Us<strong>in</strong>g the SUR Model <strong>of</strong> Tourism Dem<strong>and</strong> for Neighbour<strong>in</strong>g Regions <strong>in</strong> Sweden <strong>and</strong> Norway 111homogeneity <strong>and</strong> symmetry. This accords with the microeconomic foundations <strong>of</strong> the model<strong>and</strong> <strong>in</strong>creases the credibility <strong>of</strong> the elasticity values.Substitutability <strong>and</strong> complementarity among dest<strong>in</strong>ations are <strong>in</strong>dicated by positive <strong>and</strong>negative relative price elasticities, respectively. Clear conclusions about thecomplementarity or substitutability among dest<strong>in</strong>ations are not usually obta<strong>in</strong>ed <strong>in</strong> studiesus<strong>in</strong>g the ISUR model, which have produced few well def<strong>in</strong>ed relative price effects.However, the results <strong>in</strong> this study seem consistent <strong>and</strong> also co<strong>in</strong>cide with a prioriexpectations. Hence they are taken as an <strong>in</strong>dication <strong>of</strong> the relative magnitudes <strong>and</strong>directions <strong>of</strong> changes <strong>in</strong> dem<strong>and</strong>.<strong>The</strong> ma<strong>in</strong> purpose <strong>of</strong> this chapter is to estimate the dem<strong>and</strong> for tourism to two neighbour<strong>in</strong>gregions <strong>in</strong> Sweden <strong>and</strong> Norway from five different countries: namely, Denmark, the UK,Switzerl<strong>and</strong>, Japan, <strong>and</strong> the US. Monthly time series data from 1993:01 to 2006:12 is collectedfrom Statistics Sweden for this purpose. For each visit<strong>in</strong>g country, We specify a separateequation with the relevant <strong>in</strong>formation <strong>in</strong>cluded <strong>in</strong> each equation. We conduct severaldiagnostic tests <strong>in</strong> order to specify the five equations for SW:6 <strong>in</strong> Sweden <strong>and</strong> NWT <strong>in</strong>Norway. We then estimate these equations us<strong>in</strong>g Zellner’s ISUR, which takes <strong>in</strong>toconsideration any possible correlation between the equations <strong>and</strong> hence is more efficientthan other s<strong>in</strong>gle equation estimation methods, such as OLS.<strong>The</strong> results also <strong>in</strong>dicates that CPI, some lagged dependent variables, <strong>and</strong> several monthlydummy variables represent<strong>in</strong>g seasonal effects have a significant impact on the number <strong>of</strong>visitors to SW6 <strong>in</strong> Sweden <strong>and</strong> NWT <strong>in</strong> Norway. <strong>The</strong> results also show that the relative price<strong>and</strong> exchange rate have a significant effect on <strong>in</strong>ternational tourism dem<strong>and</strong> for somecountries. However, although we could view this conclusion as support<strong>in</strong>g a theoreticalframework that describes tourism dem<strong>and</strong> model variable relationships, our dem<strong>and</strong>system lacks a travel cost variable. Nonetheless, our results could also have importantimplications for the decision-mak<strong>in</strong>g process <strong>of</strong> government tourism agencies <strong>in</strong> bothcountries when consider<strong>in</strong>g <strong>in</strong>fluential factors <strong>in</strong> their long run plann<strong>in</strong>g.6. Appendix6.1 Diagnostic tests6.1.1 <strong>The</strong> Cusum testThis test is used for time series <strong>and</strong> checks for structural changes. In the Cusum testRecursive Residuals (RR) calculated by the Kalman Filter are used.I now describe the construction <strong>of</strong> recursive residuals <strong>and</strong> the Kalman filter technique. <strong>The</strong>recursive residuals can be computed by forward or backward recursion. Only forwardrecursion is described, backward recursion be<strong>in</strong>g analogous.Given N observations, consider the l<strong>in</strong>ear model (2 . 2 . 1) but with the correspond<strong>in</strong>gvector <strong>of</strong> coefficient expressed as t , imply<strong>in</strong>g that the coefficients may vary over time t.<strong>The</strong> hypothesis to be tested is = = , . . ., = = . <strong>The</strong> OLS estimator based on Nobservations is:b = ( X'X ) -1 X'y ,where X is a N by k matrix <strong>of</strong> observations on the regressors, <strong>and</strong> y is an N by 1 vector <strong>of</strong>observations for the dependent variable. Suppose that only r observations are used toestimate . <strong>The</strong>n for r > k, where k is the number <strong>of</strong> <strong>in</strong>dependent variables,
112Advances <strong>in</strong> Econometrics - <strong>The</strong>ory <strong>and</strong> Applicationsb r = ( X r 'X r ) -1 X r 'y r ,r = k+1, . . ., N .Us<strong>in</strong>g b r , one may "forecast" y r at sample po<strong>in</strong>t r, correspond<strong>in</strong>g to the vector X r <strong>of</strong> theexplanatory variables at that po<strong>in</strong>t.Recursive residuals are now derived by estimat<strong>in</strong>g equation (2 . 2 . 1) recursively <strong>in</strong> the samemanner, that is by us<strong>in</strong>g the first k observations to get an <strong>in</strong>itial estimate <strong>of</strong> , <strong>and</strong> thengradually enlarg<strong>in</strong>g the sample, add<strong>in</strong>g one observation at a time <strong>and</strong> re-estimat<strong>in</strong>g ateach step. In this way, it is possible to get (N-k) estimates <strong>of</strong> the vector , <strong>and</strong>correspond<strong>in</strong>gly (N-k-1) forecast errors <strong>of</strong> the type:Wr yr Xrbr 1r = k+1, . . ., Nwhere b r-1 is an estimate <strong>of</strong> based on the first r - 1 observations. It can be shown that,under the null hypothesis, these forecast errors have mean zero <strong>and</strong> variance 2 d r2, whered r is a scalar function <strong>of</strong> the explanatory variables, equal to [ 1 + X r '(X' r-1 X r-1 ) -1 X r ] 1/2 .<strong>The</strong>n the quantity:y X bWr[1 X ( X X ) X ]r r r1' ' 1/2r r1 r1)rr = k+1, . . ., Ngives a set <strong>of</strong> st<strong>and</strong>ardized prediction errors, called "recursive residuals". <strong>The</strong> recursiveresiduals are <strong>in</strong>dependently <strong>and</strong> normally distributed with mean zero <strong>and</strong> constant variance 2 . As a result <strong>of</strong> a change <strong>in</strong> the structure over time, these recursive residuals will no longerhave zero mean, <strong>and</strong> the CUSUM <strong>of</strong> these residuals can be used to test for structural change.CUSUM <strong>in</strong>volves the plot <strong>of</strong> the quantity:rV W / *rtk1r = k+1, . . ., N,where * is the estimated st<strong>and</strong>ard deviation based on the full sample.<strong>The</strong> test f<strong>in</strong>ds parameter <strong>in</strong>stability if the cumulative sum goes outside the area between thetwo error bounds. Thus, movements <strong>of</strong> V t outside the error bounds are a sign <strong>of</strong> parameter<strong>in</strong>stability.6.1.2 <strong>The</strong> Breusch-Godfrey-test<strong>The</strong> Breusch-Godfrey test can be separated <strong>in</strong>to several stages:1. Run an OLS on:y X y it t ti tt
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