- Email: [email protected]
Forecasting the Dutch heavy truck market
International Journal of Forecasting North-Holland
4 (1988) 57-79
57
R.M.J. HEUTS Tilburg Urkersity,
5000 LE Tilburg, The Netherlands
J.H.J.M. BRONCKERS DA F Trucks Limited, 5602 BB Eindhoven, The Netheriands
A bstract:
In this paper it is investigated whether multivariate time series models can improve the forecasting performance of Holt-Winters models and univariate ‘naive’ ARIMA-models. which are easier to construct, but do not take into account lead-lag relationships. This research deals with the Dutch truck market performance in the light of overall economic developments. A S-variate model is built, containing two truck sales series and three economic indicators. It is found that a substantial reduction in residual variance can be found by using a multivariate model. In the case of one truck series (rigids) this leads to uniformly better forecasts. For the other output series (artics), no improvement in forecasting accuracy was found. Holt-Winters models do not seem to improve any of the sophisticated models. Uni- and multivariate ARIMA-models, ment, Forecasting performance.
Holt-Winters,
Truck market modelling. Align-
1. Introduction AS with most durables, industrial goods are highly sensitive to economic activity. This is especially the case with truck sales. Commercial vehicles, unlike cars which form part of final demand, are principally investment goods. Demand for new commercial vehicles shows much of the volatility traditionally associated with demand for investment goods. As for trucks the market is evidently full-grown. New demand is mostly demand for replacement. Together with the economy, the total road goods transport changes and therefore the number of newly sold vehi4es. Evidently. replacement of vehicles may be advanced or postponed in phase with business cycles. Therefore, actual scrapping mostly deviates from latent scrapping. the latter being determined simply by the age structure of the goods vehicle fleet. oreover. the relationship between forecasted total haulage requirement and vehicles fleet in operation (and thus newly sold trucks) is complicated by trends in vehicle carrying capacity and operating efficiency. Often this is represented by a function of time with downward slope. A number empirical studies of the heavy truck ma&et of various European countries have been performed in ronckers (1982) and Bronckers (1983). In &is article we investigate the relationship be en the registrations of heavy rigids and utch economic indicators: newly articulate trucks (artics) in the ~N:cthc”r 0169-2070/M/$3.50
I_ 1988. Elsevier Science Publishers
B.V. (North-Holland)
R. M. J. Heuts and J. H. J. M. Bronckers
58
/ tbrecasting
the Dutch truck market
registered passengers curs in the Netherlands, the production index of building materials and the index of industrial production. The next section of this article reviews some theoretical aspects of multivariate time series analysis. In section 3 empirical results are given, both for univariate and multivariate models. Forecasting performance of all models is discussed in section 4. Section 5 contains a number of conclusive remarks. For details we refer to a technical report [Bronckers and Heuts (1984)j.
2. Multivariate ARIMA-models An important class of discrete univariate models is the class of so-called ARIMA-models which are extensively analysed by. e.g., Box and Jenkins (1976). However, when m series { xl,[ ). 1 -yz., >* * . * ’ { -xn,.I > are considered simultaneously, it is necessary to allow for dynamic relationships that may exist between the series, for possibilities of feedback and for correlations between the shocks affecting the series. A useful class of models is obtained by direct generalization of the univariate models to
0) where @JB)=I-@,B... - QpBP, O,(B) = I - 0, B.. . pqB4 are matrix polynomials in the backshift operator B, the Q’s and O’s are m X m matrices, X, = X, - p is the vector of deviations from some origin p, which is the vector of means if the series are weakly stationary, and Ai = (a,,,, . . . , am.,) is a sequence of vector-valued shocks, distributed as multivariate normal N(0, C,), that allows for contemporaneous correlation between the elements a,,,, . . . . a,,, I, but which are mutually uncorrelated at non-simultaneous times. We shall assume that the zeros of the determinantal polynomials PP(t)l and IO,(=)1 are on or outside the unit circle. In order to ensure the existence of a unique weakly stationary solution of (1) we must restrict the range of (!?I+,..., $) to 2$(m), where S,(m) :=
(@*,...,@#et
@p(z)#O,(~I
=l]
[see Tigelaar (1982)]. To make sure that Xl can appropriately be written in the infinite AR- and MA-form. we suppose that the zeros of the determinantal polynomials 1@Jz) 1 and 1@J z) 1 are outside the unit circle. It should be noted that 2, is a vector of stationary series, obtained by appropriately transforming and differencing the original time series. While multiple time series models are naturally more complicated to handle and experience in their use is more limited, they provide a potential means of improving on results from univariate models. For example, information about 1,, in addition to that contained in its own past, may be available from other related series ( g2., }, . . . , (Z,,, , }. When this is so, improved forecasts should be possible. Parallelling the univariate case [see, e.g., Box (1978j], when the model is invertible, the forecasting procedure runs as follows: q+,=
j(I) + q(l), n”),f
,
,+,-,’
where
Lp=
(2)
R. M.J. Hews and J. H.J. M. Bt-onckers / Forecasting the Dutch truck market
59
The II;‘) ‘s and q,‘s are m X m matrices obtained from the relations Qb(B)rk(B)=O,(B),
4(B)=I+\k,B+qrzB2+...,
@q(B)IT(B)=@p(B),
Kl(B)=I-fl,B-&B2-...,
and
(3) (4)
In (2) &I)
is the conditional expectation of z,+,
it(l) =E(rS,+,I&, u,(l) =
ff+...>,
2r+,-il(r),I=
and
l,...
is the vector of forecast errors made at origin C. The error vector U,(I) is normally distributed with zero mean and covariance matrix I-1
Cov(U,(I)) =
c
i=O
?qz,?&‘.
(5)
With respect to the identifiability conditions for the parameters we refer to Hannan (1969) and Tigelaar (1982). It should be noted that the identiiiability of Hannan is based on an infinite sample size, opposite to that of Tigelaar. The objective of the model building strategy is to make an initial guess of the structure of a multioariate stochastic model. This so-called identification procedure will be based on various correlation and partial correlation functions. During the last years two methods of identification have been proposed. The first method is based on two sets of statistics (:h= cross correlation and partial cross correlation matrices) calculated from the stationary series x’,, obtained by appropriately transforming and differencing the original time series (non-prewhitzning method). The second one is based on the same statistics calculated from the residuals after fitting univariate models to each time series separately (prewhitening method). Two objections can be raised against the correlating of non-prewhitening series. Sample cross correlations for two autocorrelated processes (1) can suggest two-sided influence or feedback, although the underlying relationships are one-sided; (2) can be heavily correlated, thus showing spurious patterns. owever, Jenkins and Alavi (1981) show that multivariate models containing an autoregressive ture can cause difficulties in the prewhitening procedure. One runs the risk especially of overparameterization. Maravall (1981) also points to the problem of overparameterization as a drawback of the two-sided prewhitening procedure. Zellner (1979), Sims (1977) and Schwert (1979) suspect the filtering of the data to bias the results of the analysis towards seiies independence. .lenk+llJ and Alavi (1981) therefore recommend two ways of identification: cross correlating ihe prewhitened series as well as the original series. as used for this article, is based on identification without The computer package WMTS, whit prewhitening, as proposed by Tiao and ox (1981). The aim of the modelling approach of 80x and Tiao is to employ statistics which can be readily calculated and allow to select a subclass of models for further consideration. Specific statistics in I ox-Tiao concept are
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
cross correlation matrices of the original data or the matrices of appropriately differenced data. The sample cross correlations $,,(I) are estimates of the theoretical values p,,( 1)‘s. They are particularly useful in detecting low order moving average models since p, ,( !) = 0 for I > q, where q is the order of the MA-model. (ii) Estimates of the partial cross correlation matrices obtained by fitting stepwise autoregressive models of increasing order. Sample estimates of the partial cross correlation matrices P(I) are useful in identifying the order of an autoregressive model. In the WMTS computer package, the estimates P( 1) are obtained by fitting autoregressive models of I = 1, 2,. . . successively. (iii) A likelihood ratio statistic, M(I), is used to help determine the order of the ar;toregressive part of the model. It corresponds to testing the null hypothesis @, = 0 against the alternative @,z 0, when an autoregressive model of order I is fitted. Under the null-hypothesis M(1) follows a x2 distribution with ,I’ degrees of freedom. Diagnostic checking of the residual series { a,}, where (9
Sample
a, = &&&...
4$&+e,a,_,
+ . . . +&&.
should be performed by plotting the standardized residual series against time, as well as examining cross correlation matrices of the residuals { 2, } against their two standard error confidence limits. The problem of estimating the parameters in a multivariate time series model is more difficult than in the univariate case. However, Hillmer and Trao (1979) have developed procedures which give a close approximation to the exact likelihood estimates in multivariate ARIMA-models. For more details we refer to Tiao et al. (1989).
3. I. Uniuariare models In this section we examine the relationship between LM sales of new heavy rigids and artics in the Netherlands and three economic indicators: new passen,zr car sales, the production of building materials and total industrial production. The demand cp&private cars is a generally accepted leading economic indicator in most Western countries. i*urthermcre, from publications of the Dutch Central Bureau of Statistics it can be seen that during the last years approximately 20% of the yearly road goods transport exists of building materials. Except for this sector other categories are important, such as food and final products. Indicators for these categories ilily be the prr luction index of food and beverages, retail sales, private consumption and industrial production. From these the latter is taken into our analysis. In fig. 1 the following registration data are shown: newly registered rigids ( _qr) and artics (x2.,) above 9 tons G.V.W. (Gross Vehicle Weight) and newly registered private cars (s,.,). Also the plots of the production of building materials (x4.,) and of industrial production ( xs,J in the Netherlands are shown. Together with the raw data, their 12 monthly moving averages are plotted. Each series consists of 168 observations, ranging from Janua.y 1968 up to and including december 1982. In the data series x,., an outlier due to a registration error was detected for May 1971, which was adjusted. The behaviour of the sample ACF PACF suggests a nu arameter estimation i iagnostic checking
R. M.J. Hem and J. H. J M. Bronckers / Forecasting the Dutch truck market
61
Table 1 Univariate models (PACK-program). Series
Fitted models
Residual
Box-Ljung’s Xl00
variance
diagnostic Q’=’ (d.o.f.)
-2 0,
1 2 3 4 5
0.5227 E-l (0.04) (0.07) 0.2638 (l-O.22 B -0.32 B2)(ln x2,, -5.16) = t2,, (0.07) (0.07) (0.09) (1 - 012 B -0.26 B’)(l-0.54 B” -0.37 B24)(ln x3,, -10.51) = Q,,, 0.4010 E-l (0.07) (0.08) \CI+s) (0.08) (0.40) 0.1285 E-l (1 - ti.&! B)( 1 - 0.84 B*‘)(ln x~,~- 4.69) = Z4,, (0.10 (0.07) (0.04) 0.7005 E-3 VI712 111Ys,, = ‘; -0.49 B)(l -0.59 B”) = P,,, (0.07) (0.07) -
(1 - 0.51 B)(ln x],~ - 6.25) = Q,,,
NEH
REGISTRRTIONS
RIGIDS
L 3
TONS
17.39 (17)
24.1 I
16.76 (16)
16.76
17.73 (14)
64.27
19.50 (16)
66.30
16.66 (16)
35.29
GVW
NETHERLRNDS
1
67
6
69
70
71.
72
73
74
Fig. 1. Graphs of series 1 to series 5 and their 12 monthly
75
76
i
77
moving average.
I
78
.’
I
79
80
1
81
82
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
NS FIRTICS
t
9 TOhS
GVW
NETtiEALRNUS
I
67
88
69
70
1
1
71
72
I
73
I
74
I
I
75
76
I
I
77
78
I
79
I
80
1
81
92
TINE Fig. 1 (continued).
els in table 1. For series xs .t we used a testing procedure of Hasza and Fuller (1982) for nonstationary parameter specifications for that particular seasonal model. However, it can be demonstrated that the testing procedure of Hasza and Fuller has some drawbacks [see Bronckers and Heuts (1984)]. For diagnostic testing Ljung and Box (1978) developed the following test statistic: Q=n(n+2)
E
(n-k)-‘i$
k=l
with n the number of observations and m the number of autocorrelations use . Simulation studies by Ljung and Box (1978) and Ljung (1982) suggest that the ower of the diagnostic test increases for smaller values of m when the alternative is a low order model. This motivated our choice of m = 18. _3’.*.
rrltirariate mode/s
One way to detect a non-stationary latent variable is to difference the time series to the same degree as in the univariate analysis. The approach suggested by Hillmer et al. (1983) is to fit a model
R.M.J. Heuts and J.H.J. M. Bronckers / Forecasting the Dutch truck market
BILE
63
REGISTRRTIBNS
NETHERLANDS O-
o
68
69
70
71
n-7.71 72
1
73
74
75
76
77
78
I
79
I
80
I
81
92
TIPIE Fig. 1 (continued).
This will be the case if two independent trendy series are cross-correlated. However, since only series 5 is non-stationary, this risk does not exist in this application. Cross-correlating the five undifferenced series showed several interactions. However, near unit roots appeared in the diagonal polynomial of series 5. This forced us to take a regular and a seasonal difference for th;c eU.80-+ps “IIIC. . This is in accordance with the result in table 1. Because these differences obviously influence the interactions of series 5 with the other series, we re-estimated the cross correlations. The most relevant ones can be found in fig. 2. When fitting multivariate stochastic models to two or more time series, it may be necessary to shift a series xl,, in time by d time units to produce a time series x~,,+~, before relating it to another . . time series x~,~. arameter d is called the alignment parameter and the resulting model related to the . . time series ~i,~+~ and x2.1 is called an aligned model. It is indicated that there a contemporaneous dependencies, which cannot be modelled with help from the W package. In order to take these effects into account, we applied an alignment procedure. Usually, alignment is applied to center the cross-correlation function around lag zero in order to obtain a more parsimonious representation of the model. In this article, it is Ksed to mode! explicitly correlatioils at simultaneous time, which is normally acco information was noise series, being a consequence of model fitting. This e forecasti oses CTSta
R. M.J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
64
WERRGE
DAILY
MRTERXQLS
67
68
69
70
71
72
PRtZDUCTlffN
,ETC.
73
IN
74
THE
75
(3F BUILDING NETiiERLANDS
76
77
78
79
80
81
82
Fig. 1 (continued).
The original sample cross correlation function suggested contemporaneous and led to the following alignment: +t, X2.t and X3.v *
x2.t
=
x2,t+1’
-G.t
=
X3,t+2
dependencies
between
7
efore entering the forecasting stage, the model should be re-aligned. According to the ideas in section 1 the stepwise autoregression up to order 6 (nonseasonal) and up to seasonal order 2 (period 12) have been fitted (see table 2). From this the following model type is suggested:
(6) with
4 = (qt,
X&v $3
qt,
+.tY.
The (i, j)-th
element from the respective matrices will be tween brackets
R. M. J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
03’ r?l
I
I
67
1
68
I
69
I
70
I
71
I
72
I
I
73
74
1
75
1
76
I
1
77
78
i
79
I
80
65
I
81
82
TIME Fig. 1 (continued).
of the sample variance-covariance correlation matrix k a.
2, =
ii,=
0.284 0.102 -0.661* -0.146 -0.820
* 10-r * 10-l 1o-4 * 1O-2 * 1o-4
.
0.177 -0.300 * 1o-2 0.348 * 1o-2
1 .O@O 0.144 1 .OOO 0.002 - 0.033 - 0.025 0.076 - 0.018 0.127
he multivariate
matrix t,,
which is given together with its corresponding
0.471 * 10-l -0.103 * 1o-2 0.536 * 1O-3
0.120 * 10-l 0.615 * 1O-3
sample
0.735 * 1O-3
1 JO0 0.091
0.207
1.000
model gives an extra reduciion of residual variance in comparison ) this extra reduction i (rigids) of 34.‘&. For s it is shown that the est’ eters in model (6) are i
with the
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
66
c. c. I5
C. C. F.
SERIES
SERIES
1 - SERIES
2
- SERIES
Fig. 2. Plots of relevant cross-correlation
2
3
C. 6. F.
C. C. F.
SERIES
SERIES
I
- SEBIES
4 - SERIES
3
1
functions.
The significant interactions in the multivariate time series model can be interpreted as follows:
R. M. J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
67
Table 2 Summary results of the stepwise autoregression for the 5-variate model. (Plus sign for values greater than 2. minus sign for values less than 2, dot for in between values.) Lag
Standardized partial AR coefficients
1
5.76 0.98 1.51 0.82 - 0.65
8.58 4.01 0.20 1.17 0.55
- 0.21 4.12 2.59 0.86 0.34
1.58 0.44 - 2.23 1.02 0.42
-
2
0.02 1.56 0.07 0.73 - 1.79
-1.44 2.52 - 0.09 1.06 0.77
2.93 0.01 2.88 0.31 - 0.47
2.03 0.40 0.23 - 1.59 - 0.76
0.13 0.00 1.59 - 0.31 - 0.21
3
- 0.75 0.89 - 0.39 0.83 - 1.42
-1.00 2.30 0.32 0.69 - 0.20
- 1.78 - 1.79 0.71 - 0.34 -0.12
- 0.96 - 0.38 5.67 - 2.24 0.05
0.39 1.37 - 1.33 - 1.52 - 0.28
2.77 0.04 - 0.51 0.86 - 0.80
0.76 0.15 0.34 -0.19 1.70
0.89 - 0.91 - 2.59 0.52 - 1.85
-
-
1.40 - 0.02 0.63 4.65 1.62
2.30 1.90 - 1.09 0.38 1.24
- 1.55 0.:7 0.93 -0.18 - 0.48
- 1.66 1.44 1.50 3.82 - 2.04
- 1.54 - 0.40 1.53 - 0.15 0.?4
1.72 0.38 - 0.99 1.46 0.30
0.98 - 0.05 -1.36 - 1.54 - 0.57
2.00 -0.12 - 1.60 - 0.32 1.08
0.47 - 0.43 - 0.55 4.14 0.71
- 1.11 -0.51 0.29 - 0.55 - 0.86
0.99 0.23 2.06 2.15 2.08
0.70 0.92 8.00 - 0.74 - 0.16
0.08 1.81 - 1.53 11.44 - 0.53
- 0.61 0.57 - 0.16 1.22 - 3.76
1.21 0.28 0.19 0.20 1.91
- 0.74 - 0.61 3.91 - 0.73 - 1.17
0.74 1.73 - 1.33 2.93 - 0.49
- 1.37 - 1.08 0.91 1.01 - 3.54
4
5
6
12
24
-
-
-
1.89 0.56 0.80 1.37 1.25
1.00 1.37 - 2.14 0.39 1.52
-
There is a contemporaneous
0.47 0.95 6.27 2.04 0.07
0.34 1.15 0.06 0.06 6.77
0.73 0.22 0.37 0.25 1.42
Significance
Residual variances
Chi-square test M(l)
++* * ++* . . +-. . . . . . . . .
184.41
. -
0.298E - 01 O.l67E+OO O.l05E+OO 0.378E - 01 0.847E-03
* +-f-s +. . . . +. . . . . . . . . .
0.271E-01 O.l57E+OO 0.966E - 01 0.367E - 01 0.819E-03
38.24
0.259E - Oi 0.144E + 00 0.762E - 01 0.345E - 01 0.806E - 03
51.08
0.241 E 0.142E + 0.570E 0.330E 0.760E -
01 00 01 01 03
52.25
0.220E - 03 O.l36E+OO 0.545E-01 0.255E-01 0.717E- 03
54.34
0.206E - 01 O.l35E+OO 0.522E-01 0.219E - 01 0.703E - 03
31.45
O.l99E-01 O.l30E+OO 0.325E - 01 0.107E - 01 0.590E - 03
134.37
O.l90E-01 O.l21E+OO 0.276E - 01 0.993E - 02 0.509E - 03
45.55
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p ~11between the registrations depend_nl,
reduction index of building materials has a lead OF 5
of rigids and private cars
onths on the registration of rigids
R. M.J. Hews and J. H.J. M. Bronckers
68
Table 3 Exact maximum likelihood estimates
and standard
deviations
/ Forecasting
for the Svariate
the Dutch truck market
ARMA-model
(6).
Parameter
@,(L 1)
w9 2)
@P,G 2)
@,(2? 3)
@*(3,3)
@,(4,4)
value (st. dev)
0.35 (0.06)
0.29 (0.03)
0.28 (0.08)
0.50 (0.13)
0.29 (0.08)
0.43 (0.07)
Parameter
@#,a
@,(3* 3)
@2(1,3)
@A29 2)
@dL2)
@5(4,1)
value (st. dev)
0.21 (0.09)
0.24 (0.09)
0.17 (0.06)
0.29 (0.06)
(0.03)
0.09 (0.02)
Parameter
@,,(3,3)
@,2(4r 4)
@24(3, 3)
@,(5,5)
@,2(5,5)
Q,(l)
%(2)
value (st. dev)
0.54 (0.08)
0.79 (0.04)
0.24 (0.08)
0.48 (0.08)
(0.08)
1.78 (0.41)
1.09 (0.38)
-
0.07
0.54
There is no significant correlation between industrial production and the other series. This is caused by the transformation (1 - I?)(1 - B12)x,. Cross correlating the original series, however, exhibited significant interactions.
INDEX BUILDING
REGISTRATIONS OF RIGIDS
REGISTRATIONS
REGIST-RATIONS
R. M.J. Heuts and J._H.J. M. Bronckers
00
0
/ Foreswing
0
d’
0
0
[he DuicA :rwk
00
X
0
0
rt
d
%l
s d I
0
murket
R. M. J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch tnrck market
70
The production index of building materials can be seen as a leading indicator for the Dutch truck market development: an increase of 1% of this index is followed by an increase of 0.09% of the new registrations of rigids after 5 months. At the same time the registration volume of artics and passenger cars will incline. If the volume of newly registered rigids grows by 1% at the same time the volume of newly registered artics will rise by 0.29% and 4 months later by 0.07% and that of passenger cars by 0.17% simultaneously. An increase of 1% of the artics registration volume implies also a coincident increase of new passenger car registrations of 0.50%. In figure 3 the interactions are illustrated by a flow chart.
L EtW x--_-m
HBL T-WINTERS
----
MUL TIVWIFITE
UNIVRRIFITE
Fig. 4a. Forecasting performance for the series of newly ARIMA-model as well as for the I-Jolt-Winters-model.
registered
rigids,
both
for the univariate
and
multivariate
R. M. J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
71
4. Forecasting performance The forecasts of the univariate and multivariate models were also judged against Holt-Winters models as a naive comparison of the more sophisticated models. We use.4 the Holt-Winters exponential smoothing method as described in the SAS/ETS manual with an additive linear trend and a multiplicative seasonal factor. The three smoothing parameters were obtained by a grid search procedure. To test the forecasting performance, several accuracy measures can be defined. We examined the following four measures of forecasting accuracy: (root) mean square error (( R)M.SE), mean absolute percentage error (MME), mean percentage error (MPE), and mean absolute deviation (MAD).
YERV
,?BWL
U TE PERCEN SERIES
1
Q- I o
I
12
3
I
4
I
5
x
YBL T-WINTERS
UNI I/RRIF)TE ----
Fig. 4a (continued).
MU1 TI VRRIflTE
ERRCW
I
I
F
I
7
L EQD x
TRGE
I
I
8
9
I
IO
I
11
I
,’2
4
13
R. M.J. Hews and J. H. J. M. Bronckers / Forecasting the Dutch truck market
72
MAPE is less sensitive to large forecast errors or to measurement errors in the actual data. The MPE measure of overall bias in the forecast errors. A major disadvantage of the MPE is that its value may be quite small even though individual errors may be quite large. To calculate the individual forecasting errors, twelve additional observations were available. The computational procedure was as follows. From forecasting origin 163 (July 1981), forecasts up to twelve months ahead were generated. After that this forecasting origin was shifted six times, and each time forecasts up to twelve months ahead were calculated. The final origin was observation number 168. In doing so, we obtained for each lead time six forecasts and forecasting errors. From these we computed the MAPE, RMSE, MPE, MAD for each lead time. This gives us a truthworthy indication of forecasting accuracy for each period ahead. In figs. 4a and 4b these results are shown for the main series: the rigids and artics series. The post-sample accuracy measures for these series are also given in tabular form in tables 5 and 6. is a
I'?E?V RBSL7iUTE SERIES
LlEVIRTIm' I
I
0
o/
I
I
0123456789
1
I
I
I
L ERD m--_-m
HOL T-WINTERS UNIVFIRIf7TE
_---
Fig. da (continued).
MLIL TI VFIRIFITE
I
I
I
1
10
I
11
I
?2
1
13
R. M.J. Hem
MEQN
ROOT
and J. H.J. M. Bronckers / Forecasting the Dutch truck market
SWARE SERIFS
73
ERRLTR
1
-
~
-.-_ --r-----T---11 2
-7---r--l--
3
4
5
6
7
8
9
11
10
L EAD XL----+
I
12
I
13
HO1 T -WINTFRS UNI VRRIRTE
----
MU1 TIVRRIRTE
Fig. 4a (continued). Table 5 Forecasting Forecast periods ahead
performance Univariate MAPE
1
15.2 14.8 18.6 19.6 21.1 15.7 14.7 Id.1 21.3 21.0 22.4 29.5
ARIM. MPE
2 3 4 5 6 7 8 9 10 11 12
of rigids series (series1).
-
Multivariate MAD
ML4PE
MPE
MAD
Holt-Winters RMSE
M.4PE
MPE
M4D
RMSE
--
..1’4 0
59.U
lV.8 - 12.7 - 14.7 - 16.1 - 11.0 - Cz. 7 - 12.3 -- 20.0 - 19.8 - 21.2 - 28.3
62.0 83.5 86 2 92.5 74.8 70.2 HI.8 89.8 88.8 93.5 113.3
-
RMSE
ARIMA
79.4 70.6 92.2 93.0 99.6 77.2 75.3 89.6 98.5 97.6 102.7 126.1
14.7 11.2 12.0 15.9 16.3 9.7 10.0 18.1 18.4 17.3 20.1 28.0
- 14.7 - 9.2 - 5.4 - 7.4 -9.1 - 4.0 - 2.6 - 7.5 - 15.2 - 15.6 - 1U.l - 26.0
62.9 4u.o 53.8 73.8 74.5 4u.u 50.2 84.9 79.2 73.7 84.3 109.4
69.0 50.6 69.9 90.3 uu.l 57.1 64.3 93.4 84.5 79.9 93.u 12O.U
13.3 13.5 16.6 17.9 20.6 18.3 19.0 17.8 14.3 15.7 21.4 18.8
- 1.6 4.0 9.6 10.u 11.0 18.3 19.0 17.8 14.3 15.7 21.4 1u.n
84.7 90.7 127.7 135.0 150.5 146.5 150.7 135.5 99.7 111.8 167.0 141.3
108.3 106.2 166.5 160.6 169.2 172.5 179.0 154.u llU.5 137.8 208.5 177.2
Bronckers / Forecasting the Dutch truck market
R.M.J. Hew’s andJ.H.J.M.
74
SERIFS
1
0
I
1
I2
jyc--iw
I
I
I
3
4
5
2
1
6
I
I
7
8
I
9
1
I
IO
11
I
I2
i
13
L ERD
HOLT-WINTERS Uh'IVF)RIRTE
----
MUL TIVRRIGTE
Fig. 4b. Forecasting performance for the series of newly registered artics, both for the univariate ARIMA-model as well as for the Holt-Winters-model.
and multivariate
Table 6 Forecasting performance of artics series (series 2). Forecast periods ahead 1 2 3 4 5 6 7 8 9
10 11 12 -
Univariate ARIMA
Multivariate ART: -2
Holt-Winters
MAPE
MPE
MAD
RMSE
MAPE
MPE
MAD
RMSE
MAPE
MPE
MAD
RMSE
41.7 39.4 35.6 37.9 39.9 26.0 23.2 19.5 142 14.8 14.6 22.0
-22.2 -8.9 0.9 3.2 3.8 26.0 23.2 17.2 11.1 11.2 11.1 -5.8
64.7 67.5 69.8 76.7 79.7 74.5 64.0 52.7 35.5 37.3 36.8 37,5
79.0 82.2 81.2 84.1 g5.0 81.3 71.2 63.2 45.3 47.7 47;l 48.8
30.6 43.3 40.1 41.0 33.0 28.6 25.5 168 15s 13.1 25.1 24.7
7.6 10.7 11.2 13.5 33.0 28.6 21.7 15.4 15.0 13.3 - 5.2 0.2
61.9 91.1 87.4 91.5 92.4 78.3 67.8 42.2 39.1 41.1 43.2 52.8
73.1 98.0 97.7 99.8 95.7 86.3 77.7 52.4 48.5 54.0 55.1 70.7
30.0 33.8 36.0 38.9 45.1 34.1 33.1 31.2 27.6 26.8 28.2 21.5
-8.1 0.2 8.1 11.4 13.5 34.1 33.1 31.2 27.6 26.8 28.2 18.6
51.0 64.0 77.8 88.3 99.3 98.0 91.8 81.2 67.2 67.2 67.0 47.7
70.4 78.3 89.8 98.3 100.6 102.3 97.0 85.0 71.7 79.2 71.6 53.9
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
SERIES
%I
I
I
I
1
I
I
75
2
I
0123456789
I
I
I
I
I
10
11
12
13
L EF7D m--_?w
----
ML
T-WINTERS
UNI
L;RRIR
TE
MUL TI VRRIRTE
Fig. 4b (continued).
Rigid series From fig. 4a it can be observed that the overall forecast performance of the multivariate model is clearly better than that of the univariate one for all forecast lead times, whereas the univariate model performs better than Holt-Winters. The negative bias that was experienced in the univariate case, is still present, but is significantly reduced by the multivariate model. As can be seen from the MPE-plot, the Holt-Winters forecasts are positively biased. A rtic
series
In contradiction to the rigid series there is no evident improvement in forecasting accuracy when the multivariate model is applied to the artic series. We experienced that both the univariate and
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
76
IZ~‘EAX RBWL
(I/‘?-E PERCEA’ Tt? YE SERIES
ERR,?/?
2
I
-1
Q-i 0
12
3
4
~---_w
HOL T-WINTERS
----
MUL TI VRRIRTE
5
1
I
I
7 6 : EQD
8
9
I
I
I
10
I1
12
1 13
UNI Vk?RIF! TE
Fig. 4b (continued).
multivariate model lead to large errors at short leads and to smaller errors at longer leads. This might be due to the fact that all forecasts started around a turning point of the series. The autoregressive nature of the univariate model leads to extrapolation towards the direction of the most recent observation, which is opposite to the actual development (see graph of artic series). In the multivariate case the artic series is correlated with the rigid series and, indirectly, with the series of production of building materials. Here the autoregressive structure has similar consequences. As only most recent observations are used, the downward trend in the mentioned series cannot account for the upturn of the artic series at short notice. On very short leads (1 and 2) the Holt-Winters model outperformed the more sophisticated models, but clearly worsened at longer term forecasts. For all other series we found slightly better results for the multivariate time series model.
R.M.J. Heuts and J. H.J.M. Bronckers / Forecasting the Dutch truck market
Q-
-I Oi23456
I
m-x
I
I
I
I
I
1
I
8
9
1
20
I
21
I
22
23
HLILT-WINTERS UNIVRRIF)TE
----
MULTI VQRIRTE
Fig. 4b (continued).
5. Conclusions
In this article, univariate models for the series of newly registered rigids and artics were extended to a multivariate one, incorporating a number of economic indicators. Merely for reasons of interpretation, we choosed for AR structures in our models. Using a multivariate model extra reduction in residual variance could be obtained. A better forecasting performance was found for the rigid series, but not for the artic series. Holt-Winters models have been used as a naive comparison to the more complex models. Although no method for calculating exact forecast confidence limits is available, Holt-Winters models are easy to use. In general, both the univariate and multivariate ARIMA-models outperformed the Halt-Winters procedure, although slightly better results were found by the latter for the artic series on very short forecast leads.
78
R. M.J. Heuts and J. H.J.M. Bronckers / Forecasting the Dutch truck market
The latter can be blamed on both the choice of the forecast origins and the autoregressive model structure. The origins for the artic series were all positioned in the neighbourhood of a cyclical trough. Apparently, the AR-structure of our model failed to predict the direction of the series as the trend of the most recent observations was extrapolated. This disability to predict turning points was even greater for the multivariate model, as can be seen from fig. 4b. However, for the rigid series uniformly higher forecast accuracy was achieved, since the forecasts were not positioned in the neighbourhood of a turning point. Although more work should be done on this matter, it is felt that in the case of cyclical time series, which economic data often are, an AR-model structure is inferior to an MA-structure in terms of forecasting accuracy in the neighbourhood of turning points. Moreover, this inferiority may grow with the dimension of the model. In general, our results support the view of Armstrong (1984) that complex multivariate models can provide better fits to historical data, but that this superiority does not need to hold for forecasting. This may be due to the fact that multivariate time series models are more sensitive to changes in structure than univariate ones. In case of structural changes, the probability of unnecessary parameters in a complex model is large. In this context, Ledolter and Abraham (1981) and Bunn (1980) show that unnecessary parameters in a model will increase the mean square forecast error. References Armstrong, J.S.. 1984. Forecasting by extrapolation: Conclusions from 25 years of research, Interfaces 14, 52-66.
BOX,G.E.P. and G.M. Jenkins. 1976. Time series analysis: Forecasting anb control. revised ed.. (Holden-Day. San Francisco. CA). Bronckers. J.H.J.M., 1982, The market for heavy commercial vehicles in the Netherlands (in Dutch). Master thesis (Department of Econometrics, Tilburg University, Tilburg). Bronckers, J.H.J.M. 1983, Forecasting the demand for heavy commercial vehicles in Western Europe. Internal report (Sales Department HQ, DAF Trucks, Eindhoven). Bronckers. J.H.J.M. and R.M.J. Heuts, 1984. Forecasting the number of registrations of the Dutch heavy truck market: A multivariate time series analysis, Invited paper presented at the workshop Dynamic Models and Time Series Analysis, Erasmus University, Rotterdam. Bunn. D.W., 1979, The synthesis of predictive models in marketing research, Journal of Marketing Research 16. 280-283. Hannan. E.J.. 1969. The identification of vector mixed autoregressive moving average systi:rns, Biometrika 57. 223-225. Hasza. D.P. and W.A. Fuller, 1982, Testing for nonstationary parameter specifications in seasonal time series models. The Annals of Statistics 10, 1209-1216. Hillmer. S.C. and G.C. Tiao, 1979, Likelihood function of stationary multiple autoregressive moving average models, Journal of the American Statistical Association 74. 652-660. Hillmer. S.C.. D.F. Larcker and D.A. Schroeder, 1983, Forecasting accounting data: A multiple time series analysis, Journal of Forecasting 2.389-404. Jenkins. G.M. and A. Alavi. 1981. Some aspects of modelling and forecasting multivariate time series, Journal of Time Series Analysis 2, l-47. Ledoher. J.. 1978. A I iultivariate time series approach to modelling macroeconomic stquences. Empirical Economics 2. 225-243. Ledolter. J. and B. Abraham. 1981, Parsimony and its importance in time series forecasting, Technometrics 23. 411-414. Ljung. GM.. 1982. Testing the adequacy of a fitted autoregressive moving average model, American Statistical Association, Proceedings of the Business and Economic Statistics Section, 200-203. Ljung. GM. and G.E.P. Box. 1978, On a measure of lack of fit in time series models, Biometrika 66. 265-270. Maravall. A.. 1981. A note on identification of multivariate time series models, Journal of Econometrics 16. 237-247. Raise. T. and D. Tjostheim, 1984. Theory and practice of multivariate AKMA forecasting, Journal of Forecasting 3. 209-317. Schwert. G.W.. 1979. Tests of causality: The message in the innovations, in: K. Brunner and A.H. Meltxer. eds.. Carnegie-Rochester Conference Series on Public Policy, Vol. 10 (North-Holland, Amsterdam). Sims. C.. 1977, Comment on relationships - and the lack theory - between economic time series, with special reference to money and interest rates. Journal of the American Statistical Association 72, 23-24. Tiao, G.C. and G.E.P. Box, 1981, Model ing multiple time series with applications, Journal of the American Statistical Association 76, 602-616.
R. M. J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
79
Tiao, G.C. and G.E.P. Box, M.R. Grupe, G.D. Hudak, W.R. Bell and I. Chang, 1980, The Wisconsin multiple time series program, User’s guide to the computer program (Madison, WI). Tigelaar, H.H., 1982, Identification and informative sample size, Mathematical Centre tracts no. 147 (Mathematical Centre. Amsterdam). Tjestheim, D. and J.E. Paulsen, 1983, Bias of some commonly used time series estimates, Biometrika 70.389-400. Umashankar, S. and J. Ledolter, 1983, Forecasting with diagonal multiple time series models: An extension of multivariate models, Journal of Marketing Research 20, 58-63. Zellner, A., 1979, Causality and econometrics, in: K. Brunner and A.H. Meltzer, eds., Carnegie-Rochester Conference Series on Public Policy, Vol. 10 (North-Holland, Amsterdam).
Biogwphy: R.M.J. HEUTS obtained his Ph.D. in econometrics at Tilburg University and is currently
assistant professor in management science at that university. His current research interests include time series analysis, portfolio analysis and production-inventory modelling. He has published in the Australian Journal of Statistics, Zeitschrift ftir Operations Research, Computational Statistics Quarterly and others. J.H.J.M. BRONCKERS obtained his M.Sc. degree in econometrics at Tiiburg University and is currently manager of the Quantitative Market Analysis Department of DAF Trucks in Eindhoven, The Netherlands.
4 (1988) 57-79
57
R.M.J. HEUTS Tilburg Urkersity,
5000 LE Tilburg, The Netherlands
J.H.J.M. BRONCKERS DA F Trucks Limited, 5602 BB Eindhoven, The Netheriands
A bstract:
In this paper it is investigated whether multivariate time series models can improve the forecasting performance of Holt-Winters models and univariate ‘naive’ ARIMA-models. which are easier to construct, but do not take into account lead-lag relationships. This research deals with the Dutch truck market performance in the light of overall economic developments. A S-variate model is built, containing two truck sales series and three economic indicators. It is found that a substantial reduction in residual variance can be found by using a multivariate model. In the case of one truck series (rigids) this leads to uniformly better forecasts. For the other output series (artics), no improvement in forecasting accuracy was found. Holt-Winters models do not seem to improve any of the sophisticated models. Uni- and multivariate ARIMA-models, ment, Forecasting performance.
Holt-Winters,
Truck market modelling. Align-
1. Introduction AS with most durables, industrial goods are highly sensitive to economic activity. This is especially the case with truck sales. Commercial vehicles, unlike cars which form part of final demand, are principally investment goods. Demand for new commercial vehicles shows much of the volatility traditionally associated with demand for investment goods. As for trucks the market is evidently full-grown. New demand is mostly demand for replacement. Together with the economy, the total road goods transport changes and therefore the number of newly sold vehi4es. Evidently. replacement of vehicles may be advanced or postponed in phase with business cycles. Therefore, actual scrapping mostly deviates from latent scrapping. the latter being determined simply by the age structure of the goods vehicle fleet. oreover. the relationship between forecasted total haulage requirement and vehicles fleet in operation (and thus newly sold trucks) is complicated by trends in vehicle carrying capacity and operating efficiency. Often this is represented by a function of time with downward slope. A number empirical studies of the heavy truck ma&et of various European countries have been performed in ronckers (1982) and Bronckers (1983). In &is article we investigate the relationship be en the registrations of heavy rigids and utch economic indicators: newly articulate trucks (artics) in the ~N:cthc”r 0169-2070/M/$3.50
I_ 1988. Elsevier Science Publishers
B.V. (North-Holland)
R. M. J. Heuts and J. H. J. M. Bronckers
58
/ tbrecasting
the Dutch truck market
registered passengers curs in the Netherlands, the production index of building materials and the index of industrial production. The next section of this article reviews some theoretical aspects of multivariate time series analysis. In section 3 empirical results are given, both for univariate and multivariate models. Forecasting performance of all models is discussed in section 4. Section 5 contains a number of conclusive remarks. For details we refer to a technical report [Bronckers and Heuts (1984)j.
2. Multivariate ARIMA-models An important class of discrete univariate models is the class of so-called ARIMA-models which are extensively analysed by. e.g., Box and Jenkins (1976). However, when m series { xl,[ ). 1 -yz., >* * . * ’ { -xn,.I > are considered simultaneously, it is necessary to allow for dynamic relationships that may exist between the series, for possibilities of feedback and for correlations between the shocks affecting the series. A useful class of models is obtained by direct generalization of the univariate models to
0) where @JB)=I-@,B... - QpBP, O,(B) = I - 0, B.. . pqB4 are matrix polynomials in the backshift operator B, the Q’s and O’s are m X m matrices, X, = X, - p is the vector of deviations from some origin p, which is the vector of means if the series are weakly stationary, and Ai = (a,,,, . . . , am.,) is a sequence of vector-valued shocks, distributed as multivariate normal N(0, C,), that allows for contemporaneous correlation between the elements a,,,, . . . . a,,, I, but which are mutually uncorrelated at non-simultaneous times. We shall assume that the zeros of the determinantal polynomials PP(t)l and IO,(=)1 are on or outside the unit circle. In order to ensure the existence of a unique weakly stationary solution of (1) we must restrict the range of (!?I+,..., $) to 2$(m), where S,(m) :=
(@*,...,@#et
@p(z)#O,(~I
=l]
[see Tigelaar (1982)]. To make sure that Xl can appropriately be written in the infinite AR- and MA-form. we suppose that the zeros of the determinantal polynomials 1@Jz) 1 and 1@J z) 1 are outside the unit circle. It should be noted that 2, is a vector of stationary series, obtained by appropriately transforming and differencing the original time series. While multiple time series models are naturally more complicated to handle and experience in their use is more limited, they provide a potential means of improving on results from univariate models. For example, information about 1,, in addition to that contained in its own past, may be available from other related series ( g2., }, . . . , (Z,,, , }. When this is so, improved forecasts should be possible. Parallelling the univariate case [see, e.g., Box (1978j], when the model is invertible, the forecasting procedure runs as follows: q+,=
j(I) + q(l), n”),f
,
,+,-,’
where
Lp=
(2)
R. M.J. Hews and J. H.J. M. Bt-onckers / Forecasting the Dutch truck market
59
The II;‘) ‘s and q,‘s are m X m matrices obtained from the relations Qb(B)rk(B)=O,(B),
4(B)=I+\k,B+qrzB2+...,
@q(B)IT(B)=@p(B),
Kl(B)=I-fl,B-&B2-...,
and
(3) (4)
In (2) &I)
is the conditional expectation of z,+,
it(l) =E(rS,+,I&, u,(l) =
ff+...>,
2r+,-il(r),I=
and
l,...
is the vector of forecast errors made at origin C. The error vector U,(I) is normally distributed with zero mean and covariance matrix I-1
Cov(U,(I)) =
c
i=O
?qz,?&‘.
(5)
With respect to the identifiability conditions for the parameters we refer to Hannan (1969) and Tigelaar (1982). It should be noted that the identiiiability of Hannan is based on an infinite sample size, opposite to that of Tigelaar. The objective of the model building strategy is to make an initial guess of the structure of a multioariate stochastic model. This so-called identification procedure will be based on various correlation and partial correlation functions. During the last years two methods of identification have been proposed. The first method is based on two sets of statistics (:h= cross correlation and partial cross correlation matrices) calculated from the stationary series x’,, obtained by appropriately transforming and differencing the original time series (non-prewhitzning method). The second one is based on the same statistics calculated from the residuals after fitting univariate models to each time series separately (prewhitening method). Two objections can be raised against the correlating of non-prewhitening series. Sample cross correlations for two autocorrelated processes (1) can suggest two-sided influence or feedback, although the underlying relationships are one-sided; (2) can be heavily correlated, thus showing spurious patterns. owever, Jenkins and Alavi (1981) show that multivariate models containing an autoregressive ture can cause difficulties in the prewhitening procedure. One runs the risk especially of overparameterization. Maravall (1981) also points to the problem of overparameterization as a drawback of the two-sided prewhitening procedure. Zellner (1979), Sims (1977) and Schwert (1979) suspect the filtering of the data to bias the results of the analysis towards seiies independence. .lenk+llJ and Alavi (1981) therefore recommend two ways of identification: cross correlating ihe prewhitened series as well as the original series. as used for this article, is based on identification without The computer package WMTS, whit prewhitening, as proposed by Tiao and ox (1981). The aim of the modelling approach of 80x and Tiao is to employ statistics which can be readily calculated and allow to select a subclass of models for further consideration. Specific statistics in I ox-Tiao concept are
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
cross correlation matrices of the original data or the matrices of appropriately differenced data. The sample cross correlations $,,(I) are estimates of the theoretical values p,,( 1)‘s. They are particularly useful in detecting low order moving average models since p, ,( !) = 0 for I > q, where q is the order of the MA-model. (ii) Estimates of the partial cross correlation matrices obtained by fitting stepwise autoregressive models of increasing order. Sample estimates of the partial cross correlation matrices P(I) are useful in identifying the order of an autoregressive model. In the WMTS computer package, the estimates P( 1) are obtained by fitting autoregressive models of I = 1, 2,. . . successively. (iii) A likelihood ratio statistic, M(I), is used to help determine the order of the ar;toregressive part of the model. It corresponds to testing the null hypothesis @, = 0 against the alternative @,z 0, when an autoregressive model of order I is fitted. Under the null-hypothesis M(1) follows a x2 distribution with ,I’ degrees of freedom. Diagnostic checking of the residual series { a,}, where (9
Sample
a, = &&&...
4$&+e,a,_,
+ . . . +&&.
should be performed by plotting the standardized residual series against time, as well as examining cross correlation matrices of the residuals { 2, } against their two standard error confidence limits. The problem of estimating the parameters in a multivariate time series model is more difficult than in the univariate case. However, Hillmer and Trao (1979) have developed procedures which give a close approximation to the exact likelihood estimates in multivariate ARIMA-models. For more details we refer to Tiao et al. (1989).
3. I. Uniuariare models In this section we examine the relationship between LM sales of new heavy rigids and artics in the Netherlands and three economic indicators: new passen,zr car sales, the production of building materials and total industrial production. The demand cp&private cars is a generally accepted leading economic indicator in most Western countries. i*urthermcre, from publications of the Dutch Central Bureau of Statistics it can be seen that during the last years approximately 20% of the yearly road goods transport exists of building materials. Except for this sector other categories are important, such as food and final products. Indicators for these categories ilily be the prr luction index of food and beverages, retail sales, private consumption and industrial production. From these the latter is taken into our analysis. In fig. 1 the following registration data are shown: newly registered rigids ( _qr) and artics (x2.,) above 9 tons G.V.W. (Gross Vehicle Weight) and newly registered private cars (s,.,). Also the plots of the production of building materials (x4.,) and of industrial production ( xs,J in the Netherlands are shown. Together with the raw data, their 12 monthly moving averages are plotted. Each series consists of 168 observations, ranging from Janua.y 1968 up to and including december 1982. In the data series x,., an outlier due to a registration error was detected for May 1971, which was adjusted. The behaviour of the sample ACF PACF suggests a nu arameter estimation i iagnostic checking
R. M.J. Hem and J. H. J M. Bronckers / Forecasting the Dutch truck market
61
Table 1 Univariate models (PACK-program). Series
Fitted models
Residual
Box-Ljung’s Xl00
variance
diagnostic Q’=’ (d.o.f.)
-2 0,
1 2 3 4 5
0.5227 E-l (0.04) (0.07) 0.2638 (l-O.22 B -0.32 B2)(ln x2,, -5.16) = t2,, (0.07) (0.07) (0.09) (1 - 012 B -0.26 B’)(l-0.54 B” -0.37 B24)(ln x3,, -10.51) = Q,,, 0.4010 E-l (0.07) (0.08) \CI+s) (0.08) (0.40) 0.1285 E-l (1 - ti.&! B)( 1 - 0.84 B*‘)(ln x~,~- 4.69) = Z4,, (0.10 (0.07) (0.04) 0.7005 E-3 VI712 111Ys,, = ‘; -0.49 B)(l -0.59 B”) = P,,, (0.07) (0.07) -
(1 - 0.51 B)(ln x],~ - 6.25) = Q,,,
NEH
REGISTRRTIONS
RIGIDS
L 3
TONS
17.39 (17)
24.1 I
16.76 (16)
16.76
17.73 (14)
64.27
19.50 (16)
66.30
16.66 (16)
35.29
GVW
NETHERLRNDS
1
67
6
69
70
71.
72
73
74
Fig. 1. Graphs of series 1 to series 5 and their 12 monthly
75
76
i
77
moving average.
I
78
.’
I
79
80
1
81
82
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
NS FIRTICS
t
9 TOhS
GVW
NETtiEALRNUS
I
67
88
69
70
1
1
71
72
I
73
I
74
I
I
75
76
I
I
77
78
I
79
I
80
1
81
92
TINE Fig. 1 (continued).
els in table 1. For series xs .t we used a testing procedure of Hasza and Fuller (1982) for nonstationary parameter specifications for that particular seasonal model. However, it can be demonstrated that the testing procedure of Hasza and Fuller has some drawbacks [see Bronckers and Heuts (1984)]. For diagnostic testing Ljung and Box (1978) developed the following test statistic: Q=n(n+2)
E
(n-k)-‘i$
k=l
with n the number of observations and m the number of autocorrelations use . Simulation studies by Ljung and Box (1978) and Ljung (1982) suggest that the ower of the diagnostic test increases for smaller values of m when the alternative is a low order model. This motivated our choice of m = 18. _3’.*.
rrltirariate mode/s
One way to detect a non-stationary latent variable is to difference the time series to the same degree as in the univariate analysis. The approach suggested by Hillmer et al. (1983) is to fit a model
R.M.J. Heuts and J.H.J. M. Bronckers / Forecasting the Dutch truck market
BILE
63
REGISTRRTIBNS
NETHERLANDS O-
o
68
69
70
71
n-7.71 72
1
73
74
75
76
77
78
I
79
I
80
I
81
92
TIPIE Fig. 1 (continued).
This will be the case if two independent trendy series are cross-correlated. However, since only series 5 is non-stationary, this risk does not exist in this application. Cross-correlating the five undifferenced series showed several interactions. However, near unit roots appeared in the diagonal polynomial of series 5. This forced us to take a regular and a seasonal difference for th;c eU.80-+ps “IIIC. . This is in accordance with the result in table 1. Because these differences obviously influence the interactions of series 5 with the other series, we re-estimated the cross correlations. The most relevant ones can be found in fig. 2. When fitting multivariate stochastic models to two or more time series, it may be necessary to shift a series xl,, in time by d time units to produce a time series x~,,+~, before relating it to another . . time series x~,~. arameter d is called the alignment parameter and the resulting model related to the . . time series ~i,~+~ and x2.1 is called an aligned model. It is indicated that there a contemporaneous dependencies, which cannot be modelled with help from the W package. In order to take these effects into account, we applied an alignment procedure. Usually, alignment is applied to center the cross-correlation function around lag zero in order to obtain a more parsimonious representation of the model. In this article, it is Ksed to mode! explicitly correlatioils at simultaneous time, which is normally acco information was noise series, being a consequence of model fitting. This e forecasti oses CTSta
R. M.J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
64
WERRGE
DAILY
MRTERXQLS
67
68
69
70
71
72
PRtZDUCTlffN
,ETC.
73
IN
74
THE
75
(3F BUILDING NETiiERLANDS
76
77
78
79
80
81
82
Fig. 1 (continued).
The original sample cross correlation function suggested contemporaneous and led to the following alignment: +t, X2.t and X3.v *
x2.t
=
x2,t+1’
-G.t
=
X3,t+2
dependencies
between
7
efore entering the forecasting stage, the model should be re-aligned. According to the ideas in section 1 the stepwise autoregression up to order 6 (nonseasonal) and up to seasonal order 2 (period 12) have been fitted (see table 2). From this the following model type is suggested:
(6) with
4 = (qt,
X&v $3
qt,
+.tY.
The (i, j)-th
element from the respective matrices will be tween brackets
R. M. J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
03’ r?l
I
I
67
1
68
I
69
I
70
I
71
I
72
I
I
73
74
1
75
1
76
I
1
77
78
i
79
I
80
65
I
81
82
TIME Fig. 1 (continued).
of the sample variance-covariance correlation matrix k a.
2, =
ii,=
0.284 0.102 -0.661* -0.146 -0.820
* 10-r * 10-l 1o-4 * 1O-2 * 1o-4
.
0.177 -0.300 * 1o-2 0.348 * 1o-2
1 .O@O 0.144 1 .OOO 0.002 - 0.033 - 0.025 0.076 - 0.018 0.127
he multivariate
matrix t,,
which is given together with its corresponding
0.471 * 10-l -0.103 * 1o-2 0.536 * 1O-3
0.120 * 10-l 0.615 * 1O-3
sample
0.735 * 1O-3
1 JO0 0.091
0.207
1.000
model gives an extra reduciion of residual variance in comparison ) this extra reduction i (rigids) of 34.‘&. For s it is shown that the est’ eters in model (6) are i
with the
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
66
c. c. I5
C. C. F.
SERIES
SERIES
1 - SERIES
2
- SERIES
Fig. 2. Plots of relevant cross-correlation
2
3
C. 6. F.
C. C. F.
SERIES
SERIES
I
- SEBIES
4 - SERIES
3
1
functions.
The significant interactions in the multivariate time series model can be interpreted as follows:
R. M. J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
67
Table 2 Summary results of the stepwise autoregression for the 5-variate model. (Plus sign for values greater than 2. minus sign for values less than 2, dot for in between values.) Lag
Standardized partial AR coefficients
1
5.76 0.98 1.51 0.82 - 0.65
8.58 4.01 0.20 1.17 0.55
- 0.21 4.12 2.59 0.86 0.34
1.58 0.44 - 2.23 1.02 0.42
-
2
0.02 1.56 0.07 0.73 - 1.79
-1.44 2.52 - 0.09 1.06 0.77
2.93 0.01 2.88 0.31 - 0.47
2.03 0.40 0.23 - 1.59 - 0.76
0.13 0.00 1.59 - 0.31 - 0.21
3
- 0.75 0.89 - 0.39 0.83 - 1.42
-1.00 2.30 0.32 0.69 - 0.20
- 1.78 - 1.79 0.71 - 0.34 -0.12
- 0.96 - 0.38 5.67 - 2.24 0.05
0.39 1.37 - 1.33 - 1.52 - 0.28
2.77 0.04 - 0.51 0.86 - 0.80
0.76 0.15 0.34 -0.19 1.70
0.89 - 0.91 - 2.59 0.52 - 1.85
-
-
1.40 - 0.02 0.63 4.65 1.62
2.30 1.90 - 1.09 0.38 1.24
- 1.55 0.:7 0.93 -0.18 - 0.48
- 1.66 1.44 1.50 3.82 - 2.04
- 1.54 - 0.40 1.53 - 0.15 0.?4
1.72 0.38 - 0.99 1.46 0.30
0.98 - 0.05 -1.36 - 1.54 - 0.57
2.00 -0.12 - 1.60 - 0.32 1.08
0.47 - 0.43 - 0.55 4.14 0.71
- 1.11 -0.51 0.29 - 0.55 - 0.86
0.99 0.23 2.06 2.15 2.08
0.70 0.92 8.00 - 0.74 - 0.16
0.08 1.81 - 1.53 11.44 - 0.53
- 0.61 0.57 - 0.16 1.22 - 3.76
1.21 0.28 0.19 0.20 1.91
- 0.74 - 0.61 3.91 - 0.73 - 1.17
0.74 1.73 - 1.33 2.93 - 0.49
- 1.37 - 1.08 0.91 1.01 - 3.54
4
5
6
12
24
-
-
-
1.89 0.56 0.80 1.37 1.25
1.00 1.37 - 2.14 0.39 1.52
-
There is a contemporaneous
0.47 0.95 6.27 2.04 0.07
0.34 1.15 0.06 0.06 6.77
0.73 0.22 0.37 0.25 1.42
Significance
Residual variances
Chi-square test M(l)
++* * ++* . . +-. . . . . . . . .
184.41
. -
0.298E - 01 O.l67E+OO O.l05E+OO 0.378E - 01 0.847E-03
* +-f-s +. . . . +. . . . . . . . . .
0.271E-01 O.l57E+OO 0.966E - 01 0.367E - 01 0.819E-03
38.24
0.259E - Oi 0.144E + 00 0.762E - 01 0.345E - 01 0.806E - 03
51.08
0.241 E 0.142E + 0.570E 0.330E 0.760E -
01 00 01 01 03
52.25
0.220E - 03 O.l36E+OO 0.545E-01 0.255E-01 0.717E- 03
54.34
0.206E - 01 O.l35E+OO 0.522E-01 0.219E - 01 0.703E - 03
31.45
O.l99E-01 O.l30E+OO 0.325E - 01 0.107E - 01 0.590E - 03
134.37
O.l90E-01 O.l21E+OO 0.276E - 01 0.993E - 02 0.509E - 03
45.55
. . . .
. . .
I
.
+.
.
.
.
.
.
.
.
.
+.
.
.
f.
.
.
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.
.
.
-.
.
.
.
.
.
.
.
.
--.
.
.
.
.
.
.
.
.
.
.
.
-
+-
.
-.
.
-.
.
f..
.
.
.
.
.
+-
.
.
.
.
.
.
.
.
.
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+.
.
.
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_+.
.
-.
f.
.
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.
.
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-
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+.
.
-
p ~11between the registrations depend_nl,
reduction index of building materials has a lead OF 5
of rigids and private cars
onths on the registration of rigids
R. M.J. Hews and J. H.J. M. Bronckers
68
Table 3 Exact maximum likelihood estimates
and standard
deviations
/ Forecasting
for the Svariate
the Dutch truck market
ARMA-model
(6).
Parameter
@,(L 1)
w9 2)
@P,G 2)
@,(2? 3)
@*(3,3)
@,(4,4)
value (st. dev)
0.35 (0.06)
0.29 (0.03)
0.28 (0.08)
0.50 (0.13)
0.29 (0.08)
0.43 (0.07)
Parameter
@#,a
@,(3* 3)
@2(1,3)
@A29 2)
@dL2)
@5(4,1)
value (st. dev)
0.21 (0.09)
0.24 (0.09)
0.17 (0.06)
0.29 (0.06)
(0.03)
0.09 (0.02)
Parameter
@,,(3,3)
@,2(4r 4)
@24(3, 3)
@,(5,5)
@,2(5,5)
Q,(l)
%(2)
value (st. dev)
0.54 (0.08)
0.79 (0.04)
0.24 (0.08)
0.48 (0.08)
(0.08)
1.78 (0.41)
1.09 (0.38)
-
0.07
0.54
There is no significant correlation between industrial production and the other series. This is caused by the transformation (1 - I?)(1 - B12)x,. Cross correlating the original series, however, exhibited significant interactions.
INDEX BUILDING
REGISTRATIONS OF RIGIDS
REGISTRATIONS
REGIST-RATIONS
R. M.J. Heuts and J._H.J. M. Bronckers
00
0
/ Foreswing
0
d’
0
0
[he DuicA :rwk
00
X
0
0
rt
d
%l
s d I
0
murket
R. M. J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch tnrck market
70
The production index of building materials can be seen as a leading indicator for the Dutch truck market development: an increase of 1% of this index is followed by an increase of 0.09% of the new registrations of rigids after 5 months. At the same time the registration volume of artics and passenger cars will incline. If the volume of newly registered rigids grows by 1% at the same time the volume of newly registered artics will rise by 0.29% and 4 months later by 0.07% and that of passenger cars by 0.17% simultaneously. An increase of 1% of the artics registration volume implies also a coincident increase of new passenger car registrations of 0.50%. In figure 3 the interactions are illustrated by a flow chart.
L EtW x--_-m
HBL T-WINTERS
----
MUL TIVWIFITE
UNIVRRIFITE
Fig. 4a. Forecasting performance for the series of newly ARIMA-model as well as for the I-Jolt-Winters-model.
registered
rigids,
both
for the univariate
and
multivariate
R. M. J. Heuts and J. H. J. M. Bronckers / Forecasting the Dutch truck market
71
4. Forecasting performance The forecasts of the univariate and multivariate models were also judged against Holt-Winters models as a naive comparison of the more sophisticated models. We use.4 the Holt-Winters exponential smoothing method as described in the SAS/ETS manual with an additive linear trend and a multiplicative seasonal factor. The three smoothing parameters were obtained by a grid search procedure. To test the forecasting performance, several accuracy measures can be defined. We examined the following four measures of forecasting accuracy: (root) mean square error (( R)M.SE), mean absolute percentage error (MME), mean percentage error (MPE), and mean absolute deviation (MAD).
YERV
,?BWL
U TE PERCEN SERIES
1
Q- I o
I
12
3
I
4
I
5
x
YBL T-WINTERS
UNI I/RRIF)TE ----
Fig. 4a (continued).
MU1 TI VRRIflTE
ERRCW
I
I
F
I
7
L EQD x
TRGE
I
I
8
9
I
IO
I
11
I
,’2
4
13
R. M.J. Hews and J. H. J. M. Bronckers / Forecasting the Dutch truck market
72
MAPE is less sensitive to large forecast errors or to measurement errors in the actual data. The MPE measure of overall bias in the forecast errors. A major disadvantage of the MPE is that its value may be quite small even though individual errors may be quite large. To calculate the individual forecasting errors, twelve additional observations were available. The computational procedure was as follows. From forecasting origin 163 (July 1981), forecasts up to twelve months ahead were generated. After that this forecasting origin was shifted six times, and each time forecasts up to twelve months ahead were calculated. The final origin was observation number 168. In doing so, we obtained for each lead time six forecasts and forecasting errors. From these we computed the MAPE, RMSE, MPE, MAD for each lead time. This gives us a truthworthy indication of forecasting accuracy for each period ahead. In figs. 4a and 4b these results are shown for the main series: the rigids and artics series. The post-sample accuracy measures for these series are also given in tabular form in tables 5 and 6. is a
I'?E?V RBSL7iUTE SERIES
LlEVIRTIm' I
I
0
o/
I
I
0123456789
1
I
I
I
L ERD m--_-m
HOL T-WINTERS UNIVFIRIf7TE
_---
Fig. da (continued).
MLIL TI VFIRIFITE
I
I
I
1
10
I
11
I
?2
1
13
R. M.J. Hem
MEQN
ROOT
and J. H.J. M. Bronckers / Forecasting the Dutch truck market
SWARE SERIFS
73
ERRLTR
1
-
~
-.-_ --r-----T---11 2
-7---r--l--
3
4
5
6
7
8
9
11
10
L EAD XL----+
I
12
I
13
HO1 T -WINTFRS UNI VRRIRTE
----
MU1 TIVRRIRTE
Fig. 4a (continued). Table 5 Forecasting Forecast periods ahead
performance Univariate MAPE
1
15.2 14.8 18.6 19.6 21.1 15.7 14.7 Id.1 21.3 21.0 22.4 29.5
ARIM. MPE
2 3 4 5 6 7 8 9 10 11 12
of rigids series (series1).
-
Multivariate MAD
ML4PE
MPE
MAD
Holt-Winters RMSE
M.4PE
MPE
M4D
RMSE
--
..1’4 0
59.U
lV.8 - 12.7 - 14.7 - 16.1 - 11.0 - Cz. 7 - 12.3 -- 20.0 - 19.8 - 21.2 - 28.3
62.0 83.5 86 2 92.5 74.8 70.2 HI.8 89.8 88.8 93.5 113.3
-
RMSE
ARIMA
79.4 70.6 92.2 93.0 99.6 77.2 75.3 89.6 98.5 97.6 102.7 126.1
14.7 11.2 12.0 15.9 16.3 9.7 10.0 18.1 18.4 17.3 20.1 28.0
- 14.7 - 9.2 - 5.4 - 7.4 -9.1 - 4.0 - 2.6 - 7.5 - 15.2 - 15.6 - 1U.l - 26.0
62.9 4u.o 53.8 73.8 74.5 4u.u 50.2 84.9 79.2 73.7 84.3 109.4
69.0 50.6 69.9 90.3 uu.l 57.1 64.3 93.4 84.5 79.9 93.u 12O.U
13.3 13.5 16.6 17.9 20.6 18.3 19.0 17.8 14.3 15.7 21.4 18.8
- 1.6 4.0 9.6 10.u 11.0 18.3 19.0 17.8 14.3 15.7 21.4 1u.n
84.7 90.7 127.7 135.0 150.5 146.5 150.7 135.5 99.7 111.8 167.0 141.3
108.3 106.2 166.5 160.6 169.2 172.5 179.0 154.u llU.5 137.8 208.5 177.2
Bronckers / Forecasting the Dutch truck market
R.M.J. Hew’s andJ.H.J.M.
74
SERIFS
1
0
I
1
I2
jyc--iw
I
I
I
3
4
5
2
1
6
I
I
7
8
I
9
1
I
IO
11
I
I2
i
13
L ERD
HOLT-WINTERS Uh'IVF)RIRTE
----
MUL TIVRRIGTE
Fig. 4b. Forecasting performance for the series of newly registered artics, both for the univariate ARIMA-model as well as for the Holt-Winters-model.
and multivariate
Table 6 Forecasting performance of artics series (series 2). Forecast periods ahead 1 2 3 4 5 6 7 8 9
10 11 12 -
Univariate ARIMA
Multivariate ART: -2
Holt-Winters
MAPE
MPE
MAD
RMSE
MAPE
MPE
MAD
RMSE
MAPE
MPE
MAD
RMSE
41.7 39.4 35.6 37.9 39.9 26.0 23.2 19.5 142 14.8 14.6 22.0
-22.2 -8.9 0.9 3.2 3.8 26.0 23.2 17.2 11.1 11.2 11.1 -5.8
64.7 67.5 69.8 76.7 79.7 74.5 64.0 52.7 35.5 37.3 36.8 37,5
79.0 82.2 81.2 84.1 g5.0 81.3 71.2 63.2 45.3 47.7 47;l 48.8
30.6 43.3 40.1 41.0 33.0 28.6 25.5 168 15s 13.1 25.1 24.7
7.6 10.7 11.2 13.5 33.0 28.6 21.7 15.4 15.0 13.3 - 5.2 0.2
61.9 91.1 87.4 91.5 92.4 78.3 67.8 42.2 39.1 41.1 43.2 52.8
73.1 98.0 97.7 99.8 95.7 86.3 77.7 52.4 48.5 54.0 55.1 70.7
30.0 33.8 36.0 38.9 45.1 34.1 33.1 31.2 27.6 26.8 28.2 21.5
-8.1 0.2 8.1 11.4 13.5 34.1 33.1 31.2 27.6 26.8 28.2 18.6
51.0 64.0 77.8 88.3 99.3 98.0 91.8 81.2 67.2 67.2 67.0 47.7
70.4 78.3 89.8 98.3 100.6 102.3 97.0 85.0 71.7 79.2 71.6 53.9
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
SERIES
%I
I
I
I
1
I
I
75
2
I
0123456789
I
I
I
I
I
10
11
12
13
L EF7D m--_?w
----
ML
T-WINTERS
UNI
L;RRIR
TE
MUL TI VRRIRTE
Fig. 4b (continued).
Rigid series From fig. 4a it can be observed that the overall forecast performance of the multivariate model is clearly better than that of the univariate one for all forecast lead times, whereas the univariate model performs better than Holt-Winters. The negative bias that was experienced in the univariate case, is still present, but is significantly reduced by the multivariate model. As can be seen from the MPE-plot, the Holt-Winters forecasts are positively biased. A rtic
series
In contradiction to the rigid series there is no evident improvement in forecasting accuracy when the multivariate model is applied to the artic series. We experienced that both the univariate and
R. M.J. Heuts and J. H.J. M. Bronckers / Forecasting the Dutch truck market
76
IZ~‘EAX RBWL
(I/‘?-E PERCEA’ Tt? YE SERIES
ERR,?/?
2
I
-1
Q-i 0
12
3
4
~---_w
HOL T-WINTERS
----
MUL TI VRRIRTE
5
1
I
I
7 6 : EQD
8
9
I
I
I
10
I1
12
1 13
UNI Vk?RIF! TE
Fig. 4b (continued).
multivariate model lead to large errors at short leads and to smaller errors at longer leads. This might be due to the fact that all forecasts started around a turning point of the series. The autoregressive nature of the univariate model leads to extrapolation towards the direction of the most recent observation, which is opposite to the actual development (see graph of artic series). In the multivariate case the artic series is correlated with the rigid series and, indirectly, with the series of production of building materials. Here the autoregressive structure has similar consequences. As only most recent observations are used, the downward trend in the mentioned series cannot account for the upturn of the artic series at short notice. On very short leads (1 and 2) the Holt-Winters model outperformed the more sophisticated models, but clearly worsened at longer term forecasts. For all other series we found slightly better results for the multivariate time series model.
R.M.J. Heuts and J. H.J.M. Bronckers / Forecasting the Dutch truck market
Q-
-I Oi23456
I
m-x
I
I
I
I
I
1
I
8
9
1
20
I
21
I
22
23
HLILT-WINTERS UNIVRRIF)TE
----
MULTI VQRIRTE
Fig. 4b (continued).
5. Conclusions
In this article, univariate models for the series of newly registered rigids and artics were extended to a multivariate one, incorporating a number of economic indicators. Merely for reasons of interpretation, we choosed for AR structures in our models. Using a multivariate model extra reduction in residual variance could be obtained. A better forecasting performance was found for the rigid series, but not for the artic series. Holt-Winters models have been used as a naive comparison to the more complex models. Although no method for calculating exact forecast confidence limits is available, Holt-Winters models are easy to use. In general, both the univariate and multivariate ARIMA-models outperformed the Halt-Winters procedure, although slightly better results were found by the latter for the artic series on very short forecast leads.
78
R. M.J. Heuts and J. H.J.M. Bronckers / Forecasting the Dutch truck market
The latter can be blamed on both the choice of the forecast origins and the autoregressive model structure. The origins for the artic series were all positioned in the neighbourhood of a cyclical trough. Apparently, the AR-structure of our model failed to predict the direction of the series as the trend of the most recent observations was extrapolated. This disability to predict turning points was even greater for the multivariate model, as can be seen from fig. 4b. However, for the rigid series uniformly higher forecast accuracy was achieved, since the forecasts were not positioned in the neighbourhood of a turning point. Although more work should be done on this matter, it is felt that in the case of cyclical time series, which economic data often are, an AR-model structure is inferior to an MA-structure in terms of forecasting accuracy in the neighbourhood of turning points. Moreover, this inferiority may grow with the dimension of the model. In general, our results support the view of Armstrong (1984) that complex multivariate models can provide better fits to historical data, but that this superiority does not need to hold for forecasting. This may be due to the fact that multivariate time series models are more sensitive to changes in structure than univariate ones. In case of structural changes, the probability of unnecessary parameters in a complex model is large. In this context, Ledolter and Abraham (1981) and Bunn (1980) show that unnecessary parameters in a model will increase the mean square forecast error. References Armstrong, J.S.. 1984. Forecasting by extrapolation: Conclusions from 25 years of research, Interfaces 14, 52-66.
BOX,G.E.P. and G.M. Jenkins. 1976. Time series analysis: Forecasting anb control. revised ed.. (Holden-Day. San Francisco. CA). Bronckers. J.H.J.M., 1982, The market for heavy commercial vehicles in the Netherlands (in Dutch). Master thesis (Department of Econometrics, Tilburg University, Tilburg). Bronckers, J.H.J.M. 1983, Forecasting the demand for heavy commercial vehicles in Western Europe. Internal report (Sales Department HQ, DAF Trucks, Eindhoven). Bronckers. J.H.J.M. and R.M.J. Heuts, 1984. Forecasting the number of registrations of the Dutch heavy truck market: A multivariate time series analysis, Invited paper presented at the workshop Dynamic Models and Time Series Analysis, Erasmus University, Rotterdam. Bunn. D.W., 1979, The synthesis of predictive models in marketing research, Journal of Marketing Research 16. 280-283. Hannan. E.J.. 1969. The identification of vector mixed autoregressive moving average systi:rns, Biometrika 57. 223-225. Hasza. D.P. and W.A. Fuller, 1982, Testing for nonstationary parameter specifications in seasonal time series models. The Annals of Statistics 10, 1209-1216. Hillmer. S.C. and G.C. Tiao, 1979, Likelihood function of stationary multiple autoregressive moving average models, Journal of the American Statistical Association 74. 652-660. Hillmer. S.C.. D.F. Larcker and D.A. Schroeder, 1983, Forecasting accounting data: A multiple time series analysis, Journal of Forecasting 2.389-404. Jenkins. G.M. and A. Alavi. 1981. Some aspects of modelling and forecasting multivariate time series, Journal of Time Series Analysis 2, l-47. Ledoher. J.. 1978. A I iultivariate time series approach to modelling macroeconomic stquences. Empirical Economics 2. 225-243. Ledolter. J. and B. Abraham. 1981, Parsimony and its importance in time series forecasting, Technometrics 23. 411-414. Ljung. GM.. 1982. Testing the adequacy of a fitted autoregressive moving average model, American Statistical Association, Proceedings of the Business and Economic Statistics Section, 200-203. Ljung. GM. and G.E.P. Box. 1978, On a measure of lack of fit in time series models, Biometrika 66. 265-270. Maravall. A.. 1981. A note on identification of multivariate time series models, Journal of Econometrics 16. 237-247. Raise. T. and D. Tjostheim, 1984. Theory and practice of multivariate AKMA forecasting, Journal of Forecasting 3. 209-317. Schwert. G.W.. 1979. Tests of causality: The message in the innovations, in: K. Brunner and A.H. Meltxer. eds.. Carnegie-Rochester Conference Series on Public Policy, Vol. 10 (North-Holland, Amsterdam). Sims. C.. 1977, Comment on relationships - and the lack theory - between economic time series, with special reference to money and interest rates. Journal of the American Statistical Association 72, 23-24. Tiao, G.C. and G.E.P. Box, 1981, Model ing multiple time series with applications, Journal of the American Statistical Association 76, 602-616.
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Biogwphy: R.M.J. HEUTS obtained his Ph.D. in econometrics at Tilburg University and is currently
assistant professor in management science at that university. His current research interests include time series analysis, portfolio analysis and production-inventory modelling. He has published in the Australian Journal of Statistics, Zeitschrift ftir Operations Research, Computational Statistics Quarterly and others. J.H.J.M. BRONCKERS obtained his M.Sc. degree in econometrics at Tiiburg University and is currently manager of the Quantitative Market Analysis Department of DAF Trucks in Eindhoven, The Netherlands.