3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. Truncated standard normal distribution for a = 0.5, 0, and 0.5. CDS Mphil Econometrics

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1 Lmted Dependent Varable Models: Tobt an Plla N 1 CDS Mphl Econometrcs Introducton Lmted Dependent Varable Models: Truncaton and Censorng Maddala, G Lmted Dependent and Qualtatve Varables n Econometrcs. Cambrdge Unversty Press. Truncaton A truncated dstrbuton s the part of an untruncated dstrbuton that s above or below some specfed value. If a contnuous random varable x has pdf f(x) and a s a constant, then the densty of the truncated RV s f(x) f(x x > a) = Pr ob(x > a) 3 4 a = 0.5 a = 0.5 a = 0 Truncated standard normal dstrbuton for a = 0.5, 0, and 0.5 Truncaton Truncaton occurs when some observatons on both the dependent varable and regressors are lost. For example, ncome may be the dependent varable and only low-ncome people are ncluded n the sample. In effect, truncaton occurs when the sample data s drawn from a subset of a larger populaton

2 Censorng censorng occurs when the value of an observaton s only partally known. One of the earlest attempts to analyse a statstcal problem nvolvng censored data: Danel Bernoull's 1766 analyss of smallpox morbdty and mortalty data to demonstrate the effcacy of vaccnaton. Censored Regresson Model Censorng occurs when data on the dependent varable s lost (or lmted) but not data on the regressors. When the dependent varable s censored, values n a certan range are all transformed to (or reported as) a sngle value. 7 8 Censored Regresson Model For example, people of all ncome levels may be ncluded n the sample, but for some reason the ncome of hgh-ncome ncome people may be top-coded as, say, Rs100,000. A defect n the sample An Example A labor supply model estmates the relatonshp between hours worked by employees and characterstcs of employees such as age, educaton and famly status. For people who are unemployed, t s not possble to observe the number of hours they would have worked had they had employment. 9 Stll we know age, educaton and famly status for those observatons. 10 Another Example Suppose we are nterested n fndng out the amount of money a HH spends on a house n relaton to soco-economc varables. Many HHs may not have purchased house: Zero expendture for them CDS Mphl Econometrcs CDS Mphl Econometrcs 2

3 Another Example Suppose we are nterested n studyng how much an ndvdual desredto gve to charty. For many people the amount we observe s zero,.e. they gve nothng to charty. For others, we observe the actual amount they contrbuted. Censored & Truncated Regresson Model Truncated regresson models are used for data, where whole observatons are mssng so that the values for the dependent and the ndependent varable are unknown Censored & Truncated Regresson Model Censored regresson models are used for data, where only the value for the dependent varable (hours of work for example) s unknown whle the value of the ndependent varable (age, educaton, famly status) s stll avalable Tobt Model Censored regresson? or Truncated regresson? Orgnal Tobt model suggested by James Tobn ( ) CDS Mphl Econometrcs 18 3

4 Some examples n the emprcal lterature TobtModel The structural equaton n the Tobt model s: where u N(0, σ 2 ) y = x β + u Analyze a dependent varable that s zero for a sgnfcant fracton of the observatons. y s a latent varable that s observed for values greater than τ and censored otherwse Tobt Model TobtModel = xβ u In the typcal y + The observed y s defned by the followng measurement equaton In the typcal Tobt model, we assume that τ = 0.e. the data are censored at 0. y = y, f y > τ τ y, f y τ Thus, we have y = y, f y > 0 0, f y 0 21 CDS Mphl Econometrcs Tobt Model TobtModel y = y, f y > 0 0, f y 0 y = x β + u ~ N 0, 2 ( σ ) Ths model contans a Probt model for y beng zero or postve and a standard Regresson model for the postve values of y. u The Probt model may, for example, descrbe the nfluence of explanatory varables on the decson whether or not to donate to charty, whle the Regresson model measures the effect of the explanatory varables on the sze of the amount for donatng ndvduals

5 Tobt Model Why Use the Tobt Model? Why not just use the observatons for whch y > 0 and estmate the model usng OLS? The answer: f you do, your parameter estmates wll be based and nconsstent. The degree of bas wll also ncrease as the number of observatons that take on the value of zero ncreases. y = Why Use the Tobt Model? y, f 0, f y 0 y = x β + u > 0 2 ( σ ) u ~N0, Neglectng the truncaton can lead to based estmates of α and β E[y y > 0,x] = φ[(xβ)/ xβ + σ Φ[(x β)/ φ = pdf and Φ = cdf: p(y > 0) Why Use the Tobt Model? E[y truncaton] = µ + σλ(α) E[y y > 0,x] = φ[(xβ)/ xβ + σ Φ[(x β)/ The last term on the RHS [σλ(α) ] : the nverse Mlls rato / hazard functon for the std N dstrbuton. 27 Inverse Mlls Rato Named after John P. Mlls, The rato of the Probablty Densty Functon over the Cumulatve Dstrbuton Functon of a dstrbuton. If x s a random varable dstrbuted normally wth mean µand varance σ 2, then where αs a constant, ϕdenotes the standard normal pdf, and Φs the standard normal cdf. α µ ϕ σ E[x x > α] = µ + σ α µ Φ σ ϕ(z) = µ + σ Φ(z) 28 Why Use the Tobt Model? Why Use the Tobt Model? Consder for example, the amount a person gves to charty. Suppose the true relatonshp between the amount a person wantsto gve to charty and that person s ncome s 29 CDS Mphl Econometrcs 5

6 Why Use the Tobt Model? The lower ncome people would actually lke to gve negatve amounts (.e. get money back!). Why Use the Tobt Model? In realty, we do not observe ndvduals makng negatve contrbutons. What we observe s they gve nothng. The red lne ndcates the true regresson lne for the relatonshp between ncome and donatons 31 The observed data looks lke ths: 32 Why Use the Tobt Model? Why Use the Tobt Model? If we smply estmated the model by OLS, OLS tends to underestmate the magntude of the slope. OLS regresson lne OLS regresson lne True relatonshp the parameter estmates would be based downwards. True relatonshp the parameter estmates would be based downwards Interpretng Tobt Estmates A bt more complex than nterpretng estmated coeffcents from the OLS model. In partcular, the estmated coeffcents represent the margnal effect of x on y. That s : margnal effect of x on the latent varable y not on the observed varable y. E[y x] = βk xk 35 Interpretng Tobt Estmates What we want to explan s the observed amount of chartable contrbutons not the desred amount of chartable contrbutons. Thus, what we want s the expected value of y condtonal on y beng greater than zero: E[y y > 0, x]. 36 6

7 Interpretng Tobt Estmates What we want s the expected value of y condtonal on y beng greater than zero: E[y y > 0, x]. That s: φ[(xβ)/ E[y y > 0,x] = xβ + σ = Φ[(xβ)/ φ(z) β + σ Φ(z) The desred margnal effects are the dervatve of ths functon wth respect to x. x Interpretng Tobt Estmates We have E[y y > 0,x] = x E(y) = E(y y > 0) P(y > 0) f(x) f(x x > a) = Prob(x > a) E[y x] = βk xk φ(z) β + σ Φ(z) φ(z) E(y) = Xβ + σ Φ(z) Φ(z) E(y) = xβφ(z) + σφ(z) E(y) = βkφ(z) xk Estmaton Heckman Alternatve Method of maxmum lkelhood Olsen s (1978) reparameterzaton smplfes ML estmaton. James Heckman has proposed a smple alternatve to the ML method: J. J. Heckman, Sample Selecton Bas as a Specfcaton Error, Econometrca, vol. 47, pp Conssts of a two-step estmatng procedure: Step 1: estmate the probablty of, say, a consumer ownng a house, on the bass of the probt model Heckman Alternatve Heckman Alternatve Step 2: estmate the model y = y, f 0, f y 0 > 0 by addng to t the nverse Mlls rato or the hazard rate that s derved from the probt estmate. E[y y > 0,x] = y = x β + u φ[(xβ)/ xβ + σ Φ[(x β)/ The Heckman procedure yelds consstent estmates of the parameters, but they are not as effcent as the ML estmates. An Example of Tobt model: IMR

8 43 44 Extramartal Educaton Of the 601 responses, 451 ndvduals had no extramartal affars, and 150 ndvduals had one or more affars. Ray Far Model: OLS 451 observatons lyng along the horzontal axs. a a censored sample, a tobt model may be approprate Ray Far Model: Tobt Compare OLS & Tobt estmates OLS Tobt

9 Ray Far Model: Tobt Ray Far Model: Tobt Tobt: Coeffcent Estmates & Margnal Effects Tobt n Stata Statstcs Lnear models & related Censored regresson Tobt regresson Also Compare.. Statstcs Postestmaton (1) Tests: Test parameters (2) Margnal effects or elastctes