Binomial Distribution: Tossing a coin m times. p = probability of having head from a trial. y = # of having heads from n trials (y = 0, 1,..., m).

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1 [7] Count Data Models () Some Dscrete Probablty Densty Functons Bnomal Dstrbuton: ossng a con m tmes p probablty of havng head from a tral y # of havng heads from n trals (y 0,,, m) m m! fb( y n) p ( p) p ( p) y y!( m y)! E(y) mp; var(y) mp(-p) y y y m y Posson Dstrbuton: Let p µ/m for a bnomal dstrbuton y µ µ f p( y) lm m fb( y m) e, y 0,,, y! E(y) var(y) µ LDV-38

2 Dgresson to the Chocolate Chp Cookes problem: Wsh to put at least one CC on a cooke wth 95% he CC machne locates CC s n a cooke followng a Posson Dstrbuton wth µ Pr(y 0) < 005 e -µ < 005 µ > -ln(005) 995 You need a machne wth µ at least 3 Geometrc Dstrbuton: y # of trals untl havng a head f ( y) ( p) y p, y, g E(y) /p; var(y) (-p)/p egatve Bnomal Dstrbuton: y # of trals untl havng r heads y r fn( y) p ( p) r y r E(y) r/p; var(y) r(-p)/p, y r, r+, LDV-39

3 () Basc Posson Regresson Model Assume y d wth Posson(µ ), where µ ep( β) EX: y # of vstng doctors β µ y e β y e ( µ ) e ( e ) f( y ) y! y! ln ( ) β f y e + y ( β ) ln( y!) ( ) { ( ) β l β y β e ln( y!) } l ( β ) β Σ ( Σ y ) he Posson ML estmator of β can be vewed as a GMM estmator based on ( ) E ( y ) 0 hs moment condton s vald as long as E( y ) It means that the Posson ML estmator of β s consstent even f the Posson assumpton s ncorrect ( ) ( B β y β Σ ) ; H ( β ) Σ e If the Posson assumpton s truly correct, use [ ( ˆ )] H βpoi ML or [ B ( ˆ β )] POI ML as an estmate of Cov( ˆ β ) POI ML LDV-40

4 If you are not sure, use [ H ( ˆ β )] POI ML ( ˆ B βpoi ML) [ ( ˆ H βpoi ML)] For the measures of goodness of ft, see Greene In fact, we can estmat by LLS appled to y ε + wth heteroskedastc error terms Dgresson to LLS wth Heteroskedastc Errors: y h(,β) + ε Let H(,β) h (, β ) β he LLS estmator of β ( ˆL β ) mnmzes: ( (, )) Σ y h β ( ) Cov( ˆ β ) Σ H (, ˆ β ) H (, ˆ β ) where L L L Σ t L L e H(, ˆ β ) H(, ˆ β ) ( ˆ ˆ t H( t, βl) H( t, βl) ) Σ End of Dgresson e y h(, ˆ β ) ML LDV-4

5 () Compound Posson Model (egatve Bnomal Model): Hausman, Hall, Grlches (HHG, ECO, 984), and Cameron and rved (C, JAE, 986) β α α Assume that the y follow Posson(λ ), whereλ e µ e, µ, Ee ( α ), and the e α follow a Gamma dstrbuton: fgamma ( η) θ ( θ ) θ θη θ e η, Γ + where 0 < η < and θ t Γ ( θ ) t e dt 0 Here, α s an unobservable ndvdual effect Dgresson to Gamma dstrbuton: he most general form of the Gamma densty functon s gven: f y y e ( αβ ) α y / β gen gamma( αβ, ), Γ α where y s a contnuous postve random varable ( y > 0) E(y) αβ and var(y) αβ f ( η) s obtaned by settng y η, α θ and β /θ gamma Choose α θ and β /θ to make E(η) [a normalzaton] End of Dgresson LDV-4

6 ote β+ α β e β+ α y e u β y e ( e ) e ( e u ) f( y, u) y! y! hen, where f ( y ) f( y, u ) f ( u ) du gamma 0 r Γ ( θ + y ) θ y r ( r), Γ ( y + ) Γ( θ ) θ, Γ () ( ) ( ) + θ s s Γ s, and Γ () s ( s )! f s s an nteger hus, when θ s a postve nteger, ( θ + y ) f y r r θ θ θ ( ) ( ) θ + y ( ), hs s the form of the negatve bnomal dstrbuton Compound Posson egatve bnomal dstrbuton! Snce (θ+y ) follows eg-bn, E(θ+y ) θ/(-r ) e β var(y ) var(θ+y ) θ(-r )/r + θ E(y ) If we allow θ to vary over and set θ α E(y ) ; var(y ) (+α) e β + e θ β, we have hs model s called eg-bn model (HHG) LDV-43

7 If we set θ /α, E(y ) ; var(y ) ( ) ( ) e β αe β e β α e β + + hs model s called eg-bn model (HHG) Comment: Posson, eg-bn and eg-bn assume that E(y ) ep( β) If ths mean specfcaton s correct, the Posson, eg-bn, eg-bn MLE are all consstent as long as the true dstrbuton belongs to the lnear eponental famly [see Gourerou, Monfort and rognon (ECO, 984)] (3) estng Posson: H o : E(y ) var(y ) H a : E(y ) (Posson); β β, but var(y ) e α ( e ) s + and α 0 [If s, H a eg-bn If s, H a eg-bn ] For gven s, under H o, ( { }) E {( y µ ) y} E ( y µ ) y 0; µ µ var {( y µ ) y}, µ LDV-44

8 where µ hen, under H o, CL mples: L {( y ) y} Σ µ µ ( s ) Σ ( µ ) s µ (0,) Snce µ s unobservable, we need to use ˆ ˆ µ, where ˆ β s the Posson ML estmator But, we stll can show that under H o, ˆ L {( y ˆ ) y} Σ ˆ µ µ ( s ) Σ ( ˆ µ ) s ˆ µ (0,) H o may hold even f the y do not follow Posson For such cases, use ˆ L {( y ˆ ) y} Σ ˆ µ µ s ˆ µ { y} s Σ ( ˆ µ ) Σ ( y ˆ µ ) ( ) ˆ µ LDV-45

9 (4) Posson Model for Panel Data Assume y t d wth Posson(λ t ), where λ t ep( t β+α ) µ t ep(α ) Fed Effects Model reat α as parameters Surprsngly, MLE s consstent! λt yt e ( λt ) f( yt t ) y! t ln f ( y ) λ + y ( β + α ) ln( y!) t t t t t t (,,,, ) { ( ) t β α l βαα α t yt + t β α e ln( yt!)} l ( β, α,, α ) α t { yt e α µ t} Σ y t ln µ t t α j Σt j Σ Σ + Σ 0 Substtute these solutons nto l : l ( β ) Σ Σ y ln p ( β ), c t t t where p t ep( t β ) Σ ep( β ) t t c he MLE estmator of β based on l ( β ) s consstent even f the true dstrbuton of y t s not Posson as long long as E(y t t,α ) ep( t β)ep(α ) [Wooldrdge (JEC, 999)] For correct LDV-46

10 covarance matr of the ML estmator of β, use the robust form [( H ) B ( H ) ] Random Effects Model Assume that the e α follow a Gamma Dstrbuton f( y, y,, u ) f( y, α ), t t t µ tu y e ( µ tu ) t yt! yt ( µ ) ( ) t Γ θ +Σ t t yt Σ ( ( )!) ( ) t Γ θ y t t Σt µ t gamma 0 f ( y,, y, ) f( y,, y,,, u ) f ( u ) du where Q θ θ +Σ µ θ Q ( Q) y t t l βθ f y y t (, ) Σ (,,,, ) t Σ Can use the Hausman test to determne whether RE or FE s correct t y t LDV-47