Econometrics 2 - Lecture 2
Models with Limited Dependent Variables


Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
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Hackl, Econometrics 2, Lecture 2
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Cases of Limited Dependent Variable
nTypical situations: functions of explanatory variables are to be explained
nDichotomous dependent variable, e.g., ownership of a car (yes/no), employment status
(employed/unemployed), etc.
nOrdered response, e.g., qualitative assessment (good/average/bad), working status
(full-time/part-time/not working), etc.
nMultinomial response, e.g., trading destinations (Europe/Asia/Africa), transportation means
(train/bus/car), etc.
nCount data, e.g., number of orders a company receives in a week, number of patents granted to a
company in a year
nCensored data, e.g., expenditures for durable goods, duration of study with drop outs
n
n
n
n
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Example: Car Ownership and Income
nWhat is the probability that a randomly chosen household owns a car?
nSample of N=32 households
qProportion of car owning households:19/32 = 0.59
nEstimated probability for owning a car: 0.59
nBut: the probability will differ for rich and poor!
nThe sample data contains income information:
qYearly income: average EUR 20.524, minimum EUR 12.000, maximum EUR 32.517
qProportion of car owning households among the 16 households with less than EUR 20.000 income: 9/16
= 0.56
qProportion of car owning households among the 16 households with more than EUR 20.000 income:
10/16 = 0.63
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Car Ownership and Income, cont’d
nHow can probability – or prediction – of car ownership take the income of a household into
account?
nNotation: N households
qdummy yi for car ownership; yi =1: household i has car
qincome xi2
nFor predicting yi  – or of P{yi =1} – , a model is needed that takes the income into account
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Modelling Car Ownership
nHow is car ownership related to the income of a household?
1.Linear regression yi = xi’β + εi = β1+ β2xi2 + εi
nWith E{εi|xi} = 0, the model yi = xi’β + εi gives
n P{yi =1|xi} = xi’β
n due to E{yi|xi} = 1*P{yi =1|xi} + 0*P{yi =0|xi} = P{yi =1|xi}
nModel yi = xi’β + εi: xi’β can be interpreted as P{yi =1|xi}!
nProblems:
qxi’β not necessarily in [0,1]
qError terms: for a given xi
nεi has only two values, viz. 1- xi’β and xi’β
nV{εi |xi} = xi’β(1- xi’β), heteroskedastic, dependent upon β
nModel for y actually is specifying the probability that y = 1 as a function of x
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Modelling Car Ownership, cont’d
2.Use of a function G(xi,β) with values in the interval [0,1]
n P{yi =1|xi} = E{yi|xi} = G(xi,β)
nThe probability that yi =1, i.e., the household owns a car, depends on the income (and other
characteristics, e.g., family size)
nUse for G(xi,β) the standard logistic distribution function
q
q
n L(z) fulfils limz→ -∞ L(z) = 0, limz→ ∞ L(z) = 1
nInterpretation:
qFrom P{yi =1|xi} = pi = exp{xi’β}/(1+exp{xi’β}) follows
q
q
qAn increase of xi2 by 1 results in a relative change of the odds pi/(1- pi) by β2 or by 100β2%;
cf. the notion semi-elasticity
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Car Ownership and Income, cont’d
nE.g., P{yi =1|xi} = 1/(1+exp(-zi)) with z = -0.5 + 1.1*x, the income in EUR 1000 per month
nIncreasing income is associated with an increasing probability of owning a car: z goes up by 1.1
for every additional EUR 1000
nFor a person with an income of EUR 1000, z = 0.6 and the probability of owning a car is
1/(1+exp(-0.6)) = 0.65
nThe standard logistic distribution function, with z on the horizontal and F(z) on the vertical
axis
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x
z
P{y =1|x}
1000
0.6
0.646
2000
1.7
0.846
3000
2.8
0.943

Odds
nThe odds in favour of an event is the ratio of a pair of numbers, the first (the second)
representing the relative likelihood that the event will happen (will not happen)
nIf p is the probability in favour of the event, the probability against the event therefore being
1-p, the odds of the event are the quotient
n
nOdds are read as “1 to p/(1-p)” or “1:p/(1-p)”
n
n
n
n
nThe logarithm of the odds of the probability p is called the logit of p
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p
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
odds
1:9
1:4
1:2.3
1:1.5
1:1
1:0.67
1:0.43
1:0.25
1:0.11
p/(1-p)
0.11
0.25
0.43
0.67
1
1.5
2.33
4
9

Odds: Example
nExample: the odds that a randomly chosen day of the week is a Sunday are 1:6 (say “one to six”)
because p = P{Sunday} = 1/7 = 0.143, p/(1-p) = (1/7)/(6/7) = 1/6; the odds are 1:6
nIn bookmakers language: odds are not in favour but against
nThe bookmaker would say
qThe odds that a randomly chosen day of the week is a Sunday are 6:1
qThe odds that Czech Republic men's national ice hockey team wins the World Championship is 2:1;
i.e., the probability is considered to be 0.333
q
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
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Binary Choice Models
nModel for probability P{yi =1|xi}, function of K (numerical or categori-cal) explanatory variables
xi and unknown parameters β, such as
n E{yi|xi} = P{yi =1|xi} = G(xi,β)
nTypical functions G(xi,β): distribution functions (cdf’s) F(xi’β)
nProbit model: standard normal distribution function; V{z} = 1
n
n
nLogit model: standard logistic distribution function; V{z}=π2/3=1.812
q
q
nLinear probability model (LPM)
n
n
q
q
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Linear Probability Model (LPM)
nAssumes that
n P{yi =1|xi} = xi’β  for  0 ≤ xi’β ≤ 1
n but sets
n P{yi =1|xi} = 0  for  xi’β < 0
n P{yi =1|xi} = 1  for  xi’β > 1
nTypically, the model is estimated by OLS, ignoring the probability restrictions
nStandard errors should be adjusted using heteroskedasticity-consistent (White) standard errors
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Probit Model: Standardization
nE{yi|xi} = P{yi =1|xi} = G(xi,β): assume G(.) to be the distribution function of N(0, σ2)
n
n
nGiven xi, the ratio β/σ2 determines P{yi =1|xi}
nStandardization restriction s2 = 1: allows unique estimates for β
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Probit vs Logit Model
nDifferences between the probit and the logit model:
qShape of distribution is slightly different, particularly in the tails.
qScaling of the distribution is different: The implicit variance for ei in the logit model is p2/3
= (1.81)2, while 1 for the probit model
qProbit model is relatively easy to extend to multivariate cases using  the multivariate normal or
conditional normal distribution
nIn practice, the probit and logit model produce quite similar results
qThe scaling difference makes the values of b not directly comparable across the two models, while
the signs are typically the same
qThe estimates in the logit model are roughly a factor p/Ö3 »1.81 larger than those in the probit
model
q
q
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Interpretation of Coefficients
nFor assessing the effect of changing xk the
nCoefficient bk
nis of interest, but also related characteristics such as
nSign of bk
nSlope, i.e., the “average” marginal effect ¶F(xi’b)/¶xik
q
q
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Binary Choice Models: Marginal Effects
nLinear regression models: βk is the marginal effect of a change in xk
nFor E{yi|xi} = F(xi’β):
n
n
n with density function f(.)
nThe effect of changing the regressor xk depends upon xi’β, the shape of F, and βk
nThe marginal effect of changing xk
qProbit model: ϕ(xi’β) βk, with standard normal density function ϕ
qLogit model: L(xi’β)[1 - L(xi’β)] βk
qLinear probability model
q
q
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Binary Choice Models: Slopes
nInterpretation of the effect of a change in xk
n“Slope”, i.e., the gradient of E{yi|xi} at the sample means of the regressors
n
n
nFor a dummy variable D: marginal effect is calculated as the difference of probabilities
P{yi =1|x(d),D=1} – P{yi =1|x(d),D=0}; x(d) stands for the sample means of all regressors except D
nFor the logit model:
q
q
q
qThe coefficient βk is the relative change of the odds when increasing xk by 1 unit
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Binary Choice Models: Estimation
nTypically, binary choice models are estimated by maximum likelihood
nLikelihood function, given N observations (yi, xi)
n L(β) = Πi=1N P{yi =1|xi;β}yi P{yi =0|xi;β}1-yi
n          = Πi F(xi’β)yi (1- F(xi’β))1-yi
nMaximization via the log-likelihood function
n ℓ(β) = log L(β) = Si yi log F(xi’β) + Si (1-yi) log (1-F(xi’β))
nFirst-order conditions of the maximization problem
n
n
nei: generalized residuals
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Generalized Residuals
nThe first-order conditions allow to define generalized residuals
nFrom
n
n
nfollows that the generalized residuals ei can assume two values:
qei = f(xi’b)/F(xi’b) if yi =1
qei = - f(xi’b)/(1-F(xi’b)) if yi =0
n b are the estimates of β
nGeneralized residuals are orthogonal to each regressor; cf. the first-order conditions of OLS
estimation
n
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Estimation of Logit Model
nFirst-order condition of the maximization problem
n
n
n
n gives [due to P{yi =1|xi} = L(xi,β)]
n
n
nFrom Si    xi = Siyixi follows – given one regressor is an intercept –:
qThe sum of estimated probabilities Si     equals the observed frequency Siyi
nSimilar results for the probit model, due to similarity of logit and probit functions
n
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Properties of ML Estimators
nConsistent
nAsymptotically efficient
nAsymptotically normally distributed
nThese properties require that the assumed distribution is correct
nCorrect shape
nNo autocorrelation and/or heteroskedasticity
nNo dependence between errors and regressors
nNo omitted regressors
n
n
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Goodness-of-Fit Measures
nConcepts
nComparison of the maximum likelihood of the model with that of the naïve model, i.e., a model with
only an intercept, no regressors
qPseudo-R2
qMcFadden R2
nIndex based on proportion of correctly predicted observations
qHit rate
n
n
n
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McFadden R2
nBased on log-likelihood function
nℓ(b) = ℓ1: maximum log-likelihood of the model to be assessed
nℓ0: maximum log-likelihood of the naïve model, i.e., a model with only an intercept; ℓ0 ≤ ℓ1 and
ℓ0, ℓ1 < 0
qThe larger ℓ1 - ℓ0, the more contribute the regressors
qℓ1 = ℓ0, if all slope coefficients are zero
qℓ1 = 0, if yi is exactly predicted for all i
nPseudo-R2: a number in [0,1), defined by
n
n
n
nMcFadden R2: a number in [0,1], defined by
n
nBoth are 0 if ℓ1 = ℓ0, i.e., all slope coefficients are zero
nMcFadden R2 attains the upper limit if ℓ1 = 0
n
n
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Naïve Model: Calculation of ℓ0
nMaximum log-likelihood function of the naïve model, i.e., a model with only an intercept: ℓ0
nLog-likelihood function (cf. urn experiment)
n log L(p) = N1 log(p) + (N – N1) log (1-p)
n with N1 = Siyi, i.e., the observed frequency
nMaximum likelihood estimator for p is N1/N
nMaximum log-likelihood of the naïve model
n ℓ0 = N1 log(N1/N) + (N – N1) log (1 – N1/N)
n
n
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Hit Rate
nComparison of correct and incorrect predictions
nPredicted outcome
n ŷi = 1 if xi’b > 0
n     = 0 if xi’b ≤ 0
nCross-tabulation of actual and predicted outcome
nProportion of incorrect predictions
n wr1 = (n01+n10)/N
nHit rate: 1 - wr1
q     proportion of correct predictions
nComparison with naive model:
qPredicted outcome of naïve model
q ŷi = 1 if    = N1/N > 0.5, ŷi = 0 if     ≤ 0.5 (for all i)
qRp2= 1 – wr1/wr0
q with wr0 = 1 -     if      > 0.5, wr0 =     if      ≤ 0.5 in order to avoid Rp2 < 0
n
n
n
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ŷ = 0
ŷ  = 1
Σ
y = 0
n00
n01
N0
y = 1
n10
n11
N1
Σ
n0
n1
N

Example: Effect of Teaching Method
nStudy by Spector & Mazzeo (1980); see Greene (2003), Chpt.21
nPersonalized System of Instruction: new teaching method in economics; has it an effect on student
performance in later courses?
nData:
qGRADE (0/1): indicator whether grade was higher than in principal course
qPSI (0/1): participation in program with new teaching method
qGPA: grade point average
qTUCE: score on a pretest, entering knowledge
n32 observations
n
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Effect of Teaching Method, cont’d
nLogit model for GRADE, GRETL output
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Model 1: Logit, using observations 1-32
Dependent variable: GRADE
  Coefficient Std. Error z-stat Slope*
const -13.0213 4.93132 -2.6405
GPA 2.82611 1.26294 2.2377 0.533859
TUCE 0.0951577 0.141554 0.6722 0.0179755
PSI 2.37869 1.06456 2.2344 0.456498
Mean dependent var 0.343750 S.D. dependent var 0.188902
McFadden R-squared 0.374038 Adjusted R-squared 0.179786
Log-likelihood -12.88963 Akaike criterion 33.77927
Schwarz criterion 39.64221 Hannan-Quinn 35.72267
*Number of cases 'correctly predicted' = 26 (81.3%)
f(beta'x) at mean of independent vars = 0.189
Likelihood ratio test: Chi-square(3) = 15.4042 [0.0015]
Predicted
                   0    1
  Actual 0  18    3
         1    3    8

Effect of Teaching Method, cont’d
nLogit model for GRADE, actual and fitted values of 32 observations
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Effect of Teaching Method, cont’d
nComparison of the LPM, logit, and probit model for GRADE
nEstimated models: coefficients and their standard errors
n
n
n
n
n
n
n
n
nCoefficients of logit model: due to larger variance, larger by factor √(π2/3)=1.81 than that of
the probit model
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LPM
Logit
Probit
coeff
s.e.
coeff
s.e.
coeff
s.e.
const
-1.498
0.524
-13.02
4.931
-7.452
2.542
GPA
0.464
0.162
2.826
1.263
1.626
0.694
TUCE
0.010
0.019
0.095
0.142
0.052
0.084
PSI
0.379
0.139
2.379
1.065
1.426
0.595

Effect of Teaching Method, cont’d
nGoodness of fit measures for the logit model
nWith N1 = 11 and N = 32
n ℓ0 = 11 log(11/32) + 21 log(21/32) = - 20.59
nAs     = N1/N = 0.34 < 0.5: the proportion wr0 of incorrect predictions with the naïve model is
n wr0 =      = 11/32 = 0.34
nFrom the GRETL output: ℓ0 = -12.89, wr1 = 6/32
nGoodness of fit measures
nRp2 = 1 – wr1/wr0 = 1 – 6/11 = 0.45
nMcFadden R2 = 1 – (-12.89)/(-20.59) = 0.374
n
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Example: Utility of Car Owning
nLatent variable yi*: utility difference between owning and not owning a car; unobservable (latent)
nDecision on owning a car
qyi* > 0: in favor of car owning
qyi* ≤ 0: against car owning
nyi* depends upon observed characteristics (like income) and unobserved characteristics εi
n yi* = xi’β + εi
nObservation yi = 1 (i.e., owning car) if yi* > 0
n P{yi =1} = P{yi* > 0} = P{xi’β + εi > 0} = 1 – F(-xi’β) = F(xi’β)
n last step requires a symmetric distribution function F(.)
nLatent variable model: based on a latent variable that represents underlying behavior
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Latent Variable Model
nModel for the latent variable yi*
n yi* = xi’β + εi
n yi*: not necessarily a utility difference
nεi‘s are independent of xi’s
nεi has standardized distribution
qProbit model if εi has standard normal distribution
qLogit model if εi has standard logistic distribution
nObservations
qyi = 1 if yi* > 0
qyi = 0 if yi* ≤ 0
nML estimation
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Binary Choice Models in GRETL
nModel > Nonlinear Models > Logit > Binary
nEstimates the specified model using error terms with standard logistic distribution
nModel > Nonlinear Models > Probit > Binary
nEstimates the specified model using error terms with standard normal distribution
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Multiresponse Models
nModel for explaining the choice between discrete outcomes
nExamples:
a.Working status (full-time/part-time/not working), qualitative assessment (good/average/bad), etc.
b.Trading destinations (Europe/Asia/Africa), transportation means (train/bus/car), etc.
nMultiresponse models describe the probability of each of these outcomes, as a function of
variables like
qperson-specific characteristics
qalternative-specific characteristics
nTypes of multiresponse models (cf. above examples)
qOrdered response models: outcomes have a natural ordering
qMultinomial (unordered) models: ordering of outcomes is arbitrary
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Example: Credit Rating
nCredit rating: numbers, indicating experts’ opinion about (a firm’s) capacity to satisfy financial
obligations, e.g., credit-worthiness
nStandard & Poor's rating scale: AAA, AA+, AA, AA-, A+, A, A-, BBB+, BBB, BBB-, BB+, BB, BB-, B+,
B, B-, CCC+, CCC, CCC-, CC, C, D
nVerbeek‘s data set CREDIT
qCategories “1“, …,“7“ (highest)
qInvestment grade with alternatives “1” (better than category 3) and “0” (category 3 or less, also
called “speculative grade“)
nExplanatory variables, e.g.,
qFirm sales
qEbit, i.e., earnings before interest and taxes
qRatio of working capital to total assets
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Ordered Response Model
nChoice between M alternatives
nObserved alternative for sample unit i: yi
nLatent variable model
n yi* = xi’β + εi
n with K-vector of explanatory variables xi
q yi = j  if γj-1 < yi* ≤ γj for j = 0,…,M
nM+1 boundaries γj, j = 0,…,M, with γ0 = -∞, …, γM = ∞
nεi‘s are independent of xi’s
nεi typically follow the
qstandard normal distribution: ordered probit model
qstandard logistic distribution: ordered logit model
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Example: Willingness to Work
n„How much would you like to work?“
nPotential answers of individual i: yi = 1 (not working), yi = 2 (part time), yi = 3 (full time)
nMeasure of the desired labour supply
nDependent upon factors like age, education level, husband‘s income
nOrdered response model with M = 3
n yi* = xi’β + εi
n with
q yi = 1  if yi* ≤ 0
q yi = 2  if 0 < yi* ≤ γ
q yi = 3  if yi* > γ
nεi‘s with distribution function F(.)
nyi* stands for “willingness to work” or “desired hours of work”
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Willingness to Work, cont’d
nIn terms of observed quantities:
n P{yi = 1 |xi} = P{yi* ≤ 0 |xi} = F(- xi’β)
n P{yi = 3 |xi} = P{yi* > γ |xi} = 1 - F(γ - xi’β)
n P{yi = 2 |xi} = F(γ - xi’β) – F(- xi’β)
nUnknown parameters: γ and β
nStandardization: wrt location (γ = 0) and scale (V{εi} = 1)
nML estimation
nInterpretation of parameters β
nWrt yi*: willingness to work increases with larger xk for positive βk
nWrt probabilities P{yi = j |xi}, e.g., P{yi = 3 |xi} increases and P{yi = 1 |xi} decreases with
larger xk for positive βk
n
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Example: Credit Rating
nVerbeek‘s data set CREDIT: 921 observations for US firms' credit ratings in 2005, including firm
characteristics
nRating models:
1.Ordered logit model for assignment of categories “1“, …,“7“ (highest)
2.Binary logit model for assignment of “investment grade” with alternatives “1” (better than
category 3) and “0” (category 3 or less, also called “speculative grade“)
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Credit Rating, cont’d
nVerbeek‘s data set CREDIT
nRatings and characteristics for 921 firms: summary statistics
n
n
n
n
n
n
n
n
n_____________________
nBook leverage: ratio of debts to assets
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Credit Rating, cont’d
nVerbeek, Table 7.5.
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Ordered Response Model: Estimation
nLatent variable model
n yi* = xi’β + εi
n with explanatory variables xi
q yi = j  if γj-1 < yi* ≤ γj for j = 0,…,M
nML estimation of β1, …, βK and γ1, …, γM-1
nLog-likelihood function in terms of probabilities
nNumerical optimization
nML estimators are
qConsistent
qAsymptotically efficient
qAsymptotically normally distributed
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Multinomial Models
nChoice between M alternatives without natural order
nObserved alternative for sample unit i: yi
n“Random utility” framework: Individual i
nattaches utility levels Uij to each of the alternatives, j = 1,…, M,
nchooses the alternative with the highest utility level
nUtility levels Uij, j = 1,…, M, as a function of characteristics xij
n Uij = xij’β + εij
nerror terms εij follow the Type I extreme value distribution:
n
n
n for j = 1, …, M
nand Σj P{yi = j} = 1
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Variants of the Logit Model
nFor setting the location: constraint xi1’b = 0 or exp{xi1’b} = 1
nConditional logit model: for j = 1, …, M
n
n
nAlternative-specific characteristics xij
nE.g., mode of transportation is affected by travel costs, travel duration, etc.
nMultinomial logit model: for j = 1, …, M
n
n
nPerson-specific characteristics xi
nE.g., mode of transportation is affected by income, gender, etc.
n
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Multinomial Logit Model
nThe term “multinomial logit model” is also used for both the
nthe conditional logit model
nthe multinomial logit model (see above)
nand also the mixed logit model: combines
qAlternative-specific characteristics and
qPerson-specific characteristics
n
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Independence of Errors
nIndependence of the error terms εij implies independent utility levels of alternatives
nIndependence assumption may be restrictive
nExample: High utility of alternative „travel with red bus“ implies high utility of „travel with
blue bus“
nImplies that the odds ratio of two alternatives does not depend upon the number of alternatives:
“independence of irrelevant alternatives” (IIA)
n
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Multiresponse Models in GRETL
nModel > Nonlinear Models > Logit > Ordered...
nEstimates the specified model using error terms with standard logistic distribution, assuming
ordered alternatives for responses
nModel > Nonlinear Models > Logit > Multinomial...
nEstimates the specified model using error terms with standard logistic distribution, assuming
alternatives without order
nModel > Nonlinear Models > Probit > Ordered...
nEstimates the specified model using error terms with standard normal distribution, assuming
ordered alternatives
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Models  for Count Data
nDescribe the number of times an event occurs, depending upon certain characteristics
nExamples:
nNumber of visits in the library per week
nNumber of misspellings in an email
nNumber of applications of a firm for a patent, as a function of
qFirm size
qR&D expenditures
qIndustrial sector
qCountry, etc.
qSee Verbeek‘s data set PATENT
n
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Poisson Regression Model
nObserved variable for sample unit i:
n yi: number of possible outcomes 0, 1, …, y, …
nAim: to explain E{yi | xi }, based on characteristics xi
n E{yi | xi } = exp{xi’β}
nPoisson regression model
n
n
n with λi = E{yi | xi } = exp{xi’β}
n y! = 1x2x…xy, 0! = 1
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Poisson Distribution
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C:\Users\PHackl\Documents\O'trie\_Brno_SS\800px-Poisson-Verteilung.PNG

Poisson Regression Model: The Practice
nUnknown parameters: coefficients β
nFitting the model to data: ML estimators are
nConsistent
nAsymptotically efficient
nAsymptotically normally distributed
nEquidispersion condition
nPoisson distributed X obeys
n E{X} = V{X} = λ
nIn many situations not realistic
nOverdispersion
nRemedies: Alternative distributions, e.g., negative Binomial, and alternative estimation
procedures, e.g., Quasi-ML, robust standard errors
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Count Data Models in GRETL
nModel > Nonlinear Models > Count data…
nEstimates the specified model using Poisson or the negative binomial distribution
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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Tobit Models
nTobit models are regression models where the range of the (continuous) dependent variable is
constrained, i.e., censored from below
nExamples:
nExpenditures on durable goods as a function of income, age, etc.: a part of units does not spend
any money on durable goods
nHours of work as a function of qualification, age, etc.
nExpenditures on alcoholic beverages and tobacco
nTobit models
nStandard Tobit model or Tobit I model; James Tobin (1958) on expenditures on durable goods
nGeneralizations: Tobit II to V
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Example: Expenditures on Tobacco
nVerbeek‘s data set TOBACCO: expenditures on tobacco in 2724 Belgian households, Belgian household
budget survey of 1995/96
nModel:
n yi* = xi’b + ei
nyi*: optimal expenditures on tobacco in household i
nxi: characteristics of the i-th household
nei: unobserved heterogeneity (or measurement error or optimization error)
nActual expenditures yi
n yi = yi* if yi* > 0
n      =  0 if yi* ≤ 0
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The Standard Tobit Model
nThe latent variable yi* depends upon characteristics xi
n yi* = xi’b + eI
n with error terms (or unobserved heterogeneity)
n ei ~ NID(0, s2), independent of xi
nActual outcome of the observable variable yi
n yi = yi* if yi* > 0
n      =  0 if yi* ≤ 0
nStandard Tobit model or censored regression model
nCensoring: all negative values are substituted by zero
nCensoring in general
qCensoring from below (above): all values left (right) from a lower (an upper) bound are
substituted by the lower (upper) bound
nOLS produces inconsistent estimators for b
n
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The Standard Tobit Model, cont’d
nStandard Tobit model describes
1.The probability P{yi = 0} as a function of xi
n P{yi = 0} = P{ei £ - xi’b } = 1 - F(xi’b/s)
2.The distribution of yi given that it is positive, i.e., the truncated normal distribution with
expectation
n E{yi | yi > 0} = xi’b + E{ei | ei > - xi’b} = xi’b + s l(xi’b/s)
n   with l(xi’b/s) = f(xi’b/s) / F(xi’b/s) ³ 0
nAttention! A single set b of parameters characterizes both expressions
nThe effect of a characteristic
qon the probability of non-zero observation and
qon the value of the observation
n have the same sign!
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The Standard Tobit Model: Interpretation
nFrom
n P{yi = 0} = 1 - F(xi’b/s)
n E{yi | yi > 0} = xi’b + s l(xi’b/s)
n follows:
nA positive coefficient bk means that an increase in the explanatory variable xik increases the
probability of having a positive yi
nThe marginal effect of xik upon E{yi | yi > 0} is different from bk
nThe marginal effect of xik upon E{yi} is bkP{yi > 0}
qIt is close to bk if P{yi > 0} is close to 1, i.e, little censoring
nThe marginal effect of xik upon E{yi*} is bk
n
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The Standard Tobit Model: Estimation
nOLS produces inconsistent estimators for b
1.ML estimation based on the log-likelihood
n log L1(b, s2) = ℓ1(b, s2) = SiϵI0 log P{yi = 0} + SiϵI1 log f(yi)
n with appropriate expressions for P{.} and f(.), I0 the set of censored observations, I1 the set
of uncensored observations
nFor the correctly specified model: estimates are
nConsistent
nAsymptotically efficient
nAsymptotically normally distributed
2.Truncated regression model: ML estimation based on observations with yi > 0 only:
n ℓ2(b, s2) = SiϵI1[ log f(yi) - log P{yi > 0}]
nEstimates based on ℓ1 are more efficient than those based on ℓ2
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Example: Model for Budget Share for Tobacco
nVerbeek‘s data set TOBACCO: Belgian household budget survey of 1995/96
nBudget share wi* for expenditures on tobacco corresponding to maximal utility: wi* = xi’b + eI
n xi: log of total expenditures (LNX) and various characteristics like
qnumber of children £ 2 years old (NKIDS2)
qnumber of adults in household (NADULTS)
qAge (AGE)
nActual budget share for expenditures on tobacco
n wi = wi* if wi* > 0,
n     = 0  otherwise
n2724 households
n
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Model for Budget Share for Tobacco
nTobit model,
nGRETL output
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Model 2: Tobit, using observations 1-2724
Dependent variable: SHARE1 (Tobacco)
             coefficient            std. error         t-ratio        p-value
  ----------------------------------------------------------
  const         -0,170417        0,0441114       -3,863      0,0001   ***
  AGE            0,0152120      0,0106351        1,430       0,1526
  NADULTS   0,0280418       0,0188201       1,490       0,1362
  NKIDS        -0,00295209    0,000794286   -3,717      0,0002   ***
  NKIDS2      -0,00411756    0,00320953     -1,283      0,1995
  LNX            0,0134388        0,00326703     4,113      3,90e-05 ***
  AGELNX    -0,000944668   0,000787573   -1,199     0,2303
  NADLNX     -0,00218017    0,00136622     -1,596     0,1105
  WALLOON  0,00417202     0,000980745    4,254     2,10e-05 ***
Mean dependent var   0,017828   S.D. dependent var   0,021658
Censored obs              466   sigma                0,024344
Log-likelihood       4764,153   Akaike criterion    -9508,306
Schwarz criterion   -9449,208  Hannan-Quinn        -9486,944

Model for Budget Share for Tobacco, cont’d
nTruncated regres-
nsion model,
nGRETL output
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Model 7: Tobit, using observations 1-2724 (n = 2258)
Missing or incomplete observations dropped: 466
Dependent variable: W1 (Tobacco)
             coefficient             std. error    t-ratio   p-value
  ---------------------------------------------------------
  const          0,0433570         0,0458419        0,9458   0,3443
  AGE            0,00880553       0,0110819       0,7946    0,4269
  NADULTS   -0,0129409        0,0185585      -0,6973   0,4856
  NKIDS         -0,00222254     0,000826380   -2,689     0,0072  ***
  NKIDS2       -0,00261220      0,00335067    -0,7796   0,4356
  LNX             -0,00167130      0,00337817    -0,4947   0,6208
  AGELNX     -0,000490197    0,000815571  -0,6010    0,5478
  NADLNX      0,000806801     0,00134731    0,5988    0,5493
  WALLOON  0,00261490       0,000922432   2,835      0,0046  ***
Mean dependent var   0,021507   S.D. dependent var   0,022062
Censored obs                0   sigma                0,021450
Log-likelihood       5471,304   Akaike criterion    -10922,61
Schwarz criterion   -10865,39   Hannan-Quinn        -10901,73

Two Models for Budget Share for Tobacco, Comparison
nEstimates (coeff.) and standard errors (s.e.) for some coefficients
n of the Tobit (2724 observations, 644 censored) and the truncated regression model (2258
uncensored observations)
n
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constant
NKIDS
LNX
WALL
Tobit model
coeff.
-0,1704
-0,0030
0,0134
0,0042
s.e.
0,0441
0,0008
0,0033
0,0010
Truncated
regression
coeff.
0,0433
-0,0022
-0,0017
0,0026
s.e.
0,0458
0,0008
0,0034
0,0009

Specification Tests
nVarious tests based on
ngeneralized residuals
n l(- xi’b/s) if yi = 0
n ei/s if yi > 0 (standardized residuals)
n with l(-xi’b/s) = - f(xi’b/s) / F(-xi’b/s), evaluated for estimates of b, s
nand “second order” generalized residuals corresponding the estimation of s2
nTests
nfor normality
nfor omitted variables
nTest for normality is standard test in GRETL‘s Tobit procedure: consistency requires normality
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Contents
nLimited Dependent Variable Cases
nBinary Choice Models
nBinary Choice Models: Estimation
nBinary Choice Models: Goodness of Fit
nApplication to Latent Models
nMultiresponse Models
nMultinomial Models
nCount Data Models
nThe Tobit Model
nThe Tobit II Model
n
n
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An Example: Modeling Wages
nWage observations: available only for the working population
nModel that explains wages as a function of characteristics, e.g., the person‘s age
nTobit model: for a positive coefficient of age, an increase of age
qincreases wage
qincreases the probability that the person is working
qNot always realistic!
nTobit II model: allows two separate equations
qfor labor force participation and
qfor the wage of a person
nTobit II model is also called “sample selection model”
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Tobit II Model for Wages
nWage equation describes the wage of person i
n wi* = x1i’b1 + e1i
n with exogenous characteristics (age, education, …)
nSelection equation or labor force participation
n hi* = x2i’b2 + e2i
nObservation rule: wi actual wage of person i
n wi = wi*, hi = 1 if hi* > 0
n wi not observed, hi = 0 if hi* £ 0
n hi: indicator for working
nDistributional assumption for e1i, e2i
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Tobit II Model for Wages, cont’d
nSelection equation: a binary choice model; probit model needs standardization (s22 = 1)
nCharacteristics x1i and x2i may be different; however,
qIf the selection depends upon wi*: x2i is expected to include x1i
qBecause the model describes the joint distribution of wi and hi given one set of conditioning
variables: x2i is expected to include x1i
qSign and value of coefficients of the same variables in x1i  and x2i can be different
nSpecial cases
qIf s12 = 0, sample selection is exogenous
qIf x1i’b1 = x2i’b2 and e1i = e2i, the Tobit II model coincides with the Tobit I model
q
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Tobit II Model for Wages: Wage Equation
nExpected value of wi, given sample selection:
n E{wi | hi =1} = x1i’b1 + s12 l(x2i’b2)
n with the inverse Mill’s ratio or Heckman’s lambda
n l(x2i’b2) = f(x2i’b2) / F(x2i’b2)
nHeckman’s lambda
qPositive and decreasing in its argument
qThe smaller the probability that a person is working, the larger the value of the correction term
l
nExpected value of wi only equals x1i’b1 if s12 = 0: “no sample selection” error
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Tobit II Model: Log-likelihood Function
nLog-likelihood
n ℓ3(b1,b2,s12,s12) = SiϵI0log P{hi=0} + SiϵI1 [log f(yi|hi=1)+log P{hi=1}]
n = SiϵI0 log P{hi=0} + SiϵI1 [log f(yi) + log P{hi=1|yi}]
n with
n P{hi=0} = 1 - F(x2i’b2)
n
n
n
n
n and using f(yi|hi = 1) P{hi = 1} = P{hi = 1|yi} f(yi)
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Tobit II Model: Estimation
nMaximum likelihood estimation, based on the log-likelihood
n ℓ3(b1,b2,s12,s12) = SiϵI0 log P{hi=0}+SiϵI1 [log f(yi|hi=1)+log P{hi=1}]
nTwo step approach (Heckman, 1979)
1.Estimate the coefficients b2 of the selection equation by standard probit maximum likelihood: b2
2.Compute estimates of Heckman’s lambdas: li = l(x2i’b2) = f(x2i’b2) / F(x2i’ b2) for i = 1, …, N
3.Estimate the coefficients b1 and s12 using OLS
q wi = x1i’b1 + s12 li + ηi
nGRETL: procedure „Heckit“ allows both the ML and the two step estimation
n
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Tobit II Model for Budget Share for Tobacco
nHeckit  ML
nestimation,
nGRETL output
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Model 7: ML Heckit, using observations 1-2724
Dependent variable: SHARE1
Selection variable: D1
              coefficient        std. error           t-ratio        p-value
  -------------------------------------------------------------
  const          0,0444178       0,0492440       0,9020      0,3671
  AGE           0,00874370     0,0110272       0,7929      0,4278
  NADULTS  -0,0130898      0,0165677       -0,7901    0,4295
  NKIDS        -0,00221765    0,000585669   -3,787      0,0002    ***
  NKIDS2      -0,00260186    0,00228812     -1,137      0,2555
  LNX            -0,00174557    0,00357283     -0,4886    0,6251
  AGELNX    -0,000485866  0,000807854    -0,6014   0,5476
  NADLNX     0,000817826   0,00119574     0,6839     0,4940
  WALLOON  0,00260557    0,000958504    2,718       0,0066    ***
  lambda        -0,00013773   0,00291516     -0,04725   0,9623
Mean dependent var   0,021507    S.D. dependent var   0,022062
sigma                0,021451   rho                 -0,006431
Log-likelihood       4316,615   Akaike criterion    -8613,231
Schwarz criterion   -8556,008  Hannan-Quinn        -8592,349

Tobit II Model for Budget Share for Tabacco, cont’d
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nHeckit  ML
nestimation,
nGRETL output
n
Model 7: ML Heckit, using observations 1-2724
Dependent variable: SHARE1
Selection variable: D1
Selection equation
              coefficient        std. error           t-ratio        p-value
  -------------------------------------------------------------
  const         -16,2535        2,58561         -6,286        3,25e-010 ***
  AGE            0,753353       0,653820       1,152         0,2492
  NADULTS   2,13037         1,03368          2,061        0,0393    **
  NKIDS         -0,0936353    0,0376590     -2,486       0,0129    **
  NKIDS2       -0,188864      0,141231       -1,337       0,1811
  LNX             1,25834          0,192074       6,551        5,70e-011 ***
  AGELNX     -0,0510698    0,0486730      -1,049       0,2941
  NADLNX     -0,160399       0,0748929     -2,142       0,0322    **
  BLUECOL   -0,0352022    0,0983073      -0,3581    0,7203
  WHITECOL  0,0801599    0,0852980      0,9398     0,3473
  WALLOON   0,201073      0,0628750      3,198        0,0014    ***

Models for Budget Share for Tabacco
const.
NKIDS
LNX
WALL
Tobit model
coeff.
-0,1704
-0,0030
0,0134
0,0042
s.e.
0,0441
0,0008
0,0033
0,0010
Truncated
regression
coeff.
0,0433
-0,0022
-0,0017
0,0026
s.e.
0,0458
0,0008
0,0034
0,0009
Tobit II model
coeff.
0,0444
-0,0022
-0,0017
0,0026
s.e.
0,0492
0,0006
0,0036
0,0010
Tobit II
selection
coeff.
-16,2535
-0,0936
1,2583
0,2011
s.e.
2,5856
0,0377
0,1921
0,0629
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Estimates and standard errors for some coefficients of the
   standard Tobit, the truncated regression and the Tobit II model

Test for Sampling Selection Bias
nError terms of the Tobit II model with s12 ≠ 0: standard errors and test may result in misleading
inferences
nTest of H0: s12 = 0 in the second step of Heckit, i.e., fitting the regression wi = x1i’b1 + s12
li + ηi
nt-test on the coefficient for Heckman’s lambda
nTest results are sensitive to exclusion restrictions on x1i
n
n
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Tobit Models in GRETL
nModel > Nonlinear Models > Tobit
nEstimates the Tobit model; censored  dependent variable
nModel > Nonlinear Models > Heckit
nEstimates in addition the selection equation (Tobit II), optionally by ML- and by two-step
estimation
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Your Homework
1.Verbeek‘s data set CREDIT contains credit ratings of 921 US firms, as well as characteristics of
the firm; the variable rating has categories “1“, …,“7“ (highest) . Generate the variable GF (good
firm) with value 1 if rating > 4 and 0 otherwise, and the more detailed variable CR (credit rating)
with CR = 1 if rating < 3, CR = 2 if rating = 3, CR = 3 if rating = 4, and CR = 4 otherwise.
a.Estimate a binary logit model for the assignment of the GF ratings, and an ordered logit model
for assignment CR.
b.Compare the effects of the regressors in the models, based on coefficients and slopes.
c.Compare the hit rates of the models based on GF and on CR?
2.People buy for yi* of an investment fund, with yi* = xi’b + ei with eI ~ N(0,1); xi consists of
an intercept and the variables age and  income. The dummy di = 1 if yi* > 0 and di = 0 otherwise.
n
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Your Homework, cont’d
a.Derive the probability for di = 1 as function of xi.
b.Derive the log-likelihood function of the probit model for di.
3.Verbeek‘s data set TOBACCO contains expenditures on alcohol in 2724 Belgian households, taken
from the Belgian household budget survey of 1995/96, as well as other characteristics of the
households; for the expenditures on alcohol, the dummy D1=1 if the budget share for alcohol SHARE1
differs from 0, and D1=0 otherwise.
a.Model the budget share for alcohol, using (i) a Tobit model, (ii) a truncated regression, and
(iii) a Tobit II model, using the household characteristics AGE, LNX, NKIDS, and the dummy
FLANDERS.
b.Compare the effects of the regressors in the models, based on coefficients and slopes.
c.Compare the results for FLANDERS with that for the WALLOON.
n
2.
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