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Censored Regression Models

Censored Regression Models. In contrast to the truncated distribution, with a Censored Random Variable , values in certain range are transformed to single value or or y* is the latent variable and y is observed. Truncated: y = y* when y * ≥ τ. Camp Randall example.

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Censored Regression Models

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  1. Censored Regression Models • In contrast to the truncated distribution, with a Censored Random Variable, values in certain range are transformed to single value or or • y* is the latent variable and y is observed Truncated: y = y* when y* ≥ τ Camp Randall example Gould et al., (1989)

  2. Censored Regression Models • Example: y = y* when τ ≥ 0 Distribution of y* PDF(y*) f(y*) Pr(y*<0) Pr(y*>0) 0 y*

  3. Censored Regression Models y* ≡ latent variable y ≡ observed variable Distribution of y PDF(y) Pr(y*<0) Pr(y*>0) 0 Y Note: When y*>0, y has the PDF of y*

  4. Censored Regression Models y* ≡ latent variable y ≡ observed variable • Assume y*~N(,2) and • This implies • If y* > 0 → y has the PDF of y* transform to standard normal RV ≡ Z Standard normal CDF

  5. Censored Regression Models • The entire distribution of y is a mixture of discrete and continuous components. • The total probability of observing all values is 1 as required • Under truncated model we rescaled the second part to generate a PDF whose integral = 1 • Under censoring we assign the full probability in the censored region to the censoring point • In the above case the censoring point was 0

  6. Censored Regression Models Pr(y*≤τ) Pr(Y* > τ) τ • Mean and variance of a censored RV • Theorem 19.3, Greene p. 847

  7. Censored Regression Models • Moments of a censored RV λ is the IMR = standard normal CDF truncaton results standard normal RV

  8. Censored Regression Models • Moments of a censored RV • Expected value of y, the observed value Pr. of being at the limit = Pr(y* < τ) limit value Expect value if below limit is τ Pr. of being above the limit Expected value if above limit value, E(y|y* > τ), conditional value Truncation results

  9. Censored Regression Models • Moments of a censored RV • With τ = 0 → Prob of being ≤ 0 Prob. of being above 0 E(y|y*>0)

  10. Censored Regression Models • Moments of a censored RV • The above mapping implies that with the above censoring framework, E(y) equals • The probability of being above the limit • Times the expected value of y given that one is above the limit • In terms of the unconditional variance of y IMR as defined above

  11. Censored Regression Models • Assume we have the following latent regression model: • Referred to as the Tobit Model • yi (the observed RV) has the following • Distribution when yi* > 0: • Probability mass function (PMF) when yi* ≤ 0: Pr(yi* ≤ 0) = CDF(yi* = 0)

  12. Censored Regression Models Distribution of y, a Censored RV Non-censored portion of normal PDF of y* Pr(y*≤0) 0

  13. Censored Regression Models • Note the following: Std. Normal CDF Remember, εi~N(0,σ2)

  14. Censored Regression Models • Similar to the above: due to symmetry Std. Normal CDF w/E(εi)=0

  15. Censored Regression Models • 3 Expected Values of Interest • E(y*|X) → Latent Variable • E(y|X) → Observed Data • E(y|X,y* > 0) → Non-Censored Data (Conditional Expectation) • Also interested in the probability of being above (or below) the limit • What are the marginal impacts of changes in X on each model component? • McDonald and Moffitdecomposed the marginal impacts

  16. Censored Regression Models • M &M show a common error that previously existed in the literature • It is not correct to assume that the β coefficients measure the marginal effects for observations above the limit, E(y|y*>0) • From their results: β is the marginal effect only when z = infinity → Φ(z)=1 and φ(z)=0 z ≡ Xβ/σ 16

  17. Censored Regression Models • The Likelihood Function for the Tobit model as the sum of: • The PDF of ε when y* > τ • The PMF of ε when y* ≤ τ Pr(y*≤τ) Noncensored portion of the distribution of y* τ

  18. Censored Regression Models εi = yi* − Xiβ • With εi~N(0,σ2) the Tobitlikelihood function given the above mappingcan be represented as: std. normal PDF Error term dist. std. normal CDF

  19. Censored Regression Models εi = yi*−Xiβ • This implies the following sample log- likelihood function: ln[(1/σ)φ(ε|yiXi)] ln(1−Pr(εi<Xiβ))= ln(Pr(εi<−Xiβ))=ln(Pr(yi*≤0)) Error term dist. −Xiβ 19 19

  20. Censored Regression Models • The sample log- likelihood function can be simplified to the following: where dt = 1 when yt > 0, 0 otherwise • Remember, bothβ and σ2 are parameters to be estimated T1 = no. of nonzero obs. 20

  21. Censored Regression Models • Estimation of Censored Regression (Tobit) Model • Lets look at a flowchart of an estimation algorithm 21

  22. Censored Regression Models ML Est. of Censored (Tobit) Regression Least Squares Starting Values Full/Trunc. Data Define the Tobit LLF Dep. & Exog. Variables Numerical Gradients (jacobian) Numerical Hessian (hessian) Maximum Likelihood Proc. [Newton Raphson] Function to Calc. Asymptotic Analytical Covariance Matrix Estimates of , σ2, LLF Function for Elasticity Impacts on Pr(y* > 0) Functions for Elasticity Impacts on E(y) and E(y|y* > 0) Elasticity Impacts and Standard Deviations

  23. Censored Regression Models • The information matrix for the Tobit model can be represented via the following: i = obs. (K+1) x (K+1) 23

  24. Censored Regression Models • Estimation of Censored Regression (Tobit) Model • Canadian FAFH expenditures • Same data used in truncated example except we include the portion of the sample with 0 FAFH expenditures • 9,767 HH’s, 21.2% with $0 expenditures • Dependent variable is bi-weekly FAFH expenditures • Exogenous Variables: HHInc, Kids Present?, FullTime? Provincial Dummy Variables • Overview of MATLAB code • Supporting equations for parameter variance calculations • Proc returns information matrix

  25. Censored Regression Models • Implications of heteroscedasticity under the Tobit model • Summary of the Tobit Model Value if above limit Prob(being above the limit)

  26. Censored Regression Models • Lets assume the “true” latent model is heteroscedastic w/parameters σ0i2, β0 • The “misspecified” model is: • The use of σ instead of σ0i is the source of the estimation bias if one assumes homoscedasticity • The direction of the bias depends on the assumption about the variance of the error term

  27. Censored Regression Models • Compare this result with the CRM • What is the E(yi) when εi~N(0,σi2) and y does not have a censored distribution? • E(yi)=Xiβ versus the censored (at 0) model where

  28. Censored Regression Models • Implications of Heteroscedasticity under the Tobit model • Remember the LLF for the homoscedastic Tobitmodel is: • How would you modify the above to account for heteroscedasticity given that σ2 is a parameter to estimate?

  29. Censored Regression Models Homoscedastic Heteroscedastic (T + K) parameters 29

  30. Censored Regression Models • Why don’t we replace σ2 with σt2 where we have σt2 a function of another set of exogenous variables? • σt2 =exp(Dtα) > 0 • Estimate the α parameter vector along with β’s via the following: ln(exp(Dtα) ) σt2 ln(Prob(yt= 0)) S + K parameters

  31. Censored Regression Models • Similar to the homoscedastic specification we can simplify the above to the following: where dt=1 if yt> 0, 0 otherwise 31

  32. Censored Regression Models • Lets estimate the FAFH model we previously estimated using the homoscedastic Tobit code but this time assuming the following heteroscedastic specification: • σt2=exp(γ0 + γ1Incomet + γ2Fulltimet + γ3Quebect) • Note, you do not necessarily need to use variables that are part of the X-matrix

  33. Censored Regression Models • As with other specifications, the marginal impacts get a little more complicated under the heteroscedastic Tobit model • Remember the various probability and expected value expressions • A particular exogenous variable could impact both numerator and denominator of the standardization ratio, Xtβ/σt • In the above example Incomet impacts both FAFH expenditures and the error variance (e.g. in both X and D matrices) exp(Dα)

  34. Censored Regression Models • To evaluate the income elasticities under the heteroscedasticTobit all we need to do is to recognize that ∂(Xtβ/σt)/∂X ≠ (β/σt) if income is an explanatory variable in the variance expression • In general we have: • If an exogenous variable (Wi ) is in bothXt and Dt we have the following: Using quotient rule Remember σt2=exp(Dtα) i = variable 34

  35. Censored Regression Models • Given the above, we have: 35

  36. Censored RegressionModels ML Est. of HeteroscedasticTobit Model Homoscedastic Tobit Starting Values (NR) Define Heteroscedastic Likelihood (Het_Tobit_LLF) Dep. & Exog. Variables Error Covariance Structure Numerical Gradiant & Hessian (jacobian, hessian) Maximum Likelihood Proc. (Newton Raphson) Estimates of Θ, ΣΘ Unrestricted LLF Restricted LLF (α2=α3=… =αS= 0) Likelihood Ratio Test 2*(LLFU−LLFR) Homoscedasticity

  37. Censored Regression Models • We have assumed the following error variance structure: σt2=exp(γ0 + γ1Incomet + γ2Fulltimet + γ3Quebect) • Lets review the structure of the R code for estimating a Tobit model where εt~N(0,σt2 ) and the above error variance specification • Overview of Heteroscedastic TobitMATLAB Code 37

  38. Censored Regression Models • Lets compare the elasticity estimates wrt a change in income under homoscedastic and heteroscedastic specifications 38

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