Post on 17-Jul-2016
description
Poisson Distribution• A Poisson distribution is given by:
Where, is the average number of occurrences in a specified interval
..2,1,0,!
]Pr[ yy
eyY
y
Assumptions:• Independence• Prob. of occurrence In a short interval is proportional to the length of the interval• Prob. of another occurrence in such a short interval is zero
Poisson Model• The dependent variable is a count variable taking
small values (less than 100).• It has been proposed that the count dependent
variable follows a Poisson process whose parameters are determined by the exogenous variables and the coefficients
• Justified when the variable considered describes the number of occurrences of an event in a give time span eg. # of job-related accidents=f(factory charact.), ship damage=f(type, yr.con., pd.op.)
Specification of the Model• The primary equation of the model is
• The most common formulation of this model is the log-linear specification:
• The expected number of events per period is given by
..2,1,0,!
]Pr[ ii
yi
ii yy
eyYii
'ln ii x
'
]|[ ixiii exyE
Specification….
• The major assumption of the Poisson model is :
• Later on when we do diagnostic testing, we will test this assumption. It is called testing for over-dispersion (if Var[y]>E[y]) or under- dispersion (if Var[y]<E[y])
• Thus:ii
i
ii
xxyE
]|[
]|[]|['
iix
iii xyVarexyE i
Estimation• We estimate the model using MLE. The
Likelihood function is non linear:
• The parameters of this equation can be estimated using maximum likelihood method
• Note that the log-likelihood function is concave in and has a unique maxima. (Gourieroux[1991])
n
iiii
n
iiiii yxyeyxyL i
1
'
1
' ]!ln[]!ln[ln'
x
0]][['
1
i
n
ii yeL
i
xx
Estimation….• The Hessian of this function is:
• From this, we can get the asymptotic variance- covariance matrix of the ML estimator:
• Finally, we use the Newton-Raphson iteration to find the parameter estimates:
)()(1
)()1( iiii gH
]['
'
1
'2
ieLH
n
iii
xxx
1
1
' ]][[)ˆ(var'
ien
iiiasy
xxx
Interpretation of the coefficients• Once we obtain the parameter estimates, i.e.
estimates , we can calculate the conditional mean:
Which gives us the expected number of eventsper period.• Further, if xik is the log of an economic variable,
i.e. xik = logXki, can be interpreted as an elasticity
ik
iik X
yElog
][logˆ
ik
'iei
x
Diagnostic Testing• As we had mentioned before, a major assumption
of the Poisson model is:]|[]|[ '
iiiii xyVarexyE i x
• Here the diagnostic tests are concerned with checking for this assumption
1. Cameron and Trivedi (1990) test H0 : Var (yi) = H1 : Var (yi) = + g( ), usually g( )= or
Test for over or under dispersion is =0 iniiiii yy
^22
^
)(
We check the t-ratio for
i
i ii i 2
i
Diagnostic Testing…• An alternative approach is by Wooldridge(1996)
which involves regressing the square of standardized residuals-1 on the forecasted value and testing alpha = 0 in the following test equation
ii
i
iiy
1)( 2
• In case of miss-specification, we can compute QMLestimators, which are robust – they are consistent estimates as long as the conditional mean in correctlyspecified, even if the distribution is incorrectlyspecified.
Diagnostic Testing…• With miss-specification, the std errors will not be
consistent. We can compute robust std errors using Huber/White (QML) option or GLM , which corrects the std errors for miss-specification.
• For Poisson, MLE are also QMLE• The respective std errors are:
n
ii
ii
MLGLM
yKN 1
22
2
)(1
)(var)(var
Where,
And,
1'
1
ˆˆ)(var
HggHi
iiQML