Palgrave Handbook of Econometrics: Applied Econometrics

Ruey S. Tsay 1009

To overcome this weakness, alternative innovational distributions have been
proposed in the literature. Engle and Russell (1998) entertain the Weibull distri-

bution for (^) iand Zhang, Russell and Tsay (2001) consider the generalized Gamma
distribution. The probability density function of a standardized Weibull random
variableXis:
f(x|α)=
{
α
[

(
1 +^1 α
)]α
xα−^1 exp
{
−
[

(
1 +α^1
)
y
]α}
,ifx≥0,
0 otherwise,
(21.7)
where theαis referred to as the shape parameter and(.)is the usual Gamma
function. The mean and variance ofXareE(X)=1 and Var(X)=( 1 + 2 /α)/[( 1 +
1 /α)]^2 −1. The hazard function ofXis:
h(x|α)=α
[

(
1 +
1
α
)]α
xα−^1.
Consequently, ifα>1, the hazard function is a monotonously increasing func-
tion ofx.If0<α<1, then the hazard function is a monotonously decreasing
function ofx.
The probability density function of a generalized Gamma random variableX
withE(X)=1 is:
f(x|α,κ)=
⎧
⎪⎨
⎪⎩
αxκα−^1
λκα(κ)
exp
[
−
(x
λ
)α]
,ifx>0,
0 otherwise,
(21.8)
whereλ=(κ)/(κ+ 1 /α)withα>0 andκ>0. Bothαandκare shape parameters
so that the hazard function ofXbecomes more flexible than that of a Weibull
distribution.
If (^) iof a duration model follows the standardized Weibull distribution with prob-
ability density functionf(x|α)in equation (21.7), the conditional density function
ofxigivenFi− 1 is:
f(x,α)=α
[

(
1 +
1
α
)]αxα− 1
i
ψiα
exp
⎧
⎪⎨
⎪⎩
−
⎡
⎣

(
1 +^1 α
)
xi
ψi
⎤
⎦
α⎫⎪
⎬
⎪⎭
, (21.9)
which can be used to obtain the conditional log-likelihood function of the data
for estimation.
If (^) iof a duration model follows the generalized Gamma distribution withE(
i)=
1 in equation (21.8), the conditional density function ofxigivenFi− 1 is:
f(xi|α,κ)=
αxκαi −^1
(ψiλ)κα(κ)
exp
[
−
(
xi
ψiλ
)α]
, (21.10)
where, again,λ=(κ)/(κ+ 1 /α). This density function can be used to perform
conditional maximum likelihood estimation of the model.
In what follows, we refer to the duration model in equations (21.1)–(21.2) as
the WACD(p,q) or GACD(p,q) model if the innovation (^) ifollows the standardized
Weibull or generalized Gamma distribution, respectively.