1. Empirical Economics (2005) 30:277–308
DOI 10.1007/s00181-005-0241-0
The forecasting ability of a cointegrated VAR
system of the UK tourism demand for France,
Spain and Portugal
Maria M. De Mello1, Kevin S. Nell2
1
CETE* - Centro de Estudos de Economia Industrial do Trabalho e da Empresa, Faculdade de
Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal
(e-mail: mmello@fep.up.pt)
2
´
´
Faculdade de Economia e Gestao, Universidade Catolica, Centro Regional do Porto, Rua
Diogo Botelho, 1327, 4169-005 Porto, Portugal(e-mail: knell@porto.ucp.pt)
First version received: September 2002/Final version received: September 2003
Abstract. This paper uses the vector autoregressive (VAR) methodology as
an alternative to Deaton and Muellbauer’s Almost Ideal Demand System
(AIDS), to establish the long-run relationships between I(1) variables:
tourism shares, tourism prices and UK tourism budget. With appropriate
testing, the deterministic components and sets of exogenous and endogenous
variables of the VAR are established, and Johansen’s rank test is used to
determine the number of cointegrated vectors in the system. The cointegrated
VAR structural form is identified and the long-run structural parameters are
estimated. Theoretical restrictions such as homogeneity and symmetry are
tested and not rejected by the VAR structure. The fully restricted cointegrated
VAR model reveals itself a theoretically consistent and statistically robust
means to analyse the long-run demand behaviour of UK tourists, and an
accurate multi-step forecaster of the destinations’ shares when compared with
unrestricted reduced form and first differenced VARs, or even with the
structural AIDS model.
Key words: Tourism demand, cointegration, equal forecasting accuracy tests
JEL classification: C5, D1
We would like to thank Professors O. O’Donnel, M. Mendes de Oliveira and T. Sinclair, for
helpful discussions and suggestions. We are also grateful to two anonymous referees for helpful
comments. The usual disclaimer applies.
*Research Center supported by Fundacao para a Ciencia e a Tecnologia, Programa de
¸ ˜
ˆ
Financiamento Plurianual through the Programa Operacional Ciencia, Tecnologia e Inovacao
ˆ
¸ ˜
(POCTI), financed by FEDER and Portuguese funds.

2. 278
M. M. De Mello, K. S. Nell
1. Introduction
Witt and Witt (1992, 1995), Sinclair and Stabler (1997) and Song and Witt
(2000), observe that the ‘questionable quality’ of most empirical results in
early tourism demand studies and the poor forecasting performance of their
models, may be linked with the lack of an explicit formulation for dynamics
underlying tourism demand behaviour. With few exceptions, the modelling of
dynamics in tourism research has been confined to single equation error
correction specifications, based on Engle and Granger’s (1987) two-stages
approach (e.g. Kulendran 1996; Kulendran and King 1997; Kim and Song
1998; Vogt and Wittayakorn 1998; Song et al. 2000). These studies have been
regarded as an important step forward in the path to construct more reliable
models to explain tourists’ demand behaviour, which seems to be dynamic in
nature. Yet, a single equation framework does not allow for modelling and
testing other features of tourism demand requiring a multi-equation structure.
In recent studies, Deaton and Muellbauer’s (1980a, b) Almost Ideal Demand System (AIDS) has been used for modelling demand and testing consumer theory restrictions such as homogeneity and symmetry. Syriopoulos
and Sinclair (1993), Papatheodorou (1999) and De Mello et al. (2002) are
examples of applications of this approach to tourism demand contexts.
However, an AIDS specification is constructed within a static framework,
includes an assumed endogenous-exogenous division of variables and usually
employs nonstationary time series for its parameter estimation. When dealing
with nonstationary data, failure to establish cointegration often means the
non-existence of a steady state relationship among the variables. Hence,
estimation results obtained with static AIDS models can be deemed spurious
and statistical inference invalid, if the usual assumption of exogenous
regressors does not hold and/or no cointegrated relationships exist. Thus,
there seems to be a risk involved in the estimation of static systems with
nonstationary data, which regress endogenous variables on assumed exogenous variables, if their statistical validity is not sanctioned with appropriate
testing and cointegration analysis. Given that the number of cointegrated
vectors is unknown and simultaneously determined variables may exist,
empirical analysis must go one step further and specify econometric models
which can be efficiently estimated and validly tested within a system of
equations approach.
The crucial next step towards obtaining consistent parameter estimates
and reliable forecasts of one specific origin tourism demand for several destinations, needs dynamics and a system structure. Few studies in tourism
research specify dynamic systems to model demand. For example, Kulendran
and King (1997), Song and Witt (2000) and Kulendran and Witt (2001) use
unrestricted vector autoregressive systems. Lyssioutou (1999) and De Mello
and Sinclair (2000) use dynamic AIDS systems. None, to the best of our
knowledge, compares the estimates and forecasting performance of a static
system with those of a dynamic VAR system in an application for tourism
demand.
This paper contributes an empirical basis for the validation or otherwise, of estimation, inference and forecasting procedures conducted within
a static AIDS approach. Using De Mello et al.’s (2002) study of UK
tourism demand for France, Spain and Portugal over the period 1969–

3. The forecasting ability of a cointegrated VAR system
279
1997, we compare this study’s structural estimates and forecasting results
with those of a cointegrated structural VAR we construct for the same
countries and data set.
In the process of identifying the structural form of the cointegrated VAR,
we use Sims’ (1980) approach to model the reduced form relationships between destinations’ tourism shares and their determinants; apply Johansen’s
(1988) reduced rank test to establish the existence of cointegrated vectors;
employ techniques included in Pesaran and Shin (2002), Garratt et al. (2000)
and Pesaran et al. (2000) to specify a cointegrated VAR with exogenous I(1)
variables, and exactly-identify the long-run coefficients in accordance with the
theoretical principles underlying Deaton and Muellbauer’s structural model.
The empirical results obtained support the cointegrated structural VAR
(CSV) and the AIDS structural system as statistically robust and theoretically
consistent specifications, producing similar estimates for the long-run
parameters. However, the CSV supplies accurate multi-step ahead dynamic
forecasts for all destination tourism shares, out performing the AIDS,
unrestricted reduced form VAR and unrestricted first differenced VAR
models.
The paper proceeds as follows. Section 2 establishes the integration order
and the appropriate lag-length of the variables in the system, and specifies the
unrestricted VAR for UK tourism demand. Section 3 determines the number
of cointegrated vectors and presents the cointegrated structural VAR estimates. Section 4 presents the analysis of forecast accuracy obtained with
CSV, unrestricted and differenced VAR and the AIDS models. Section 5
concludes. Appendix A gives the full derivation of Deaton and Muellbauer’s
AIDS system and its properties.
2. VAR Modelling of the UK tourism demand
When analysing the existence of long-run relationships among non-stationary series, and there are doubts about the exogenous nature of some
regressors, one appropriate modelling strategy consists of starting by
treating all variables as endogenous within a reduced-form VAR. Next,
exogeneity tests for the set of variables in doubt can be carried out. Once
the endogenous-exogenous division is established, the reduced rank test
can be used to establish the number of cointegrated vectors. Then, estimates of the long-run coefficients can be assessed by imposing exactlyidentifying restrictions to the VAR. After identifying the VAR structural
form, additional restrictions can also be implemented to test its compatibility with specific theories. Following these steps, we start by specifying a
reduced-form VAR of the UK tourism demand for France, Spain and
Portugal, using data on the variables of vector Vt = [WF, WS, WP PF,
PS, PP, E], for the period 1969–1997. WF, WS and WP denote the UK
tourism expenditure shares of France, Spain and Portugal, respectively;
PF, PS and PP denote tourism prices in France, Spain and Portugal; E
stands for the UK real per capita tourism budget. All variables’ definition
and data sources are described in Appendix A, according to the definitions
and data sources used by De Mello et al. (2002).

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Fig. 1. Variables in levels
2.1. Order of integration of the variables included in the VAR
We start by determining whether the time series in Vt are stationary, and the
appropriate lag-length of the VAR.1 Figures 1 and 2 present a set of graphs
showing how the variables levels and first differences evolved over time.
Figure 1 shows the plots of the level variables (WF, WS, WP PF, PS, PP, E),
and Fig. 2 the plots of their first differences (DWF, DWS, DWP DPF, DPS,
DPP, DE).
Some features can be readily spotted from the plots. For instance, the
typical oscillatory movement of the first difference variables seem to indicate
stationarity and hence the presence of a unit root in their levels. The plots also
show that both the price variables level and first difference behave peculiarly
in a sample sub-period that can roughly be placed at 1973–1986. Events
justifying such behaviour can be linked to the 1970’s oil crises, mid 1970’s
Portuguese revolution and first democratic elections in Portugal and Spain,
Spain joining EFTA in 1980, and Portugal and Spain joining the European
Union (EU) in 1986. Some of these events are likely to have produced breaks
in the data which may have repercussions on the conventional unit-root tests
preformed below. For some series, these breaks may not have a strong enough impact to affect the tests results. For others, however, a clear conclusion
based on conventional tests may be difficult. Table 1 shows the unit root test
statistics of Dickey-Fuller (1979, 1981) (DF) and Augmented Dickey-Fuller
(ADF) for the level variables and their first differences, and MacKinnon’s
(1991) critical values at the 5% level. It also shows the Akaike (1973, 1974)
1
All estimations and statistical tests were computed with Pesaran and Pesaran (1997) Microfit 4.0.

5. The forecasting ability of a cointegrated VAR system
281
Fig. 2. Variables in first differences
Information Criterion (AIC) and Schwarz (1978) Bayesian Criterion (SBC)
for the lag-length selection of each test equation.
The tests clearly indicate all level variables as non-stationary and, except
for DPP, all first difference variables as stationary. Hence, according to the
DF and ADF tests, all level variables, except PP, are I(1). Although DPP and
PP would be considered I(0) and I(1) respectively, if Charemza and Deadman’s (1997) 5% critical values or MacKinnon’s (1991) 10% (instead of 5%)
critical values were used, we applied the non-parametric correction of the t
statistic proposed in Phillips and Perron (1988), using White (1980) and
Newey and West (1987) covariance matrix, to remove doubts that could
persist about the integration order of PP and DPP.2 This procedure confirmed
DPP as I(0) and PP as I(1), at the 5% level. Thus, we consider all level
variables in Vt to be I(1).
2.2. Determination of the order of the VAR
The sample size available, does not allow for the VAR lag-length (p) to
exceed two. Hence, we used the AIC and SBC criteria and the adjusted (for
small samples) Likelihood Ratio (LR) test for selecting the VAR order with
2
Conventional unit root tests may not be powerful enough for testing stationarity in series that
present mean shifts over time and uncertainty associated with the points at which such shifts may
have occurred. As this seems to be the case for PP variable, the Phillips-Perron (1988) procedure
was used for it enables computing a covariance matrix that ‘adjusts’ the t statistic which is
compared with the critical value (À2.975). For PP variable, the adjusted t ratio (À2.9039) offers
the same conclusion of non-stationarity as the DF test. For DPP variable however, the adjusted t
ratio (À3.9587) clearly indicates stationarity, contradicting the DF test result.

6. 282
M. M. De Mello, K. S. Nell
Table 1. Unit root DF and ADF tests for variables WF, WS, WP, PF, PS, PP and E
Variable
Test
Statistic
AIC criterion
SBC criterion
Critical value
WF
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
DF
ADF(1)
À2.100
À2.008
À5.163
À3.284
À1.965
À1.756
À5.187
À3.510
À1.738
À1.458
À5.429
À4.827
À1.869
À2.311
À3.435
À3.417
À2.083
À2.428
À3.788
À2.968
À1.418
À2.605
À2.480
À2.245
À2.362
À1.624
À3.696
À2.675
54.21*
50.81
49.71*
46.36
52.09*
48.71
48.08*
44.78
86.03*
81.64
81.49*
78.25
32.98*
32.24
30.59*
28.69
33.50*
31.74
29.78*
27.24
31.74
34.17*
31.81*
29.16
20.51*
18.48
18.07*
16.94
52.87*
48.87
48.42*
44.47
50.76*
46.77
46.72*
42.89
84.70*
79.69
80.20*
76.36
31.64*
30.30
29.23*
26.80
32.17*
29.80
28.48*
25.35
30.41
32.23*
30.51*
27.27
19.18*
16.53
16.77*
15.06
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
À2.971
À2.975
À2.975
À2.980
DWF
WS
DWS
WP
DWP
PF
DPF
PS
DPS
PP
DPP
E
DE
the maximum lag-length permitted. Table 2 presents the selection results. The
LR test rejects order zero, but cannot reject a first order VAR. The SBC
criterion clearly indicates an order one VAR. The AIC criterion indicates p to
be two, but by a very small margin. So, we select a VAR(1). Yet, it is sensible
to confirm this decision by checking for residuals serial correlation in the
equations.
2.3. The unrestricted VAR specification of the UK demand for tourism
The reduced form of a first order unrestricted VAR of UK tourism demand
for France, Spain and Portugal (hereafter denoted PUREVAR) can be
written as:
zt ¼ A0 þ A1 ztÀ1 þ t
ð1Þ
Table 2. AIC and SBC criteria and adjusted LR test for selecting the order of the VAR
Order (p)
AIC
SBC
Adjusted LR test
2
1
0
296.91
296.59
185.90
246.37
269.37
182.01
—
v2 (36) = 37.67 (0.393)
v2 (72) = 366.02 (0.000)

7. The forecasting ability of a cointegrated VAR system
283
where z’t = [WFt , WSt , PPt , PSt , PFt , Et] ; A0 is a (6 Â 1) intercept vector;
A1 is a (6 Â 6) parameter matrix and t is a vector of well behaved disturbances. To avoid perfect collinearity brought about by the share variables
WFt, WSt and WPt, which sum up to unity, one of the share equations is
omitted. The estimation results are invariant whichever equation is excluded
and, by the adding-up property (see Appendix A), all coefficient estimates of
the omitted equation can be retrieved from the coefficient estimates of the
remaining ones. We omit the share equation for Portugal (WPt).
The statistical quality of the PUREVAR is accessed by estimating (1) and
computing relevant diagnostic statistics. Table 3 shows the estimation results
(t ratios in brackets), the AIC and SBC criteria and a set of statistics adjusted R2, F statistic and v2 statistic for diagnostic tests of serial correlation, functional form, error normality and heteroscedasticity (p values in
brackets). There is no evidence of serial correlation in the PUREVAR. Hence,
the lag-length selected seems to be adequate. Yet, the diagnostic tests indicate
problems in the functional form and error normality for some equations,
suggesting that the present specification may need to be improved.
As showed in A.2 of Appendix A, the structure of the AIDS model for UK
tourism demand specifies the shares WF, WS and WP, as the only endogenous variables. Changes in these variables are explained by a set of assumed
exogenous regressors including tourism prices (PF, PS and PP) and the UK
real per capita tourism expenditure (E). In a VAR specification, we are willing
to question the assumed exogeneity of the price variables implicit in the AIDS
structure. However, there seems to be no obvious theoretical or empirical
basis for challenging the multi-stage budgeting process underlying the rationality of an AIDS expenditure share system, which sets variable E as an
exogenous determinant of the demand shares. The reasons are as follows. In a
VAR, all variables are endogenous implying that a bi-directional cause-effect
relationship between them should exist. In a tourism demand context, however, even if it is reasonable to consider that changes in the UK real per capita
expenditure affect the tourism shares of important UK holiday destinations
such as France, Spain or Portugal, it does not seem realistic to expect that
changes in these shares influence the way in which UK consumers allocate
their budgets. Yet, more than a statement may be needed to assist this claim,
and empirical evidence is given to support this line of reasoning.
From the estimation results of Et equation in Table 3, we can infer that
the 99% of this variable’s variations explained by the model lie exclusively on
its own lagged value. No other variable in that equation is individually or
jointly significant. In fact, the F statistic for joint significance of all explanatory variables excluding Et-1 is 0.86. In addition, the estimation results for
WFt and WSt indicate that the lagged value of Et does not affect significantly
the current values of these shares.
To investigate further the link between Wit and Et, we analyse the relationships between the error term of a conditional model for Wit and the
stochastic disturbance of the assumed data generating process (d.g.p.) of Et.
Consider that the ith share equation is
Wit ¼ a0 þ a1 Et þ a2 WitÀ1 þ uit
where i ¼ F ; S; P
ðaÞ
and that Et is a stochastic variable with underlying d.g.p.
Et ¼ b1 EtÀ1 þ et ;
b1 1 and t ! N ð0; r2 Þ
ðbÞ

8. 0.9518
0.6820
À0.0712
0.4620
À0.3195
À0.0071
À0.2630
WFt
(2.10)
(1.44)
(À1.14)
(4.85)
(À2.38)
(À0.79)
(À0.60)
Equations
AIC
SBC
Adjusted R2
F statistic
Serial correlation
Functional form
Normality
Heteroscedasticity
61.24
56.58
0.789
17.88
0.78(0.38)
3.81(0.05)
1.02(0.60)
0.39(0.53)
Section criteria and diagnostic statistics
WFtÀ1
WStÀ1
PPtÀ1
PStÀ1
PFtÀ1
EtÀ1
Intercept
Regressors
58.82
54.15
0.824
22.12
2.48(0.12)
8.66(0.00)
1.56(0.46)
1.19(0.28)
(À1.57)
(À1.48)
(7.05)
(À0.49)
(À1.29)
(À1.23)
(1.57)
31.99
27.33
0.771
16.11
0.73(0.39)
1.85(0.17)
0.56(0.75)
1.33(0.25)
À2.0208
À1.9954
1.2525
À0.1341
À0.4922
À0.0312
1.9533
À0.5617
À0.2932
0.1062
À0.4622
0.3157
0.0000
0.8449
(À1.14)
(À0.57)
(1.56)
(À4.45)
(2.16)
(0.00)
(1.77)
PPt
WSt
Table 3. Estimation results and statistical performance of the unrestricted PUREVAR
(À0.75)
(À0.48)
(1.10)
(2.54)
(À0.55)
(À0.10)
(0.57)
29.77
25.11
0.562
6.79
0.19(0.67)
7.39(0.01)
4.82(0.09)
0.28(0.60)
À1.0503
À0.7034
0.2108
0.7453
À0.2276
À0.0027
0.7627
PSt
(À1.32)
(À0.78)
(3.06)
(À0.01)
(0.17)
(À1.35)
(1.01)
34.10
29.44
0.612
8.09
0.37(0.54)
0.00(0.98)
11.65(0.00)
0.03(0.87)
À1.3501
À0.9772
0.5033
À0.0014
0.0598
À0.0318
1.1671
PFt
(À0.15)
(À0.61)
(À1.62)
(0.14)
(1.09)
(22.40)
(0.59)
18.12
13.45
0.991
511.34
0.58(0.45)
0.18(0.67)
1.66(0.44)
0.21(0.65)
À0.3106
À1.3497
À0.4728
0.0602
0.6837
0.9315
1.1990
Et
284
M. M. De Mello, K. S. Nell

9. The forecasting ability of a cointegrated VAR system
285
If uit and t are uncorrelated we can say that EV(uit, s ) = 0 for all t, s (where
EV stands for expected value, not to be confused with variable E). Then, it is
possible to treat Et as if it was fixed, that is, Et is independent of uit such that
EV(Et,uit) = 0. Hence, Et can be treated as exogenous in terms of (a), and can
be said to Granger-cause Wit. Equation (a) is a conditional model since Wit is
conditional on Et, with Et being determined by the marginal model (b).3
‘‘A variable cannot be exogenous per se’’ (Hendry 1995, p.164). A variable
can only be exogenous with respect to a set of parameters of interest. Hence,
if Et is deemed to be exogenous with respect to parameters aj (j = 0, 1, 2) in
(a), the marginal model (b) can be neglected and the conditional model (a) is
complete and sufficient to sustain valid inference. Hence, knowledge of the
marginal model will not significantly improve the statistical or forecasting
performance of the conditional model. Following this line of reason, we run
regression (a) for the expenditure shares of France, Spain and Portugal and
regression (b) as a representation of the d.g.p. of Et. We retrieve the residual
series of these four regressions, namely uFt , uSt , uPt standing for the residual
series of (a) for, respectively, France, Spain and Portugal, and et standing for
the residuals of (b). We then regress the current and lagged values (up to the
fifth lag) of the residuals uit (i = F, S, P) on the current and lagged (up to the
fifth lag) values of t . The estimation results of all regressions indicate no
significant relationship linking the current or lagged residuals of the conditional models to the current and lagged residuals of the marginal model.
These results can be viewed as an indication that knowledge of the marginal
model does not improve the statistical or forecasting performance of the
conditional (on Et-1) equations for WFt, WSt, PPt, PSt and PFt in the VAR.
To supply further empirical support for our claim that feedback effects
might be absent in the relationships between Et and Wit, (i = F, S), we also
use the causality concept proposed by Granger (1969). The block Granger
non-causality test is a multivariate generalisation of the Granger-causality
test that can be used to establish if one or more variables should or should not
integrate the set of endogenous variables in a VAR. This test uses the LR
statistic to provide a measure of the extent to which lagged values of a set of
variables (say Et), are important in predicting another set of variables (say
Wit), once lagged values of the latter (Wit-1) are included in the model. The
LR statistic for block non-causality of Et, testing the null of zero-value
coefficients for Et-1 in the block equations WFt, WSt, PPt, PSt and PFt, is
v2 (5)=10.578 (third test in Table 4). The null is not rejected at the 5% level
critical value (11.071). Supported by these results, we exclude Et from the set
of endogenous variables in the VAR.
Our quest for a correct specification of a VAR explaining the UK tourism
demand for France, Spain and Portugal must also take under consideration
other aspects. For instance, there is cause to believe that events in the 1970s
(change of political regimes in Portugal and Spain and the oil crises) may
have affected the time path of variables included in the VAR. Additionally,
the integration of Spain and Portugal in the EU in 1986, is likely to have
3
As noted in Harris (1995), if (b) is reformulated as Et ¼ b1 EtÀ1 þ b2 WitÀ1 þ t , EV(Et, uit ) = 0
still holds. But, as past values of Wit now determine Et, Et can only be considered weakly
exogenous in (a). Its current value still causes Wit but not in the Granger sense, since lags of Wit
now determine Et.

10. 286
M. M. De Mello, K. S. Nell
Table 4. LR tests for form specification of the PUREVAR
Model
ML
LR ! v2 (i)
H0: Non-significance
of intercept (INT)
U: WF WS PP PS PF E INT
R: WF WS PP PS PF E
351.68
342.05
v2 (6) = 18.05
C. V. (5%)
Result
Rejected
12.59
H0: Non-significance
of dummy variables D1 D2 D3
U: WF WS PP PS PF E
D1 D2 D3 INT
R: WF WS PP PS PF E INT
385.40
351.68
v2 (18) = 67.43
H0: Block non-causality
of E without dummy variables
U: WF WS PP PS PF E INT
R: WF WS PP PS PF E INT
351.68
346.39
v2 (5) = 10.58
H0: Block non-causality
of E with dummy variables
U: WF WS PP PS PF E
D1 D2 D3 INT
R: WF WS PP PS PF
E D1 D2 D3 INT
Rejected
28.87
Not rejected
11.07
Not rejected
385.40
381.21
v2 (5) = 8.37
11.07
affected the UK tourism demand for its neighbours. To account for the 1970s
events we add dummy D1 (= 1 in 1974–1981, and zero otherwise). To account for the Iberian countries integration in the EU, we add dummy variables D2 (=1 in 1982–1988 and zero otherwise) and D3 (=1 in 1989–1997
and zero otherwise), splitting the integration process into two sub-periods: the
integration period (1982–1988) and post-integration period (1989–1997). The
dummies are assumed to be exogenous.
Table 4 presents the LR tests performed to establish the final form of the
VAR (hereafter ‘WHOLEVAR’). The null hypotheses are: non-significance of
the intercept; non-significance of D1, D2 and D3; block non-causality of Et-1
in the specification without D1, D2 and D3; block non-causality of Et-1 in the
specification with D1, D2 and D3.
The first column of Table 4 presents the null for each test and shows the
variables entering the VAR (time subscripts are omitted for simplicity) under
the ‘‘unrestricted’’ (U), and ‘‘restricted’’ (R) null. In each case, the set of
endogenous variables is separated from the set of deterministic components
and exogenous variables by the symbol ‘’. The second column presents the
maximum value of the likelihood function (ML) for the unrestricted and
restricted alternatives. In the third column, the LR statistic is computed.
Under the null, LR is asymptotically distributed as v2 with degrees of freedom
(i) equal to the number of restrictions. The null is rejected if LR is larger than
the relevant critical value. The LR test for block Granger non-causality of Et
performed on the VAR with the dummies, confirms the results of the similar
test performed on the VAR without the dummies. The LR statistic shows now
the value of 8.37, which is well below the 5% critical value (11.07), further
supporting the null of statistically insignificant coefficients of Et-1 in the block
equations.

11. The forecasting ability of a cointegrated VAR system
287
To evaluate the quality of the WHOLEVAR, Table 5 includes the same
diagnostic tests and selection criteria used for the PUREVAR. The results
show the superior quality of the WHOLEVAR as compared with that of the
PUREVAR.
Although WHOLEVAR is statistically more robust than PUREVAR, the
latter is a general model, while the former is a partial system conditioned on
exogenous variables. We are interested in the economic interpretation of
structural parameters, which is only possible if the underlying structural
model is identified from the reduced-form. The PUREVAR may not be an
ideal means to conduct economic analysis, for an a-theoretical reduced-form
VAR is unlikely to produce results interpretable within the limits of sensible
economic assumptions. Yet, it might be interesting to compare the predictive
accuracy of the reduced-form with that of the structural VAR. Hence, we use
the PUREVAR for forecasting purposes only.
3. Johansen’s reduced rank test
The next step is to determine the number of cointegrated vectors using Johansen’s rank test. Besides establishing the process for cointegration rank
testing, Johansen’s approach (Johansen 1988, 1991, 1995, 1996; Johansen and
Juselius 1990, 1992) provides a general framework for identification, estimation and hypothesis testing in cointegrated systems. Yet, Johansen’s
‘empirical process’ to exactly-identify the long-run coefficients may not always be adequate, particularly in contexts where theory provides strong,
sensible and testable restrictions (Pesaran and Shin 2002). In these cases, the
cointegrated vectors must be subject to identifying restrictions suggested by
theory and relevant a priori information, rather than to some normalisation
process that does not consider the theoretical and empirical framework within
which the phenomenon evolves. This is particularly important in a tourism
demand context involving a system of equations, which regress tourism shares
on destination prices, and a per capita tourism budget. In this case, the
number of long-run relationships theory predicts is the number of share
equations in the system. Hence, if theory is correct, the cointegration tests
involving variables WFt, WSt, PPt, PSt, PFt and Et, should indicate that the
VAR relevant long-run relationships are those established by the share
Table 5. AIC and SBC selection criteria and diagnostic tests for the WHOLEVAR
Selection criteria
and diagnostic tests
Equations
WFt
AIC
SBC
Adjusted R2
F statistic
Serial correlation
Functional form
Normality
Heteroscedasticity
WSt
PPt
PSt
PFt
69.79
63.13
0.892
25.87
0.35(0.55)
0.04(0.84)
1.28(0.53)
0.10(0.75)
65.69
59.03
0.899
27.64
0.02(0.90)
0.21(0.64)
0.63(0.73)
0.01(0.92)
32.27
25.61
0.788
12.16
0.04(0.85)
0.57(0.45)
0.08(0.96)
0.68(0.41)
27.28
20.62
0.508
4.10
0.21(0.64)
7.24(0.01)
3.50(0.17)
0.26(0.61)
33.69
27.02
0.623
5.96
0.10(0.66)
0.14(0.71)
11.64(0.03)
0.03(0.86)

12. 288
M. M. De Mello, K. S. Nell
equations. In addition, the identification process of the structural equations
should confirm their steady-state form as that of the equations of an AIDS
model. Thus, for both the PUREVAR and WHOLEVAR, we expect to find
exactly two cointegrated vectors and to identify the structural parameters
with restrictions that match those of the normalisation process used to
identify the share equations of an AIDS system. If this is the case, we can
confirm that two long-run relations exist in the system, and inference based
on their estimates is valid. Then, using a cointegrated structural VAR, we can
subject its long-run relations to further restrictions such as homogeneity and
symmetry, and contribute an empirical basis for confirming the rationale of
consumer theory principles underlying an AIDS system.
The cointegrated VAR with endogenous and exogenous I(1) variables,
intercept and no trend, can be given by the general model (Pesaran and
Pesaran 1997, pp. 429–433):
Dyt ¼ a0y À Py ztÀ1 þ
pÀ1
X
i¼1
Ciy DztÀi þ et
ð2Þ
À
Á0
where zt = yt0 ; x0t is the (mÂ1) vector of variables; yt = [WF, WS, PF, PS,
PP]¢ is the (myÂ1) vector of endogenous variables and xt = [E, D1, D2, D3]¢
is the (mxÂ1) vector of exogenous variables. Cointegration analysis concerns
the estimation of (2) when the rank of P ¼ ay b0 is, at most, my. ay is the
(myÂr) speed of adjustment matrix, and b is the (mÂr) long-run coefficient
matrix. The (rÂ1) vector b0 zt defines the cointegrated relationships in (2). If
(mÀ1) cointegrated vectors in b. So,
Py has reduced rank, there are r
cointegration testing amounts to determining the number of linearly independent columns (r) in P. The null that at most r vectors exist can be tested
with the eigenvalue trace (ktrace ) and/or maximum eigenvalue (kmax ) statistics.
Table 6 includes the results of these tests.
For the PUREVAR, at the 5% level, both kmax and ktrace statistics reject
r = 0 and r = 1 (statistic valuecritical value), but cannot reject r = 2
(statistic value critical value) and the SBC criterion also supports r=2
cointegrated vectors. For the WHOLEVAR, at the 5% level, the kmax statistic
suggests two vectors while the ktrace statistic does not reject the hypothesis of
only one. This disagreement is not uncommon, particularly in cases of small
samples and added dummy variables. Yet, we have enough evidence supporting the choice of r = 2. We have the kmax statistic clearly rejecting the
existence of only one in favour of two vectors; we have the SBC criterion
selecting the model with two vectors; we have theory suggesting the existence
of two, and not one, long-run relationships, and we have the unmistakable
support of both test statistics and selection criterion in the more general
PUREVAR. Given these results, we proceed by setting r = 2 for both VAR
models.
To identify the structural form of the cointegrated vectors, we use the
exact-identifying restrictions implicit in the share equations of the AIDS
model. Given the following notation for the long-run coefficient matrixes of
the PUREVAR (bPURE ) and WHOLEVAR (bWHOLE )
!
bà bà bà bà bà bà bÃ
11
21
31
41
51
61
71
b0PURE ¼
Ã
Ã
Ã
Ã
Ã
Ã
Ã
b12 b22 b32 b42 b52 b62 b72

13. The forecasting ability of a cointegrated VAR system
289
Table 6. Tests for the cointegration rank of PUREVAR and WHOLEVAR models
Eigen values
^
kmax
H0
mÀr
r
Purevar
k1=0.9201
k2=0.7489
k3=0.5889
k4=0.3273
k5=0.1838
k6=0.0958
Wholevar
k1=0.8798
k2=0.7734
k3=0.5833
k4=0.2767
k5=0.2489
^
ktrace
kmax critical
5%
10%
ktrace critical
5%
SBC
10%
r
r
r
r
r
r
=
=
=
=
=
=
0
1
2
3
4
5
m
m
m
m
m
m
À
À
À
À
À
À
r
r
r
r
r
r
=6
=5
=4
=3
=2
=1
70.76
38.69
24.89
11.10
5.69
2.82
40.53
34.40
28.27
22.04
15.87
9.16
37.65
31.73
25.80
19.86
13.81
7.53
153.96
83.20
44.50
19.61
8.51
2.82
102.56
75.98
53.48
34.87
20.18
9.16
97.87
71.81
49.95
31.93
17.88
7.53
274.71
290.09
292.78
291.89
287.45
283.63
r
r
r
r
r
=
=
=
=
=
0
1
2
3
4
m
m
m
m
m
À
À
À
À
À
r
r
r
r
r
=5
=4
=3
=2
=1
59.31
41.57
24.51
9.07
8.01
46.77
40.91
34.51
27.82
20.63
43.80
38.03
31.73
25.27
18.24
142.48
83.16
41.59
17.08
8.01
119.77
90.60
63.10
39.94
20.63
114.38
85.34
59.23
36.84
18.24
265.82
272.15
272.94
268.54
259.74
b0WHOLE ¼
b11
b12
b21
b22
b31
b32
b41
b42
b51
b52
b61
b62
b71
b72
b81
b82
b91
b92
!
b101
;
b102
the restrictions which identify the cointegrated vectors as share equations of
an AIDS system in both models are:
'
b11 ¼ bà ¼ À1 b12 ¼ bà ¼ 0
11
12
H:
b21 ¼ bà ¼ 0 b22 ¼ bà ¼ À1
21
22
Table 7 shows the coefficients estimates of the two cointegrated vectors in
PUREVAR and WHOLEVAR (asymptotic t ratios in brackets). The ‘third
vector’ relates to the coefficients of the share equation for Portugal (WP),
retrieved from the coefficient estimates of the other two equations with the
adding-up property.
There is a sharp difference between the estimates of the cointegrated
WHOLEVAR and PUREVAR models, both in magnitude, expected signs
and statistical relevance. For instance, at the 5% level, the PUREVAR
estimates indicate all coefficients as irrelevant in the share equation for
Portugal, and only the price of Spain and intercept as significant in the
share equations for France and Spain. In the equation for Portugal, the
own-price and intercept estimates have ‘wrong’ signs and implausible
magnitudes. In the equation for France, an implausible magnitude is also
the case for the intercept. By contrast, the WHOLEVAR estimates are
generally significant, present the expected signs and magnitudes and give
plausible information about how events represented by the dummy variables affected the UK demand for these destinations. For instance, the
coefficients of D1 indicate that the oil crises and political changes in Portugal and Spain affected negatively UK tourists’ preferences for these
destinations, favouring France instead. The coefficients of D2 indicate that
Spain and Portugal’s integration in the EU caused UK tourism flows to
divert from France to the Iberian Peninsula, although favouring more
Spain than Portugal. The D3 coefficients indicate a recovery of the share

14. WF
WS
PP
PS
PF
E
D1
D2
D3
INT
Variables
0.8309 (2.20)
(À0.86)
(2.09)
(À0.48)
(À1.12)
0.3818 (1.97)
(0.60)
(À2.97)
(1.20)
(0.51)
(0.95)
(À0.97)
(À0.33)
(1.45)
À0.2127 (À0.95)
0.3184
À0.2277
À0.0875
0.0525
À1
0
À0.0027
0.2256
À0.3394
0.0153
0.0380
À0.0565
0.0574
0.3687
(À0.05)
(4.68)
(À3.59)
(2.33)
(5.27)
(À3.65)
(5.17)
(16.26)
0
À1
0.1781
À0.5937
0.3119
0.0159
À1
0
À0.4965
0.8214
À0.2244
À0.0684
0
À1
0.1090
À0.3075
0.3044
À0.0183
À0.0154
0.0528
À0.0590
0.5443
(1.69)
(À5.22)
(2.57)
(2.29)
(À1.74)
(2.78)
(À4.34)
(19.50)
Vector 2 (WS)
Vector 1 (WF)
Vector 2 (WS)
Vector 1 (WF)
‘Vector’ 3 (WP)
Cointegrated WHOLEVAR
Cointegrated PUREVAR
Table 7. Long-run coefficients estimates of the exactly-identified share equations
À0.1062
0.0820
0.0350
0.0030
À0.0226
0.0037
0.0016
0.0870
(À3.28)
(2.90)
(0.56)
(0.78)
(À5.18)
(0.41)
(0.24)
(6.33)
‘Vector’ 3 (WP)
290
M. M. De Mello, K. S. Nell

15. The forecasting ability of a cointegrated VAR system
291
for France at the expense of Spain’s share in the post-integration period
(the opening of the channel tunnel in 1994 may also have contributed to
this result).
The reason for the different results obtained with the PUREVAR and
WHOLEVAR is simple. We showed previously that there is statistical support for considering Et as exogenous and including D1, D2 and D3 as relevant regressors. The WHOLEVAR incorporates these features, but the
PUREVAR does not. Hence, the implausible estimates of the PUREVAR
should be expected, and we can confirm that it is not an appropriate model
for supplying reliable information on the long-run demand behaviour of UK
tourists. Consequently, we carry on the analysis with the cointegrated
WHOLEVAR model.
Tourism studies using AIDS systems seldom show well-defined cross-price
effects among destinations. So, clear conclusions on degree and direction of
destinations’ competing behaviour are not usually obtained. Also, homogeneity and symmetry, which are basic premises of consumer demand theory,
have often been rejected with static AIDS models. To test these hypotheses
within the framework of a cointegrated VAR, we focus on the structural
equations of the WHOLEVAR. According to theory, homogeneity and
symmetry should hold and, according to De Mello et al.’s (2002) assumption
about the competitive behaviour of neighbouring destinations, price changes
in France (Portugal) should not affect UK demand for Portugal (France),
while price changes in Spain should affect significantly UK tourism demand
for both France and Portugal.4
In Table 8, the LR test results indicate that null cross-price effects between
France and Portugal, homogeneity and symmetry, and all these hypotheses
simultaneously, cannot be rejected. These results underline the importance of
well-defined structural models for delivering empirical support to theoretical
assumptions and helping understand long run demand behaviour. Table 9
shows the coefficient estimates of the cointegrated structural WHOLEVAR
(hereafter CSV) under homogeneity, symmetry and null cross-price effects
(asymptotic t ratios in brackets).
Given the log-linear form of the CSV, the impacts that price and
expenditure changes have on UK demand are better evaluated through the
corresponding elasticities. We compute the expenditure and uncompensated
own- and cross-price elasticities, using the CSV long-run coefficient estimates
and the formulae given in A2 of Appendix A.5 Table 10 presents these and
the corresponding estimates of De Mello et al.’s (2002) model.6
4
The significance and signs of the cross-price effects provide information about the competing
behaviour of tourism destinations. Positive and negative signs indicate substitutability and
complementarity, respectively. The non-significance of cross-price effects between France and
Portugal is a reasonable expectation given differences in size, origin proximity, product diversity
and features of the UK demand for both countries (e.g. visit periodicity and average length of
stay). See De Mello et al. (2002, p. 518).
5
Except for the cross-price elasticity of France (Portugal) in the share of Portugal (France)
which, as predicted, are statistically irrelevant, all the other CSV elasticities are statistically
significant at the 5% level.
6
We use De Mello et al.’s elasticities estimates for the period (1980–1997), denoted by the
authors as the ‘‘second period’’, because these correspond to more recent behaviour of UK
tourism demand.

16. 292
M. M. De Mello, K. S. Nell
Table 8. Tests of over-identifying restrictions on the cointegrated WHOLEVAR
Hypothesis
LR ! v2 (i)
5% Critical value
Null cross-price effects
Homogeneity and symmetry
Homogeneity, symmetry
and null cross-price effects
v2 (2)=0.36
v2 (3)=7.28
v2 (4)=7.53
5.99
7.81
9.49
The estimated elasticities of the CSV and AIDS are similar. The expenditure elasticities are close to unity for all destinations in both models and,
except in the CSV equation for Spain, the own-price elasticity estimates are
close to À2. The CSV and AIDS cross-price elasticities also give similar
indications: insignificant cross-price effects between the equations for Portugal and France indicating that these two destinations do not compete between them; significant cross-price effects between Spain and France and
Spain and Portugal indicating destination substitutability and bilateral
competitive behaviour. The magnitudes of the elasticities, showing that the
UK demand for Portugal or France is more sensitive to price-changes in
Spain than that for Spain is to price changes in its neighbours, suggest a more
stable and persistent demand for Spain than that for its neighbouring competitors. The negligible sensitivity of UK demand for Spain to price changes
in Portugal, but considerable sensitivity of UK demand for Portugal to price
changes in Spain, denote substantial differences in market attractiveness and
size between the two Iberian countries. The significance and magnitudes of
these cross-price effects indicate Spain and its smaller neighbour as uneven
competitors for similar demand niches, and Spain as a clearly preferred
destination. France and Spain, however, are both ‘tourism giants’ and similarly popular destinations. Although UK demand for Spain is less sensitive to
price changes in France (0.523) than that for France is to price changes in
Spain (0.793), their close scale reflects Spain and France ‘neck and neck’
competition for UK tourists. This competitive behaviour can also be detected
in Figure 1, by the mirror-like movements of France and Spain’s tourism
shares. The cross-price effects between Spain and its neighbours, its relatively
Table 9. Long-run coefficients estimates of the CSV
Variables
CSV
Vector 1 (WF)
WF
WS
PP
PS
PF
E
D1
D2
D3
INT
Vector 2 (WS)
‘Vector’ 3 (WP)
À1
0
0
0.2891
À0.2891
0.0091
0.0354
À0.0373
0.0429
0.3849
0
À1
0.0895
À0.3785
0.2891
À0.0121
À0.0115
0.0360
À0.4184
0.5230
À0.0895 (À5.80)
0.0895 (5.80)
0
0.0030 (0.95)
À0.0239 (À7.05)
0.0013 (0.20)
À0.0011 (À0.27)
0.092 (11.69)
(4.79)
(À4.79)
(1.16)
(3.83)
(À2.36)
(4.56)
(13.17)
(5.80)
(À5.97)
(4.79)
(À1.38)
(À1.13)
(2.02)
(À3.95)
(16.96)

17. The forecasting ability of a cointegrated VAR system
293
Table 10. Expenditure and uncompensated own- and cross-price elasticities estimates
Equations
Models
Expenditure
elasticities
Own-price
elasticities
Cross-price elasticities
PP
WP
WS
WF
CSV
AIDS
CSV
AIDS
CSV
AIDS
1.039
0.947
0.979
1.150
1.026
0.808
À2.158
À1.797
À1.057
À1.933
À1.817
À1.901
PS
PF
X
X
0.161
0.124
À0.002
0.017
1.137
0.830
X
X
0.793
1.077
À0.017
0.019
0.523
0.658
X
X
low own-price elasticity and close to unity expenditure elasticity, indicate
Spain as a preferred destination for UK tourists. Yet, the trends of the shares
in Fig. 1 show that Spain is losing ground to its neighbours in more recent
years. If this tendency persists, it is possible that France becomes the favourite
destination relative to Spain, and Portugal can see its UK tourism share
increased beyond the 10% level.
As the CSV model fully complies with the theoretical predictions underlying the AIDS model, the similarity of their estimates and the cointegration
analysis implemented with the VAR, confirming the share equations as the
only meaningful long-run relations, further support the AIDS approach as an
adequate means of explaining the long-run behaviour of UK tourism demand. Yet, the analysis is not complete without assessing the forecasting
ability of the CSV compared to that of the AIDS and PUREVAR models.
4. The Forecasting accuracy of VAR and AIDS models
As in Clements and Hendry (1998, p.139), we consider that ‘‘the practical
importance of imposing cointegration restrictions for forecast accuracy in small
samples is worthy of study’’. Thus, using a small data set to compare the
predictive accuracy of an unrestricted VAR with that of a cointegrated
structural VAR and AIDS models, offers an empirical contribution for research in this area. Yet, given the widespread use of first difference models in
forecasting and the fact that a differenced VAR could reveal itself more
robust to structural breaks than the PUREVAR (hereafter PV), we include in
our comparison exercise forecasts obtained from a first difference unrestricted
VAR (hereafter DV), reflecting the sort of benchmark models forecasters
generally use. The difference forecasts obtained with the DV are transformed
back into levels for comparison with the level forecasts of the other models.
Within a tourism demand context, the relevance of distinguishing between
short and long run forecast accuracy cannot be overstressed. Frequently, this
activity involves long term investments of substantial dimensions, for which
the ‘go-ahead’ decisions have to be carefully pondered in view of major losses
that can occur if demand levels do not fulfil expectations. On the other hand,
not predicting increasing tourism demand can be equally costly, as tourists’
disappointment has long-term memory effects. Thus, although short run
forecasts are important, it seems that accurate long run prediction of demand

18. 294
M. M. De Mello, K. S. Nell
levels is a crucial matter for sustained success in tourism business. Unfortunately, data series available in this area are seldom long enough to obtain
reliable long run forecasts.
To assess the predictive ability of the four models, we estimate them for
the period 1969–1993, leaving the last years (1994-1997) as the forecasting
evaluation period.7 We also consider recursive estimation of the models
throughout the evaluation period, estimating them up to 1993, then to 1994,
etc. Due to the scarcity of observations, the evaluation period has to be short.
This imposes limits to the prediction horizon within which we can sensibly
decide which model is the best forecaster. One-step prediction is important,
not only because some tests for equal accuracy have optimal properties within
this horizon, but also because it makes available the full set of four forecasting observations. Two-step prediction would leave us with three observations, three-step with two, and four with one. Hence, to evaluate short-run
accuracy, we use one-step ahead forecasts obtained with the models estimated for the period 1969-1993 (‘regular’ one-step forecasts) and one-step
forecasts obtained with the models estimated recursively throughout the
evaluation period (‘recursive’ one-step forecasts). To evaluate long-run
accuracy, the only possible ‘longer’ run we have available with the full set of
four observations is four multi-step ahead forecasts, that is, one-step ahead
forecast for 1994, two-step for 1995, three for 1996 and four for 1997. See A6
in Appendix A, for a clear distinction between h-step and multi-step forecasts.
Table 11 reports, for one- and multi-step forecast errors, a set of ‘traditional’ statistics to evaluate forecasting performance: the mean squared error
(MSE), the root mean squared error (RMSE) and the mean absolute percentage error (MPE). As a visual aid of forecast accuracy, plots of the actual
and forecasted levels of the models, for ‘regular’ one-step and multi-step
forecasts are given in Figures 3 and 4, respectively.
4.1. One-step ahead forecast accuracy
From Table 11 and Figure 3, we can immediately perceive some features of
the models forecasting quality. Given the measures’ ranking consistency for
all models, all destination shares, and both ‘regular’ and ‘recursive’ one-step
errors, we focus the analysis on the MPE. PV is the worse forecaster for
France and Spain’s shares (MPE8%) and the best for Portugal’s share
(MPE4%). CSV and DV are the best forecasters for, respectively, the
shares of France (MPE3%) and Spain (MPE2%). For all share equations, DV, CSV and AIDS seem to have fairly similar performances. But, is
their accuracy statistically equal?
7
We estimated DV, PUREVAR, WHOLEVAR and AIDS for the period 1969-1993. The AIDS,
PUREVAR and DV coefficient estimates for this period were the basis to compute their forecasts.
In the cases of the PUREVAR and WHOLEVAR, we re-applied Johansen rank test to determine
the number of cointegrated vectors. For both models and with both kmax and ktrace at the 5%
level, we obtained evidence of two cointegrated vectors. Thus, we confirmed the existence of two
cointegrated vectors, even when the sample is reduced to 1969-1993. After identifying the
WHOLEVAR structural form and imposing homogeneity, symmetry and null cross-price effects
restrictions, we computed one-step and multi-step ahead forecasts.

19. 0.0019
0.0441
11.06%
0.0001
0.0122
2.77%
0.0003
0.0159
3.66%
0.0056
0.0752
18.79%
MSE
RMSE
MPE
0.0006
0.0251
6.33%
0.0009
0.0302
6.30%
‘Multi-step’ ahead forecast errors
MSE
RMSE
MPE
0.0000
0.0068
1.54%
0.0001
0.0089
1.87%
‘Recursive’ one-step ahead forecast errors
0.0002
0.0142
2.99%
‘Regular’ one-step ahead forecast errors
MSE
RMSE
MPE
AIDS
0.0002
0.015
3.54%
0.0002
0.0151
3.55%
0.0003
0.0168
3.90%
0.0015
0.0385
6.84%
0.0001
0.0121
1.95%
0.0001
0.0112
1.57%
DV
CSV
DV
PV
SPAIN
FRANCE
Table 11. Forecasting quality summary statistics
0.0009
0.0294
4.40%
0.0061
0.0782
14.78%
0.0021
0.0461
8.80%
PV
0.0000
0.0094
1.10%
0.0002
0.0150
2.04%
0.0002
0.0150
2.17%
CSV
0.0002
0.0133
2.12%
0.0002
0.0133
2.13%
0.0003
0.0165
2.84%
AIDS
0.0003
0.0164
17.90%
0.0000
0.0063
5.68%
0.0000
0.0059
5.59%
DV
PORTUGAL
0.0000
0.0058
6.11%
0.0000
0.0036
3.87%
0.0000
0.0031
3.30%
PV
0.0000
0.0071
7.46%
0.0000
0.0079
8.25%
0.0000
0.0064
6.57%
CSV
0.0000
0.0079
8.93%
0.0000
0.0053
8.87%
0.0000
0.0053
5.21%
AIDS
The forecasting ability of a cointegrated VAR system
295

20. 296
M. M. De Mello, K. S. Nell
Fig. 3. Actual levels and ‘regular’ one-step forecasts for France, Spain and Portugal shares
Equal accuracy of two competing forecast series, i and j, can be judged by
testing the significance of the difference (dij) between economic losses associated with forecast error series ei and ej. While in many cases, economic loss
may be poorly accessed by ‘traditional’ measures, in a tourism demand
context we may assume that the loss related with prediction failure is a
symmetric function of the forecast error, since over-forecasting can be as
costly as under-forecasting. Hence, we allow time t loss associated with a

21. The forecasting ability of a cointegrated VAR system
297
Fig. 4. Actual levels and multi-step forecasts for France, Spain and Portugal shares
series of n forecasts to be a direct function of the forecast error, g(e), with
MSE as the standard measure of forecast quality such that g(e)=e2. The null
of equal accuracy of two competing h-step forecast series i and j is E(dijt) ¼ 0,
where dijt ¼ g(eit)Àg(ejt); t ¼ 1,…,n. For testing the null, we use Harvey
et al.’s (1997) S1* test, which is a modified version of the Diebold and
Mariano’s (1995) S1 test, and a simple encompassing test proposed in Hendry
(1986) and Clements and Hendry (1998).

22. 298
M. M. De Mello, K. S. Nell
To facilitate interpretation of the tests, we first provide a brief overview
of the different test statistics, and mention some conclusions that the authors report in their studies. Diebold and Mariano’s (1995) contribution on
comparative prediction accuracy, proposes and evaluates a set of explicit
tests for the null of equal accuracy between two forecast series, which are
valid for a very wide class of loss functions (not needing to be quadratic,
symmetric, or even continuous), and for forecast errors that can be nonGaussian, nonzero mean, serially correlated and contemporaneously correlated. In particular, the asymptotic test statistic S1 can handle a serially
correlated loss differential better than the other proposed statistics. However, the authors also recognise that although S1 is robust to serial and
contemporaneous correlation in large samples, it can be oversized in small
samples. Harvey et al. (1997) show that this problem becomes more acute
as the forecast horizon, h, increases, and explore the possibility of alleviating the problem through modifications of the Diebold-Mariano (DM)
procedure. By employing an approximately unbiased estimator for the
variance of the mean of d [var(d)], the authors propose the modified DM
È
à É1=2
, where S1 is the
test statistic S1Ã ¼ S1 Â n þ 1 Ã 2h þ nÀ1 hðh À 1Þ =n
À
v^rðdÞ À1=2 . The other modification of the DM test
original DM statistic, d a
that Harvey et al. (1997) suggest, is to compare S1* with critical values
from the Student’s T(n-1) distribution, rather than from the N(0;1) used
for the S1 statistic. The authors report ‘dramatic improvements’ occurring
for small samples with S1* relative to S1, but concede that the original
version has advantages in small samples when the error terms are normally
distributed.
The forecast-encompassing tests investigate whether a model (Mb), can
explain the forecast errors of another model (Ma) and vice-versa. Given
the regression eat=afbt+et of Ma’s forecast errors (ea) on Mb’s forecasts
(fb), the test consists in testing the null of a=0. If the null is rejected, Mb
forecast-encompasses Ma. Given that the possible presence of heteroscedasticity and/or autocorrelation in the regression errors could invalidate
the conclusions of the significance tests (Harvey et al., 1998), these are
preformed using robust standard errors supplied by the Newey and West
(1987) consistent covariance matrix with a Parzen truncation lag of
one. The results of S1* and encompassing tests (involving series i
encompassing j and series j encompassing i), are reported in Table 12.8
The results in Table 12 indicate similar conclusions for the ‘regular’ and
‘recursive’ forecast errors and basically support what was already hinted by
the quality measures of Table 11. For Portugal’s share, S1* does not detect
significant accuracy differences between the four models (MPEs ranging
from 3.3% to 6.6% for the ‘regular’ errors, and from 3.8% to 8.9% for
the ‘recursive’ errors). Yet, the encompassing tests show that DV and PV
forecast-encompass AIDS and that AIDS forecast-encompasses CSV. For
France and Spain’s shares, the hints of the quality measures also seem to
8
The tests are preformed for one-step errors only, because they have optimal properties with
h-step, and not multi-step horizons. Since multi-step and one-step are structurally different basis
to evaluate accuracy, comparison between one- and multi-step forecasting performances should
be interpreted with prudence.

23. Test
Statistic
distribution
S1*
i encompasses
j encompasses
S1*
i encompasses
j encompasses
S1*
i encompasses
j encompasses
j
i
j
i
j
i
T(3)
T(3)
T(3)
France
Spain
Portugal
S1*
i encompasses
j encompasses
S1*
i encompasses
j encompasses
S1*
i encompasses
j encompasses
j
i
j
i
j
i
T(3)
T(3)
T(3)
‘Recursive’ one-step ahead forecast errors
France
Spain
Portugal
‘Regular’ one-step ahead forecast errors
Shares
2.35
2.35
2.35
2.35
2.35
2.35
10% Critical
values
Table 12. Tests for equal one-step forecasting accuracy
0.47
1.88
À1.76
À2.30
À7.79
À3.26
À2.33
8.93
4.99
0.46
1.79
À1.04
À3.20
À13.78
À2.36
À2.94
9.13
2.80
DV(i) versus
PV (j)
À0.27
2.21
À1.84
À0.23
À1.78
À3.26
0.69
1.05
4.76
À0.07
2.02
À1.10
À0.43
À1.49
À2.39
0.18
0.80
2.68
DV(i) versus
CSV(j)
À0.46
À11.08
À1.72
À0.83
À0.003
À3.18
0.15
1.21
4.94
0.29
À2.95
À1.03
À0.98
0.33
2.33
À0.83
0.23
2.77
DV(i) versus
AIDS(j)
Competing forecast series i versus j
1.47
1.78
2.29
À2.41
À8.43
À1.85
À2.52
8.20
1.03
1.04
1.80
2.18
À2.27
À11.78
À1.45
À2.21
10.82
0.86
CSV(i) versus
PV (j)
À1.51
2.06
À11.30
À2.31
À7.16
0.01
À2.31
9.86
1.29
0.45
1.89
À2.99
À2.51
À13.06
0.32
À2.43
9.28
0.23
AIDS(i) versus
PV (j)
0.02
2.43
À11.56
0.15
À1.72
À0.01
À0.83
1.09
1.19
0.20
2.16
À3.00
À0.20
À1.45
0.32
À0.52
0.83
0.24
CSV(i) versus
AIDS(j)
The forecasting ability of a cointegrated VAR system
299

24. 300
M. M. De Mello, K. S. Nell
be confirmed by S1*, which does not reject equal accuracy between DV,
CSV and AIDS models (MPEs ranging from 1.57% to 3.90% for the
‘regular’ errors, and from 1.87% to 3.66% for the ‘recursive’ errors).
Based on S1* solely, a clear conclusion of equal (or otherwise) accuracy
between either of these three models and PV is not offered, because the
statistic values are in the vicinity of the critical value. However, the
encompassing tests unmistakably show for the shares of France and Spain,
that DV, CSV and AIDS forecast-encompass PV and that CSV forecastencompasses DV. Thus, for one-step forecasts, the main conclusion from
evidence gathered in Tables 11 and 12 is that DV, CSV and AIDS models
are equally accurate forecasters and that DV performs better the role of
benchmark than PV, at least for Spain and France’s shares. Nevertheless, a
far more interesting debate can be associated with the performance of PV,
which imposes neither integration nor cointegration, and DV, which imposes integration, relative to that of CSV, which imposes both. Christoffersen and Diebold (1998) state that these models differences in forecast
accuracy ‘‘may simply be due to the imposition of integration, irrespective of
whether cointegration is imposed’’ (p.455). This contradicts several previous
analyses (e.g., Engle and Yoo 1987; Clements and Hendry 1998) that
attribute those differences to the imposition of cointegration and not
integration. Surely the DV model imposes integration, and the evidence
provided by S1* shows that it performs as well as CSV. However, the
debate relates to long-horizon forecasts, and evidence obtained from onestep horizon and a sample of four forecast observations certainly is not
sufficient to add any substantial contribution to the debate. Nevertheless,
this is an important issue to which further consideration will be devoted in
future research.
4.2. Multi-step ahead forecast accuracy
From Table 11 and Fig. 4, we can again perceive some forecast quality features of the models, given their consistent ranking by all quality measures, in
all destination shares. Once again we focus on MPE. DV is now the worse
forecaster for all shares. For Spain and France’s shares, CSV (MPEs2%)
and AIDS (MPEs3.6%) perform similarly, but better than PV and DV
(MPEs ranging from 4.40% to 6.84%). For Portugal’s share,
CSV(MPE=7.46%) and PV (MPE=6.11%) seem to have similar performance, but much better than that of DV (MPE=17.9%). This can be confirmed in Figure 4, where the point forecasts of CSV and PV almost overlap
for the whole forecasting range. Figure 4 also shows that the DV forecasts
tend to diverge from the actual values as we move away from 1993, although
being remarkably accurate for 1994, and that PV completely misses the
turning points of the actual shares for all destinations, while CSV misses only
one and the AIDS model misses two in each destination. Based on these
results, we conclude that for multi-step forecasting, the PV and DV models
cannot be considered accurate forecasters, when compared with structural
models such as the AIDS or CSV.
Given that in tourism research, data availability on the explanatory
variables is much wider than that on the dependent variables (namely
tourism expenditure of specific origins in specific destinations), the

25. The forecasting ability of a cointegrated VAR system
301
forecasting evaluation periods are usually short, which prevents clear
conclusions with tests of equal accuracy for longer horizons than one- or
two-steps ahead. However, multi-step forecasts for four-, five- or even
eight-step ahead are easily obtainable if four, five or eight observations are
left out of the estimation period, to be object of forecasting evaluation.
Yet, tests specifically tailored for multi-step accuracy comparison between
competing forecasts have not been given as much attention as h-step ones,
and this is a subject worthy of attention, given the importance that such
tests could have for sensible decision-making in tourism business.
5. Conclusion
De Mello et al.’s (2002) AIDS model for UK tourism demand is a static
system of equations that includes nonstationary variables and assumes exogeneity for all its regressors. These features can risk the validity of estimation, inference and forecasting procedures, if no cointegrated
relationships exist and/or exogeneity assumptions do not hold. As an
alternative, we specified a reduced-form VAR for the same data, establishing its lag-length, deterministic components and endogenous/exogenous
division of variables with appropriate tests, and used Johansen’s rank test to
determine the number of cointegrated vectors in the VAR. Theory underlying a system of equations regressing two destination shares on destination
prices and an origin tourism budget, predicts the existence of exactly two
long-run relationships. Thus, in a VAR with the same variables, two cointegrated vectors should be accounted for. The rank test provided evidence
to support this prediction. The theory underlying an AIDS model for UK
tourism demand also establishes the structural form of the long-run relations it predicts. Hence, the structural parameters of the cointegrated VAR
should be exactly-identified with restrictions matching those of the normalisation process that identifies the share equations in an AIDS system.
This was accomplished and the resulting cointegrated VAR was then subjected to the additional restrictions of homogeneity, symmetry and null
cross-price effects between the equations of Portugal and France. These
restrictions were not rejected. Consequently, evidence was obtained on the
ability of the cointegrated VAR to comply with theoretical predictions
underlying consumers’ demand behaviour and destinations’ competitive
conduct. Finally, the structural coefficients of the restricted cointegrated
VAR (CSV) were used to compute the expenditure, own- and cross-price
elasticities of UK demand. These estimates, similar to the corresponding
ones of the AIDS model, proved to be statistically relevant and empirically
plausible. Besides its consistency and statistical robustness, the CSV model
also shows an excellent predictive ability. Indeed, when compared with the
reduced-form PUREVAR (PV), differenced VAR (DV) and AIDS models,
the quality criteria indicate the CSV as the best predictor in the longer-run
of multi-step forecasting. For one-step forecasts however, tests of equal
accuracy indicate the CSV, AIDS and DV as equally precise forecasters.
Given the modelling simplicity and estimation ease of DV, its accurate onestep forecasts recommend its use in short-run forecasting of tourism demand levels. Yet, if the interest is longer run prediction, the evidence obtained tends to favour the CSV or AIDS models rather than DV or PV.

26. 302
M. M. De Mello, K. S. Nell
Appendix A
A1. Derivation of Deaton and Muellbauer’s (1980a, 1980b) AIDS model
Let x be the exogenous budget or total expenditure which is to be spent,
within a given period, on some or all of n goods. These goods are bought in
nonnegative quantities qi at given prices pi ; i ¼ 1; . . . ; n. Let
q ¼ ðq1 ; q2 ; . . . ; qn Þ be the quantity vector of the n goods purchased, and
p ¼ ðp1 ; p2 ; . . . ; pn Þ the price vector. The consumer’s budget constraint is
n
P
pi qi ¼ x. Given utility function uðqÞ, the consumer maximises utility,
i¼1
subject to the budget constrain:
n
X
p i qi ¼ x
max uðqÞ; subject to
ðA1Þ
i¼1
The solution for this maximisation problem leads to the Marshallian
(uncompensated) demand functions qi ¼ gi ðp; xÞ. Alternatively, the consumers’ problem can be defined as the minimum total expenditure necessary to
attain a specific utility level uà , at given prices:
n
X
min
pi qi ¼ x subject to uðqÞ ¼ uÃ
ðA2Þ
i¼1
The solution for this minimisation problem leads to the Hicksian (compensated) demand functions qi ¼ hi ðp; uÞ. Therefore, a cost function can be defined as
n
X
pi hi ðp; uÞ ¼ x
ðA3Þ
C ðp; uÞ ¼
i¼1
Given total expenditure x and prices p, the utility level u* is derived from the
solution in (A1). Solving (A3) for u, an indirect utility function is obtained
such that u ¼ vðp; xÞ. The AIDS model specifies a cost function, which is used
to derive the demand functions for the commodities. The derivation process
can be summarised in the following three steps:
p;u
1) @ Cðpi Þ ¼ hi ðp; uÞ is derived establishing the Hicksian demand functions.
@
2) solving (A3) for u, the indirect utility function is obtained, such that
u ¼ vðp; xÞ.
3) hi ½p; vðp; xÞŠ ¼ gi ðp; xÞ is retrieved stating the Hicksian and the Marshallian demand functions as equivalent.
These demand functions have the following properties:
P
pi gi ðp; xÞ ¼ x; all budget shares sum to unity;
1. Adding-up: Ri pi hi ðp; uÞ ¼
2. Homogeneity: hi ðp; uÞ ¼ hiiðhp; uÞ ¼ gi ðp; xÞ ¼ gi ðhp; hxÞ 8h 0 ; a proportional change in all prices and expenditure has no effect on the quantities purchased;
@h ðp;uÞ
j
i ðp;u
3. Symmetry: @h@p Þ ¼ @p ; 8i 6¼ j ; consumer’s choices are consistent;
j
i
i ðp;u
4. Negativity: The (nxn) matrix of elements @h@p Þ is negative semi-definite,
j
i ðp;u
that is, for any n vector n , the quadratic form Ri Rj ni nj @h@p Þ 0 , i.e., a
j
rise in prices results in a fall in demand as required for normal goods.
The AIDS model specifies the cost function:

27. The forecasting ability of a cointegrated VAR system
303
ln Cðp; uÞ ¼ aðpÞ þ u:bðpÞ
ðA4Þ
where aðpÞ ¼ a0 þ Ri ai ln pi þ 1 Ri Rj cij ln pi ln pj and bðpÞ ¼ b0
2
The derivative of (A4) with respect to ln pi is:
Q
i
b
pi i .
@ ln C ðp; uÞ
b
¼ ai þ Rj cij ln pj þ ubi b0 Pi pi i :
@ ln pi
(A5)
As C ðp; uÞ ¼ x , ln C ðp; uÞ ¼ ln x;
ðA6Þ
then ln x ¼ aðpÞ þ u:bðpÞ
Solving (A6) for u we obtain u ¼ ½ln x À aðpÞŠ=½bðpÞŠ
ðA7Þ
Substituting (A7) in (A5) we have
@ ln C ðÞ @C ðÞ pi
pi
p i qi
¼ hi ð Þ
¼
¼ wi
¼
@ ln pi
@pi C ðÞ
C ð Þ
x
¼ ai þ Rj cij ln pj þ bi ½ln x À að pÞŠ:
If a price index P is defined such that ln P ¼ aðpÞ, then
@ ln C ðp; uÞ
¼ ai þ Rj cij ln pj þ bi ½ln x À ln P Š
@ ln pi
or wi ¼ ai þ Rj cij ln pj þ bi lnðx=P Þ;
1
where ln P ¼ a0 þ Rk ak ln pk þ Rk R‘ cà ln pk ln p‘
k‘
2
Equations (A8) are the basic equations of the AIDS model.
In a tourism demand context, there are n alternative destinations demanded
by tourists of a given origin. The dependent variable wi, stands for destination i
share of the origin’s tourism budget allocated to the n destinations This share’s
variability is explained by tourism prices (p) in i and alternative destinations j
and by the per capita expenditure (x) allocated to the set of n destinations,
deflated by price index P. The model has the following properties:
1. adding-up restriction requiring that all budget shares sum up to unity:
X
X
X
ai ¼ 1;
bi ¼ 0;
cij ¼ 0; for all j;
i
i
i
2. homogeneity restriction requiring that a proportional change in all prices
P
cij ¼ 0 , for
and expenditure has no effect on the quantities purchased:
j
all i;
3. symmetry restriction requiring consumer consistent choices: cij ¼ cji , for all
i, j;
4. negativity restriction requiring that a rise in prices result in a fall in demand, i.e., negative own-price elasticities for all destinations.
The restrictions imposed on a and c comply with these assumptions and
ensure that in (A8), P is defined as a linear homogeneous function of individual prices. If prices are relatively collinear, then P is ‘‘approximately
proportional to any appropriately defined price index, for example, the one used
P
by Stone, the logarithm of which is
wk ln pk ¼ ln P Ã’’ (Deaton and
Muellbauer 1980a, p.76). Hence, the deflator P in (A8) can be substituted by
the Stone price index ln P* such that,

28. 304
M. M. De Mello, K. S. Nell
ln P Ã ¼
X
wiB ln pi
ðA9Þ
i
where wiB is the budget share of destination i in the year base. With this
simplification for P, system (A8) can be rewritten and estimated in the following form:
x
X
cij ln pj þ bi ln Ã
wi ¼ aà þ
ðA10Þ
i
P
j
A2. Expenditure, own- and cross-price elasticities
Expenditure and price elasticities cannot be directly accessed in (A10), given
its log-linear form, but their values can be retrieved from (A10) coefficients
using the following formulae:
Expenditure elasticity : ei ¼
1 dwi
b
þ1¼ i þ1
wi d ln x
wi
Uncompensated own-price elasticity : eii ¼
1 dwi
c
wB
À 1 ¼ ii À bi i À 1
wi d ln pi
wi
wi
Uncompensated cross-price elasticity : eij ¼
wjB
cij
1 dwi
¼ À bi
wi d ln pj wi
wi
Compensated own-price elasticity : e ¼ eii þ wiB ei ¼
ii
cii
þ wiB À 1
wi
Compensated cross-price elasticity : e ¼ eij þ wB ei ¼
ij
i
cii
þ wB
j
wi
where wjB is destination j’s share (j=1,…,n) in the year base and wi is destination i’s sample average share (i=1,…,n).
A3. The AIDS model of the UK tourism demand for France, Spain and Portugal
The AIDS model assumes that consumers allocate their budget to commodities in a multi-stage budgeting process implying independent preferences. Thus, for the AIDS model of UK tourism demand, it is assumed that
the UK tourism expenditure allocated to France, Spain and Portugal is
separable from that allocated to other destinations, and that the decision to
spend money in those countries is made in several stages. First, UK tourists
allocate their budget to tourism and other goods; then to tourism in France,
Spain and Portugal and other destinations; finally they decide between
France, Spain or Portugal. The AIDS system reproduces this last stage using
the following form:
8
WPt ¼ aP þ cPP PPt þ cPS PSt þ cPF PFt þ bP Et þ upt
WSt ¼ aS þ cSP PPt þ cSS PSt þ cSF PFt þ bS Et þ ust
:
WFt ¼ aF þ cFP PPt þ cFS PSt þ cFF PFt þ bF Et þ uft

29. The forecasting ability of a cointegrated VAR system
305
A4. Variables’ definition
The variables in the AIDS model of UK tourism demand for France, Spain
and Portugal are the UK tourism budget shares allocated to these destinations WP, WS and WF; destination prices PP, PS, PF and UK real per capita
tourism budget E. Each Wi, i = F S P, is defined as:
Wi ¼ EXPi =ðEXPF þ EXPS þ EXPP Þ
where EXPi is the nominal tourism expenditure allocated by UK tourists to
destination i. The effective price of tourism in destination i is defined as:
Pi ¼ ln½ðCPIi =CPIUK Þ=Ri Š
where CPIi is destination i consumer price index, CPIUK is UK consumer
price index, Ri is the exchange rate between i and the UK, defined as number
of currency units of country i per unit of UK currency. The per capita UK
real tourism expenditure allocated to all destinations is:
!
#
X
Ã
EXPi =UKP =P
E ¼ ln
i
where UKP is UK population and lnP* is the Stone index defined in
Eq. (A10).
A5. Data sources
The UK tourism expenditure data, disaggregated by destinations and measured in £ million sterling, were obtained from Business Monitor MA6 (19701993), continued as Travel Trends (1994–1998). The UK population, price
indexes and exchange rates were obtained from the International Financial
Statistics (IMF) Yearbooks (1984, 1990 and 1998).
A6. Multi-step and h-step ahead forecasts
Suppose that, with observations of Yt and Xt ðt ¼ 1; . . . ; T Þ in the period
^
^
^
^
1969–1993, we estimate Yt ¼ b1 þ b2 Xt þ b3 YtÀ1 . The 1-step ahead forecasts of
Y are computed assuming that X values are available up to
T þ j; ðj ¼ 1; 2; . . . ; mÞ and Y values are available up to T+jÀ1. Hence, in
computing Y 1-step forecast for 1994, X1994 and Y1993 are known; for 1995,
X1995 and Y1994 , are known, and so on. 2-step ahead forecasts are computed
assuming that X values are known up to T+j, but Y values are known only
up to T+jÀ2. 3-step forecasts are computed assuming knowledge of XT þj
and YT þj À 3, and so on. We denote the 1-step YT þj forecast, given YT þjÀ1 , as
^
^
YTfþj=YT þjÀ1 ; 2-step YT þ j forecast, given YT þ j À 2, as YTfþj=YT þjÀ2 ; etc. The
1-step forecasts of Y for 1994-1997 are:

30. 306
M. M. De Mello, K. S. Nell
^
^
^
^f
Y1994=Y1993 ¼ b1 þ b2 X1994 þ b3 Y1993 ;
^
^
^
^f
Y1995=Y1994 ¼ b1 þ b2 X1995 þ b3 Y1994 ;
^
^
^
^f
Y1996=Y1995 ¼ b1 þ b2 X1996 þ b3 Y1995 ;
^
^
^
^f
Y1997=Y1996 ¼ b1 þ b2 X1997 þ b3 Y1996 ;
gathering four 1-step forecasts.
Computing a 2-step ahead forecast for Y1994, is the same as computing
1-step ahead, since X1994 and Y1993 are know. Yet, for 1995, Y1994 is not
known. So, we have to use a forecast of Y1994 to insert into the forecasting
equation. The same has to be done for the 2-step forecasts of Y for 1996
and 1997, but updating the forecasts according to knowledge of YT þjÀ2 .
Hence, the 2-step ahead forecasts of Y for 1994–1997 are:
^
^
^
^f
Y1994=Y1993 ¼ b1 þ b2 X1994 þ b3 Y1993
^f
Y1995=Y1993
^f
Y1996=Y1994
^f
Y1997=Y1995
(the same as the 1st forecast in the 1-step forecasts)
^ ^
^ ^
¼ b þ b X1995 þ b Y f
1
2
3 1994=Y1993
^ ^
^ ^f
¼ b1 þ b2 X1996 þ b3 Y1995=Y1994 ;
^ ^
^ ^f
¼ b1 þ b2 X1997 þ b3 Y1996=Y1995 ;
^ ^
^
^f
where Y1995=Y1994 ¼ b1 þ b2 X1995 þ b3 Y94
^ ^
^
^f
where Y1996=Y1995 ¼ b1 þ b2 X1996 þ b3 Y95 ;
gathering only three ‘true’ 2-step forecasts.
The 3-step forecasts of Y for 1994–1997 are:
^
^
^
^f
Y1994=Y1993 ¼ b1 þ b2 X1994 þ b3 Y1993
^f
Y1995=Y1993
(the same as the 1st forecast in the1-step forecasts)
^
^
^ ^
¼ b þ b X1995 þ b Y f
1
2
3 1994=Y1993
(the same as the 2nd forecast in the 2-step forecasts)
^
^
^ ^f
^f
Y1996=Y1993 ¼ b1 þ b2 X1996 þ b3 Y1995=Y1993
^
^
^ ^f
^f
Y1997=Y1994 ¼ b1 þ b2 X1997 þ b3 Y1996=Y1994
^
^
^ ^f
^f
where Y1996=Y1994 ¼ b1 þ b2 X1996 þ b3 Y1995=Y1994 ¼
^
^
^ ^
^
¼ b1 þ b2 X1996 þ b3 b1 þ b2 X1995 þ Y1994 ;
gathering only two ‘true’ 3 -step forecasts
The 4-step forecasts of Y for 1994-1997 are:
^
^
^
^f
Y1994=Y1993 ¼ b1 þ b2 X1994 þ b3 Y1993
^f
Y1995=Y1993
(the same as the 1st forecast in the 1-step forecasts)
^
^
^ ^f
¼ b1 þ b2 X1995 þ b3 Y1994=Y1993
(the same as the 2nd forecast in the 2-step forecasts)

31. The forecasting ability of a cointegrated VAR system
307
^
^
^ ^f
^f
Y1996=Y1993 ¼ b1 þ b2 X1996 þ b3 Y1995=Y1993
^f
Y1997=Y1993
(the same as the 3rd forecast in the 3-step forecasts)
^
^
^ ^
¼ b1 þ b2 X1997 þ b3 Y f
1996=Y1993
gathering just one ‘true’ four-step ahead forecast. We call this last set of
forecasts multi-step ahead since there is one of each h-step.
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