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Tobit Regression Models Using R

Tobit Regression Models Using R. Nhat Ho. Outline. Theoretical Background What is Tobit Regression? Standard Tobit Model Estimation Method When we should not use Tobit model? Implementation in R using CensReg package with real data and simulated data. What is Tobit Regression?.

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Tobit Regression Models Using R

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  1. Tobit Regression Models Using R Nhat Ho

  2. Outline • Theoretical Background • What is Tobit Regression? • Standard Tobit Model • Estimation Method • When we should not use Tobit model? • Implementation in R using CensReg package with real data and simulated data

  3. What is Tobit Regression? • Consider the following variables: • number of hour work • expenditure on durable goods • number of extramarital affairs • Data generating process produced large percentage of zeroes and continuous non-zero values Clearly violate the normality and linearity assumption Source: Amemiya, T. (1984). Tobit models: A survey. Journal of econometrics, 24(1), 3-61.

  4. Application to Related Field • Marketing: Amount purchased of one brand of goods by the household in month t • Source: Elrod, T., & Winer, R. S. (1982). An empirical evaluation of aggregation approaches for developing market segments. The Journal of Marketing, 65-74. • Finance: Lease to book value of equity (many firms does not lease in anyone year) • Source:Ang, J., & Peterson, P. P. (1984). The leasing puzzle. The Journal of Finance,39(4), 1055-1065.

  5. Standard Tobit Model • yi*is the unobserved (“latent”) variable • yiis the observed variable • xiare explanatory variables • β are the unknown parameters • εi is normally distributed drawn fromN(0, σ2) Source: Henningsen, A. (2010). Estimating Censored Regression Models in R using the censReg Package. R package vignettes.

  6. What is Tobit Regression? • You want a model that can produce that kind of data • The latent variable, yi*, was used as a device to serve that purpose • No one actually believe in this latent variable • .

  7. Estimation Method • Maximum Likelihood • Parameters Estimation: β and σ • βestimate the linear effect of x onthe expected y* , the latent variable. • βdoes measure effect on the expected value of y, just not a linear effect Source: Amemiya, T. (1984). Tobit models: A survey. Journal of econometrics, 24(1), 3-61.

  8. Estimation Method • The Log Likelihood function for the Tobit Regression: • The first part correspond to the probability that an observations has value 0 • The second part correspondent to the standard model for observation that has value greater than 0 • Without 0s, the Likelihood Function is the same as the standard model No different than OLS • Few O’s, the classical model and Tobit model likelihood differ very little `Use OLS

  9. When we should not use Tobit model? • A model is good if it produces data that look like what you actually observe • If Tobit model can’t produce data that look like the data you observe than it’s not a good model • Ex (from by Dr. Westfall): • Your data have 50% zeros, and the remaining 50% are roughly normally distributed with mean 100 and standard deviation 10? Tobit model can’t produce data like this! • Heteroskedastic errors make the coefficient of Tobit model badly biased • Non-normality: asymptotic bias was found.

  10. Implementation in R • “censReg” package: • censReg (Y ~ X, left = 0, right =Inf, data = data) • default: left (default to 0), right (default to Infinity) • No left-censored: left = -inf Source: Henningsen, A. (2010). Estimating Censored Regression Models in R using the censReg Package. R package vignettes http://artax.karlin.mff.cuni.cz/r-help/library/AER/html/tobit.html

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