Forecasting traffic accidents using disaggregated data

Forecasting traffic accidents using disaggregated data

A. García-Ferrer∗

Dept. de Análisis Económico: Economía Cuantitativa.

Av. Tomás y Valiente, 5. Facultad de Económicas Universidad Autónoma de Madrid.

Campus de Cantoblanco. Carretera de Colmenar km. 15. 28049 Madrid, Spain.

e-mail: [email protected]. Tel. nr. 34-91-4974811. Fax nr. 34-91-4974091

A. de Juan

Dept. de Análisis Económico: Economía Cuantitativa.

Av. Tomás y Valiente, 5. Facultad de Económicas Universidad Autónoma de Madrid.

Campus de Cantoblanco. Carretera de Colmenar km. 15. 28049 Madrid, Spain.

e-mail: [email protected]. Tel. nr. 34-91-4974100. Fax nr. 34-91-4974091

P. Poncela

Depto. de Análisis Económico: Economía Cuantitativa.

Av. Tomás y Valiente, 5. Facultad de Económicas Universidad Autónoma de Madrid.

Campus de Cantoblanco. Carretera de Colmenar km. 15. 28049 Madrid, Spain.

e-mail: [email protected]. Tel. nr. 34-91-4975521. Fax nr. 34-91-4974091

November 10, 2005

∗Corresponding author

1

Forecasting traffic accidents using disaggregated data

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Abstract

Traffic accidents, measured monthly, present different characteristics when the aggregate is com-

pared to its individual components. When disaggregated data are used, the effects of policy variables,

calendar events, and different seasonal behavior should be clearly understood and their coefficients

properly estimated. In this paper, we compare the empirical performance of various models in as-

sessing the effects of policy variables, legal changes, and traffic security campaigns. In addition,

aggregated versus disaggregated forecasts of the main accident variables are compared in order to

examine the robustness of forecasting improvement when using disaggregated data. In particu-

lar, we test the robustness of this improvement against the specification of the model, information

set, type of measure of forecasting accuracy, and forecast year. Overall, we conclude that forecast

combinations based on disaggregated models display better performance.

Keywords: accuracy criteria, disaggregation, forecast combination, time series, traffic acci-

dents.

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1 Introduction

The effects of disaggregation on forecasting the aggregate have been studied extensively over the past

two or three decades. From a theoretical standpoint, Tiao and Guttman (1980) provided guidelines for

conditions, with no guarantee of gain in forecast efficiency (measured by the minimum mean squared

error), when employing the component series rather than using the aggregate. Wei and Abraham (1981)

compared forecast efficiency based on the aggregate series, univariate component series and joint mul-

tiple time series. Kohn (1982) defined the aggregate as any linear combination of the observed series

and analyzed the equality of forecasts between the aggregated and disaggregated approaches. Later,

Lütkepohl (1984, 1985) established the equality of forecasts when sets of linear combinations of the

observed time series are considered. More recently, Clark (2000) examined the problem of forecasting

an aggregate composed of cointegrated disaggregates. In addition, this author established conditions

under which forecasts of an aggregate variable obtained from a disaggregated Vector Error Correction

Model are equal to those obtained from an aggregated univariate time series model. In the case of

common trends, Poncela and Garcia-Ferrer (2005) derived conditions that guarantee the equality of

forecasts between the aggregated and disaggregated approaches using unobserved component models.

The empirical counterpart to this issue has shown mixed results (being case dependent). Nevertheless,

recent results regarding Gross Domestic Product growth rates (Zellner and Tobias, 2000) and several

European macroeconomic variables (Marcellino et al., 2003) indicated that, generally, ”it pays to dis-

aggregate”. Most studies on short-term predictions show that considerable gains in efficiency, based on

mean-square-error-type criteria, can be obtained when using models based on disaggregated data. How-

ever, as the prediction horizon increases, the gain in efficiency from using disaggregated data diminishes

substantially (Koreisha and Fang, 2004).

The existence of many empirical aggregate relationships tends to "obscure" some of the under-

lying characteristics of the individual series when data on traffic accidents are analyzed. For instance, in

the case of monthly seasonal data, a different behavior between the aggregate variable and its individ-

ual components is often observed. Given this situation, the estimation of both the aggregate seasonal

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pattern and the calendar effects may be highly biased if these elements differ substantially among the

individual components. In some cases, aggregated calendar effects seem to be negligible as a result of

offsetting individual effects of different signs (Garcia-Ferrer and del Hoyo, 1987). These unexpected

results may not only provide misleading conclusions for policy, but also, considerable deterioration in

the forecasting performance of alternative models.

Spain is one of the European countries with the highest rate of road accidents (Page, 2001).

Roughly speaking, Spain had the same number of motor fatalities in 2003 that it had thirty years ago.

This correspondence occurred in spite of huge infrastructure investments, considerable technological

improvement, and generous spending on numerous traffic security campaigns. Traffic authorities in

other European countries regularly set targets that are aimed at reducing road casualties. However, in

Spain, most of these efforts thus far have been in vain. This situation is especially apparent when we

compare the circumstances in Spain to those observed in other European countries. For instance, in

1987 the targets in Great Britain were aimed at reducing the number of fatalities and serious injuries

by one-third, compared with the average for 1981 to 1985, by 2000. This target was surpassed; road

fatalities fell by 39% and serious injuries by 45%. The success in Great Britain was made possible

through legislation changes, improved infrastructure, and vehicle crash protection (Raeside, 2004).

In this paper, we compare the forecasting performance of a large number of econometric models in

order to address the issue of setting realistic targets. Specifically, we apply these models to monthly

Spanish traffic accident variables during the years 1975 to 2003. This paper has two main objectives.

The first objective is to evaluate and compare the empirical performance of various models in assessing

the effects of policy variables, legal changes and traffic security campaigns. When dealing with this

issue, disaggregation is crucial in order to avoid misleading conclusions for policy decisions. The second

objective is to compare both individual and various combination forecasts of the main accident variables.

We also evaluate whether or not the forecasting improvement when using disaggregated data is robust

to the specification of the model, information set in each model, type of measure of forecasting accuracy,

and forecast year. In Section 2 we discuss the definitions and characteristics of the data used in the

study. The methodologies used to analyze the aggregate relationships between traffic accidents and

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real economic activities are briefly described in Section 3. In addition, the empirical results of this

study are presented in Section 3. Beginning with an analysis of the data, we estimate several univariate

and intervention models in order to provide starting points for the construction of causal econometric

models. In Section 4, we analyze the predictive performance of models that focus their attention on

disaggregated data and assess the value of disaggregation in terms of forecasting. A conclusion of the

study is provided in Section 5.

2 Description of the data

The empirical implementation of alternative models requires information on both accident rates and

economic variables. As a result, the data used in this paper have been separated into three groups:

1. The following variables related to accident rates [monthly data from January 1975 to December

2003 (348 observations)] were measured: (i) number of accidents with injured passengers-ACC ; (ii)

number of killed passengers-FAT1 ; and (iii) number of injured passengers-INJ. The data from each

series have also been disaggregated into urban and road series. To differentiate between the urban and

road series, we have added a U or R to the acronyms used for the aggregated series. For instance, urban

accidents will be denoted as ACCU. Occasionally, public attention is focused on road accident rates and

less importance is placed on urban accident rates. Since, ACCU and INJU represented 52% and 46%

of ACC and INJ, respectively, in Spain in 2003, this view is misleading. On the other hand, in the case

of total fatalities, FATR included 85% of FAT.

2. The following variables representing the stock of vehicles and its variation [monthly data from

January 1975 to December 2003 (348 observations)] were measured: (i) the stock or number of vehicles

and (ii) new registration of vehicles-NUVE. Important traffic variables, such as automobile and road

conditions and the total number of miles on the vehicle, were not available in the monthly database.

1Although after 1996, Spain has accepted the definition of the Vienna Convention and considers a crash-related fatality

as an individual who dies at the scene of the crash or within 30 days following the crash, we will still use the former

definition that only includes deaths within the first 24 hours after the accident. The series using the 30-day time frame

are too short and present some inconsistencies due to the way the data are collected.

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3. The following variables representing the evolution of economic activity were measured: (i) index

of industrial production (IPI) from January 1975 to December 2003; (ii) gasoline consumption in million

of liters (GAS) from January 1975 to December 2003; and (iii) diesel consumption in million of liters

(DIESEL) from January 1982 to December 2003 (264 observations).

* Insert Figure 1 around here

Plots of the main variables are shown in Figure 1. All variables show a growing trend characteristic

of nonstationary behavior, as well as strong and possibly varying seasonality. Seasonality is further

complicated by the presence of ”moving” holidays that differ from year to year. The evaluation of these

calendar effects requires further comment. Dependent on the series and its level of aggregation, two

important intervention variables [i.e. the alternation of the Easter (EAST) holidays between March

and April and the change in the number of working days (NWD)] will lead to results of magnitudes of

different signs. In particular, both the seasonal factors and the calendar effects may work in opposite

directions when distinguishing between urban and non-urban accidents. While all variables related to

road accidents tend to increase during Easter and over the weekends, the opposite tendency exists with

urban traffic. In Figure 2, we have plotted total monthly ACC and its individual components, ACCR

and ACCU. Specifically, for each of the three series we have plotted 12 annual series (e.g. one series for

January, one series for February, etc). As Figure 2 shows, the weights of the seasonal factors clearly differ

between urban and non-urban accidents. Therefore, in order to verify whether or not the explanations

for the different types of accidents have any specific common variables, it is important to make a further

urban/non-urban disaggregation.

*Insert Figure 2 around here

3 Methodologies and empirical results

In this paper, we will use two different methodologies and three levels of information sets. When our

information set is exclusively restricted to the past history of the forecast variable, the dynamic harmonic

6

regression (DHR) model will be used. As we enlarge the information set (including exogenous economic

variables), variants of the intervention ARIMA model and dynamic transfer functions will be considered.

The theoretical frameworks are briefly sketched in the next subsections and the empirical results are

then presented.

3.1 The dynamic harmonic regression (DHR) model

The DHR model, developed by Young et al. (1999), belongs to the Unobserved Component type and is

formulated within the state space framework. The DHR model is based on a spectral approach under

the hypothesis that the observed time series can be decomposed into several DHR components whose

variances are concentrated around certain frequencies. In the univariate case, the DHR model can be

written as a special case of the univariate unobserved component model. This DHR model has the

general form:

yt = Tt + St + et; t = 0, 1, 2, . . . , (1)

where yt is the observed time series, Tt is the trend or low-frequency component, St is the seasonal

component, and et is an irregular component normally distributed with zero mean and variance σ2e.

Model (1) is appropriate for dealing with data that exhibits a pronounced trend and seasonality,

such as is the monthly variables used in this paper. The low frequency component Tt and its derivative

Dt can be described by the following second order generalized random walk model, Tt

Dt

=

α β

0 γ

Tt−1

Dt−1

+ η1t

η2t

(2)

where ηt = [ η1t η2t]0 is a white noise vector with zero mean and covariance matrix Q. For simplicity

it is assumed that Q is a diagonal matrix diag (q11, q22), with unknown elements q11 and q22. The

generalized random walk model subsumes as special cases (Young, 1994): the very well known random

walk (α = 1, β = γ = 0, q22 = 0); the smooth-random walk (0 < α < 1, β = γ = 1, q11 = 0) and the

integrated random-walk (α = β = γ = 1, q11 = 0). Using our current data, the Linear DHR (LDHR)

algorithm (Bujosa et al, 2005) identifies an integrated random walk model for all trends. In this case,

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Tt = Tt−1 +Dt−1

Dt = Dt−1 + η2t

(3)

where Tt and Dt can be interpreted as the level and slope (derivative) of a time variable trend. The

variance of η2t (q22) is the only unknown in (3) and can be estimated through the noise variance ratio

(NVR), the ratio between q22 and the variance of the irregular component σ2e

NVRT =q22σ2e. (4)

In order to estimate the trend, the series is smoothed with the Kalman filter and the NVRT is permitted

to work as a smoothing parameter. Very low NVRT values are indicative of near deterministic linear

trends. Near the limit, when NVRT = 0, the estimated trend is linear. On the other hand, for large

NVRT values, the estimated trend mimics the original time series yt. Additionally, we assume that the

seasonal component in (1) can be represented by

St =NsXj=1

ajt cos(ωjt) + bjt sin(ωjt) (5)

where the regression coefficients ajt, bjt, j = 1 . . . Ns are time variable to handle non-stationary season-

ality. As seen in the previous trend model, time variation in ajt and bjt may follow any variant within

the generalized random walk framework. The LDHR algorithm (briefly sketched in the Appendix),

identifies that parameter variation in aj and bj is modelled as a random walk process, where ξ1t and

ξ2t are again i.i.d with zero expected values and variances σ2ξ1and σ2ξ2 respectively. ajt

bjt

= ajt−1

bjt−1

+ ξ1t

ξ2t

(6)

In the estimation of the DHR models, the trend’s NVR is updated during estimation, and the NVR

values associated with the main seasonal frequency (and its harmonics) are also estimated recursively.

The autoregressive spectral estimates have peaks at 12, 6, 4, 3, 2.4 and 2 (6th harmonic) months. As a

result, the annual frequencies used in the subsequent analysis will be 0, 112 ,

16 ,

14 ,

13 ,

512 and

12 . The NVR

estimates for all accident rates, using the 1975.1-2000.12 sample, are shown in Table 1.

8

* Insert Table 1 around here

Although there are some differences in the trend NVR estimates, all series exhibit smooth trends.

In general, the three fatality series show the smoothest trends. However, the numerical differences with

the other series do not indicate a different trend behavior in the long run. Similar comments apply

to the remaining frequencies where, in general, the main seasonal frequency (H:12) shows the highest

values. However, note the atypical behavior of urban fatalities (FATU). Urban fatalities show estimates

of zero in three seasonal frequencies (12, 6 and 3), which is indicative of low power at these frequencies.

As shown later, this series is particularly difficult to forecast given its anomalies.

3.2 Causal econometric models

In order to asses both the effects of interventions and economic variables, our general econometric causal

model can be written as:

φ(L)Φ(Ls)∇d∇Dyt =kXi=1

vi(L)∇d∇Dxit +mXj=1

∂j(L)∇d∇Dzjt + θ(L)Θ(Ls)at (7)

where xit are intervention variables, vi(L) includes the dynamic model for the i-th intervention variable,

θ(L) and Θ(Ls) are the regular and seasonal moving average operators, and φ(L) and Φ(Ls) are the

regular and seasonal autoregressive operators. The process at is assumed to be white noise, and ∇D and

∇d allow for seasonal and non-seasonal differencing. The usual stationarity and invertibility conditions

apply to the autoregressive and moving average operators. As indicated by Peña (2001), most outlier

specifications in the literature (additive, innovational, level shifts and transitory changes) are embedded

in equation (7) under proper parametrization of the polynomial operator vi(L). On the other hand, zjt is

a vector of exogenous economic variables, and ∂j(L) represents the dynamic model for the j-th exogenous

variable. Since most economic variables are affected by the same intervention variables as those affecting

the accident outputs (particularly EAST and NWD), we first estimate intervention ARIMA models and

later present the empirical results of the dynamic transfer functions.

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3.2.1 Testing for regular and seasonal stationarity

At times, time series analysis and econometrics have followed different formal procedures when dealing

with regular and seasonal unit roots. While the traditional Box-Jenkins approach relies on results from

estimated auto- and partial correlation functions to decide on orders of differencing, many econometri-

cians prefer to base such a decision on formal statistical tests. In order to avoid further discrepancies

concerning the ”correct” order of integration, both alternatives have been used in this paper. When

estimated auto- and partial correlation functions are used, both regular and seasonal differences are

identified for accident and economic variables. This finding is also confirmed through the unit root test

procedure (OCBS) proposed by Osborn et al. (1988). The results of this test with the main aggregate

traffic and economic variables are shown in Table 22. The critical values in Table 2 have been taken

from Rodrigues and Osborn (1997). Perhaps due to the erratic behavior of its urban component, the

OCBS test rejects the null of a seasonal unit root only for fatalities and for the DIESEL variable. At

this stage, the econometric literature on differencing and forecasting accuracy can help when deciding

the appropriate orders of differencing. For instance, Franses (1991) and Kawasaki and Franses (2004),

found empirical evidence indicating that models with a small number of seasonal roots provide more

accurate one-step-ahead forecasts. On the other hand, Clements and Hendry (1997) found that the use

of annual seasonal differences could improve forecast accuracy even when these differences are not sup-

ported by the seasonal unit root test provided by Hylleberg et al. (1990). Sánchez and Peña (2001) have

confirmed this later result. These authors have shown that, for forecasting purposes, overdifferencing is

a better alternative due to the greater parsimony of overdifferenced models. Taking this finding and the

forecasting objective of this paper into account, we decided to use both regular and seasonal differences

in the specification of intervention and transfer function models in the subsequent sections.

*Insert Table 2 around here2The results with the disaggregated variables are available from the authors upon request.

10

3.2.2 Assessing the effects of the intervention variables

Detailed definitions of the main intervention variables are included in Table 3. A summary of the

estimation results of the IARIMA models (hereafter INT), using the identification approach of Chen

and Liu (1993) and implemented in the SCA software, is shown in Table 43. The following comments

are worth mentioning:

* Insert Table 3 around here

1. As anticipated earlier, the effects of the Easter holiday on road and urban accident rates were

positive and negative respectively. This differing behavior may render insignificant coefficients in some

of the aggregates, as evidenced with the ACC estimate. In addition, the Easter effect is negative in the

case of IPI, but the effect is positive in the remaining economic variables. Results obtained with traffic

variables indicate that the Easter effect is always larger (in absolute value) for road, rather than urban

variables. Results found using road variables range from 6.2% (ACCR) to 9.1% (INJR), while findings

using urban variables range from -2.1% (INJU) to -3.35% (ACCU).

2. The effect of the number of working days (NWD) is negative and significant for the aggregate

and road variables, and positive (but insignificant) for urban variables. Once again, road variables show

the largest magnitudes, ranging from -1.25% (ACCR) to -1.74% (INJR), and contain small numerical

differences amongst themselves. Once more, empirical results obtained with this intervention variable

are in agreement with our previous expectations. It should be noted that the NWD is a negative proxy

for weekends, as all road accident rates tend to increase over the weekend. Similarly, the opposite

influence that weekends have on the economic variables was equally expected on a priori basis.

3. Given that their effects are important in a large number of traffic variables, the remaining

intervention variables have been included in Table 4. The largest magnitude corresponds to ALF,

therefore affecting both FAT and FATR. This intervention (additive outlier, AO, type) identified the

increase (around 35%) in fatalities observed when the Alfaques camping accident took place in July 1978.

3The estimation period for all variables except DIESEL is 1975.1-2000.12. Complete estimation results are available

from the authors upon request.

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An increase in accident rates due to extreme rainfall during these particular periods was identified by

the other AO intervention variables of JAN84 and NOV93. Finally, the intervention variable JUN92

corresponds to a level shift in all traffic variables as a consequence of the June 1992 Traffic Security

Plan. This plan established the use of mandatory seatbelts on car passengers and implemented other

legal norms. The permanent effect of this plan was a considerable reduction in accident rates, especially

in FATR (-18.7%). However, this effect was lower than the effect found by Harvey and Durbin (1986)

in the UK4.

* Insert Table 4 around here

3.2.3 Dynamic transfer function models

Both the explanatory ability and the forecasting performance of the univariate models may be improved

through the use of additional information from other related variables. The identification and estimation

of dynamic relationships between our output variables and the corresponding economic inputs, measured

by general economic activity and the degree of car utilization, are necessary for this improvement. This

process involves the identification of the components of the zjt vector and the corresponding ∂j(L) lag

structure in equation (7). After prewhitening, the cross-correlation functions between the univariate

model’s residual series are used to identify alternative transfer function models for each output in the

single input/single output case. This information is later used in the single output/multiple inputs

equations. To avoid the usual problem of orthogonality among inputs in these equations, the character-

istics of the input variables are taken into account (e.g., Liu and Hanssens, 1982). This identification

approach is preferable to the alternatives, suggested by the usual Akaike’s or Schwartz’s information

criteria, where longer lags are common and models are highly parametrized.

* Insert Table 5 around here4 Interestingly, empirical results on this variable are rather robust to alternative dynamic model specifications and

always range from -15% to -18% of a permanent effect.

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A summary of alternative specifications is included in Table 5. The empirical results for the main

aggregates are shown in Tables 6a, 6b and 6c5. As in the case of the intervention models in Table 4, all

variables are in logs and are affected by the usual ∇∇12 transformation. In addition to some variables

that correspond to important outliers in the series (always AO type), all causal models include the

corresponding intervention variables (when needed) used in Table 4. The empirical results of the main

intervention variables change very little in comparison with those shown in Table 4. In regards to the

effects of the exogenous variables, the following comments are worth mentioning:

1. All estimated coefficients show the expected signs and the dynamic responses in most cases are

either contemporaneous (lag 0) or short leading (1 and 2 lags). Interestingly, the total effects (dynamic

gains) of some inputs change little among alternative specifications. For instance, the dynamic gain of

IPI in the three equations for ACC (models CM1, CM4 and CM5) varies from .119 to .155. Also, the

dynamic gain in the same models for FAT varies from .295 to .337. Similar comments apply to NUVE

and GAS in the ACC and INJ equations.

2. Although most estimated models do not show strong evidence of misspecification, the Ljung-

Box statistics, (LBQ(24)), in some models, indicate the presence of residual autocorrelation at higher

lags, possibly as a consequence of undetected outliers. This situation mainly affects model CM5 for

the INJ and ACC variables. Additionally, the gain in terms of statistical fitting (with respect to the

INT models) measured by residual variance reduction is marginal. This marginal gain indicates only a

modest improvement (between 3-5%) in the explanatory power of the transfer function models.

* Insert Tables 6a, 6b and 6c around here

3. The previous results are not at all surprising if we take into consideration the frequency content

of the dynamic response of our input variables. This can be shown by taking into account the square

of the spectral gains G2i (ω) (0 < ω < π) for IPI and NUVE in some of the estimated models in Table

6. When using monthly data, π represents two months’ effects and π2 and

π6 four months and one year,

5The remaining results of the causal models for urban and road accidents rates are available from the authors upon

request. In order to avoid severe multicollinearity effects, please note that we have not included the DIESEL input in

model CM5 due to its high correlation with IPI (rDI = .81).

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respectively. As Figure 3 shows, in both cases, G2i (ω) reaches its maximum at ω = 0, indicative of a

long-run effect on accident rates. On the other hand, their minimum values are reached around ω = π,

implying that, in the short-run, their effects are very small. These results are in agreement with those

found by Garcia-Ferrer et al. (2004) regarding the existence of common long-run cycles between accident

rates and economic variables. In addition, these authors found other idiosyncratic short cycles, different

from those found in the economic indicators, in some accident rates.

* Insert Figure 3 around here

4. We have found a varying lead-lag relationship, depending on the cycle and the particular es-

timation sample, when using different sampling intervals for estimation. This is a clear indication of

potential instability on the estimated coefficients. This finding might also have important effects on

the forecasting ability of causal econometric models when compared to default univariate alternatives.

The potential instability of the estimated coefficients may be due to different causes that range from

technological advances, which have improved the quality and security of vehicles, to improvements in

road infrastructure. The sample used in this paper is large, so changes in the environment are likely to

be important.

The likelihood ratio test, proposed in Tsay and Wu (2003), was used to verify the constant-coefficient

transfer function model. The test is based on the following generalization of the traditional constant-

coefficient transfer function model (equation 7) into a functional coefficient framework

Yt = c(St) + υ(St;L)Zjt +kXi=1

δi(L)Xit +Nt

where Yt = ∇d∇Dyt;Zjt = ∇d∇Dzjt;Xit = ∇d∇Dxit;Nt = θ(L)Θ(Ls)φ(L)Φ(Ls)at; St denotes the state vector

at time t, which may be the status of an economy or an environment. In our case, we have used a

simple time index St = t/(T/2), where T is the total number of observations. This variable serves

as a proxy for technology development, growth in the Spanish population and the increase in road

kilometers. The function c(.) is a smooth function of St and υ(St, L) =∞Xj=0

υj(St−j)Lj where υj(St−j)

is a smooth function of St−j . Hence, we use the following first-order Taylor approximation for the

functional coefficients: c(St) = c0 + c1St and υj(St−j) = υ0,j + υ1,jSt−j . Thus, the functional transfer

14

function model becomes

Yt = c0 + c1St +∞Xl=0

υ0,lZj,t−l +∞Xl=0

υ1,lSt−lZj,t−l +kXi=1

δi(L)Xit +Nt

Let Z∗jt = St × Zjt; the functional coefficient model can be rewritten as

Yt = c0 + c1St +∞Xl=0

υ0,lZj,t−l +∞Xl=0

υ1,lZ∗j,t−l +

kXi=1

δi(L)Xit +Nt

Using a convenient re-parametrization, the model becomes:

Yt = c0 + c1St +ω0(L)

δ0(L)Zj,t +

ω1(L)

δ1(L)Z∗j,t +

ω2(L)

δ2(L)Xi,t +Nt (8)

where c0 and c1 are constants, ωs(L) =nXi=0

ωs,iLi and δs(L) = 1−

mXl=0

δs,jLj , s = 0, 1, 2. It is assumed

that ωs(L) and δs(L) satisfy the usual conditions in transfer function models. This new equation can

be easily estimated and provides enough flexibility to model the varying dynamic relationships between

the exogenous and accident variables.

The null hypothesis that all the added parameters associated to St and Z∗j,t are zero can be tested here

by means of a likelihood ratio statistic. The parameters included in (7) are denoted Ω0, and Ω = [Ω0,Ω1]

are the parameters included in (8). These designations are made in order to test H0 : Ω1 = 0 with the

statistic

λT = 2hlnL(bΩ)− lnL(bΩ0)i

where λT converges in distribution to χ2k under the null hypothesis, being k the number of elements in

Ω1.

The results of the stability tests for the transfer function models of the main accident variables are

shown in Table 7. The instability of the coefficients is confirmed in 8 out of the 15 cases analyzed in

this paper6. The estimated coefficients for ACC models are the most unstable. As a matter of fact,

the estimated coefficients for all variables in model CM3 are the only coefficients that shows stability.

Another important finding is that the null for model CM4 (including IPI and NUVE as inputs) is

6 Similar results have been found for disaggregated variables. The estimated functional transfer function models and

the stability tests for disaggregated variables are available from the authors upon request.

15

always rejected. The dynamic relationships between accident variables with IPI and NUVE as inputs

seem to vary over time. These relationships seem to be affected by changes in the environment, economic

conditions, and technology.

Insert Table 7 around here

The performance of individual forecasts is often found to be unstable (e.g. Stock and Watson, 2004);

whether a predictor worked well depended on the current economic shocks and institutional and policy

particulars. Surprisingly, however, some simple forecast combination -the median and the trimmed mean

of the panel of forecasts- were stable and reliably outperformed individual forecasts. This issue will be

explored in the following section.

4 Forecasting

The main purpose of this section is to assess the predictive performance of the models that have earlier

been analyzed. In particular, we use the same exogenous variables to examine the predictive accuracy

of aggregated versus disaggregated models. In addition, we examine the improvement in prediction that

occurs when we use more elaborate models with larger information sets. The choice of the forecasting

period is directly related to the beginning of the traffic authorities’ public announcements of targets

aimed at reducing traffic accidents. Therefore, for the nine accident rates for 2001, 2002 and 2003, we

obtained forecasts one to twelve periods ahead. The estimation of the first forecasting period ends in

December 2000, while the estimation period is extended for an additional year during the second and

third forecasting periods. The forecasting periods are particularly difficult to predict. As a matter of

fact, most variables displayed not just one, but two turning points in their annual growth rates during

the forecasting horizon. Additionally, some accident rates (e.g., ACCR, INJ and INJR) again showed

large and positive annual growth rates in 2003. This finding confirmed the cyclical characteristic of

these variables during the forecasting period.

Forecast results are assessed using the four accuracy criteria of aggregate root mean square error

(RMSE), mean absolute prediction error (MAPE), annual percentage error (APE), and forecasted annual

16

growth rates (FGR). In order to avoid the finding that the evaluation of different forecasting techniques

may depend upon the choice of a particular accuracy measure, the adoption of a variety of error measures

has been repeatedly stressed in the literature (Fildes, 1992). Although conflicts among these evaluations

are not desirable, they are only an indication of the different goals that exist in alternative prediction

exercises. However, for longer forecasting horizons (beyond the usual one step-ahead period) the use

of the RMSE or MAPE may not only be dangerous, but misleading (García-Ferrer and Queralt, 1997).

In this case, we contend that the FGR and APE criteria, particularly when evaluating targets based

on annual growth rates, become more relevant. A summary of the individual forecasting results is

shown in Table 8, where the medians of the individual models for each forecasting period and different

accuracy criteria 7 are presented. The last three columns in the table show the observed growth rates

(OGR) in order to make the FGR results self-contained. Considerable heterogeneity in forecast accuracy,

particularly for FATU, is shown by the median results in Table 8. Also, the forecasting information

provided by the alternative accuracy measures for FATU in 2001 is rather contradictory. While APE

and FGR indicate superb forecasting performance, MAPE and RMSE show the opposite result. Similar

contradictions (although less severe) are also found with other variables in the table.

Insert Table 8 around here

4.1 Assessing the effects of disaggregation

Our forecasting results confirm the gain in forecasting efficiency when using disaggregated models for

ACC, FAT and INJ. We can summarize the 252 forecasting outcomes, using the individual results (not

shown), in Tables 9 and 10. These tables present results by both predictive criteria and characteristics of

the model. Findings indicate that disaggregated models perform better in 145 cases (57.5%), aggregated

models perform better in 93 cases (36.9%), and results are identical in 14 cases (5.6%) (mainly in MAPE).

However, as shown in Tables 9 and 10, results change considerably according to the accuracy criteria, the

7The models are as follows: DHR, INT, CM1, CM2, CM3, CM4, and CM5. We also have seven other models of

the main aggregates (ACC, INJ and FAT) that correspond to the disaggregated alternatives of these models. Individual

forecasting results are available from the authors upon request

17

forecast year, and the characteristics of the model. For instance, if we concentrate on RMSE, the ratio

between the best performers among aggregated and disaggregated models is almost one (30/28), while

it deteriorates considerably to 19/44 when APE or FGR are used as measures of predictive accuracy.

When the four criteria are considered together, the ratio between the best performers among aggregated

and disaggregated models also changes according to the forecast year (i.e., 22/59 in 2001, 34/47 in 2002

and 37/39 in 2003). Previous findings confirm the possibility of conflicts among alternative criteria and

the need for the user to explicitly specify their accuracy preferences when comparing forecasts. Also,

as shown in Table 10, remarkable differences occur when results are presented in relation to models.

Results indicate that the DHR, CM2 and CM5 models overwhelmingly favor the use of disaggregated

models. Results for INT suggest a similar, although less robust, preference. However, the evidence in

the remaining cases is mixed. In any case, the advantages of using disaggregated models will also be

highlighted when dealing with the use of forecasts combination in subsection 4.3.

Insert Tables 9 and 10 around here

4.2 Forecasting comparisons using different information sets

At this stage, it would be interesting to verify whether the use of more elaborated models (using larger

information sets) improves forecasting behavior when compared to the benchmark alternative (i.e.,

aggregate DHR) that only uses the past history of the relevant accident variables. The conditional

nature of the forecasts of our intervention and econometric models, where the future values of inputs

are known, must be explicitly acknowledged when dealing with this issue 8.

In regards to overall comparisons among the different models, we have summarized the individual

forecasting results in Table 11. This table shows the number of times that each method ranks first for

each one of the traffic variables. However, care must be exercised when making an interpretation of the

8Given the deterministic feature of the inputs, this is not a problem for intervention models. Nevertheless, it is a real

problem for econometric models with stochastic inputs when the forecasting horizon is beyond the lead shown by the

exogenous leading variables. However, given the delay between the public availability of the final accident figures and the

early publication of economic indicators (6-8 months), this problem has small practical consequences in this study.

18

results in this table. The reader should be aware that, although it is useful to report forecast performance

based on various measures, it might be misleading to synthesize the performance of different methods

by aggregating the results of various measures.

Insert Table 11 around here

With this caveat in mind, we attempt to summarize the main results that have emerged from

the previous tables. First, as might be expected, one model does not dominate the others under all

forecasting criteria and forecasting periods. For instance, the DHR model performs almost as well as

the other two top performing models (i.e., INT and CM5) in 2001. However, CM5 emerges as the

best alternative in 2002 and INT produces the best forecasts in 2003. Second, similar differences are

found when results are examined according to series. For example, both INT and CM5 are the best

models for ACC, while INT, CM3 and CM5 show the best performance for fatalities. Both the INT

and DHR models share the lead for the INJ series. Third, neither the CM1 or CM2 model seems to

perform particularly well in terms of forecasting. Fourth, our comparisons of the overall results among

the different methods tend to confirm earlier findings of Fildes et al. (1998), who found that simple

methods (i.e. DHR and INT in this study) appear to perform as well as more complex procedures,

although the relative performance depends upon the accuracy measure used. Finally, there does not

seem to be a link between the instability of the individual model and the corresponding forecast accuracy.

4.3 Forecasts combination

The lack of a systematic dominance of one particular method or model of forecasting can lead to

an improvement in forecast accuracy. This improvement may be achieved by combining alternative

forecasting methods. In general, benefits in forecasting accuracy are optimized when the forecasts

of the models that are combined differ substantially. A question susceptible to both theoretical and

empirical research is the choice of weighting schemes and the choice and number of models to include

(e.g. Newbold and Harvey, 2002). When a record of past performance is available, the theory of

combination forecasting suggests that methods that put a stronger emphasis on forecasts that display

19

better performances will perform better than simple combination forecasts. Also, further gains might

be obtained by introducing time variation in these weights, or by discounting observations in the distant

past. However, as indicated by Clemen (1989) and Stock and Watson (2004), it is difficult to find a

procedure superior to that of using the equal weights (mean) or the median benchmarks in this process.

The median APE and FGR of all accident rates forecasts are provided in Columns 8 to 13 in Table 8.

For instance, in terms of APE in 2001, the largest error (a turning point year) occurs in ACCU (3.46%)

and the lowest error corresponds to ACCR (.03%). In 2002, the largest error occurs in FATU (4.7%),

while the lowest error corresponds to INJU (-.16%). Finally, in 2003 (another turning point year) the

largest error occurs in FATU (-6.1%), and the lowest figure corresponds to ACC (.03%). In general, the

median APEs indicate good forecasting performance, with figures considerably below (between two and

five times) the standard errors of the historical annual growth rates.

However, at this stage of combining forecasts for the main aggregates, it would be worthwhile to check

if a different strategy is suggested by the advantages of disaggregation found in the previous section. In

Table 12, we show the mean and the median APEs for 2001, 2002 and 2003 and the corresponding three

different strategies. The median and the mean of the forecasts of all the models (14 in total) correspond

to columns 4, 7 and 10 in this table. It should be noted that these figures for the median coincide with

those found in columns 8 to 13 in Table 8. In addition, columns 2, 5 and 8 in Table 12 correspond

to the mean and median of the seven aggregated models, while columns 3, 6 and 9 correspond to the

mean and median of the seven disaggregated models. The advantage of using simple combinations of

disaggregated models for accidents and injured passengers can easily be seen in this study. However,

given the high share of the road component (85%) in fatalities and the anomalies associated its urban

component, this advantage is not shown for fatalities.

*Insert Table 12 around here

20

5 Conclusions and policy issues

The traffic agencies in many developed countries periodically set road safety targets. Although these

targets are only mildly defined in Spain, various governments have openly encouraged the target of a

"drastic" reduction in accident rates by 2010. In order to maintain an effective use of resources, it

is important to monitor the progress that is made towards these targets. This paper, using seasonal

monthly data of recent Spanish accident rates, presents a methodology for analyzing the forecasting

performance of a large number of econometric models. In conclusion, the disaggregation of accident

rates becomes crucial for the improvement of forecast accuracy and monitoring progress towards these

targets.

The presence and evaluation of the calendar effects in the accident series requires separate analyses for

urban and non-urban accidents. As a matter of fact, the effects of two important intervention variables

(e.g. moving Easter holidays and changes in the number of working days) show different signs on road

and urban accident rates. This difference may be a consequence of the different seasonal factor weights

between these two groups. This opposite behavior explains why the coefficients of the calendar effects

in some of the aggregate variables are not statistically significant. In general, however, the magnitude

of the calendar effects is larger for road than for urban variables. Among the remaining intervention

variables, the June 1992 Traffic Security Plan, which established the use of mandatory seatbelts on car

passengers, induced a permanent reduction in fatalities. This reduction ranged from -15% to -18%,

depending on the estimates of alternative models. Other specific campaigns that were undertaken by

the Spanish Dirección General de Tráfico agency did not seem to have had noticeable success in reducing

accident rates over a long period of time.

Theoretically, both the explanatory ability and the forecasting performance of univariate and inter-

vention models may be improved by using dynamic econometric models with a larger information set.

The identification and estimation of the dynamic relationships between our output variables and the

corresponding economic inputs, measured by general economic activity and the degree of car utilization,

is necessary for this improvement. A considerable number of specification searches and the necessity of

21

using several combinations of alternative models are implied by such a process. In addition to the finding

that most estimated econometric models do not show strong evidence of misspecification, all estimated

coefficients show the expected signs and are statistically significant. Despite these findings, the gains

(measured by residual variance reduction) in terms of statistical fitting (with respect to univariate mod-

els) are very marginal. Also, when using different sampling intervals, we have found a varying lead-lag

relationship that depends on the cycle and the particular estimation period. This is an indication of a

potential instability in the estimated coefficients, and possibly the reason why the forecasting ability of

these models is not as promising as we had anticipated.

The predictive performance of the models (assessed using different accuracy criteria) for the last

three years of the sample (2001-2003) provides some interesting findings. Our results confirm the

gain in forecasting efficiency when using disaggregated models, although results change considerably

depending on the accuracy criteria, the forecast year, and the characteristics of the models. Moreover,

our overall comparisons between the different methods tend to confirm earlier findings suggesting that

simple methods appear to perform as well as more complex procedures. However, once again, the relative

performance depends on the accuracy measure used. In general, the median percentage errors indicate

a good forecasting performance. Also, the forecasts combination using disaggregated models becomes a

promising alternative for accidents and injuries rates. Nevertheless, given the predominance of the road

component for fatalities, this finding does not apply to forecasts of fatalities.

Results of this paper confirm how, in the short- and medium-term, individual or combined models

can be used when forecasting accident rates with monthly data. Individual and combined models can

also be used to test the effectiveness of previous traffic security plans and may serve as useful tools

when controlling expected outcomes from traffic agencies. However, our results do not predict a drastic

reduction in accident rates in Spain in the near future. As a matter of fact, our forecasts, using various

single or combined alternatives, show that it is unlikely that the government targets will be achieved.

22

6 Appendix: LDHR algorithm

In the DHR model Tt and St consist of a number of DHR components, spjt , with the general form

spjt = ajt cos(ωjt) + bjt sin(ωjt), where pj and ωj = 1/pj are the period and the frequency associated

with each jth DHR component respectively. Tt is the zero frequency term (Tt ≡ s∞t = a0t), while the

seasonal component is St =PRj=1 s

pjt ; where j = 1, . . . , R are the seasonal frequencies. Hence, the

complete DHR model is

ydhrt =RXj=0

spjt + et =

RXj=0

©ajt cos(ωjt) + bjt sin(ωjt)

ª+ et.

The oscillations of each DHR component, spjt , are modulated by ajt and bjt which are stochastic

Time Variable Parameters that follow an AR(2) stochastic process with at least one unit root

(1− αjL)(1− βjL)ajt = ξjt−1(1− αjL)(1− βjL)bjt = ξjt−1

0 ≤ αj ,βj ≤ 1, ξj ∼ w.n. N(0,σ2j);

therefore, non-stationarity is allowed in the various components.

This DHR model can be considered a straightforward extension of the classical harmonic regression

model, in which the gain and phase of the harmonic components can vary as a result of estimated

temporal changes in the parameters ajt and bjt.9

The method for optimizing the hyper-parameters of the model (i.e., the variances σ2dhr =£σ20,σ

21, . . . ,σ

2R

¤0of the processes ξj , j = 0 . . . , R, and the variance σ2e of the irregular component) was formulated

by Young et al. (1999) in the frequency domain, and is based upon expressions for the pseudo-spectrum

of the full DHR model:

fdhr¡ω,σ2

¢=

RXj=0

σ2jSj(ω) + σ2e; σ2 =

£σ2dhr,σ

2e

¤0where σ2jSj(ω) are the pseudo-spectra of the DHR components s

pj , and σ2e is the variance of the irregular

9The main difference between the DHR model and its related techniques, such as Harvey’s structural model, (Harvey,

1989) lies in the formulation of the unobserved component model (for the periodic components) and the method of

optimizing the hyper-parameters.

23

component. In the optimization processes it is seek the vector σ2 that minimizes10

min[σ2]∈RR+1

°°fy(ω)− fdhr ¡ω,σ2¢°° , (9)

where fy(ω) is the spectrum of the observed time series. The DHR components follow non-stationary

ARMA processes; therefore, in order to find an Ordinary Least Squares (OLS) solution of Equation (9)

it is needed to eliminate the unit modulus AR roots of fdhr¡ω,σ2

¢. The DHR components spjt are

stochastic processes of the form

ϕj(L)spjt = θj(L)ξjt−1 , ξjt t ∼ w.n. N(0,σ2ξjt );

and the pseudo-spectrum of the complete DHR model is

fdhr¡ω,σ2

¢=

RXj=1

σ2jθj(e

−iω)θj(eiω)ϕj(e

−iω)ϕj(eiω)+ σ2e.

If Equation (9) is multiplied by the function Ψ(ω) = Φ(e−iω)Φ(eiω), where Φ(L) is the minimum order

polynomial with all unit modulus AR roots of the complete DHR model, the algorithm minimizes

minσ2∈RR+2

°°Ψ(ω) · fy(ω)−Ψ(ω) · fdhr ¡ω,σ2¢°° ; (10)

but, because in (10) all the unit modulus AR roots cancel, now the minimization problem can be solved

by OLS.

Finally, fy(ω) can be substituted in (10) by the estimated AR-spectrum bfy(ω) = bσ2 ¡φy(e−iω)φy(eiω)¢−1 ,where φy(B) is an AR polynomial fitted to the series, and bσ2 is the residual variance of the fitted ARmodel. The size, shape and location of the spectral peaks of bfy(ω) are used to identify the models ofeach of the DHR components spj .

Once the models of the DHR components have been identified, and the hyper-parameters σ2dhr =£σ20,σ

21, . . . ,σ

2R

¤0have been optimized by OLS, the DHR components Tt, St, and et can be estimated

using the Kalman Filter and the Fixed Interval Smoothing.

10Young et al. (1999) modified the problem using the residual variance bσ2 from the fitted AR model, as an estimation of

σ2e . These authors then divided the residual variance by bσ2 in order to find the vector NVR = [1, NV R0, . . . , NV RR],

where NVRj = σ2j/bσ2.

24

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28

8 Tables and figures

Variable Trend H : 12b H : 6 H : 4 H : 3 H : 2.4 H : 2 bσACCa 7.49 37.0 3.50 6.96 3.76 7.07 .461 .0255

ACCR 4.27 27.0 5.69 7.79 3.93 12.3 1.46 .0374

ACCU 8.81 14.5 7.91 7.88 6.30 4.62 1.56 .0285

INJ 5.27 36.7 3.11 8.08 1.88 9.65 .646 .0316

INJR 1.48 27.0 7.41 6.54 2.40 8.85 4.12 .0567

INJU 8.67 14.9 7.10 6.93 6.69 4.89 2.05 .0431

FAT 1.02 18.6 12.2 8.99 4.08 13.4 .150 .0578

FATR .911 26.2 2.76 11.2 4.98 18.8 .320 .0637

FATU .623 .000 .000 25.3 .000 10.6 16.2 .1032

Table 1: NVR estimates for all variables. aAll NVR estimates coefficients are multiplied by 103. bH:X stands for

harmonic where X is the annual frequency.

F1−12 Conclusion Lag Structure(a)

ACC 18.92 I(1,1) at α = .05 1-2, 12-13, 23-24, 35-36

FAT 55.90 I(1,0) at α = .05(b) 1-2, 11-13, 23-24, 35-36

INJ 22.36 I(1,1) at α = .05 1-2, 12-13, 23-24, 35-36

IPI 8.26 I(1,1) at α = .01 1-2, 10-13, 23-24

NUVE 17.97 I(1,1) at α = .01 1-2, 10-16, 23-25

GAS 3.61 I(1,1) at α = .01 1-3, 10-16, 23-25, 35-36

DIESEL 39.19 I(1,0) at α = .01(b) 1-4, 10-13, 23-24, 35-36

Table 2: Unit root test results. (a) The lag structure is chosen in order to obtain white noise residuals. (b)

Conclusion based on t statistic from OCBS et al. (1988).

29

Name Definition Facts Affected variables

EAST 1 in easter month Easter effects ALL

0 otherwise

NWD Number of working days Trading days effects ALL

per month

ALF 1 in 1978.7 Alfaques camping accident FAT, FATR

0 otherwise

JAN84 1 in 1984.1 Unusual rainy month ACC, ACCR, INJ, INJR

0 otherwise

NOV93 1 in 1993.11 Unusual rainy month ACCR, INJ, INJR

0 otherwise

JUN92 0 before 1992.6 Traffic Law takes effect ALL

1 after 1992.6

Table 3: Definition of the main intervention variables

30

Intervention variables

EAST NWD ALF JAN84 NOV93 JUN92

Accident Variables

ACC .0087 -.0056∗ — .1120∗ — -.1262∗

ACCR .0620∗ -.0125∗ — .1236∗ -.1242∗ -.1481∗

ACCU -.0335∗ .0005 — — — -.1093∗

FAT .0511∗ -.0126∗ .3199∗ — — -.1540∗

FATR .0649∗ -.0153∗ .3709∗ — — -.1871∗

FATU -.0284 .0050 — — — -.0967∗

INJ .0330∗ -.0112∗ — .1228∗ -.0948∗ -.1491∗

INJR .0909∗ -.0174∗ — .1388∗ -.1514∗ -.1705∗

INJU -.0214∗ -.0030 — — — -.1440∗

Economic variables

IPI -.0290∗ .0145∗

NUVE .0385 .0297∗

CGAS .0681∗ .0065∗

CDIESEL .0347∗ .0029

Table 4: Summary of the estimated IARIMA models. (*) significant coefficient at α = 0.05. All variables are

in logs and with regular and seasonal differences

Model Output Variables Input Variables Estimation period

CM1 ACC, FAT, INJ IPI 1975.1-2000.12

CM2 ACC, FAT, INJ NUVE 1975.1-2000.12

CM3 ACC, FAT, INJ GAS, DIESEL 1982.1-2000.12

CM4 ACC, FAT, INJ IPI, NUVE 1975.1-2000.12

CM5 ACC, FAT, INJ IPI, NUVE, GAS 1975.1-2000.12Table 5: Alternative causal models’ specifications for accident rates

31

Causal model CM1- IPI Causal model CM2 - NUVE

ACC FAT INJ ACC FAT INJ

EAST .018(.008)

.070(.024)

.047(.010)

FEB78 - .125(.037)

- .118(.036)

ALF .349(.074)

.335(.072)

APR82 - .217(.071)

JUN83 .149(.074)

JAN84 .099(.030)

.104(.030)

JUL90 - .134(.036)

- .105(.035)

JUN92 - .131(.032)

- .144(.055)

- .149(.035)

- .125(.031)

- .165(.045)

- .150(.035)

NOV93 - .084(.030)

- .094(.036)

- .096(.035)

FEB94 - .167(.044)

APR95 .097(.030)

.103(.036)

.095(.030)

MAR97 .063(.030)

.108(.037)

IPI .155L(.057)

.337L(.127)

.382L(.075)

+ .204L2(.074)

NUVE .075L(.018)

+ .041L2(.018)

.126L(.037)

.092L(.022)

θ1 .439(.052)

.728(.038)

.522(.049)

.452(.052)

.80(.036)

.548(.051)

θ3 - .148(.059)

θ4 .154(.058)

θ12 .741(.041)

.913(.018)

.702(.041)

.738(.041)

.773(.045)

.786(.037)

φ12 - .193(.067)bσa .0377 .0803 .0445 .0375 .0777 .0432

LBQ(12) 8.8 8.8 9.9 13.2 9.9 10.2

LBQ(24) 18.6 28.2 28.0 24.9 30.6 37.8

T 312 312 312 312 312 312

Table 6a) Estimated CM1 and CM2 models for aggregated variables (in logs and regular and seasonal

differences). Standard errors in parenthesis. LBQ: Ljung-Box Q statistics; bσa = residual standard error; T =

estimation size.

32

Causal model CM3 - CGAS + CDIESEL Causal model CM4 - IPI + NUVE

ACC FAT INJ ACC FAT INJ

FEB78 - .119(.036)

ALF .355(.075)

JAN86 - .079(.035)

JAN84 .100(.030)

JUL90 - .116(.034)

- .126(.035)

JUN92 - .132(.033)

- .128(.044)

- .165(.034)

- .123(.031)

- .145(.053)

- .144(.034)

NOV93 - .082(.031)

- .083(.034)

- .089(.035)

FEB94 - .109(.068)

- .142(.072)

APR95 .096(.031)

.123(.034)

.096(.030)

.101(.035)

MAR97 .063(.031)

.105(.035)

.062(.030)

.111(.035)

IPI .119L(.060)

.295L(.132)

.280L(.075)

+ .189L2(.071)

NUVE .066L(.019)

+ .042L2(.018)

.079L(.041)

.076L(.022)

CGAS .139(.070)

.355(.149)

.099(.091)

CDIESEL .177L(.060)

.582L(.109)

.209L(.072)

θ1 .408(.060)

.793(.045)

.483(.068)

.454(.052)

.729(.040)

.507(.047)

θ12 .848(.035)

.831(.036)

.738(.057)

.728(.042)

.810(.040)

.726(.038)

φ12 .244(.068)

.195(.065)

φ24 .195(.084)bσa .0375 .0742 .0418 .0373 .0804 .0425

LBQ(12) 13.5 12.9 10.3 14.5 9.8 11.4

LBQ(24) 27.2 32.1 21.0 26.2 26.6 34.1

T 228 228 228 312 312 312

Table 6b) Estimated CM3 and CM4 models for aggregated variables (in logs and regular and seasonal

differences). Standard errors in parenthesis. LBQ: Ljung-Box Q statistics; bσa = residual standard error; T =

estimation size.

33

Causal model CM5 - IPI + NUVE + CGAS

ACC FAT INJ

FEB78 - .120(.038)

ALF - .0888(.0295)

.359(.075)

JUN83 .131(.073)

JUL90 - .119(.038)

JUN92 - .124(.031)

- .133(.050)

- .162(.036)

NOV93 - .085(.030)

- .075(.038)

APR95 .097(.030)

MAR97 .058(.031)

IPI .122L(.060)

.318L(.131)

.135L(.075)

NUVE .056L(.019)

+ .047L2(.018)

.067L(.040)

.057L(.023)

CGAS .136(.056)

.268(.121)

.173(.070)

θ1 .459(.050)

.695(.058)

.537(.050)

θ2 .113(.057)

θ12 .778(.037)

.931(.014)

.715(.058)

θ24 .180(.056)bσa .0370 .0789 .0447

LBQ(12) 14.9 11.4 18.3

LBQ(24) 44.1 29.3 53.9

T 312 312 312

Table 6c) Estimated CM5 model for aggregated variables (in logs and regular and seasonal differences).

Standard errors in parenthesis. LBQ: Ljung-Box Q statistics; bσa = residual standard error; T = estimation

size.

34

ACC FAT INJ

λT k λT k λT k

CM1 99.82(a) 3 1.62 3 94.30(a) 4

CM2 100.40(a) 4 1.52 3 0.192 3

CM3 1.972 4 8.99(a) 4 4.932 4

CM4 101.15(a) 5 37.24(a) 4 94.67(a) 5

CM5 102.09(a) 6 3.146 5 0.306 5

Table 7: Results for the stability test. (a) Reject H0 : Ω1 = 0 at α = 0.01

MAPE RMSE APE FGR OGR

2001 2002 2003 2001 2002 2003 2001 2002 2003 2001 2002 2003 2001 2002 2003

ACC .043 .018 .021 .048 .024 .026 2.7 .7 .03 1.3 -1.3 1.6 -1.3 -2.0 1.6

ACCR .033 .039 .022 .040 .045 .028 .03 1.9 -1.0 1.7 .6 5.0 1.7 -1.4 6.0

ACCU .058 .025 .041 .063 .034 .048 3.5 -.4 .5 -.4 -2.8 -1.6 -3.8 -2.5 -2.1

FAT .060 .066 .054 .079 .084 .064 2.4 1.1 -3.4 -2.1 -2.2 -2.4 -4.5 -3.3 1.1

FATR .060 .067 .053 .079 .086 .064 2.7 .6 -3.1 -1.5 -1.8 -2.0 -4.2 -2.5 .7

FATU .109 .126 .192 .131 .174 .229 .3 4.7 -6.1 -5.3 -3.3 -2.9 -6.0 -8.0 3.4

INJ .041 .026 .027 .048 .033 .032 1.8 1.1 -.3 1.7 -.8 2.1 -.1 -1.8 2.5

INJR .041 .050 .034 .052 .060 .037 -.5 1.7 -1.9 .6 .3 5.3 1.1 -1.4 7.3

INJU .050 .025 .046 .054 .034 .059 1.8 -.2 1.5 .4 -2.3 -.9 -1.4 -2.2 -2.3

Table 8: Medians for accuracy forecasting criteria. MAPE: Mean Average Percentage Error; RMSE: Root

Mean Squared Error; APE: Annual Percentage Error; FGR: Forecasted Growth Rate; OGR: Observed Growth

Rate.

35

Best Aggregated Best Disaggregated Matches

2001 2002 2003 2001 2002 2003 2001 2002 2003

MAPE 4 10 11 15 8 6 2 3 4

RMSE 7 12 11 13 9 6 1 0 4

APE 5 6 8 16 15 13 0 0 0

FGR 6 6 7 15 15 14 0 0 0

Total 22 34 37 59 47 39 3 3 8

93 145 14

Table 9: Summary of forecasting results: Aggregated versus Disaggregated by criteria. ”Matches” stands for

situations when the forecasting results are identical for aggregated and disaggregated models.

Best Aggregated Best Disaggregated Matches

2001 2002 2003 2001 2002 2003 2001 2002 2003

DHR 1 0 6 11 12 5 0 0 1

INT 5 5 2 7 7 8 0 0 2

CM1 5 7 7 7 5 4 0 0 1

CM2 0 1 8 10 10 4 2 1 0

CM3 7 5 2 5 5 9 0 2 1

CM4 4 10 6 7 2 3 1 0 3

CM5 0 6 5 12 6 7 0 0 0

Table 10: Summary of forecasting results: Aggregated versus Disaggregated by criteria. ”Matches” stands for

situations when the forecasting results are identical for aggregated and disaggregated models.

36

Variables

Models Accidents Fatalities Injured Sum

DHR 6 2 10 18

INT 16 9 11 36

CM1 2 2 0 4

CM2 1 3 2 6

CM3 2 8 4 14

CM4 3 5 6 14

CM5 10 8 5 23Table 11: Summary of Forecasting results by models and variables. Number of times that each method ranks

first for the different estimated models and variables, for all criteria.

MEAN

2001 2002 2003

A D A+D A D A+D A D A+D

ACC 3.03 2.17 2.60 .32 .34 .33 .09 .00 .03

FAT 2.77 2.85 2.81 1.35 1.40 1.38 -3.29 -3.68 -3.49

INJ 2.58 1.11 1.85 .89 .78 .84 -.26 -.34 -.30

MEDIAN

ACC 3.11 2.05 2.68 .69 .56 .67 .17 .00 .03

FAT 2.39 2.46 2.43 .80 1.12 1.10 -3.43 -3.43 -3.43

INJ 2.52 1.36 1.84 1.23 .73 1.05 -.47 -.22 -.28

Table 12: Mean and median APE. A: APE of aggregated variables; D: APE of disaggregated variables. Bold

figures indicate the lowest average/median APE for each variable and year.

37

4000

6000

8000

10000

12000 Accidents

4000

8000

12000

16000

20000Injured

200

400

600

800

1000Fatalities

20

40

60

80

100

120

IPI

0

40000

80000

120000

160000

200000

1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

NUVE

Figure 1: Main variables in the model

38

4000

6000

8000

10000

12000

AC C M e a n s b y S e a s o n

Jan F e b M ar Ap r M ay Jun Jul Au g S e p O ct Nov D e c

S e a s o n a l F a c to r W e ig h ts fo r AC C

1000

2000

3000

4000

5000

6000

AC C R M e a n s b y S e a s o n

Jan F e b M ar Ap r M ay Jun Jul Au g S e p O ct Nov D e c

S e a so n a l F a c to r W e ig h ts fo r AC C R

2000

3000

4000

5000

6000

AC C U M e a n s b y S e a s o n

Jan F e b M ar Ap r M ay Jun Jul Au g S e p O ct Nov D e c

S e a s o n a l F a c to r W e ig h ts fo r AC C U

Figure 2: Seasonal Factor Weights for Accident Variables

39

.00

.01

.02

.03

.04

.05

.06

ππππππππ//// 2222

Square Gain

Square of spectral gain for NUVE in CM2 - variable ACC

ωωωω.00

.05

.10

.15

.20

.25

.30

ππππππππ//// 2222

Square Gain

Square of Spectral Gain for IPI in CM1 - variable INJ

ωωωω

.000

.002

.004

.006

.008

.010

.012

ωωωωππππππππ//// 2222

Square Gain

Square of Spectral Gain for NUVE in CM4 - variable ACC

.00

.04

.08

.12

.16

.20

.24

ωωωωππππππππ//// 2222

Square Gain

Square of Spectral Gain for IPI in CM4 - variable INJ

.000

.002

.004

.006

.008

.010

.012

ωωωωππππππππ//// 2222

Square Gain

Square of Spectral Gain for NUVE in CM5 - variable ACC

Figure 3: Square of the Spectral Gains

40