j. david cummin and alwys n powell€¦ · j. david cummin and alwys n powell 1. introduction when...

Astin Bulletin 11 (1980) 91-106

THE PERFORMANCE OF ALTERNATIVE MODELS FORFORECASTING AUTOMOBILE INSURANCE PAID CLAIM COSTS

J. DAVID CUMMINS and ALWYN POWELL

1. INTRODUCTION

When setting or modifying insurance rates on the basis of observed data,adjustments should be made for inflation between the average claim date ofthe observation period and the average claim date covered by policies to whichthe new rates will apply. In automobile insurance ratemaking in the UnitedStates, the adjustment is made as follows 1:

(1) La = U x (i + p)&

where La = trended losses and loss adjustment expenses incurred,Lo — observed losses and loss adjustment expenses incurred during the

experience period,(5 = the predicted rate of inflation per quarter, andk = the number of quarters for which losses are being trended.

The inflation coefficient, (3, is an exponential growth factor that is estimatedusing the following equation 2:

(2) lnj^ = x

where y% = average paid claim costs for the year ending in quarter t,t = time, in quarters, t = 1, 2, . . . , 12, and

et = a random disturbance term.

The estimation period for the equation is the twelve quarters immediatelypreceding the first forecast period, and the estimation is conducted utilizingordinary least squares. Annual averages of claim costs are used as the dependentvariable to smooth random and seasonal fluctuations.

The use of equation (2) can be expected to lead to two major problems:l. Since the equation merely extrapolates past inflation rates into the future,

the forecasts are likely to be seriously inaccurate when the inflation rate inthe forecast period is substantially different from that in the preceding twelvequarters.

1 In some cases, an adjustment also is made for frequency. The formula in such casesis: La = L0(i + J3s)

fc . (i + (3/)*, where (3, = the predicted rate of change in severityand (3/ = the predicted rate of change in frequency. This article is limited to an analysisof severity, which is the predominant source of claims cost inflation.

2 Trend factors are discussed in COOK (1970). When frequency trend factors are em-ployed, they also are obtained through exponential trending.

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0,2 J. DAVID CUMMINS AND ALWYN POWELL

2. Forecasts based on four quarter moving averages are likely to respondslowly to changing inflation rates. Thus, premium rates will be unresponsive,contributing to the underwriting cycle.

An alternative approach to forecasting average paid claim costs is to usean econometric model which predicts claim costs as a function of relevanteconomic variables. This approach has become feasible in recent years asprogress has been made in developing forecasting models for major economicaggregates in most industrialized nations 3. The purpose of this article is todevelop and test the accuracy of econometric models for predicting automobileliability insurance paid claim costs. The estimation and testing are conductedusing claim cost data for the United States.

2. EQUATION SPECIFICATION AND METHODOLOGY

Among the advantages of the exponential trend model (equation (2)) are itsfamiliarity and simplicity. These are important qualities because rate increasesin most states must receive regulatory approval. Since regulators are likelyto be suspicious of highly technical models, any methodology proposed as areplacement for exponential trending should be as simple as possible. Hence,the approach taken in this article is to begin by utilizing ordinary least squares(OLS) to estimate equations with the simplest possible specification. Teststhen are conducted to determine whether the resulting equations depart fromthe assumptions underlying the OLS technique. Where departures are found,the equations are reestimated with the appropriate adjustments.

The three basic specifications used in the research are the following:

(3) yt = ct. + $xt + zt

(4) In yt = a + (3 In xt + st

(5) yt = cL

where yt = average paid claim costs in quarter t,xt = a price or wage index, andS{ = the random disturbance term.

Linear and log-linear equations in which two economic indices appear asexplanatory variables also were estimated, and dummy variables for seasonalitywere tested with each type of equation. Since the same estimation methodsapply to (3) and (4), the examples pertaining to these equations in the followingdiscussion are based only on equation (3).

3 Models of the U.S. economy are reviewed in KLEIN and BURMEISTER (1976). Formodels applicable to other countries, see WAELBROECK (1976).



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AUTOMOBILE INSURANCE CLAIM COSTS 93

The classical ordinary least squares model incorporates the following as-sumptions about £(4:

(6) E(et) = o

(7) E[(xt-Xt)et] = o

(8) £(s«) = a*

(g) E{tt . zt~i) = o, for i = l , 2, . . . t.

If the assumptions are satisfied, ordinary least squares yields parameterestimates for equations (3) and (4) that are best linear unbiased, i.e., that havethe smallest variances among all linear unbiased estimators. It is not necessaryto assume normality in order to achieve this result, although the usual tests ofsignificance for the coefficients will be correct only asymptotically if normalityis not present. Even if all of the assumptions are satisfied, the ordinary leastsquares estimators for equation (5) are biased although they are consistentand asymptotically efficient.

As assumptions (6) and (7) rarely cause problems, attention is focused onassumptions (8) and (9), i.e., homoskedasticity and the absence of serialcorrelation, respectively. If either assumption is violated, ordinary leastsquares estimators of the parameters in equations (3) and (4) are unbiased andconsistent but are inefficient both in finite samples and asymptotically. Forequation (5), violation of assumption (8) implies that ordinary least squaresestimators are consistent but asymptotically inefficient, while violation ofassumption (9) renders the OLS estimators both inconsistent and asymptotical-ly inefficient.

Because of the potential importance of assumptions (8) and (9) for forecastaccuracy, tests are conducted to determine whether they are satisfied in theestimated relationships. For equations (3) and (4), the Durbin-Watson dstatistic is used to test for serial correlation, while the hypothesis of homo-skedasticity is tested using a procedure suggested by GOLDFELD and QUANDT

(1965) 5. Because the Durbin-Watson d statistic is inappropriate for equationswith lagged dependent variables among the explanatory variables, an alter-native procedure suggested by DURBIN (1970) is used to test for serial correla-tion in equation (5) 6. The Goldfeld-Quandt heteroskedasticity test also isapplied to equation (5).

4 The discussion of estimation theory in this section draws heavily on KMENTA (1971).5 The Goldfeld-Quandt test has been shown to be at least as powerful as other major

heteroskedasticity tests. See HARVEY and PHILLIPS (1974) and HARRISON and MCCABE

(1979)-6 DURBIN proposed two tests—one based on the statistic h and the other on regressions

involving the OLS residuals of the equation being tested. The latter test is used herebecause the h statistic has been shown to be biased in small samples. See SPENCER (1975).



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94 J- DAVID CUMMINS AND ALWYN POWELL

If the tests indicate that autocorrelation is present in equation (3) or (4),the approach is to hypothesize that e< = pst-i + Ut, where ut satisfies (6)through (9). The equation then is reestimated by solving the following problem:

T

(10) Minimize 2 [yt - pyt-i - a(l - p) - $(xt - pxt-1)]2

(a .P .p) ( - «

Minimizing (10) is asymptotically equivalent to maximum likelihood estimationif Ut is normally distributed 7.

A method suggested by GLEJSER (1969) is used to adjust for heteroskedasti-city. This technique is based on the hypothesis that st — P{z)wt, where P(z)is a function of a mathematical variable z, and wt satisfies (6) through (9).The variable z is taken to be (one of) the independent variable (s) in the re-gression, in this case xt. The estimation is conducted as follows:

(1) Estimate equation (3) or (4) using OLS and obtain the residuals, sj =

yt-x- $xt.(2) Regress [ s* | on simple functions of the independent variable such as

To + Yixt> To + Yixl< To + Ti^2» e^c- ar*d choose as the functional form P(xt) theequation with the highest explanatory power (R2).

(3) Transform the regression equation by dividing each term by P(xt).E.g., if the best regression for | e* | is yo+yi*2, the transformed variables are

equal to y't = ytl(y0 + r>«), x't = *(/(YO + Y>?)> a n d c't = ^/(Yo + Yi**).

where c't is the transformed constant term.(4) Estimate specification (3) or (4) using the transformed variables and

OLS. For example, equation (3) becomes: y't = a.c't+ $x't + wt.If the hypothesis is correct, the error term of the transformed equation

satisfies (6) through (9). However, the correction will be only approximate forfinite sample sizes.

When the error terms in equation (3) or (4) are characterized by both auto-correlation and heteroskedasticity, the estimation problem is more difficult.The ideal solution would be to use a generalized nonlinear estimation procedurewhich would correct simultaneously for both problems. Since computerprograms to perform this type of estimation were not readily available to theauthors, two alternative approaches were adopted, based on extensions ofGlejser's method.

Method 1. The following error structure is hypothesized: sj = pst-i + Ut,where ut = P{xt)wt) E{utzt-i) = E(utut-i) = E(wtwt-i) = o, for i = 1, 2,. . . , t; and E(w2

t) = a2. Under this hypothesis, the first step in the estimationis to solve the minimization problem in expression (10). The absolute valuesof the estimated residuals from this equation then are regressed on various

' Minimization was conducted using the Time Series Processor (TSP) program. SeeHALL, HALL, and BECKETTI (1977).



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rAUTOMOBILE INSURANCE CLAIM COSTS 95

functions of xt to determine P(xt). Finally, OLS is used to estimate the fol-lowing equation:

(U) y t h t - i = « ( 1 P ) + $(xtpxt-i) + ut

P(xt) P(xt) P(xt) P(xt)

where p = the estimate of p obtained in the first stage by solving the mini-mization problem in (io), and

P(xt) = the functional form of xt which provides the highest explanatorypower in regressions of the absolute residuals from (io).

Method 2. In this case, the error structure is assumed to be: zt =P{xt)\put-i + wi\, w h e r e E(ut-iWt) = E(wtwt-i) = o , i = 1 , 2 , ...,t; a n d

E(w2t) = a2. The estimation procedure is to obtain OLS estimates of a and [3,

define %t = yt — u— fi%t, and regress | et | on various functions of xt. Denotingthe best of these functions as P{xt), the final step is to minimize the followingexpression with respect to a, (3, and p:

( yP(xt) P(xi) P{xt)

Equations estimated through both methods were used in the forecastingexperiments with the choice between the two based on forecast accuracy.

Efficient estimation of equations with lagged dependent variables amongthe regressors when the residuals are autocorrelated or heteroskedastic is adifficult econometric problem requiring specialized computer programs notavailable to the authors. Accordingly, no adjustments were made whenequation (5) was characterized by these problems, and the OLS versions wereused in the forecasting experiments.

3. THE DATA AND THE FORECASTING EXPERIMENTS

The data used in estimating the forecasting equations are automobile bodilyinjury and property damage liability insurance average paid claim costsreported by the Insurance Services Office (ISO). The ISO is the leading ratingbureau in the United States, maintaining automobile insurance data for morethan one hundred contributing insurance companies. Pooled data from allcontributing companies are used by ISO to develop trend factors that affectthe rates of a subset of contributors (others price independently). A credibilityweighted average of state and national data is used in the trending processfor rates applicable to a particular state. The forecasts in this article are basedon the national averages.

The data period for automobile property damage liability claim costs isthe first quarter of 1954 (1954.1) to the second quarter of 1978 (1978.2). For



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96 J. DAVID CUMMINS AND ALWYN POWELL

bodily injury liability, the period is 1964.1 to 1978.2. The property damagefigures are for total limits coverage, while the bodily injury data are for$ 10,000 basic limits coverage (i.e., claims in excess of $ 10,000 appear in theaverages as $ 10,000). Each series is obtained by dividing the total dollaramount of claim payments in a given quarter by the total number of claims inthe quarter. The claims cost series, converted to index form, are presentedin the Appendix 8.

Several price and wage indices were tested as independent variables. Amongthose with the highest explanatory power are the implicit price deflator forgross national product, the implicit price deflator for autos and parts, andwage indices for the service sector and for the private sector of the economy.Actual values of the deflators and indices are available in published U.S.government sources 9, while predicted values were obtained from WhartonEconometric Forecasting Associates (WEFA), one of the leading U.S. fore-casting firms 10.

The initial screening of predictor variables was based on summary regressionstatistics such as the coefficient of determination (i?2) u . After selecting thetwo or three best variables, subsequent evaluations of variables and specifica-tions were based on predictive accuracy. The approach taken in the experimentswas to forecast average paid claim costs for periods of eight quarters beginningat one-year intervals from 1971 through 1977 12. The third quarter of eachyear was arbitrarily selected as the initial forecast period. The goal in theexperiments was to reproduce as accurately as possible the results that wouldhave been obtained if the forecasting equations actually had been employedin the periods under consideration. Accordingly, the econometric models wereestimated using actual values of the dependent and independent variables forthe period ending in the quarter prior to the initial forecast period, and theforecasts were generated using predicted values of the independent variables.

8 Transforming data into indices affects the coefficients of any regressions based onthe transformed data but not the values of the essential test statistics. KMENTA (1971),PP- 377-379-

9 The full title of the autos and parts deflator is the implicit deflator, personal consump-tion expenditures, durables, autos and parts. It and the GNP deflator are available in theSurvey of Current Business (Washington, D.C.: U.S. Government Printing Office, monthly).The wage indices are compensation per man hour, private sector and compensation perman hour, service sector. They are constructed from series available in the Survey ofCurrent Business and in Employment and Earnings (Washington, D.C.: U.S. GovernmentPrinting Office, monthly).

10 WEFA forecasts are released quarterly. A summary version of the forecast is presentedeach quarter in The Wharton Magazine, which is published at the University of Penn-sylvania, Philadelphia, Pa.

11 i?2 is not a good indicator of predictive accuracy. However, it was considered ac-ceptable as an initial screening statistic.

12 Data availability limited the last forecast period to four quarters.



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The accuracy of the forecasts is determined by reference to the followingstatistics:

(13) TPCE = (ys~ yB

\x 100

MAPE = V l ^ ^ J x 100^-—i Vt

RMSPE = IOO x 1/ y {p.—

where TPCE = total predicted change error,MAPE = mean absolute percentage error,

RMSPE = root mean square percentage error,yt = the predicted value of average paid claim costs in quarter t

of the forecast period,yt = actual average paid claim costs in quarter t of the forecast

period, andyo = the actual value of average paid claim costs in the quarter

immediately preceding the forecast period.

The TPCE is the most important measure of forecasting accuracy forratemaking purposes because it measures the percentage by which predictedclaim costs exceed (fall short of) actual claim costs for the period to which thenew rates would apply.

Tests with the ordinary least squares versions of equations (3), (4), and (5)revealed little difference among alternative predictor variables in terms ofgoodness-of-fit or forecast accuracy. Since the private sector wage rate per-formed at least as well as the other variables during most forecast periods,it is used exclusively in the reported regressions. Equations with more thanone economic explanatory variable were not much better than the univariateequations in terms of goodness-of-fit and, in fact, often yielded unrealisticsigns for the coefficients because of multicollinearity. Accordingly, the resultspresented below are based on equations with one economic regressor. The bestforecasts were obtained with equations estimated over the maximum availableobservation period.

4. RESULTS: AUTOMOBILE PROPERTY DAMAGE LIABILITY PAID CLAIM COSTS

For automobile property damage liability paid claim costs, the OLS versionof equation (5) yielded forecasts which generally were more accurate than thosefrom the OLS versions of equations (3) and (4). Tests revealed that all threespecifications are subject to autocorrelation and heteroskedasticity in nearly



https://doi.org/10.1017/S0515036100006681



every estimation period 13. Since it was not feasible to adjust equation (5) forthese problems due to program limitations, the next-best OLS model, equation(3), was chosen for adjustment. The exponential trend equations and the OLSversion of equation (5) are presented in Table 1. Equation (3), adjusted forautocorrelation, is shown in the top panel of Table 2, while the bottom panelshows this specification adjusted for both autocorrelation and heteroskedasti-city according to Method 1 14. Dummy variables for seasonality are included inthe equations presented in Table 2 but not in the OLS version of equation (5)because preliminary results indicated that the inclusion of seasonal dummiesin the OLS equations leads to less accurate forecasts.

The forecast results for property damage liability paid claim costs areshown in Table 3. The table gives the TPCE, the MAPE, and the RMSPE forthe exponential trend model (ET), the OLS model with lagged endogenousregressor (LE), equation (3) adjusted for autocorrelation (AR), and the samespecification adjusted for both autocorrelation and heteroskedasticity (ARH).To provide a benchmark for analyzing the forecast errors, a shift index tomeasure unanticipated inflation has been included in the table. The index isthe ratio of the (quarterly) geometric mean rate of claims cost inflation duringeach forecast period to the geometric mean inflation rate during the precedingtwelve quarters.

In terms of TPCE, all three econometric models outperform the exponentialtrend model in five of the seven forecast periods. As expected, the performanceof the trend model is worst when unanticipated inflation is highest. Amongthe econometric models, those adjusted for both autocorrelation and heteroske-dasticity do best during periods characterized by high unanticipated inflation(e.g., the forecast periods beginning in 1973.4, 1974.4, and 1975.4). The OLSmodel also does quite well during these periods, suggesting its use as a rela-tively simple replacement for the exponential trend model.

5. RESULTS: AUTOMOBILE BODILY INJURY LIABILITY PAID CLAIM COSTS

The exponential time trend and best OLS equations for automobile bodilyinjury liability claim costs are presented in Table 4. In this case, equation (3)is the best OLS model. The hypothesis of homoskedasticity is rejected for thismodel during the last five estimation periods. The Durbin-Watson test in-dicates the presence of autocorrelation during the last four periods and givesinconclusive results for the period ending in 1973.2. Since the behavior ofthe test statistics when autocorrelation and heteroskedasticity are presentsimultaneously is not clear, the equations for all periods were reestimated

13 All significance test results reported in this article are at the 5 percent level.14 For the first two sample periods, P{xt) = y0 + yj%t '• while for the last five periods



https://doi.org/10.1017/S0515036100006681


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https://doi.org/10.1017/S0515036100006681



TABLE 2 : AUTOREGRESSIVE EQUATIONS FOR AUTOMOBILE PROPERTY DAMAGE LIABILITY

AVERAGE PAID CLAIM COSTS:

EQUATIONS IN LOWER PANEL ADJUSTED FOR HETEROSKEDASTICITY

Equation forForecastBeginning in

1971-3

1972.3

1973-3

1974-3

1975-3

1976.3

1977-3

1971-3

1972-3

1973-3

1974-3

1975-3

1976.3

1977-3

yt

85.6807(13.780)

77.2711(21.109)

76.0740(23-796)

72.2098

(29-452)

72.7631

(31.229)

72.4086

(35-250)

73-4537(35-232)

89.6291

(37-599)

77.9910(33-828)

75-5438(38.015)

71.0324

(43-322)

72.0044

(45-473)

72.2156

(49-337)

72.7016

(51-657)

= p*-1 + *(.

Yi

— 9.3200(-11.828)

— 9.8140(-10.518)

—10.2579

(-10.725)

— 10.6073

(-9-39O)

— 10.8620( — 10.101)

— 10.7907(-10.588)

-11-5857(-11.108)

-14.1917(-17.047)

-10.7988

(-10.874)

— 9.4062(-11.164)

-8.3723(-10.736)

-9-3375(—10.980)

-10.1239

(-H-175)

-10.3194

(-11-635)

-P) + P K -

Ya

-2.3364(—2.960)

-2-7595(-2.940)

-2.7396

(-2.835)

-3.0991(-2.688)

-35125(-3-196)

-3-511O(-3-359)

-3.6506(-3-396)

— 0.7009(-0.879)

-2.6878

(-2.778)

-2.8493(-3-498)

-3.1814(-4.244)

-3-3O19(-4.012)

-33716(-3-826)

-3.4609(-4.016)

M + fA-

Ya

-5-9115(-7.464)

-5-8231(-6.211)

-5-5684

(-5784)

-5.6290

(-4-938)

-57196(-5-238)

-5.6970

(-5.489)

-6.1675

(-5-77O)

-6.4039

(-7.4297)

-5-7983(-5-938)

-5-7251(-6.979)

-5-9O34(-7.786)

-5-9526

(-7-179)

-58504(-6.614)

-6.1045(-7.050)

HYaSa + Y3S3

P

•950495(64-735)

.924875

(50.785)

.918491(49.380)

.897217

(38.963)

.895484

(39-829)

•893974

(41-125)

.892990

(40.288)

•95O495

•924875

.918491

.897217

.895484

•893974

.892990

R*

•9983

.9980

•9982

•9977

.9981

.9985

.9987

•9751

.8983

•8030

.5889

•7483

•8724

.8612

Key: yt = average paid claim costs in quarter t; wt — private sector wages per man-hour in quarter t; and St = 1.0 in quarter i of each year, o otherwise.Note: The estimation period for each equation begins in 1954.1 and ends in the quarterpreceding the first forecast period. The number in parentheses below each estimatedcoefficient is the ratio of the coefficient to its estimated standard deviation. The constantterm a was estimated in an earlier version of the equations but was found to be insig-nificant.



https://doi.org/10.1017/S0515036100006681


TA

BL

E 3

: R

ESU

LT

S O

F E

IGH

T Q

UA

RT

ER

FO

RE

CA

STS

OF

AU

TO

MO

BIL

E P

RO

PER

TY

DA

MA

GE

LIA

BIL

ITY

AV

ER

AG

E P

AID

CLA

IM C

OST

S

Fir

st P

erio

dof

For

ecas

t

1971

-3

1972

-3

1973

-3

1974

-3

1975

-3

1976

.3

1977

-3 °

Abs

olut

e A

vera

ges

ET

To

tal

Pre

dict

edC

hang

e E

rro

rs a

LE

AR

10.8

%

5.4

%

3.7

%

5-4

-10

.3

-15

-6

-8-7

-6.9

-1.6 8-

5

7-3

1.8

-2.4 1.6

-4-3

-3-O

%

3-7°

/

6.0 o-3

0.6 °-5

-5°

-3-4

D

2.8

%

AR

H

4-3%

6-3

0 0 0

-5°

-3-6 2-

7%

Mea

n A

bsol

ute

Per

cent

. E

rro

rs a

ET

6.0%

5-7

1.1

5-6

5-6

2.9

1.2 4-o%

LE

3-8%

2.4 3°

2-9 i-7

1-9

1-9

2-5/

0

AR

2-5%

1.9

2.1 1.2

1.8

2.0

2.0 1-9

/o

AR

H

2-4%

1-9

2.0 1-4

1-5

2.1

2.2 1-9%

Roo

t M

ean

Squa

rePe

rcen

t. E

rror

s a

ET

6-9%

5-9

i-7

6.4

5-9

3-2 1.2 4-5%

LE

4-4%

3-2

3-7

3-1 1.9

2.4

2.2 3-o%

AR

2-7%

2.4

2.9 1.6

2.0

2-5

2-3 2.3%

AR

H

2-7%

2-5

2.9 1.8

1.8

2-5 2-5 2-4%

Shi

ftIn

dex

t>

51

%

61

12

6

20

1

15

0

13

8

11

5

C O y 0 0 w

Key

: E

T

=

expo

nent

ial

tren

d m

odel

, L

E

=

econ

omet

ric

mod

el w

ith

lagg

ed

endo

geno

us r

egre

ssor

, A

R

=

auto

regr

essi

ve m

odel

wit

ho

ut

hete

rosk

edas

tici

ty a

djus

tmen

t, A

RH

=

au

tore

gres

sive

mod

el w

ith

hete

rosk

edas

tici

ty

adju

stm

ent.

No

te:

The

eco

nom

etri

c fo

reca

sts

(col

umns

lab

elle

d L

E,

AR

, an

d A

RH

) ar

e ba

sed

on p

redi

cted

val

ues

of t

he

pri

vat

e se

ctor

wag

e ra

tepr

omul

gate

d b

y W

har

ton

Eco

nom

etri

c F

orec

asti

ng A

ssoc

iate

s, I

nc.

The

equ

atio

ns w

ere

esti

mat

ed u

sing

rev

ised

val

ues

of t

he

wag

era

te s

erie

s, w

hich

dif

fer

slig

htly

in

mos

t ca

ses

from

th

e pu

blis

hed

valu

es a

vail

able

at

the

tim

e th

e W

har

ton

fore

cast

s w

ere

mad

e. T

hus,

the

pred

icte

d va

lue

of t

he

wag

e ra

te,

wt,

for

use

in t

he

clai

m c

ost

fore

cast

s is

wt

= w

oc .

u>

twl^

ov>

whe

re w

oc

= t

he

curr

ent

(rev

ised

)va

lue

of w

for

the

peri

od p

rece

ding

th

e fi

rst

fore

cast

per

iod,

wop

=

th

e (p

reli

min

ary)

val

ue o

f w

for

the

peri

od p

rece

ding

th

e fi

rst

fore

-ca

st p

erio

d at

th

e ti

me

the

Wh

arto

n fo

reca

st w

as m

ade,

and

wt w

=

th

e W

har

ton

fore

cast

of

w f

or p

erio

d t.

a T

ota

l pr

edic

ted

chan

ge e

rror

(T

PC

E),

mea

n ab

solu

te p

erce

ntag

e er

ror

(MA

PE

), a

nd r

oo

t m

ean

squa

re p

erce

ntag

e er

ror

(RM

SP

E)

are

defi

ned

in e

quat

ions

(13

), (

14),

and

(15

) in

th

e te

xt.

b T

he s

hift

ind

ex i

s co

mpu

ted

as f

ollo

ws:

/

=

[(yg

/yo)

1 '8 — 1

] -H

[{y

ajy-

i 2)1/

12—

1].

whe

re y

8 =

act

ual

ave

rage

pai

d cl

aim

co

sts

inth

e la

st q

uar

ter

of t

he

fore

cast

per

iod,

y0

=

actu

al c

laim

cos

ts i

n th

e q

uar

ter

prec

edin

g th

e fo

reca

st p

erio

d, a

nd y

. 12

=

act

ual

cla

imco

sts

thir

teen

qu

arte

rs p

rio

r to

th

e fo

reca

st p

erio

d.c

Thi

s fo

reca

st p

erio

d ru

ns

from

19

77.3

th

rou

gh

1978

.2.

n > o o 3



https://doi.org/10.1017/S0515036100006681


TA

BL

E 4

: E

XPO

NE

NT

IAL

TR

EN

D A

ND

LIN

EA

R

OL

S P

RE

DIC

TIO

N

EQ

UA

TIO

NS

FOR

AU

TO

MO

BIL

E B

OD

ILY

IN

JUR

Y L

IAB

ILIT

Y

AV

ER

AG

E P

AID

CLA

IM C

OST

S

Ke

y:

yt

=

aver

age

pai

d cl

aim

co

sts

in q

ua

rte

r t;

yt

=

aver

age

pai

d cl

aim

co

sts

for

the

yea

r en

din

g in

qu

art

er

t;t=

ti

me,

in

qu

art

ers

;a

nd

wt

=

pri

va

te s

ecto

r w

ages

per

man

-ho

ur

in q

ua

rte

r t.

No

te:

Th

e es

tim

atio

n p

erio

d fo

r ea

ch t

ime

tre

nd

equ

atio

n is

th

e tw

elv

e q

ua

rte

rs p

rece

din

g th

e fi

rst

fore

cast

p

erio

d.

Fo

r ea

ch

lin

ear

eco

no

met

ric

equ

atio

n,

the

esti

mat

ion

per

iod

beg

ins

in

1964

.1 a

nd

end

s in

th

e q

ua

rte

r p

rece

din

g th

e fi

rst

fore

cast

per

iod

. T

he

nu

mb

ers

in p

aren

thes

es b

elo

w t

he

esti

mat

ed

coef

fici

ents

ar

e ^-

stat

isti

cs.

a N

um

be

rs

in

this

co

lum

n ar

e D

urb

in-W

atso

n d

stat

isti

cs.

Rej

ecti

on

of t

he

hy

po

thes

is

of

no

seri

al c

orr

elat

ion

(at

the

5 p

erce

nt

lev

el)

is i

nd

icat

ed

by

an a

ster

isk

. T

he

test

fo

r 19

73.3

is

inco

ncl

usi

ve.

b T

his

co

lum

n p

rese

nts

th

e G

OL

DF

EL

D-Q

UA

ND

T (1

965)

tes

t st

atis

tics

fo

r h

eter

osk

edas

tici

ty.

Th

e h

yp

oth

esis

of

h

om

osk

edas

tici

tyis

rej

ecte

d if

th

e st

atis

tic

val

ue

exce

eds

Fai

Vl, V

2.

Th

e d

egre

es o

f fr

eed

om

p

aram

eter

s, v

x an

d v 2

, ar

e g

iven

in

par

enth

eses

bel

ow

th

e st

a-ti

stic

s. A

t th

e 5

per

cen

t le

vel

th

e h

yp

oth

esis

is

reje

cted

fo

r ea

ch o

f th

e la

st f

ou

r eq

uat

ion

s (1

97

4.3

th

rou

gh

19

77

.3).

o to

Eq

ua

tio

n fo

rF

ore

cast

Beg

inn

ing

in

1971

-3

1972

-3

1973

-3

1974

-3

1975

-3

1976

-3

1977

-3

a

6-99

43(1

868.

45)

7-O

458

(844

-845

)

7-13

O3

(5O

5-19

3)

7.19

64(6

03.5

96)

7-19

37(1

210.9

5)

7-2

514

(112

5-23

)

7-33

°4(1

162.

57)

lny t

=

a

+ (3

/

P

0.0

14

7(2

8.93

8)

0.0

171

(15.

070)

0.0

116

(6-0

53)

0.0

09

4(5

-772

)

0.0

174

(21-

525)

0.0

200

(22.

812)

0.0

202

(23-

549)

R* .988

.958

.786

.769

•979

.981

.982

yt

a

- 14

7603

(-2

.92

9)

-85

-92

3(-

1.8

84

)

— 1

4.90

8(-

0.3

60

)

27.2

017

(0.7

12)

104.4

25

(2.9

07)

107.

879

(3-5

90)

112.

189

(4-3

47)

= i +

fo

34

0.0

42

(24-

356)

321.8

92

(26.

432)

30

1.2

99

(28.

304)

28

9.3

16

(30.

657)

26

8.3

13

(31-

695)

26

7.4

02

(39

-6i8

)

26

6.2

64

(48.

278)

i?2

•955

• 95

6

•957

•959

.958

.970

•978

DW

*

2.0

88

1.82

7

1.46

1

1-35

6*

1.2

08*

1-25

5*

1.3

09*

GQ

b

1-35

(10,

10)

1-93

(12,

12)

2.29

(H.1

4)

3-25

(16,1

6)

2-93

(18,

18)

3-37

(20,2

0)

2.67

(22,2

2)

d 0 r>rMMIN en t-1 ALW1 O <



https://doi.org/10.1017/S0515036100006681



TABLE 5 : AUTOREGRESSIVE EQUATIONS FOR AUTOMOBILE BODILY INJURY LIABILITYAVERAGE PAID CLAIM COSTS:

EQUATIONS IN LOWER PANEL ADJUSTED FOR HETEROSKEDASTICITY

Equation forForecast —Beginning in

1971-3

1972-3

1973-3

1974-3

1975-3

1976-3

1977-3

1971-3

1972-3

1973-3

1974-3

1975-3

19763

1977-3

a

— 110.946(-1-559)

N.E.

N.E.

127.914(1-556)

288.274(2-435)

235-516(3.021)

221.158(3-538)

- 142-4O5(-2.253)

N.E.

N.E.

106.858(1-252)

234.868

231-793(2.849)

220.898

= PW

344-294(17-381)

313-435(41-365)

313.612(39.736)

284.169(16.527)

252.929(12.912)

260.034(19.667)

263.160(25.981)

353-898(18.752)

317-833(35-971)

317-593(37-473)

290.600

(15-467)

264.464(11.848)

261.226(18.589)

262.653(29-542)

i ( i - * ) + &(«A

Yi

-41-1317(-3-O37)

— 40.3019

(-2-773)

-41.3901(-3-070)

-38.1477(—2.902)

-45.5498(-3-449)

-46.9951(-3.621)

-49.726(-4.069)

-38.1636(-3-239)

-42.1668(-3-578)

-42.2427(-3-789)

— 40.8001(-3-276)

-45-5516(-3-6655)

— 46.6890(-3-645)

— 51.6890(-4.190)

Y2

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

t) + Tx5l + Yr

Ys

-74.0772(-4-774)-70.9714(-4-473)

-70.3248(-5-O15)

-81.546(-5.908)

-90.9309(-6.458)

-83.5657(-6.206)

-84.430(-6.732)

— 79.2642(—6.201)

-84.5401(-7.021)

-83.5804(-7-436)

-82.9638(-6.396)

— 89.5063(-6.896)

-84.1028

(-6-343)

-83.8710(-6.609)

S2 + Y3S3

P

.440689(2.145)

•527511(3-129)

.586522(4.208)

.620412

(5-oi5)

•712105(6.303)

•654109

(5-897)

•638744(6.030)

.414891(2.464)

•56165(4-371)

•57197(4-8i5)

.63102(5-i69)

.71128(6.390)

.66127(6.016)

•61937(5.717)

i?2

•9763

.9711

.9767

.9807

.9822

.9862

.9901

.966

•969

•978

.872

.852

.980

•995

Key: yt = average paid claim costs in quarter t; wt = private sector wages per man-hour in quarter t; and S( = 1.0 in quarter i of each year, o otherwise.

Note: The estimation period for each equation begins in 1964.1 and ends in the quarterpreceding the first forecast period. The number in parentheses below each estimatedcoefficient is the ratio of the coefficient to its estimated standard deviation. N.E. = foundto be insignificant in earlier versions of the equations and thus omitted from the finalversion.



https://doi.org/10.1017/S0515036100006681


TA

BL

E 6

: R

ES

UL

TS

O

F E

IGH

T Q

UA

RT

ER

F

OR

EC

AS

TS

O

F A

UT

OM

OB

ILE

B

OD

ILY

IN

JUR

Y L

IAB

ILIT

Y A

VE

RA

GE

PA

ID C

LA

IM C

OS

TS

Fir

st P

erio

dof

For

ecas

t

1971

-3

1972

.3

1973

-3

1974

-3

1975

-3

1976

.319

77.3

c

Abs

olut

e A

vera

ges

Tot

al P

redi

cted

Cha

nge

Err

ors a

ET

10

.1%

-8.4

-4.6

-4.6

-5-o 2.4

3-9

5-6%

OL

S A

R

3-4%

5

-6%

0.1

0.4

3.2

6.1

5-3

6.3

0.7

-0

.4

2-5

2.8

i-5

2.6

-7

A 0

/ 0

=0

/

2-4

/o

3-5

/o

AR

H

7-o%

1.6

7.6

7-3

1-4

2.9

2-5

4-3%

ET

Mea

n A

bsol

ute

Per

cent

. E

rror

s

OL

S

2.6%

3-7

%

5-9

2.2 6.8

2.2

0.5

0.8

4.6

4-5

5-3

2.4

2.0 1.8

2 n

0/

1

c0/

3-%

3

-5/0

AR

4-3%

2.4 4-3

4.6 1-5

1.2 i-7

2.9

%

a

AR

H

5-2%

2.7

5-1

5-3

1.9

1.2 1.6

3-3%

Roo

t M

ean

Squ

are

Per

cent

. E

rror

s a

ET

2.8

%

6.0

2.6 7.2

2.4

0.7 1.1 3-3%

OL

S

4-5%

5-7

5-4

5-6

2.8

2.4

2.2 4-1

%

AR

5-o%

3-4

5°

5-i

2.0 1.6

2.0 3-4%

AR

H

6.0

%

3-6

5-7

5-6

2.2 1.6

2.0 3-8%

Shi

ftIn

dex

*>

63

%9

0

12

6

16

1

17

1 85 80

(—i d % d d g w d

Key

: E

T =

ex

pone

ntia

l tr

end

mod

el,

OL

S =

or

dina

ry l

east

squ

ares

mod

el,

AR

=

aut

oreg

ress

ive

mod

el w

itho

ut

hete

rosk

edas

tici

tyad

just

men

t,

AR

H

=

auto

regr

essi

ve m

odel

wit

h he

tero

sked

asti

city

ad

just

men

t.

No

te:

The

eco

nom

etri

c fo

reca

sts

(col

umns

lab

elle

d O

LS

, AR

, an

d A

RH

) ar

e ba

sed

on p

redi

cted

val

ues

of t

he

priv

ate

sect

or w

age

rate

prom

ulga

ted

by

Wh

arto

n E

cono

met

ric

For

ecas

ting

Ass

ocia

tes,

Inc

. T

he e

quat

ions

wer

e es

tim

ated

usi

ng r

evis

ed v

alue

s of

the

wag

era

te s

erie

s, w

hich

dif

fer

slig

htly

in

mos

t ca

ses

from

th

e pu

blis

hed

valu

es a

vail

able

at t

he

tim

e th

e W

har

ton

fore

cast

s w

ere

mad

e. T

hus,

the

pred

icte

d va

lue

of

the

wag

e ra

te w

t, fo

r us

e in

the

clai

m c

ost

fore

cast

s is

wt

= w

oc •

wtw

lwop

> w

here

woc =

th

e cu

rren

t (r

evis

ed)

valu

e of

w f

or t

he

peri

od p

rece

ding

th

e fi

rst

fore

cast

per

iod,

wop =

th

e (p

reli

min

ary)

val

ue o

f w

for

the

peri

od p

rece

ding

th

e fi

rst

fore

cast

per

iod

at

the

tim

e th

e W

har

ton

fore

cast

was

mad

e, a

nd w

t w =

th

e W

har

ton

fore

cast

of

w fo

r pe

riod

t.

a T

otal

pre

dict

ed c

hang

e er

ror

(TP

CE

), m

ean

abso

lute

per

cent

age

erro

r (M

AP

E),

and

roo

t m

ean

squa

re p

erce

ntag

e er

ror

(RM

SP

E)

are

defi

ned

in e

quat

ions

(13

), (

14),

and

(15

) in

the

tex

t.b

The

shi

ft i

ndex

is

com

pute

d as

foll

ows:

I

=

[(y s

/y0)

1/s

— 1

] -r-

\_{

yjy

-n)lin

—

1].

whe

re y

s =

act

ual

aver

age

paid

cla

im c

osts

in

the

last

qu

arte

r of

the

fore

cast

per

iod,

y0 =

ac

tual

cla

im c

osts

in

the

qu

arte

r pr

eced

ing

the

fore

cast

per

iod,

and

;y_

12 =

act

ual

clai

mco

sts

thir

teen

qua

rter

s pr

ior

to t

he

fore

cast

per

iod.

c T

his

fore

cast

per

iod

runs

fro

m

1977

.3 t

hrou

gh 1

978.

2.

O



https://doi.org/10.1017/S0515036100006681



with adjustment for autocorrelation and for both autocorrelation and hetero-skedasticity, using Method 2 15. These results are shown in Table 5.

The forecasting errors for the four sets of bodily injury equations are pre-sented in Table 6. Using the TPCE as the measure of accuracy, the OLSequations give the best results in five of the seven forecast periods and theexponential trend model is superior in two. In one of these periods, however,the trend line underpredicts while the econometric models overpredict. In theother, the trend line is better than the OLS equation by only o.l percentagepoint. When MAPE and RMSPE are used to gauge forecasting accuracy, theexponential trend equation is superior to the econometric equations in four ofthe seven periods. However, two of the periods when at least one of the econ-ometric equations is better are those with the highest unanticipated inflation.

6. SUMMARY AND CONCLUSIONS

The results of this study indicate that econometric models can be used to fore-cast automobile insurance paid claim costs more accurately than the exponentialtrend models now in use in the United States. The superiority of the econometricmodels is especially evident during periods of high unanticipated inflation.Although adjustments for seasonality, autocorrelation, and heteroskedasticityoften improve forecasting accuracy, even the OLS models perform quite well.Adoption of the econometric approach may enable insurance companies toeliminate some of the uncertainty associated with claims cost inflation.

REFERENCES

COOK, C. (1970). Trend and Loss Development Factors, Proceedings of the CasualtyActuarial Society, 52, 1-14.

DURBIN, J. (May 1970). Testing for Serial Correlation in Least-Squares Regression WhenSome of the Regressors Are Lagged Dependent Variables, Econometrica, 38, no. 3,410-421.

GLEJSER, H. (1969). A New Test for Heteroskedasticity, Journal of the American Statis-tical Association, 64, 316-323.

GOLDFELD, S. M. and R. E. QUANDT. (1965). Some Tests for Homoskedasticity, Journalof the American Statistical Association, 60, 539-547.

HALL, B., HALL, R., and S. BECKETTI. (1977). Time Series Processor, Version 3.5, Pri-vately published. Stanford, Calif.

HARRISON, M. J. and B. P. M. MCCABE. (1979). A Test for Heteroskedasticity Based onOrdinary Least Squares Residuals, Journal of the American Statistical Association,74, 494-499-

HARVEY, A. C. and G. D. A. PHILLIPS. (1974). A Comparison of the Power of Some Tests forHeteroskedasticity in the General Linear Model, Journal of Econometrics, 2, 307-316.

KLEIN, L. R. and E. BURMEISTER. (1976). Econometric Model Performance: Comparative Si-mulation Studies of the U.S. Economy, University of Pennsylvania Press, Philadelphia.

KMENTA, J. (1971). Elements of Econometrics, The Macmillan Co., New York.SPENCER, B. G. (1975). The Small Sample Bias of Durbin's Test for Serial Correlation,

Journal of Econometrics, 3, 249-254.WAELBROECK, J., ed. (1976). The Models of Project LINK, North-Holland Publishing Co.,

Amsterdam.15 For the first three sample periods, P{xt) = ~X\x\\ while for the last four periods

H*t) = T0+7V4



https://doi.org/10.1017/S0515036100006681


AP

PE

ND

IX:

PRIN

CIP

AL

DA

TA S

ER

IES

Qua

rter

1954-1

1954-2

1954

-319

54-4

1955-1

1955-2

1955

-319

55-4

1956.1

1956.2

1956

-319

56.4

1957-1

1957-2

1957

-319

57-4

19

58

11958.2

1958

-319

58.4

1959-1

1959-2

1959

-319

59-4

1960.1

1960.2

1960.3

1960.4

1961.1

1961.2

1961.3

1961.4

1962.1

1962.2

PD

1.0

0.9

90

30.9

709

1.0

097

1.0

097

1.0

291

1.0

194

1.0

680

1.0

680

1.0

777

1.0

777

1.1

456

1-1

553

1.1

845

1.1

845

1.2

427

1.2

330

1.2

427

1.2

427

1.2

718

1.2

524

1.2

913

1.2

816

1-33

981.

3204

1.3

204

1-33

O1

1.36

891.3

010

1-34

951-3

592

1.4

078

1.36

891.

3981

WR

1-95

1.96

1.98

2.0

2.0

12.0

3

2.0

62.0

8

2.1

2

2-1

52.1

92.2

22.2

52.2

8

2.3

1

2-33

2-35

2-37

2.4

0

2.4

22.4

42.4

62.4

8

2-5

1

2-55

2.5

8

2-5

92.6

22.6

3

2.6

5

2.67

2.6

9

2-73

2-75

Qua

rter

1962.3

1962.4

1963.1

1963.2

1963

319

634

1964.1

1964.2

1964

-31964.4

1965.1

1965.2

1965

319

65.4

1966.1

1966.2

1966

.319

66.4

1967.1

1967.2

1967

319

67.4

1968.1

1968.2

1968

.319

68.4

1969.1

1969.2

1969

319

69.4

1970.1

1970.2

1970.3

1970.4

PD

1.4

078

1.44

661.4

272

1.4

660

1-47

571-5

146

1-48

541-

5437

1-55

341.6

214

1.6

117

1.67

961.6

990

1.7

961

1.78

641.

8447

1.84

471-9

515

1.9

417

1.9

903

2.0

291

2.1

068

2.1

262

2.2

136

2.2

330

2.3

592

2-3

592

2.4

466

2.4

951

2.6

117

2.5

631

2.67

962-

7379

2.8

350

BI

1.0

1-0

341

0.9

330

1.0

364

1.0

4

1.0

811

1.0

129

1.0

999

1.0

764

1.1

340

1.1

011

1-1

915

1.2

045

1.2

468

1-2

233

1.2

667

1.2

785

1.3

619

1.2

808

1.3

690

1359

61.4

101

1.4

148

1.4

160

1.43

481.5

018

1.4

606

1.5

018

WR

2.7

6

2.7

92.8

22.8

4

2.8

62.9

0

2.94

72.9

72

3.0

10

3.0

26

3.0

40

3.06

73.1

06

3-1

41

3-18

53-2

41

3.2

92

3-33

13-

384

3-43

33-

472

3-51

53.6

09

3.66

93.7

21

3.78

93-8

42

3-9O

43

.96

64-

°35

4.0

98

4.16

84-

253

4.30

6

Qu

arte

r

1971-1

19

71

2

1971-3

1971.4

1972.1

1972.2

1972-3

1972.4

1973-1

1973-2

1973

-319

73-4

1974-1

1974-2

1974

-319

74-4

1975-1

1975-2

1975

-319

75-4

1976.1

1976.2

1976

-319

76.4

1977-1

1977-2

1977

-319

77-4

1978.1

1978.2

PD

2.78

642.9

029

2-9

515

3.0

291

2.9

417

2.9

320

3-o

i94

3-1

165

3.08

743-

i845

3-18

453-

3495

3-23

3°3.2

621

3.40

783.6

117

3.67

963-

7476

3.80

58

;3-

9417

3.98

064.

0874

4.18

45

:4-

4175

4.40

78

:4-5

922

4-75

734.9

612

4-8

83

55-

1456

BI

•527

6•7

O74

1-6

273

1.6

592

L.6

228

•578i

•585

2.6

804

1.7

004

1-74

741-

6439

1-77

O9

1.86

961-9

389

1.8

284

1.9

130

1.88

481-

9965

2.03

64

2-16

332.

1293

2.17

742.

1598

2.29

962.

2985

2-39

252.

3678

2.4

618

2.3

960

2.4

982

WR

4-39

74.

468

4-54

O4-

57O

4.6

79

4-73

74-

797

4.85

85.0

06

5.0

90

5-17

95.2

72

5-38

45-

538

5-67

55-

837

5-99

96.

097

6.1

91

6.30

56-

439

6.57

86.

707

6.8

42

7.0

07

7.1

18

7.2

40

7-35

57613

7-75

8

O ON

'—,

> <; 5 0 a a g s; w > 0 > t-1 < z 0 W f

Key

: BI

and P

D =

aut

omob

ile

bodi

ly i

njur

y an

d p

rop

erty

dam

age

liab

ilit

y pa

id c

laim

cos

ts,

resp

ecti

vely

. The

se s

erie

s ar

e ex

pres

sed

as i

ndic

es w

ith

the

earl

iest

per

iod

= 1.0

. WR

= t

he

pri

vat

e se

ctor

wag

e ra

te.



https://doi.org/10.1017/S0515036100006681


j. david cummin and alwys n powell€¦ · j. david cummin and alwys n powell 1. introduction when...

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