Econometric Analysis of Panel Data
Professor William Greene
Phone: 212.998.0876
Office: KMC
778
Home page:www.stern.nyu.edu/~wgreene
Email: [email protected]
URL for course web page:
www.stern.nyu.edu/~wgreene/Econometrics/PanelDataEconometrics.htm
Assignment 5
Nonlinear Models
Part I.
Weibull Regression Model
In class, we examined a ‘loglinear,’ exponential regression model,
i
i
i
i
y
1
f(y 
,1)
exp
⎛
⎞
=
−
⎜
⎟
θ
θ
⎝
⎠
i
x
,
θ
i
= exp(
x
i
′β
) = E[y
i

x
i
]
The Weibull model is an extension of the exponential model which adds a shape parameter,
γ
;
1
i
i
i
i
i
y
y
f(y 
, )
exp
γ
γ−
γ
⎛
⎞
⎡
⎤
γ
⎜
⎟
γ
=
−
⎢
⎥
⎜
⎟
θ
θ
⎣
⎦
⎝
⎠
i
x
E[y
i

x
i
]=
Γ
[(
γ
+1)/2]
θ
i
=
.5*sqr(
π
)
if
γ
= 2.
The exponential model results when
γ
= 1.
(This distribution looks like, but is not the gamma
distribution we discussed in class.)
An interesting special case is the Rayleigh distribution, which has
γ
= 2.
The resulting density is
2
i
i
i
2
i
i
2y
y
f(y 
,2)
exp
⎛
⎞
⎡
⎤
⎜
⎟
=
−
⎢
⎥
⎜
⎟
θ
θ
⎣
⎦
⎝
⎠
i
x
One of the interesting things about the Rayleigh distribution is that E[y
x
i
]= .5
π
θ
i
(compared to
θ
i
for the exponential.
.5
π
is approximately equal to 0.866.)
One difference is the variance.
The
variance of the exponential variable is
θ
i
2
.
The variance of the Rayleigh variable is [
Γ
(2) 
Γ
2
(1.5)]
θ
i
2
.
Department of Economics
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