Journal of Hospitality & Tourism Research, Feb 2008; vol. 32 - Sage

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Comparing Forecasting Models in Tourism Rachel J. C. Chen, Peter Bloomfield and Frederick W. Cubbage Journal of Hospitality & Tourism Research 2008; 32; 3 originally published online Nov 8, 2007; DOI: 10.1177/1096348007309566 The online version of this article can be found at: http://jht.sagepub.com/cgi/content/abstract/32/1/3

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COMPARING FORECASTING MODELS IN TOURISM Rachel J. C. Chen University of Tennessee Peter Bloomfield Frederick W. Cubbage North Carolina State University This study uses three major U.S. national parks as applications of statistically selecting appropriate methods to forecast attendance. Forecasting methods assessed include Naïve 1, Naïve 2, single moving average (SMA), single exponential smoothing (SES), Brown’s, Holt’s, autoregressive integrated moving average (ARIMA), derived time series crosssection regression (TSCSREG), and time series analysis with explanatory variable models. The mean absolute percentage error (MAPE) is used to measure the accuracy of forecasting methods. Based on the MAPE values, SMA produces the most accurate forecasting, followed closely by ARIMA, Brown’s, and Naïve 1 models. Holt’s and TSCSREG models produce the next most accurate forecasting, followed by SES, time series analysis with explanatory variable model, and Naïve 2. Methods used in this article are readily transferable to other hospitality and tourism data sets with annual visitation figures. Merits and limits of the proposed forecasting methods are discussed. KEYWORDS:

mean absolute percentage error; forecasting methods

Several U.S. national parks were established in the late 1800s and early 1900s. The main mission of the U.S. National Park Service (USNPS) is “to conserve the scenery and the natural and historic objects and . . . to provide for the enjoyment of the same in such manner and . . . leave them unimpaired for the enjoyment of future generations” (National Park Service, 2006, para. 1). Most national parks initially pursued extensive and aggressive strategic promotion and infrastructure development to entice more visitors and public support. Information provided by the Statistical Office of the USNPS (National Park Service, 2002) shows that during the 1920s and 1930s, the volume of visitors increased steadily and increased dramatically after World War II. Visitor numbers fluctuated from year to year because of many factors such as bad weather, construction, and economic recession. Overall, the visitation series of the National Park Service revealed mostly upward trends. The total annual visitation of USNPS units

Authors’ note: We thank the editor and three referees for their helpful comments. Journal of Hospitality & Tourism Research, Vol. 32, No. 1, February 2008, 3-21 DOI: 10.1177/1096348007309566 © 2008 International Council on Hotel, Restaurant and Institutional Education


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JOURNAL OF HOSPITALITY & TOURISM RESEARCH

was estimated to exceed 22 million in 1920s, 100 million in 1930s, 199 million in 1940s, and 590 million in 1950s. The visitation reached 1,278 million in 1960s. During the 1960s, “Mission 66” (an initiative to construct more national park facilities across the United States by 1966) sought to accommodate the increased number of visitors. More funding was allocated to support road and facility upgrades in the 1970s. From the 1970s to 1980s, visitation rose 36% (National Park Service, 2002). During the 1990s, more than 2,724 million recreation visits were made to the USNPS units (e.g., national parks, national historical sites, national parkways, and national monuments). As national parks have become more popular destinations, to provide better services, monitor park resources, and maintain visitor safety, understanding the future demand of national park users has become critical. STUDIES IN PROJECTING VISITATION

Forecasting demand has attracted considerable interest. Studies that have documented the visitation forecasts are numerous and include Archer (1994), Barry and O’Hagan (1972), Chen (2006), Geurts (1982), Gonzalez and Moral (1995), Lim (1999), Sheldon (1993), Smeral, Witt, and Witt (1992), Summary (1987), Uysal and Crompton (1985), and Van Doorn (1984). More recent publications on new developments in forecasting applications used various techniques, including cointegration analysis/error correction model (Kulendran & Wilson, 2000; Kulendran & Witt, 2003), vector autoregressive (AR) model (Fernando & Ramos, 2003), and time varying parameter modeling (Song & Witt, 2000). Cummings and Busser (1994) reviewed forecasting studies and concluded that the formulation, interpretation, and evaluation of forecasts are critical skills for destination managers. Archer (1994) foresees the need for forecasts in hospitality and tourism industries in the following way: Forecasting should be an essential element in the process of management. No manager can avoid the need for some form of forecasting: a manager must plan for the future in order to minimize the risk of failure or, more optimistically, to maximize the possibilities of success. In order to plan, he must forecast. . . . Forecasts are needed for marketing, production, and financial planning. (p. 105)

In choosing a forecasting technique, attributes such as accuracy of the forecasting methods, cost of generating the forecasts, and efficiency of producing the forecasts need to be considered. Mentzer and Kahn (1994) indicated that the major criteria used to choose forecasting methods for various applications (e.g., business growth, services/goods trends) were accuracy, credibility, and ease of use. An inappropriate forecasting method, such as incorrect construction of equations, can produce poor forecasts. Any policy based on such biased estimates will be misguided. Seeking appropriate forecasting methods, coupled with accessible historical data, can help decision makers develop feasible management strategies to plan for the future. Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

Chen et al. / FORECASTING MODELS IN TOURISM

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TYPES OF FORECASTING METHODS

Classifications and some of the techniques of forecasting methods were discussed in Archer (1994), Uysal and Crompton (1985), and Witt and Witt (1995). In general, quantitative and qualitative approaches are two major types of forecasting methods. Qualitative methods are also called “judgmental methods.” Qualitative approaches, such as the Delphi method, allow the expert opinions of selected participants to be expressed under certain specified conditions. The assumptions, advantages, and limitations underlying the use of qualitative methods were reported in Moeller and Shafer (1994). Quantitative forecasting methods can be classified into two categories: causal methods and time series methods (e.g., basic, intermediate, and advanced extrapolative methods). Causal methods, including regression analysis and structural models, establish methodologies for identifying relationships between dependent and independent variables. However, the most common difficulty of applying the causal methods is how to statistically determine the independent variables that affect the forecast variables. Thus, the reliability of final forecast outputs will depend on the quality of other variables (Chen, 2006; Uysal & Crompton, 1985). Furthermore, those independent variables themselves must typically be forecast to estimate the forecast for the relevant dependent variable. This is often difficult at best. Time series methods offer concepts and techniques that facilitate specification, estimation, and evaluation, often yielding more accurate forecasting results than causal quantitative approaches (Chen, 2006). The most important feature of the time series methods is that observations made at different time points are not assumed to be statistically independent. Using suitable time series methods, accompanied with visitation figures in the hospitality and tourism industry, the benefits of accurate forecasts for marketing strategies at different time intervals (short, medium, and long term), such as scheduling, promoting, and investing, can be achieved. METHODS CURRENTLY USED BY THE USNPS TO PREDICT VISITATION

The USNPS currently uses a spreadsheet format to estimate visitation. Using annual visitation data from the past 5 years, annual forecasts for the next 2 years are estimated. Essentially, this methodology uses a trend model for appropriate park visitation data sets, and when visitation numbers have no discernible trend for the past 5 years, the most recent annual figure is used for the forecast. The slope, intercept, and root mean square error of a linear trend fitted to each data series are estimated (National Park Service, 2002). The accuracy of the method used by the National Park Service is measured as the expected average error percentage in the forecast. When the figure is less than 20%, a trend model is used to forecast the next 2 years. When it is higher than 20%, the most recent total visitation is used for the forecast. In contrast, this study used univariate time series and time series analysis with explanatory Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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variable approaches that can also identify relationships between dependent and independent variables to generate national park visitation forecasts. PURPOSES OF THE STUDY

Although there is extensive literature on the implementation of forecasting models, most of these forecasts are applied to country-based data sets. Studies concerned with forecasting the trends for destination visitations are rare in the literature (Chen, 2006; Chen, Bloomfield, & Fu, 2003). Indeed, although researchers have explored the use of time series methods to model visitor volumes, the majority of the discussions focus on implementing basic (naive, single moving average [SMA]) and intermediate (single exponential smoothing [SES], double exponential smoothing, and autoregression) extrapolative methods rather than advanced time series methods. Therefore, this study adds to the literature by using advanced time series methods (including the autoregressive integrated moving average [ARIMA] method, time series cross-section regression method, and time series analysis with explanatory variable model) to forecast park visitor volumes. This research focused on examining different statistical techniques and applying them to the National Park Service data in the United States. Two objectives of this study were to (a) implement basic, intermediate, and advanced extrapolative methods to the USNPS data and (b) evaluate the advantages and disadvantages of using basic, intermediate, and advanced time series methods. METHOD

This study used Naïve 1, Naïve 2, SMA, SES, Brown’s, Holt’s, ARIMA model, derived time series cross-section regression (TSCSREG) methods and time series analysis with explanatory variable model. The construction of the park forecasting models consisted of three stages: specification, estimation, and evaluation. The specification stage focused on testing, generating, and selecting an appropriate model. The estimation stage used the selected model to predict future visitation. Finally, the evaluation phase investigated various models over time to see if there were any needs to modify models and make appropriate adjustments to improve the reliability of estimates (Chen, 2006; Chen et al., 2003). Basic Extrapolative Methods

(1) Naïve 1. The Naïve 1 method simply states that the forecast value for this period (t) is equal to the available actual value of the last period (t – 1) (see Table 1). (2) Naïve 2. The Naïve 2 method for period t is obtained by multiplying the current visitor numbers with the growth rate between the previous visitation in time period, t – 1, and the current visitation figures in time period, t (Newbold & Bos, 1994). The equation of the Naïve 2 method is displayed in Table 1. Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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Table 1 Equations of Basic Time Series Forecasting Methods Method/Equations

Definition

(1) Naïve 1 Ft = At − 1

where Ft = forecast visitation at time t; At – 1 = actual visitor number at time t – 1.

(2) Naïve 2 Ft = At – 1 [1 + (At – 1 − At – 2)/At – 2] (3) Single Moving Average Mt – 1 = Ft = [(At – 1 + At – 2 + At – 3 + . . . + At – n)/n]

(4) Single Exponential Smoothing Ft = Ft – 1 + α(At – 1 – Ft – 1)

where Ft = forecast visitation at time t; At – 1 = actual visitor number at time t – 1. where Mt – 1 = moving average at time t – 1; Ft = forecasted value for next period; At – 1 = actual value at period t – 1; n = number of terms in the moving average. where Ft = forecasted value for next period at time t; α = smoothing constant (0 < α < 1); At – 1 = actual visitor number at time t – 1.

(3) SMA. Based on adding the previous observations together and dividing by the number of observations, the SMA method allows the produced average figures to forecast future values. One assumption of the SMA method (see Table 1) is that all selected previous data points have the same weight on the forecast value (Kendall, Stuart, & Ord, 1983). (4) SES. The SES method (see Table 1) allows forecasters to determine the influence of recent observation on the forecast values. The SES equation states that the forecast for the current period (t) is equal to the forecast for the previous period (t – 1) plus a smoothing constant (α) multiplied by the error that the forecasting model produced for the previous period (t – 1). The previous values are weighted by the smoothing constant, which must take a value between 0 and 1 and is set by the forecaster. The lower the smoothing constant, the more weight it gives to the ones prior to the last value (Gardner, 1985; Moore, 1989). Intermediate Extrapolative Methods

(5) Brown’s method. Brown’s method (see Table 2) has the capability to capture increasing or decreasing linear trends well and is also called “double exponential smoothing method” (Brown, 1963). (6) Exponential smoothing adjusted for trend: Holt’s method. Holt’s twoparameter method is one kind of exponential smoothing technique frequently used to handle a linear trend. The trend and slope can be smoothed by this technique through using different smoothing constants for each. The difference between Holt’s and Brown’s approach is that the latter uses only one smoothing Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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Table 2 Equations of Intermediate Time Series Forecasting Methods Method/Equations

Definition

(5) Brown’s Yt = Yt – 1 + α(At – 1 – Yt – 1) Y’t = Y’t – 1 + α(Yt – Y’t – 1) Ct = Yt + (Yt – Y’t) Tt = [(1 – α)/α]*(Yt – Y’t) Vt + h = Ct + hTt

(6) Holt’s Method Vt = αAt + (1 – α)(Vt – 1 + Ψt – 1) Ψt = β(Vt – Vt – 1) + (1 – β)Ψt – 1 Ft + h = Vt + hΨt

where Yt = single exponential smoothing series at time t; α = smoothing constant (0 < α < 1); At – 1 = actual visitor number at time t – 1; Y’t = double exponential smoothing series at time t; Ct = the intercept of the Y’ forecast series at time t; Tt = the slope of the Y’ forecast series at time t; Vt + h = forecast visitation at time t + h; h = the number of time periods ahead. Vt = new smoothed value; α = smoothing constant for the data (0 ≤ α ≤ 1); At = new observation or actual value of series in period t; β = smoothing constant for estimating the trend (0 ≤ β ≤ 1); Ψt = trend estimate in period t; h = periods to be forecast; Ft + h = forecast for h periods into the future.

constant and that the estimated trend values are very sensitive to random influences. Unlike Brown’s technique, more flexibility is given by Holt’s technique in selecting the smoothing constant rates (Makridakis, Wheelwright, & Hyndman, 1998). Three main equations of Holt’s method are expressed in Table 2. Advanced Time Series Forecasting Methods

(7) ARIMA. The Box-Jenkins (Box & Jenkins, 1970) univariate method is considered a highly sophisticated technique than the other techniques used in this study. The mathematical statement of an autoregressive moving average (ARMA) model reveals how an independent variable Vt is related to its own past values {Vt – 1, Vt – 2, Vt – 3, . . .}. The ARMA models described here are all stationary and can therefore be used in a straightforward manner to construct forecasts for stationary time series. They can, however, also be used to construct forecasts for some nonstationary time series. For instance, if the first differences of the time series under a study are stationary, an ARMA model may be fitted to them and used to forecast the differences. These forecasted differences may then be accumulated to produce forecasts for the values of the original series. When the d’th differences (d = times of differencing) of a time series have ARMA structure, the time series is said to have ARIMA structure (see Table 3). An ARIMA model is a refined curve-fitting device that uses present and past values of dependent variables to forecast future values. When a series contains no growth or decline, it is known to be stationary. A stationary series is defined as having a constant sample mean, variance, and autocorrelation function (ACF) over Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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Table 3 Equations of Advanced Time Series Forecasting Methods Method/Equations

Definition

(7) Autoregressive Integrated Moving Average (ARIMA) Φp(B)∇dVt = Θq(B)εt

(8) Derived Time Series Cross Section Regression Yi,t = αi + βit + Ui,t Uˆ i,t = Yi,t – (αi + βit) Uˆ i,t+1 = ρi*Ui,t Yî,t+1 = αi + βi(t+1) + Uˆ i,t+1 Uˆ i,t+2 = (ρi)2*Ui,t Yî,t+2 = αi + βi(t+2) + Uî,t+2 (9) Models With Explanatory Variables ln(Yt) = β0 + β1ln(X1t) + β2ln(X2t) + . . . + βkln(Xkt) + Zt

where Vt = dependent variable (e.g., number of visitors in time t); Φp = regression coefficients; B = the backshift operator, where BhVt = Vt – h,; ∇d = for the dth difference of the series Vt; Θq = coefficient, or called weights; εt = residual terms. Where Y = actual value; α = intercept; β = coefficient of year; U = error term; Uˆ = forecast error term; ρ = estimated matrix value; Yˆ = forecast value; t = time period (i.e., year).

where Yt = dependent variable (e.g., number of visitors in time t); Xkt are economic and socioeconomic factors; and Zt is an ARIMA time series.

time. After the second or third time lag, the autocorrelation (AC) coefficients of stationary series data drop to 0, while those of a nonstaionary series are significantly different from 0 for several time periods. Differencing is used to transform a nonstationary series into a series with stationarity. For nonseasonal data, it is rarely differenced more than twice. Differencing is a process involving the creation of a new stationary series, derived from successive changes in an original nonstationary series, Vt. The inspection of the AC structure is employed to determine the degree of differencing. To distinguish the difference between an ARIMA process and an ARMA process, it is important to note two primary components of an ARIMA process: first, an invertible moving average (MA) error structure whose adherence may not be strictly maintained while using the ARIMA method. Second, to achieve stationarity, a nonstationary AR component needs to be differenced up to d times. The purpose of Box, Jenkins, and Reinsel (1994) was to provide a modeling approach uniquely representative of the data structure. Three separate stages are included: (a) model identification, (b) model parameter estimation and testing of model adequacy, and (c) model forecasting. Model identification is based on the behavior of three estimated functions: the ACF, the inverse ACF (IACF), and the partial ACF (PACF). When the function of ACF drops off to 0 after lag q, it means an MA (q) model. If the function of IACF and PACF drop off to 0 after lag p, it indicates an AR (p) model. In Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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addition, the ACF is commonly used for monitoring very slow decay and identifying nonstationarity. After one or two substantial drops in the ACF, the slow dying off may occur. When the ACF dies off very slowly, a unit root is indicated. Once the time series has been identified, the parameter estimates can be obtained. For determining the adequacy of the model, two aspects are involved: (a) the existence of white noise errors and (b) the significance of the parameter estimates. The existence of white noise errors can be obtained by checking the Q statistic, and the significance of the parameter estimates is obtained through the use of t tests. The problems of overfitted and misspecified models need to be investigated by checking the possible convergence and unstable parameter estimates. Because an ACF exists in every AR and MA time series, based on the ACF inspection procedure, a forecaster is able to identify whether the computed data series is AR, MA, or mixed ARMA, as well as by observing the orders of p and q of the ARIMA components. To check if the residuals of the estimated models exhibit white noise, the Q statistics are employed. Commonly, Akaike information criterion is used at the stage of model selection. This statistic tells us whether the specific model could be the most appropriate choice to produce the best fitted forecasts. The actual visitation figures for three selected national parks show that the overall trends seem to be increasing, suggesting the existence of nonstationary sample means. Thus, a first-differenced approach is applied. For more information about the calculation and limitations of the ARIMA method, readers are referred to Box et al. (1994). (8) TSCSREG procedure. When the panel data sets (e.g., in this study, other 15 national park visitation data sets were intergraded into the model) consist of time series observations on each of several cross-sectional units (unit means the annual visitation figure in the case study), the TSCSREG procedure is a suitable approach. Because the error structures of the model influence the performance of the model regression parameters, the examination of the contemporaneous correlation between cross-sections is critical. In this study, the TSCSREG procedure uses the Parks method to estimate the first-order AR model with contemporaneous correlation between cross-sections. The covariance matrix is estimated by a two-stage procedure leading to the estimation of model regression parameters by generalized least squares. The TSCSREG procedure requires that the time series for each cross-section have the same number of observations and cover the same time range. The input data set used in the TSCSREG procedure must be sorted by cross-section and by time within each cross-section. The next step is to invoke the procedure and specify the number of crosssections in the data set and the number of time series observation in each crosssection. Based on the forms of the available data sets and objectives of this study, six derived equations were used for the TSCSREG procedure (see Table 3). (9) Time Series Analysis With Explanatory Variable Model. While including other explanatory variables, in this study, the transformed log model with time series errors was used and is expressed in Table 3. Differencing takes care of Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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nonstationarity in the mean, and logarithmic transformation is taken to ensure variance stationarity. One of the most important aspects of the time series analysis with explanatory variable analysis is the process of calculating hypothesized relationships and incorporating these economic relationships into forecast models. The variables included in the forecasting procedures were selected based on suggestions and results from past studies (Uysal & Crompton, 1985). Because of the lack of the cross-sectional component, which often contains the socioeconomic and demographic indicators needed in such a study, demographic and socioeconomic indicators such as age, education, occupation, marital status, and number of children could not be used. This study used factors that had been included as explanatory variables in previous studies and were available as an independent economic series compiled by public agencies (including the U.S. Bureau of Labor Statistics under the Department of Labor as well as the Bureau of Census and the Bureau of Economic Analysis under the Department of Commerce). Annual per capita disposable personal income, unemployment, gross domestic product, personal consumption expenditure, nation-wide population, and four Consumer Price Index (CPI) categories—food, transportation, lodging, and services—were incorporated into the model. Using a time series analysis with an explanatory variable approach and based on suggestions of previous studies (Kulendran & Witt, 2003), the t value and the parameters’ signs were used as a variable selection criterion. That is, when a variable appears to be statistically significant at the 5% level and has a correct coefficient sign, the forecaster will include this explanatory variable. In addition, to ensure the statistical acceptability of the method to be employed, one of the main requirements of the models is that the independent variables must not be correlated with each other. As Table 4 shows, the highest correlation between the independent variables is less than 0.5 in absolute value. Error Magnitude Measurement

The error magnitude measures allow forecasters to evaluate and compare the performance of various models across different time periods. It relates to forecast error and is defined as follows: εt = At – Ft

(1)

where ε = the forecast error, At = the actual number of visitors in period t, Ft = the forecast value in period t (t = time period, i.e., a month, quarter, or year). In a survey of visitation trend forecasts, Witt and Witt (1995, pp. 447-490) found that “accuracy is the most important forecast evaluation criterion.” Because of its clear and straightforward definition, the mean absolute percentage error (MAPE) was selected for this study. MAPE. The MAPE is written as follows: (2)

MAPE = = Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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Table 4 Correlation of Independent Variables

APCDPI UNEMP GDP PCE NWP FOOD TRAN LODG SERV

APCDPI

UNEMP

GDP

PCE

NWP

FOOD

TRAN

LODG

SERV

1.00 –0.19 0.18 0.23 0.15 0.12 0.07 0.14 0.07

–0.19 1.00 –0.12 –0.23 0.06 0.11 0.17 0.13 0.07

0.18 –0.12 1.00 0.05 0.03 0.13 –0.08 0.16 0.11

0.23 –0.23 0.05 1.00 0.08 0.14 0.19 0.12 0.15

0.15 0.06 0.03 0.08 1.00 0.11 0.18 0.15 0.07

0.12 0.11 0.13 0.14 0.11 1.00 0.13 0.08 0.10

0.07 0.17 –0.08 0.19 0.18 0.13 1.00 0.13 0.12

0.14 0.13 0.16 0.12 0.15 0.08 0.13 1.00 0.06

0.07 0.07 0.11 0.15 0.07 0.10 0.12 0.06 1.00

Note: APCDPI = annual per capita disposable personal income; UNEMP = unemployment; GDP = gross domestic product; PCE = personal consumption expenditure; NWP = nationwide population; FOOD = food; TRAN = transportation; LODG = lodging; SERV = services.

where n = number of time periods, et = forecast error in time period t, At = actual number of visitors in time period t. Lower MAPE values are better because they indicate that smaller percentage errors are produced by a forecasting model. The following interpretation of MAPE values was suggested by Lewis (1982) as follows: Less than 10% is highly accurate forecasting, 10% to 20% is good forecasting, 20% to 50% is reasonable forecasting, and 50% or more is inaccurate forecasting. DATA FOR MODELING AND FORECASTING The Reliability and Availability of Forecast and Explanatory Variables

Visitation reports supplied by the USNPS statistical office showed significant changes regarding the data collection procedures in 1960 throughout different national parks. A new definition of the “visit” was adopted in 1960, that is “the entry of any person into an area administered by the National Park Service such that he makes some use of the services or facilities provided therein by the Service” (National Park Service, 1964, p. 1). Since then, the National Park Service began calculating the annual visits consistently and efficiently. For example, reentries (visitors who make more than one entry into a park during a brief period of time) were excluded from the visitation calculation. Thus, for consistency, this study obtained data sets from 1960 to 2000 for the selected national parks. Data Sources and Periods

Three selected national park data sets used in this study were the Great Smoky Mountains National Park, Yellowstone National Park, and Yosemite National Park. The national park data sets referred to annual visitation figures and were supplied by the National Park Service. Each of these data sets consisted of 41 years of annual attendance figures (1960 to 2000). Based on the purpose of this Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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study, seasonal elements (for example, quarterly or monthly data sets) were not included into the various forecasting models. It is likely that the annual data series on the three selected national park visitation figures exhibited trend. Data from 1960 to 1995 were used to estimate next 5-year forecasts (1996 to 2000), the estimated models with 1960-1998 data were used to make next 2year forecasts (1999 to 2000), and data from 1960 to 1999 were used to make next 1-year prediction (2000). Forecasting Performance: Implication of MAPE

As mentioned previously, the study obtained data sets from 1960 to 2000. It then excluded the last 1, 2, and 5 years of those data sets as a subset and projected for those years. Note that MAPE, as calculated here, was dominated by one-step out-of-sample forecast errors. Table 5 displays the results of MAPE and summarizes the performances of the various forecasting methods for next 1-year, next 2-year, and next 5-year forecasting horizons. A smaller MAPE value indicates that a better forecasting model produces more accurate values of prediction. In the case of the Great Smoky Mountains National Park, examination of the MAPE values in Table 5 reveals that ARIMA was the best among the nine techniques when next 1-year forecast was estimated. The MAPE values of the Naïve 1 and Holt’s methods were also small. The TSCSREG performed worst among other models. When next 2-year forecasts were considered, the time series analysis with explanatory variable model performed best for the Great Smoky Mountains National Park, whereas the SES was the worst among other techniques. When next 5-year forecasts were estimated, the ARIMA model was the best and the Naïve 2 performed the worst among others. In the case of the Yellowstone National Park, Holt’s was the best among the listed techniques when next 1-year forecast was estimated. The MAPE values of the ARIMA and TSCSREG were also small. The Naïve 2 performed worst among other models. When next 2-year forecasts were considered, the TSCSREG method performed best for the Yellowstone National Park, while Naïve 2 was the worst among other techniques. When next 5-year forecasts were estimated, the SMA model was the best and the Naïve 2 performed worst among others. In the case of the Yosemite National Park, SES was the best among the listed techniques when next 1-year forecast was estimated. The MAPE values of the Naïve 1 and ARIMA models were small as well. The Holt’s model performed worst among other models. When next 2-year forecasts were considered, the SES performed best for the Yosemite National Park, while the Naïve 2 was the worst among other techniques. When next 5-year forecasts were estimated, the Naïve 1 model was the best and the SES performed worst among others. In summary, based on the MAPE values of next 1-, next 2-, and next 5-year forecasts of the three national parks, SMA produced highly accurate forecasting (MAPE values were less than 10%) nine times; the Naïve 1, Brown’s, and ARIMA models produced highly accurate forecasting eight times and produced good forecasting once (the MAPE value was between 10% and 20%); Holt’s and TSCSREG models produced highly accurate forecasting seven times and Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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Table 5 Forecasting Performance Evaluated by the Mean Absolute Percentage Error (MAPE) Model/Park {Estimation Period}: Forecasting Period (1) Naïve 1 {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (2) Naïve 2 {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (3) Single Moving Average {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (4) Single Exponential Smoothing (SES) {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (5) Brown’s Method {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (6) Holt’s Method {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (7) Autoregressive Integrated Moving Average (ARIMA) {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (8) Derived Time Series Cross Section Regression (TSCSREG) {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000 (9) Time Series Analysis With Explanatory Variable Model {1960-1999}: 2000 {1960-1998}: 1999-2000 {1960-1995}: 1996-2000

Great Smoky Mountains

Yellowstone

Yosemite

0.27% (2) 7.17% (7) 4.89% (3)

7.49% (6) 3.97% (4) 4.58% (4)

0.39% (2) 10.46% (3) 4.22% (1)

7.35% (7) 4.29% (5) 11.48% (8)

11.21% (8) 5.48% (8) 23.65% (8)

8.99% (7) 14.34% (8) 4.73% (4)

3.79% (5) 8.68% (8) 4.97% (4)

5.67% (5) 4.46% (5) 2.72% (1)

5.54% (5) 9.45% (2) 4.34% (2)

8.49% (8) 10.43% (9) 6.59% (7)

7.69% (7) 3.89% (3) 10.53% (7)

0.11% (1) 0.22% (1) 13.68% (8)

4.39% (6) 7.09% (6) 4.72% (2)

5.26% (4) 3.78% (2) 5.56% (6)

6.74% (6) 10.62% (4) 4.57% (3)

1.89% (3) 2.99% (2) 6.17% (6)

0.12% (1) 4.82% (7) 3.18% (2)

11.16% (8) 12.07% (6) 6.56% (6)

0.20% (1) 3.19% (3) 4.29% (1)

3.47% (2) 4.76% (6) 3.42% (3)

2.28% (3) 13.84% (7) 6.75% (7)

15.15% (9) 3.52% (4) 5.83% (5)

4.95% (3) 3.57% (1) 5.09% (5)

3.36% (4) 11.99% (5) 5.14% (5)

3.44% (4) 2.52% (1) —

— — —

— — —

Note: Figures in parenthesis denote rankings. For example, in the case of the Great Smoky Mountains National Park, based on the examination of the MAPE values, ARIMA (ranked 1) was the best and TSCSREG (ranked 9) performed worst among other models when next 1-year forecast was calculated. Time series analysis with explanatory variable model (ranked 1) had the lowest MAPE and SES (ranked 9) had the highest MAPE among all the forecasting methods when next 2-year forecasts were calculated. ARIMA was the best and Naïve 2 performed worst among other models when next 5-year forecast was calculated. Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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produced good forecasting twice (the MAPE values were between 10% and 20%); SES produced highly accurate forecasting six times and produced good forecasting three times; and Naïve 2 produced highly accurate forecasting five times, produced good forecasting three times, and provided reasonable forecasting once (the MAPE value was between 20% and 50%). ARIMA models. Based on the examination of the ACF, the needs for differencing procedures for the forecasted series are confirmed in the case of the three selected national parks. The final ARIMA (2, 1, 3) forecast model for the Great Smoky Mountains National Park’s 1-year-ahead series is denoted as follows: (1 – 0.52B + 0.8B2) ∇dVt = (1 – 0.47B + 0.86B2 – 0.35B3) εt

The final ARIMA (2, 1, 3) forecast model for the Great Smoky Mountains National Park’s 2-year-ahead series is written as follows: (1 – 0.51B + 0.8B2) ∇dVt = (1 – 0.45B + 0.87B2 – 0.33B3) εt

The final ARIMA (2, 1, 2) forecast model for the Great Smoky Mountains National Park’s 5-year-ahead series is denoted by the following: (1 – 0.49B + 0.92B2) ∇dVt = (1 – 0.25B + 0.88B2) εt

The final ARIMA (0, 1, 2) forecast model for the Yellow Stone National Park’s 1-year-ahead series is denoted by the following: Vt = 39,613.8 + (1 – 0.32B – 0.33B2)εt

The final ARIMA (0, 1, 1) forecast model for the Yellowstone National Park’s 2-year-ahead series is written as follows: Vt = 43,672.7 + (1 – 0.48B)εt

The final ARIMA (0, 1, 2) forecast model for the Yellowstone National Park’s 5-year-ahead series is denoted by the following: Vt = 40,795.6 + (1 – 0.33B – 0.31B2)εt

The final ARIMA forecast model for the Yosemite National Park’s 1-year-, 2year-, and 5-year-ahead series is denoted by the following: Vt + 1 = Vt + MU Vt + 2 = Vt + 1 + MU

where MU = estimated mean. For the time series analysis with explanatory variable model, neither the presence of autocorrelated errors nor relationships between the dependent variable and the explanatory variables were found for next 5-year forecasts of the Great Smoky Mountains National Park and for next 1-, next 2-, and next 5-year Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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forecasts of the Yellowstone National Park and Yosemite National Park. In the case of the Great Smoky Mountains National Park, for next 1-year forecast, the model is written as follows: Ln(Vt) = 4.93 – 0.6ln(Food) + Zt

For next 2-year forecasts, the model for the Great Smoky Mountains National Park is written as follows: Ln(Vt) = 4.75 – 0.6ln(Food) + Zt

CONCLUSION

This study was intended to provide validated and understandable methods for projecting park visitation trends. The study compared the accuracy of the different forecasting models for the three U.S. national parks. Forecasting periods were estimated in the equation of the MAPE (see Equation 2) to compare the performances of various forecasting methods. The MAPE values indicated that the ARIMA and SES models appeared in the first position twice; Naïve 1, TSCSREG, time series analysis with explanatory variable, Holt’s, and SMA models appeared in the first position once; Naïve 2 appeared in the last position five times; SES appeared in the last position twice; and Holt’s and TSCSREG models appeared in the last position once. Advantages and Disadvantages of the Proposed Forecasting Methods

The advantages of the Naïve methods are that they have the capability to generate forecasts by using short previous observations. For example, based on the equations of Naïve methods, Naïve 1 only needs one previous observation and Naïve 2 only needs at least two previous observations to produce the next prediction. While using the MA method, each one of the values entering the averaging process receives equal weight. When wide variations around a trend are detected under a time series, then the longer SMA, the better it will provide a smooth trend. That is, the SMA model generates more accurate forecasts with a series with little variation than one with volatility (Makridakis et al., 1998). Because the three selected national park visitation figures provide smooth series, the authors use the average of the previous two values to serve as our forecast for the next period. When series include seasonal patterns, the SES cannot be used. That is, forecasters may only consider SES for dealing with monthly or quarterly time series where seasonality has been removed. It is clear that the three selected national park annual visitation series were upward without stationary in its mean. We, therefore, achieve stationarity of a trended series by differencing it. The exponential smoothing methods average previous observations. Both MA and SES require that the data series are stationary for the forecasting to make sense. If the series are trended, then the forecasting techniques will consistently under- or Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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overforecast. The main difference between the exponential smoothing methods and the MA method is that the exponential smoothing methods treat the averaged observations progressively (Makridakis et al., 1998; Moore, 1989). For example, more recent observations get higher weight while less recent data receive less weight in an exponential manner. Different exponential smoothing methods have different ways to treat the trend within the data sets. The parameter of the SES is fixed, and the method is better for horizontal trend data sets. We vary our smoothing constant in 1/10th or 51/100th increments between 0 and 1 to see if we can achieve a smaller MAPE to obtain the best exponential smoothing model. The difference between Holt’s and Brown’s exponential smoothing methods is that the former uses two parameters and the latter uses one single parameter. Ideally, Holt’s two parameters may provide more flexibility and generate more accurate forecasts than Brown’s one parameter. In the case of the U.S. national parks, however, based on the MAPE values, the results indicated that Brown’s exponential smoothing method sometimes outperformed Holt’s. Overall, the basic time series methods (Naïve 1, Naïve 2, and SMA) are easy to use and capable of producing accurate, good, and reasonable forecasts. By using the spreadsheet format set up by forecasters, the national park managers will gain sufficient information using these methods. Ease of using the intermediate time series methods (SES, Brown’s, and Holt’s methods) depends on a person’s statistical knowledge, training, and degree of familiarity in employing a spreadsheet package. In the case of the three selected national parks, the results of these methods generated accurate and good forecasts. In moving to the use of the ARIMA model, a good understanding of the BoxJenkins univariate approach and the theory of a differencing approach is required. Thus, an experienced forecaster is necessary to contribute to the model’s accuracy. The main limitation of the ARIMA method is that it requires longer historical data sets than other methods to gain reliable forecasts (Chen, 2006). The main aim of the TSCREG employed here is to compile other national parks’ visitation information into a panel data to see how each data set of various national parks influenced each other’s visitation trends. The TSCSREG is complicated. A forecaster needs to be familiar with the structure of compiling data sets, and then, based on the output of estimating pooled series through a statistical package (e.g., SAS), more forecasting equations are constructed by the analyst. In this study, this method produced accurate and good prediction. The time series method with explanatory variable method needs more data sources than other methods. The availability of good-quality data sets and understanding of the micro- or macro-economic environment make this method more difficult to use. In this method, the relationship between the forecast and explanatory variables in the data could be found by checking the p significant values, t values, and “right” coefficient signs. The effects from the CPI food category played a minor role toward the Great Smoky Mountain National Park’s next 1- and next 2-year forecasts. For 1% decreases in the food category, total visitor volumes would rise by 0.6% for both Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009

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next 1- and next 2-year forecasts. The results indicated that this particular factor had a weak influence on the visitation. Other selected independent variables showed no influence on the forecast variable in the case of national park case studies. Future research must use care in including relevant, accessible, and meaningful factors throughout various time periods across different cases. Moreover, future findings may vary by selecting different explanatory variables across different destinations. Limitations

In the tourism literatures, demographic characteristics (i.e., income, age, gender, and race) have effects on projecting participation. For example, Cheek and Burch (1976) noted that visitors with average or high levels of income traveled more miles and spent much more money while traveling than people with low incomes. Higher income individuals also have been reported to have higher levels of participation in various activities (Howard & Crompton, 1980), museums and zoos (Holzer, Scott, & Bixler, 1998; Hood, 1993), physical-oriented activities (Shaw, Bonen, & McCabe, 1991), and artistic events (Robinson, 1994). McAvoy (1979) found that the elderly were more likely to be involved in passive leisure activities and less likely to engage in outdoor recreation. Godbey and Blazey (1983) reached similar conclusions and particularly pointed out that older park users tended to socialize with others and relax while in parks instead of being involved in sport-related recreation activities. In addition, Rubenstein (1987) showed that the earlier experience of the older persons had greater effects on their destination decision-making process than it did for younger groups. Several studies have identified gender differences in levels of various activity participation (Firestone & Shelton, 1994; Jackson & Henderson, 1995; Toth & Brown, 1997). Differences in motivation and behavior between males and females were found in these studies, although they also found that these two groups had many similarities. Some studies reported that the levels of use and visits to parks among Black, White, Asian, and Hispanic people were different (Cheek, Field, & Burdge, 1976). In addition to demographic characteristics, visitor behaviors were influenced by visitor personality types in terms of levels of activity involvement, destination preferences, and previous experience (Nickerson & Ellis, 1991; Ross, 1994). The literature cited above reported relationships between demographic/ behavioral variables and visitation/participation. Future forecasting research may include those mentioned variables (if they were available) to predict visitation and/or participation in the hospitality and tourism industries. In summary, accurate forecasting results have the capability of shaping public or private tourism-related policies to national parks well into the next decade. The intention of analyzing the comparative accuracy of the forecast models in this study was to use the three best known U.S. national parks as applications in directing destination managers on how the evaluation process works in selecting appropriate forecasting methods. We conclude that the methods used in this article are readily transferable to most hospitality and tourism data sets with annual visitation figures. The time series methods examined here did provide Downloaded from http://jht.sagepub.com at SAGE Publications on May 1, 2009


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Submitted September 19, 2002 First Revision Submitted October 28, 2003 Final Revision Submitted December 8, 2006 Accepted December 12, 2006 Refereed Anonymously Rachel J. C. Chen, PhD (e-mail: rchen@utk.edu), is a Dollywood Professor and associate professor in the Department of Retail, Hospitality, and Tourism Management at the University of Tennessee (247 Jessie Harris Bldg., Knoxville, TN 37996-1911). Peter Bloomfield, PhD (e-mail: bloomfld@stat.ncsu.edu), is a professor in the Department of Statistics at North Carolina State University (Raleigh, NC 27695). Frederick W. Cubbage, PhD (e-mail: fred_cubbage@ncsu.edu), is a professor in the Department of Forestry and Environmental Resources at North Carolina State University (Raleigh, NC 27695).
