Notes on Count Data Duration Model高级计量Notes(美国A&M大学,甘犁教授)

Count Data and Duration Analysis Li Gan Nov 2007

Count Data Example: # of times someone is arrested during a given year; # of doctor office visits, etc. # of fish caught.

We are interested in how the number of doctor office visits is affected by insurance, etc, among other factors.

E(y|x) = xβ The obvious problem of this simply regression is similar to those of discrete choice models, i.e., the predicted value of y could be negative.

Poisson regression models with cross section Data The density of y given x is determined by its mean: μ(x) = E(y|x):

()()()()!exp|yxx

xyf

yμμ−

=

where y! is y factorial. Often, μ(x) = E(y|x) = exp(xβ). Note that Poisson distribution is restrictive in the sense that: Var(y|x) = E(y|x) A weaker assumption allows the variance-mean ratio to be any positive constant: Var(y|x) = σ2E(y|x)

The case σ2>1 is called overdispersion.

Let E(y|x)=m(x,β). Let ci represent the individual heterogeneity. Assume: yi |xi,ci ~ Poisson[cim(xi,β)]

If we further assume that ci is independent of xi, and has gamma distribution with unit mean and var(ci) = η2. Then the distribution of yi |xi can be shown to be negative binomial.

1 MLE is often used to estimate Poisson models. If we let μ(x)=m(x, β), the likelihood function for Poisson is:

()()()()βββ,,logiiiixmxmyl−=

The likelihood for negative binomial is:

()()

()

()()

()⎟

⎟

⎠

⎞

⎜⎜⎝⎛Γ+Γ+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+=−−−−

−−

2222

222log

,

,log,log,ηηβηββη

ηηηβiiiiii

yxmxmyxml

where ()()∫

∞

−−=Γ

01expdzzzrr

2Duration Models Examples: the time when something dies – a person, a firm, etc. Or when something good happen – the time that somebody finds job (from unemployed state), gets married, gets tenured, etc. Obviously we are not sure about when something would die, so it is a random variable for us.

Let T be such a random variable. F(t) = Pr(T≤t), t ≥ 0 The survivor function is defined the probability that something has not yet died, or the probability that something surviving past time t.

S(t) = 1 – F(t) = Pr(T>t) The hazard function is the probability of dying in the interval [t, t+h) given survival up until time t. The hazard function for T is defined as:

()()htThtTt

th≥+<≤=→|Prlim0λ

Example (unemployment duration: If T is the length of time unemployed, measured in weeks, then λ(20) is roughly the probability of becoming employed between weeks 20 and 21, conditional on having been unemployed through week 20. Note λ(t)is NOT the density of T, which is f(t).

()()

()()()()()()tS

tftFtf

tFhtFhtFhtThtTtthh=−=−⋅−+=≥+<≤=→→111)(lim

|Prlim

00λ

Because the derivative of S(t) is –f(t), we have: ()()dttSdtlog

−=λ

and, using F(0) = 0, we can integrate to get: , and ()()0,exp10≥⎟⎠⎞⎜⎝⎛−−=∫tdsstF

t

λ

3 ()()().exp0⎟⎠⎞⎜⎝⎛−=∫tdssttfλλ

Therefore, all probabilities can be computed using the hazard function. For example, for points a1 < a2,

()()()()⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−=−

−=≥≥∫2

1exp11|Pr1212aadssaFaFaTaTλ, and

()()()()()⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−=−

−=≥<≤∫2

1exp11|Pr112121aadssaFaFaFaTaTaλ

Shape of λ(t): (1) the simplest possible form: λ(t) = λ F(t) = 1 – exp(-λt)

(2) (Weibull distribution): If T has a Weibull distribution, its cdf is given by: F(t) = 1 – exp(-γtα) f(t) = γαtα-1exp(-γtα)

λ(t) = f(t)/(1-F(t)) = γαtα-1

(3) log-logistic hazard function ()α

α

γγα

λ

ttt+=−1

1

When α = 1, the hazard is monotonically decreasing from γ at t = 0 to zero as t Æ

infinity. When α < 1, the hazard is also monotonically decreasing to zero as t Æ

infinity,

but the hazard is unbounded as t approaches to zero. When α = 1, the hazard is increasing until t = [(α -1)/γ]1- α, and then it decreases to zero.

Straightforward integration gives:

()()αγλtdsst+=

∫

1log

0 So

F(t) = 1 – (1+γtα)-1, and

f(t) = γαtα-1(1+γtα)-2

4