tobit — Tobit regression - Stata

1 Tobit regressionDescriptionQuick startMenuSyntaxOptionsRemarks and examplesStored resultsMethods and formulasReferencesAlso seeDescriptiontobitfits models for continuous responses where the outcome variable is censored. Censoringlimits may be fixed for all observations or vary across startTobit regression ofyonx1andx2, specifying thatyis censored at the minimum ofytobit y x1 x2, llAs above, but where the lower-censoring limit is zerotobit y x1 x2, ll(0)As above, but specify the lower- and upper-censoring limitstobit y x1 x2, ll(17) ul(34)As above, but wherelowerandupperare variables containing the censoring limitstobit y x1 x2, ll(lower) ul(upper)

2 MenuStatistics>Linear models and related>Censored regression > Tobit regression12 Tobit Tobit regressionSyntaxtobitdepvar[indepvars] [if] [in] [weight] [,options]optionsDescriptionModelnoconst antsuppress constant termll[(varname|#)]left-censoring variable or limitul[(varname|#)]right-censoring variable or limitoffset(varname)includevarnamein model with coefficient constrained to 1constraints(constraints)apply specified linear constraintsSE/Robustvce(vcetype)vcetypem ay beoim,opg,robust,clusterclustvar,bootstr ap,orjackknifeReportinglevel(#)set confidence level; default islevel(95)nocnsreportdo not display constraintsdisplayoptionscontrol columns and column formats, row spacing, line width,display of omitted variables and base and empty cells, andfactor-variable labelingMaximizationmaximizeoptionscontr ol the maximization process; seldom usedcollinearkeep collinear variablescoeflegenddisplay legend instead of statisticsindepvarsmay contain factor variables; see[U] Factor contain time-series operators; see[U] Time-series ,bootstrap,by,collect,fmm,fp,jackknife,n estreg,rolling,statsby,stepwise, andsvyareallowed; see[U] Prefix commands.

3 For more details, see [BAYES]bayes: tobitand [FMM]fmm: are not allowed with thebootstrapprefix; see [R] are not allowed with thejackknifeprefix; see [R] ()and weights are not allowed with thesvyprefix; see [SVY] ,fweights,iweights, andpweights are allowed; see[U] not appear in the dialog [U] 20 Estimation and postestimation commandsfor more capabilities of estimation Model noconstant; see [R]Estimation [(varname|#)]andul[(varname|#)]indicate the lower and upper limits for censoring, re-spectively. Observations withdepvar ll()are left-censored; observations withdepvar ul()are right-censored; and remaining observations are not censored.

4 You do not have to specify thecensoring values. If you specifyll, the lower limit is the minimum ofdepvar. If you specifyul,the upper limit is the maximum Tobit regression 3offset(varname),constraints(constraints ); see [R]Estimation options. SE/Robust vce(vcetype)specifies the type of standard error reported, which includes types that are derived fromasymptotic theory (oim,opg), that are robust to some kinds of misspecification (robust), thatallow for intragroup correlation (clusterclustvar), and that use bootstrap or jackknife methods(bootstrap,jackknife); see [R]vceoption. Reporting level(#),nocnsreport; see [R]Estimation :noci,nopvalues,noomitted,vsquish,noempt ycells,baselevels,allbaselevels,nofvlabe l,fvwrap(#),fvwrapon(style),cformat(%fmt ),pformat(%fmt),sformat(%fmt), andnolstretch; see [R]Estimation options.

5 Maximization maximizeoptions:difficult,technique(algo rithmspec),iterate(#),[no]log,trace,grad ient,showstep,hessian,showtolerance,tole rance(#),ltolerance(#),nrtolerance(#),no nrtolerance, andfrom(initspecs); see [R]Maximize. These options areseldom following options are available withtobitbut are not shown in the dialog box:collinear,coeflegend; see [R]Estimation and a linear regression model for a censored continuous outcome. Censoring occurs whenthe dependent variable is observed only within a certain range of values. When it is not, we knowonly that it is either above (right-censoring) or below (left-censoring) the censoring value.

6 Censoringdiffers from truncation. When the data are truncated, we do not observe either the dependent variableor the covariates; see [R] may result from study design or may be a result of how the outcome is of data may occur, for example, in income surveys that top code the highest incomecategory. Any respondent that earns the censoring limit or more reports only the value at the limit,and we do not know the respondent s true income. Left-censoring arises naturally when measurementsare obtained from an instrument or a laboratory procedure that has a limit of detection. If we observea value at the measurement limit, we know the true value is at the limit or below censoring limits to be the same for all observations or to vary from observation to (1958) originally conceived the Tobit model as one of consumption of consumer durableswhere purchases were left-censored at zero.

7 Contemporary literature treats this and similar cases as acorner solution model. See Wooldridge (2020, sec. ), Long (1997, 196 210), and Maddala andLahiri (2006, 333 336) for an introduction to the Tobit model. Wooldridge (2010, chap. 17 and 19)provides an advanced treatment of censored regression models. Cameron and Trivedi (2010, chap. 16)discuss the Tobit model using Stata Tobit model can be written as the latent regression modely=x + with a continuousoutcome that is either observed or unobserved. Following Cong (2000), the observed outcome forobservationiis defined as4 Tobit Tobit regressiony i={yiifa < yi< baifyi abifyi bwhereais the lower-censoring limit andbis the upper-censoring limit.}

8 The Tobit model assumesthat the error term is normally distributed; N(0, 2I). Depending on the problem at hand, thequantity of interest in a Tobit model may be the censored outcome,y i, or the uncensored outcome,yi. In the measurement instrument scenario above, we may wish to predict the values that fall belowthe measurement threshold. By contrast, in the consumption of consumer durables scenario above,the latent variable is an artificial construct and the variable of interest is the observed 1: Constant-censoring limitUniversity administrators want to know the relationship between high school grade point average(GPA) and students performance in fictional data on a cohort of 4,000college students.

9 CollegeGPA(gpa2) and high schoolGPA(hsgpa) are measured on a continuousscale between zero and four. The outcome of interest is the student s collegeGPA. But, for reasonsof confidentiality,GPAs below are reported as In other words, the outcome is censored onthe believe thatGPAis also a function of the logarithm of income of the student s parents (pincome)and whether or not the student participated in a study-skills program while in college (program).. use (High school GPA and performance in college). Tobit gpa2 hsgpa pincome program, llRefining starting values:Grid node 0: log likelihood = full model:Iteration 0: log likelihood = 1: log likelihood = 2: log likelihood = 3: log likelihood = 4: log likelihood = 5: log likelihood = regression Number of obs = 4,000 Uncensored = 2,794 Limits: Lower = 2 Left-censored = 1,206 Upper = +inf Right-censored = 0LR chi2(3) = > chi2 = likelihood = Pseudo R2 = Std.

10 Err. t P>|t| [95% conf. interval] .0128699 .633399 ..0074568 .3013103 ..0147468 .5265297 ..0478484 ( ).161703 .0044004 .1533019 .1705645tobitreports the coefficients for the latent regression model. Thus, we can interpret the coefficientsjust as we would the coefficients fromOLS. For example, participation in a study-skills programincreases the expected uncensoredGPAby Tobit regression 5 Example 2: Tobit model for a corner solutionSuppose that we are interested in the number of hours married women spend working for wages,and we treat observations recording zero hours as observed, per the corner-solution approach discussedWooldridge (2010, chap.)