Bài 8: Mô hình Tobit

 prices, when controlled by government  working hours, stays zero for unemployed  loan, remains zero for rejected application... Example: non-censored/truncated variable..[r]

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THE TOBIT MODEL

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Censored & Truncated data

 Sometimes we can’t observe values lower or

beyond a certain level, for example:

 dividend, which remains zero until profit reaches a

certain level

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Censored & Truncated data

 y is censored if:

 we can observe all values of y, but

 only in a certain interval, values beyond the interval

are recorded as a constant (example: 0)

 y >= k : censored from below  y <= k : censored from above

 y is truncated if we can only observe it in the

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OLS, censored/truncated variable and the Tobit model

 OLS with censored/truncated dependent variables

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The Tobit model

 Tobit model

 Notes:

 We can observe y*, but can’t observe y  y is the latent variable

*

y x if a y b

y a if y a

b if y b

  

     

 

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Estimation of the Tobit model

 Log-likelihood function (d is dummy: = censored)

 2    2  

2

1

log log log 2 1 1

2

N N

i i i

i i

y X X

L   d  d 

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Case study: credit card balance

 Dependent variable: balance of credit card (USD)  Explanatory variables:

 interest charged (%)  age (year)

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Credit card balance and interest 0 10 00 00 20 00 00 30 00 00 40 00 00 ba la nce

5 10 15 20 25 30

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Tobit model in Stata

right-censored observations 1886 uncensored observations

Obs summary: 1018 left-censored observations at balance<=0

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Hypotheses testing

Prob > F = 0.0000 F( 3, 2900) = 118.33 ( 3) [model]edu = 0

( 2) [model]male = 0 ( 1) [model]age = 0 test age male edu

Prob > F = 0.0000 F( 1, 2900) = 18.21 ( 1) [model]interest = 0

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Marginal effects of Tobit model Three types of marginal effects after Tobit:

 Type 1: the betas indicate how latent

variable y change when regressors x changes.

 Type 2: indicates how y* changes

when x changes, provided that y* is within the boundaries.

 Type 3: indicates how the observed

variable y* changes when x changes.

y x      * *  |

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Marginal effects (Type 2) at average

edu -694.8761 115.1101 -6.04 0.000 -920.4877 -469.2645 male 2610.969 709.4244 3.68 0.000 1220.523 4001.415 age -371.4221 21.45434 -17.31 0.000 -413.4718 -329.3723 interest -277.4943 65.10832 -4.26 0.000 -405.1043 -149.8844 dy/dx Std Err z P>|z| [95% Conf Interval] Delta-method

dy/dx w.r.t : interest age male edu

Expression : E(balance|balance>0), predict(e(0,.)) Model VCE : OIM

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Marginal effects (Type 2) at specific value

edu = 12

at : interest = 12 dy/dx w.r.t : interest age male edu

Expression : E(balance|balance>0), predict(e(0,.)) Model VCE : OIM

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Marginal effects (Type 3) at average

dy/dx w.r.t : interest age male edu

Expression : E(balance*|balance>0), predict(ystar(0,.)) Model VCE : OIM

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Marginal effects (Type 3) at specific value

edu = 12

at : interest = 12 dy/dx w.r.t : interest age male edu

Expression : E(balance*|balance>0), predict(ystar(0,.)) Model VCE : OIM

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Applications of Tobit model

Mayer & Walker (1996) An Empirical Analysis of the

Choice of Payment Method in Corporate Acquisitions

Quarterly J of Bus and Econ 35 (3): 48-65

 Sample: 261 acquisitions 1979-90 Fortune 500

 Dependent variable: % cash financing the acquisition

 independent variables:

 preference of manager on control

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Min&Kim (2003) Modeling Credit Card Borrowing Southern Economic Journal 70(1): 128-43.

 Data: US Survey of Consumer Finance 1998, 2904 inds

 dependent variable: individual credit card balance

 independent variables:

 interest rate charged  income

 liquid assets  taste

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Amuedo-Dorantes (2006) Money Transfer among Banked and Unbanked Mexican Immigrants

Southern Economic J 73(2): 374-401.

 Data: 2928 Mexican immigrants in the US  dependent variable: remittances

 independent variable:

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