Tobit

ZUFE Weicai Ni 4
outline
y* xb u , u N (0, s 2 ) 1: y max(0, y* )
Xb P ( y 0 | x ) 1 Φ ( s ) 2: P ( y 0 | x) Φ ( X b ) s
d (c)Φ (c) Φ (c) (c) 2 dc Φ (c )
' ' 2
c (c)Φ (c) (c) Φ 2 (c ) c
(c )
2 Φ (c) Φ ZUFE Weicai Ni (c )

(c )
2
c 2
9
About Байду номын сангаасnverse Mills ratio
ZUFE Weicai Ni 6
dz=

c
z ( z )dz
Alternative proof
E ( Z | Z c)

ydFZ |Z c ( y )
FZ |Z c ( y ) P ( Z y | Z c ) P ( Z y, Z c) ( ( y ) (c)) I ( y c) P( Z c) 1 (c ) ( ( y ) (c)) I ( y c) E ( Z | Z c) yd 1 (c )
c
( z)
1 (c ) 1 (c) 1 z2 2 1 z2 2 z2 ) c z 2 e dz c 2 e d( 2 1 (c ) 1 (c) 2 1 c (0 e 2 ) (c) 2 1 (c ) 1 (c )
17.2 THE TOBIT MODEL
Condition: Let y be a variable that is essentially continuous over strictly positive values but that takes on zero with positive probability. The Tobit model is most easily defined as a latent variable model: y* = xb + u, u|x ~ Normal(0,s2) (17.18) But we only observe y = max(0, y*) (17.19) we have absorbed the intercept into x for notational simplicity.
ZUFE Weicai Ni 11
《MUS》P523！
E ( y | y 0, x) xb s ( xb / s ) / Φ ( xb / s ) xb s ( xb / s ) (17.24)
Equation（ 17.24）suggests why the OLS regression yi on x i yields an inconsistent estimate-there will be "omitted variables"


c
yd ( y )
1 (c )

(c )
1 (c )
7
ZUFE Weicai Ni
compute E(y|y>0,x)
E ( y | y 0, x) E ( y | y 0, x)
* *
E ( xb u | xb u 0, x) xb E (u | u xb ) xb s E (u / s | u / s xb / s ) xb s ( xb / s ) / [1-Φ (- xb / s )](?) xb s ( xb / s ) / Φ ( xb / s ) xb s ( xb / s ) (17.24)
s
[( y X i b ) / s ] (17.20)
3 : y 0, discrete P( y 0 | x) P( y 0 | x) P(u X b | x)

P(
u
s

X b
s
| x) Φ (
X b
s
) 1 Φ (
Xb
s
)
2
ZUFE Weicai Ni
ZUFE Weicai Ni
13
note
This equation shows that when y follows a Tobit model, E(y|x) is a nonlinear function of x and β , which makes partial effects difficult to obtain. This is one of the costs of using a Tobit model. Once we have estimates of β and σ, we can be sure that predicted values for y, that is , estimates of E(y|x) are positive (for all values of β and σ). [Homework!]
E ( y | y 0, x) xb s 3: E ( y | x) E ( y | y 0, x) xb s
E ( y | y 0, x) b j {1 c 2 } , where c xb / s x j 4: E ( y |, x) b j x j ZUFE Weicai Ni
MLE [ to be improved]
1:we can obtain the log-likelihood function for each observation i: li ( b , s ) 1( yi 0) log[1 ( xi b / s )] 1 1( yi 0) log{ [( yi xi b ) / s ]} (17.22) s 2:The loglikelihood for a random sample of size n is obtained by summing (17.22) n across all i.
L( b , s ) li ( b , s )
i 1
ZUFE Weicai Ni
3
Interpreting the Tobit Estimates
Important to realize that b estimates the effect of x on y*, the latent variable, not y Unless the latent variable y* is what’s of interest, can’t just interpret the coefficient
i
Φi
.
The missing variable can be generated by a probit model that models the probability of the outcome that y >0. i Let d i =1 denote the outcome that y* >0,and let d i 0 otherwise. i The probit estimator can provide a consistent estimate of i A linear regression of y i on x i and ( ˆ i Φi
ZUFE Weicai Ni 1
the density of y
2 : y 0, (continous ) 1
(2s 2 ) 1 2 exp[ ( yi X i b ) 2 / (2s )]
y , y 0 1: y * 0, y 0 2
* *
y 0, P ( yi 0 | X i ) 1 Φ ( X i b / s ) (17.21) P ( yi 0 | X i ) Φ ( X i b / s )
i
Φi
.
) will provide an estimate of b .
12
ZUFE Weicai Ni
compute E(y|,x)
Given E(y|y>0,x), we can easily find E(y|x):
E ( y | x) 0 P( y 0 | x) E ( y | y 0, x) P ( y 0 | x) Φ ( xb / s ) E ( y | y 0, x) [by (17.21), (17.24)] Φ ( xb / s ) [ xb s ( xb / s )] Φ ( xb / s ) xb s ( xb / s ) (17.25)
5
preliminary:
if z～N (0,1), then E ( z | z c) (c) / [1 (c)]
f ( z) ( z) proof : (1) f ( z | z c) P ( z c ) 1 (c )
c
(2) E ( z | z c) z f ( z | z c)dz z
ZUFE Weicai Ni 8
the inverse Mills ratio
(c) = (c)/(c)
is called the inverse Mills ratio[or is called no selected hazard]; it is the ratio between the standard normal pdf and standard normal cdf, each evaluated at c.
1: accounting: information 2: survival analysis : hazard
ZUFE Weicai Ni
10
Importance of (17.23)
1:It shows that the expected value of y conditional on y >0 is equal to xβ, plus a strictly positive term, which is σ times the inverse Mills ratio evaluated at xβ/σ. 2:This equation also shows why using OLS only for observations where yi >0 will not always consistently estimate ; essentially, the inverse Mills ratio is an omitted variable, and it is generally correlated with the elements of x.
What happens if we want to estimate the expected value of y as a function of x? In Tobit models, two expectations are of particular interest: E(y|y>0,x), which is sometimes called the “conditional expectation” because it is conditional on y>0, and E(y|x), which is, unfortunately, called the “unconditional expectation.”