Modelling and Estimating Individual and Firm Effects with Count Panel Data*

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1 Forthcomng n Astn Bulletn Modellng and Estmatng Indvdual and Frm Eects wth Count Panel Data* Jean-Franços Angers a, Dense Desjardns b Georges Donne b and Franços Guertn c 9 January 208 Abstract We propose a new parametrc model or the modellng and estmaton o event dstrbutons or ndvduals n derent rms. The analyss uses panel data and takes nto account ndvdual and rm eects n a non-lnear model. Non-observable actors are treated as random eects. In our applcaton, the dstrbuton o accdents s aected by observable and non-observable actors rom vehcles, drvers, and leets o vehcles. Observable and unobservable actors are sgncant to explan road accdents, whch means that nsurance prcng should take nto account all these actors. A xed eects model s also estmated to test the consstency o the random eects model. Keywords: Indvdual eect, rm eect, non-lnear model, panel data, Posson, Drchlet, nsurance prcng, R code. JEL codes: C23, C25, C55, G22. a Department o Mathematcs and Statstcs, Unversty o Montreal, Montreal b Canada Research Char n Rsk Management, HEC Montréal, Montreal c Calcul Québec, Unversty o Montreal, Montreal Contact author: Georges Donne, Canada Research Char n Rsk Management, HEC Montréal, 3000, Chemn de la Côte-Sante-Catherne, room 4.454, Montreal (Qc), Canada, H3T 2A7. georges.donne@hec.ca. *Acknowledgements We thank the Socété de l assurance automoble du Québec or provdng the data and Helm Jedd or hs very useul comments. We also thank Vctore Mchal or her excellent assstance n programmng the R code.

2 . INTRODUCTION Snce the early 980s, several researchers have proposed derent models to account or correlaton resultng rom temporal repettons o observatons. Indeed, the use o panel-type ndvdual data has become popular n many economc applcatons n the elds o labor economcs, health economcs, rm productvty, patents, transportaton, and educaton (Hausman and Wse, 979; Goureroux et al, 984; Hausman et al, 984; Cameron and Trved, 996; Hsao, 986; Baltag, 995; Donne et al, 997, 998). In the doman o count-data applcatons, the ground-breakng contrbuton s the artcle o Hausman et al (984) that proposes a Maxmum Lkelhood Method (MLE) or estmatng the parameters. In ths artcle we extend Hausman et al s (984) parametrc model to add a rm eect to the ndvdual eect n the estmaton o event dstrbutons, and we apply the model to the accdent dstrbutons o trucks belongng to leets o vehcles. 2 To our knowledge there s no non-lnear econometrc model n the lterature that estmates ndvdual and rm eects wth panel data. The matchng o longtudnal ndvdual and rm data s very mportant n envronments where the observed results (here accdents) are a uncton o both partes characterstcs (here, ndvdual and rm) and unobserved actons. For nsurance companes, knowng all the sources o accdents nvolvng vehcles belongng to a leet s essental to develop a ar prcng scheme that takes nto account the neglgence o each actor. Ths s also mportant or the regulator, whch has to compute the optmal nes o derent nractons (drver, leet owner) that aect accdent dstrbutons (Fluet, 999). In our applcaton, we estmate the dstrbuton o vehcle accdents or derent leets over tme, by rst decomposng the explanatory actors nto heterogeneous actors lnked to vehcles and ther drvers, then nto heterogeneous actors lnked to leets, and nally nto resdual actors. See also Gouréroux et al (984) who propose a pseudo-mle treatment o the data. 2 On nsurance applcatons wth non-parametrc models, see Pnquet (203), Fardlha et al (206), and Desjardns et al (200). On accdent dstrbuton estmaton or nsurance prcng see Purcaru and Denut (2003), Boucher, Denut, and Gullen (2008), Boucher and Denut (2006), Angers et al (2006), Frangos and Vrontos (200), Frees and Valdez (20), and Cameron and Trved (203a). Another class o models uses the herarchcal credblty approach wth random eects n lnear models (Norberg, 986). It can be shown that the Negatve Bnomal model s a knd o herarchcal model (see Secton 2.2 o ths artcle). All these contrbutons do not consder separately trucks and leets eects. 2

3 Factors lnked to vehcles and drvers and those lnked to leets can be correlated. For example, a neglgent manager may not spend enough money on mechancal repar o hs trucks and mght ask hs employees to drve too ast. However, the employees may also exceed the speed lmt wthout normng the manager. As mentoned elsewhere, the modellng o Hausman et al (984) s not drectly applcable to the ex-post calculaton o nsurance premums usng a Bayesan model (Angers et al, 2006). However the extended model we propose can be used or the nsurance prcng o vehcles that ncludes ndvdual and rm eects. Our model can also be appled to any count modellng wth random ndvdual and common eects. We may thnk o derent prncpal-agent output such as operatonal rsk events n banks, nnovatons n teams, deaths n hosptals, arlne accdents, or any other event nvolvng many agents workng or derent prncpals under asymmetrc normaton (Holmstrom, 982; Laont and Martmort, 200). Ths research uses parametrc models exclusvely. Frst, we want to compare our results wth those o Hausman et al (984), who use parametrc estmaton methods n ther study. It s well known that parametrc models nvolve explct assumptons about the statstcal dstrbuton o the data and, thus, hypothess testng nvolves estmaton o the key parameters o the chosen dstrbuton. Gven that nonparametrc models are dstrbuton-ree, they could be appled n a broader range o stuatons even where the parametrc condtons o valdty are not met. In our case we study accdent dstrbutons. The parametrc Posson amly s known to satsy the necessary condtons o valdty or accdent data. Another advantage o the nonparametrc test s ts ablty to handle varous data types even measured mprecsely or they comprse outlers, anomales wdely known to serously aect parametrc tests. However, the oremost advantage o usng parametrc models s the statstcal power o the estmatons when the assumptons are satsed. Under these crcumstances, parametrc tests produce more accurate and precse estmates than do nonparametrc tests. Thereore, snce our data set meets the sample sze requrements, s very precse, and does not contan outlers that could not saely be removed rom the dataset, we nd t reasonable to consder the above statstcal power argument as a thrd argument n avor o the use o parametrc models. 3

4 Lastly, parametrc models are very convenent when we wsh to obtan close-orm expressons or rsk and premum orecastng. In secton 2, we propose a short lterature revew o count data models, and secton 3 develops our econometrc model. Sectons 4 and 5 present the data and the results o our estmatons. We also analyze our results based on varous statstcal perormance crtera, ncludng accdents predcton or the next year. Fnally, we compare our random eects estmators wth those obtaned rom a xed eects model and we test the consstency o the random eects model. Secton 6 concludes the paper. 2. LITERATURE 2. BASIC COUNT DATA MODELS Our presentaton s based on trucks accdents but the model can be appled to any other event nvolvng count data. Most o the econometrc models appled to count varables that takes nonnegatve values start rom the Posson dstrbuton, where the probablty o truck o leet beng nvolved n y t observable accdents (or clams) n perod t s estmated. By denton o the Posson law, the mathematcal expectaton o the number o accdents s equal to the varance, E ( Y ) Var ( Y ) = = λ where Y t s the random varable representng the t t t number o accdents o truck, leet n perod t and λ t ( > 0) s the Posson parameter equal to the mean and the varance o the dstrbuton. In act, the parameter exp( X ) vector Xt ( x t,,xtp) λ = β, where the = represents the p characterstcs o truck o leet observed n perod t t t and β s a vector o parameters to be estmated. The exponental orm o λ t ntroduces a nonlnear relatonshp between accdents and control varables ncluded n X t. X t can contan contnuous varables and such varables can be non-lnear. For example, x t 2 can be the klometers drven and x t3 the square o the klometers drven. Moreover, X t can contan categorcal varables wth a xed number o possble values such as sze o the leet or number o trac volatons. These varables can also ntroduce non-lnear eects between accdents and derent 4

5 observable categores. All these characterstcs are applcable n the usual Generalzed Lnear Model (GLM) settng. Posson model can also ncorporate data that are collected spatally by ntroducng a spatal autocorrelaton term or n a Generalzed Addtve Model (GAM) settng by addng smooth unctons. It s not clear however that such extensons would sgncantly mprove our results or the type o basc data we used where spatal or strong non-lnear eects are absent. The Posson model s an equdsperson model. Ths modellng mplctly supposes that the dstrbuton o accdents can be explaned entrely by observable heterogenety. To take nto account the overdsperson property n the data, we can suppose that the parameter λ t has a Xt random term such that λ = e β+ε =αγ wth α = e ε Xt and γ = and where α s the t t t e β random ndvdual specc eect or truck. Suppose that α ollows a gamma dstrbuton o parameter (, ) where ( ) δ δ, we obtan the negatve bnomal dstrbuton 3 (NB2): ( ) ( yt ) ( ) ( yt ) δ y Γ δ + t δ γ t t γt δ = δ +γt δ +γt P Y, Γ δ Γ + Γ s the gamma uncton. The mean remans equal to ( ) t, () exp X β and the varance to mean rato s equal to ( +δ ) δ. (Hausman et al, 984; Cameron and Trved, 986; Boyer, Donne, and Vanasse, 992). Ths modellng does not smply allow or overdsperson. It also lets us consder unobserved or latent heterogenety that s absent rom the Posson model. Unobserved heterogenety s very mportant or prcng nsurance premums under asymmetrc normaton (Donne and Vanasse, 989, 992). The above modellng s approprate or ndependent observatons, meanng those wthout ndvdual and tme eects, and cannot be approprate or panel data. 4 3 For an analyss o the Posson lognormal mxture see Greene (2005). 4 Note that the Posson model can also be estmated wth panel data. We do not consder ths possblty here. See Cameron and Trved (203b) or a detaled analyss. 5

6 2.2 Takng tme nto account Let us now consder data that contan observatons where the same unt (ndvdual or truck, or example) s observed over several successve perods but wthout rm or group eects. There are two treatments or panel data n the lterature, the xed eects and the random eects model. In ths secton we lmt our dscusson to the random eects Negatve Bnomal (NB) model appled to short perods o tme where the number o perods s xed and the number o ndvduals s large. Hausman et al (984) propose an extenson o the model expressed by equaton (), whch s not desgned to take nto account repettons o observatons over tme. The new model s a herarchcal model that comes drectly rom the Posson model. Thus accordng to the NB2 model o parameters αγ t and φ, where Y t would be dstrbuted α and φ vary across ndvduals. α s the random rm specc eect and φ s an addtonal random eect that permt the random rm specc eect to vary over tme (Hausman et al, 984). Suppose that ( ) +α φ ollows a beta dstrbuton o parameters (a,b), we obtan a closed orm soluton or the random eects negatve bnomal model: Γ a+ ( ) γt Γ b yt a b + Γ + t t Γ ( yt +γt ) P( Y,,Y T ) =. Γ ( a) Γ ( b) t Γ ( γ t ) Γ ( yt + ) Γ a+ b+ γ t + yt t t (2) The NB2 model can also be estmated wth ndvdual dummes (or other methods) n a xed eects verson. The β parameters can be nconsstent however because o the ncdental parameters problem, but some contrbutons have shown that the nconsstency may be not mportant (Allson and Waterman, 2002; Green, 2004). Estmatng the random eects model n (2) can also yeld nconsstent random eects estmators because α and the vector o observable ndvdual characterstcs may be correlated. We can apply the Hausman (978) test statstc to determne whether or not we should reject the null hypothess that the ndvdual eects are not correlated wth the varables n the regresson component. The model n (2) s sutable or estmatng parameters wth ndvdual eects but cannot take nto account 6

7 the rm or the leet eect when ndvdual observatons belong to derent rms wth common characterstcs that can aect accdent dstrbutons. 3. Methodology: Takng tme and rm eects nto account smultaneously We now move on to the generalzaton o the model, whch wll allow us to account, smultaneously, or the ndvdual eect, the rm eect and the tme eect. 5 We are nterested n observatons that have common characterstcs because they belong to the same rm, or example: workers n a rm, vehcles n a leet, patents n a hosptal or chldren attendng the same school. max We consder a set I,...,I o dates and a collecton T I For each we may reer to t as = o ndvduals, a partton reers to the same element o T whenever j. max I,..., I,..., I F o I, a set T =,...,T where T s a subset o T. I s the number o trucks n leet.,2,...,t, keepng n mnd that T = and Tj = does not necessarly The vector Xt ( x t,,x tk,,xtp) = stll represents the p characterstcs o ndvdual rom rm observed n perod t. Here we can have many derent rms over a gven number o perods. For example, the vector may contan specc normaton about the vehcle or the drver and other specc normaton about the leet. β s a vector o p parameters to be estmated. Let α be the random eects assocated wth leet (.e. the rsk or non-observable characterstcs attrbutable to the leet), whereas θ ( ) s the random eects o truck o leet where I = θ ( ) =, I beng the number o vehcles n leet. Fnally η ()t s the tme random eects o perod t o truck o leet such that η ()t = where T s the number o perods or truck. Our model has an embedded structure whch explans why the two above summatons are equal to one. The random varable T t= α s ndependent o other regressors ncludng the x tk, k =,, p, whle the θ ( ) are dependent between themselves or a gven leet and the η ()T are 5 Angers et al (206) propose a model wth ndvdual and rm eects but ther model cannot be appled to panel data. 7

8 dependent between themselves or the truck o a gven leet. Fnally, the perods n T are not necessarly consecutve. An ndvdual or a truck can leave the rm and come back. Model assumptons: Let us suppose that t t ( ( ) ()t ) 0 Xt λ =γ αθ η > wth γ =. We post that: t e β I ) α ollows a gamma dstrbuton o parameters T κ, κ ; 2) the vector = ( θ θ θ ) θ,,, ( ) = ollows a Drchlet dstrbuton o parameters ( ν, ν,, ν 3) the vector ( ) ( )2 ( )I ( δ, δ,, δ ) () ()2 ()T () () ()2 ()T ( ) ( )2 ( )I ); and η = ( η, η,, η ) ollows a Drchlet dstrbuton o parameters where T s the number o perods o vehcle. The Drchlet dstrbutons have been proposed because they naturally generalzes the beta dstrbuton already used n the Negatve Bnomal model. They allow us to dstrbute the whole leet eects on all trucks. Moreover, they permt to obtan an analytcal soluton or the model. It s clear that we can use other dstrbutons than the Drchlet or that purpose. Suppose that the varable X on + or propertes as the Drchlet. X Xn =,...,n ollows any densty. Then,..., X X wll have the same Usng a general dstrbuton wll add non-necessary complextes, however. Suppose, or example, that Z X = X + X 2 wth the ollowng dstrbuton: X ollowng an unorm dstrbuton over the nterval (0,). We wll obtan 0.5 (z) = 2 ( z) 0 < z (z) = 2 z 0.5 < z <. Even wth such an easy case, computatons wll become much more complex than by usng a Drchlet dstrbuton. Proposton: The jont dstrbuton o accdents o all vehcles n leet s gven by: 8

9 I I I y Γ S T t 0 + T κ κ I T Γ ν( ) ( ) ( κ ) = γ t = = I I T = I ( ) I I t yt = = S0+ T κ Γ + T Γ κ ( ) = Γ S g +ν κ +γ ( ) 2 = = ( ) P Y,,Y I = I ( S ( ) ) Γ +ν ( ( ) ) I T I T Γ ( y +δ ) Γ δ γ +ν + κ +ν t ()t ()t g I I = t= = t= g2 g I ( ) ( ) T I 2F S ( ),S0 T, S ( ), T = = = Γδ 2 ( ()t ) Γ S + δ()t κ +γg = t= = t= = Γν γ. (3) Proo: The condtonal dstrbuton o the number o accdents or all the vehcles n leet s gven by: I T ( I ) ( ) T α θ( ) η () = t λt P Y,,Y,, P Y = I I T e λ y t ( λt ) ( y ) Γ + = t= t t = t= I T I T y t = ( t ) = t= ( yt ) λ Γ + = t= I T λt = t = e. (4) Snce λ t =γt ( αθ ( ) η ()t ) then: I T I T I I T T I T y y t yt y y t t t t ( λ ) ( ) ( ) t t = = = = γt α ( θ( ) ) ( η()t ) (5) = t= = t= = = t= Let S T = y and S = y = S, equaton (5) can be rewrtten as ollows t= t I T I 0 t = t= = I T I T I I T yt yt S S y 0 t ( λ t ) = ( γt ) ( α ) ( θ( ) ) ( η()t ). = t= = t= = = t= Moreover, the summaton I T = t= λ t n equaton (4) can be wrtten as I T. α θ γ η ( ) t ()t = t= 9

10 Wrtng the general orm o the jont dstrbuton o the number o accdents or all the vehcles n leet as: ( ) ( ) ( ) I I P Y,,Y = P Y,,Y η η dη (6) I T I T () () () wth ( ) ( ) ( ) P Y,,Y η = P Y,,Y θ, η θ dθ (7) I T I () I T I ( ) () ( ) ( ) P Y,,Y θ, η = P Y,,Y α, θ, η α dα, (8) and ( ) ( ) ( ) I I I ( ) () 0 and ntegratng equaton (8) wth respect to α, we obtan y ( t ) ( y ) t I ( ) ( ) ( ) I T I I T I γ S yt T κ = θ( ) η()t κ Γ S0 + T κ = t= t = = t= = Γ +. (9) I I S0 T I T + κ = Γ T κ κ + θ( ) γt η()t = = t= By replacng P( Y,,Y I ) T θ( ), η() n equaton (7) by ts value gven n (9) and by replacng the I densty uncton ( θ( ) ) by the densty o a parametrc Drchlet dstrbuton o parameters ( ν, ν,, ν ), we obtan the ollowng expresson: ( ) ( )2 ( )I ( I ) T η() P Y,,Y I I I I I T T κ yt I S ( ) 0 T = Γ + κ κ Γ ν( ) ( γt η()t ) ( θ( ) ) = = = t= = = I T I I I T Γ ( yt + ) Γ T κ Γ( ν( ) ) = t= = = κ + θ γ η = t= S +ν( ) ( ) t ()t I S0+ Tκ = dθ ( ) (0) We must estmate the multdmensonal ntegral o equaton (0) to obtan the model parameters. We analyze two possbltes. 0

11 ) All trucks o the same leet have dentcal a pror rsk Ths rst possblty, whch greatly smples the estmatons, s to suppose that all the γ t o the I vehcles are dentcal and equal to γ, or all perods. Under ths hypothess, the multdmensonal ntegral o equaton (0) s reduced to: I S +ν( ) ( θ( ) ) Γ ( S +ν( ) ) = = I dθ ( ) = I I I 0 T S + T κ S 0+ Tκ = ( κ +γ ) = Γ ( S +ν( ) ) κ +γ θ( ) η()t = = t= I () and the jont dstrbuton o the number o accdents at perod t or the I vehcles n leet s gven by the ollowng expresson: ( I ) T η() P Y,,Y I I I I I T T κ yt I S ( ) 0 T = Γ + κ κ Γ ν( ) ( γt η()t ) Γ S +ν( = = = t= ( ) ) = = I T I I I I S0+ Γ ( yt + ) Γ T κ Γ( ν Tκ ( ) ) ( κ +γ ) ( S ) = t= = = = Γ +ν( ) = (2) Further, by replacng P( Y,,Y I ) T η() n equaton (6) by the expresson n (2) and by I replacng the densty uncton ( η() ) by the densty o a parametrc Drchlet dstrbuton o parameters ( δ, δ,, δ () ()2 ()T ), we obtan the ollowng approxmaton or (6), the jont dstrbuton o the number o accdents or all the vehcles n all leets:

12 ( I ) T P Y,,Y I = I I I T κ I S ( ) 0 T = y Γ + κ κ Γ ν( ) I t ( ) ( ) T Γ S +ν( ) γ t = = = I I ( ) I t yt = = Γ + S0+ Tκ T I Γ κ Γ( ν( ) ) ( κ +γ ) = Γ ( S +ν( ) ) = = = I I T T Γ ( yt +δ()t ) Γ δ()t = t= = t= I T I T Γ( δ()t ) Γ S + δ()t = t= = t= (3) Ths s an approxmaton because the man workng hypothess or ths rst scenaro supposes mplctly that all the vehcles n the leet represent dentcal a pror rsks, whch s probably a very strong hypothess because, as we shall see, several varables dstngushng vehcles and drver behavor are sgncant n estmatng the probabltes o accdents o derent vehcles. Another possblty s to dvde the vehcles nto homogeneous rsk groups, as nsurers do when classyng rsks. ) Trucks belong to derent groups Under ths second possblty, we suppose that γ t = γ t =,, T where γ = T T = γ t. We can separate the vehcles nto two groups (hgh rsk and low rsk) and dene G =,, g as the set o all vehcles o the rst group wth γ = g g γ = g, and G 2 = g+,, I, as the set o all vehcles o the second group wth γ = g2 I γ = g + g 2. The ntegral o equaton (0) thus becomes: 2

13 g I S +ν( ) S +ν( ) ( θ( ) ) ( θ( ) ) dθ. (4) = = g + I S0+ T g I κ = κ +γg θ ( ) +γg 2 θ( ) = = g + ( ) Let v = θ I χ ( θ ) ( ) ( ), u = v = and w = χ ( θ ) 2 ( ) v χ = truck belongs to group s ( s, 2). s 0 otherwse. The ntegral o equaton (4) can be rewrtten as ollows: = and S +ν( ) S +ν( ) ( vu ) ( ( v) w ) v u ( v) w (( κ +γ ) v+ ( κ +γ )( v) ) g g2 Sg +ν( )g S I +ν ( )I g I g I I S0+ Tκ = ( ) g I g v v dudwdv. By ntegratng we obtan: I ( S ( ) ) Γ +ν g I I γg γ = 2 g = I I 2F S ( ),S0 T, S ( ), +ν + κ +ν S0+ T κ = = = κ +γg 2 Γ S ( ) = +ν( ) κ +γg 2 =. (5) Thus, by replacng the ntegral n equaton (4) by ts value gven n (5) we obtan the ollowng approxmaton or P( Y,,Y I ) T η() n (0): I I I I I T I T κ yt Γ S T ( ) = + κ κ Γ ν ( γ η ) Γ ( S +ν ) 2 0 ( ) t ()t ( ) = = = t= = I I T I I I S0+ T κ ( yt ) T ( ( ) ) S Γ + Γ κ Γ ν Γ +ν( ) ( κ +γ ) = g2 = t= = = = g I I γg γ 2 g F S +ν ( ),S0 + T κ, S +ν( ), = = = κ +γg2 (6) where 2 F s a hypergeometrc uncton whose value s equal to: 3

14 wth [ ] g [ ] [ ] I γg γ 2 g S +ν ( ) S0 + T κ κ +γ = = g 2 +, (7) I [ ] =! S +ν( ) = ( ) ( ) h = h h+ h+ +, beng an ncreasng actoral uncton. Further, by replacng P( Y,,Y I ) T η() n equaton (6) by the expresson n (6) and by I replacng the densty uncton ( η() ) by the densty o a parametrc Drchlet dstrbuton o parameters ( δ, δ,, δ () ()2 ()T ), we obtan (3), whch completes the proo. Ths procedure n estmatng the ntegral can be generalzed to several homogeneous groups, but t s not obvous that the precson ganed would be greater than that correspondng to a Monte Carlo approxmaton o the multvarate ntegral o equaton (0), whch s not presented here. 6 In Secton 5, we present the econometrc results obtaned rom equaton (3). 3.3 Parameters estmaton Let ν ( ) =ν and δ ()t =δ t. We can apply the maxmum lkelhood method to estmate the unknown parameters,,, νκ δ and β o the log lkelhood correspondng uncton o equaton (3). 7 In the applcaton, presented n Secton 5, we wll apply the quas-newton method o estmaton (known as a varable metrc algorthm). We use the package Optm avalable n R (see Appendx D or more detals). The ntal values o the vector β are the maxmum lkelhood estmates o the NB2 model, and the ntal values or νκ,, δ parameters are set to one. To determne the varance-covarance matrx o the asymptotc dstrbuton, we solve the Hessan 6 Ths thrd possblty o estmatng the ntegral n (0) by the Monte Carlo method s presented n Angers et al (2006). It s shown that the results are very smlar to the two groups method. 7 We could have used the Monte Carlo method wth mportance samplng to perorm the parameters estmaton. However, snce the lkelhood uncton s hghly skewed and gven the large number o parameters to estmate, we have chosen to use the maxmum lkelhood method. 4

15 matrx at νκ ˆ, ˆ, δ and ˆβ. The sze o the data s qute large; to reduce the computaton tme ˆ drastcally, we compute the log lkelhood uncton wth a homemade C program nsde the R system. To dvde the trucks o a leet nto two homogeneous groups as shown n Secton 3.2, we take the maxmum lkelhood estmates ( ˆβ ) o the NB2 model to estmate γ ˆ = e β or all the vehcles. Gven that a truck has an estmate by year o ollow-up, we calculated ts mean ˆ γ = sorted ˆγ or = and calculated the derence ( ˆ ˆ + ) γc where c s such that arg max ( ˆ ˆ + ),, I choose a cut-o pont ˆγ <γ or s n group 2 c ˆγ γ or all the observatons o the truck. c t X ˆ t T T = γˆ t. We γ γ or =,, I. Ater, we γ γ. The truck s n group For example, or a leet o 8 trucks and 20 observatons (truck-years) as shown n Table, c = arg max ( ) = 6. Then the cuto pont γ c = γ ˆ 6 = All observatons o truck to truck 5 wll thereore be n group (low rsk group ) and all the others wll be n group 2 (hgh rsk group). I we use the medan or the mean nstead o the maxmum derence then the cut-o pont wll be respectvely (( )/2) and , and truck 5 wll change to group 2. However, t s more approprate to be n group because ˆγ 5 s nearer to those n group. Table Example o group dvson, leet o 8 trucks and 20 truck-years Truck Year ˆγ t 5 ˆγ Derence Group

16 Mean DATA The Socété de l assurance automoble du Québec (SAAQ) provded the les or our data set. The SAAQ s n charge o montorng whether vehcles engaged n road transportaton o people or merchandse comply wth applcable laws and regulatons. The SAAQ s also the nsurer or bodly njures lnked to trac accdents or ndvduals and leets o vehcles and collects normaton on all truck accdents. Our startng pont s the whole populaton o carrers regstered n a SAAQ le on July 997. To be n that le o carrers, a carrer must be the owner o a vehcle that meets derent admnstratve condtons. Lnked to each carrer, the data contan: () normaton on volatons (wth convctons) commtted by the carrer durng the perod, ether or non-complance wth the Hghway Saety Code s provsons on mechancal nspecton; wth rules on vehcles and ther equpment; wth codes on drvng and hours o servce or or oversze or poor load securement, etc., and (2) normaton dentyng the carrer. We also have access to normaton on vehcles regstered n Quebec or the perod o January, 990 to December 3, 998. We can lnk vehcles to carrers n each perod. From the authorzaton status, we obtaned normaton descrbng vehcles and plates. For each plate number, we have data coverng the perod drawn rom the les on mechancal nspecton o vehcles and rom the record o drvers volatons wth convcton and demert ponts or speedng, alure to stop at a red lght or stop sgn, and llegal passng, and or accdents. 6

17 The type o nsurance coverage we consder s or property damages o the trucks. The premum or these losses s pad by the owner o the leet to a prvate nsurer. A truck can be drven by derent drvers and a drver can drve derent trucks durng a polcy perod. We assume that the owner knows who drves each truck o the leet at any pont n tme. The nsurer does not observe the saety actons o both the drver and the leet owner and there s also asymmetrc normaton on preventon between the drver and the owner o the leet. In the applcaton o the model, the ndvduals are the trucks and ther accdents are uncton o both observable and non-observable characterstcs or actons rom drvers, vehcles and leets. The owner observes accdents and the drvers trac volatons commtted whle drvng a truck o the company. The choce and the descrpton o the varables used n ths study are presented n Appendx A. 5. RESULTS 5. Descrptve statstcs 5.. By leet We have 7,542 leets wth at least two trucks wth a ollow-up o at least two perods. In December 3, 998, the average number o years each carrer s n the sample s seven, the mnmum s one year and three months, and the maxmum s 20 years and 9 months. Table 2 shows the dstrbuton o leets accordng to ther man economc sector: 76.75% o 7,542 carrers are ndependent truckng rms, 3.09% are bulk publc truckng rms and 8.70% are general publc truckng rms. The sector s unknown or only a ew rms. In addton, a ew leets also transport passengers or are short-term leasng rms. 7

18 Table 2 Dstrbuton o rm s man actvty Frm s man actvty N % Unknown (sect_00) Transportng passengers (sect_4) General publc truckng (sect_05), Independent truckng (sect_07) 3, Short-term leasng rm (sect_08) Bulk publc truckng (sect_06) 2, Total 7, We note n Table 3 that approxmately 4% o 7,542 leets have over 20 trucks. On average, a truck has 3.87 observaton perods rangng rom 3.38 (a truck rom a leet o sze 2) to 4.30 (a truck rom a leet o 0 to 20 trucks). Table 3 Sze o leet dstrbuton Sze o leet N % Average observaton perod per truck 2 6, , to 5 3, to 9 2, to 20, to More than We note n Table 4 that a quarter o the 7,542 carrers have eght years o ollow-up. 8

19 Table 4 Number o years o ollow-up o the rm Number o years o ollow-up N % 2 3, , , , , , , Total 7, Table 5 shows that there are 3,629 leets or whch we have two consecutve years o ollow-up, whch s 99.5% (3,629/3,649) o leets wth two observaton perods (Table 4). Ths percentage vares rom (3 perods) to 87.7% (7 perods). The hgher the number o years o ollow-up, the hgher the percentage o carrers wth absences durng the reportng perod. Table 5 Number o consecutve years o ollow-up o the leets by year o ollow-up start, Quebec 99 to 997 Year o ollow-up start Number o years o ollow-up Total ,92 3, , , , , , ,440 4,440 Total 8,22,929,52,360,370,263,92 6, By truck-years There are 43,037 trucks n 99. Ths number ncreased to 63,749 n 996. It decreases to 52, 392 n 998 or a total o 456,77 truck-years, 5% o whch had an accdent durng one year (Table 6). In 99, nearly 86 out o 00 vehcles had no accdent; ths percentage rses to 88 out o 00 n 997. Other statstcs are presented n Appendx B. 9

20 Table 6 Number o truck accdents dstrbuton accordng to year o observaton Number o truck % (by year o observaton) accdents Total and more Number o trucks 43,037 55,388 57,795 59,347 6,97 63,749 62,552 52, ,77 Means truck crash Estmaton o the models For comparson we rst estmate the Hausman (994) random eects model that cannot take nto account the rm-specc eect. The results are presented n columns 2 and 3 o Table 7. Several varables measure observable heterogenety. Some o these varables (type o uel, number o cylnders, number o axles, type o vehcle used) are characterstcs concernng vehcles, whereas others (sector, leet sze, etc.) have to do wth the leet. We also nclude the number o volatons o the truckng standards the year beore the accdents and the number o volatons o the road saety code leadng to demert ponts the year beore the accdents. The rst group o volatons s more related to leet behavor, and the second group s more related to drvers behavor. Almost all coecents are sgncant at %. Table C. (see appendx C), present the correspondng estmates o the Posson model n columns 2 and 3 and the NB2 model n columns 4 and 5. The estmate o δ s equal to wth the standard error o The mpled varance to mean rato ( +δ ) δ s 2.23, whch s greater than. Thus, the NB2 model speccaton allows or overdsperson n accdents dstrbuton so we reject the Posson model. Otherwse, the coecents o the observable characterstcs are very stable between the two models. All these results do not control or rm-specc eect so the seral correlaton o resduals may be a problem havng panel data. Columns 4 and 5 n Table 7 present the estmates o our Gamma-Drchlet model when we add random rm-specc eect. The estmated coecents are also very stable between the two models n the table, wth the excepton o the year varables. In the Hausman model, the year coecents relect the statstcs provded or truck accdents n Table 6, where the leet eect s not present. When we look at Table B3 n 20

21 Appendx B, we see that the average truck accdent by leet sze does not have the same pattern durng many years, as n Table 6. Snce the year varables n the Gamma-Drchlet model are or trucks o a gven leet, ths may explan the derence. In Table C.2, we present three other estmatons o the Gamma-Drchlet model by omttng derent categorcal varables, ncludng the year varables. We observe that all parameters, ncludng the random eects parameters, are stable. Table 7 Estmaton o the parameters o the dstrbuton o the number o annual truck accdents or the perod (leet o two trucks or more and trucks wth two perods or more), wth the Hausman and Gamma- Drchlet models. Hausman model Gamma-Drchlet model Explanatory varables Standard Standard Coecent Coecent error error Constant * Number o years as carrer at 3 December * * Sector o actvty n 998 Other sector * General publc truckng 0.003* * Bulk publc truckng Reerence group Reerence group Prvate truckng 0.574* * Short-term rental rm * * Sze o leet 2 Reerence group Reerence group * * to * * to * * to * * to * * More than * * Number o days authorzed to drve n prevous year.6878* * Number o volatons o truckng standards n year beore For overload 0.26* * 0.05 For excessve sze 0.456*** *** For poorly secured cargo * * For alure to respect servce hours * * For alure to pass mechancal nspecton * * For other reasons * ** Type o vehcle use Commercal use ncludng transport o goods * * wthout C.T.Q. permt Transport o other goods ** * Transport o "bulk" goods Reerence group Reerence group 2

22 Explanatory varables 22 Hausman model Coecent Standard error Gamma-Drchlet model Standard Coecent error Type o uel Desel Reerence group Reerence group Gas * * Other * * Number o cylnders to 5 cylnders 0.359* * to 7 cylnders * * or more than 0 cylnders Reerence group Reerence group Number o axles 2 axles (3,000 to 4,000 kg) * * axles (more than 4,000 kg) -0.75* * axles * * axles * * axles * * axles or more Reerence group Reerence group Number o volatons wth demert ponts year beore For speedng * * For drvng wth suspended lcense * * For runnng a red lght * * For gnorng stop sgn or trac ocer * * For not wearng a seat belt * * Observaton perod ** *** * * * * * * * * Reerence group Reerence group â * ˆb.8274* ˆν * ˆκ * ˆδ * Number o observatons: 456,7 456,7 * sgncant at %; ** sgncant at 5%; *** sgncant at 0% The random eects parameters are sgncant n both models. Let us concentrate on the Gamma- Drchlet model proposed n ths artcle. The sgncance o the three random eects parameters means that the random eects assocated wth the leets (or the non-observable rsk o the leets) ( ˆκ ), as well the random eects o the trucks ncludng the drvers ( ˆν ) and the random tme eect ( ˆδ ) sgncantly aect the truck dstrbuton o accdents even when we control or many observable characterstcs.

23 Suppose we are proposng a parametrc model to rate nsurance or vehcles belongng to a leet. Accordng to the results n Table 7, ths premum wll be a uncton o observable characterstcs o the vehcle and leet o the vehcle, as well a uncton o volatons o the road-saety code commtted by drvers and carrers. 8 Ths wll not be enough because many unobservable characterstcs o trucks, drvers and carrers also aect the trucks dstrbuton o accdents. The premums wll also have to be adjusted usng the parameters o the random eects so as to account or the mpact that the unobservable characterstcs or actons o carrer, truck and drvers and even tme can have on the truck accdent rate. Ths orm o ratng makes t possble to vsualze the mpact (observable and non-observable) o behavors o owners and drvers on the predcted rate o accdents, and consequently on premums under potental moral hazard (see Angers et al, 2006, or more detals). We show below how we can predct the number o truck accdents. 5.3 Ft statstcs o derent models accordng to leet sze We now analyze the perormance o the models. Model ts are based on the log lkelhood statstcs as well on other measures o normaton crtera such as the Akake's normaton crteron (AIC) and the Bayesan normaton crteron (BIC). One advantage o usng these two normaton crteron measures s that they can compare non-nested models. Table 8 Ft statstcs o the two models wth two data sets Statstcs Hausman model Gamma-Drchlet model Hausman model Gamma-Drchlet model 7,542 leets havng more than truck 5,423 leets havng more than 4 trucks Log L -97, ,6.8-55, , BIC 394, , , ,33.4 AIC 394, , , ,672.4 Number o trucks,06,06 79,609 79,609 Number o observatons 456,77 456,77 336, ,772 Number o parameters Bayesan Inormaton Crteron (BIC) = 2ln L + k ln(n) ; Akakes Inormaton Crteron ( AIC) = 2ln L + 2k where k and N are the number o parameters and observatons respectvely, LogL = Log Lkelhood rato. 8 It can also be a uncton o observable characterstcs o the drvers but we do not consder them here. 23

24 For two models estmated rom the same data set, the model wth the smaller BIC and AIC s preerable. We note n Table 8 that the Gamma-Drchlet model s preerred to the Hausman model wathever the leet sze. The same results were obtaned or leets wth more than two trucks and leets wth more than three trucks. Because the large majorty o trucks belong to leets that have more than two trucks, t s clear that our model permts better estmaton o accdent dstrbutons than the Hausman model does. Detaled estmaton results o the models wth leets havng more than our trucks are presented n Appendx C Predcted numbers o accdents In order to check how the Gamma-Drchlet model perorms n predctng the number o truck accdents per leet at tme t+, we assess an out-o-sample perormance o the model n 998 and we compare ts orecastng perormance wth the observed accdents n 998. From a methodologcal pont o vew, we proceed as ollows: We partton the orgnal sample perod nto two subsamples: an estmaton sample or the perod and a orecastng sample or the year 998. The estmatng sample conssts o 6,344 leets wth at least two trucks and 393,634 trucks wth a ollow-up o at least two perods rom 99 to 997. We obtan the coecent estmates or the Gamma-Drchlet model presented n Table C.3, where we observe that the coecents are smlar to those presented n Table 7. We should menton that the year varable s contnuous n table C.3 to smply the computaton o the predctve accdent dstrbuton. Ths modcaton does not aect the estmaton results. One nterestng eature o the Bayesan parametrc model s to compute a parametrc predctve dstrbuton o accdents, P( Y ) T +,,Y I T I + Y,,Y,,Y,,Y T I I T I ( T ) ITI T I ITI P Y,,Y Y,,Y,,Y,,Y + + =, whch s equal to: ( T ) T+ I ITI I TI + P( Y,,Y T,,Y I,,YI T ) P Y,,Y,Y,,Y,,Y,Y I (8) 24

25 Usng the Gamma-Drchlet model, we obtan: ( T ) + ITI + T I ITI P Y,,Y Y,,Y,,Y,,Y y t+ T + I ( γ ) Γ ( y + ) t+ I I I t+ Γ S0 + Tκ + S0 + κ Γ T κ T + = = = I t+ I I = T + Γ S0 + Tκ Γ T κ + κ = = = I t+ I I κ Γ ( ) ( S +ν( ) ) κ Γ S +ν ( ) + y = T + = = I t+ I I t+ S t+ 0 + κ Γ ( S ) ( S ( ) ) ( ) S = +ν + 0 Γ +ν g2 = = ( κ +γ ) t+ I I T * * Γδ ( ()T ) + S ( y ) + + = * ( ) t+ I I T I T Γ ( yt ) S ()t + +δ Γ + δ ()T+ = = t= = t= I = * Γ δ ()t +δ()t + T Γ + δ ()t + T +δ()t Γ δ()t t= t= = = t+ g I I I t+ t+ t+ g2 g 2F ( S +ν ( ) ) + S g,s0 + Tκ + S 0 + κ, ( S +ν ( ) ) + S 0, = = = = κ +γg2 γ γ g I I g2 g 2F ( S +ν ( ) ),S0 + T κ, ( S +ν( ) ), = = = κ +γg2 where * means that the truck s present n the orecastng sample; γ γ (9) t I + s the number o trucks o leet n the orecastng sample; γ are calculated or the orecastng sample wth the coecent estmates or the years 99 to 997 T + presented n Table C.3; S 0 s the total number o trucks accdents o leet beore t+. t S + 0 t S + g s the total number o truck accdents o leet at t+; s the total number o truck accdents n group o leet at t+. The orecastng sample conssts o 8,40 leets wth 2 trucks or more or a total o 4,64 trucks. In Table 9, we observe that out o the 8,40 leets, 5,670 o them had no accdent n 998 (.e. 67.5%). The average predctve probablty o havng zero accdent s equal to 68.9% or the same year, n supposng that the number o accdents o truck at tmet +, y +, s equal to zero or T 25

26 all trucks o leet and so on or all leets. Consequently there s 68.9% chance that a leet wll have no accdent durng the next year. The average predctve probablty that a leet has accdent durng the next year s 8.6% whle the observed one s 9.%. For two accdents, the respectve probabltes are 6.% and 6.5%, whle or three accdents, they are 2.6% and 2.7%. Results or more than three accdents are avalable rom the authors. Detals o the computatons are presented n Appendx E. We used a pared t-test to compare the observed percentages and the predcted ones rom the Gamma-Drchlet model. Frst we need to check whether the derences between the two percentages ollow a Normal dstrbuton (.e. Shapro-Wlk test o normalty). In Table 9 we observe large p-values or the normalty test, thus, we do not reject the Normal dstrbuton. Moreover, snce the p-values o pared t-tests are greater than 0.05, we do not reject H0 that the mean derence between the observed and the predcted percentages o accdents do not der rom zero at the 5% level o sgncance. Table 9 Percentage o 8,40 leets havng no accdent, accdent, 2 accdents or 3 accdents n orecastng sample and the average predctve probablty o havng n accdents rom the Gamma-Drchlet model by sze o leet and or all rms. Sze o leet % o rms wth 0 % o rms wth % o rms wth 2 % o rms wth 3 accdent accdent accdents accdents Observed Gamma- Gamma- Gamma- Gamma- Observed Observed Observed Drchlet Drchlet Drchlet Drchlet to to to to More than All rms Shapro-Wlks normalty test p-value Pared t-test d p-value

27 5.5 Fxed and random eects models We now analyze the consstency o the random eects estmators or the Gamma-Drchlet model. One mportant assumpton n the random eects model s that the random eects are uncorrelated wth the observed explanatory varables used n the estmatons. One way to very the consstency o the random eects model s to compare the results wth those obtaned rom a xed eects model by applyng the Hausman (978) test. In the Gamma-Drchlet model, we assume that Y t ollows a Posson dstrbuton wth parameter explanatory varables: log t 0 t ( ) ( ) t λ t. Let λ t be a log-lnear uncton o the λ = β + βx + ξz + α + θ + η (20) where x t represents the tme-varyng explanatory varables, z represents the tme-nvarant explanatory varables, α denotes the rm eects, θ ( ) the truck eects wth θ( ) = and η ( ) the tme eects wth η( ) t =. T t= In the random eects model, α s assumed to be an ndependent and dentcally dstrbuted (d) random varable ollowng the Gamma dstrbuton mplyng no correlaton wth the other regressors. In the xed eects model, such an assumpton s not needed because α s estmated usng dummy varables. In the Gamma-Drchlet model, the vector θ ( ) ollows a Drchlet dstrbuton. Hence, ts components are not ndependent rom one another. The same stuaton holds or the vector η ( ). When the random eects model s correctly speced, both the xed and the random eects estmators would be consstent. The derence between the two estmators can be used as the bass or a Hausman test. Cameron and Trved (203b) propose the ollowng representaton o the test: where T H s the Hausman test statstc, T ( ˆ β β ) V ˆ[ β ˆ β ] ( ˆ β β ) = (2) H RE FE FE RE RE FE βfe are the estmated parameters obtaned rom the xed eects model and ˆRE β are the estmated parameters obtaned rom the random eects model. To 27 I =

28 estmate the varance term Vˆ[ β ˆ β ] we can use a panel bootstrap method that resamples over the rms o the sample: FE RE ( b) ( b) ( b) ( b) ( )( ) B ˆ Vˆ[ β β ] = β ˆ β β ˆ β (22) FE RE FE RE FE RE B b= where β and ( b ) FE ˆ ( b ) β RE are the estmates obtaned rom the bth bootstrap replcaton (see Appendx F or more detals). I T < χ then, at the 5% level o sgncance, we do not reject the null 2 H p,0,05 hypothess that the random eects are uncorrelated wth the regressors and there s no need to use the xed eects estmaton. In the xed eects model, all characterstcs that are not tme-varyng are captured by the xed eects varables and have to be removed rom the model. So we carry out the Hausman test only on the coecents o the tme-varyng varables. We estmated the xed eects Posson regresson model wth the conventonal Posson model usng 5,423 dummy varables or the leets o our trucks and more. 9 Greene (2004) has demonstrated the computatonal easblty o ths approach. Table 0 shows the estmated coecents and standard devatons o the tme-varyng varables o both the xed eects Posson model and the Gamma-Drchlet random eects model. The estmates o the two models are lkewse qute smlar, wth ew exceptons. We must menton that the Gamma-Drchlet model has a constant term whle the Posson model does not by constructon. Moreover, as or Table 7, the coecents o the year varables der between the two models. These derences, agan, seem to be explaned by the presence o the leets eects n the Gamma-Drchlet model. But the man pont n ths secton s to test the random eects are uncorrelated wth the regressors. As Equaton (2) above shows, the test conssts n veryng whether the coecents between the two regressons are statstcally derent. 9 We used ths group o leets to reduce the number o dummes. It s clear that the methodology can be used or all groups o leets. 28

29 Table 0 Estmaton o the parameters o the dstrbuton o the number o annual truck accdents or the perod (leets o our trucks or more and wth trucks havng at least two perods) o the xed eects Posson model wth 5,423 dummy varables (coecents not presented here) and the Gamma Drchlet model. Fxed eects Posson model Gamma-Drchlet model Explanatory varables Standard Standard Coecent Coecent error error Constant -.96 * Number o volatons o truckng standards n prevous year For overload 0.584* * For excessve sze * * For poorly secured cargo 0.232* * For alure to obey servce hours * * For alure to pass mechancal nspecton 0.583* * For other reasons 0.233* * Number o volatons wth demert ponts n prevous year For speedng * * 0.06 For drvng under suspenson * * For runnng a red lght * * For gnorng a stop sgn or trac agent * * For not wearng a seat belt 0.29* * Sze o leet 5 trucks Reerence group Reerence group 6 to 9 trucks to 20 trucks * * to 50 trucks * More than 50 trucks 0.084* * Observaton perod * * * * * * * * * * * Reerence group Reerence group ˆν.7438* ˆκ * ˆδ * Log L -50, ,255.6 Number o observatons: 336, ,772 * Sgncant at %. 29

30 Fgure presents the values o Hausman test statstc, TH, based on the number o bootstrap replcatons (or the bootstrap varance matrx estmated n the Hausman test). We observe that ater 300 replcatons, TH < χ where 2 22,0,05 2 χ 22,0,05 = So, at 5% or any lower level o sgncance, we do not reject the null hypothess that the random eects are uncorrelated wth the regressors. Consequently, there s no statstcal derence between the coecents o the Gamma- Drchlet model and those o the xed eects Posson model presented n Table 0. Fgure : TH values o the Hausman test s based on the number o bootstrap replcatons or the rm eects 6. CONCLUSION In ths artcle, we propose a new parametrc model wth random eects or the estmaton o accdents dstrbuton n the presence o ndvdual and rm eects. Non-observable actors are treated as random eects. A Posson xed eects model s estmated to very the consstency o the random eects model. We do not reject the null hypothess that the random eects are uncorrelated wth the regressors. Ths type o model can be used to compute nsurance premums or drvers or vehcles belongng to a leet because the characterstcs and the management behavor o the leets can aect the accdent rate o vehcles and ther drvers. For example, the manager o a gven leet may have a 30

31 hgh rsk appette and ask ther drvers to drve aster or to work more than the regulated number o hours durng a week. He may also ask them to transport poorly secured cargo. A prcng rule that ncludes the observable and non-observable characterstcs o all partes that aect accdent dstrbutons should consequently be arer, and ntroduce the approprate ncentves o all partes under asymmetrc normaton. Our results show that the Gamma-Drchlet model perorms well n predctng out-o-sample accdents. The methodology developed n ths study can be appled to estmatng event dstrbutons n many other domans than nsurance prcng. Snce 2004, banks are regulated by Basel II or keepng captal or operatonal rsk. The operatonal rsk o derent banks s a uncton o the observable characterstcs and the non-observable behavor o the personnel and o the management. A smlar envronment s present or the deault rsk o derent rms or or the accdent rsk o any publc nsttuton or transportaton rm ncludng arlne accdents. Other domans o applcatons nclude the alure or success rate o hosptals, unverstes, or any nsttuton wth prncpal-agent stuatons wth teams. In ths study, we used a parametrc model to estmate accdent dstrbutons. The man motvaton was to obtan explct parameter estmates or the nsurance prcng o vehcles that ncludes ndvdual and rm eects. Snce we have a very large dataset, we could also have used the Classcaton and Regresson Tree (CART) approach whch does not requre any ex-ante relatonshp between dependent and ndependent varables (Chang and Chen, 2005). It would be nterestng to extend our analyss to such data mnng technques and see ther advantages and dsadvantages wth respect to our prcng objectves. REFERENCES Allson, P.D., Waterman, R.P., Fxed-eects negatve bnomal regresson models. Socologcal Methodology 32, Angers J-F., Desjardns D., Donne G., Guertn F., Vehcle and leet random eects n a model o nsurance ratng or leets o vehcles. Astn Bulletn 36, Baltag B.H., 995. Econometrc Analyss o Panel Data. Wley, Chchester. 3