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 nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 2 Example nExplain whether a household owns a car: explanatory power have nincome nhousehold size netc. nRegression for describing car-ownership is not suitable! nOwning a car has two manifestations: yes/no nIndicator for owning a car is a binary variable nModels are needed that allow to describe a binary dependent variable or a, more generally, limited dependent variable Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 3 Cases of Limited Dependent Variables nTypical situations: functions of explanatory variables are used to describe or explain nDichotomous or binary 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 4 Example: Car Ownership and Income nWhat is the probability that a randomly chosen household owns a car? nSample of N=32 households, among them 19 households with car 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 contain 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 5 Car Ownership and Income, cont’d nHow can a model for the 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 of i-th household: xi2 nFor predicting yi – or estimating the probability P{yi =1} – , a model is needed that takes the income into account Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 6 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} nThe systematic part of yi = xi’β + εi, xi’β, is P{yi =1|xi}! nModel for y is specifying the probability for y = 1 as a function of x nProblems: qxi’β not necessarily in [0,1] qError terms: for a given xi nεi can take on only two values, viz. 1- xi’β and xi’β nV{εi |xi} = xi’β(1- xi’β), heteroskedastic, dependent upon β Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 7 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) nStandard logistic distribution function q q n L(z) fulfils limz→ -∞ L(z) = 0, limz→ ∞ L(z) = 1 nBinary choice model: q P{yi =1|xi} = pi = L(xi’β) = [1 + exp{-xi’β}]-1 qCan be written using the odds ratio pi/(1- pi) for the event {yi =1|xi} q q qInterpretation of coefficients β: An increase of xi2 by 1 results in a relative change of the odds ratio pi/(1- pi) by β2 or by 100β2%; cf. the notion semi-elasticity Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 8 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 x 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)) = 0.646 nStandard logistic distribution function L(z), with z on the horizontal and L(z) on the vertical axis Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 9 x z P{y =1|x} 1 0.6 0.646 2 1.7 0.846 3 2.8 0.943 Odds, Odds Ratio nThe odds or the odds ratio (in favour) of event A is the ratio of the probability that A will happen to the probability that A will not happen nIf the probability of success is 0.8 (that of failure is 0.2), the odds of success are 0.8/0.2 = 4; we say, “the odds of success are 4 to 1” nIf the probability of event A is p, that of “not A” therefore being 1-p, the odds or the odds ratio of event A is the ratio p/(1-p) nWe say the odds (ratio) of A is “p/(1-p) to 1” or “1 to (1-p)/p” n n n n nThe logarithm of the odds p/(1-p) is called the logit of p n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 10 p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 p/(1-p) 0.11 0.25 0.43 0.67 1 1.5 2.33 4 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 Betting Odds nThe probability of success is 0.8 nThe odds of success are 4 to 1 nBetting odds for success are 1:4 qThe bookmaker is prepared to pay out a prize of one fourth of the stake and return the stake as well, to anyone who places a bet on success Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 11 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 12 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) = F(z) 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 13 Linear Probability Model (LPM) nAssumes that n P{yi =1|xi} = xi’β for 0 ≤ xi’β ≤ 1 n but sets restrictions 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 14 Probit Model: Standardization nE{yi|xi} = P{yi =1|xi} = F(xi’β): assume F(.) 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 15 Probit vs Logit Model nDifferences between the probit and the logit model: qShapes of distribution are slightly different, particularly in the tails. qScaling of the distributions 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 of b in the logit model are roughly a factor p/Ö3 »1.81 larger than those in the probit model q q Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 16 Marginal Effects of Binary Choice Models nLinear regression model E{yi|xi} = xi’β: the marginal effect ¶E{yi|xi}/¶xik of a change in xk is βk nFor E{yi|xi} = F(xi’β) n n nThe marginal effect of changing xk qProbit model: ϕ(xi’β) βk, with standard normal density function ϕ qLogit model: exp{xi’β}/[1 + exp{xi’β}]2 βk qLinear probability model: βk if xi’β is in [0,1] nIn general, the marginal effect of changing the regressor xk depends upon xi’β, the shape of F, and βk; the sign is that of βk q Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 17 Interpretation of Binary Choice Models nThe effect of a change in xk can be characterized by the n“Slope”, i.e., the “average” marginal effect or the gradient of E{yi|xi} for 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 ratio when increasing xk by 1 unit q Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 18 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 19 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 of 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 20 Generalized Residuals nThe first-order conditions Sieixi = 0 define the generalized residuals n n nThe generalized residuals ei can assume two values, depending on the value of yi: 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 21 Estimation of Logit Model nFirst-order condition of the maximization problem n n n n gives [due to P{yi =1|xi} = pi = L(xi,β)] n n nFrom Si xi = Siyixi follows – given that the model contains 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 22 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 23 Example: Effect of Teaching Method nStudy by Spector & Mazzeo (1980); see Greene (2003), Chpt.21 nPersonalized System of Instruction: a 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 pre-test, entering knowledge n32 observations n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 24 mean min max GPA 3.12 2.06 4.00 TUCE 21.9 12 29 Effect of Teaching Method, cont’d nLogit model for GRADE, GRETL output Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 25 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 nEstimated logit model for the indicator GRADE n P{GRADE = 1} = p = L(z) = exp{z}/(1+exp{z}) n with n z = −13.02 + 2.826*GPA + 0.095*TUCE + 2.38*PSI n = log {p/(1-p)} = logit{p} n nRegressors qGPA: grade point average qTUCE: score on a pre-test, entering knowledge qPSI (0/1): participation in program with new teaching method nSlopes qGPA: 0.53 qTUCE: 0.02 qDifference P{GRADE =1|x(d),PSI=1} – P{GRADE =1|x(d), PSI=0}: 0.49; cf. Slope 0.46 q q n n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 26 Effect of Teaching Method, cont’d nLogit model for GRADE, actual and fitted values of 32 observations Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 27 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 – correlations – between errors and regressors nNo omitted regressors n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 28 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 29 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 or hit rates qRp2 n n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 30 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 1 if ℓ1 = 0 n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 31 Naïve Model: Calculation of ℓ0 nMaximum log-likelihood function of the naïve model, i.e., a model with only an intercept: ℓ0 nP{yi =1} = p for all i (cf. urn experiment) nLog-likelihood function 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 32 Goodness-of-fit Measure Rp2 nComparison of correct and incorrect predictions nPredicted outcome n ŷi = 1 if F(xi’b) > 0.5, i.e., if xi’b > 0 n = 0 if F(xi’b) < 0.5, i.e., 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 for all i (!), if = N1/N > 0.5; ŷi = 0 for all i if ≤ 0.5 qwr0 = 1 - if > 0.5, wr0 = if ≤ 0.5 qGoodness-of-fit measure: Rp2= 1 – wr1/wr0; may be negative! q n n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 33 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 pre-test, entering knowledge n32 observations n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 34 Effect of Teaching Method, cont’d nLogit model for GRADE, GRETL output Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 35 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 36 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 nVery similar slopes Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 37 LPM Logit Probit coeff slope coeff slope coeff slope const -1.498 -13.02 -7.452 GPA 0.464 0.464 2.826 0.534 1.626 0.533 TUCE 0.010 0.010 0.095 0.018 0.052 0.017 PSI 0.379 0.379 2.379 0.456 1.426 0.464 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: ℓ1 = -12.89, wr1 = 6/32 nGoodness-of-fit measures nMcFadden R2 = 1 – (-12.89)/(-20.59) = 0.374 npseudo-R2 = 1 - 1/[1 + 2(-12.89 + 20.59)/32) = 0.325 nRp2 = 1 – wr1/wr0 = 1 – 6/11 = 0.45 n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 38 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 39 Modelling Utility nLatent variable yi*: utility difference between owning and not owning a car; unobservable (latent) nDecision on owning a car qyi* > 0: in favour of car owning qyi* ≤ 0: against car owning nyi* depends upon observed characteristics (e.g., 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 distribution function F(.) with symmetric density nLatent variable model: based on a latent variable that represents the underlying behaviour Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 40 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 a 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 41 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 42 Multi-response Models nModels 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. nMulti-response models describe the probability of each of these outcomes, as a function of variables like qperson-specific characteristics qalternative-specific characteristics nTypes of multi-response models (cf. above examples) qOrdered response models: outcomes have a natural ordering qMultinomial (unordered) models: ordering of outcomes is arbitrary Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 43 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 44 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 follows the qstandard normal distribution: ordered probit model qstandard logistic distribution: ordered logit model Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 45 Example: Willingness to Work nMarried females are asked: „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” Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 46 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*(= xi’β + εi): willingness to work increases with larger xk for positive βk nWrt probabilities P{yi = j |xi}, e.g., for positive βk qP{yi = 3 |xi} = P{yi* > γ |xi} increases and qP{yi = 1 |xi} P{yi* ≤ 0 |xi} decreases with larger xk Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 47 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“) Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 48 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 49 Credit Rating, cont’d nVerbeek, Table 7.5. Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 50 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 51 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 52 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 max{Ui1, ..., UiM} nUtility levels Uij, j = 1,…, M, as a function of characteristics xij n Uij = xij’β + εij = μij + εij nerror terms εij follow the Type I extreme value distribution: leads to n n n for j = 1, …, M nand Σj P{yi = j} = 1 nFor setting the location: constraint xi1’b = μi1 = 0 or exp{μi1} = 1 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 53 Variants of the Logit Model nConditional logit model: for j = 1, …, M n n nAlternative-specific characteristics xij nE.g., mode of transportation (by car, train, bus) is affected by the travel costs, travel time, etc. of the individual i 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 54 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 for the mixed logit model: it combines qalternative-specific characteristics and qperson-specific characteristics nNumber of parameters nconditional logit model: vector b with K components nmultinomial logit model: vectors b2, ..., bM, each with K components n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 55 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 other alternatives: “independence of irrelevant alternatives” (IIA) n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 56 Multi-response 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 57 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 58 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 visits of a customer in the supermarket 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 59 Example: Patents and R&D Expenditures nVerbeek‘s data set PATENTS: number of patents (p91), expenditures for R&D (logrd91), sector of industry, and region; N = 181 nQuestion: Is the number of patents depending of R&D expenditures, sector, region? Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 60 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 61 Poisson Distribution Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 62 C:\Users\PHackl\Documents\O'trie\_Brno_SS\800px-Poisson-Verteilung.PNG Poisson Regression Model: Estimation nUnknown parameters: coefficients β nEstimates of β allow assessing how exp{xi’β} = E{yi | xi } is affected by xi nFitting the model to data: ML estimators for β are nConsistent nAsymptotically efficient nAsymptotically normally distributed Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 63 Patents and R&D Expenditures nVerbeek‘s data set PATENTS: number of patents (p91), expenditures for R&D (log_rd91), sector of industry, and region; N = 181 nQuestion: Is the number of patents depending of R&D expenditures, sector, region? nModel: n E{yi | xi } = exp{xi’β} nyi: number of patents in company i in year 1991 nxi: characteristics of company i: intercept, R&D expenditures in1991, dummy for sector (aerosp, chemist, computer, machines, vehicles), region (US, Europe, Japan) n nVariable p91: mean: 73.6, std.dev.: 150.9 n Overdispersion ? Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 64 Patents and R&D Expenditures nPoisson regression model for p91, GRETL output Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 65 Convergence achieved after 8 iterations Model 1: Poisson, using observations 1-181 Dependent variable: p91 coefficient std. error z p-value ------------------------------------------------------------------------------------------------ const −0.873731 0.0658703 −13.26 3.72e-040 *** log_rd91 0.854525 0.00838674 101.9 0.0000 *** aerosp −1.42185 0.0956448 −14.87 5.48e-050 *** chemist 0.636267 0.0255274 24.92 4.00e-137 *** computer 0.595343 0.0233387 25.51 1.57e-143 *** machines 0.688953 0.0383488 17.97 3.63e-072 *** vehicles −1.52965 0.0418650 −36.54 2.79e-292 *** japan 0.222222 0.0275020 8.080 6.46e-016 *** us −0.299507 0.0253000 −11.84 2.48e-032 *** Mean dependent var 73.58564 S.D. dependent var 150.9517 Sum squared resid 1530014 S.E. of regression 94.31559 McFadden R-squared 0.675242 Adjusted R-squared 0.674652 Log-likelihood −4950.789 Akaike criterion 9919.578 Schwarz criterion 9948.365 Hannan-Quinn 9931.249 Overdispersion test: Chi-square(1) = 18.6564 [0.0000] Poisson Regression Model: Overdispersion 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 66 Count Data Models in GRETL nModel > Nonlinear Models > Count data… nEstimates the coefficients β of the specified model using Poisson (Poisson) or the negative binomial (NegBin 1, NegBin 2) distribution nPerforms overdispersion test for Poisson regression Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 67 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 68 Tobit Models nTobit models are regression models where the range of the (continuous) dependent variable is constrained, i.e., censored from below nExamples: nHours of work as a function of age, qualification, etc. nExpenditures on alcoholic beverages and tobacco nHoliday expenditures as a function of the number of children nExpenditures on durable goods as a function of income, age, etc.: a part of units does not spend any money on durable goods nTobit models nStandard Tobit model or Tobit I model; James Tobin (1958) on expenditures on durable goods nGeneralizations: Tobit II to V Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 69 Example: Expenditures on Tobacco nVerbeek‘s data set TOBACCO: expenditures on tobacco and alcoholic beverages 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 (latent) 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 70 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 71 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{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! Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 72 E{y[i] | y[i]*[ ]> 0}: See Greene, Theorem 22.3 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} can be shown to be 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 (due to yi* = xi’b + ei) Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 73 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 74 The Standard Tobit Model: Estimation nOLS produces inconsistent estimators for b; alternatives: 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|yi > 0)] = SiϵI1[ log f(yi) - log P{yi > 0}] nEstimates based on ℓ1 are more efficient than those based on ℓ2 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 75 Example: Model for Budget Share for Tobacco and Alcohol nVerbeek‘s data set TOBACCO: Belgian household budget survey of 1995/96; expenditures for tobacco and alcoholic beverages nBudget share wi* for expenditures on alcoholic beverages 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 alcohol (SHARE1, W1) n wi = wi* if wi* > 0, n = 0 otherwise n2724 households n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 76 Model for Budget Share nBudget share wi* for expenditures on alcoholic beverages n wi* = xi’b + eI n regressors xi: qlog of total expenditures (LNX) and qhousehold characteristics: AGE, NADULTS, NKIDS, NKIDS2 qinteractions AGELNX (=LNX*AGE), NADLNX (=LNX*NADULTS) nActual budget share for expenditures on alcohol (SHARE1, W1) n wi = wi* if wi* > 0, n = 0 otherwise nAttention! Sufficiently large change of income will create positive w* for any household! n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 77 Model for Budget Share for Alcohol nTobit model, nGRETL output Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 78 Model 2: Tobit, using observations 1-2724 Dependent variable: SHARE1 (alcohol) 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 Alcohol, cont’d nTruncated regres- nsion model, nGRETL output Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 79 Model 7: Tobit, using observations 1-2724 (n = 2258) Missing or incomplete observations dropped: 466 Dependent variable: W1 (alcohol) 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 Models for Budget Share for Alcohol, 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 80 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 nTests nfor normality nfor omitted variables nTests 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 to the estimation of s2 nTest for normality is standard test in GRETL‘s Tobit procedure: consistency requires normality Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 81 Contents nLimited Dependent Variable Cases nBinary Choice Models nBinary Choice Models: Estimation nBinary Choice Models: Goodness of Fit nApplication to Latent Models nMulti-response Models nMultinomial Models nCount Data Models nThe Tobit Model nThe Tobit Model: Estimation nThe Tobit II Model n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 82 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, gender, education, etc. nLow value of education increases probability of no wage qFrom a sample of wages the effect of education might be underestimated q“Sample selection bias” 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! nTobacco consumption: Abstention from smoking may be a person’s attitude not depending on factors which determine smoking intensity n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 83 Modeling Wages, cont’d nTobit II model: allows two separate equations: nEquation for labor force participation of a person nEquation for the wage of a person nTobit II model is also called “sample selection model” Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 84 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: usually normality with Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 85 Model for Wages: Selection Equation nSelection equation hi* = x2i’b2 + e2i: probit model for binary choice; 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 qx2i should contain variables not included in x1i qSign and value of coefficients of the same variables in x1i and x2i are not the same nSpecial cases qIf s12 = 0, sample selection is exogenous qTobit II model coincides with Tobit I model if x1i’b1 = x2i’b2 and e1i = e2i q Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 86 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, consistent OLS estimates of the wage equation Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 87 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) Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 88 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 89 Tobit II Model for Budget Share for Alcohol nHeckit ML nestimation, nGRETL output n nD1: dummy, n 1 if SHARE1 > 0 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 90 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 Alcohol, cont’d Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 91 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 92 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 nGRETL: t-test on the coefficient for Heckman’s lambda nGRETL: Heckit-output shows rho, estimate for r12 from s12 = r12s1 nTest results are sensitive to exclusion restrictions on x1i n n Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 93 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 Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 94 Your Homework 1.People buy for yi* assets of an investment fund, with yi* = xi’b + ei, ei ~ N(0, s2); xi consists of a “1” for the intercept and the variable income. The dummy di = 1 if yi* > 0 and di = 0 otherwise. a.Derive the probability for di = 1 as function of xi. b.Derive the log-likelihood function of the probit model for di, i = 1,...,N. c.Derive the ML estimator of the probability for di = 1 as function of xi of the logit model. 2.Verbeek‘s data set TOBACCO contains expenditures on tobacco in 2724 Belgian households, taken from the household budget survey of 1995/96, as well as other characteristics of the households; for the expenditures on tobacco, the dummy D2=1 if the budget share for tobacco (SHARE2) differs from 0, and D2=0 otherwise. Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 95 Your Homework, cont’d a.Model the budget share for tobacco, using (i) a Tobit model, (ii) a truncated regression, and (iii) a Tobit II model; using the household characteristics LNX, AGE, NKIDS, the interaction LNX*AGE, and the dummy FLANDERS; in addition BLUECOL for the selection equation. b.Compare the effects of the regressors in the three models, based on coefficients and t-statistics. c.Discuss the effect of the variable FLANDERS. n 2. Mar 18, 2016 Hackl, Econometrics 2, Lecture 2 96