The Time Varying Parameter (TVP)

The application of the TVP method to tourism demand forecasting has found popularity among tourism researchers only in recent years; these studies include Song and Wong, 2003; Witt et al. ,2003; Song et al. 2003a; Li et al. 2005; Li et al. ,2006 and Shen et al., 2008. Traditionally, econometric models of tourism demand are usually based on the search for structural stability and a belief that the future will be similar to the past by assuming that the coefficients of the model are constant over time. However, with changes in expectations and tastes by tourists in the process of making decisions the coefficients can vary systematically over time. To overcome the limitations of the traditional fixed- parameter models, the TVP model has been developed, based on the Kalman (1960) filter technique which relaxes the restriction on the parameter constancy and takes account of the possibility of parameter changes over time, and in this way may improve forecasting accuracy.

51

According to the general to specific approach, if a dependent variable is determined by k

explanatory variables, the data generating process (DGP) may be written as an autoregressive distributed lag model (ADLM) of the form:

_{t i t} p i i i jt k j p i ji t x y y =

α

β

φ

ε

∑

∑∑

Where p is the lag length, which is determined by the type of data used and normally decided by the Aikake Information Criterion (AIC) and Schwarz-Bayesian Criterion (SBC) statistics; k is the number of explanatory variables and εt is the error term, which is assumed

to be white noise: normally distributed with zero mean and constant variance σ2

The TVP model is a special case of the ADLM specification. The TVP approach uses a recursive estimation process in which the more recent information is weighted more heavily than the information obtained in the past. With the restriction p = 0 imposed on the coefficients in equation (1), the TVP model is specified in a state space form as follows:

. As quarterly data are used in this study, a lag length of four is adopted.

yt= xtβt + ut , β (2) t = Фβt-1 + Rtet where: , (3) yt x

: a vector of tourism demand,

t

β

: a row vector of k explanatory variables,

t

Ф : a k k matrix initially assumed to be known,

52 Rt

u

: a k g matrix,

t : a residual with zero mean and constant covariance matrix Ht

e

, and

t : a g 1 vector of serially uncorrelated residuals with zero mean and constant

covariance matrix Qt .

Equation (2) is the measurement equation or system equation, and equation (3) is called the transition equation or state equation, and the assumptions in both equations are that the initial vector β0 has a mean of b0 and a covariance matrix P0, and the residual terms ut and et are not correlated.

If the components of the matrix Ф equal unity, the transition equation (3) becomes a random

walk:

βt = βt-1 + Rtet

If the transition equation is a random walk, the parameter vector β

(4)

t is said to be non-

stationary.

Another possible form of the transition equation is:

βt = μ – Ф(βt-1 - μ )+ Rtet

where μ is the mean of β

(5)

t and a stationary process is indicated.

The transition equation is determined by experimentation using the goodness of fit and the predictive power of the model. Once the state space (SS) model is formulated, a

53

convenience algorithm, known as the Kalman Filter (KF), can be used to estimate the SS model (for more detail see Harvey (1987)). The KF produces the optimal estimator using each observation, resulting in the final values to be used for forecasting of b, P and y.

In this study, the TVP forecasts are obtained using the EVIEWS 6.0 software.

3.2.3 The Naïve 1

The Naïve 1 method simply states that the current period’s actual is the next period’s forecast. This simple forecasting method can be used as a benchmark in comparing with forecasting models. Naïve 1is calculated using the following equation:

Ft = At-1

where,

,

Ft

A

= Forecasting value at time t, t = Observed value at time t.

Additionally, the forecasts when summed from the regional flows are compared on the basis of forecast accuracy with the forecast of the total flow for the whole national flow data. This tests the more general issue of the accuracy of regionally disaggregating tourism time series. It is important to determine whether regional based forecasts can be accurately derived, and apart from the absolute measures of forecast accuracy (such as MAPE and RMSE) this issue is also a relative question. Comparison is also made of the forecast growth rates regionally and nationally, to compare the relative accuracy of the different databases and the different types of data.

54