Uni Rodeo and Economic Loss Analysis

Transcription

1 Do Propery-Casualy Insurance Underwriing Margins Have Uni Roos? Sco E. Harringon* Moore School of Business Universiy of Souh Carolina Columbia, SC 98 (83) Tong Yu College of Business Adminisraion Universiy of Rhode Island Kingson, RI 88 (4) Revised April * Conac auhor. An earlier version of his paper was presened a he Annual Meeing of he American Risk and Insurance Associaion in Balimore.

2 Do Propery-Casualy Insurance Underwriing Margins Have Uni Roos? Absrac A growing lieraure analyzes deerminans of insurance prices using ime series daa on insurer underwriing margins. If he variables analyzed are saionary, convenional regression models may be appropriaely used o es hypoheses. Based on pre-ess for a uni roo, several sudies have insead used coinegraion analysis o analyze he long run relaionship beween purporedly non-saionary underwriing margins and macroeconomic variables. We apply a baery of uni roo ess o invesigae wheher underwriing margins are saionary under differen assumpions concerning deerminisic componens in he daa generaing process (DGP). When linear and/or quadraic rends are included in he assumed DGPs, he ess rejec he null hypohesis of a uni roo for loss raios, expense raios, combined raios, and economic loss raios during for many of he individual lines examined and for all lines combined. Consisen wih prior work on wheher macroeconomic variables have uni roos, a simulaion of es power for underwriing margins during he sample period demonsraes ha nonrejecions of he null hypohesis of a uni roo could easily reflec low power. The overall findings sugges ha convenional regression mehods can be used appropriaely o analyze underwriing margins afer conrolling for deerminisic influences and ransforming any non-saionary regressors.. Inroducion A large lieraure examines ime-series variaion in propery-casualy insurance underwriing margins, such as loss raios and combined raios (see Harringon and Niehaus,, for a review). Many sudies provide evidence ha hisorical underwriing margins have been cyclical. A smaller bu imporan lieraure esimaes he relaionship beween underwriing margins, measures of insurance capaciy, and macroeconomic variables o es heories of he deerminans of insurance prices (e.g., Gron, 994; Winer, 994). Economeric modeling of underwriing margins may grow in imporance as heoreical work on insurance price flucuaions progresses. A criical issue in ime series regression analyses is wheher he underwriing margins and relevan explanaory variables are saionary. If underwriing margins or one or more regressors are non-saionary, hen leas squares regression resuls are essenially

3 meaningless (see, e.g., Enders, 995, pp. 6 ff. for deailed discussion). An excepion o his disressing conclusion arises if he regression disurbance is noneheless saionary, which requires ha he regressand and regressors be coinegraed (inegraed of he same order) and implies a linear long run relaionship among he variables (Engle and Granger, 987). Leas squares regression hus provides meaningful inferences only when he regressand and regressors are eiher all saionary or coinegraed. In eiher case, wheher underwriing margins have uni roos is of considerable imporance in applied work. Tesing for uni roos, iniially developed by Granger and Newbold (974), exploded following Nelson and Plosser s (98) evidence ha many economic ime series had uni roos. Such esing evenually became sandard operaing procedure in economeric sudies using ime series daa. Early ime series analyses of underwriing resuls are ofen silen on saionariy (see, e.g., Venezian, 985; Cummins and Oureville, 987; Dohery and Kang, 988). A few sudies analyze firs differences in underwriing margins, which is appropriae if margins are difference saionary. Several more recen sudies employ coinegraion analysis and error correcion models o analyze shor and long run relaionships beween underwriing margins, ineres raes, and oher macroeconomic variables (Haley, 993, 995; Grace and Hochkiss, 995; Choi and Thisle, 997). Haley (993) repors ha propery-casualy insurer underwriing profi margins are coinegraed wih ineres raes wih a negaive long run relaionship. Grace and Hochkiss (995) presen evidence wih quarerly daa ha combined raios, he shor- Two variables are inegraed of he same order if hey mus be differenced he same number of imes o become saionary. Dickey and Fuller (979, 98) developed empirical mehods o es for uni roos. Engle and Granger (987) iniiaed coinegraion analysis and error correcion models.

4 erm ineres rae, he CPI, and real GDP are coinegraed. Choi and Thisle (997) repor ha underwriing margins are coinegraed wih annual Treasury yields. 3 These sudies generally fail o rejec he null hypohesis ha underwriing margins have a uni roo using an augmened Dickey-Fuller (ADF) es (Dickey and Fuller, 979, 98), wihou considering possible rend or mean and rend in he DGP. However, wheher underwriing margins (and he oher variables analyzed) are non-saionary has no been horoughly explored. The ADF es permis several alernaive assumpions regarding deerminisic componens in he DGP for a given series and ime period: () he DGP includes neiher mean nor rend; () i includes a mean bu no rend; (3) i includes boh mean and rend; and (4) i includes linear and quadraic rend. Wes (987) suggess ha failure o consider rend could be problemaic because uni roo es power is grealy reduced when rend is incorrecly omied from he model (also see DeJong, e al., 99a). Conversely, incorrec inclusion of a deerminisic rend also reduces power (Dickey, 984). More generally, uni roo ess are known o have relaively low power, especially for samples wih fewer han observaions (DeJong, e al., 99a; Rudebusch, 993; Ellio, e al., 996; Hwang and Schmid, 996). Alhough Nelson and Plosser s (98) resuls suggesed ha many macroeconomic ime series had uni roos, subsequen sudies ofen sugges ha macroeconomic series are rend saionary, aribuing Nelson and Plosser s resuls o low power ess (e.g., DeJong and Whieman, 99; DeJong, e al., 99a; Rudebusch, 993; Diebold and Senhadji, 996). DeJong, e al. (99a) show ha ess for uni roos generally will have low power agains near uni roo alernaives and 3 Wih he excepion of Haley s (995) subsequen analysis of by-line underwriing resuls, coinegraion analyses of underwriing margins sugges a long run relaionship beween underwriing margins and 3

5 vice versa. Ellio, e al. (996) and Hwang and Schmid (996) show ha sandard ADF ess end o have low power compared wih ess wih asympoic power funcions close o asympoic power bound. They propose ADF ess based on generalized leas squares demeaned and derended daa (GLS-ADF ess). Aya and Burridge () show ha correcly including quadraic rend in he es equaion can increase he power of uni roo ess. Several sequenial ess for uni roos have been proposed when deerminisic componens of he DGP are unknown (Perron, 988; Doldado e al., 99; Enders, 995, and, using GLS-ADF ess, Aya and Burridge, ), bu here is no consensus concerning he mos powerful uni roo es or he suiabiliy of sequenial ess. The choice among differen uni roo ess remains an imporan empirical issue wih criical implicaions for appropriae empirical models and esimaion procedures. We apply a baery of ess for uni roos in propery-casualy insurance underwriing margins under differen assumpions concerning deerminisic componens in he DGPs for he sample period. Our moivaion is hree-fold. Firs, inuiion, heory, and sylized facs abou sof and hard markes in propery-casualy insurance provide relaively lile indicaion ha underwriing margins will be non-saionary afer allowing for possible rend. They sugges ha he effec of shocks will be emporary, which augurs agains a uni roo. If, for example, underwriing margins are rend saionary hey flucuae around a deerminisic rend hen he impac of shocks will disappear relaively quickly. If, on he oher hand, underwriing margins are difference saionary hey have a uni roo hen he effec of shocks is permanen. Second, previous work on propery-casualy insurance has no carefully considered he issues of power and inclusion of deerminisic componens in he DGP when esing ineres raes ha is broadly consisen wih basic heory and inuiion. 4

6 for a uni roo, especially wheher underwriing margins may be rend saionary. Third, wheher convenional regression mehods can be appropriaely used in analyses of underwriing margins is of considerable pracical imporance in esing hypoheses implied by developing heories on he causes of insurance price volailiy. In conras wih sudies of underwriing margin sudies ha ignore possible deerminisic componens in he DGPs, we consider DGPs ha include mean, linear rend, and/or quadraic rend. As DeJong, e al. (99a, p. 93) noe, a rend in a series need no lierally be par of he daa generaing process, bu may be viewed as a subsiue for a complicaed and unknown funcion of populaion, capial accumulaion, echnological progress, ec. Allowing for rends in underwriing margins is sensible because a variey of facors migh produce rends in propery-casualy underwriing margins during he laer half of he h cenury. Such facors include increased price compeiion in conjuncion wih he gradual breakdown of bureau raing sysems under prior approval regulaion, he long run growh of direc wriers wih lower underwriing expenses, gradual increases in he lengh of he claims ail, and changes in echnology. A some risk of over-fiing, we consider DGPs wih quadraic rend in addiion o linear rend models. This choice reflecs Aya and Burridge s () findings concerning possible power increases if he rue DGP includes quadraic rend and he possibiliy of diminishing rends in underwriing margins over ime due, for example, o slower growh of direc wriers. 4 4 An alernaive approach is o allow for regime shifs in DGP. Following Peron (989), we applied ess ha allowed for discree regime shifs during he sample period and generally obained resuls similar o hose repored. Given he lack of an obvious break poin or poins for regime shifs, allowing for gradually changing rend wih a quadraic specificaion is sensible and perhaps no more likely o resul in daa snooping bias han he regime shif approach. 5

7 We apply he uni roo ess o loss raios, expense raios, combined raios, and economic loss raios (see, e.g., Winer, 994) for nine propery-casualy insurance lines and for all lines combined using indusry-wide daa from Bes s Aggregaes and Averages for he period When eiher linear or quadraic rends are included in he model, he null hypohesis ha combined raios and economic loss raios have a uni roo is rejeced for mos lines, and, perhaps more imporan, for all lines combined using boh ADF ess and GLS-ADF ess. While ess ha allow for linear rend only in he DGPs of convenional loss raios and expense raios ofen fail o rejec he null hypohesis of a uni roo, allowing for quadraic rend frequenly causes rejecion of he null hypohesis for hose series. We conduc a Mone Carlo simulaion o provide evidence of es power in he specific conex of underwriing margins during he sample period. Our goal is o shed addiional ligh on wheher he failure o rejec he null hypohesis of a uni roo for some of he hisorical series could plausibly reflec low power and on he relaive power of possibly misspecified es equaions. We firs esimae for each daa series AR() regression models ha include mean, linear rend, and quadraic rend for he sample period. We nex simulae samples for each series using he parameer esimaes from hose regressions and saionary disurbances. We hen apply he uni roo ess o he simulaed series, which are saionary by design wih esimaed DGPs consisen wih he hisorical daa. Our simulaions show es power is low for some lines and es equaions. I generally is greaes when he ADF or GLS-ADF es includes eiher linear or quadraic rend, and i generally is negligible when neiher mean nor rend are included. 6

8 Our overall resuls sugges ha underwriing margins, especially combined raios and economic loss raios, generally are saionary around linear or quadraic rends. They provide suppor for using convenional leas squares mehods o esimae models of underwriing margins as a funcion of lagged underwriing margins, relevan deerminisic regressors, and oher saionary (or rend saionary) variables. They sugges ha coinegraion analysis may be neiher relevan nor necessary. Indeed, our main resuls imply ha combined raios and economic loss raios should no be coinegraed wih anyhing. 5 We emphasize ha our findings do no imply ha anyhing goes in regression model specificaion. Appropriae procedure will involve esing regressors for uni roos and will consider possible deerminisic componens in heir DGPs. If underwriing margins are (rend) saionary, he inclusion of any non-saionary regressors would sill make leas squares regression nonsensical. A possible sraegy in ha even would be o include relevan deerminisic componens and ransform (e.g., difference) he nonsaionary regressors o achieve saionariy, assuming ha he ransformed regressors would sill be consisen wih he underlying heory used o moivae esable hypoheses. 6 Secion describes he daa and uni roo ess. Secion 3 repors resuls of he uni roo ess. The power of differen uni roo ess for saionary series parameerized by he hisorical daa is analyzed in Secion 4. Secion 5 concludes. 5 Noe ha if ineres raes were non-saionary, basic insurance pricing heory implies ha he loss raio, expense raio, and combined raio would likely be non-saionary as well. However, Garcia and Perron (996) and Malliaropulos () provide evidence ha ineres raes are rend saionary. We applied ADF and GLS-ADF ess o he nominal 3-monh Treasury bill rae during our sample period allowing for quadraic rend. Boh ess rejeced he null hypohesis of a uni roo a he. level. 7

9 . Daa and Mehodology We es he hypohesis of a uni roo for loss raios, expense raios, combined raios, and economic loss raios for nine propery-casualy insurance lines and all lines combined using annual indusry-wide daa during from Bes s Aggregaes and Averages. The loss raio is he raio of losses and loss adjusmen expenses incurred in a line o ne premiums earned in ha line. The expense raio is he raio of underwriing expenses o ne premiums wrien. The combined raio is he sum of he loss raio and expense raio. 7 The economic loss raio is an esimae of he presen discouned value of incurred losses as a fracion of premiums ne of expenses. Following Winer (994), we calculae economic loss raios as D x loss raio / ( expense raio), where D is a line specific discoun facor calculaed wih 5-year Treasury bond yields and claim payou facors esimaed from Schedule P. 8 Figure plos he all lines expense raio, loss raio, combined raio, and economic loss raio during he sample period. I suggess an overall downward rend for he expense raio and upward rends for he oher variables. Table summarizes he individual lines analyzed. We include auo liabiliy, auo physical damage, homeowners, commercial muli-peril, workers compensaion, and 6 Enders (995, pp. 6-) provides a succinc overview of he implicaions of non-saionary regressands or regressors for regression resuls. Chaper 5 of his book includes useful discussion of he shor and long run dynamics of he regressand in a valid regression. 7 Because he combined raio is he sum of he loss raio and expense raio, esing for a uni roo in he combined raio is equivalen o esing for coinegraion of he loss and expense raio wih a resricion on he coinegraing vecor. We hank Jan Breuer and an anonymous referee for poining his ou. 8 The discoun facors used are lised in he Appendix. We use Winer s esimaed payou facors for all lines combined. Daa o esimae payou facors for individual lines were obained from Schedule P, Par 3A of he 99 issue of Bes s Aggregaes and Averages. We average he esimaes for acciden years Five-year Treasury bond raes were obained from he S. Louis Federal Reserve Bank's FRED daabase. 8

10 "oher" liabiliy (someimes called general or miscellaneous liabiliy). 9 Because Bes s Aggregaes and Averages began reporing separae resuls for medical malpracice, privae passenger auo liabiliy, and privae passenger auo physical damage in he 97s, we creaed and analyzed hree addiional series: () oher liabiliy excluding medical malpracice afer 977, () privae passenger auo liabiliy including commercial auo liabiliy prior o 978, and (3) privae passenger auo physical damage including commercial auo physical damage prior o 978. We apply seven uni roo ess: () an ADF es wih no mean or rend, () an ADF es wih mean only, (3) an ADF es wih mean and rend, (4) a GLS demeaned ADF es, (5) a GLS-ADF es wih mean and rend, (6) an ADF es wih quadraic rend, and (7) a GLS-ADF es wih quadraic rend. Based on preliminary ess and prior evidence on cyclical paerns in underwriing margins, we apply he ess assuming ha he underlying DGPs follow an AR() process. Augmened Dickey-Fuller (ADF) Tess To moivae he ADF es in he presen conex, consider he AR() process: y = ρ y + ρ y + ε, () where he disurbance erm, ε, is whie noise, i.e., idenically and independenly wih 9 Early ediions of Bes s Aggregaes and Averages repor underwriing experience for muual insurers separaely from sock insurers. We aggregaed muual and sock insurers underwriing margins by aking an earned premium weighed-average. Tess wihin he Dickey-Fuller framework assume ha errors are whie noise (Dickey, 976; Dickey and Fuller, 979, 98). Said and Dickey (984) show ha he ADF es can be used when he error process is a moving average. Phillips and Perron (988) developed modified ess ha allow for weak serial dependence among errors. Dufour and King (99) and DeJong, e al. (99b) invesigae he impac of serial correlaion in disurbances on he performance of ADF ess. We applied Phillips-Perron ess o he all lines combined raio and rejeced he null hypohesis of a uni roo a he. level, compared wih he.5 level for he ess repored below. 9

11 mean zero and finie variance ( ε ~ iid(, σ ε ) ). Deducing y from boh sides, his process can be rewrien as: y = a y + a y + ε, () wih a = ρ + ρ and a = ρ. Saionariy requires ρ + ρ (see Enders, 995). The ADF es is a es of H : < a = agains he alernaive a <. Equaion () illusraes ha one lagged difference in y should be included in he ADF es equaion for an AR() process. Allowing for mean, rend, and quadraic rend in he DGPs gives: y y = a = a + a y + a y + ε (3) + a y + a y + a3 + ε (4) y = a + a y + a y + a3 + a4 + ε (5) As discussed above and emphasized in he lieraure on esing for uni roos, including irrelevan deerminisic componens or excluding relevan componens reduces es power. Preliminary ess based on he Schwarz crierion favored he AR() specificaion over AR() and AR(3) models. We conduced uni roo ess assuming an AR(3) process in preliminary work and obained similar resuls o hose repored. We focus on ess of wheher a =. Dickey and Fuller (98) provide F-saisics for esing he join hypohesis ha a = a = a 3 =. We esed his join hypohesis for he underwriing margins for all lines combined. Broadly consisen wih he resuls repored below, he null hypohesis was rejeced for he combined raio and he economic loss raio bu no for he loss raio and expense raio. As noed in he inroducion, Doldado, e al. (99) and ohers sugges sequenial procedures o es for uni roos using ADF ess when he form of he DGP is unknown. (Aya and Burridge () sugges an alernaive sequenial mehod based on he GLS-ADF procedure.) Doldado, e al. (also see Enders, pp ) sugges applying he ADF es o he leas resricive model (i.e., he model wih mean and rend) firs. If he null hypohesis of a uni roo is no rejeced, esing proceeds o more resricive models. They also argue ha if he evidence suggess ha he correc model includes a mean or rend componen, he null hypohesis of a uni roo can be esed using he sandard normal disribuion raher han applying he empirical cumulaive disribuion for he Dickey-Fuller es. Perron (988), however, shows ha he resuling es is inconsisen (is power does no approach one as he sample size increases). Given he lack of agreemen abou preferred sequenial ess, we simply repor resuls for each es equaion. Noe,

12 GLS-ADF Tess Ellio, e al. (996) and Hwang and Schmid (996) compare he asympoic power funcion of he ADF es o ha of he upper bound of he limiing power funcions for he family of Neyman-Pearson ess. They find ha powers of ADF ess are lower han hose of he limiing power funcions when deerminisic componens (mean or rend) are included in he DGP. These sudies inroduce a poenially more powerful uni roo es: he GLS version of he ADF es. Two seps are involved in a GLS-ADF es. The firs sep is o remove deerminisic componens from he DGP using a GLS regression o esimae he coefficiens of he deerminisic componens. For a DGP wih mean, he mean is removed using: y * = y a, (6) ˆ where â is he GLS esimaed mean of he DGP. When boh mean and rend are included in he DGP, hey are removed using: y * = y a a. (7) ˆ ˆ3 where â o and â 3 are he GLS esimaes of he mean and rend coefficiens. Similarly, mean, rend, and quadraic rend are removed using: aˆ3 aˆ4 y * = y aˆ. (8) o The parameers of (6), (7) and (8) are esimaed wih GLS using weighs designed o maximize he value of he power funcion of a uni roo es (see he Appendix). The second sep is o conduc an ADF es on he derended and/or derended series using he saisic on he coefficien of * y in he following ordinary leas squares regression: however, ha he null hypohesis of a uni roo in underwriing margins is ofen rejeced for he leas

13 y * = a y * + a y * + ε (9) where a = ρ + ρ and a = ρ. 3. Tes Resuls Table repors -saisics for he coefficiens on he lagged underwriing margin from applying ADF and GLS-ADF ess o he loss raio, expense raio, combined raio, and economic loss raio under alernaive assumpions regarding he underlying DGPs. 3 The sample period is (wo years are los given inclusion of he lagged margins in he es equaions). Panel shows es saisics for loss raios. For he es equaion wihou mean or rend (column ), he null hypohesis of a uni roo in he loss raio canno be rejeced for any line of business. Resuls when he assumed DGP for loss raios includes a mean are repored in columns and 3. The null hypohesis again is seldom rejeced. The GLS- ADF resuls are similar o hose for he ADF ess. Panel, columns (4) and (5) show es resuls for loss raio models ha include boh mean and rend. Alhough he null hypohesis of a uni roo in loss raios is seldom rejeced a he five-percen level, he null hypohesis of a uni roo is rejeced for all lines combined a he. level by boh he ADF and GLS-ADF ess. I is rejeced a eiher he.5 or. level for four lines using he ADF and ADF-GLS ess. 4 Columns 6 and 7 in Panel show es resuls for he loss raio equaion ha allows for quadraic rend. The null hypohesis is rejeced a resricive model. 3 Criical values were obained from Fuller (996). 4 We also rejeced he hypohesis of a uni roo for homeowners insurance when a dummy variable was included for 99 o conrol he effecs of Hurricane Andrew on he loss raio, alhough convenional criical values for ADF ess may no be accurae when anoher regressor is included.

14 he.5 level for he all lines loss raio wih he ADF-GLS es and for six individual lines a he.5 or. level wih ha es. Panel of Table repors es saisics for uni roos in expense raios. The null hypohesis of a uni roo generally canno be rejeced unless he es equaion includes quadraic rend. When he es equaion includes mean and linear rend, he null hypohesis is rejeced only for commercial muli-peril insurance. When quadraic rend is included, however, he null hypohesis is rejeced a he.5 level for all lines combined and for five individual lines. The ADF and GLS-ADF ess yield similar resuls. Tes resuls for combined raios are repored in Panel 3 of Table. When a mean componen is included in he DGP, he null hypohesis of a uni roo is rejeced a he.5 level for four lines and a he. level for anoher hree lines and all lines combined. Moreover, when boh mean and rend are included, he null hypohesis is rejeced a he. level in all cases and a he.5 level for five lines (four lines wih he ADF es) and all lines combined. 5 The null hypohesis is rejeced for fewer individual lines when quadraic rend is included in he combined raio es equaion. A possible explanaion is ha any quadraic componens in loss raios and expense raios are parially offseing and ha inclusion of quadraic rend reduces power when i has lile explanaory power. 6 The es resuls for economic loss raios (Panel 4) are similar o hose for combined raios. 5 Again, when a dummy variable was included for year 99, we rejeced he null hypohesis of a uni roo in he combined raio for homeowners insurance. 6 The possibiliy ha loss and expense raios could have uni roos bu be coinegraed, so ha he combined raio is saionary, is inconsisen wih he evidence ha loss and expense raios for many of he series do no have uni roos. Tha he null hypohesis of a uni roo is rejeced for he combined raio and eiher he loss or expense raio bu no boh for a series may no be surprising given possibly low power, sampling error, and he large number of ess conduced. For example, he null is rejeced for he AL expense raio and combined raio for a leas one specificaion bu no for he AL loss raio. The simulaion analysis suggess ha power is low for he AL loss raio (see Table 4). 3

15 4. Tes Power This secion describes and presens resuls of simulaions o provide evidence of uni roo es power for underwriing margins during he sample period. Our goal is o shed addiional ligh on wheher he failure o rejec he null hypohesis of a uni roo for some of he hisorical series could plausibly reflec low power and on he relaive power of possibly misspecified es equaions. We firs esimae parameers of a DGP ha allows for possible mean, linear rend, and quadraic rend for each daa series: y = a ρ y + ρ y + a + a + ε. () Table 3 shows he resuls. The esimaes of ρ are posiive, while he esimaes of ρ are negaive (wih he excepion of homeowners). The sum of he esimaes is less han one for each series. The esimaed coefficiens for rend in he loss raio equaions (Panel ) are posiive and significan for five of he series, including all lines combined. The coefficiens for he quadraic rend erm are negaive and no infrequenly significan, consisen wih a diminishing rend in loss raios over ime. The esimaed rend coefficiens for expense raios (Panel ) are generally negaive whereas he esimaed coefficiens for quadraic rend are posiive. Thus, here is some evidence of a diminishing downward rend in expense raios. The opposie signs compared wih he loss raio equaions are no surprising given ha equilibrium loss raios should be negaively relaed o expense raios. The esimaed coefficiens for linear and quadraic rend for he combined raios (Panel 3) are insignifican for mos lines and all lines combined. However, equaions wih linear bu no quadraic rend (no repored) generally yielded posiive and 4

16 significan coefficiens for rend. The resuls herefore sugges ha any quadraic rends in loss and expense raios ended o offse and ha he inclusion of quadraic rend in he combined raio equaions makes i difficul o disinguish beween rend and quadraic rend ( and are very highly correlaed) and plausibly reduces power when quadraic rend is included in he uni roo es equaions (recall Panel 3 of Table ). The resuls of esimaing equaion () wih economic loss raios (Table 3, Panel 4) are similar o hose for combined raios. We hen simulaed samples for each series using: y = ˆ + ν, () y where ŷ is he prediced value from equaion () and ν is drawn from a normal disribuion wih mean zero and sandard deviaion equal o he esimae of σˆ ε from equaion (). Fory-four observaions are included in each simulaed series. The iniial wo values of he disurbance used o generae he lagged underwriing margins for he firs observaion (corresponding o 955) were drawn from a normal disribuion wih mean equal o he average value of he firs five observaions in he hisorical daa series and sandard deviaion given by (see Hamilon, 994): 7 σ u = ρ ρ σ ^ ε ρ ρ ( ) ρ () Given ha he sum of he esimaed auocorrelaion coefficiens for equaion () was less han for each series and ha he simulaed disurbances are saionary, each simulaed series is saionary wih a DGP ha reflecs he parameer esimaes using he 7 These wo iniial observaions ( y and y ) are used o simulae he remaining daa during he sample period. They are no included in he uni roo ess. The 44-year sample period for he uni roo ess corresponds o he number of observaions in he hisorical daa. 5

17 hisorical daa. We can herefore apply he uni roo ess o each series o examine he empirical power of each es o rejec he null hypohesis of a uni roo when i is false. Noe ha each es equaion excep he quadraic is misspecified, wih he degree of misspecificaion depending on he rue unknown parameers and sampling error in he esimaes of equaion (). The procedure herefore provides evidence of he power of differen, poenially misspecified ess o rejec he null hypohesis of a uni roo for known saionary series wih characerisics similar o he hisorical underwriing margins and he exen o which any of he ess have reasonable power for a given series. 8 Table 4 shows he empirical rejecion raes of each es a he.5 es size (significance level) for, samples of each series. ADF ess assuming no mean or rend in he DGP for underwriing margins have lile power, which may accoun for he failure of earlier sudies o rejec he null hypohesis and herefore assume ha underwriing margins have uni roos. Power increases subsanially when he model allows for a non-zero mean and increases furher when boh mean and rend are included. The ADF and GLS-ADF ess have similar power. For he equaions ha allow for linear bu no quadraic rend, es power is relaively high for combined raios (less so for economic loss raios), bu low for loss raios and especially expense raios. When quadraic rend is included, es power is ofen much higher for loss raios and especially expense raios compared wih he linear rend equaions. Power is low, however, for some lines, regardless of he es equaions. 8 Rudebusch (993) uses a similar approach o examine he power of uni roo ess ha are consisen wih he assumed DGP. Also see Diebold and Senhadji (996). 6

18 5. Conclusions Our overall resuls provide evidence ha propery-casualy underwriing margins are saionary and imply ha researchers migh fail o rejec he null hypohesis of a uni roo if hey do no allow for deerminisic componens in he DGP. When mean and rend are included in he assumed DGP, he null hypohesis ha combined raios and economic loss raios have a uni roo during is rejeced for mos of he business lines analyzed and for all lines combined using boh ADF and GLS-ADF ess. Including quadraic rend in he assumed DGP frequenly leads o he rejecion of he null hypohesis for loss raios and expense raios as well. Our Mone Carlo simulaions indicae ha es power is relaively low for some lines of business and generally very low when neiher mean nor rend are included in he es equaion. Our findings sugges ha i is neiher necessary nor appropriae in empirical work o assume ha underwriing margins are difference saionary or o employ coinegraion analysis. They sugges insead ha convenional regression mehods can be used o esimae economeric models of he deerminans of underwriing margins in levels, allowing for possible linear and quadraic rend, and provided ha he relevan explanaory variables are saionary (or rend saionary). Appropriae mehodology will involve esing regressors for uni roos and will consider possible deerminisic componens in heir DGPs. We emphasize ha our findings do no imply ha anyhing goes in model specificaion. Inclusion of non-saionary regressors will produce meaningless resuls even if underwriing margins are (rend) saionary. If an explanaory variable is no saionary, i may be possible achieve saionariy by differencing wihou violaing he underlying heory used o generae esable hypoheses. 7

19 Appendix Claims Payou Facors. The esimaed claims payou facors used o calculae he economic loss raios are shown below (see Table for line definiions): Line Year Year Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 ALL AL PPAL HO CMP WC OL OLMM The facors for all lines combined (ALL) are hose repored in Winer (994). The individual lines facors are hree-year averages of facors esimaed for acciden years from Schedule P, Par 3A, as repored in he 99 issue of Bes s Aggregaes and Averages. The GLS-ADF Tes. Based on Ellio, e al (996), he GLS esimaor for a, in equaion (8) is obained by regressing he vecor, ^ ' y T ρ y T y, y ρ y..., on he vecor, ', ρ,..., ρ, where ρ = 7. / T and T is he number of observaions used for he uni roo es. The asympoic power funcions under his es are close o upper bound of all possible power funcions (he power envelope). Specifically, A his poin, he asympoic power funcion of he GLS-ADF es is angen o is power envelope a a power of 5 percen when he size is se a a 5 percen level. King (988) suggess ha a es s power funcion is close o he power envelope over a considerable range under his seup. When boh mean and rend componens are boh included in he DGP, ^ ^ [ a 3 a, ]' in equaion (9) are esimaed by regressing, [ y, y ρ y,..., y T ρo yt ]' on he marix 8

20 ,, ρ, ρ,...,..., ρ T ( T ) ρ ', where ρ equals 3.5 / T. When a quadraic rend is included in he DGP, ^ ^ ^ [ a 4 a, a 3, ]' in equaion () are esimaed by regressing, [ y, y ρ y,..., y T ρ o yt ]' on he marix,,, ρ, ρ, 4 ρ,...,...,..., ρ T ( T ) ρ, where ρ T ( T ) ρ ' equals 8.5 / T. 9

21 References Aya, Leila, and R. Burridge,, Uni Roo Tess in he Presence of Uncerainy Abou he Non-Sochasic Trend, Journal of Economerics, 95: Choi, Seungmook and Paul Thisle, 997, A Srucural Approach o Underwriing Cycles in he Propery-Liabiliy Insurance Indusry," Working Paper, Wesern Michigan Universiy. Cummins, J. David, and Francois Oureville, 987, An Inernaional Analysis of Underwriing Cycles In Propery-Liabiliy Insurance, Journal of Risk and Insurance, 54: DeJong, David N. and Charles H. Whieman, 99, The Case for Trend-Saionariy is Sronger han We Though, Journal of Applied Economerics, 6: DeJong, David N., John C. Nankervis, N.E. Savin, and Charles H. Whieman, 99a, Inegraion Versus Trend Saionary in Time Series, Economerica, 6: DeJong, David, N., John C. Nankervis, N.E. Savin and Charles H. Whieman, 99b, The Power Problems of Uni Roo Tess in Time Series wih Auoregressive Errors, Journal of Economerics, 53: Dickey, D.A., 984, Powers of Uni Roo Tess, In Proceedings of he Business and Economics Secion, American Saisical Associaion, pp Doldado, Juan, Tim Jenkinson, and Simon Sosvilla-Rivero, 99, Coinegraion and Uni Roos, Journal of Economic Survey, 4: Dickey, David and Wayne A. Fuller, 979, Disribuion of he Esimaes for Auoregressive Time Series wih a Uni Roo, Journal of he American Saisical Associaion, 74: Dickey, David and Wayne A. Fuller, 98, Likelihood Raio Saisics for Auoregressive Time Series wih a Uni Roo, Economerica, 49: Diebold, Francis X. and A.S. Senhadji, 996, The Uncerain Uni Roo in Real GNP: Commen, American Economic Review, 86: Dohery, Neil and Ham Bin Kang, 988, Ineres Raes and Insurance Price Cycles, Journal of Banking and Finance, : Dufour, Jean-Marie and Maxwell L. King, 99, Opimal Invarian Tess for he Auocorrelaion Coefficien in Linear Regressions wih Saionary or Nonsaionary AR() Errors, Journal of Economerics, 47: Ellio, G., T.J. Rohenberg, and J.H. Sock, 996, Efficien Tess for an Auoregressive Uni Roo, Economerica, 64: Enders, Waler, 995, Applied Time Series Economerics, New York: John Wiley and Sons. Engle, Rober and C. W. Granger, 987, Co-inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, 55: Fuller, Wayne A., 996, Inroducion o Saisical Time Series, New York: John Wiley and Sons. Garcia, R. and Perron, P., 996, An Analysis of he Real Ineres Rae Under Regime Shifs, Review of Economics and Saisics, 78: -5. Grace, Marin and Julie Hochkiss, 995, Exernal Impacs on he Propery-Liabiliy Insurance Cycle, Journal of Risk and Insurance, 6:

22 Granger, C. W. and P. Newbold, 974, Forecasing Economic Time Series, New York: Academic press. Gron, Anne, 994, Evidence of Capaciy Consrains in Insurance Markes, Journal of Law and Economics, 37: Haley, Joseph, 993, A Coinegraion Analysis of he Relaionship Beween Underwriing Margins and Ineres Raes: , Journal of Risk and Insurance, 6: Haley, Joseph, 995, A By-Line Coinegraion Analysis of Underwriing Margins and Ineres Raes in he Propery-Liabiliy Insurance Indusry, Journal of Risk and Insurance, 6: Hamilon, James D., 994, Time Series Analysis, Princeon Universiy Press, Princeon, New Jersey. Harringon, Sco E. and Greg Niehaus,, Volailiy and Underwriing Cycles, in Georges Dionne, ed., The Handbook of Insurance Economics, Boson: Kluwer Academic. Hwang, Jaeyoun and Peer Schmid, 996, Alernaive Mehods of Derending and he Power of Uni Roo Tess, Journal of Economerics: King, M. L., 988, Towards a Theory of Poin Opimal Tesing, Economeric Reviews, 6: Malliaropulos, Dimirios,, A Noe on Nonsaionariy, Srucural Breaks, and he Fisher Effec, Journal of Banking and Finance, 4: Nelson, Charles R. and Charles I. Plosser, 98, Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implicaions, Journal of Moneary Economics, : Perron, Pierre, 988, Trends and Random Walks in Macroeconomic Time Series: Furher Evidence from a New Approach, Journal of Economic Dynamics and Conrol, : Perron, Pierre, 989, The Grea Crash, he Oil Price Shock, and he Uni Roo Hypohesis, Economerica, 57: Rudebusch, Glenn D., 993, The Uncerain Uni Roo in Real GNP, American Economic Review, 83: Said, Said E. and David A. Dickey, 984, Tesing for Uni Roos in Auoregressive-Moving Average Models of Unknown Order, Biomerika, 7: Venezian, Emilio, 985, Raemaking Mehods and Profi Cycles in Propery and Liabiliy Insurance, Journal of Risk and Insurance, 5: 99-. Wes, K., 987, A Noe on he Power of Leas Squares Tess for a Uni roo, Economics Leers, 4: Winer, Ralph A., 994, The Dynamics of Compeiive Insurance Markes, Journal of Financial Inermediaion, 3:

23 Figure All lines Loss Raios, Expense Raios, Combined Raios, and Economic Loss Raios ( ) 4 8 Percen Combined Raios Loss Raios Expense Raios Economic Loss Raios

24 Table Lines of Business Analyzed Variable Descripion Sample Period Noe ALL All lines combined AL Auo liabiliy APD Auo physical damage PPAL PPAPD Privae passenger auo liabiliy Privae passenger auo physical damage HO Homeowners CMP Commercial muliple peril WC Workers compensaion Includes commercial auo liabiliy prior o Includes commercial auo physical damage prior o 978 OL Oher liabiliy Includes medical malpracice prior o 978 OLMM Oher liabiliy and medical malpracice

25 Table Uni Roo Tes Saisics for Alernaive DGPs No Mean or Trend Mean Only Mean and Trend Quadraic Trend GLS- GLS- GLS- ADF ADF ADF ADF ADF ADF ADF () () (3) (4) (5) (6) (7) Panel : Loss Raios ALL ** -3.5** -3.66** -3.87* AL APD * -.89** -4.5* -4.9* -5.74* -5.89* PPAL ** PPAPD.8-4.5* -.79** -4.39* -4.* 6.9* -6.47* HO ** -3.38** * CMP * * -3.89* WC.3 -.7** OL ** OLMM ** ** ** Panel : Expense Raios ALL * -4.49* AL * -5.34* APD -.79** PPAL * -5.* PPAPD -.98* HO -.7* * CMP * -3.38* -3.8* -4.44* -3.89* -4.* WC OL * -4.5* OLMM * -3.8** Panel 3: Combined Raio ALL. -.8** -.83** -4.4* -4.* -3.9* -3.79** AL.7-3.7* -3.3* -3.38** -3.5* APD.7-4.9* -4.3* -4.36* -4.47* -6.* -6.* PPAL. -3.8* 3.* ** PPAPD * -4.47* -4.45* -4.33* -6.4* -6.4* HO.8 -.7** -.7** -3.** -3.4** CMP ** -.78** -4.9* -4.* -3.94* -4.* WC ** -.79** -3.5** -3.3** OL ** -3.7** -3.67** -3.79* OLMM * -3.7* ** Panel 4: Economic Loss Raios ALL.8-3.* -.6** -3.4** -3.57* -4.8* -3.8** AL * -.93* ** -4.* -3.7** PPAL * -.99* ** -3.56** -3.5 HO CMP ** -4.* -4.43* -4.6* WC.7-3.5* -3.47* -3.56* -3.66* -3.86* -4.* OL ** -.76** -3.46** -3.73* -3.6** -3.86* OLMM ** * -3.5* -3.66** -3.88* See Table for line descripions. The esimaion period is Values repored are -saisics for he lagged underwriing margin. From Fuller (996), he.5 and. criical values (wih 5 observaions) are: neiher mean nor rend, -.95 and -.6; mean only, -.93 and.6; linear rend, -3.5 and -3.8; and quadraic rend, and * Significan a.5 level. ** Significan a. level. 4

26 Table 3 Esimaed Parameers and Residual Sandard Deviaions for AR() Model of Underwriing Margins (in percen) wih Mean, Linear, and Quadraic Trend: y = a + y + ρ y ρ + a + a + ε 3 4 ^ a ^ 3 a Panel : Loss Raios ALL 3.85*.5* -.6**.79* -.3** AL 7.8*.9 -..* -.48* APD 43.79*.74* -.*.77* -.53* PPAL 5.64*.7 -..* -.43* PPAPD 48.36*.9* -.4*.7* -.55** HO 4.4* CMP 8.8* * -.3* WC 5.76* * -.55* OL 6.98*.* -..97* -.37* OLMM 3.5*.6* -.*.99* -.34* Panel : Expense Raios ALL 6.39* -.5*.4*.5* -.5* AL 5.3* -.*.3*.7* -.66* APD 5.5* -.3*.3*.7* PPAL 7.79* -.4*.3*.6* -.6* PPAPD 5.9* -.6*.3*.65* HO 6.7* -.43*.6*.33* CMP 5.6* * -.49* WC 4.86* -.*.3*.47* -.66* OL 5.7* -.3*.5*.3* -.6* OLMM 3.5* -.3*.4*.6* -.57* ^ 4 a ^ ρ ρ ^ σ ^ ε 5

27 Table 3 Coninued ^ a ^ 3 a Panel 3: Combined Raios ALL 49.5* * -.37* AL 6.9* * APD 7.3* **.8* -.57* PPAL 4.88* * -.5* PPAPD 76.*.47 -.*.74* -.58* HO 69.9* CMP 44.58* * -.37* WC 5.64*.4 -..* -.49* OL 36.67* * -.4* OLMM 3.3*.76** -..99* -.37* Panel 4: Economic Loss Raios ALL 47.7* -.3.8*.83* -.36* AL 33.7* -.35*.7*.8* -.5* PPAL 3.* * -.5* HO 6.6* * CMP 57.8* -.83*.5*.78* -.37* WC 3.7* * -.59* OL 37.36* * -.3* OLMM 35.97*..5.85* -.3* See Table for line descripions; -saisics are repored below he esimaed coefficiens. The esimaion period is * Significan a.5 level. ** Significan a. level. ^ 4 a ^ ρ ρ ^ σ ^ ε 6

28 Table 4 Simulaed Powers of Uni Roo Tess a.5 Tes Size No Mean or Trend Mean Only Mean and Trend Quadraic Trend ADF ADF GLS-ADF ADF GLS-ADF ADF GLS-ADF () () (3) (4) (5) (6) (7) Panel : Loss Raios ALL % % % 66% 6% 74% 77% AL APD PPAL PPAPD HO CMP WC OL OLMM Panel : Expense Raios ALL AL APD PPAL PPAPD HO CMP WC OL OLMM Panel 3: Combined Raios ALL AL APD 97 PPAL PPAPD HO CMP WC OL OLMM Panel 4: Economic Loss Raios ALL AL PPAL HO CMP WC OL OLMM See Table for line descripions. Values shown are percenages of saionary samples of 44 observaions where he null hypohesis of uni roo is rejeced. 7