Empirical Economics (2005) 30:277–308
DOI 10.1007/s00181-005-0241-0

The forecasting ability of a cointegrated VAR
system ...
278

M. M. De Mello, K. S. Nell

1. Introduction
Witt and Witt (1992, 1995), Sinclair and Stabler (1997) and Song and Witt...
The forecasting ability of a cointegrated VAR system

279

1997, we compare this study’s structural estimates and forecast...
280

M. M. De Mello, K. S. Nell

Fig. 1. Variables in levels

2.1. Order of integration of the variables included in the V...
The forecasting ability of a cointegrated VAR system

281

Fig. 2. Variables in first differences

Information Criterion (AI...
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

...
The forecasting ability of a cointegrated VAR system

283

where z’t = [WFt , WSt , PPt , PSt , PFt , Et] ; A0 is a (6 Â 1...
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...
The forecasting ability of a cointegrated VAR system

285

If uit and t are uncorrelated we can say that EV(uit, s ) = 0 f...
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-si...
The forecasting ability of a cointegrated VAR system

287

To evaluate the quality of the WHOLEVAR, Table 5 includes the s...
288

M. M. De Mello, K. S. Nell

equations. In addition, the identification process of the structural equations
should confi...
The forecasting ability of a cointegrated VAR system

289

Table 6. Tests for the cointegration rank of PUREVAR and WHOLEV...
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....
The forecasting ability of a cointegrated VAR system

291

for France at the expense of Spain’s share in the post-integrat...
292

M. M. De Mello, K. S. Nell

Table 8. Tests of over-identifying restrictions on the cointegrated WHOLEVAR
Hypothesis

...
The forecasting ability of a cointegrated VAR system

293

Table 10. Expenditure and uncompensated own- and cross-price el...
294

M. M. De Mello, K. S. Nell

levels is a crucial matter for sustained success in tourism business. Unfortunately, data...
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...
296

M. M. De Mello, K. S. Nell

Fig. 3. Actual levels and ‘regular’ one-step forecasts for France, Spain and Portugal sha...
The forecasting ability of a cointegrated VAR system

297

Fig. 4. Actual levels and multi-step forecasts for France, Spai...
298

M. M. De Mello, K. S. Nell

To facilitate interpretation of the tests, we first provide a brief overview
of the differe...
Test

Statistic
distribution

S1*
i encompasses
j encompasses
S1*
i encompasses
j encompasses
S1*
i encompasses
j encompas...
300

M. M. De Mello, K. S. Nell

be confirmed by S1*, which does not reject equal accuracy between DV,
CSV and AIDS models ...
The forecasting ability of a cointegrated VAR system

301

forecasting evaluation periods are usually short, which prevent...
302

M. M. De Mello, K. S. Nell

Appendix A
A1. Derivation of Deaton and Muellbauer’s (1980a, 1980b) AIDS model
Let x be t...
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...
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 ye...
The forecasting ability of a cointegrated VAR system

305

A4. Variables’ definition
The variables in the AIDS model of UK ...
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 ...
The forecasting ability of a cointegrated VAR system

307

^
^
^ ^f
^f
Y1996=Y1993 ¼ b1 þ b2 X1996 þ b3 Y1995=Y1993
^f
Y19...
308

M. M. De Mello, K. S. Nell

International Monetary Fund (1984-1998) International financial statistics yearbook. Inter...
Empirical economics
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Empirical economics

  1. 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. 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. 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).
  4. 4. 280 M. M. De Mello, K. S. Nell 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. 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. 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. 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. 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. 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. 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. 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. 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 bà 11 21 31 41 51 61 71 b0PURE ¼ à à à à à à à b12 b22 b32 b42 b52 b62 b72
  13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Ã É1=2 , where S1 is the test statistic S1à ¼ S1  n þ 1 à 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. 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. 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. 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. 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 uà 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; xފ 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. 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 aðpފ=½bðpފ 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 pފ: 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. 28. 304 M. M. De Mello, K. S. Nell ln P à 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 à 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. 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 à 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. 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)
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