empirical economics
TRANSCRIPT
The forecasting ability of a cointegrated VARsystem of the UK tourism demand for France,Spain and Portugal
Maria M. De Mello1, Kevin S. Nell
2
1 CETE* - Centro de Estudos de Economia Industrial do Trabalho e da Empresa, Faculdade deEconomia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal(e-mail: [email protected])2 Faculdade de Economia e Gestao, Universidade Catolica, Centro Regional do Porto, RuaDiogo Botelho, 1327, 4169-005 Porto, Portugal(e-mail: [email protected])
First version received: September 2002/Final version received: September 2003
Abstract. This paper uses the vector autoregressive (VAR) methodology asan 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 appropriatetesting, the deterministic components and sets of exogenous and endogenousvariables of the VAR are established, and Johansen’s rank test is used todetermine the number of cointegrated vectors in the system. The cointegratedVAR structural form is identified and the long-run structural parameters areestimated. Theoretical restrictions such as homogeneity and symmetry aretested and not rejected by the VAR structure. The fully restricted cointegratedVAR model reveals itself a theoretically consistent and statistically robustmeans to analyse the long-run demand behaviour of UK tourists, and anaccurate multi-step forecaster of the destinations’ shares when compared withunrestricted reduced form and first differenced VARs, or even with thestructural AIDS model.
Key words: Tourism demand, cointegration, equal forecasting accuracy tests
JEL classification: C5, D1
Empirical Economics (2005) 30:277–308DOI 10.1007/s00181-005-0241-0
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
Financiamento Plurianual through the Programa Operacional Ciencia, Tecnologia e Inovacao
(POCTI), financed by FEDER and Portuguese funds.
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 inearly tourism demand studies and the poor forecasting performance of theirmodels, may be linked with the lack of an explicit formulation for dynamicsunderlying tourism demand behaviour. With few exceptions, the modelling ofdynamics in tourism research has been confined to single equation errorcorrection specifications, based on Engle and Granger’s (1987) two-stagesapproach (e.g. Kulendran 1996; Kulendran and King 1997; Kim and Song1998; Vogt and Wittayakorn 1998; Song et al. 2000). These studies have beenregarded as an important step forward in the path to construct more reliablemodels to explain tourists’ demand behaviour, which seems to be dynamic innature. Yet, a single equation framework does not allow for modelling andtesting other features of tourism demand requiring a multi-equation struc-ture.
In recent studies, Deaton and Muellbauer’s (1980a, b) Almost Ideal De-mand System (AIDS) has been used for modelling demand and testing con-sumer theory restrictions such as homogeneity and symmetry. Syriopoulosand Sinclair (1993), Papatheodorou (1999) and De Mello et al. (2002) areexamples 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 usuallyemploys nonstationary time series for its parameter estimation. When dealingwith nonstationary data, failure to establish cointegration often means thenon-existence of a steady state relationship among the variables. Hence,estimation results obtained with static AIDS models can be deemed spuriousand statistical inference invalid, if the usual assumption of exogenousregressors does not hold and/or no cointegrated relationships exist. Thus,there seems to be a risk involved in the estimation of static systems withnonstationary data, which regress endogenous variables on assumed exoge-nous variables, if their statistical validity is not sanctioned with appropriatetesting and cointegration analysis. Given that the number of cointegratedvectors is unknown and simultaneously determined variables may exist,empirical analysis must go one step further and specify econometric modelswhich can be efficiently estimated and validly tested within a system ofequations approach.
The crucial next step towards obtaining consistent parameter estimatesand reliable forecasts of one specific origin tourism demand for several des-tinations, needs dynamics and a system structure. Few studies in tourismresearch specify dynamic systems to model demand. For example, Kulendranand King (1997), Song and Witt (2000) and Kulendran and Witt (2001) useunrestricted vector autoregressive systems. Lyssioutou (1999) and De Melloand Sinclair (2000) use dynamic AIDS systems. None, to the best of ourknowledge, compares the estimates and forecasting performance of a staticsystem with those of a dynamic VAR system in an application for tourismdemand.
This paper contributes an empirical basis for the validation or other-wise, of estimation, inference and forecasting procedures conducted withina static AIDS approach. Using De Mello et al.’s (2002) study of UKtourism demand for France, Spain and Portugal over the period 1969–
278 M. M. De Mello, K. S. Nell
1997, we compare this study’s structural estimates and forecasting resultswith those of a cointegrated structural VAR we construct for the samecountries 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 be-tween 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 thetheoretical 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 theoreticallyconsistent specifications, producing similar estimates for the long-runparameters. However, the CSV supplies accurate multi-step ahead dynamicforecasts for all destination tourism shares, out performing the AIDS,unrestricted reduced form VAR and unrestricted first differenced VARmodels.
The paper proceeds as follows. Section 2 establishes the integration orderand the appropriate lag-length of the variables in the system, and specifies theunrestricted VAR for UK tourism demand. Section 3 determines the numberof cointegrated vectors and presents the cointegrated structural VAR esti-mates. Section 4 presents the analysis of forecast accuracy obtained withCSV, unrestricted and differenced VAR and the AIDS models. Section 5concludes. Appendix A gives the full derivation of Deaton and Muellbauer’sAIDS system and its properties.
2. VAR Modelling of the UK tourism demand
When analysing the existence of long-run relationships among non-sta-tionary series, and there are doubts about the exogenous nature of someregressors, one appropriate modelling strategy consists of starting bytreating all variables as endogenous within a reduced-form VAR. Next,exogeneity tests for the set of variables in doubt can be carried out. Oncethe endogenous-exogenous division is established, the reduced rank testcan be used to establish the number of cointegrated vectors. Then, esti-mates of the long-run coefficients can be assessed by imposing exactly-identifying restrictions to the VAR. After identifying the VAR structuralform, additional restrictions can also be implemented to test its compati-bility with specific theories. Following these steps, we start by specifying areduced-form VAR of the UK tourism demand for France, Spain andPortugal, 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 UKtourism expenditure shares of France, Spain and Portugal, respectively;PF, PS and PP denote tourism prices in France, Spain and Portugal; Estands for the UK real per capita tourism budget. All variables’ definitionand data sources are described in Appendix A, according to the definitionsand data sources used by De Mello et al. (2002).
The forecasting ability of a cointegrated VAR system 279
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 theappropriate lag-length of the VAR.1 Figures 1 and 2 present a set of graphsshowing 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, thetypical oscillatory movement of the first difference variables seem to indicatestationarity and hence the presence of a unit root in their levels. The plots alsoshow that both the price variables level and first difference behave peculiarlyin a sample sub-period that can roughly be placed at 1973–1986. Eventsjustifying such behaviour can be linked to the 1970’s oil crises, mid 1970’sPortuguese revolution and first democratic elections in Portugal and Spain,Spain joining EFTA in 1980, and Portugal and Spain joining the EuropeanUnion (EU) in 1986. Some of these events are likely to have produced breaksin the data which may have repercussions on the conventional unit-root testspreformed below. For some series, these breaks may not have a strong en-ough impact to affect the tests results. For others, however, a clear conclusionbased on conventional tests may be difficult. Table 1 shows the unit root teststatistics 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)
1All estimations and statistical tests were computed with Pesaran and Pesaran (1997) Microfit 4.0.
Fig. 1. Variables in levels
280 M. M. De Mello, K. S. Nell
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, exceptfor DPP, all first difference variables as stationary. Hence, according to theDF and ADF tests, all level variables, except PP, are I(1). Although DPP andPP would be considered I(0) and I(1) respectively, if Charemza and Dead-man’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 tstatistic proposed in Phillips and Perron (1988), using White (1980) andNewey and West (1987) covariance matrix, to remove doubts that couldpersist about the integration order of PP and DPP.2 This procedure confirmedDPP as I(0) and PP as I(1), at the 5% level. Thus, we consider all levelvariables 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) toexceed two. Hence, we used the AIC and SBC criteria and the adjusted (forsmall 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 thatpresent mean shifts over time and uncertainty associated with the points at which such shifts mayhave occurred. As this seems to be the case for PP variable, the Phillips-Perron (1988) procedurewas used for it enables computing a covariance matrix that ‘adjusts’ the t statistic which iscompared with the critical value (�2.975). For PP variable, the adjusted t ratio (�2.9039) offersthe same conclusion of non-stationarity as the DF test. For DPP variable however, the adjusted tratio (�3.9587) clearly indicates stationarity, contradicting the DF test result.
Fig. 2. Variables in first differences
The forecasting ability of a cointegrated VAR system 281
the maximum lag-length permitted. Table 2 presents the selection results. TheLR test rejects order zero, but cannot reject a first order VAR. The SBCcriterion clearly indicates an order one VAR. The AIC criterion indicates p tobe two, but by a very small margin. So, we select a VAR(1). Yet, it is sensibleto confirm this decision by checking for residuals serial correlation in theequations.
2.3. The unrestricted VAR specification of the UK demand for tourism
The reduced form of a first order unrestricted VAR of UK tourism demandfor France, Spain and Portugal (hereafter denoted PUREVAR) can bewritten as:
zt ¼ A0 þ A1zt�1 þ �t ð1Þ
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 �2.100 54.21* 52.87* �2.971ADF(1) �2.008 50.81 48.87 �2.975
DWF DF �5.163 49.71* 48.42* �2.975ADF(1) �3.284 46.36 44.47 �2.980
WS DF �1.965 52.09* 50.76* �2.971ADF(1) �1.756 48.71 46.77 �2.975
DWS DF �5.187 48.08* 46.72* �2.975ADF(1) �3.510 44.78 42.89 �2.980
WP DF �1.738 86.03* 84.70* �2.971ADF(1) �1.458 81.64 79.69 �2.975
DWP DF �5.429 81.49* 80.20* �2.975ADF(1) �4.827 78.25 76.36 �2.980
PF DF �1.869 32.98* 31.64* �2.971ADF(1) �2.311 32.24 30.30 �2.975
DPF DF �3.435 30.59* 29.23* �2.975ADF(1) �3.417 28.69 26.80 �2.980
PS DF �2.083 33.50* 32.17* �2.971ADF(1) �2.428 31.74 29.80 �2.975
DPS DF �3.788 29.78* 28.48* �2.975ADF(1) �2.968 27.24 25.35 �2.980
PP DF �1.418 31.74 30.41 �2.971ADF(1) �2.605 34.17* 32.23* �2.975
DPP DF �2.480 31.81* 30.51* �2.975ADF(1) �2.245 29.16 27.27 �2.980
E DF �2.362 20.51* 19.18* �2.971ADF(1) �1.624 18.48 16.53 �2.975
DE DF �3.696 18.07* 16.77* �2.975ADF(1) �2.675 16.94 15.06 �2.980
Table 2. AIC and SBC criteria and adjusted LR test for selecting the order of the VAR
Order (p) AIC SBC Adjusted LR test
2 296.91 246.37 —1 296.59 269.37 v2 (36) = 37.67 (0.393)0 185.90 182.01 v2 (72) = 366.02 (0.000)
282 M. M. De Mello, K. S. Nell
where z’t = [WFt ,WSt , PPt , PSt , PFt , Et] ; A0 is a (6 � 1) intercept vector;A1 is a (6 � 6) parameter matrix and �t is a vector of well behaved distur-bances. To avoid perfect collinearity brought about by the share variablesWFt, WSt and WPt, which sum up to unity, one of the share equations isomitted. The estimation results are invariant whichever equation is excludedand, by the adding-up property (see Appendix A), all coefficient estimates ofthe omitted equation can be retrieved from the coefficient estimates of theremaining ones. We omit the share equation for Portugal (WPt).
The statistical quality of the PUREVAR is accessed by estimating (1) andcomputing 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 correla-tion, functional form, error normality and heteroscedasticity (p values inbrackets). There is no evidence of serial correlation in the PUREVAR. Hence,the lag-length selected seems to be adequate. Yet, the diagnostic tests indicateproblems 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 UKtourism demand specifies the shares WF, WS and WP, as the only endoge-nous variables. Changes in these variables are explained by a set of assumedexogenous regressors including tourism prices (PF, PS and PP) and the UKreal per capita tourism expenditure (E). In a VAR specification, we are willingto question the assumed exogeneity of the price variables implicit in the AIDSstructure. However, there seems to be no obvious theoretical or empiricalbasis for challenging the multi-stage budgeting process underlying the ratio-nality of an AIDS expenditure share system, which sets variable E as anexogenous determinant of the demand shares. The reasons are as follows. In aVAR, all variables are endogenous implying that a bi-directional cause-effectrelationship between them should exist. In a tourism demand context, how-ever, even if it is reasonable to consider that changes in the UK real per capitaexpenditure affect the tourism shares of important UK holiday destinationssuch as France, Spain or Portugal, it does not seem realistic to expect thatchanges in these shares influence the way in which UK consumers allocatetheir 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 thatthe 99% of this variable’s variations explained by the model lie exclusively onits own lagged value. No other variable in that equation is individually orjointly significant. In fact, the F statistic for joint significance of all explan-atory variables excluding Et-1 is 0.86. In addition, the estimation results forWFt and WSt indicate that the lagged value of Et does not affect significantlythe current values of these shares.
To investigate further the link between Wit and Et, we analyse the rela-tionships between the error term of a conditional model for Wit and thestochastic disturbance of the assumed data generating process (d.g.p.) of Et.Consider that the ith share equation is
Wit ¼ a0 þ a1Et þ a2Wit�1 þ uit where i ¼ F ; S; P ðaÞand that Et is a stochastic variable with underlying d.g.p.
Et ¼ b1Et�1 þ et; b1<1 and �t ! Nð0; r2Þ ðbÞ
The forecasting ability of a cointegrated VAR system 283
Table
3.Estim
ationresultsandstatisticalperform
ance
oftheunrestricted
PUREVAR
Regressors
Equations
WFt
WSt
PPt
PSt
PFt
Et
WFt�
10.9518(2.10)
�0.5617(�
1.14)
�2.0208(�
1.57)
�1.0503(�
0.75)
�1.3501(�
1.32)
�0.3106(�
0.15)
WSt�
10.6820(1.44)
�0.2932(�
0.57)
�1.9954(�
1.48)
�0.7034(�
0.48)
�0.9772(�
0.78)
�1.3497(�
0.61)
PPt�
1�0.0712(�
1.14)
0.1062(1.56)
1.2525(7.05)
0.2108(1.10)
0.5033(3.06)
�0.4728(�
1.62)
PSt�
10.4620(4.85)
�0.4622(�
4.45)
�0.1341(�
0.49)
0.7453(2.54)
�0.0014(�
0.01)
0.0602(0.14)
PFt�
1�0.3195(�
2.38)
0.3157(2.16)
�0.4922(�
1.29)
�0.2276(�
0.55)
0.0598(0.17)
0.6837(1.09)
Et�
1�0.0071(�
0.79)
0.0000(0.00)
�0.0312(�
1.23)
�0.0027(�
0.10)
�0.0318(�
1.35)
0.9315(22.40)
Intercept
�0.2630(�
0.60)
0.8449(1.77)
1.9533(1.57)
0.7627(0.57)
1.1671(1.01)
1.1990(0.59)
Sectioncriteria
anddiagnostic
statistics
AIC
61.24
58.82
31.99
29.77
34.10
18.12
SBC
56.58
54.15
27.33
25.11
29.44
13.45
Adjusted
R2
0.789
0.824
0.771
0.562
0.612
0.991
Fstatistic
17.88
22.12
16.11
6.79
8.09
511.34
Serialcorrelation
0.78(0.38)
2.48(0.12)
0.73(0.39)
0.19(0.67)
0.37(0.54)
0.58(0.45)
Functionalform
3.81(0.05)
8.66(0.00)
1.85(0.17)
7.39(0.01)
0.00(0.98)
0.18(0.67)
Norm
ality
1.02(0.60)
1.56(0.46)
0.56(0.75)
4.82(0.09)
11.65(0.00)
1.66(0.44)
Heteroscedasticity
0.39(0.53)
1.19(0.28)
1.33(0.25)
0.28(0.60)
0.03(0.87)
0.21(0.65)
284 M. M. De Mello, K. S. Nell
If uit and �t are uncorrelated we can say that EV(uit, �s) = 0 for all t, s (whereEV stands for expected value, not to be confused with variable E). Then, it ispossible to treat Et as if it was fixed, that is, Et is independent of uit such thatEV(Et,uit) = 0. Hence, Et can be treated as exogenous in terms of (a), and canbe said to Granger-cause Wit. Equation (a) is a conditional model since Wit isconditional on Et, with Et being determined by the marginal model (b).3
‘‘A variable cannot be exogenous per se’’ (Hendry 1995, p.164). A variablecan 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) iscomplete and sufficient to sustain valid inference. Hence, knowledge of themarginal model will not significantly improve the statistical or forecastingperformance of the conditional model. Following this line of reason, we runregression (a) for the expenditure shares of France, Spain and Portugal andregression (b) as a representation of the d.g.p. of Et. We retrieve the residualseries of these four regressions, namely uFt , uSt , uPt standing for the residualseries of (a) for, respectively, France, Spain and Portugal, and et standing forthe residuals of (b). We then regress the current and lagged values (up to thefifth lag) of the residuals uit (i = F, S, P) on the current and lagged (up to thefifth lag) values of �t. The estimation results of all regressions indicate nosignificant relationship linking the current or lagged residuals of the condi-tional models to the current and lagged residuals of the marginal model.These results can be viewed as an indication that knowledge of the marginalmodel does not improve the statistical or forecasting performance of theconditional (on Et-1) equations for WFt, WSt, PPt, PSt and PFt in the VAR.
To supply further empirical support for our claim that feedback effectsmight be absent in the relationships between Et and Wit, (i = F, S), we alsouse the causality concept proposed by Granger (1969). The block Grangernon-causality test is a multivariate generalisation of the Granger-causalitytest that can be used to establish if one or more variables should or should notintegrate the set of endogenous variables in a VAR. This test uses the LRstatistic to provide a measure of the extent to which lagged values of a set ofvariables (say Et), are important in predicting another set of variables (sayWit), once lagged values of the latter (Wit-1) are included in the model. TheLR statistic for block non-causality of Et, testing the null of zero-valuecoefficients for Et-1 in the block equations WFt, WSt, PPt, PSt and PFt, isv2(5)=10.578 (third test in Table 4). The null is not rejected at the 5% levelcritical value (11.071). Supported by these results, we exclude Et from the setof endogenous variables in the VAR.
Our quest for a correct specification of a VAR explaining the UK tourismdemand for France, Spain and Portugal must also take under considerationother 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) mayhave 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
3As noted in Harris (1995), if (b) is reformulated as Et ¼ b1Et�1 þ b2Wit�1 þ �t, EV(Et, uit ) = 0still holds. But, as past values of Wit now determine Et, Et can only be considered weaklyexogenous in (a). Its current value still causes Wit but not in the Granger sense, since lags of Witnow determine Et.
The forecasting ability of a cointegrated VAR system 285
affected the UK tourism demand for its neighbours. To account for the 1970sevents we add dummy D1 (= 1 in 1974–1981, and zero otherwise). To ac-count for the Iberian countries integration in the EU, we add dummy vari-ables D2 (=1 in 1982–1988 and zero otherwise) and D3 (=1 in 1989–1997and zero otherwise), splitting the integration process into two sub-periods: theintegration period (1982–1988) and post-integration period (1989–1997). Thedummies are assumed to be exogenous.
Table 4 presents the LR tests performed to establish the final form of theVAR (hereafter ‘WHOLEVAR’). The null hypotheses are: non-significance ofthe 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 thespecification with D1, D2 and D3.
The first column of Table 4 presents the null for each test and shows thevariables entering the VAR (time subscripts are omitted for simplicity) underthe ‘‘unrestricted’’ (U), and ‘‘restricted’’ (R) null. In each case, the set ofendogenous variables is separated from the set of deterministic componentsand exogenous variables by the symbol ‘&’. The second column presents themaximum value of the likelihood function (ML) for the unrestricted andrestricted 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 thanthe 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 similartest performed on the VAR without the dummies. The LR statistic shows nowthe value of 8.37, which is well below the 5% critical value (11.07), furthersupporting the null of statistically insignificant coefficients of Et-1 in the blockequations.
Table 4. LR tests for form specification of the PUREVAR
Model ML LR ! v2 (i) C. V. (5%) Result
H0: Non-significanceof intercept (INT) RejectedU: WF WS PP PS PF E & INT 351.68 v2 (6) = 18.05R: WF WS PP PS PF E 342.05 12.59
H0: Non-significanceof dummy variables D1 D2 D3 RejectedU: WF WS PP PS PF E &D1 D2 D3 INT 385.40 v2 (18) = 67.43R: WF WS PP PS PF E & INT 351.68 28.87
H0: Block non-causalityof E without dummy variables Not rejectedU: WF WS PP PS PF E & INT 351.68 v2 (5) = 10.58R: WF WS PP PS PF & E INT 346.39 11.07
H0: Block non-causalityof E with dummy variables Not rejectedU: WF WS PP PS PF E &D1 D2 D3 INT 385.40 v2 (5) = 8.37R: WF WS PP PS PF &E D1 D2 D3 INT 381.21 11.07
286 M. M. De Mello, K. S. Nell
To evaluate the quality of the WHOLEVAR, Table 5 includes the samediagnostic tests and selection criteria used for the PUREVAR. The resultsshow the superior quality of the WHOLEVAR as compared with that of thePUREVAR.
Although WHOLEVAR is statistically more robust than PUREVAR, thelatter is a general model, while the former is a partial system conditioned onexogenous variables. We are interested in the economic interpretation ofstructural parameters, which is only possible if the underlying structuralmodel is identified from the reduced-form. The PUREVAR may not be anideal means to conduct economic analysis, for an a-theoretical reduced-formVAR is unlikely to produce results interpretable within the limits of sensibleeconomic assumptions. Yet, it might be interesting to compare the predictiveaccuracy of the reduced-form with that of the structural VAR. Hence, we usethe PUREVAR for forecasting purposes only.
3. Johansen’s reduced rank test
The next step is to determine the number of cointegrated vectors using Jo-hansen’s rank test. Besides establishing the process for cointegration ranktesting, Johansen’s approach (Johansen 1988, 1991, 1995, 1996; Johansen andJuselius 1990, 1992) provides a general framework for identification, esti-mation and hypothesis testing in cointegrated systems. Yet, Johansen’s‘empirical process’ to exactly-identify the long-run coefficients may not al-ways be adequate, particularly in contexts where theory provides strong,sensible and testable restrictions (Pesaran and Shin 2002). In these cases, thecointegrated vectors must be subject to identifying restrictions suggested bytheory and relevant a priori information, rather than to some normalisationprocess that does not consider the theoretical and empirical framework withinwhich the phenomenon evolves. This is particularly important in a tourismdemand context involving a system of equations, which regress tourism shareson destination prices, and a per capita tourism budget. In this case, thenumber of long-run relationships theory predicts is the number of shareequations in the system. Hence, if theory is correct, the cointegration testsinvolving variables WFt, WSt, PPt, PSt, PFt and Et, should indicate that theVAR relevant long-run relationships are those established by the share
Table 5. AIC and SBC selection criteria and diagnostic tests for the WHOLEVAR
Selection criteriaand diagnostic tests
Equations
WFt WSt PPt PSt PFt
AIC 69.79 65.69 32.27 27.28 33.69SBC 63.13 59.03 25.61 20.62 27.02Adjusted R2 0.892 0.899 0.788 0.508 0.623F statistic 25.87 27.64 12.16 4.10 5.96Serial correlation 0.35(0.55) 0.02(0.90) 0.04(0.85) 0.21(0.64) 0.10(0.66)Functional form 0.04(0.84) 0.21(0.64) 0.57(0.45) 7.24(0.01) 0.14(0.71)Normality 1.28(0.53) 0.63(0.73) 0.08(0.96) 3.50(0.17) 11.64(0.03)Heteroscedasticity 0.10(0.75) 0.01(0.92) 0.68(0.41) 0.26(0.61) 0.03(0.86)
The forecasting ability of a cointegrated VAR system 287
equations. In addition, the identification process of the structural equationsshould confirm their steady-state form as that of the equations of an AIDSmodel. Thus, for both the PUREVAR and WHOLEVAR, we expect to findexactly two cointegrated vectors and to identify the structural parameterswith restrictions that match those of the normalisation process used toidentify the share equations of an AIDS system. If this is the case, we canconfirm that two long-run relations exist in the system, and inference basedon their estimates is valid. Then, using a cointegrated structural VAR, we cansubject its long-run relations to further restrictions such as homogeneity andsymmetry, and contribute an empirical basis for confirming the rationale ofconsumer 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 andPesaran 1997, pp. 429–433):
Dyt ¼ a0y �Pyzt�1 þXp�1
i¼1CiyDzt�i þ et ð2Þ
where zt = y0t ; x0t
� �0is the (m�1) vector of variables; yt = [WF, WS, PF, PS,
PP]¢ is the (my�1) vector of endogenous variables and xt = [E, D1, D2, D3]¢is the (mx�1) vector of exogenous variables. Cointegration analysis concernsthe estimation of (2) when the rank of P ¼ ayb
0 is, at most, my. ay is the(my�r) speed of adjustment matrix, and b is the (m�r) long-run coefficientmatrix. The (r�1) vector b0 zt defines the cointegrated relationships in (2). IfPy has reduced rank, there are r � (m�1) cointegrated vectors in b. So,cointegration testing amounts to determining the number of linearly inde-pendent columns (r) in P. The null that at most r vectors exist can be testedwith 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 rejectr = 0 and r = 1 (statistic value>critical value), but cannot reject r = 2(statistic value < critical value) and the SBC criterion also supports r=2cointegrated vectors. For the WHOLEVAR, at the 5% level, the kmax statisticsuggests two vectors while the ktrace statistic does not reject the hypothesis ofonly one. This disagreement is not uncommon, particularly in cases of smallsamples and added dummy variables. Yet, we have enough evidence sup-porting the choice of r = 2. We have the kmax statistic clearly rejecting theexistence of only one in favour of two vectors; we have the SBC criterionselecting the model with two vectors; we have theory suggesting the existenceof two, and not one, long-run relationships, and we have the unmistakablesupport of both test statistics and selection criterion in the more generalPUREVAR. Given these results, we proceed by setting r = 2 for both VARmodels.
To identify the structural form of the cointegrated vectors, we use theexact-identifying restrictions implicit in the share equations of the AIDSmodel. Given the following notation for the long-run coefficient matrixes ofthe PUREVAR (bPURE) and WHOLEVAR (bWHOLE)
b0PURE ¼b�11 b�21 b�31 b�41 b�51 b�61 b�71b�12 b�22 b�32 b�42 b�52 b�62 b�72
� �
288 M. M. De Mello, K. S. Nell
b0WHOLE ¼b11 b21 b31 b41 b51 b61 b71 b81 b91 b101
b12 b22 b32 b42 b52 b62 b72 b82 b92 b102
� �;
the restrictions which identify the cointegrated vectors as share equations ofan AIDS system in both models are:
H :b11 ¼ b�11 ¼ �1 b12 ¼ b�12 ¼ 0b21 ¼ b�21 ¼ 0 b22 ¼ b�22 ¼ �1
� �
Table 7 shows the coefficients estimates of the two cointegrated vectors inPUREVAR and WHOLEVAR (asymptotic t ratios in brackets). The ‘thirdvector’ relates to the coefficients of the share equation for Portugal (WP),retrieved from the coefficient estimates of the other two equations with theadding-up property.
There is a sharp difference between the estimates of the cointegratedWHOLEVAR and PUREVAR models, both in magnitude, expected signsand statistical relevance. For instance, at the 5% level, the PUREVARestimates indicate all coefficients as irrelevant in the share equation forPortugal, and only the price of Spain and intercept as significant in theshare equations for France and Spain. In the equation for Portugal, theown-price and intercept estimates have ‘wrong’ signs and implausiblemagnitudes. In the equation for France, an implausible magnitude is alsothe case for the intercept. By contrast, the WHOLEVAR estimates aregenerally significant, present the expected signs and magnitudes and giveplausible information about how events represented by the dummy vari-ables affected the UK demand for these destinations. For instance, thecoefficients of D1 indicate that the oil crises and political changes in Por-tugal and Spain affected negatively UK tourists’ preferences for thesedestinations, favouring France instead. The coefficients of D2 indicate thatSpain and Portugal’s integration in the EU caused UK tourism flows todivert from France to the Iberian Peninsula, although favouring moreSpain than Portugal. The D3 coefficients indicate a recovery of the share
Table 6. Tests for the cointegration rank of PUREVAR and WHOLEVAR models
Eigen values H0 kmax kmax critical ktrace ktrace critical SBC
r m � r 5% 10% 5% 10%
Purevar
k1=0.9201 r = 0 m � r =6 70.76 40.53 37.65 153.96 102.56 97.87 274.71k2=0.7489 r = 1 m � r =5 38.69 34.40 31.73 83.20 75.98 71.81 290.09k3=0.5889 r = 2 m � r =4 24.89 28.27 25.80 44.50 53.48 49.95 292.78
k4=0.3273 r = 3 m � r =3 11.10 22.04 19.86 19.61 34.87 31.93 291.89k5=0.1838 r = 4 m � r =2 5.69 15.87 13.81 8.51 20.18 17.88 287.45k6=0.0958 r = 5 m � r =1 2.82 9.16 7.53 2.82 9.16 7.53 283.63Wholevar
k1=0.8798 r = 0 m � r =5 59.31 46.77 43.80 142.48 119.77 114.38 265.82k2=0.7734 r = 1 m � r =4 41.57 40.91 38.03 83.16 90.60 85.34 272.15k3=0.5833 r = 2 m � r =3 24.51 34.51 31.73 41.59 63.10 59.23 272.94
k4=0.2767 r = 3 m � r =2 9.07 27.82 25.27 17.08 39.94 36.84 268.54k5=0.2489 r = 4 m � r =1 8.01 20.63 18.24 8.01 20.63 18.24 259.74
The forecasting ability of a cointegrated VAR system 289
Table
7.Long-runcoeffi
cients
estimatesoftheexactly-identified
share
equations
Variables
CointegratedPUREVAR
CointegratedWHOLEVAR
Vector1(W
F)
Vector2(W
S)
‘Vector’3(W
P)
Vector1(W
F)
Vector2(W
S)
‘Vector’3(W
P)
WF
�1
0�1
0WS
0�1
0�1
PP
�0.4965(�
0.86)
0.1781(0.60)
0.3184(0.95)
�0.0027(�
0.05)
0.1090(1.69)
�0.1062(�
3.28)
PS
0.8214(2.09)
�0.5937(�
2.97)
�0.2277(�
0.97)
0.2256(4.68)
�0.3075(�
5.22)
0.0820(2.90)
PF
�0.2244(�
0.48)
0.3119(1.20)
�0.0875(�
0.33)
�0.3394(�
3.59)
0.3044(2.57)
0.0350(0.56)
E�0.0684(�
1.12)
0.0159(0.51)
0.0525(1.45)
0.0153(2.33)
�0.0183(2.29)
0.0030(0.78)
D1
0.0380(5.27)
�0.0154(�
1.74)
�0.0226(�
5.18)
D2
�0.0565(�
3.65)
0.0528(2.78)
0.0037(0.41)
D3
0.0574(5.17)
�0.0590(�
4.34)
0.0016(0.24)
INT
0.8309(2.20)
0.3818(1.97)
�0.2127(�
0.95)
0.3687(16.26)
0.5443(19.50)
0.0870(6.33)
290 M. M. De Mello, K. S. Nell
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 tothis result).
The reason for the different results obtained with the PUREVAR andWHOLEVAR is simple. We showed previously that there is statistical sup-port for considering Et as exogenous and including D1, D2 and D3 as rele-vant regressors. The WHOLEVAR incorporates these features, but thePUREVAR does not. Hence, the implausible estimates of the PUREVARshould be expected, and we can confirm that it is not an appropriate modelfor supplying reliable information on the long-run demand behaviour of UKtourists. Consequently, we carry on the analysis with the cointegratedWHOLEVAR model.
Tourism studies using AIDS systems seldom show well-defined cross-priceeffects among destinations. So, clear conclusions on degree and direction ofdestinations’ competing behaviour are not usually obtained. Also, homoge-neity and symmetry, which are basic premises of consumer demand theory,have often been rejected with static AIDS models. To test these hypotheseswithin the framework of a cointegrated VAR, we focus on the structuralequations of the WHOLEVAR. According to theory, homogeneity andsymmetry should hold and, according to De Mello et al.’s (2002) assumptionabout the competitive behaviour of neighbouring destinations, price changesin France (Portugal) should not affect UK demand for Portugal (France),while price changes in Spain should affect significantly UK tourism demandfor both France and Portugal.4
In Table 8, the LR test results indicate that null cross-price effects betweenFrance and Portugal, homogeneity and symmetry, and all these hypothesessimultaneously, cannot be rejected. These results underline the importance ofwell-defined structural models for delivering empirical support to theoreticalassumptions and helping understand long run demand behaviour. Table 9shows 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 andexpenditure changes have on UK demand are better evaluated through thecorresponding elasticities. We compute the expenditure and uncompensatedown- and cross-price elasticities, using the CSV long-run coefficient estimatesand the formulae given in A2 of Appendix A.5 Table 10 presents these andthe 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 competingbehaviour of tourism destinations. Positive and negative signs indicate substitutability andcomplementarity, respectively. The non-significance of cross-price effects between France andPortugal is a reasonable expectation given differences in size, origin proximity, product diversityand features of the UK demand for both countries (e.g. visit periodicity and average length ofstay). 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 statisticallysignificant at the 5% level.6 We use De Mello et al.’s elasticities estimates for the period (1980–1997), denoted by theauthors as the ‘‘second period’’, because these correspond to more recent behaviour of UKtourism demand.
The forecasting ability of a cointegrated VAR system 291
The estimated elasticities of the CSV and AIDS are similar. The expen-diture elasticities are close to unity for all destinations in both models and,except in the CSV equation for Spain, the own-price elasticity estimates areclose to �2. The CSV and AIDS cross-price elasticities also give similarindications: insignificant cross-price effects between the equations for Por-tugal and France indicating that these two destinations do not compete be-tween them; significant cross-price effects between Spain and France andSpain and Portugal indicating destination substitutability and bilateralcompetitive behaviour. The magnitudes of the elasticities, showing that theUK demand for Portugal or France is more sensitive to price-changes inSpain than that for Spain is to price changes in its neighbours, suggest a morestable and persistent demand for Spain than that for its neighbouring com-petitors. The negligible sensitivity of UK demand for Spain to price changesin Portugal, but considerable sensitivity of UK demand for Portugal to pricechanges in Spain, denote substantial differences in market attractiveness andsize between the two Iberian countries. The significance and magnitudes ofthese cross-price effects indicate Spain and its smaller neighbour as unevencompetitors for similar demand niches, and Spain as a clearly preferreddestination. France and Spain, however, are both ‘tourism giants’ and simi-larly popular destinations. Although UK demand for Spain is less sensitive toprice changes in France (0.523) than that for France is to price changes inSpain (0.793), their close scale reflects Spain and France ‘neck and neck’competition for UK tourists. This competitive behaviour can also be detectedin Figure 1, by the mirror-like movements of France and Spain’s tourismshares. The cross-price effects between Spain and its neighbours, its relatively
Table 8. Tests of over-identifying restrictions on the cointegrated WHOLEVAR
Hypothesis LR ! v2 (i) 5% Critical value
Null cross-price effects v2 (2)=0.36 5.99Homogeneity and symmetry v2 (3)=7.28 7.81Homogeneity, symmetryand null cross-price effects
v2 (4)=7.53 9.49
Table 9. Long-run coefficients estimates of the CSV
Variables CSV
Vector 1 (WF) Vector 2 (WS) ‘Vector’ 3 (WP)
WF �1 0WS 0 �1PP 0 0.0895 (5.80) �0.0895 (�5.80)PS 0.2891 (4.79) �0.3785 (�5.97) 0.0895 (5.80)PF �0.2891 (�4.79) 0.2891 (4.79) 0E 0.0091 (1.16) �0.0121 (�1.38) 0.0030 (0.95)D1 0.0354 (3.83) �0.0115 (�1.13) �0.0239 (�7.05)D2 �0.0373 (�2.36) 0.0360 (2.02) 0.0013 (0.20)D3 0.0429 (4.56) �0.4184 (�3.95) �0.0011 (�0.27)INT 0.3849 (13.17) 0.5230 (16.96) 0.092 (11.69)
292 M. M. De Mello, K. S. Nell
low own-price elasticity and close to unity expenditure elasticity, indicateSpain as a preferred destination for UK tourists. Yet, the trends of the sharesin Fig. 1 show that Spain is losing ground to its neighbours in more recentyears. If this tendency persists, it is possible that France becomes the favouritedestination relative to Spain, and Portugal can see its UK tourism shareincreased beyond the 10% level.
As the CSV model fully complies with the theoretical predictions under-lying the AIDS model, the similarity of their estimates and the cointegrationanalysis implemented with the VAR, confirming the share equations as theonly meaningful long-run relations, further support the AIDS approach as anadequate means of explaining the long-run behaviour of UK tourism de-mand. Yet, the analysis is not complete without assessing the forecastingability 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 practicalimportance of imposing cointegration restrictions for forecast accuracy in smallsamples is worthy of study’’. Thus, using a small data set to compare thepredictive accuracy of an unrestricted VAR with that of a cointegratedstructural VAR and AIDS models, offers an empirical contribution for re-search in this area. Yet, given the widespread use of first difference models inforecasting and the fact that a differenced VAR could reveal itself morerobust to structural breaks than the PUREVAR (hereafter PV), we include inour comparison exercise forecasts obtained from a first difference unrestrictedVAR (hereafter DV), reflecting the sort of benchmark models forecastersgenerally use. The difference forecasts obtained with the DV are transformedback into levels for comparison with the level forecasts of the other models.
Within a tourism demand context, the relevance of distinguishing betweenshort and long run forecast accuracy cannot be overstressed. Frequently, thisactivity involves long term investments of substantial dimensions, for whichthe ‘go-ahead’ decisions have to be carefully pondered in view of major lossesthat 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 runforecasts are important, it seems that accurate long run prediction of demand
Table 10. Expenditure and uncompensated own- and cross-price elasticities estimates
Equations Models Expenditureelasticities
Own-priceelasticities
Cross-price elasticities
PP PS PF
WP CSV 1.039 �2.158 X 1.137 �0.017AIDS 0.947 �1.797 X 0.830 0.019
WS CSV 0.979 �1.057 0.161 X 0.523AIDS 1.150 �1.933 0.124 X 0.658
WF CSV 1.026 �1.817 �0.002 0.793 XAIDS 0.808 �1.901 0.017 1.077 X
The forecasting ability of a cointegrated VAR system 293
levels is a crucial matter for sustained success in tourism business. Unfortu-nately, data series available in this area are seldom long enough to obtainreliable long run forecasts.
To assess the predictive ability of the four models, we estimate them forthe period 1969–1993, leaving the last years (1994-1997) as the forecastingevaluation period.7 We also consider recursive estimation of the modelsthroughout 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 sensiblydecide which model is the best forecaster. One-step prediction is important,not only because some tests for equal accuracy have optimal properties withinthis horizon, but also because it makes available the full set of four fore-casting observations. Two-step prediction would leave us with three obser-vations, three-step with two, and four with one. Hence, to evaluate short-runaccuracy, we use one-step ahead forecasts obtained with the models esti-mated for the period 1969-1993 (‘regular’ one-step forecasts) and one-stepforecasts obtained with the models estimated recursively throughout theevaluation period (‘recursive’ one-step forecasts). To evaluate long-runaccuracy, the only possible ‘longer’ run we have available with the full set offour observations is four multi-step ahead forecasts, that is, one-step aheadforecast for 1994, two-step for 1995, three for 1996 and four for 1997. See A6in 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 ‘tradi-tional’ statistics to evaluate forecasting performance: the mean squared error(MSE), the root mean squared error (RMSE) and the mean absolute per-centage error (MPE). As a visual aid of forecast accuracy, plots of the actualand forecasted levels of the models, for ‘regular’ one-step and multi-stepforecasts 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 ofthe models forecasting quality. Given the measures’ ranking consistency forall models, all destination shares, and both ‘regular’ and ‘recursive’ one-steperrors, we focus the analysis on the MPE. PV is the worse forecaster forFrance and Spain’s shares (MPE>8%) and the best for Portugal’s share(MPE<4%). CSV and DV are the best forecasters for, respectively, theshares of France (MPE<3%) and Spain (MPE<2%). For all share equa-tions, DV, CSV and AIDS seem to have fairly similar performances. But, istheir 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 determinethe 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 twocointegrated vectors, even when the sample is reduced to 1969-1993. After identifying theWHOLEVAR structural form and imposing homogeneity, symmetry and null cross-price effectsrestrictions, we computed one-step and multi-step ahead forecasts.
294 M. M. De Mello, K. S. Nell
Table
11.Forecastingquality
summary
statistics
FRANCE
SPAIN
PORTUGAL
DV
PV
CSV
AID
SDV
PV
CSV
AID
SDV
PV
CSV
AID
S
‘Regular’one-step
aheadforecast
errors
MSE
0.0002
0.0019
0.0001
0.0003
0.0001
0.0021
0.0002
0.0003
0.0000
0.0000
0.0000
0.0000
RMSE
0.0142
0.0441
0.0122
0.0168
0.0112
0.0461
0.0150
0.0165
0.0059
0.0031
0.0064
0.0053
MPE
2.99%
11.06%
2.77%
3.90%
1.57%
8.80%
2.17%
2.84%
5.59%
3.30%
6.57%
5.21%
‘Recursive’
one-step
aheadforecast
errors
MSE
0.0003
0.0056
0.0001
0.0002
0.0001
0.0061
0.0002
0.0002
0.0000
0.0000
0.0000
0.0000
RMSE
0.0159
0.0752
0.0089
0.0151
0.0121
0.0782
0.0150
0.0133
0.0063
0.0036
0.0079
0.0053
MPE
3.66%
18.79%
1.87%
3.55%
1.95%
14.78%
2.04%
2.13%
5.68%
3.87%
8.25%
8.87%
‘Multi-step’aheadforecast
errors
MSE
0.0006
0.0009
0.0000
0.0002
0.0015
0.0009
0.0000
0.0002
0.0003
0.0000
0.0000
0.0000
RMSE
0.0251
0.0302
0.0068
0.015
0.0385
0.0294
0.0094
0.0133
0.0164
0.0058
0.0071
0.0079
MPE
6.33%
6.30%
1.54%
3.54%
6.84%
4.40%
1.10%
2.12%
17.90%
6.11%
7.46%
8.93%
The forecasting ability of a cointegrated VAR system 295
Equal accuracy of two competing forecast series, i and j, can be judged bytesting the significance of the difference (dij) between economic losses asso-ciated with forecast error series ei and ej. While in many cases, economic lossmay be poorly accessed by ‘traditional’ measures, in a tourism demandcontext we may assume that the loss related with prediction failure is asymmetric function of the forecast error, since over-forecasting can be ascostly as under-forecasting. Hence, we allow time t loss associated with a
Fig. 3. Actual levels and ‘regular’ one-step forecasts for France, Spain and Portugal shares
296 M. M. De Mello, K. S. Nell
series of n forecasts to be a direct function of the forecast error, g(e), withMSE as the standard measure of forecast quality such that g(e)=e2. The nullof equal accuracy of two competing h-step forecast series i and j is E(dijt) ¼ 0,where dijt ¼ g(eit)�g(ejt); t ¼ 1,…,n. For testing the null, we use Harveyet al.’s (1997) S1* test, which is a modified version of the Diebold andMariano’s (1995) S1 test, and a simple encompassing test proposed in Hendry(1986) and Clements and Hendry (1998).
Fig. 4. Actual levels and multi-step forecasts for France, Spain and Portugal shares
The forecasting ability of a cointegrated VAR system 297
To facilitate interpretation of the tests, we first provide a brief overviewof the different test statistics, and mention some conclusions that the au-thors report in their studies. Diebold and Mariano’s (1995) contribution oncomparative prediction accuracy, proposes and evaluates a set of explicittests for the null of equal accuracy between two forecast series, which arevalid for a very wide class of loss functions (not needing to be quadratic,symmetric, or even continuous), and for forecast errors that can be non-Gaussian, nonzero mean, serially correlated and contemporaneously cor-related. In particular, the asymptotic test statistic S1 can handle a seriallycorrelated loss differential better than the other proposed statistics. How-ever, the authors also recognise that although S1 is robust to serial andcontemporaneous correlation in large samples, it can be oversized in smallsamples. Harvey et al. (1997) show that this problem becomes more acuteas the forecast horizon, h, increases, and explore the possibility of allevi-ating the problem through modifications of the Diebold-Mariano (DM)procedure. By employing an approximately unbiased estimator for thevariance of the mean of d [var(�d)], the authors propose the modified DM
test statistic S1� ¼ S1 nþ 1� 2hþ n�1hðh� 1Þ� �
=n 1=2
, where S1 is the
original DM statistic, �d varð�dÞ� ��1=2
. The other modification of the DM testthat Harvey et al. (1997) suggest, is to compare S1* with critical valuesfrom the Student’s T(n-1) distribution, rather than from the N(0;1) usedfor the S1 statistic. The authors report ‘dramatic improvements’ occurringfor small samples with S1* relative to S1, but concede that the originalversion has advantages in small samples when the error terms are normallydistributed.
The forecast-encompassing tests investigate whether a model (Mb), canexplain the forecast errors of another model (Ma) and vice-versa. Giventhe 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, Mbforecast-encompasses Ma. Given that the possible presence of heterosce-dasticity and/or autocorrelation in the regression errors could invalidatethe conclusions of the significance tests (Harvey et al., 1998), these arepreformed using robust standard errors supplied by the Newey and West(1987) consistent covariance matrix with a Parzen truncation lag ofone. The results of S1* and encompassing tests (involving series iencompassing 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 bythe quality measures of Table 11. For Portugal’s share, S1* does not detectsignificant accuracy differences between the four models (MPEs rangingfrom 3.3% to 6.6% for the ‘regular’ errors, and from 3.8% to 8.9% forthe ‘recursive’ errors). Yet, the encompassing tests show that DV and PVforecast-encompass AIDS and that AIDS forecast-encompasses CSV. ForFrance 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 withh-step, and not multi-step horizons. Since multi-step and one-step are structurally different basisto evaluate accuracy, comparison between one- and multi-step forecasting performances shouldbe interpreted with prudence.
298 M. M. De Mello, K. S. Nell
Table
12.Tests
forequalone-step
forecastingaccuracy
Shares
Test
Statistic
distribution
10%
Critical
values
Competingforecast
series
iversusj
DV(i)versus
PV
(j)
DV(i)versus
CSV(j)
DV(i)versus
AID
S(j)
CSV(i)versus
PV
(j)
AID
S(i)versus
PV
(j)
CSV(i)versus
AID
S(j)
‘Regular’one-step
aheadforecast
errors
Portugal
S1*
T(3)
2.35
0.46
�0.07
0.29
1.04
0.45
0.20
iencompasses
j1.79
2.02
�2.95
1.80
1.89
2.16
jencompasses
i�1.04
�1.10
�1.03
2.18
�2.99
�3.00
Spain
S1*
T(3)
2.35
�3.20
�0.43
�0.98
�2.27
�2.51
�0.20
iencompasses
j�13.78
�1.49
0.33
�11.78
�13.06
�1.45
jencompasses
i�2.36
�2.39
2.33
�1.45
0.32
0.32
France
S1*
T(3)
2.35
�2.94
0.18
�0.83
�2.21
�2.43
�0.52
iencompasses
j9.13
0.80
0.23
10.82
9.28
0.83
jencompasses
i2.80
2.68
2.77
0.86
0.23
0.24
‘Recursive’
one-step
aheadforecast
errors
Portugal
S1*
T(3)
2.35
0.47
�0.27
�0.46
1.47
�1.51
0.02
iencompasses
j1.88
2.21
�11.08
1.78
2.06
2.43
jencompasses
i�1.76
�1.84
�1.72
2.29
�11.30
�11.56
Spain
S1*
T(3)
2.35
�2.30
�0.23
�0.83
�2.41
�2.31
0.15
iencompasses
j�7.79
�1.78
�0.003
�8.43
�7.16
�1.72
jencompasses
i�3.26
�3.26
�3.18
�1.85
0.01
�0.01
France
S1*
T(3)
2.35
�2.33
0.69
0.15
�2.52
�2.31
�0.83
iencompasses
j8.93
1.05
1.21
8.20
9.86
1.09
jencompasses
i4.99
4.76
4.94
1.03
1.29
1.19
The forecasting ability of a cointegrated VAR system 299
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) accuracybetween either of these three models and PV is not offered, because thestatistic values are in the vicinity of the critical value. However, theencompassing tests unmistakably show for the shares of France and Spain,that DV, CSV and AIDS forecast-encompass PV and that CSV forecast-encompasses DV. Thus, for one-step forecasts, the main conclusion fromevidence gathered in Tables 11 and 12 is that DV, CSV and AIDS modelsare equally accurate forecasters and that DV performs better the role ofbenchmark than PV, at least for Spain and France’s shares. Nevertheless, afar more interesting debate can be associated with the performance of PV,which imposes neither integration nor cointegration, and DV, which im-poses integration, relative to that of CSV, which imposes both. Christof-fersen and Diebold (1998) state that these models differences in forecastaccuracy ‘‘may simply be due to the imposition of integration, irrespective ofwhether cointegration is imposed’’ (p.455). This contradicts several previousanalyses (e.g., Engle and Yoo 1987; Clements and Hendry 1998) thatattribute those differences to the imposition of cointegration and notintegration. Surely the DV model imposes integration, and the evidenceprovided by S1* shows that it performs as well as CSV. However, thedebate relates to long-horizon forecasts, and evidence obtained from one-step horizon and a sample of four forecast observations certainly is notsufficient to add any substantial contribution to the debate. Nevertheless,this is an important issue to which further consideration will be devoted infuture research.
4.2. Multi-step ahead forecast accuracy
From Table 11 and Fig. 4, we can again perceive some forecast quality fea-tures of the models, given their consistent ranking by all quality measures, inall destination shares. Once again we focus on MPE. DV is now the worseforecaster for all shares. For Spain and France’s shares, CSV (MPEs<2%)and AIDS (MPEs<3.6%) 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 perfor-mance, but much better than that of DV (MPE=17.9%). This can be con-firmed in Figure 4, where the point forecasts of CSV and PV almost overlapfor the whole forecasting range. Figure 4 also shows that the DV forecaststend to diverge from the actual values as we move away from 1993, althoughbeing remarkably accurate for 1994, and that PV completely misses theturning points of the actual shares for all destinations, while CSV misses onlyone and the AIDS model misses two in each destination. Based on theseresults, we conclude that for multi-step forecasting, the PV and DV modelscannot be considered accurate forecasters, when compared with structuralmodels such as the AIDS or CSV.
Given that in tourism research, data availability on the explanatoryvariables is much wider than that on the dependent variables (namelytourism expenditure of specific origins in specific destinations), the
300 M. M. De Mello, K. S. Nell
forecasting evaluation periods are usually short, which prevents clearconclusions with tests of equal accuracy for longer horizons than one- ortwo-steps ahead. However, multi-step forecasts for four-, five- or eveneight-step ahead are easily obtainable if four, five or eight observations areleft out of the estimation period, to be object of forecasting evaluation.Yet, tests specifically tailored for multi-step accuracy comparison betweencompeting forecasts have not been given as much attention as h-step ones,and this is a subject worthy of attention, given the importance that suchtests 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 staticsystem of equations that includes nonstationary variables and assumes ex-ogeneity for all its regressors. These features can risk the validity of esti-mation, inference and forecasting procedures, if no cointegratedrelationships exist and/or exogeneity assumptions do not hold. As analternative, we specified a reduced-form VAR for the same data, estab-lishing its lag-length, deterministic components and endogenous/exogenousdivision of variables with appropriate tests, and used Johansen’s rank test todetermine the number of cointegrated vectors in the VAR. Theory under-lying a system of equations regressing two destination shares on destinationprices and an origin tourism budget, predicts the existence of exactly twolong-run relationships. Thus, in a VAR with the same variables, two co-integrated vectors should be accounted for. The rank test provided evidenceto support this prediction. The theory underlying an AIDS model for UKtourism demand also establishes the structural form of the long-run rela-tions it predicts. Hence, the structural parameters of the cointegrated VARshould be exactly-identified with restrictions matching those of the nor-malisation process that identifies the share equations in an AIDS system.This was accomplished and the resulting cointegrated VAR was then sub-jected to the additional restrictions of homogeneity, symmetry and nullcross-price effects between the equations of Portugal and France. Theserestrictions were not rejected. Consequently, evidence was obtained on theability of the cointegrated VAR to comply with theoretical predictionsunderlying consumers’ demand behaviour and destinations’ competitiveconduct. Finally, the structural coefficients of the restricted cointegratedVAR (CSV) were used to compute the expenditure, own- and cross-priceelasticities of UK demand. These estimates, similar to the correspondingones of the AIDS model, proved to be statistically relevant and empiricallyplausible. Besides its consistency and statistical robustness, the CSV modelalso shows an excellent predictive ability. Indeed, when compared with thereduced-form PUREVAR (PV), differenced VAR (DV) and AIDS models,the quality criteria indicate the CSV as the best predictor in the longer-runof multi-step forecasting. For one-step forecasts however, tests of equalaccuracy indicate the CSV, AIDS and DV as equally precise forecasters.Given the modelling simplicity and estimation ease of DV, its accurate one-step forecasts recommend its use in short-run forecasting of tourism de-mand levels. Yet, if the interest is longer run prediction, the evidence ob-tained tends to favour the CSV or AIDS models rather than DV or PV.
The forecasting ability of a cointegrated VAR system 301
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 innonnegative quantities qi at given prices pi; i ¼ 1; . . . ; n. Letq ¼ ðq1; q2; . . . ; qnÞ be the quantity vector of the n goods purchased, andp ¼ ðp1; p2; . . . ; pnÞ the price vector. The consumer’s budget constraint isPn
i¼1piqi ¼ x. Given utility function uðqÞ, the consumer maximises utility,
subject to the budget constrain:
max uðqÞ; subject toXn
i¼1piqi ¼ x ðA1Þ
The solution for this maximisation problem leads to the Marshallian(uncompensated) demand functions qi ¼ giðp; xÞ. Alternatively, the consum-ers’ problem can be defined as the minimum total expenditure necessary toattain a specific utility level u�, at given prices:
minXn
i¼1piqi ¼ x subject to uðqÞ ¼ u� ðA2Þ
The solution for this minimisation problem leads to the Hicksian (compen-sated) demand functions qi ¼ hiðp; uÞ. Therefore, a cost function can be de-fined as
C p; uð Þ ¼Xn
i¼1pihi p; uð Þ ¼ x ðA3Þ
Given total expenditure x and prices p, the utility level u* is derived from thesolution in (A1). Solving (A3) for u, an indirect utility function is obtainedsuch that u ¼ vðp; xÞ. The AIDS model specifies a cost function, which is usedto derive the demand functions for the commodities. The derivation processcan be summarised in the following three steps:
1) @ C p;uð Þ@ pi
¼ hi p; uð Þ is derived establishing the Hicksian demand functions.
2) solving (A3) for u, the indirect utility function is obtained, such thatu ¼ vðp; xÞ.
3) hi½p; vðp; xÞ� ¼ giðp; xÞ is retrieved stating the Hicksian and the Mar-shallian demand functions as equivalent.
These demand functions have the following properties:
1. Adding-up: Ripihi p; uð Þ ¼P
ipigi p; xð Þ ¼ x; all budget shares sum to unity;
2. Homogeneity: hi p; uð Þ ¼ hi hp; uð Þ ¼ gi p; xð Þ ¼ gi hp; hxð Þ 8h > 0 ; a pro-portional change in all prices and expenditure has no effect on the quan-tities purchased;
3. Symmetry: @hi p;uð Þ@pj¼ @hj p;uð Þ
@pi; 8i 6¼ j ; consumer’s choices are consistent;
4. Negativity: The (nxn) matrix of elements @hi p;uð Þ@pj
is negative semi-definite,
that is, for any n vector n , the quadratic form RiRjninj@hi p;uð Þ@pj� 0 , i.e., a
rise in prices results in a fall in demand as required for normal goods.
The AIDS model specifies the cost function:
302 M. M. De Mello, K. S. Nell
ln Cðp; uÞ ¼ aðpÞ þ u:bðpÞ ðA4Þ
where a pð Þ ¼ a0 þ Riai ln pi þ 12RiRjcij ln pi ln pj and b pð Þ ¼ b0
Qi
pbii .
The derivative of (A4) with respect to ln pi is:
@ ln C p; uð Þ@ ln pi
¼ ai þ Rjcij ln pj þ ubib0Pipbii : (A5)
As C p; uð Þ ¼ x, lnC p; uð Þ ¼ ln x; then ln x ¼ aðpÞ þ u:bðpÞ ðA6Þ
Solving (A6) for u we obtain u ¼ ln x� a pð Þ½ �= b pð Þ½ � ðA7ÞSubstituting (A7) in (A5) we have
@ lnC �ð Þ@ ln pi
¼ @C �ð Þ@pi
pi
C �ð Þ ¼ hi �ð Þpi
C �ð Þ ¼piqi
x¼ wi
¼ ai þ Rjcij ln pj þ bi ln x� a pð Þ½ �:If a price index P is defined such that ln P ¼ aðpÞ, then
@ lnC p; uð Þ@ ln pi
¼ ai þ Rjcij ln pj þ bi ln x� ln P½ �
or wi ¼ ai þ Rjcij ln pj þ bi ln x=Pð Þ;
where ln P ¼ a0 þ Rkak ln pk þ1
2RkR‘c
�k‘ ln pk ln p‘
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 ishare of the origin’s tourism budget allocated to the n destinations This share’svariability is explained by tourism prices (p) in i and alternative destinations jand 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
i
ai ¼ 1;X
i
bi ¼ 0;X
i
cij ¼ 0; for all j;
2. homogeneity restriction requiring that a proportional change in all pricesand expenditure has no effect on the quantities purchased:
Pj
cij ¼ 0 , forall i;
3. symmetry restriction requiring consumer consistent choices: cij ¼ cji, for alli, j;
4. negativity restriction requiring that a rise in prices result in a fall in de-mand, i.e., negative own-price elasticities for all destinations.
The restrictions imposed on a and c comply with these assumptions andensure that in (A8), P is defined as a linear homogeneous function of indi-vidual prices. If prices are relatively collinear, then P is ‘‘approximatelyproportional to any appropriately defined price index, for example, the one usedby Stone, the logarithm of which is
Pwk ln pk ¼ ln P�’’ (Deaton and
Muellbauer 1980a, p.76). Hence, the deflator P in (A8) can be substituted bythe Stone price index ln P* such that,
The forecasting ability of a cointegrated VAR system 303
ln P � ¼X
i
w Bi ln pi ðA9Þ
where w Bi 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 fol-lowing form:
wi ¼ a�i þX
j
cij ln pj þ bi lnx
P �
� �ðA10Þ
A2. Expenditure, own- and cross-price elasticities
Expenditure and price elasticities cannot be directly accessed in (A10), givenits log-linear form, but their values can be retrieved from (A10) coefficientsusing the following formulae:
Expenditure elasticity : ei ¼1
�wi
dwi
d ln xþ 1 ¼ bi
�wiþ 1
Uncompensated own-price elasticity : eii ¼1
�wi
dwi
d ln pi� 1 ¼cii
�wi� bi
w Bi
�wi� 1
Uncompensated cross-price elasticity : eij ¼1
�wi
dwi
d ln pj¼
cij
�wi� bi
w Bj
�wi
Compensated own-price elasticity : e�ii ¼ eii þ w Bi ei ¼
cii
�wiþ w B
i � 1
Compensated cross-price elasticity : e�ij ¼ eij þ wBi ei ¼
cii
�wiþ wB
j
where w Bj is destination j’s share (j=1,…,n) in the year base and �wi is desti-
nation 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 com-modities in a multi-stage budgeting process implying independent prefer-ences. Thus, for the AIDS model of UK tourism demand, it is assumed thatthe UK tourism expenditure allocated to France, Spain and Portugal isseparable from that allocated to other destinations, and that the decision tospend money in those countries is made in several stages. First, UK touristsallocate their budget to tourism and other goods; then to tourism in France,Spain and Portugal and other destinations; finally they decide betweenFrance, Spain or Portugal. The AIDS system reproduces this last stage usingthe following form:
WPt ¼ aP þ cPP PPt þ cPSPSt þ cPF PFt þ bP Et þ upt
WSt ¼ aS þ cSP PPt þ cSSPSt þ cSF PFt þ bSEt þ ust
WFt ¼ aF þ cFP PPt þ cFSPSt þ cFF PFt þ bF Et þ uft
8<
:
304 M. M. De Mello, K. S. Nell
A4. Variables’ definition
The variables in the AIDS model of UK tourism demand for France, Spainand Portugal are the UK tourism budget shares allocated to these destina-tions WP, WS and WF; destination prices PP, PS, PF and UK real per capitatourism budget E. Each Wi, i = F S P, is defined as:
Wi ¼ EXPi= EXPF þ EXPS þ EXPPð Þwhere EXPi is the nominal tourism expenditure allocated by UK tourists todestination i. The effective price of tourism in destination i is defined as:
Pi ¼ ln CPIi=CPIUKð Þ=Ri½ �where CPIi is destination i consumer price index, CPIUK is UK consumerprice index, Ri is the exchange rate between i and the UK, defined as numberof currency units of country i per unit of UK currency. The per capita UKreal tourism expenditure allocated to all destinations is:
E ¼ lnX
i
EXPi=UKP
!=P �
" #
where UKP is UK population and lnP* is the Stone index defined inEq. (A10).
A5. Data sources
The UK tourism expenditure data, disaggregated by destinations and mea-sured in £ million sterling, were obtained from Business Monitor MA6 (1970-1993), continued as Travel Trends (1994–1998). The UK population, priceindexes and exchange rates were obtained from the International FinancialStatistics (IMF) Yearbooks (1984, 1990 and 1998).
A6. Multi-step and h-step ahead forecasts
Suppose that, with observations of Yt and Xtðt ¼ 1; . . . ; T Þ in the period1969–1993, we estimate Yt ¼ b1 þ b2Xt þ b3Yt�1. The 1-step ahead forecasts ofY are computed assuming that X values are available up toT þ j; ðj ¼ 1; 2; . . . ;mÞ and Y values are available up to T+j�1. Hence, incomputing 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 computedassuming that X values are known up to T+j, but Y values are known onlyup to T+j�2. 3-step forecasts are computed assuming knowledge of XTþj
and YTþj � 3, and so on. We denote the 1-step YTþj forecast, given YTþj�1, as
Y fTþj=YTþj�1
; 2-step YT þ j forecast, given YT þ j� 2, as Y fTþj=YTþj�2
; etc. The
1-step forecasts of Y for 1994-1997 are:
The forecasting ability of a cointegrated VAR system 305
Y f1994=Y1993
¼ b1 þ b2X1994 þ b3Y1993;
Y f1995=Y1994
¼ b1 þ b2X1995 þ b3Y1994;
Y f1996=Y1995
¼ b1 þ b2X1996 þ b3Y1995;
Y f1997=Y1996
¼ b1 þ b2X1997 þ b3Y1996;
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 notknown. So, we have to use a forecast of Y1994 to insert into the forecastingequation. The same has to be done for the 2-step forecasts of Y for 1996and 1997, but updating the forecasts according to knowledge of YTþj�2.Hence, the 2-step ahead forecasts of Y for 1994–1997 are:
Y f1994=Y1993
¼ b1 þ b2X1994 þ b3Y1993
(the same as the 1st forecast in the 1-step forecasts)
Y f1995=Y1993
¼ b1þ b2X1995þ b3Yf1994=Y1993
Y f1996=Y1994
¼ b1þ b2X1996þ b3Yf1995=Y1994
; where Y f1995=Y1994
¼ b1þ b2X1995þ b3Y94
Y f1997=Y1995
¼ b1þ b2X1997þ b3Yf1996=Y1995
; where Y f1996=Y1995
¼ b1þ b2X1996þ b3Y95;
gathering only three ‘true’ 2-step forecasts.The 3-step forecasts of Y for 1994–1997 are:
Y f1994=Y1993
¼ b1 þ b2X1994 þ b3Y1993
(the same as the 1st forecast in the1-step forecasts)
Y f1995=Y1993
¼ b1 þ b2X1995 þ b3Y f1994=Y1993
(the same as the 2nd forecast in the 2-step forecasts)
Y f1996=Y1993
¼ b1 þ b2X1996 þ b3Y f1995=Y1993
Y f1997=Y1994
¼ b1 þ b2X1997 þ b3Y f1996=Y1994
where Y f1996=Y1994
¼ b1 þ b2X1996 þ b3Yf1995=Y1994
¼
¼ b1 þ b2X1996 þ b3 b1 þ b2X1995 þ Y1994
� �;
gathering only two ‘true’ 3 -step forecastsThe 4-step forecasts of Y for 1994-1997 are:
Y f1994=Y1993
¼ b1 þ b2X1994 þ b3Y1993
(the same as the 1st forecast in the 1-step forecasts)
Y f1995=Y1993
¼ b1 þ b2X1995 þ b3Y f1994=Y1993
(the same as the 2nd forecast in the 2-step forecasts)
306 M. M. De Mello, K. S. Nell
Y f1996=Y1993
¼ b1 þ b2X1996 þ b3Yf1995=Y1993
(the same as the 3rd forecast in the 3-step forecasts)
Y f1997=Y1993
¼ b1 þ b2X1997 þ b3Yf1996=Y1993
gathering just one ‘true’ four-step ahead forecast. We call this last set offorecasts multi-step ahead since there is one of each h-step.
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