -1.01651 persontrip Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1837.3183 Prob > chi2 = 0.0000 Wald chi2(3) = 283.49Negative Binomial with Endogenous Stratification Number of obs = 919
. nbstrat persontrip Travelcost educat income, nolog
Slide16-37Principles of Econometrics, 3rd Edition
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
Slide16-38Principles of Econometrics, 3rd Edition
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
16.7 Limited Dependent Variables
16.7.1 Censored Data
Figure 16.3 Histogram of Wife’s Hours of Work in 1975
Slide16-39Principles of Econometrics, 3rd Edition
16.7.1 Censored Data
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).
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.
Slide16-40Principles of Econometrics, 3rd Edition
(16.30) 1 2|E y x x
16.7.1 Censored Data
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).
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.
Slide16-41Principles of Econometrics, 3rd Edition
(16.30) 1 2|E y x x
Censoring versus Truncation
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
16.7.2 A Monte Carlo Experiment
We give the parameters the specific values and
Assume
Slide16-43Principles of Econometrics, 3rd Edition
(16.31)
1 29 and 1.
*1 2 9i i i i iy x e x e
2~ 0, 16 .ie N
*
* *
0 if 0;
if 0.
i i
i i i
y y
y y y
16.7.2 A Monte Carlo Experiment
Create N = 200 random values of xi that are spread evenly (or
uniformly) over the interval [0, 20]. These we will keep fixed in
further simulations.
Obtain N = 200 random values ei 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 yi using
Slide16-44Principles of Econometrics, 3rd Edition
*
* *
0 if 0
if 0
i
i
i i
yy
y y
16.7.2 A Monte Carlo Experiment
Figure 16.4 Uncensored Sample Data and Regression FunctionSlide16-45
16.7.2 A Monte Carlo Experiment
Figure 16.5 Censored Sample Data, and Latent Regression Function and Least Squares Fitted Line
Slide16-46Principles of Econometrics, 3rd Edition
16.7.2 A Monte Carlo Experiment
Slide16-47Principles of Econometrics, 3rd Edition
(16.32a)ˆ 2.1477 .5161
(se) (.3706) (.0326)i iy x
(16.32b)ˆ 3.1399 .6388
(se) (1.2055) (.0827)i iy x
(16.33) ( )1
1 NSAM
MC k k mm
E b bNSAM
OLS for all the 200 observations predicts:
OLS for only the 100 positive observations (y >0) predicts:
Our Monte Carlo experiment resamples 200 times and on average predicts on average:
16.7.2 A Monte Carlo Experiment
Slide16-48Principles of Econometrics, 3rd Edition
(16.32a)ˆ 2.1477 .5161
(se) (.3706) (.0326)i iy x
(16.32b)ˆ 3.1399 .6388
(se) (1.2055) (.0827)i iy x
(16.33) ( )1
1 NSAM
MC k k mm
E b bNSAM
OLS for all the 200 observations predicts:
OLS for only the 100 positive observations (y >0) predicts:
Our Monte Carlo experiment resamples 200 times and on average predicts on average:
16.7.3 Maximum Likelihood Estimation
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 yi = 0 is:
Slide16-49Principles of Econometrics, 3rd Edition
1 20 [ 0] 1i i iP y P y x
1
221 2 21 2 1 22
0 0
1, , 1 2 exp
2i i
ii i
y y
xL y x
16.7.3 Maximum Likelihood Estimation
The maximum likelihood estimator is consistent and asymptotically
normal, with a known covariance matrix.
Using the artificial data the fitted values are:
Slide16-50Principles of Econometrics, 3rd Edition
(16.34)10.2773 1.0487
(se) (1.0970) (.0790)i iy x
16.7.3 Maximum Likelihood Estimation
Slide16-51Principles of Econometrics, 3rd Edition
16.7.3 Maximum Likelihood Estimation
Slide16-52Principles of Econometrics, 3rd Edition
You can run this experiment yourselves in GRETL
open "c:\Program Files\gretl\data\poe\tobit.gdt"smpl 1 200genr xs = 20*uniform()loop 1000 --progressive genr y = -9 + 1*xs + 4*normal() genr yi = y > 0#which is a handy command to generate dummies! genr yc = y*yi ols yc const xs --quiet genr b1s = $coeff(const) genr b2s = $coeff(xs) store coeffs.gdt b1s b2sendloop
16.7.3 You can repeat the experiment using only the positive values:
Slide16-53Principles of Econometrics, 3rd Edition
open "c:\Program Files\gretl\data\poe\tobit.gdt"genr xs = 20*uniform()genr idx = 1matrix A = zeros(1000,3)loop 1000 --quiet smpl --full genr y = -9 + 1*xs + 4*normal() smpl y > 0 --restrict ols y const xs --quiet genr b1s = $coeff(const) genr b2s = $coeff(xs) matrix A[idx,1]=idx matrix A[idx,2]=b1s matrix A[idx,3]=b2s genr idx = idx + 1endloop
# The matrix A contains all 1000 sets of coefficients # bb finds the column mean of A
matrix bb = meanc(A) bb
16.7.3 You can repeat the experiment using only the positive values:
Slide16-54Principles of Econometrics, 3rd Edition
# The matrix A contains all 1000 sets of coefficients # bb finds the column mean of A
matrix bb = meanc(A) bb
Note that the first cell refers to the average of the “case” number (500.5, since there are 1000 cases numbered 1 to 1000)
16.7.4 Tobit Model Interpretation
Because the cdf values are positive, the sign of the coefficient does
tell the direction of the marginal effect, just not its magnitude. If
β2 > 0, as x increases the cdf function approaches 1, and the slope of
the regression function approaches that of the latent variable model.
Slide16-55Principles of Econometrics, 3rd Edition
(16.35) 1 2
2
|E y x x
x
16.7.4 Tobit Model Interpretation
Figure 16.6 Censored Sample Data, and Regression Functions for Observed and Positive y values
Slide16-56Principles of Econometrics, 3rd Edition
16.7.5 An Example
Slide16-57Principles of Econometrics, 3rd Edition
(16.36)1 2 3 4 4 6HOURS EDUC EXPER AGE KIDSL e
2 73.29 .3638 26.34
E HOURS
EDUC
16.7.5 An Example
Slide16-58Principles of Econometrics, 3rd Edition
Tobit example in GRETL
#Tobit
open "c:\Program Files\gretl\data\poe\mroz.gdt"
tobit hours const educ exper age kidsl6
genr H_hat = $coeff(const)+$coeff(educ)*mean(educ) +$coeff(exper)*mean(exper) \
+$coeff(age)*mean(age)+$coeff(kidsl6)*1
genr z = cnorm(H_hat/$sigma)
genr pred = z*$coeff(educ)
smpl hours > 0 --restrict
ols hours const educ exper age kidsl6
smpl --full
ols hours const educ exper age kidsl6
Slide16-59Principles of Econometrics, 3rd Edition
16.7.6 Sample Selection
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 (not
original author), Nobel Prize winning econometrician James
Heckman.
Slide16-60Principles of Econometrics, 3rd Edition
16.7.6a The Econometric Model
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.
Slide16-61Principles of Econometrics, 3rd Edition
(16.37)*1 2 1, ,i i iz w u i N
(16.38)
*1 0
0 otherwise
i
i
zz
16.7.6a The Econometric Model
The second equation is the linear model of interest. It is
Slide16-62Principles of Econometrics, 3rd Edition
(16.39)
(16.40)
1 2 1, ,i i iy x e i n N n
(16.41)
*1 2| 0 1, ,i i i iE y z x i n
1 2
1 2
ii
i
w
w
16.7.6a The Econometric Model
The estimated “Inverse Mills Ratio” is
The estimating equation is
Slide16-63Principles of Econometrics, 3rd Edition
(16.42)
1 2
1 2
ii
i
w
w
1 2 1, ,i i i iy x v i n
16.7.6b Heckit Example: Wages of Married Women
Slide16-64Principles of Econometrics, 3rd Edition
(16.43) 2ln .4002 .1095 .0157 .1484
(t-stat) ( 2.10) (7.73) (3.90)
WAGE EDUC EXPER R
1 1.1923 .0206 .0838 .3139 1.3939
(t-stat) ( 2.93) (3.61) ( 2.54) ( 2.26)
P LFP AGE EDUC KIDS MTR
1.1923 .0206 .0838 .3139 1.3939
1.1923 .0206 .0838 .3139 1.3939
AGE EDUC KIDS MTRIMR
AGE EDUC KIDS MTR
OLS yields
Probit on dummy indicating “being in the labour force” yields:
From here predict the inverse Mill’s ratio:
16.7.6b Heckit Example: Wages of Married Women
The maximum likelihood estimated wage equation is
The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method.
Slide16-65Principles of Econometrics, 3rd Edition
(16.44)
ln .8105 .0585 .0163 .8664
(t-stat) (1.64) (2.45) (4.08) ( 2.65)
(t-stat-adj) (1.33) (1.97) (3.88) ( 2.17)
WAGE EDUC EXPER IMR
ln .6686 .0658 .0118
(t-stat) (2.84) (3.96) (2.87)
WAGE EDUC EXPER
Heckit example in GRETL
#Heckit
open "c:\Program Files\gretl\data\poe\mroz.gdt“
genr kids = (kidsl6+kids618>0)
logs wage
list X = const educ exper
list W = const mtr age kids educ
probit lfp W
genr ind = $coeff(const) + $coeff(age)*age + $coeff(educ)*educ + $coeff(kids)*kids + $coeff(mtr)*mtr
#Predict the inverse Mill’s ratio:
genr lambda = dnorm(ind)/cnorm(ind)
ols l_wage X lambda
heckit l_wage X ; lfp W --two-step
Slide16-66Principles of Econometrics, 3rd Edition
Keywords
Slide 16-67Principles 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
Further models
Survival analysis (time-to-event data analysis)
Multivariate probit (biprobit, triprobit, mvprobit)
References
Hoffmann, 2004 for all topicsLong, S. and J. Freese for all topicsCameron and Trivedi’s book for count
data
Agresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley.