Lecture "Advanced Econometrics (Part II) - Chapter 6: Models for count data" presentation of content: Poisson regression model, goodness of fit, overdispersion, negative binomial regression model, too many zeros data.
AMBIENT/
Chủ đề:
Nội dung Text: Lecture Advanced Econometrics (Part II) - Chapter 6: Models for count data
- Advanced Econometrics - Part II Chapter 6: Models for count data
Chapter 6
MODELS FOR COUNT DATA
A count variable is a variable that takes on non-negative integer values:
• There is no natural upper bound
• The outcome will be zero for at least some members of the population
Y is count variable, X is a vector of explanatory variables. It is better to model E (Y X )
directly and to choose functional forms that ensure possibility for any value of X and any
parameter value.
When Y has no upper bound, the most popular of these is the exponential function
E (Y X ) = exp( Xβ )
I. POISSON REGRESSION MODEL:
• The basic Poisson regression model assumes that Y given X = ( X 1 , X 2 ,..., X k ) has a
Poisson distribution.
• The Poisson regression model specifies that each Yi is drawn from a Poisson
distribution with parameter λi , which is related to the regressor X i .
e − λi λYii
Pr ob(Y = Yi X i ) = (Yi ! = 1 × 2 × ... × Yi )
Yi !
λi and X i are related as: ln λi = X i β or λi = e X β i
The expected number of events is given by: E[Yi X i ] = Var[Yi X i ] = λi = e X i β
(Poisson distribution properties)
∂E[Yi X i ]
So: = λi β
∂X i
• With the parameter estimate in hand, this vector can be computed using any data vector
desired.
• In principle, the Poisson model is simply a non-linear regression, but it is easier to
estimate the parameters with maximum likelihood techniques. The log-likelihood
function is:
Nam T. Hoang
UNE Business School 1 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
n n
ln ∑ [-λi + Yi ( X i β ) − ln(Yi !)] =
L == ∑ [-eXi β + Y ( X i β ) − ln(Yi !)]
=i 1 =i 1
The likelihood equations are:
∂ ln L n
= ∑ (Yi − λi ) X i = 0
∂β i =1
n
= ∑ (Yi − e X i β ) X i = 0
i =1
The Hessian is:
∂ 2 ln L n
∂β∂β '
= − ∑
i =1
λi X i X i'
The Hessian is negative definite for all X and β . Newton-Raphson method is a simple
n
algorithm for this model and will converge rapidly. At convergence ∑ λˆi X i X i' is an
i=1
estimator of the asymptotic covariance matrix for β . λˆi = exp( X i βˆ ) .
λˆi is the prediction for observation i → λˆi = exp( X i βˆ ) estimated variance of λˆi will be
λˆi2 X i'VX i , where V is the estimated asymptotic covariance matrix for βˆ ,
−1
n
V = ∑ λˆi X i X i'
i =1
II. GOODNESS OF FIT:
2
Y − λˆ
n
∑ i i
i =1 λˆi
R2 = 1 − 2
n
Yi − Y
∑
i =1 Y
This measure compares the fit of the model with a model of only one constant term.
Note: Yi is integer, the prediction λˆi = e X i β is continuous.
ˆ
III. OVERDISPERSION:
Poisson model has been criticized because of its implicit assumption that the variance of
Yi equals it's mean. Many extensions of Poisson model that relax this assumption have
been proposed.
Test for over dispersion:
Nam T. Hoang
UNE Business School 2 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
H o : Var[Yi ] = E[Yi ]
H A : Var[Yi ] = E[Yi ] + αg ( E[Yi ])
(Yi − λˆi ) 2 − Yi
Regress: Z i =
λˆi 2
Most of count model with overdispersion (variance exceeds the mean) specify
overdispersion to be the form:
Var[Yi X i ] = E[Yi ] + αg ( E[Yi ])
Where α is unknown parameter, g(.) is a known function most commonly g ( µ ) = µ 2 ,
or g ( µ ) = µ .
Test: Ho : α = 0
H A : α ≠ 0 (or α > 0)
Can be carried out by running the regression:
(Yi − λˆi ) 2 − Yi g (λˆi )
=α + ui
λˆ i λˆ i
Where ui is an error term. The reported t-statistic for α is asymptotically normal under
H o : α = 0 (Cameron & Trivedi 1990). This test can be also used for underdispersion
α < 0 , in which case the conditional variance is less than the conditional mean.
Conditional mean & variance µi of the Poisson distribution, suppose now that the
parameter is random rather than being a completely deterministic function of regressor
Xi .
Let: µi = λi ui → ln µi = ln λi + ln ui = X i β + ε i
Xiβ εi
This distribution of Yi conditioned on X i and ui (ε i ) remain Poisson with conditional
mean and variance µi :
e − λiui (λi ui )Yi
f (Yi X i , ui ) =
Yi !
Prob(Y Y=
= i X i , ui )
∞
e − λiui (λi ui )Yi
→ Pr ob(Y = Yi X i ) = ∫ g (ui )dui
0
Yi !
Nam T. Hoang
UNE Business School 3 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
g (ui ) is density function of ui
The choice of g (ui ) defines the unconditional distribution. For mathematical convenience, a
gamma distribution is usually assumed for ui = (ε ε i ) . Assume E (ui ) = 1 (for E ( µi λi ) = λi ).
θ θ −θu θ −1
→ g (ui ) = e ui i
Γ(θ )
This density function for Yi is then
∞
e − λi ui (λi ui )Yi θ θ uiθ −1e −θ ui
Prob(
= Y Y=
i Xi ) f (Y=
i Xi ) ∫
0
Yi ! Γ(θ )
dui
Γ(θ + Yi ) λi
r Yi (1 − ri )θ where ri =
Γ(Yi + 1)Γ(θ ) λi + θ
IV. NEGATIVE BINOMIAL REGRESSION MODEL:
• The assumed equality of the conditional mean and variance is the major shortenings of
the Poisson model.
• We generalize the Poisson model by introducing an individual unobserved effect into the
conditional mean.
• Suppose now that the conditional mean & variance µi of the Poisson distribution is
random rather than being completely deterministic function of X (Because of unobserved
heterogeneity different obs may have different µi . λi is an parameter of Poisson but part of
this difference is due to a random (unobserved) component ui not only because of X i µi
is just a parameter of distribution we want E ( µi ) = e X i β we don’t want µ i = e X i β
Let: µi = λi ui
Where λi = e X i β
→ ln µi = ln λi + ln ui = X i β + ε i
The disturbance ε i reflects cross-sectional heterogeneity that normally characterizes
micro-economic data.
The distribution of Yi conditional on X i and ui remain Poisson with conditional
mean & variance µi :
e − λi ui (λi ui )Yi
Prob(Y Y=
= i X i , ui )
= f (Yi X i , ui )
Yi !
Nam T. Hoang
UNE Business School 4 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
The unconditional distribution Pr ob(Y = Yi X i ) is the expected value over ui of
f (Yi X i , ui ) .
∞
e − λi ui (λi ui )Yi
Prob(
= Y Y=
i Xi ) ∫
0
Yi !
g (ui )dui
g (ui ) is a density function of ui problem: the choice of g (ui ) ?
For mathematical convenience, a gamma distribution is usually assumed for ui .
Assume E (ui ) = 1 (for E ( µi λi ) = λi ).
θ θ −θu θ −1
→ g (ui ) = e ui i
Γ(θ )
Then:
∞
e − λi ui (λi ui )Yi θ θ uiθ −1e −θ ui
Prob(
= Y Y=
i Xi ) ∫0 Yi ! Γ(θ )
dui
∞
θ θ λYi i
Γ(Y + 1)Γ(θ ) ∫
−( λi +θ ) ui
= e uiθ +Yi −1dui
i 0
θ θ λYi Γ(Yi + θ )
i
=
Γ(Yi + 1)Γ(θ )(λi + θ )θ +Y i
Γ(Yi + θ )
= riYi (1 − ri )θi
Γ(Yi + 1)Γ(θ )
λi
where ri =
λi + θ
This is the form of the negative binomial distribution the distribution has conditional
1
mean λi and conditional variance λi 1 + λi
θ
→ E [Yi X i ] = λi = e X i β
1
Var [Yi X i ] = λi = λi 1 + λi
θ
∞
∫ t e dt
P −1 − t
Note: gamma function: Γ( P ) =
0
We have: Γ( P) = ( P − 1)Γ( P − 1)
→ Γ( P) = ( P − 1)! if P is a integer number gamma function is a generalization of the
factor function for non-integer values.
Nam T. Hoang
UNE Business School 5 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
which is the form of the negative binomial distribution.
E (Yi X i ) = λi
1
Var (Yi X i ) = λi 1 + λi
θ
λP
Note: gamma distribution: f ( x) = e −λx x P−1
Γ( P)
If E ( x) = 1 → λ = P x ≥ 0, λ > 0, P > 0
because E ( x) = P P
λ V ( x) = λ2
E ( µi λi ) = λi if E (ui ) = 1 the interpretation as in the Poisson model
E (Yi X i ) = λi
1
V (Yi X i ) = λi (1 + λi )
θ
This negative binomial model can be estimated by maximum likelihood without much
difficulty. A test of a the Poisson distribution is often carried out of testing the hypothesis
α = 1θ = 0 using the Wald test.
E (Yi X i ) = λi = e X i β
1
Var (Yi X i ) = λi 1 + λi
θ
λ
The ratio of the variance to the mean now is 1 + i > 1 , different for different
θ
observations.
The log-likelihood:
n
L = ln = ∑ ln Γ(Y + θ ) − lnΓ(Y + 1) − ln Γ(θ ) + Y ln r + θ ln(1 − r )
i =1
i i i i i
λi
ri = ; λi = e X i β
λi + θ
can be estimated by MLE easily.
Application: X it = (1, Age, Education, Income, Kids, Insurance)
Doctor visits: count data models
Nam T. Hoang
UNE Business School 6 University of New England
- Advanced Econometrics - Part II Chapter 6: Models for count data
V. TOO MANY ZEROS DATA:
In many data sets, there is large number of zero counts. Assuming Poisson or negative
binomial is then a misspecification. Alternative is the zero-Inflated Poisson model.
• A binary probability model determines whether a zero or a nonzero outcome occurs then.
• A truncated Poisson distribution describes the positive outcomes.
Prob(
= X i ) e −θ
Yi 0=
(1 − e −θ )e − λi λi j
Prob(
= Yi j=
Xi )
j !(1 − e − λi )
Prob(
= Wi ) F (Wi , γ )
Z i 1=
e − λi λi j
Prob(Y=
i j X i , Z=
i 1)=
j!
E (Yi X i ) = F × 0 + (1 − F ) × E[Y * X i , Yi* > 0 =
λi
= (1 − F )
1 − e −λi
Where Y * denote the outcome of the Poisson process in the regime 2.
Prob(Yi =0 X i ) =Prob(regime 1) + Prob(Y =0 X i , regime 2) * Prob(regime 2)
Prob(
= Yi j=
X i ) Prob(
= Yi j X i , regime 2) * Prob(regime 2)
Nam T. Hoang
UNE Business School 7 University of New England
Thêm vào BST
Báo xấu
Download
Vui lòng tải xuống để xem tài liệu đầy đủ
