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Photographs included in the original m anuscript have been reproduced xerographically in th is copy. 35 mm slides or 6" x 9" black and w hite photographic prints are available for any photographs or illustrations appearing in th is copy for an additional charge. Contact UMI directly to order. UMI Accessing the World's information since 1938 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA O rd e r N u m b e r 8814840 M odeling spatial and temporal variations in tourism-related employment in Michigan Chen, Sz-Reng, Ph.D. Michigan State University, 1988 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark 1. Glossy photographs or p ag es_____ 2. Colored illustrations, paper or print______ 3. Photographs with dark background_____ 4. Illustrations are poor copy______ 5. Pages with black marks, not original copy 6. 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UMI MODELING SPATIAL AND TEMPORAL VARIATIONS IN TOURISM-RELATED EMPLOYMENT IN MICHIGAN By Sz-Reng Chen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Park and Recreation Resources 1988 ABSTRACT MODELING SPATIAL AND TEMPORAL VARIATIONS IN TOURISM-RELATED EMPLOYMENT IN MICHIGAN By Sz-Reng Chen There were two primary focuses of this study. First, trend and seasonal patterns of monthly tourism-related employment in Michigan between January 1974 and December 1984 were identified. Second, alternative short term forecasting models for predicting monthly tourism related employment were developed and compared at both state and regional levels. Nine study regions were formed, the state of Michigan and eight sub-regions. A general analysis and forecasting procedure was applied to each study region. Multiplicative decomposition was adopted to separate the trend and seasonal components of each series. Trend and seasonal patterns were then identified from these components. Trend models were estimated from trend compo­ nents using transfer function techniques; seasonal models were estimated from seasonal components using harmonic analysis. Forecasting models were then created by combining trend and sea­ sonal models for each region. Michigan's tourism related employment grew by 25% over the 1974-1979 period, dropped 7% between 1980 and 1982, and sub­ sequently recovered at a 4% annual growth rate in 1983 and 1984. Through the year statewide employment fluctuates plus or minus 6% around its annual average. Statewide seasonal patterns are stable over the eleven years period. Regional differences were found in both growth rate and seasonal fluctuations especially between northern tourism-dependent regions and more populous southern regions. Structural and time series models were estimated and com­ pared based upon forecast accuracy for each region. They have the same seasonal component but different trend components (i.e. either a structural or a time series trend component). The per­ formance of each model, therefore, depends primarily on its trend component. All forecasting models fit the data well. Structural models based upon economic variables forecast better for three northern tourism-dependent regions. Time series models based upon time factors forecast somewhat slightly better for the other five regions. The statewide models do not generalize well to northern regions because of differences between regional and statewide patterns. nents . Differences exist in both trend and seasonal compo­ ACKNOWLEDGEMENTS At this final moment of my 25 years student life I would like to acknowledge those, without whom, the completion of my doctoral degree would not have been possible. I am grateful to Dr. Donald Holecek, my major professor, for providing continuous guidence and assistance, both academically and finacially, throughout my whole graduate study at Michigan State University. I sincerely thank Dr. Daniel Stynes, my research advisor, for his pursuing for excelence in research, encouraging guidance, and intellectual cri­ ticism of the dissertation. In addition, the guidance and criti­ cal assessment of the dissertation by Dr. Lester Manderschied and Dr. Larry Leefers is especially appreciated. A special thank to Dr. Charles Nelson, a dear friend of mine, his caring and friend­ ship has been so encouraging to me. Finally, I thank my dear wife Wun-Lih who has suffered with me for a long time, and my parents, without their yearly supports and encouragement my graduate study would have been far more difficult and not so enjoyable. TABLE OF CONTENTS LIST OF TABLES ............................................. Page vi LIST OF FIGURES ............................................ ix Chapter I. INTRODUCTION ............................................ 1 Temporal Variations ............................. Spatial Variations ................................... Forecasting Methods .................................. Michigan As An Area of Application................... Problem Statement.................................... Study Objectives ..................................... Overview of the Study ................................ 3 4 5 6 7 9 10 II. LITERATURE R E V I E W ...................................... 11 Quantitative Forecasting Models ...................... 11 Structural Models .................................... 12 Multivariate regression models .................. Gravity models .................................. 12 15 Time Series Models ................................... 15 Decomposition Methods ........................... Harmonic Analysis ............................. Box-Jenkins Methods .................. .......... 16 19 21 Combined Structural-Time-Series Models ............... 23 Averaging Forecasts from Two or More Models ..... Transfer Function Models ........................ Decomposition-Type Combined Models .............. 23 24 25 Applications to Tourism .............................. 26 Measures of T o u r i s m ............................. Seasonality versus Trend ........................ Estimation of A Trend Model ..................... Estimation of A Seasonal Model .................. Spatial Variations in Tourism Activity .......... Comparison of Forecasting Models ................ 26 27 28 29 30 30 iii Chapter Page Applications to Michigan Tourism ..................... 32 S u m m a r y ............ 37 III. METHODS .............................................. 40 Data Selection....................................... Statewide and Regional Analyses ...................... Formulation of the Study Regions ..................... Analysis and Forecast approach ....................... 41 42 43 45 Identification of Trend and Seasonal Components ...... 48 Decomposition Methods ........................... Choice of Additive vs Multiplicative Decomposition Method ......................... 48 Model Estimation ..................................... 53 Seasonal Model .................................. Trend Model ..................................... Combined Forecasting Model ...................... Forecasting Procedure ........................... Evaluation of the Forecasting model ............. Comparison of the Forecasting Models ............ 53 54 64 65 66 69 Generalizability of the Statewide Model to the Regional Level .......................... 70 S u m m a r y .............................................. 73 IV. TREND AND SEASONAL PATTERNS ........................... 75 Trends ............................................... 75 Statewide Trends .............................. Regional Trends ................................... 75 80 Seasonality.......................................... 84 Statewide Seasonality ............................. Regional Seasonality .............................. 86 89 S u m m a r y .............................................. 93 iv 50 Chapter Page V. FORECASTING MODELS ...................................... 94 Statewide Forecasting Models ......................... 94 Statewide Trend Models ............................ 95 Statewide Seasonal Model ................. 103 Statewide Forecasting Models ...................... 105 Regional Forecasting Models .......................... Ill Regional Trend Models ............................. Ill Regional Seasonal Models .......................... 119 Regional Forecasting Models ....................... 125 Generalizability of the Statewide Model to the Regional Level ............................. 132 Summary .............................................. 140 VI. CONCLUSIONS ............................................ 142 Summary of the S t u d y ................................. Study Limitations and Future Research................ Conclusions and Discussion ........................... Application of the Results ........................... 143 145 151 155 BIBLIOGRAPHY ............................................... 159 APPENDICES ................................................. 165 A. Definitions of Harmonic Statistics Measures ....... B. Summary Tables and Figures for Chapter V .......... v 165 168 LIST OF TABLES Table Page 1. Evaluation of Decomposition Models byR e g i o n ............ 52 2. Independent Variables of the TrendModels .............. 58 3. Trends in Tourism-Related Employment in Michigan, 1974-84 ............................. 77 4. Distribution of Jobs in Tourism-Related Sectors in Michigan, 1974-84 .............................. 79 5. Trends in Tourism-Related Employment in Michigan, 1974-84, by Region ................................ 81 6. Changes in Tourism-Related Employment in Michigan, 1974-84, by R e g i o n ................................ 81 7. Changes in Tourism-Related Employment in Michigan, 1974-84, by Region andby C o u n t y ................... 85 8. Stability of Monthly Seasonal Factors by Region, 1974.07-1983.06 ................................... 87 9. Seasonal Factors for Tourism-Related Employment in Michigan, 1974-84... ............................ 88 10. Seasonal Patterns of Tourism-Related Employment in Michigan by Sectors, 1984 ...................... 88 11. Seasonal Factors for Tourism-Related Employment in Michigan, 1974-84,by Region .................... 90 12. Forecasts of the Statewide Trend Models for Tourism-Related Employment in Michigan, 1983.07-1984.06 .................................. 102 13. Forecast Accuracy of the Statewide Trend Models for Tourism-Related Employment in Michigan, 1983.07-1984.06 .................................. 102 14. Harmonic Measures of Tourism-Related Employment in Michigan, 1974-84 .............................. 104 15. Forecasts of the Statewide Seasonal Model for Tourism-Related Employment in Michigan, 1983.07-1984.06 .................................. vi 104 Page Table 16. Comparison of 1-12 Months Ahead and 1-month Ahead Forecasts of the Structural Forecasting Model for Tourism-Related Employment in Michigan, 1983.07-1984.06 ................................... 108 17. Comparison of 1-12 Months Ahead and 1-month Ahead Forecasts of the Time Series Forecasting Model for Tourism-Related Employment in Michigan, 1983.07-1984.06 ................................... 110 18. Structural Trend Models for Tourism-Related Employment in Michigan, 1974.07-1983.06, by Region ....... 112 19. Time Series Trend Models for Tourism-Related Employment in Michigan, 1974.07-1983.06, by Region ....... 115 20. Forecast Accuracy of Trend Models for Tourism-Related Employment in Michigan, 1983.07-1984.06, by Region ......................................... 118 21. Selected Harmonic Measures of Tourism-Related Employment in Michigan, 1974-84, by Region ................... 120 22. Seasonal Models for Tourism-Related Employment in Michigan, 1974-84, by Region ................... 122 23. Forecast Accuracy of Seasonal Models for Tourism-Related Employment in Michigan, 1983.07-1984.06, by R e g i o n ......................................... 124 24. Structural Forecasting Models for Tourism-Related Employment in Michigan, by Region ................. 126 25. Time Series Forecasting Models for Tourism-Related Employment in Michigan, by Region ................. 128 26. Forecast Accuracy of Forecasting Models for TourismRelated Employment in Michigan, 1983.07-1984.06, by Region ......................................... 13 0 27. Normalization of Statewide Model Components ............ 13 3 28. Normalization of Northwest Michigan ModelComponents .... 13 3 29. Forecasts of Tourism-Related Employment Index in Northwest Michigan, 1983.07-1984.06 by Model ...... 134 30. Forecast Accuracy of Four Models for Northwest Michigan, 1983.07-1984.06 ................................... 134 vii Page Table 31. Forecast Accuracy of Four Models for Tourism-Related Employment Index in Michigan, 1983.07-1984.06, by R e g i o n ......................................... 138 32. Forecast Accuracy of Four Models for the Number of Tourism-Related Jobs in Michigan, 1983.07-1984.06, by R e g i o n ......................................... 139 B1.1-B1.8. Harmonic Measures of Tourism-Related Employment in Michigan, 1974-84, byRegion .............. 168 B2.1-B2.8. Forecasts of Regional Structural Forecasting Models for Tourism-Related Employment in Michigan, 1983.07-1984.06 ............... 172 viii LIST OF FIGURES Figure Page 1. Study Regions for Tourism-Related Employment in Michigan ....................................... 44 2. A Flowchart of Analysis and Forecast Procedure ......... 47 3. Statewide Tourism-Related Employment in Michigan, 1974-84 .............................. 56 4. Cumulative Percentage Changes in Tourism-Related Employment in Michigan between 1974 (Base Year) and Subsequent Years .............................. 78 5. Cumulative Percentage Changes in Tourism-Related Employment in the State of Michigan and Southern Regions between 1974 (Base Year) and Subsequent Years .................................. 82 6. Cumulative Percentage Changes in Tourism-Related Employment in Northern Regions of Michigan between 1974 (Base Year) andSubsequent Years ...... 83 7. Seasonal Patterns of Tourism-Related Employment in Michigan, 1974-84, by Region ................... 91 8. Total and Partial Autocorrelation Functions for the Structural Regression Model for Tourism-Related Employment in Michigan............................ 97 9. Actual and Predicted Tourism-Related Employment in Michigan, 1974.08-1984.06....................... 106 10. Predicted Index of Tourism-Related Employment in Northwest Michigan, 1983.07-1984.06, by Test Model ..................................... 136 B1.1-B1.8. Actual and Predicted Tourism-Related Employment in Michigan, 1974.08-1984.06, by Region ..... 176 ix CHAPTER I INTRODUCTION Tourism can be defined as " the science, art, and business of attracting and transporting visitors, accommodating them, and graciously catering to their needs and wants” (McIntosh, 1977). The tourism industry is characterized by variations in size, location, functions, types of organization, and the range of ser­ vices provided (Schmoll, 1977). State and regional officials have experienced an increasing interest in the potential tourism holds for generating revenues, raising the level of employment, and contributing to the overall economic development of a partic­ ular geographic area. Some areas are heavily dependent upon tourism and their levels of economic activity fluctuate with the ups and downs of tourism demand. To ease economic instability, it is necessary to have a sufficient understanding of past trends and the forces which cause change in tourism activity. Then, appropriate policies can be developed to respond to anticipated changes in the area. Therefore, systematic monitoring, tracking, and forecasting of tourism activity is highly desirable. Tourism activity varies both spatially and temporally (Stynes and Pigozzi, 1983). For example, temporal variations can be observed in Mexican tourism to the United States which reached a peak at 3.8 million visitors in 1981, experienced a rapid decline for the next two years, and had a projected significant increase in 1984 by 21 percent and in 1985 by 9 percent (U.S. 2 Travel Data Center, 1984). Different geographic areas provide different types of tourism activity that may reach their peaks in different seasons. For example, Hawaii is primarily selling its water- and cultural-related tourism activity all year long, and both s umme r and winter tourism activity prevail in states of mid­ west region, like Michigan. Both temporal and spatial variations in tourism activity must be considered in the design of tourism plans, management policies, and marketing at the national, state, and local levels. Tourism monitoring and forecasting models are generally developed at the national or state level, although many tourism development and marketing decisions are made at the local level. An under­ standing of temporal variations in tourism activity in an area is needed to develop appropriate tourism plans and policies in the area. Changes in tourism activity in an area (region or local) are sometimes difficult to track and monitor because of the lack of good tourism data. Patterns of regional or local tourism acti­ vity must often be generalized from a broader area. The applica­ bility of national and statewide travel statistics to a regional or local level may be quite limited. 3 TEMPORAL VARIATIONS Trend and seasonality are the two major temporal patterns of tourism activity. These may not be stable over time. Trend is defined as the general patterns of increase or decrease in tour­ ism activity over time. Seasonality is defined as a regular pat­ tern of tourism activity by season of the year. In the short term (3-5 years), seasonal fluctuations are more obvious than trends. Trends reveal the historical pattern of the levels of tourism activity from which we may make forecasts. Tourism activity is very seasonal and often viewed as regular patterns in tourism businesses. During the winter season a sig­ nificant portion of tourism facilities in Michigan are idle. For example, in Michigan's eastern upper peninsula, the occupancy rate of motel and hotel rooms in winter is only about one tenth of that in summer, i.e. 5.7% vs. 55.7%, (Michigan Department of Commerce, 1975). Changes in trends and seasonality of tourism activity may impact decisions in the tourism industry. If a decreasing trend in tourism activity in an area is observed, appropriate policies can be designed to counteract this downward trend to avoid losing more tourism business and/or to stimulate the area's tourism industry or develop other economic sectors. It is suspected that the patterns of seasonality, i.e. peak and valley in tourism demand, have been changing over time. For example, off-season tourism activity in northern Michigan has been much higher than previously (McIntosh, 1977). Changes in seasonal patterns of tourism activity may have a significant 4 effect on the timing and quantity of tourism services and oppor­ tunities demanded and supplied. If tourism activity could be more evenly-spread throughout the year, tourism facilities could be more efficiently utilized. At the same time, more tourism opportunities with better quality of services could be provided to the tourists, and more stable tourism employment could be achieved (BarOn, 1973). Seasonality is not a totally uncontrol­ lable factor in tourism management and development. Forecasting models for trend and seasonality can help us better monitor the trend and seasonal patterns and provide forecasts to lead to appropriate decisions in tourism development. SPATIAL VARIATIONS Both trend and seasonal patterns of tourism activity also vary between geographical areas in a country or a state. Differ­ ent geographical areas have different patterns of flow of income, employment and prevailing types of tourism activities. In Michi­ gan, such spatial differences were demonstrated by MESC (1980), Stynes (1980) and Stynes and Pigozzi (1983). For example, in northern Michigan the northwest region is the only one experien­ cing a steadily increasing trend of tourism-related employment from 1974 through 1979. The region was not significantly impacted by the 1979 energy crisis (MESC, 1980). Each region can be characterized by its location, climate, supply of tourism opportunities and related services, and prevailing types of tourism activity. Individual regional tourism plans and policies could be improved, if designed in 5 accordance with regional patterns of tourism activity. Trends and seasonality of local and regional tourism activity may be hidden in the patterns of aggregate statewide series. Since southern Michigan accounted for approximately 92 percent of Michigan tourism-related jobs in 1979 (MESC, 1980), patterns of activity in northern Michigan are difficult to iden­ tify from the statewide data. Thus, using statewide tourism forecasts to guide tourism development in individual counties and regions of the state may be ill-advised. Quantitative forecasting methods can help us to identify, analyze and compare the trends and seasonality of statewide and regional tourism activity across Michigan. FORECASTING METHODS There are three categories of quantitative forecasting methods: (1) time series methods, (2) structural methods , and (3) combined structural-time-series methods (Pindyck and Rubinfeld, 1981). Each of these three types of models has advantages and disadvantages in particular applications. Tourism activity shows strong fluctuations by season of the year. To capture seasonal fluctuations, time series methods, such as decomposition techniques (BarOn, 1975, 1979), Box-Jenkins techniques (Guerts, 1982), and harmonic analysis (Stynes and Chen, 1985), are appropriate. Compared to time series models, structural models are more responsive to changes in activity caused by changes in population, income and other socio-economic, environmental and political variables. Structural models, such as 6 multivariate linear models (Johnson and Suits, 1983) and gravity models (Cesario, 1969) are commonly employed to forecast recre­ ation and tourism activity. Wander and Van Erden (1980), Fritz et al (1984), and Stynes and Chen (1985) report that combining time series and structural models achieves higher forecasting accuracy than with either of these two models used alone. Forecasting methods should be matched with characteristics of the problem and data series. Using an appropriate forecasting method enables us to more fully exploit and utilize information of the data series and achieve study objectives. Strengths and weaknesses of different forecasting methods will be discussed more fully in Chapter II. MICHIGAN AS AN AREA OF APPLICATION Secondary data measuring tourism activity in Michigan will be used to examine and compare trends, seasonal patterns and fore­ casting models at statewide and regional levels of aggregation. Michigan has both summer and winter tourism seasons. The state also has different economic structures between regions with northern Michigan more dependent upon tourism than southern Michigan. For example, twenty out of forty one counties in northern Michigan and only one out of forty two counties in southern Michigan had more than 9 percent of their wages and sal­ ary employment attributed to tourism-related jobs (Michigan Department of Commerce, 1975). Examining and comparing spatial variations in trend and seasonal patterns of tourism activity across several regions of the state can help us better understand regional differences in tourism dependency. Finally, developing scientific and systematic forecasting procedures and models may be useful for the Michigan Travel Bureau in its effors to better monitor and forecast tourism activity in Michigan. PROBLEM STATEMENT Spatial and temporal variations in tourism activity are expected to exist in a large geographic area like a country or a state. If we find that differences in trend and seasonal pat­ terns of tourism activity are significant among sub-regions of the state and between individual regions and the state, there is a clear need to develop forecasting models at both statewide and regional levels. Forecasting models enable us to better under­ stand the trend and seasonal patterns of tourism activity and how these patterns differ across the state. Tourism forecasts can assist the development of tourism plans and policies to properly respond to the needs and wants of tourism businesses in the state and its sub-regions. Also, they can be used by the tourism industries in strategic planning and marketing. Longitudinal studies of tourism activity across Michigan are particularly lacking. A few studies have focused on either iden­ tifying and quantifying trend and/or seasonal patterns of tourism activity in Michigan or developing statewide monitoring and/or forecasting models. The Michigan Employment Security Commission (MESC, 1980) has established a data base of tourism related employment in Michigan and reported the data series' trend, cycle, and seasonal components between 1974 and 1979. These components were descrip­ tively identified and graphically presented at the state and regional levels. The patterns of these three components of tour- ism-related employment levels and comparison of them between var­ ious regions were not completely analyzed. Also, mathematical models used to monitor and forecast the trend, cycle, and sea­ sonal components of tourism activity in Michigan have not been estimated at the state or regional levels. Stynes and Pigozzi (1983), using harmonic analysis, found that seasonal patterns of service employment varies across north­ ern Michigan. Service employment was used as a surrogate for tourism-related employment, since significant positive correla­ tions with tourism activity were found in northern Michigan. Using monthly Michigan lodging tax revenue data a linear regression model (Holecek et al, 1983) and a combined structuraltime-series model (Stynes and Chen, 1985) were developed at the state level. The linear regression model provides interim esti­ mates of Michigan lodging activity 2-3 months ahead of when they are reported for the purpose of better tracking and monitoring lodging activity. The structural-time-series model incorporates both trend and seasonal components of the lodging activity to predict monthly lodging tax receipts. These studies provide some evidence of regional differences in trend and seasonal patterns of tourism activity across Michi­ gan. They also provide a basis for a more sophisticate and com­ prehensive approach to tourism analysis and forecasting based upon quantitative mathematical models. Lack of tourism informa­ tion systematically analyzed and forecasted will hinder the 9 efforts of the state and regional officials and business managers to develop sound tourism plans. This calls for a systematic approach to analyze and forecast tourism activity across Michi­ gan. STUDY OBJECTIVES The purpose of the study is to further analyze the differ­ ences in tourism activity across Michigan by developing a system­ atic forecast modeling approach. The study will extend past research in four areas: (1) identifying and quantifying both trend and seasonal patterns of tourism activity across Michigan, (2) examining spatial variations in these two patterns, (3) developing formal models to forecast tourism activity in Michigan at both the state and regional levels, and (4) comparing the model structures and forecast performance of statewide and regional models. Formally, the study objectives are: 1. Identify and quantify trend and seasonal patterns of tourism activity in the state of Michigan and its individual sub-regions. 2. Compare trend and seasonal patterns across several regions of the state. 3. Develop, estimate and evaluate alternative models for forecasting statewide tourism activity. 4. Develop, estimate and evaluate models for forecasting tourism activity at the regional level. 5. Test the generalizability of statewide models for forecasting 10 tourism activity at regional levels. OVERVIEW OF THE STUDY To achieve these objectives, this study is divided into six chapters. Following this introductory chapter, chapter II reviews relevant literature on forecasting methods and tourism forecasting studies. III. Research methods are summarized in Chapter The results are presented in the next two chapters. Chap­ ter IV summarizes descriptive results of trend and seasonal pat­ terns of tourism-related employment in Michigan. Differences in these two patterns between regions were investigated. Chapter V presents statewide and regional forecasting models for tourismrelated employment in Michigan. The final chapter summarizes the results, limitations of the study and recommendations for future research, and concludes with major findings of the study and applications of the results to tourism planning and management. CHAPTER II LITERATURE REVIEW In this chapter three types of quantitative models are introduced: (1) structural, (2) time series and (3) combined structural-time-series models. Applications of these models in tourism, and especially to Michigan tourism are reviewed. Previous findings, particularly relevant to the study, are summarized at the end of the chapter. QUANTITATIVE FORECASTING MODELS A model is a logical structure reflecting the information or situation revealed from the real world, and is implicit in every forecast or analysis of a social or a physical system (Theil, 1966). on hand? But why is an explicit model needed to analyze the data There are several advantages to formal models. First, during the model building process, the individual is forced to examine, identify, and account for the important relationships involved in a problem. The relationships that make up the model can be tested and validated by examining whether they are as hypothesized in the model specification stage. Second, a statistical measure of the confidence of the individual relationships that make up the model, and of the model as a whole can be estimated. Estimation of forecast errors is important to the user of the forecasts. 11 Third, once a model has 12 been constructed and fitted to data, the effects of small changes in individual variables in the model can be evaluated using sensitivity analyses. This is important in both understanding and using the model (Pindyck and Rubinfeld, 1981). Structural Models Structural models represent the relationships between the variable of interest and those quantifiable factors that affect the behavior of this variable. can be used to forecast. The estimated relationship Using a measure Y^ of tourism activity to illustrate, a structural relationship can be written mathematically as Yt=f(xlt' x2t* ••• > xkt ) • (2.1) where Y-j- is the level of tourism activity at a destination in time t, and is the level of kth explanatory variable affecting Y^ at time t. Two types of structural models are the most common in tourism forecasting, multivariate regression models and gravity models . Multivariate Regression Models In a multivariate regression model, it is hypothesized that a dependent variable is a function (usually linear) of a set of explanatory variables. In the case of estimating the level of tourism activity at a destination, these explanatory variables 13 may be external to the tourism system, such as per capita income, or internal, such as promotional expenditures. Additive and multiplicative models are the most popular functional forms. An additive regression model can be written as ^t33 b o+ b lx lt+ b 2x 2t+ ••• + b ixkt+et' (2 .2 ) where bj_, i-0,..., k, are parameters to be estimated and e^ is an additive error term. A multiplicative model can be written as Yt= b 0Xb l ltXb 2 2 t ...xbik t et' (2.3) where e^ is a multiplicative error term and Y^, X ^ , and b^ are defined as above. In the multiplicative form, the coefficient of variable X^ represents the elasticity of the dependent variable with respect to Xfc. One of several ways to estimate multivariate regression models is to use ordinary or weighted least squares on Equation (2.2) or the logarithmic transformation of Equation (2.3), the assumptions are summarized in Pindyck and Rubinfeld (1981, pp.76). To use a structural model for forecasting, projections of each explanatory variable are required. Unless the future values of the explanatory variables can be reliably and accurately forecasted, the forecasts produced from the model will be of limited use. Forecasts from structural models are made under the assumption that the identified structural relationships between 14 the variables will remain constant over the forecast period. The model should be re-estimated periodically to allow for changes in these structural relationships. The multivariate regression equation should be examined for multi-collinearity, autocorrelation, or heteroscedasticity as these problems may have adverse effects on the use of the model for forecasting (Archer, 1976). Multicollinearity is often present, especially with cross-sectional data (Sheldon and Var, 1985). Although the overall forecast accuracy of the model is not thereby impaired, the effects of changes in each of the intercorrelated independent variables can not be identified and forecasted. Autocorrelation can exist between the values of the error terms. The presence of serious autocorrelation has two implications. First, the estimated relationships that make up the model may be unreliable because coefficients and the degree of variance may be considerably under- or over-estimated. Second, the regression equation is not the best prediction of the dependent variable if serial correlation is present. Methods used to correct for serial correlation, such as transfer function techniques, will be introduced later in this chapter. Heteroscedasticity means that the error terms do not have equal variances. The presence of heteroscedasticity also tends to reduce forecast accuracy. Explanatory variables for forecasting models should be selected based upon: (1) logical relationship to the variable of interest, (2) availability of past and/or future values, and (3) to avoid multicollinearity. 15 Gravity Models Gravity models are developed from the principles of Newton's law of gravitation and similar in form to regression models. A gravity model specifies the nature of relationship, especially those concerning distance between origin and destination, in a more rigid form (Archer, 1980). Trip generation models are sometimes developed from gravity models (Ewing, 1983). Trip generation models are used to estimate and forecast probabilities or number of trips generated from a given origin to a specified destination (Stynes, 1983). A gravity or trip generation model incorporates a number of variables like distance between origin and destination, population size and income level of a trip generating origin, and the attraction index of a destination, etc. to explain why origins (i.e. cities, states, or countries) generate different volumes of trips (Bruges, 1980). The gravity models require origin-destination data which are generally available on a cross sectional basis for a limited number of sites or regions. Time Series Models A time series is a series of data which are historically and systematically collected, or observed, over successive increments of time (Theil, 1966). A time series model "accounts for patterns in the past movements of a particular variable, and uses that information to predict future movements of the variable" (Pindyck and Rubinfeld, 1981, pp. 470). Time series methods are 16 not concerned with explaining the reason why a series is what it is. All causal factors are considered in the aggregate, since only the past behavior of the variable of interest is used to be the basis of our prediction. When used to forecast, it is assumed that the pattern of underlying causal forces which has caused trend, seasonality or cyclical behavior of the data series will remain constant over the forecasting period (Makridakis and Wheelwright, 1983), and that an extrapolation of the trend, seasonal or cyclical pattern will yield an appropriate forecast (Swart, Var and Gearing, 1978). A number of time series methods have been applied in tourism, such as simple trend extension, exponential smoothing, moving averages, etc. Three time series methods will be discussed in some detail here: decomposition methods, harmonic analysis and Box-Jenkins models. Decomposition Methods Decomposition methods decompose a time series into a trend, a cycle, and a seasonal component. Trend is defined as the general increasing or decreasing levels of the time series over time; and seasonality defines a regular pattern of the time series by season of the year. The cycle of the time series is the periodic fluctuation with oscillations occurring between three and seven years, often called "business cycle" (Mendenhall and Reinmuth, 1974). 17 The general mathematical representation of the decomposition approach is: Yt " f (st> Tt , Ct , Et ) where Y^ is the time series values (2.4) (actual data) in period is the seasonal component (or index) in period t, Tt is the trend component in period t, is the cyclical component in period t, and Et is the error or random component in period t. A general way to present a decomposition of a mixed model of Yt into its components, defined by Raveh (1985), is Yt=(TtEt+et )St+st , where Y^ is t=l,...,N the observation of the (2.5) series in time t.T-j- represents the trend and cyclical components in time t, S-j. and s-t are the multiplicative and additive seasonal components, respectively, and E^ and e^ are the multiplicative and additive error components, respectively. The purely additive model is obtained by setting St=Et=l for all t, that is Yt=Tt+st+et . (2.6) A purely multiplicative model is obtained by setting constraints s^=et=0 for all t, that is Yt=TtEtSt . (2.7) t.St 18 A simpler mixed additive-multiplicative model with a mixed type for seasonality and a multiplicative version for the error component, adopted by Raveh (1985), is Yt=TtEtst+st =ZtSt+st where (2.8) is the seasonally adjusted series. The selection of an appropriate decomposition model form depends on the characteristics of the data series. For the case where seasonal fluctuations are independent of the trend, a purely additive model is appropriate. When fluctuations are proportional to the trend, a purely multiplicative model should be adopted. If the fluctuations are neither proportional nor independent of the level of the trend, a mixed additivemultiplicative model may be required (Raveh, 1985). For simplicity only the purely additive and multiplicative model forms will be considered here. In the decomposition process, seasonality is first removed, then trend and finally cycle, and any residual is assumed to be randomness. Bagshaw (1985), Makridakis and Hibon (1979) and Plosser (1979) recommend this prior seasonal adjustment in the forecast procedure to produce more accurate forecasts. There are several methods for separating trend and seasonal components. The n-period moving average method, Census II method, and the classical X-ll variant of the Census (Shiskin et al., 1967) all assume a multiplicative form. A non-metric technique, called Least Polytone Analysis or LPA (Raveh, 1981), can be used to select one of the three possible model forms: purely additive, 19 purely multiplicative and mixed. Harmonic Analysis Harmonic analysis is a variant of Fourier analysis which can be applied to study periodicity in a time series with regular oscillations (Rayner, 1971). Fourier analysis estimates a time series with equal intervals of observations by an infinite series of sine and cosine curves. Harmonic analysis affords a means to quantify the seasonal component of a time series with regular periodicity over time and express them mathematically as the algebraic sum of a series of simple sine curves (Horn and Bryson, 1960). Harmonic analysis uses ordinary least squares procedures to fit a series of sine curves of varying amplitudes and frequencies to a time series. Each harmonic (one sine curve) is determined independently of the others so that the characteristics of any particular harmonic, e.g. amplitude of fluctuation, and time of peaking or trough, can be mapped and examined separately. The sum of all harmonics recreates the original time series. The formula for the Fourier series commonly used in harmonic analysis, defined by Rayner (1971), is: xk ('9)- xk= 2 [ A kSIN(k0+k)] = A 0+ A1SIN(e+1)+A2SIN(2e+2) + . • .+ AkSIN(K0+]c)+ ...+An SIN(N0+(J)n) (2.9) 20 where, is the time series of interest. is the amplitude for harmonic k, measuring the magnitude of the fluctuation of sine curve k around the arithmetic mean of the data series (Aq )• <{>]£ is the phase angle which determines the time of the year at which the peak (and consequently trough) for harmonic k occurs. 0 is a portion of the one-year period that a data point repre­ sents, measured as an angle. For example, for a series with 12 monthly data points, each month is represented by 30 degrees (360/12) if we make the simplifing assumption that all months have the same number of days. One degree roughly corresponds to one day in the year. When the number of sine terms (frequencies) needed to perfectly fit the finite data, (i.e. 100 % of variation in the series is explained). This process is called harmonic analysis, otherwise Fourier analysis. Harmonic analysis is appropriate for a time series with equal interval observations and regular periodicity over time (Rayner, 1971). Harmonic analysis can fit a time series with N observations with a series of N/2 harmonics or sine curves. Each of the k harmonics is fitted to the data curve by (1 ) increasing or decreasing the value of arithmetic mean of the data series, which decides the amplitude of the kth harmonic, and (2 ) shifting the curve horizontally, according to the phase angle, (j)^, of the harmonic k, so as to change the abscissa value at which the maximum occurs (and consequently minimum) (Horn and Bryson, 1960). Statistics, such as the amplitude, relative amplitude, phase angles, peak timing, variance explained can be produced from 21 harmonic analysis to help in explaining the behavior of the variable of interest. For their definitions and derivation, see Rayner (1971) and Appendix A. Box-Jenkins Models Box and Jenkins, in 1970, postulated a time series model which is a mix of autoregressive and moving average processes. An auto-regressive process of order p, denoted as AR(p), is to generate the current observation Y^ by weighting past observations going back p periods, together with an error term in the current period. A moving average process of order q, denoted as MA(q), is to generate the current observation Y-t by weighting error terms going back q periods (Pindyck and Rubinfeld, 1981). Box-Jenkins models are sufficient to describe a wide variety of stationary series. A series is stationary when it has constant mean and variance and its underlying forces are assumed to remain constant over time, otherwise it is non-stationary (Guerts and Ibrahim, 1975). A non-stationary series can be transformed into a stationary one by differencing. The number of differences required to produce a stationary series depends on the behavior of the series. One or two differences normally suffice for most empirical series. The general form of the Box-Jenkins model can be written as <|>(B)AdYt= S +9(B)et with (j)(B)=l-(B) the autoregressive operator and 0 ( B ) the moving average operator denote polynomial function of the operator B for the autoregressive process and the moving average process respectively. B is a backward shift operator which imposes a one-period time lag each time it is applied to a variable, e.g. Bet=et-i and BY-t^t-l* p and 9g are parameters of the autoregressive and moving average components respectively. Adyt is the degree of differencing which transforms a nonstationary series to a stationary series, e^ is a normally distributed random disturbance process with a mean equal to zero and a variance equal to autoregressive component. ^ is a constant term of the Equation (2.10) is sometimes referred to as an ARIMA (Autoregressive Integrated Moving Average) model of order [p,d,q]. Box-Jenkins models are generally derived in three steps: (1) identification of a tentative model for a series, (2 ) estimation of parameters and examination of a set of diagnostic statistics and plots, and (3) generation of forecasts. If the model is not adequate, replication of the modeling process from the first step has to be performed again until an acceptable one is obtained. Then the forecasts can be computed (Pindyck and Rubinfeld, 1981). Advantages of using Box-Jenkins techniques are: (1) they can fit any and all kinds of data patterns; and (2 ) greater forecast accuracy for some types of series can be obtained. Disadvantages are: (1 ) it is difficult to use because of the complexity of the model-building processes and the difficulty in fully understanding the underlying theory; and (2 ) the accuracy depends somewhat on individual forecasters' knowledge and experience to determine the 23 best ARIMA model (Wheelwright and Makridakis, 1985) . Structural-Time-Series Models There are several ways of combining structural models with time series models. These include varying parameter models, simulation models and combined structural-time-series models. The combined models use a structural method to capture and measure the influence of individual explanatory variables on the dependent variable and time-series methods to capture historical patterns in the data or errors. Three approaches of the combined models will be reviewed: (1 ) averaging forecasts from two or more models, (2) transfer function models, and (3) decomposition-type combined models. Averaging Forecasts from Two or More Models The first approach is to average forecasts from two or more competing models to derive a more accurate forecast than with any individual model used alone (Fritz, Brandon and Xander, 1984). Weights, sometimes based upon variances of the individual models, are applied to each forecast to calculate the combined forecasts. A simple example serves to illustrate. The combined forecast is Fc = Wt FX + (1-Wt) F2 , (2 .11 ) 24 where Fc is the combined forecast, and F2 are structural forecast and time series forecasts respectively, and and (1 -Wfc) are the weights calculated for F^ and F2 respectively. For the procedure of computing the weights, see Fritz et al. Transfer Function Models The second approach is a transfer function model. A transfer function model combines a linear regression model and a Box-Jenkins model. It can be used to improve the power of the linear regression model to explain and forecast the behavior of a series. Formally, a transfer function model takes the following form: Y-t»= (a regression model) + (an ARIMA model) = (ao+aiXit+a2x 2t+ •••+akxkt) + where (B)e (®)tyt)' (2 .12 ) is the value of explanatory variable k at time t, a normally distributed error term at time t. MOMfCAl M 0 MIA OMA HA I AANNV AN AUNI N • NAIlO Tl * • • » * * • . 3 ■IN I SAI AN Af •mini iihii. C i t N f Q N • • Ml ABA IAIOM CA tH OU N M AHAN ivumiIim uaeiAMU ■ AIM 11NAN AIM! 2 ,1 1 1 1 , u , . . u I Jhou ui I Figure 1: Study Regions of Tourism-Related Employment In Michigan. 45 ANALYSIS AND FORECAST APPROACH The first four study objectives require the estimation of models for tourism activity in Michigan. The same analysis and forecast procedure will be applied to the state and each sub-region. From the review of previous tourism studies it is concluded that trend and seasonal patterns of Michigan's tourism activity and factors affecting these two patterns should be identified, modeled and forecasted at both the state and regional levels. Combined structural-time-series forecasting methods have been selected because of their ability to capture both trend and seasonal fluctuations of a time series. Also, combined models have generally outperformed alternative models in fitting existing data and in their forecast accuracy (Wander and Erden, 1980). Decomposition methods (BarOn, 1972, 1975, 1979; Weller and Kurre, 1987) are used to separate the trend and seasonal components of the series. Transfer function modeling techniques are then applied to the trend components. A seasonal model is estimated from the seasonal component using harmonic analysis. The trend and seasonal models are then re-combined to yield the final forecasting model. 46 The general approach is divided into six steps (Figure 2): 1. Assemble data series for the region. We will be dealing with 9 time series, one for the state of Michigan as a whole and employment series for eight sub-regions of the state. 2. Choose an appropriate decomposition model. We will choose from either an additive or multiplicative model. 3. Decompose the time series employing a simplified version of Census II method and then identify its trend and seasonal patterns. 4. Model trend and seasonal components and generate their forecasts. 5. Combine the seasonal and trend models to form a structural-time-series forecasting model. Produce 1-12 months ahead forecasts of the time series for the region. 6 . Evaluate the combined model based upon goodness-of-fit to the data and forecast accuracy. The first three steps analyze the trend and seasonal patterns of a data series. The next three steps develop and evaluate a forecasting model for a data series. Decomposition methods and techniques for describing the trend and seasonal patterns will be presented first, followed by procedures for estimating and evaluating the forecasting models. 47 far Step l: Step 2: Choose the agaxgriate type of deocnpasiticn model: purely additive carnultiplicative.________ ♦ Deaaipcse the serif into: (A) moving average ccnpcnent: S^ or sj- Step 3: Identify the trend from the moving averages Identify the seasonality ccnpcnent from the * » Model the seascrality fran the seasonal cmpcnait and predict it K units ahead, -1 — *» Step 4: »< _ I.- ■ < tne tzena sen the moving averages and predict than K units ahead noaei • ••i K •••» K ________ Generate the forecasting model and predict the series by oenbining the forecasts obtained in A and B. Bor exanple: Step 5: YNtl " ^H-l * %tl far a imitiplicative model, one unit ahead Step 6: |Evaluate the model's forecasting ability 1 Figure 2: A flowchart of Analysis and Forecast Procedure. 48 Identification of Trend and Seasonal Components Trend and seasonal patterns of tourism-related employment are identified using a decomposition procedure, either multiplicative or additive. Multiplicative and additive decomposition methods will be introduced first, followed by methods to choose one of these two decompositions for all data series to be encountered. Decomposition Methods After the data series for a particular region are assembled, an appropriate decomposition method, either multiplicative or additive, will be selected to decompose the data series into trend and seasonal components. The multiplicative model is Y-t = Tt*st*Et' and the additive decomposition model is Y^ * T-t+s-t+et* T-t is the trend, and s^ are the respective seasonal components, and Et and e^ are the respective error or random components. The first step in both decomposition methods is to calculate a series of moving averages, which will include both trend and cycle. A simplified version of the Census II method using a 12 -period moving average is used to separate the trend and cycle components from the seasonal component of the time series. In the rest of this study, the trend component will represent both trend and cycle components together. 49 The following is a brief description of the 12-month centered moving average method (Hall and Hall, 1980). Assume the series to be adjusted is Y, which has N observations and a periodicity of 1 2 , i.e. a monthly series with an annual seasonal period. The trend component can be derived by calculating the moving averages of Y, T(t): T(t)= (1/12)*{(1/2)[Y(t-6 )+Y(t+6 )] +Y(t-5)+ Y(t-4)+...+Y(t+5)} (3.1) The equation describes a moving average over 12 observations centered at each observation. To calculate the multiplicative seasonal factors, S^, the ratio, R(t), of the data series to its moving averages is first calculated: R(t)= Y(t)/T(t) (3.2) This ratio contains the combined seasonal and error components. Then the vector R(t) may be rewritten as a matrix F(i,j) where each row i corresponds to a year and each column j to a month [i.e. t=j+12*(i-1)]. F contains N-12 elements (omitting 6 observations at the beginning and end of the series). The multiplicative seasonal factors are calculated by averaging each column in F to eliminate random disturbances: M S(j) = (1/N) *£F(i,j), where i=l N years j=i / • •. ,1 2 . (3.3) 50 In the purely multiplicative model, the seasonality series is the series consisting of the 12 seasonal factors, S(j), repeated across the years. These 12 seasonal factors add up to 12. There are two steps to compute the additive seasonal component, denoted S f First, we subtract the moving averages from the actual data series, that is Z(t) - Y(t)-T(t) (3.4) Then, compute the averages for each month across the i years of data to eliminate the random error component, that is s(j) = (1/N) *XZ(i,j), where i=l, ..., N years j=l, ..., 12. (3.5) Zj^j is a matrix (like F(i,j) above) rewritten from Z(t) with each row i corresponding to a year and each column j to a month [i.e. t => j+12*(i-l)]. Whichever model is used, the decomposition produces two series, a trend and a seasonality series. Choice of Additive versus Multiplicative Decomposition Method A choice must be made between the multiplicative and additive decompositions. Although the purely additive form is rare in empirical data series and not recommended by many researchers (Wheelwright and Makridakis, 1985), it could not be 51 ruled out based on a visual inspection of the series. To examine which decomposition model best fits the Michigan tourism-related employment series a preliminary test was performed. The data series for each of three selected regions were used in the test: (1) the state of Michigan, (2) northwest Michigan and (3) the Mackinac Straits region. These regions were selected to represent a range of differences in the 9 series to be modeled from visual inspections of the series. The state series represents patterns typical of a region without dominant seasonality, while the northwest Michigan and Mackinac Straits regions are more typical of tourism-dependent regions with strong seasonality. Both additive and multiplicative decompositions were calculated for each of these three regions using the period from January 1974 to December 1984. Three goodness-of-fit measures for the two models were calculated. These are introduced and discussed more fully in the section on evaluation of forecasting models later in this chapter. The multiplicative decomposition was selected as it outperformed the additive form in the two tourism-oriented regions and is much more common in empirical studies (BarOn, 1972,1975,1979; Weller and Kurre, 1987). The two decompositions fit the statewide series about equally (Table 1). 52 Table 1: Evaluation of Decomposition Models by Region. Goodness-of-Fit Region Type of model Mean Absolute Error (MAE) Mean Absolute Percentage Error (MAPE) Eta-Square (%) Mult. 1,469 0.64 99.13 Add. 1,441 0.63 99.19 Mult. 145 1.66 98.27 Add. 166 1.92 97.77 Mult. 61 4.22 99.27 100 9.15 98.31 Michigan Northwest Straits Add. Note: Multi.: Multiplicative decomposition model. Add.: Additive decomposition model. 53 Model Estimation In the fourth step of the approach, the trend and seasonal models for individual regions will be estimated. Model estimation is divided into two parts: (1 ) estimating a seasonal model and (2) estimating a trend model. Transfer function techniques will be employed to estimate three alternative trend models. They incorporate time factors, economic driver variables, or both groups, as explanatory variables. BarOn (1979) has argued that short-term tourism trends are difficult to forecast using econometric methods, since there are so many changes external and/or internal to the environment of a tourist destination. Seasonal patterns, on the other hand, are relatively stable over time and much easier to predict. A simple harmonic analysis procedure will be used to estimate this component. Seasonal Model Seasonal variations in tourism-related employment will be identified and quantified into a seasonal model using harmonic analysis. Harmonic analysis is briefly explained in pp 19-21. The harmonic model is estimated from the seasonality series, i.e. 12 seasonal factors. The harmonic model generates the seasonal com­ ponent of the tourism-related employment series. With 12 points, six harmonics may be produced. 54 The seasonal equation with all six harmonics is: St=B0+Bi sin(30t+(j>i)+B2 sin(60t+2) + .. .+ Bg sin(180t+^>g) (3.6) where A S-^: estimated seasonal component of tourism-related employment in time t , B-j_: amplitude of the i-th harmonic, and (|>£: phase angle (in degrees) of the i-th harmonic, i-1 , To allow for random error and ease of interpretation, we will primarily look at the first (annual) and second (semiannual) harmonics. Other harmonics will be included if necessary to achieve an overall explanation of 98% of the seasonal variation. Xcanfl.Mp.flel Procedures for estimating trend models for the individual regions can be divided into three steps: (1 ) estimating alternative linear regression models, (2 ) checking auto­ correlation of each model, and (3) adding transfer components. Estimating Linear Regression Models Three alternative linear regression models will be tested and evaluated for each study region: (1 ) a time-series regression model, (2) a structural regression model, and (3) a structuraltime-series regression model. These three linear regression models are tested to see which type of independent variable (time 6. 55 series, structural or both) can best capture the trends of the series under certain conditions. These three models will be discussed in turn. (1) The hypothesized time-series regression model is: Tt = A 0 + Ax t + A 2 t 2 + A 3 t 3 (3.7) where A T-^: estimated trend component of tourism-related employment in period t, A^: parameters of the equation, and t: 0=January, 1974, 1 February, 1974, ..., 131=December, 1984. This time-series regression model uses three time factors (i.e. t, t2 , and t3) to fit the trend component of tourismrelated employment series encountered. Time (t) is used to cap­ ture the changing patterns of tourism and recreation activity and tastes and preferences of tourists, t 2 and t 3 are chosen because a plot of the statewide tourism-related employment data series (Figure 3) suggests a non-linear (quadratic or cubic) function. (2) The hypothesized structural regression model is: A Tt = A 0 + A x MAUNEMPt + A 2 POPt + A 3 MAGASt + A 4 MAINCt (3.8) where MAUNEMPt: seasonally adjusted statewide unemployment rate in period t, POPt: population size in period t (in 100,000 of persons), MAGASt: seasonally adjusted U. S. gasoline retail price index in period t, MAINCt: seasonally adjusted statewide personal income in period t (in billion of dollars), A^: parameters of the equation. The rationale for using structural variables to capture the variations of the trends of tourism-related employment in MICHIGAN 270 TO U R ISM —RELATED EMPLOYM ENT (T h o u sa n d s ) 260 250 240 230 220 - 210 - 200 - 190 180 H * 1974.08 1980.07 ACTUAL Figure 3: + MONTH SEASONALLY ADJUSTED Statewide Tourism-Related Employment in Michigan, 1974-84. 1984.07 57 Michigan is provided here. It is hypothesized that same economic phenomena in Michigan (such as the 1975 recession, the subsequent recovery and periods of gasoline shortage) are reflected in not only tourism related employment but also other economic measures, e.g. unemployment, population size, retail price of gasoline and personal income, etc. The assumption made here is that the other economic variables can be used as the explanatory variables to estimate a forecasting model for tourism-related employment. The relationship between these individual explanatory variables and the volume of tourism activity is supported by studies reviewed in the previous chapter (MESC 1980; Diamond 1977; Johnson and Suits 1983; and Loeb 1982). The four independent variables are summarized in Table 2. The monthly statewide unemployment rate in Michigan had a negative linear correlation with the monthly level of tourismrelated employment (R = -0.49) for the period 1974-1979 (MESC 1980). In 1979, jobs in tourism-related sectors accounted for 7.0 percent of total employment in Michigan (MESC, 1980). It is rea­ sonable to hypothesize that the higher the statewide unemployment the lower the tourism-related employment. Monthly statewide unem­ ployment data are available from a monthly federal publication (Employment and Earnings, U.S. Bureau of Labor Statistics). Quar­ terly predictions of statewide employment are produced by Univer­ sity of Michigan's econometric forecasting model (Shapiro and Ful­ ton, 1985). The size of population of a tourist-generating area has a positive relationship to the levels of tourism activity in a tourist-destination area , which in turn impacts tourism-related Table 2: Independent Variables of the trend model. Variable Time period Level Source Unemployment Rate Monthly State Employment and Earnings, U.S. Bureau of Labor Statistics. Population Annual County Current Population Reports, U.S. Bureau of the Census. Personal Income Quarterly State Michigan Statistical Abstract. Retail Gasoline Price Index Monthly U. S. Monthly Labor Review, U.S. Bureau of Labor Statistics. * Variables available on an annual or quarterly basis were estimated on a monthly basis by simple linear interpolation between the quarterly or annual figures. * Except population, all other variables are seasonally adjusted using the 12-month centered moving average technique. 59 employment levels in that area (Diamond, 1977). Population size may capture the share of tourism-related employment serving locals or visiting friends and relatives. The larger the population size of the area, the higher the expected level of tourism-related employment. Population size data are available annually from the Current Population Reports, U.S. Department of Commerce, Bureau of the Census. by linear interpolation. Monthly estimates are obtained This data series cannot account for seasonal fluctuations in population which could be a factor in some areas. The retail price of gasoline is an indicator of travel costs. Changes in the retail price of gasoline are expected to influence frequencies of travel and length of stay (Johnson and Suits, 1981). It is hypothesized that the higher the retail price of gasoline, the lower the volume of tourism activity and in turn tourism-related employment in Michigan. The monthly retail gaso­ line price index for all urban consumers is published in the Monthly Labor Review by the U. S. Bureau of Labor Statistics. Personal income has been consistently found to be an influen­ tial determinant of tourism demand. Sauran (1978) reported that in some studies the income elasticities of tourism activity are greater than unity. It is hypothesized that the higher the per­ sonal income the higher the volume of tourism activity and in turn the level of tourism-related employment. Past values of personal income by quarter are available from the Michigan Statistical Abstract, and the predicted quarterly values are provided by the University of Michigan (Shapiro and Fulton, 1985). mates are obtained by linear interpolation. Monthly esti­ 60 Each of the variables except population size are seasonally adjusted using the 12-month centered moving average technique. It should be noted that the population size variable is the only independent variable that changes across regions. The other inde­ pendent variables are measured at the national or state level and dot not vary between regions. Although these independent vari­ ables are not perfect proxies of regional variables, they will be used in the model because local or regional tourism activities are not independent of national and/or statewide economic phenomena. (3) The hypothesized structural-time-series regression model is: A Tt = A 0 + Ax MAUNEMPt + A2 POPt + A 3 MAGASt+ A 4 MAINCt + A 5 t+ A 6 t2 + A 7 t 3 where all components are (3.9) defined as earlier.In this three time factors andfour independent variables model, are included to see whether or not combining these two types of variables can improve the model's performance. These two groups of variables may be complementary or substitute for each other. Checking Autocorrelation of the Regression Models Autocorrelation of a regression model will be identified by examining the value of the Durbin-Watson statistic (DW). If DW is not close to 2 .00 , there exist problems in positive or negative autocorrelation, which may adversely affect the model's forecast accuracy (Pindyck and Rubinfeld, 1981). From some preli­ minary analyses, auto-correlation problems are anticipated. 61 Transfer function techniques can correct for this problem. Estimating Transfer Function Models Three transfer function models will be estimated: (1) the hypothesized time-series-ARMA(p,q) transfer function model: Tt = A 0 + Ax t + A 2 t 2 + A 3 t3+ (|)“1 (B)0(B)lt5t (3.10) (2) the hypothesized structural-ARMA(p,q) transfer function model: A Tt = A 0 + Ax MAUNEMPt + A 2 POPt + A 3 MAGASt + A 4 MAINCt+ (^_ 1 (B)e(B)i|t/ and (3.11) (3) The hypothesized structural-time-series-ARMA(p,q) transfer function model: A Tt = A 0 + A x MAUNEMPt + A 2 popt + A3 MAGASt + A 4 MAINCt + A 5 t + A 6 t 2 + A 7 t 3 + cjT^BjefBJty. (3.12) where nt are error terms assumed to be independent and normally distributed with a constant variance, N(0,(Ta), and <{>(B) and 6 (B) are defined as in Equation (2.10). All other components of these transfer function models are as defined earlier. 62 To add transfer components to a regression equation, the unconditional residuals (E-^) will be computed first; that is: A Et = Tt - Tt (3.11) A where T-j- and T-j. are the actual and estimated trend values respectively. Then, we apply Box-Jenkins procedures to the residual series to identify the order (p,d,q) of the autoregressiveintegrated-moving-average error process (i.e. ARIMA(p,d,q) model) for the series. Since the values of the trend series are seasonally adjusted, this residual series is usually stationary and can be modeled as low order ARMA process, i.e. as a process with p<2 and q<2 (Pindyck and Rubinfeld, 1981). An autoregressive-moving-average error model of order (p,q), ARMA(p,q), is: Et “^l Et-1 + •••+ 4>t-p Et-p + vt + •••+ 0vt-q (3.12) V-j- are the forecast errors of the unconditional residuals. This is the error we make if we compute a forecast of the unconditional errors, E^, by applying only the autoregressive part of the ARMA(p,q) model. There are two different kinds of residuals associated with the transfer function model, i.e. E-t and V f E^-p are values of E^ lagged p periods and V-j-_q are values of and lagged q periods. respectively. p and 0q are coefficients of E-j- 63 The primary tools for selecting the probable ARMA(p,q) model for are the total autocorrelation and partial autocorrelation functions. The autocorrelation function (ACF) is a measure of how much correlation there is between neighboring data points in a series. The partial autocorrelation function (PACF) is a measure of how much correlation there is between the current observation and the N^h lagged observation after netting out the effects of any other observations. For a detailed introduction of ACF and PACF see Box and Jenkins (1970). In this identification stage we compare the shapes of the sample ACF and sample PACF with the theoretical ACF and PACF of various members of the ARMA class. Generally, spikes in the ACF indicate moving average terms, and the PACF can be used to infer the order of the autoregressive process. For example, if the PACF has significant spikes at lags 1 and 2, a second-order autoregressive model would be used. For a detailed theoretical rationale for this identification procedures refer to Box and Jenkins (1970). Finally, we estimate the transfer function model. After alternative ARMA(p,q) models for each trend model are identified, alternative transfer function models can be estimated. The estimation procedure used here is a variant of a generalized least squares algorithm whose objective is to find the parameter estimates in a most efficient manner. are estimated simultaneously. All parameters of the model The procedure incorporates the estimate of the ARMA(p,q) model for the unconditional residual from the past observation into the regression model to predict the current observation. A more complete discussion of this 64 estimation algorithm can be found in Cochrane and Orcutt (1949, pp. 32-61). Transfer function models will be estimated with data for the dependent and independent variables from July 1974 to June 1983. After three transfer function models are estimated, they will be compared with each other in terms of model structures, ability to fit the observed data and forecast accuracy. Centered moving averages of tourism-related employment data between July 1974 and June 1983 will be used to evaluate the goodness-of-fit of the estimated trend models. Cases from July 1983 to June 1984 will be used to evaluate the forecasting performance of the alternative trend models. This completes the discussion of methods for esti­ mating trend and seasonal components. Forecasting models are completed by recombining the trend and seasonal parts (step 5 of Figure 1). The Combined Forcastina Model In the fifth step the selected trend model and the estimated seasonal model are combined together to generate a forecasting model for each study region. Forecasting models for individual study regions are used to predict tourism-related employment in future months. The proposed forecasting model for region i is (in the multiplicative form): A A A *i,t = Tijt * Si/t (3.13) 65 A where Yj_ ^ estimated monthly tourism-related employment m A region i in period t, t is the trend model for region i in A period t f and is the seasonal (harmonic) model for region 1 in period t. Forecasting Procedure After a forecasting model is estimated, the model will be used to predict the levels of tourism-related employment for the next 12 months from July 1983 to June 1984. The assumption made for forecasting is that the trend and seasonal patterns captured in the model remain constant over the forecasting period. The forecasting procedure has three steps: (1) forecast the future trend values, (2 ) forecast the future seasonal factors, and (3) combine the trend and seasonal forecasts to generate the complete forecasts. First, to predict the future trend values, actual moving averages of the independent variables from July 1983 to June 1984 will be entered into the model. It is assumed that the indepen­ dent variables can be perfectly forecasted. This allows us to better identify the sources of forecast errors of the model, i.e. trend, seasonal or both components, because the forecast errors attributed to the errors in predicting the independent variables can be eliminated. During the forecasting period, the trend model will not be updated by reinserting the actual moving averages of the dependent variable into the computation process to produce forecasts. The predicted trend value of a month (e.g. July 1983) will be used to produce the trend forecast for the following month 66 (August 1983). The lead time for the individual trend forecasts is from l (for July 1983) to 12 (for June 1984). Second, the future seasonal factors are simply generated from the seasonal model given the corrresponding time (t) values. It is assumed that these seasonal patterns captured by the sea­ sonal model do not change during the forecasting period. Finally, the corresponding monthly trend and seasonal fore­ casts are multiplied together to generate the complete monthly forecasts of tourism-related employment. Forecast accuracy of the model to predict for the next 12 months will be evaluated with the holdout cases from July 1983 to June 1984. It should be noted that the actual values of the dependent variable from July 1983 to December 1983 are used in the trend model estimation, due to their presence in the 12 -month centered moving average technique to calculate the moving averages. Evaluation of the Forecasting Models In the final step, the forecasting models will be evaluated and compared . Three criteria will be used to evaluate the per­ formance of the forecasting models: (1 ) signs of the coefficients, (2) fit to the data, and (3) forecast accuracy. Signs of the coef­ ficients of the model should be as hypothesized in the model spe­ cification stage. Fit to the data refers to how well the model can fit the data series from July 1974 to June 1983. Forecast accuracy refers to how well the model can predict cases that are not included in the model estimation procedure. 67 Statistics for measuring "goodness-of-fit" of a non-linear model are mean absolute error (MAE), mean absolute percentage error (MAPE), root mean square error (RMSE) and eta-square (eta2). The smaller the MAE, MAPE and RMSE and the closer to one the eta2 is, the better the model fits the data. Formal definition of these statistics are presented after introducing some notation. Let a series of observations taken at equally-spaced time intervals (e.g. months) be denoted as Y-j-, where t=l,2,...,N, and N is the total number of observations. Y denote the mean of the series Y f Let Let P-t denote the series of predicted values for the series Y^. Mean absolute error (MAE) is the average of difference between the predicted values and the actual data values, defined formally as: MAE = JjYt"^ (3.1 N MAE is selected instead of mean squared error (MSE) as it is easier to interpret and avoids giving undue influences to a few large errors. Root mean square error (RMSE) is defined as: N 2 dt ‘ pt> — ---------RMSE = N Z. (3.15) It is the root squared of the average of squared differences between the predicted values and the actual data values. 68 Mean absolute percentage error (MAPE) is defined as: N , 2 MAPE i |*t - Pti/*t ___________ *100.0 (3.16) N It measures errors in proportion to the observed values. Eta-Square, eta2 , measures the amount of variance of the dependent variable explained by a non-linear model. N „ Eta2= l - o soft - Pt>2 __________ N 2(Y Eta2 is - (3.17) 2 -Y)2 t=l A measure of forecast accuracy is Theil's UII statistic. Theil (1966) proposed two forecast accuracy measures, UI and UII. The UII measure is prefered to the UI measure because it gives more meaningful information about the accuracy of forecasting models (Bliemel, 1973). Theil's UII statistic is defined as: N 2 S (Yt - P e r UII £ £ ---------- Jc-l If UII=0, the forecast is perfect. (3.18) A forecast with UII=1 is only as accurate as a forecast of no change. If UII>1, the forecast is worse than one of no change. Each of these five measures (i.e. MAE, RMSE, MAPE, eta2 and UII) may be applied to measure goodness-of-fit or to assess the accuracy of forecasts. In the latter case, the formulas are applied to the holdout sample of cases. Different measures of forecast accuracy are used to capture different characteristics of the forecast errors (Makridakis, et al., 1983). Also, different readers may be interested in different forecast accuracy measures 69 for some particular purposes. The measures of MAE and MAPE are easy to interpret. Theil's Statistic-UII is frequently employed to measure forecast accuracy (Theil, 1966; Geurts and Ibrahim, 1975; Fujii and Mak, 1980; Guerts, 1982). The Theil's UII and RMSE measures penalize a fore­ cast much more for extreme deviations than it does for small ones because of squaring errors. In contrast, both the MAE and MAPE measures treat all errors equally, or give the same weight to each error. Comparison of the Forecasting Models By applying thees forecasting procedure to individual study regions, regional models can be derived. All forecasting models will be compared in terms of the structures of trend and seasonal components, forecast accuracy, and sources of forecast errors. In the comparison of trend models, we examine independent variables entering the equation, the signs of the coefficients, the structure of the error model, and variance explained. In the comparison of seasonal models, we examine the magnitudes of har­ monics and variance explained. Forecast accuracy of the forecast­ ing models will be examined using MAE, MAPE, RMSE, eta2 and Theil's UII statistic. Sources of forecast errors in each fore­ casting model will be identified by examining forecast accuracy of the trend and seasonal components separately. 70 GENERALIZABILITY OF THE STATEWIDE MODEL TO THE REGIONAL LEVEL The fifth study objective is to test how well statewide models forecast at the regional level. It is suspected that the statewide model may not accurately predict future changes in tourism-related employment in each sub-region of Michigan. The differences between the predicted statewide and regional patterns may be due to differences in trend, seasonality, or both. The proposed methods to test for such differences are to substitute the trend, seasonal, or both components, of the statewide model for the corresponding component(s) of a regional model to forecast tourism-related employment in that particular region. Four models for each sub-region of Michigan will be tested. Let the statewide model be represented as A A A Ys = Ts * Ss and (3.19) the model for region i as A A A Yi - Ti * Si, A (3.20) A where Yg and YR are the statewide and regional predicted values A A and Tg, Sg, A A Ti and Siare the statewide trend andseasonal i, respectively. and regionalpredicted components for thestate as awhole and region 71 To account for differences in the size of the trend component across regions the statewide and regional trend components have to be normalized first. Thus, changes in the respective regional trend index series can be compared between regions on the same basis since thay all start from 1.00 for July 1974. The seasonal component is already standardized as an index series, i.e. a seas­ onality series with 12 seasonal factors. Define the predicted statewide index series for the trend as A A T's = Tg/(Tg for July 1974) (3.21) and a predicted trend index for region i as A A T'i - Tj/(Ti for July 1974) (3.22) where Tg and T^ are the actual statewide and regional trend components. The base month for each component series is July 1974. The four alternative index series models for sub-region i are : (3.23) (2) Y2i - T'i * Sg (3.24) (3) A A Y3i = T'g * Si (3.25) (4) Y4 i - T'g * Sg (3.26) 72 where Y^i is the estimated regional index series for region i using model j, j=l,2,3,4. In the first model, regional estimates of both the trend and seasonal components are used. In the second model, the statewide seasonal component substitutes for regional seasonal component. In the third model, the regional trend is replaced by a statewide trend, and in the last model, both trend and seasonal components are the statewide estimates. Forecast accuracy of these four index series models will be evaluated and compared for each sub-region to see which model per­ forms best in predicting the actual index series, Y'^, defined as Y'i = T'i * Si (3.27) T'i - Ti/(Ti for July 1974) (3.28) where T'i is the actual trend index series for region i starting from 1.00 for July 1974. series. Si are the actual regional seasonality Notice that Y 1* will not start from 1.00 from July 1974 unless the seasonal factor for July is also 1.00. The number of tourism-related jobs in a region is obtained by multiplying the predicted index values by the level of tourismrelated employment in the base month (July 1974). Comparisons of the four models will provide measures of the errors made in apply­ ing the statewide model (patterns of tourism activity) at regional levels. 73 SUMMARY Data selected to Investigate tourism activity across Michigan are monthly tourism-related employment series from 1974 to 1984. Nine study regions including the state of Michigan and eight subregions are formed. These sub-regions were formed by grouping counties with similar magnitude and seasonal patterns of tourismrelated employment and (with one exception) adjacent geographic locations. A general research approach for analyzing and forecasting tourism activity in the individual study regions is proposed. A multiplicative decomposition is adopted to decompose the data series into a trend and a seasonality series. Trends are identi­ fied from the changes in annual averages in tourism-related employment, and seasonality is identified from the variations of the seasonality series over a 12 months period. Trend and sea­ sonal patterns for each study region will be identified and quan­ tified to achieve the first study objective. These two patterns will be compared between regions to achieve the second study objective. In the forecasting procedure, trend and seasonal models will be developed for each study region. Three approaches will be tested to estimate the trend models: (1 ) time-series, tural, and (3) structural-time-series models. (2 ) struc­ Transfer function techniques will be used to reduce autocorrelation problems. The three trend models will be evaluated and compared with each other. Harmonic analysis will be applied to the seasonality series to generate a seasonal model. The seasonal model produces monthly 74 seasonal factors for the corresponding months of the year. The trend and seasonal models will be combined together to generate combined forecasting models for tourism activity in a study region. Accuracy of the forecasting models will be evaluated. Following this forecasting procdure, statewide and regional fore­ casting models will be compared with each other in terms of the structures and accuracy (study obj ectives 3 and 4). Finally, the generalizability of the statewide forecasting model to the regional level will be tested (study objective 5). This will be carried out by substituting the trend, seasonal, or both components of the statewide model for the corresponding com­ ponent (s) of a regional model. Four alternatives will be evalu­ ated for each sub-region of Michigan. CHAPTER IV TREND AND SEASONAL PATTERNS This chapter reports the trend and seasonal patterns of tourism-related employment in Michigan and its individual sub-regions (the first objective) and spatial differences in the trend and seasonal patterns between study regions (the second objective). Trend patterns will be presented first, followed by seasonal patterns. TRENDS Trends were identified from the changes in annual averages of tourism-related employment from 1974 to 1984. The annual averages are computed by averaging the 12 monthly tourism-related employment figures for the year. Changes over time are represented as an index starting from the base year (1974) to 1984. Statewide trend patterns will be presented first, followed by regional trends. Statewide Trends Michigan's tourism-related employment increased by about 57,700 jobs (30 percent) from 1974 to 1984, while Michigan's overall employment increased by 279,000 jobs (7.8 percent) over the same period. Tourism-related employment had a larger growth 75 76 rate than Michigan's total employment (30% vs 7.8%), resulting in an increase in tourism's share of the state's jobs from 5.4 percent in 1974 to 6.3 percent in 1984. Statewide tourism-related employment was not immediately affected by the energy crisis in 1979 but did decline with the 1980-82 economic recession in the state. Tourism-related employment in Michigan grew by 25 percent over the 1974-1979 period, dropped 7 percent between 1980 and 1982, and subsequently recovered at a 4 percent annual growth rate in 1983 and 84 (Table 3). Cumulative percentage changes in tourism-related employment in Michigan for the period 1974-1984 using 1974 as the base year are plotted in Figure 4. Examination of the distribution of jobs within the nine individual tourism-related sectors and changes in the distribution of jobs in these 9 sectors may be helpful to understand the changing patterns of tourism activity in Michigan over time. From 1974 to 1984 the distribution of jobs in the 9 tourism- related sectors has remained relatively stable (Table 4). Eating and drinking places and hotel/motel sectors provide the majority of tourism-related jobs in Michigan (about 85% in 1984) and account for more than 99% of the growth since 1974. Eating and drinking places led the other sectors in creating new jobs (51,000) since 1974, followed by hotel/motel sector with 13,000 new jobs. The gas service sector lost about 3,500 jobs between 1974 and 1984. The hotel and motel sector experienced the highest growth (64%), followed by amusements (41%). Jobs in the recreational vehicle and utility sector declined by 33% from 1974 to 1984. 77 Table 3: Trends in Tourism-Related Employment 1974-84. Year 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 a"I b. c. Average3 Annual Jobs 203,942 209,223 222,856 238,893 252,154 254,384 249,097 244,349 241,583 249,421 258,596 Ratio tob 1974 values in Michigan, Cumulative Percentage0 Changes from 1974 1.00 0 1.03 1.09 1.17 1.24 1.25 3 9 17 24 25 1.22 1.20 22 20 1.18 18 1.22 22 1.27 27 Average of 12 monthly figures for each year. Column 2 divided by 203,942. Column 3 minus 1.00. 78 60 50 40 30 20 10 0 -10 -20 YEAR Figure 4: Cumulative Percentage Changes in Tourism-Belated Employment in Michigan between 1974 (Base Year) and Subsequent Years. 79 Table 4: Distribution of Jobs in Tourism-Related sectors in Michigan, 1974-84. 1974 Jobs 1974 PCT 1984 Jobs 1984 PCT Jobs 84-74 Eat and Drink Hotel/Motel Gas Service Amusements Boat Dealers Camps & Trailer Water Transp. RV & Utility Room & Board 144,764 19,607 24,745 9,903 1,584 1,041 1,140 811 71.1 9.6 73.3 51,308 12,614 -3,498 4,073 -287 0 0.0 196,702 32,221 21,247 13,976 1,279 1,063 1,062 543 114 Total 203,595 100.0 267,595 Sector 12.2 4.9 0.8 0.5 0.6 0.4 12.0 7.9 5.2 0.5 0.4 0.4 % Chang* 35 64 -14 41 -18 22 2 0.2 0.0 -78 -268 114 -7 -33 100.0 64,000 31 0 80 Regional Trends Statewide patterns are dominated by southern Michigan which accounted for 89% of tourism-related employment in 1984 and contributed approximately 86 percent of the statewide increase over the 1974-1984 period (Table 5). With the exception of the eastern upper peninsula, northern regions of the state had a 27 percent or more increase in tourism-related jobs over the 11 year period. The eastern upper peninsula has experienced a loss in tourism-related employment since 1978. The Mackinac Straits region experienced the highest growth (54%), followed by the western upper peninsula (46%) and out-state urban counties (44%). In 1984 most northern regions experienced slight increases in tourism-related employment while Wayne and Oakland counties had a 6% increase (Table 6 ). Individual study regions of southern Michigan have trend patterns similar to those of the state of Michigan (Figure 5), but vary in the magnitude of these changes. In northern Michi­ gan, trends in individual regions vary significantly (Figure 6 ). This may be due to regions responding differently to economic conditions in the state and/or experiencing different patterns of growth and development. Northeast Michigan is the only region which showed a rela­ tively flat pattern with no decline and little growth during the 1974-1984 period. Tourism-related employment in northwest Michi­ gan and out-state urban counties grew by 6% during the 1979-80 gasoline shortage. Tourism-related employment in Wayne and Oak­ land counties, and southern rural counties remained stable from 81 Tabla 5: Trends of Tourism-Related Eaployment in Michloan 1974-84, by raglen. Year Michigan Eaatam Western Mayna 8 Out-Stata South Northwest Morthaaat Mackinac Uppar Uppar Oakland Urban Rural Michigan Michigan Straits Peninsula Peninsula Annual Averages 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 203,942* 209,223 222,856 238,893 252,154 254,384 249,097 244,349 241,583 249,421 258,596 79,633 80,156 84,644 91,697 96,048 95.984 96,579 94,467 91,648 93,475 98,263 49,073 51,053 56,845 62,646 64,776 67,736 68,019 65,865 66,238 68,182 70,586 41,053 43,709 46,636 50,447 52,128 51,936 52,093 51,529 50,624 50,777 52,525 8,071 8,619 9,272 9,868 10,513 10,983 10,485 10,424 10,423 10,760 10,774 4,604 4,843 5,340 5,587 5,771 5,732 5,758 5,824 5,845 5,865 5,869 1,136 1,272 1,330 1,530 1,675 1,584 1,526 1,640 1,717 1,771 1,754 2,523 2,572 2,787 2,817 2,442 2,054 1,947 1,944 1,974 2,171 2,101 4,187 4,689 4,828 5,082 6,052 5,871 5,686 5,981 5,924 5,938 6,118 * Numbers are not equal to the sus of the annual averages of 8 sub-regions because tourismrelated jobs in the non-profit organizations era not included in tourism-related employment in the individual counties. Table 6: Changes in Tourism-Related Employment, 1974-84, by Region. Year Michigan Eastern Western Upper Wayne 8 Out-State South Northwest Northeast Mackinac Upper Oakland Urban Rural Michigan Michigan Straits Peninsula Peninsula f* . cunuifli l l v v 1974* 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 0 3 9 17 24 25 22 20 18 22 27 0 1 6 15 21 21 21 19 15 17 23 * 1974 is the base year. 0 4 16 28 32 38 39 34 35 39 44 0 6 14 23 27 27 27 26 23 24 28 r « i i B i H i y v ------ 0 7 15 22 30 36 30 29 29 33 33 0 5 16 21 25 25 25 26 27 27 27 0 12 17 35 47 39 34 44 51 56 54 0 2 10 12 -3 -19 •23 -23 •22 •14 •17 0 12 15 21 45 40 36 43 41 42 46 I 45 40 30 PERCENTAGE CHA NG E 35 75 77 79 83 YEAR a STATE + WO O URBAN A SOUTH Figure 5 c Qmulative Percentage Changes in Tourism-Related Bnployment in the State of Michigan and Southern Regions Between 1974 (Base Year) and Subsequent Years. 60 PERCENTAGE CHANGE 50 30 20 -1 0 •20 T 75 □ 77 NW Figure 6: + NE 79 O YEAR STRAIT 81 A EUP 83 X WUP Qmulative Percentage Changes in Tourism-Related Employment in Northern Regions of Michigan Between 1974 (Base Year) and Subsequent Years. 84 1978 to 1980 and then declined from 1981 to 1983. By contrast, the Mackinac Straits region and the eastern upper peninsula experienced significant declines in tourism-related employment in 1979, followed by increases from 1981 to 1983. Northern Michigan experienced larger relative changes in tourism-related employment than the state as a whole over the study period. The turning points of the trend series were reached two years earlier in the north than in southern counties, presumably due as much to gasoline price increases as to the recession. It should be noted that significant variations exist between counties within a region (Table 7). For example, although all counties in the eastern upper peninsula experienced a loss in tourism-related employment, these losses ranged from 3% for Chip­ pewa to 46% for Alger. The increase in tourism-related jobs in the Wayne and Oakland region was almost all in Oakland county. SEASONALITY Seasonal patterns of tourism-related employment were identi­ fied and quantified from the seasonality series of 12 monthly seasonal factors. In the preliminary analysis, seasonal factors for each year from 1974 to 1983 were calculated and examined for each study region. It was; found that, during this time period, regional seasonal patterns have remained relatively stable. The individual monthly seasonal factors across the years deviate from their respective means by nor more than plus or minus 2.3% for Wayne and Oakland counties and 10% for the Mackinac Strait region 85 Table 7: Changes in Tourism-Related Employment (n Michigan, 1974-84, by Region end by County. Region County Uayne & Oakland Wayne Oeklend Other Urban Counties South Rural Jobs* 84-74 X** Change 1 69 Genesee Ingham Kalamazoo Kent Macomb Washtenaw 2,957 1,597 2,527 5,985 5,165 3,283 38 21 51 52 44 58 Allegan Barry Bay 8errien Branch Calhoun Cass Clinton Eaton Gratiot Hillsdale Huron Ionia Isabella Jackson Lapeer Lenawee Livingston Mecosta Midland Monroe Montcalm Muskegon Newaygo Ottawa Saginaw Sanilac Shiawassee St. Clair St. Joseph Tuscola Van Buren 255 -52 254 •159 -34 86 •191 156 165 •35 -96 112 -43 349 1,083 491 208 1,254 249 603 213 126 888 227 1,966 1,514 -37 1,514 62 533 1 182 26 -14 10 •4 •6 2 -32 49 24 -5 -18 19 •6 29 38 83 16 127 55 55 12 18 31 68 73 31 County Jobs 84-74 Northwest Antrim 50 Mtchigen Benzie 166 Charlevoix 91 G. Traverse 1,373 Kalaska 6 -8 Lake Leelanau 86 Manistee 111 -29 Mason 18 Missaukee Osceloa 15 20 Emmet Wexford 160 X Change 9 72 11 75 3 -7 25 29 -4 16 8 2 25 Northeast Alcona Michigan Alpena Arenac Clare Crawford Gladwin Iosco Montmorency Ogemaw Oscoda Otsego' P. Isle Roscoomon 67 -24 156 -35 202 222 79 91 99 -12 73 36 323 53 •3 39 •6 60 131 17 65 44 •10 11 26 63 Mackinac Straits 323 294 64 46 •140 •17 -153 •54 CO in 429 18,200 Region •46 -3 •15 -26 -18 •5 196 325 171 24 19 1,058 194 -52 -4 47 75 31 9 38 62 49 -25 Cheboygan Mackinac Eastern Alger Upper Chippewa Peninsula Delta Luce Schoolcraft -10 31 3 55 0 25 * Actual change in tourism-related jobs ** Percentage change in tourism-related jobs. Western Baraga Upper Dickison Peninsula Gogebic Houghton Iron Kewenaw Marquette Menominee Ontonagon 86 (Table 8 ). Thus, the nine-year averages of monthly seasonal fac­ tors were computed to estimate seasonal factors for each study region and these are asuumed constant over time. Statewide Seasonality At the statewide level of aggregation, Michigan's tourismrelated employment has relatively small seasonal fluctuations. Seasonal factors range from a low of .93 in February to a high of 1.06 in June and August (Table 9). Based on an annual average of 258,596 statewide tourism-related jobs in 1984, this translates into a difference of 33,600 jobs between February and June (or August). To identify which sectors contribute most to the seasonal fluctuations of tourism-related employment in Michigan, seasonal patterns of the 9 tourism-related sectors were examined using 1984 figures (Table 10). Annual seasonal fluctuations in tour­ ism-related employment is measured by the difference between March employment and August employment. It was found that eating and drinking places contributed most to the annual seasonal fluc­ tuations (20,591 jobs or 56.7%), followed by the amusements sec­ tor (8,200 jobs or 22.6%). Camps and trailer parks sector led the other sectors in the magnitude of seasonal fluctuations of jobs within individual sectors. August employment in this sector (2,246 jobs) was almost 3 times higher than March employment (599 jobs). 87 Table 8: Stability of monthly seasonal factors by region, Month Eastern Uestern Out-state Uayne & Urban Upper Southern Northwest Northeast Mackinac Upper Peninsula Peninsula Michigan Oakland Counties Rural Michigan Michigan Straits cienr January February March April May June July August September October November December 1974.07-1983.06. 0.009 0.009 0.008 0.012 0.007 0.008 0.008 0.009 0.008 0.004 0.009 0.006 0.011 0.009 0.003 0.023 0.014 0.013 0.008 0.011 0.009 0.007 0.013 0.010 0.011 0.013 0.012 0.016 0.017 0.017 0.016 0.016 0.017 0.011 0.011 0.009 0.012 0.016 0.014 0.009 0.011 0.013 0.012 0.013 0.012 0.012 0.012 0.015 ot vi 0.019 0.020 0.032 0.028 0.021 0.015 0.017 0.016 0.018 0.012 0.020 0.024 a. Coefficient of variation » (standard deviation/mean). ,l . . I in ation 8 0.019 0.021 0.023 0.021 0.019 0.024 0.017 0.010 0.015 0.015 0.014 0.018 0.059 0.039 0.063 0.100 0.045 0.057 0.033 0.043 0.042 0.070 0.032 0.058 0.055 0.059 0.065 0.043 0.056 0.064 0.025 0.021 0.027 0.052 0.067 0.070 0.030 0.031 0.026 0.015 0.023 0.011 0.025 0.029 0.020 0.021 0.026 0.028 88 Table 9: Seasonal Factors for Tourism-Related Employment in Michigan, 1974-84. Seasonal factors Month 0.93 0.93 0.95 0.98 1.03 1.06 1.05 1.06 1.04 January February March April May June July August September October November December 1.01 0.98 0.98 1984 Tourism Related Employment 258,596 Table 10: Seasonal F&ttems of Tourism-Related Deployment in Michigan fay Sector,1984. Sector March % August (Aug-Mar)/lfer % Change % Aug-*fer % 74.0 9.3 7.9 6.7 20,591 3,153 1,311 56.7 8.7 3.6 14 8,200 22.6 0.6 518 682 1,647 123 92 1.4 1.9 4.5 0.3 0.3 80 49 275 25 118 36,317 100.0 15 Eat & Drink Hbtel/tfctel Gas Service Amusements Boat Dealers Water Tcansp. cams & Trailer RV & utility Room & Board 184,932 76.7 204,983 22,465 9.3 25,618 20,516 8.5 21,827 10,250 4.3 18,450 1,049 0.4 1,567 1,364 682 0.3 599 0.2 2,246 495 0.2 618 78 170 .0 Total 240,526 100.0 276,843 100.0 0.5 0.8 0.2 0.1 11 6 100 89 Regional Seasonality Tourism-related employment in each of the eight regions peaks in either August or September. It reaches its trough in January for eight southern metropolitan counties; in April for northwest Michigan; and in February for the other four regions (Table 11). Patterns of seasonal fluctuations of tourism-related employment in individual regions are plotted in Figure r}. In northwest Michigan and the western upper peninsula , a slight increase in tourism-related employment from December to February is observed, presumably due to winter tourist activities in these areas. Patterns of seasonality in tourism-related employment are much more evident for northern regions than for southern regions. For example, in northwest Michigan tourism-related employment fluctuates from a high of 27% above the annual average (seasonal factor=1.27) in August to an April low of 16% below the annual average (seasonal factor of 1-.16=.84). The Mackinac Straits region shows the.largest seasonal fluctuations in tourism-related employment. In these counties, August employment (seasonal fac- tor=1.92) is almost four times higher than February employment (seasonal factor=0.38). Southern regions, on the other hand, experience relatively low seasonal fluctuations in tourismrelated employment. This is likely due to the more diversified economy, fewer seasonal residents, and larger local population bases to sustain employment in tourism-related sectors throughout the year. 90 Table IT: Seasonal Factors for Tourism-Related Employment in Michigan, 1974-84, by Region. Month January February March April May June July August September October November December Michigan Other Wayne & South Oakland Urban 0.93 0.93 0.95 0.98 1.03 1.06 1.05 1.06 1.04 1.01 0.98 0.98 0.94 0.95 0.96 1.00 1.03 1.04 1.03 1.03 1.02 1.00 0.99 1.00 1984 Tourism Related 258,946 Employment 98,263 Eastern Western South Northwest Northeast Mackinac Upper Upper Rural Michigan Michigan Straits Peninsula Peninsula 0.91 0.90 0.93 0.98 1.04 1.07 1.08 1.09 1.06 1.01 0.97 0.97 0.91 0.89 0.87 0.84 0.97 1.11 1.24 1.27 1.12 1.00 0.88 0.90 0.87 0.85 0.86 0.89 1.01 1.11 1.19 1.20 1.12 1.02 0.97 0.92 0.39 0.38 0.39 0.51 1.13 1.57 1.91 1.92 1.58 1.13 0.62 0.49 0.65 0.65 0.66 0.75 1.07 1.29 1.47 1.50 1.38 1.07 0.82 0.74 0.97 0.96 0.95 0.93 0.98 1.03 1.07 1.09 1.04 0.99 0.97 1.01 66,960 51,119 9,090 5,869 1,754 2,878 6,118 0.96 0.95 0.97 1.00 1.04 1.04 1.01 1.01 1.02 1.01 0.99 0.99 WAYNE AND OAKLAND COUNTIES 2.00— l.Wl.K- 1 .93 . 1 Ou- OUT-STATE URBAN 00UNTIES 1.70. l.fiCL 1.30. l.K. 1.10. 1. 101.00. 0.90- O.K. 0.70. 0.(0. 0.50. O.K. 0.10. 0 . 20- s*?c. 0.20. Oct. be. Mar. Apr. f t? Am July bpc. Oct. lb#. KU» NORTHWEST MICHIGAN SOUTH RURAL COUNTIES 1.20> l.CG ‘ 0.90 J 0.70 0.400.30 0.20 . Apr. May Am KKU Figure 7: July Aug. fcpc. Oct. lbv. be. K20U Seasonal Patterns of Tourism-Related Employment in Michigan, 1974-84, by Region. THE MACKINAC STRAITS REGION NORTHEAST MICHIGAN 2 .0 0 I.W- THE EASTERN UPPER PENINSULA THE WESTERN UPPER PENINSULA 2« l.M 1•) i23 160 150 l.W l.» 1.20 110 1.00 0 .9 0 -a o— 0 .6 0 0 .7 0 0 .6 0 O.W 0 .4 0 0.» 0.20 J«n. F«b. **\ Htr. iot. »iy J w KUU Figure 7: (Continued). sXily Ji*. ~ S*e. Cb. ifiT 93 SUMMARY Trend and seasonal patterns of tourism-related employment exhibit significant differences across the state of Michigan. Southern regions generally follow statewide trend patterns and had a tourism-related job increase of 23 percent or more between 1974 and 1984. Significant variations in the trend patterns between northern regions are observed. The region with the high­ est growth in tourism-related employment is the Mackinac Straits region (54%), followed by the western upper peninsula (46%) and out-state urban counties (45%). Seasonal fluctuations in tour­ ism-related employment is lowest in southern rural counties, which are less populated and have relatively low levels of tour­ ism activity. The highest seasonal variation in tourism-related jobs is in Mackinac Straits region which is highly dependent upon tourism and related industries. Thus, regions differ in trend, seasonal, or both patterns of tourism activity, which makes these regions distinguishable from each other. To further test spatial variations in the trend and seasonal patterns of tourism-related employment in the state of Michigan, statewide and regional fore­ casting models will be developed and reported in the next chap­ ter. CHAPTER V FORECASTING MODELS This chapter presents forecasting models for the state of Michigan as a whole and its 8 sub-regions (study objectives 3 and 4). The generalizability of the statewide model to sub-regions of the state (objective 5) is also tested. The forecasting mod­ els predict monthly levels of tourism-related employment. To develop a forecasting model each regional series was decomposed into trend and seasonal components using a multiplicative decom­ position. A transfer function procedure was applied to the trend component to estimate trend models. The seasonal model was gen­ erated from the results of harmonic analysis on the seasonal com­ ponent. Trend and seasonal models were then recombined into a monthly forecasting model for each study region. The ability of the statewide model to forecast at the regional level was inves­ tigated and will be reported in the last section of the chapter. The statewide model will be presented and discussed first, fol­ lowed by regional forecasting models. STATEWIDE FORECASTING MODEL Three trend models were estimated: (1) a time-seriesARMA(p,q) transfer function model, (2) a structural-ARMA(p,q) transfer function model and (3) a structural-time-ARMA(p,q) transfer function model. was evaluated. Forecast accuracy of each trend model The seasonality component was estimated using 94 95 harmonic analysis. Statewide Trend Model-S The three transfer function models differ in the composition of explanatory variables of the regression component. The resid­ uals of their respective regresssions are modeled and forecasted by the same Box-Jenkins techniques. The pure time series model is estimated as a third degree polynomial with time, time-squared, and time-cubed as the independent variables. Explanatory vari­ ables in the structural model are population size, unemployment rate, retail price of gasoline, and personal income. All seven variables are tested in the combined structural-time-series model. Estimation procedures for a transfer function model are demonstrated first for the structural model. First, a structural regression equation is estimated using ordinary least squares applied to the trend component of the monthly statewide tourismrelated employment series (July 1974 to June 1983). The esti­ mated models will be presented together with standard errors in parentheses, adjusted R-square, Durbin-Watson statistic (DW), and mean absolute percentage error (MAPE) of the equation. 96 Structural Models The structural ordinary least squares (OLS) regression model is: A Tt = -802,206 + 10,395 * PSTATEt - 157 * MAUNEMt - 86 * MAGASt + (-73,451) (792) (213) (11) 1,295 * MAINCfc (51) (5.1) R2 = 0.964 MAPE= 1.05 DW =* .008 where PSTATE-t is the population size of the state of Michigan in month t. Other variables in the equation are defined on pp. 60. The unemployment coefficient is not significant at the 5% level, but its negative sign is as hypothesized. All other coefficients are significant at the 5% level and the signs of coefficients are as hypothesized. trend series. The model explains 96% of the variation in the The mean absolute percentage error (MAPE) is 1.05. The Durbin-Watson statistic is close to zero, indicating the presence of serious positive serial correlation. forecast accuracy of the regression model. This may impare The transfer function model corrects for this problem. A transfer function model is estimated by first computing the unconditional residuals (E-j.) and then applying Box-Jenkins techniques to this series. The total and partial autocorrelation functions of E^ with lags of 1 to 24 were examined to determine the orders (p and q) of the autoregressive and moving-average processes (Figure 8 ). The autocorrelation decays geometrically from its starting value. This indicates the moving-average part of the error model should be of order 1. The partial 97 Figure 8: Total and Partial Autocorrelation Functions for the Structural Regression Model for Tourism-Related Employment in Michigan, 1974.07-1984.06. Autocorrelations ' i —r »w _« .« «_ | -11Vtw—%i< v-A«f t_A Aif-« t. | 1 1Vw»AV»r * K I '******* i i ;****** i t***** [ 1t-i-i. aiXXTf | 11 » _ » | • .i-t1 i*** [ '*** . • 1 ,** i i* 1 i i 1 . i * i **: i ■kick I 1 ***! ' **'| . * **'1 1 **, . 1 •iek i ! ** i i **l I **< 1 **• 1 1 1 Partial Autocorrelations LAG i SWnWnWWWr* ******| 1 1 1 1 1 t 1 1 1 1 Jn L.| 1 ie[ 1 ; ! . i i i , t ■ i i i ' i ' > ] i. i 2. 1 3. 1 1 1 4. | 5. 1 6. 1 7. 1 8. 1 9. 1 10. 1 ' 11. | 12. i 13. 14. • 15. 16. • 17. 1 18. 1 * 19. i i i 20. i 21. i 22. i 23. i 24. i AC 0.945 0.826 0.688 0.562 0.467 0.402 0.355 0.318 0.277 0.215 0.114 -0.011 -0.134 -0.227 -0.271 -0.268 -0.237 -0.199 -0.171 -0.165 -0.179 -0.204 -0.229 -0.235 PAC i 1 ! , , , i i i 1 1 i « ' ' 1 ' 0.945 -0.622 0.033 0.040 0.046 0.035 0.020 0.008 -0.007 -0.032 -0.059 -0.052 -0.033 -0.013 0.004 0.011 0.007 0.001 -0.009 -0.017 -0.019 -0.014 -0.007 0.005 98 auto-correlation function has significant spikes at lags 1 and 2 , indicating a second-order autoregressive interpretation of the E-j. series. This suggests two possible error models: ARMA(2,1) and ARMA(1,1). These two transfer function models were estimated using the Time Series Processor Statistical Package (Hall and Hall, 1980). Estimation procedures are described on pp. 64-65. The estimated structural-ARMA(2,1) transfer function model is: A Tt = 248,878 - 74 * PSTATEt - 6.3 * MAUNEMt + 1.3 * MAGASt (64,390) (697) (10.6) (2.2) 8 * MAINCt + 1.92 * Et_! - 0.92 * Et_2 + Vt + (39) ( 0.0035) (0.0024) 0.68 * Vt_! (0.1023) (5.2) R 2 = 0.999 MAPE= 0.04 DW = 1.99. where E^-i and E^-2 are the unconditional errors in period t-1 and t- 2 , and V-j. and V^-i are the forecast errors of the E-t series in period t and t-1, respectively. Although this model explains 99.9% of the variations in the statewide trend series with a mean absolute percentage error of 0.04 and no serial correlation (Durbin-Watson statistic is close to two), none of the structural variables are significant at the 5% level. Except for the unem­ ployment variable, the signs of coefficients are opposite to those hypothesized. This model is therefore rejected. 99 The estimated structural-ARMA(l,1) transfer funtion model is: A Tt - -419,072 + 6,949 * PSTATEt - 57 * MAUNEMt - 1.7 * MAGASt + (65,210) (702) (39) (4.3) 300 * MAINCt + 0.97 * Et_!+ Vt + 0.88 * Vt_]_ (54) (0.0088) (0.1039) R 2 = 0.999 MAPE* 0.11 (5.3) DW * 1.97. This model incorporates all four independent variables and a first-order autoregressive and first-order moving-average error model, i.e. ARMA(1,1). The model explains 99.9% of the variance of the statewide trend series with a MAPE of 0.11 and no serial correlation (DW = 1.97). hypothesized. The signs of the coefficients are as Statewide population size and personal income lev­ els have positive coefficients, while statewide unemployment and U. S. retail price of gasoline have negative coefficients. All variables, except unemployment and gasoline price index, are sig­ nificant at the 5% level. All variables are retained in the trend model for forecasting and comparison purposes. Pure time series models The same procedure was repeated for the pure time series models. Although the signs of the coefficients of the estimated time-series-ARMA(1,1) model are as hypothesized, the model has a strong negative serial correlation (DW* 3.8). is not presented here. Thus, this model On the other hand, the estimated time- series-ARMA (2,1) transfer function model is: 100 Tt - 109,516 + 5,207 * t - 66 *t2 + 0.27 * t 3 + 1.77 * Et_x (8,617) (423) (6 ) (0.03) (0.0032) 0.78 * E+-.-2 + Vt + 0.63 * Vt_i. (0.0023) (0.1017) R 2 = 0.999 MAPE= 0.04 (5.4) DW = 1.99. This model includes three powers of time and a second-order autoregressive and first-order moving-average error model, i.e. ARMA(2,1). It explains almost 100% of the variation in the statewide trend series with a MAPE of 0.04. tion problem is observed. No serial correla­ All coefficients are significant at the 5% level. Structural-time-series models In the structural-time-series-ARMA(2,1) model the coeffi­ cients of t, t2 and t 3 are significant at 5% level while the coefficients of structural variables are insignificant. This model is no better than the pure time-series-ARMA(2,1) model and the structural-ARMA(l,1) model in terms of measures for goodnessof-fit to the existing data and simplicity of the model and thus not reported here. In the statewide trend models the group of four structural variables and the group of three time factors can roughly substitute for each other when these two sets of vari­ ables are entered together. 101 Forecast Accuracy of Statewide Trend Models The structural-ARMA(l,1) and the time-series-ARMA(2,1) transfer function models were then compared in terms of forecast accuracy. These two models were used to produce one to twelve months ahead forecasts from July 1983 to June 1984. For a detailed description of forecasting procedure see pp. 67. These forecasts are also compared with those of the structural regres­ sion model and the time-series regression model (Table 12). The mean absolute error of the time-series regression model is about 35 times larger than that of the time-series transfer function model (Table 13). There is not much improvement in forecast accuracy of the structural transfer function model over the structural regression model. Among the four models tested, the time-series-ARMA(2,1) transfer function model performs best in terms of all four mea­ sures of forecast error, followed by the structural-ARMA(l,1) transfer function model. These two models will be used as alter­ native trend models to build the Michigan. forecasting models for 102 Table 12: Month Forecasts of the Statewide Trend Models for TourismRelated Employment in Michigan, 1983.07-1984.06. Time-Series Time-Series Structural Structural Regression Transfer Regression Transfer 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 240,079 240,953 241,821 242,695 243,583 244,494 245,434 246,411 247,431 248,499 249,620 250,798 230,319 229,624 228,911 228,179 227,430 226,662 225,876 225,072 224,250 223,411 222,554 221,679 239,927 240,271 240,661 241,017 241,522 241,922 242,293 242,532 242,739 242,957 243,129 243,380 241,784 242,963 244,426 245,791 247,594 249,083 250,882 251,979 252,743 253,163 253,867 255,259 Actual Trend Values 240,042 240,874 241,734 242,758 243,912 245,079 246,100 246,938 247,647 248,089 248,344 248,635 Table 13: Forecast Accuracy of the Statewide Trend Models for Tourism-Related Employment in Michigan, 1983.07-1984.06. Model MAE RMSE MAPE (%) StatisticUII Time-SeriesARMA(2,1) 537 802 0.22 0.0033 Time-Series Regression 18,849 19,673 7.67 0.0803 StructuralARMA(1,1) 3,151 3,645 1.28 0.0371 Structural Regression 4,115 4,356 1.67 0.0178 103 Statewide Model for Sasonalitv The seasonal model was estimated from the statewide season­ ality series using harmonic analysis. Statistics for all 6 har­ monics for the state of Michigan are presented in Table 14. The first harmonic explains more than 95 percent of the variation in the seasonality series. Tourism-related employment in Michigan shows an obvious annual cycle. The seasonal fluctuations in tourism-related employment across the year are relatively low, varying by plus or minus 6.5 percent of the annual average or about 16,000 jobs. The annual cycle peaks August 1, and reaches its low point on February 1. The first (annual), second (semiannual) and 4th (quaterly) harmonics are included in the seasonal model, capturing more than 98 percent of the variation in the seasonality series. This model generates the seasonal component of tourism-related employ­ ment in Michigan and is: /\ st = 1 + 0.06 * SIN(30*t+255) + 0.01 * SIN(60*t+169) + (5.5) 0.01 * SIN(120*t+238) The model predicts the 12 monthly seasonal factors within plus or minus 2 percent (Table 15). All four measures of fore­ cast error are small, e.g. MAPE= 0.95. It should be noted that MAE and RMSE are small because these 12 seasonal factors vary around 1.00. When comparing forecast accuracy of the trend and seasonal models, MAPE and Statistic-UII should be used since they are relative measures. 104 Table 14: Harmonic Measures of Tourism-Related Employment in Michigan, 1974-84. Harmonic i Amplitude Ai 0 1 2 3 4 5 6 1.00 .06 .01 .01 .01 .00 .00 Phase Angle i(degrees) 255 169 139 238 237 270 Relative Amplitude 6.5 1.0 .6 .8 .3 .2 Variance Explained 95.2 2.3 .8 1.3 .3 .1 Peak Timing Aug. 1 Table 15: Forecasts of the Statewide Seasonal Model for TourismRelated Employment in Michigan, 1983.07-1984.06. Month 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 Actual Seasonal Seasonal Forecast Seasonal Error 1.06 1.07 1.06 1.01 0.98 0.98 0.94 0.93 0.94 0.98 1.03 1.07 1.05 1.05 1.04 1.01 0.99 0.98 0.94 0.93 0.96 0.97 1.02 1.06 0.01a 0.02 0.02 0.00 -0.01 0.00 0.00 -0.00 -0.02 0.01 0.01 0.01 Absolute Percentage Error (%) 0.81b 1.86 1.72 0.48 1.19 0.42 0.48 0.45 1.80 0.60 0.71 0.87 a. Column 2 minus column 3. b. jcol 4/col 2 |*100. Measures for goodness-of-fit and forecast accuracy of the model: Eta-square = 0.97 Mean absolute error (MAE) = 0.0096 Root mean square error (RMSE) = 0.0111 Mean absolute percentage error = 0.95 Statistic-UII = 0.0110 105 Statewide Forecasting Models Since a multiplicative decomposition was adopted, the trend and seasonal models are multiplied together to generate alterna­ tive forecasting models for Michigan's tourism-related employ­ ment. The structural forecasting model is: A A A Yt = Tt * St = [ -419,072 + 6,949 * PSTATEt - 57 * MAUNEMt - 1.7 * MAGASt + 300 * MAINCt + 0.97 * Et_! + Vt + 0.88 * V-t-].] * [ 1 + 0.06 * SIN(30*t+255) + 0.01 * SIN(60*t+169) + 0.01 * SIN(120*t+238) ] (5.6) This model explains 99.9% of the variations in tourism-related employment in Michigan for the period from August 1974 to June 1983 (Figure 9). The time-series forecasting model is: A A A Yt = Tt * St = [109,516 + 5,207 * t - 66 * t2 + 0.27 * t3 + 1.77 * Et_x - 0.78 * Et_2 + Vt + 0.63 * Vt_!] * [1 + 0.06 * SIN(30*t+255) + 0.01 * SIN(60*t+169) + 0.01 * SIN(120*t+238)] (5.7) This model explains 98.4% of the variations in tourism-related employment in Michigan for the same period. The plot of the forecasts of this model is similar to Figure 9. MICHIGAN 270 TO U R ISM —RELATED EM PLOYMENT (T h o u sa n d s ) 260 250 240 230 220 - 210 - 200 - 190 160 5 ACTUAL Figure 9: 1984.07 1980.07 1974.08 MONTH + PREDICTED Actual and Predicted Tourism-Belated Employment in Michigan, 1984.08-1984.06. 107 Forecast Accuracy of the Statewide Models One to twelve-months ahead forecasts of the structural forecasting model for statewide tourism-related employment for the months from July 1983 to June 1984 were produced (Table 16). The projections follow the actual statewide patterns quite closely, but consistently underpredict. The errors are not more than 3.0% for any month and their average (MAPE) is 1.54. The model forecasts well up to 12 months ahead, but there are signs that the model may be deteriorating over time. The majority of forecast errors can be attributed to the errors in predicting the trend component. The mean absolute percentage error is 1.28 for the trend model, 0-.95 for the seasonal model, and 1.54 for the combined forecasting model. To improve the forecast accuracy adaptive forecasting was employed. Adaptive forecasting is defined as updating the model by reinserting the most recent available data for the dependent and independent variables into the model to forecast for the fol­ lowing period(s) (Pindyck and Rubinfeld, 1981). This technique is expected to provide better forecast(s) because of the more upto-date information used in the adaptive process. Using adaptive forecasting the MAPE of the forecasts is reduced from 1.54 to 0.93 for the same forecasting period. The percentage error of the 12-months ahead forecast for June 1984 is three time higher than that of the one-month ahead forecast. 108 Thble 16: Qotpariscn of 1-12 Maiths Ahead and 1-ncnth Ahead Forecasts of the Structural Forecasting Model for Tturiairftelated Employment in Michigan, 1S83.07-1984.06. 1-12 Martha Ahead* Mfcnth Actual Statewide values Forecast Error - 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 254,697 257,183 255,127 245,421 239,431 239,094 229,854 229,244 232,410 243,917 256,340 265,429 2,443 4,885 4,406 3,165 -79 3,012 3,196 2,684 121 7,257 7,685 7,274 Absolute Percentage Error (%) 0.99 1.90 1.73 1.29 0.03 1.26 1.39 1.17 0.05 2.98 3.00 2.74 1-mcnth Ahead** Absolute Forecast Percentage Error (%) Error 2,443 4,659 3,437 1,990 -2,260 578 -226 -915 -4,452 2,492 2,359 1,883 0.96 1.81 1.35 0.81 0.94 0.24 0.10 0.40 1.92 1.02 0.92 0.71 * Fbur measures far 1-12 months ahead forecast accuracy are: Mean absolute error (MAE) - 3,850 Root mean square error (REE) = 4,568 Mean absolute percentage error (MAEE) = 1.54 Statistic-UU = 0.0186 ** Four measures for 1-mcnth ahead forecast accuracy are: Mean absolute error (MAE) =* 2,308 Foot mean square error (FMSE) => 2,660 Mean absolute percentage error (MAEE) = 0.93 Stafcistic-OH = 0.0108 109 One to twelve months ahead forecasts of the time-series forecasting model for the same time period were also produced using the same forecasting procedure (Table 17). The percentage errors are not more than 1.88 for any months and their average (MAPE) is 0.77. This is lower than that for the structural forecasting model (1.54). The majority of forecast errors can be attributed to the errors in predicting the seasonal component. The MAPE is 0.22 for the trend model, 0.95 for the seasonal model and 0.77 for the forecasting model. Adaptive forecasting did not improve the forecast accu­ racy of this model. The MAPE for the 1-12 months ahead forecasts (0.77) is smaller than that for the 1-month ahead forecasts (0.89) . 110 Tkble 17: Ctrpariscn of 1-12 tenths Ahead and 1-mcnth Ahead Forecasts of the Time-Series Forecasting Model far Ttaurisii-Helated Enplcyment in Michigan, 1983.07-1S84.06. 1-12 tenths Ahead* tenth Actual Statewide Values 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 254,697 257,183 255,127 245,421 239,431 239,094 229,854 229,244 232,410 243,917 256,340 265,429 Forecast Error 2,283 4,170 3,199 1,480 -2,121 504 259 -939 -4,367 1,860 1,048 -594 Absolute Beroenbage Error (%) 0.90 1.62 1.25 0.60 -0.89 0.21 0.11 -0.41 -1.88 0.76 0.41 -0.22 1-mcnth Ahead** Absolute Forecast Feraentage Error (%) Error 2,283 4,261 3,307 1,561 -2,391 53 477 -1,599 -4,671 1,949 2,165 1,575 * Four measures far 1-12 mcnth ahead forecast accuracy are: Mean absolute error (MAE) = 1,902 Rxit mean square error (IWSB) =* 2,322 Mean absolute percentage error (MAHE) = 0.77 Statistic-UII - 0.0094 ** Four measures far 1-mcnth ahead forecast accuracy are: Mean absolute error (MAE) = 2,188 Hoot mean scpare error (IMSE) =■ 2,548 Mean absolute percentage error (M&EE) = 0.89 statistic-uii = 0.0104 0.90 1.66 1.30 0.64 1.00 0.02 0.21 0.68 2.01 0.80 0.84 0.59 Ill REGIONAL FORECASTING MODELS Given the evidence in Chapter 4 that trend and seasonal patterns of tourism-related employment do vary spatially in Michigan, we proceed to further examine these variations through formal modeling at the regional level. The same forecasting procedure carried out at the state level is now repeated for each region. Regional trend models will be presented first, followed by regional seasonal models and finally regional forecasting mod­ els. Regional Trend Models Three alternative trend models were estimated for each study region from the regional moving average series . The specifica­ tions of these three alternative trend models are presented on pp. 63. Both time-series and structural regression models were estimated for each study region, followed by transfer functions. Structural Transfer Function Models All four structural variables are included in each of the structural regression models (Table 18). Their relationships to the dependent variable are as hypothesized. Some of these four independent variables are not included in the regional structural transfer function models. Each of these independent variables excluded from the models has a relationship to the dependent variable opposite to that hypothesized or is 112 Table 18: Structural Trend Models for Tourism-Related Employment in Michigan, 1974.07-1983.06, by Region. Region Model Type Ra Constant POP MAUNEN MAGAS MAINC AR(1)d AR(2)e MA(1)f •802205.65 10394.51 -157.12 -85.57 1295.27 (73451.27) (792.71) (212.93) (10.62) (51.11) R2 D-W 0.964 0.08 0.999 1.97 0.930 0.09 Michigan T-Fb R Wayne & Oakland T-F R Out-State Urban Counties T-F -419072.28 6949.34 (65120.54)c (702.22) -249718.37 (36476.94) -57.56 (38.77) -1.73 300.41 (4.31) (54.32) 95327.75 (3187.65) -1.31 (6.82) -1.20 0.81 (1.43) (25.27) -301945.27 15149.34 (23413.24) (1059.36) 46.39 -21.90 207.22 (60.21) (3.01) (22.59) -115501.31 7587.43 (31084.56) (1365.56) 46.35 (1.79) -72764.80 (13195.52) 4412.14 -185.73 -19.62 120.17 (527.81) (51.27) (2.97) (18.61) T-F 20062.57 (12861.69) 1122.55 (488.98) -34.40 (10.85) -1.70 9.05 (1.25) (15.86) -17719.62 11624.93 (1782.24) (892.17) •67.54 (6.48) -7.94 (0.49) -1.61 (5.69) -9.33 (2.62) -1.67 (0.30) 6.58 (3.76) R Northwest Michigant T-F R Northeast Michigan T-F 9446.01 (378.73) -11740.23 8701.62 (563.16) (317.10) 3120.05 (1379.26) 0.84 (0.10) 8365.67 •217.67 -22.97 858.79 (953.96) (111.65) (5.19) (55.72) R South Rural 0.97 (0.009) 1372.28 (655.68) -20.20 (2.17) 1.88 (0.01) 0.58 (0.10) 0.999 0.975 0.98 (0.01) 0.97 (0.01) 0.97 (0.01) 0.66 -14.27 (0.17) (1.85) •1.80 (1.71) -0.88 (0.01) 0.97 (0.01) 0.93 (0.10) 1.99 0.08 0.999 1.99 0.914 0.09 0.999 1.99 0.963 0.33 0.999 1.96 0.986 0.25 0.96 0.999 (0.10) 2.04 0.98 (0.10) 0.99 (0.10) 113 Table 18: (Continued) Region Model Type R Mackinac Strai ts T-F R Eastern Upper Peninsula T-F R Western Upper Peninsula T-F constant POP -2444.09 9839.78 (707.51) (2419.43) MAUNEM 5.58 (2.04) 1305.42 (116.13) -5209.56 3795.94 (1991.62) (922.05) 43.68 (6.69) -8560.32 3698.86 (5870.02) (1014.84) 4817.55 -1979.43 (1474.27) (1308.20) 6149.20 (1007.91) -565.68 (979.34) a. Regression model. b. Transfer Function model. c. Standard error in the parentheses. MAGAS MAINC •1.62 (0.14) 16.49 (0.97) -1.13 (0.12) 4.54 (1.28) -4.23 (0.39) 6.89 (2.13) AR(1)d AR(2)e 0.96 (0.02) 0.99 (0.10) 1.00 (0.00) -0.79 (0.22) •33.62 (8.33) -3.01 (0.57) 46.75 (3.64) -0.21 (0.93) -0.20 (0.19) 3.13 (3.37) 1.83 (0.01) MA(1)f 0.95 (0.10) -0.84 (0.01) 0.82 (0.11) d. AR(1): First-order autoregression, Ef1* e. AR(2): Second-order autoregression, Et.2f. MAO): First-order moving-average, Et.^. R2 D-W 0.942 0.16 0.997 1.94 0.814 0.15 0.997 1.90 0.911 0.09 0.999 1.96 114 statistically insignificant at 5 % level. Each of the structural transfer function models explains at least 99.7 percent of vari­ ance in the dependent variable. In most structural models, population size of the study region and statewide personal income levels have positive coeffi­ cients and statewide unemployment and U. S. retail price of gaso­ line have negative coefficients-. Statewide personal income and population size of the study region have a negative coefficient in the model for northwest Michigan and the western upper penin­ sula. All components of the error model part of each structural equation are statistically significant at 5% level. Wayne and Oakland counties and the western upper peninsula are the only two regions having a second-order autoregressive term. Each of the other structural models has an ARMA(1,1) transfer function struc­ ture . Time-Series Transfer Function models Three time variables (time, time-squared, and time-cubed) are included in both a regression model and a transfer function model for each study region (Table 19). tive coefficients. Both t and t3 have posi­ The quadratic time variable is negatively related to the dependent variable. Most of the coefficients of these regional trend models are significant at the 5% level. Each model has an ARMA(2,1) transfer function structure. The inclusion of the error part of the transfer function model to correct for the autocorrelation problem did not change the respective signs of coefficients of individual time variables, 115 Table 19: Time-Series Trend Models for Tourism-Related Employment In Michigan, 1974.07-1983.06, by Region. Region Model Type Ra Constant t 173981.93 (1840.58) 1809.39 (126.85) t2 -12.72 (2.42) t* AR(1)d AR(2)e MA(1)f 0.010 (0.013) R2 o-u 0.960 0.03 0.999 1.99 0.931 0.03 0.999 1.99 0.956 0.03 0.999 1.99 0.953 0.04 0.999 1.99 0.898 0.02 0.999 1.96 0.984 0.09 0.999 2.00 Michigan T-Fb R Wayne & Oakland 109515.66 5207.49 (8617.51)c (422.73) •66.14 (6.26) 0.268 (0.029) 73347.28 517.26 (854.13)c (58.87) -1.76 (1.12) -0.013 (0.006) -24.29 (3.33) 0.095 (0.015) T-F 46372.13 (4457.21) R Out-State Urban Counties T-F 41407.43 (59.52) 563.43 (11.85) -3.78 (-4.18) 0.004 (0.005) 32754.59 (1944.29) 1067.88 (9.76) -12.41 (1.78) 0.048 (0.009) R 37651.25 (346.18) 393.57 (23.86) -3.51 (0.45) 0.007 (0.002) T-F 28866.96 (1546.34) 892.46 (79.84) -11.79 (1.22) 0.049 (0.006) R 6134.40 (124.40) 97.18 (8.57) -0.96 (0.16) 0.003 (0.001) T-F 2245.08 (966.00) 290.00 (44.84) •3.84 (0.64) 0.016 (0.003) R 4116.66 (26.72) 58.86 (1.84) -0.70 (0.03) 0.003 (0.0001) T-F 3899.28 (60.16) 69.93 (3.46) •0.86 (0.06) 0.003 (0.0003) South Rural Northwest Michigan Northeast Michigan 1943.79 (221.87) 1.77 (0.003) 1.74 (0.005) 1.81 (0.009) 1.72 (0.007) 1.72 (0.007) 1.38 (0.04) -0.78 (0.002) -0.76 (0.004) -0.83 (0.006) -0.75 (0.005) -0.74 (0.005) -0.48 (0.02) 0.63 (0.10) 0.53 (0.10) 0.99 (0.10) 0.86 (0.10) 0.60 (0.10) 0.94 (0.11) 116 Table 19: (Continued). Region Model Type Constant t t2 t3 R 937.26 (33.87) 23.61 (10.11) -0.30 (0.04) 0.001 (0.0002) T-F 859.78 (166.59) 27.51 (8.70) •0.35 (0.13) 0.002 (0.0006) R Eastern Upper Peninsula T-F 2456.34 (53.45) 60.29 (3.68) -1.34 (0.07) 0.007 (0.0004) 2309.18 (192.38) 66.64 (10.88) -1.41 (0.18) 0.008 (0.0009) 3839.79 (102.35) 46.15 (7.05) -0.23 (0.13) -0.0002 (0.0007) 3488.76 (267.74) 71.07 (16.48) •0.70 (0.28) 0.0023 (0.0015) Mackinac Straits R Western Upper Peninsula T-F a. Regression model. b. Transfer function model. c. Standard error in parentheses. t is equal to time (month). AR(1)d 1.61 (0.08) 1.72 (0.04) 1.85 (0.05) AR(2)e •0.65 (0.05) -0.76 (0.03) •0.88 (0.03) MA(1)f 0.83 (0.13) 0.77 (0.11) 0.73 (0.11) d. AR(1): First-order autoregression, Et>1. e. AR(2): Second-order autoregression, E^.g. f. MA(1): First-order moving-average, Vt>1. R2 D-W 0.882 0.04 0.999 1.99 0.879 0.04 0.999 2.00 0.900 0.03 0.999 1.99 117 but did increase the percentage of variance explained up to 99.9% in each of regional trend series. Structura1-Time-Series Transfer function models The models with both structural and time variables are no better than the time-series and the structural transfer function models in terms of measure for goodness-of-fit to the existing data and the simplicity of the model , and are therefore not pre­ sented here. As was the case with the statewide model, using both the group of three time factors and the group of four inde­ pendent variables as the explanatory variables of the transfer function models did not enhance the power of the model to fit the existing data. The group of three time variables and the group of structural variables can roughly substitute for each other. Forecast Accuracy of Regional Trend Models For each sub-region, the forecast performance of the time series and the structural trend models for the months from July 1983 to June 1984 were evaluated. Differences between sub- regions were also investigated. The time-series models have smaller mean absolute errors for southern Michigan while the structural models have smaller errors for northern Michigan (Table 20). Northern regions have larger percentage changes in tourism-related employment over time than southern regions. These changes are better predicted by changes in national, statewide and regional economic phenomena which are 118 Table 20: Forecast Accuracy of Transfer Function Models for Tourism-Related Employment Trends, 1983.07-:1984.06, by Region. Region Model Type MAE RMSE MAPE Statistic­ al (%) Michigan Time-Series Structural 537 3,151 802 3,645 0.22 1.28 0.0033 0.0149 Wayne & Oakland Time-Series Structural 891 1,216 1,333 1,594 0.92 1.25 0.0118 0.0166 Out-State Urban Counties Time-Series Structural 395 703 462 818 0.59 1.05 0.0070 0.0124 South Rural Time-Series Structural 204 995 244 1,071 0.40 1.96 0.0048 0.0212 Northwest Michigan Time-Series Structural 270 75 372 95 2.94 0.82 0.0403 0.0103 Northeast Michigan Time-Series Structural 69 30 97 36 1.17 0.50 0.0164 0.0061 Mackinac Straits Time-Series Structural 58 20 77 25 3.29 1.11 0.0433 0.0145 Eastern Upper Peninsula Time-Series Structural 331 51 405 67 11.33 1.95 0.1381 0.0228 Western Upper Time-Series Structural 4 22 5 30 0.06 0.42 0.0008 0.0050 119 captured by structural variables. The western upper peninsula has the smallest MAPE of the time-series trend models, and the eastern upper peninsula has the largest MAPE. The MAPE of the individual structural models range from 0.42 for the western upper peninsula to 1.96 for southern rural counties. One to twelve months ahead structural (trend) forecasts are presented in Appendix B. Regional Models for Seasonality The seasonality series of tourism-related employment in each study region of Michigan was modeled using harmonic analysis. The first two harmonics explain at least 94% of variance in the seasonality series for each study region. The statistics for the first two harmonics are presented in Table 21. All 6 harmonics for individual regions are reported in Appendix B. Each of study regions shows a strong summer season peaking in August. For each region the first harmonic explains 70 percent or more of the seasonal variation of tourism-related employment. The Mackinac Straits region leads the other regions in the relative magnitude of the seasonal fluctuations (RA1=90.9 and RA2=17.9), followed by the eastern upper peninsula. In contrast, the lowest seasonal fluctuations in tourism-related employment is found in out-state urban counties (RA1=3.3 and RA2=1.8). The first and second harmonics are included in every sea­ sonal model. Together, they can explain 94 percent or more of the variance in the seasonality series. Some of the other four harmonics appear(s) in the respective seasonal models for the 120 T&hle 21: Selected Pfenncnic Measures of Ttaurlan-Jtelated Expicynent in Michigan, 1974-1S84, by Regim. First Harmcnic Seocnd harncnic Percent Study Regim Relative Percent Beak Amplitude Variance Timing (RA1) (VAR1) Variance Relative Percent Beak Explained by Anplitnde Variance Timing 1st & 2nd (RA2) (VAR2) (VAR1+VAR2) Michigan 6.5 95.2 Aug. 1 1.0 2.3 Jbne 5/ Dac. 5 97.5 Wsyne & Oakland 4.1 82.6 Aug. 5 1.5 11.5 May 22/ Nov 22 94.1 Out-State Urban counties 3.3 71.8 Aug. 9 1.8 21.1 92.9 South Rural 9.1 96.8 Aug. 1 1.0 1.3 JUne 5/ Dec. 5 98.1 Northwest Michigan 17.9 79.7 Aug. 6 8.9 19.5 JUly 30/ Jan. 30 99.2 Northeast Michigan 17.0 95.6 Aug. 8 3.4 3.9 JUly 19/ Jan. 19 99.4 Mackinac Straits 80.9 94.8 Aug. 2 . 17.9 4.6 J\jly 28/ Jan. 28 99.4 Eastern Upper Peninsula 43.8 96.2 Aug. 5 8.2 3.4 JUly 28/ Jan. 28 99.6 Western Upper Peninsula 5.9 71.5 Aug. 19 3.3 22.5 JUly 25/ Jan. 25 94.0 M a x 17/ Nov 17 state of Michigan, eight urban counties, and the western upper peninsula. This shows that in these individual regions tourism- related employment has not only a strong annual pattern but also other patterns of variation over a one year period (Table 22) . Forecast Accuracy of Regional. Seasonal Models Monthly seasonal factors for each region do not fluctuate much over time, and can be accurately predicted by the seasonal models (Table 23). Errors in predicting seasonal factors for southern Michigan and the western upper peninsula are smaller than those for the northern regions. Although the seasonal model for the Straits region can explain 98% of the variation in seasonal factors for the forecast period, it has the largest error among regions, followed by northwest Michigan. This is because a high value of goodness-of-fit mea­ sure (eta2) can be obtained if a series has large fluctuations across the year(s) and its predictions can closely follow the actual patterns of the series. Also, during the period of July 1983 to June 1984, the observed seasonal factors are lower in summer but higher in winter than predicted from the previous 9 years. Table 22: Seasonal Models for Tourism-Related Employment in Michigan. 1974-1984, by Region. Region Michigan Uayne & Oakland Out-State Urban Counties Constant 1.00 1.00 1.00 South Rural 1.00 Northwest Michigan 1.00 1st harmonic 2nd harmonic 3rd harmonic 4th harmonic 5th harmonic Variance Explained (X) 0.06*SIN(30*t+255) 0.01*SIN(60*t+169) (2.3X) (95.2X) 0.01*SlN(120*t+238) (1.3X) 98.8 0.04*SlN(30*t+259) 0.02*SIN(60*t+198) (11.5X) (82.6X) 0.01*SIN(120*t+230) (3.7X) 97.8 0.03*SIN(30*t+264) 0.02*SIN(60*t+207) 0.01*SIN(90*t+100) 0.01*SlN(120*t+238) (21.IX) (3.4X) (71.9X) (3.1X) 99.5 0.09*SIN(30*t+256) 0.01*SlN(60*t+170) (96.8X) (1.3X). 98.1 0.18*SIN(30*t+250) 0.09*SIN(60*t+62) (79.7X) (19.5X) 99.2 Note: Numbers in the parentheses are the percentage of variance in the seasonality series explained by the Nth harmonic of the model. Table 22: (Continued). Northeast Michigan 1.00 Mackinac Straits 1.00 Eastern Upper Peninsula 1.00 Uestern Upper Peninsula 1.00 0.17*SIN(30*t+247) (95.6X) 0.03*SIN(60*t+81) (3.9X) 99.5 0.81*SIN(30*t+253) 0.18*SIN(60*t+68) (94.8X) (4.6X) 99.4 0.44*SIN(30*t+251) 0.08*SIN(60*t+67) (96.2X) C3.4X) 99.6 0.06*SlN(30*t+236) 0.03*SlN(60*t+72) (71.5X) (22.5X) 0.01*SIN(120*t+234) (2.4X) 0.01*SIN(150*t+206) (2.OX) 98.4 Note: Numbers in the parentheses are the percentage of variance in the seasonality series explained by the Nth harninie of the model. 124 Table 23: Forecast Accuracy of Seasonal Models for TourismRelated Employment in Michigan, 1983.07-1984.06, by Region. EtaSquare MAE RMSE MAPE (%) Statistic­ al Michigan 0.0096 0.0111 0.95 0.0110 0.95 Wayne & Oakland 0.0100 0.0122 0.99 0.0121 0.91 Out-State Urban Counties 0.0099 0.0113 0.97 0.0112 0.90 South rural 0.0099 0.0123 0.97 0.0122 0.96 Northwest Michigan 0.0268 0.0332 2.50 0.0323 0.96 Northeast Michigan 0.0293 0.0327 2.93 0.0323 0.87 Mackinac Straits 0.0620 0.0683 7.76 0.0598 0.98 Eastern Upper Peninsula 0.0252 0.0321 2.55 0.0303 0.99 Western upper Peninsula 0.0098 0.0127 0.98 0.0126 0.95 Region 125 Forecasting Models for Each Region The regional trend and seasonal models are multiplied together to generate the forecasting models for each study region. Each region has two forecasting models. One is the structural forecasting model and the other is the time series forecasting model. The structural and time series forecast­ ing models are reported in Table 24 and Table 25, respectively. Forecast Accuracy of Regional Forecasting Models All regional structural forecasting models perform quite well in fitting the observed regional tourism-related employment (Figure Bl.l to Figure B1.8) and in forecasting monthly employ­ ment for the period July 1983-June 1984 (Table B2.1 to Table B2.8). Regional trend and seasonal patterns captured by the regional structural models are assumed to remain stable over the forecasting period. Among all structural models the model for the western upper peninsula has the smallest mean absolute per­ centage error (1.02) and Statistic-UII (0.0129). The model for Mackinac Straits region has the largest values of these two fore­ cast error measures, MAPE (7.75%) and Statistic-UII (0.0637). Generally, models for northern regions have larger relative forecast errors than those for southern regions (Table 26). The majority of the forecast errors can be attributed to the errors in predicting the seasonal components for northern regions, and to the errors in predicting the trend components for southern regions (see Table 20 and Table 23). For example, for Table 24: Structural Forecasting Models for Tourism-Related Employment in Michigan, by Region. Region Michigan Forecasting model f -419072.28 ♦ 6949.34 * PSTATg - 57.56 * MAUNENg - 1.73 * MAGASt + 300.41 * MAINCg ♦ 0.97 * Eg., ♦ Vg + 0.84 Vg., ] [ 1.00 ♦ 0.06 * SIN(30*t+255) + 0.01 * SIN(60*t+169) ♦ 0.01 * SIN(120*t+238)l t 95327.75 - 1.31 * MAUNENg - 1.20 * MAGAS{ ♦ 0.81 * MAINCg + 1.88 * Eg., - 0.88 * Eg_2 + Vg ♦ 0.58 * Vg., 1 * Wayne t Oakland Out-State Urban Counties ( 1.00 * 0.04 * SIN(30*t+259) + 0.02 * SIN(60*t+198) + 0.01 * SlH(120*t+230)1 t -115501.31 + 7587.43 * PURBg + 46.35 * MAINCg + 0.98 * Eg., + Vg ♦ 0.93* Vg., 1 * [ 1.00 + 0.03 * SlN(30*t+264) + 0.02 * SIN(60*t+207) ♦ 0.01 * SIN(90*t+10Q) + 0.01 *SIN(120*t+238) ] [ 20062.57 + 1122.55 * PSOUg - 34.40 * NAUNEMg - 1.70 * MAGASg + 9.05 * MAINCg + 0.97 * Eg., ♦ V, + 0.98 * Vg., ] * South Rural t 1.00 ♦ 0.09 * SIN(30*t+256) + 0.01 * SIN(60*t+170) ] [ 9446.01 - 9.33 * HAUNEHg - 1.67 * HAGASg + 6.58 * MAINCg + 0.97 * Eg., + Vg + 0.99 * V {., ] * Northuest Michigan t 1.00 + 0.18 * SIM(30*t+250) + 0.09 * SIN(60*t+62) ] Table 24: (Continued). t 3120.05 + 1372.28 * PNEt - 1.80 * MAINCt ♦ 0.97 * Et., + Vt + 0.96 * \lt.y 1 * Northeast Michigan I 1.00 ♦ 0.17 * SIN(30*t+247) + 0.03 * SIN(60*t+81) J [ 1305.42 - 1.13 * HAGASt + 4.54 * MAlNCt + 0.96 * Et_t ♦ Vt + 0.99 * Vt_,, ] * Mackinac Straits [ 1.00 + 0.81 * SIN(30*t+253) + 0.18 * SIN(60*t+68) ] Eastern Upper Peninsula [ -8560.32 + 3698.86 * PEUPt - 0.79 * HAGASt + 1.00 * Et_., + Vt + 0.95 Vt_,, 1 * Western Upper Peninsula [ 6149.20 - 565.68 * PUUPt - 0.21 * MAUNEHt - 0.20 * HAGASt + 3.13 * MAlNCt ♦ 1.83 * Et.., - 0.84 * Et_2 + Vt + I 1.00 + 0.44 * SIN(30*t+251) + 0.08 * SlN(60*t+67) ] Note: PSTAT: Population size of the state of Michigan. PURB: Population size of other urban counties. PEUP: Population size of eastern upper peninsula. PSOU: Population size of south rural counties. PNE: Population size of northeast Michigan. PUUP: Population size of western upper peninsula. 127 0.82 * Vt.t] * t 1.00 + 0.06 * SIN(30*t+236) + 0.03 * SIN(60*t*72) + 0.01 * SIN(120*t+234) + 0.01 * SlN(150*t+206) ] Table 25: Time Series Forecasting Models for Tourism-Related Employment in Michigan, by Region. Region Michigan Forecasting model t 109515.66 + 5207,49 * t - 66.14 * t2 + 0.268 * t3 ♦ 1.77 * Et., - 0.78 * Et_2 + Vt ♦ 0.63 * Vt.t ] * E 1.00 + 0.06 * S£N(30*t+255) ♦ 0.01 * SlN(60*t*169) + 0.01 * SIN(120*t+23B>] £ 46372.13 + 1943.79 * t - 24.29 * t2 ♦ 0.095 * t3 ♦ 1.74 * Et., - 0.76 * Et_2 ♦ Vt + 0.53 * Vt_, 1 * Wayne & Oakland Out-State Urban Counties £ 1.00 + 0.04 * SlN(30*t+259) + 0.02 * SIN(60«t+198> + 0.01 * SIN(120*t+230>] [ 32754.59 ♦ 1067.88 * t - 12.41 * t2 ♦ 0.048 * t3 ♦ 1.81 * Et.., - 0.83 * Et.2 + Vt ♦ 0.99 * Vt_, ] * f 1.00 ♦ 0.03 * SIN(30*t+264) ♦ 0.02 * SIN(60*t+207> ♦ 0.01 * SIN(90*t+100) ♦ 0.01 * SIN(120*t+238) ] £ 28866.96 ♦ 892.46 * t - 11.79 * t2 + 0.049 * t3 + 1.72 * EM South Rural - 0.75 * Et_2 + V( + 0.86 * Vt., ] * E 1.00 ♦ 0.09 * SIN(30*t+256) + 0.01 * SlN(60*t+170) ] [ 2245.08 ♦ 290.00 * t - 3.84 * t2 ♦ 0.016 * t3 ♦ 1.72 * Et.., - 0.74 * E{_2 + Vt + 0.60 * Northwest Michigan [ 1.00 ♦ 0.18 * SlN(30*t+250) ♦ 0.09 * SINC60*t+62) J 1 * Table 25: (Continued). ( 3899.28 ♦ 69.93 * t - 0.86 * t2 + 0.003 * t3 ♦ 1.38 * Ej., - 0.48 * Et_2 + Vt + 0.94 * Vt_., ] * Northeast Michigan [ 1.00 * 0.17 * SIN(30*t+247) + 0.03 * SIN(60*t+81) ) I 859.78 + 27.51 * t - 0.35 * MAINCt + 0.002 * t3 + 1.61 * Et_, - 0.65 * Et.2 + Vt + 0.83 * Vt., ] * Mackinac Straits Eastern Upper Peninsula Western Upper Peninsula [ 1.00 + 0.81 • SIN(30*t+253> + 0.18 • SlN(60*t+68) ] t 2309.1B + 66.64 • t - 1.41 * t2 + 0.008 * t3 + 1.72 * Et., - 0.76 * Et_2 + Vt + 0.77 Vt_, ] • [ 1.00 + 0.44 * SlN(30*t+251> + 0.08 * SIN(60*t+67) ] [ 3488.76 + 71.07 * t - 0.70 * t2 + 0.0023 * t3 ♦ 1.85 * Et.t - 0.88 * Et_2 + Vt + 0.73 * Vt_, ] * [ 1.00 + 0.06 • SlN(30*t+236) ♦ 0.03 * SlN(60*t+72> ♦ 0.01 * SlN(120*t+234) + 0.01 * SIN(150*t+206) ] Note: PSTAT: Population size of the state of Michigan. PURB: Population size of other urban counties. PEUP: Population size of eastern upper peninsula. PS0U: Population size of south rural counties. PNE: Population size of northeast Michigan. PUUP: Population size of uestern upper peninsula. 130 Table 26 : Forecast accuracy of time series and structural forecasting models for tourism-related employment, 1983.07-1984.06, by region. Region Model Type MAE RMSE MAPE (%) StatisticUII Michigan Time Series Structural 1,902 3,850 2,322 4,568 0.77 1.54 0.0094 0.0186 Wayne & Oakland Time Series Structural 1,224 1,456 1,502 1,893 1.25 1.48 0.0157 0.0197 Out-state Urban Counties Time Series Structural 562 861 711 976 0.84 1.29 0.0107 0.0147 South Rural Time Series Structural 607 1,246 724 1,345 1.19 2.48 0.0142 0.0264 Northwest Michigan Time Series Structural 410 245 448 309 4.42 2.48 0.0474 0.0327 Northeast Michigan Time Series Structural 132 191 157 207 2.20 3.24 0.0264 0.0346 Mackinac Straits Time Series Structural 126 114 155 129 8.02 7.75 0.0763 0.0637 Eastern Upper Peninsula Time Series Structural 300 88 403 107 11.59 3.25 0.1301 0.0346 Western Upper Peninsula Time Series Structural 51 62 71 79 0.84 1.02 0.0117 0.0129 131 the Mackinac Straits region, mean absolute percentage error (MAPE) for the seasonal model is 7.76% and for the trend model is 1.11%. The forecasting model has a MAPE of 7.75%. For southern rural counties, MAPE for the trend model is 1.96, for the seasonal model is 0.97, and for the 2.48. forecasting model is Thus, forecast errors for southern rural counties are attributable more to errors in predicting the trend component than the seasonal component. Forecast accuracy of each regional time series model is also evaluated. All time series models can predict within plus or minus 2.20%, except the models for three tourism-dependent regions, i.e. northwest Michigan, the Straits region and the eastern upper peninsula. For southern rural counties, the Macki­ nac Straits region and the western upper peninsula, the majority of forecast errors can be attributed to the errors in predicting the seasonal components. Errors in predicting the trend compo­ nents dominate the other five sub-regions. Structural models perform better for Northwest Michigan, the Mackinac Straits region and the eastern upper peninsula. Time series models per­ form better for the other five sub-regions. Since forecasting models tend to deteriorate over time, it is recommended that the forecasting performance of these models be monitored over time. If obvious changes in trend, seasonal or both patterns are observed, the forecasting models should be updated or re-estimated. 132 GENERALIZABILITY OF THE STATEWIDE MODELS TO THE REGIONAL LEVEL To complete the fifth and last study objective the ability of the statewide model to forecast at the regional level was tested. The test was performed by substituting the trend, seasonal, or both components of the statewide structural model for the corresponding component(s) of a regional structural model. The resulting models were then used to forecast monthly tourism-related employment in each region for the months from July 1983 to June 1984. The forecast accuracy of four models for each sub-region was evaluated. Each model predicts the values of a tourism-related employment index in the region, which can then be translated into the actual number of tourism-related jobs. The procedures are demonstrated first for northwest Michigan (NW). Statewide trned forecasts and regional observations and trend forecasts were normalized using July 1974 as the base month (Table 27-28). Four index-series models ( Equations 3.23 to 3.26 on pp. 73) were generated and evaluated. The composition of these four models are: (1) NW's trend and NW's seasonal compo­ nents, (2) NW's trend and statewide seasonal components, (3) statewide trend and NW's seasonal components, (4) statewide trend and statewide seasonal components. Forecast accuracy of these models were compared. model with both regional components performs best. The first Its monthly absolute error is not more than 5% (Table 29) and it has the smallest mean absolute error for the index, 0.021. This is translated into 186 jobs, i.e. 0.021 * 8,862, (Table 30). 133 Table 27: Normalization of Statewide Trend Model Component. Month 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 Trend Forecasts Forecasted Trend Index 239,927 240,271 240,661 241,017 241,522 241,922 242,293 242,532 242,739 242,957 243,129 243,380 1.243a 1.245 1.247 1.249 1.251 1.253 1.255 1.257 1.258 1.257 1.260 1.261 a. Column 2 divided by 192,998 (July 1974). Table 28: Nannalizaticn of Northwest Michigan Model Coipcnsnts. Month 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 Forecasted Trend Trend Forecasts Index 9,166 9,169 9,182 9,189 9,221 9,236 9,258 9,277 9,287 9,293 9,294 9,316 1.034a 1.035 1.036 1.037 1.039 1.042 1.045 1.047 1.048 1.049 1.049 1.051 Actual Trend Vdves Actual Trend Ihdex Actual Seascnal Factor 9,148 9,182 9,221 9,263 9,295 9,323 9,321 9,276 9,232 9,183 9,136 9,106 1.30(P 1.305 1.310 1.316 1.321 1.325 1.324 1.318 1.312 1.305 1.298 1.294 1.240 1.270 1.116 0.996 0.884 0.898 0.907 0.889 0.874 0.844 0.996 1.110 a. C d u m 2 divided by 7,038 (JUly 1974). b. O d u m 4 divided by 7,038 (JUly 1974). c. O d u m 5 multiplied by O d u m 6. Actual Regional Index 1.612° 1.657 1.462 1.311 1.167 1.190 1.201 1.172 1.146 1.101 1.293 1.436 134 Table 29: Forecasts of Tourism-Related Employment Index in Northwest Michigan, 1983.07-1984.06, by Model. Models Month Actual Regional Index 1 RT*RS 2 RT*SS 3 ST*RS 4 ST*SS 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 1.612 1.657 1.462 1.311 1.167 1.190 1.201 1.172 1.146 1.101 1.293 1.436 1.627 1.634 1.481 1.282 1.167 1.164 1.197 1.185 1.133 1.134 1.261 1.481 1.369 1.368 1.359 1.312 1.298 1.281 1.230 1.231 1.263 1.286 1.351 1.404 1.553 1.561 1.415 1.226 1.116 1.112 1.142 1.130 1.080 1.080 1.203 1.411 1.307 1.307 1.299 1.255 1.241 1.223 1.174 1.174 1.204 1.224 1.289 1.338 Note: RT: Regional trend component; RS: Regional seasonal component. ST: Statewide trend component; SS: Statewide seasonal component. Table 30: Forecast Accuracy of Four Models for Northwest Michigan, 1983.07-1984.06. Models Forecast 1 RT*RS 2 RT*SS 3 ST*RS 4 ST*SS Mean Absolute Error Index Number of Jobs 0.021 186a 0.111 987 0.060 0.108 530 955 Note: RT: Regional trend component; RS: Regional seasonal component. ST: Statewide trend component; SS: Statewide seasonal component. a. Row 1 multiplied by 8,862 (July 1974). 135 Substituting statewide seasonality for regional seasonality (model 2) results in larger forecast errors than sustituting statewide trends for regional trends (model 3). Forecast error of the pure statewide model is 4 times higher than the pure regional model. Monthly differences between the four sets of forecast errors are larger in the summer months than in the win­ ter months. This is because monthly differences between regional and statewide seasonal factors are larger in the summer months than in the winter months. Forecasts of the pure regional model and the statewide-trend/regional-seasonality model follow the patterns of tourism-related employment in northwest Michigan dur­ ing the forecasting period of July 1983 and June 1984, while it is not the case for forecasts of the regional-trend/statewideseasonality model and the pure statewide model (Figure 10). This shows that northwest Michigan has similar trends as the state but different seasonality. Statewide seasonality patterns are not suitable to guide tourism planning and development in northwest Michigan. On the other hand, if a larger forecast error is acceptable, the model composed of the statewide trend and the seasonality of a subregion (like northwest Michigan) is recom­ mended. Using this type of models we can save the costs of esti­ mating the regional trend models which require more regional data and complex estimation methods. In contrast, seasonal factors can be estimated with one year of regional data. This procedure was repeated for the other sub-regions of Michigan. Forecast accuracy of four models within the region and across the regions were compared. As expected, the pure regional model performs best in all regions, while the pure statewide N O R T H W E S T MICHIGAN TOURISM-RELATED EMPLOYMENT INDEX 1.4 0.9 0.8 □ ACTUAL Figure 10: + RT*RS O MONTH RT*SS A ST*RS X ST*SS Actual and Predicted Index for Tourism-Related Bnployment in Northwest Michigan. 1983.07-1984.06, by Model. 137 model performs worst. This shows that patterns of statewide tourism-related employment do not generalize well to regional levels. Over the period examined, southern regions have exper­ ienced similar seasonal patterns as the state but large differ­ ences in trends. For each of these regions, larger forecast errors are observed when substituting the statewide trend compo­ nent for the regional trend component (model 3) than when substi­ tuting the statewide seasonal component for the regional seasonal component (model 2). For example, for out-state urban counties, the regional-trend/statewide-seasonality model performs as well as the pure regional model and almost five times better than the model with statewide trend and regional seasonal components. Model 2 has a low value of mean absolute error for the employment index ,0.021, (Table 31) and corresponding actual number of tour­ ism-related jobs, 1,026, (Table 32). Northern regions, especially the Mackinac Straits region, show large differences in both trend and seasonal patterns from the state of Michigan. All of the northern regions' index-series models have larger forecast errors than southern regions' models. The Mackinac Straits region has the largest forecast errors of all four models across all sub-regions of Michigan. Except for the western upper peninsula, substituting statewide seasonal com­ ponent for regional seasonal component (model 2) results in larger forecast errors than substituting statewide trend compo­ nent for regional trend component (model 3) for all northern regions. 138 Table 31: Forecast Accuracy of Four Models of TourismRelated Employment Index, 1983.07-1984.06, by Regions. Models Region 1 RT*RS — ------ 2 RT*SS 3 ST*RS 4 ST*SS Mean Absolute Error, Job Index -- 0.032 0.048 0.057 0.063 Out-State Urban Counties 0.015 0.021 0.130 0.130 South Rural 0.040 0.043 0.055 0.053 Northwest Michigan 0.021 0.111 0.060 0.108 Northeast Michigan 0.012 0.088 0.028 0.089 Mackinac Straits 0.057 0.785 0.314 0.796 Eastern Upper Peninsula 0.022 0.246 0.258 0.322 Western Upper Peninsula 0.008 0.038 0.189 0.189 Wayne & Oakland Note: RT: Regional trend component; RS: Regional seasonal component. ST: Statewide trend component; SS: Statewide seasoanl component. 139 Table 32: Forecast Accuracy of Four Models for Number of Tourism-related Employment Jobs, 1983.07-1984.06, by Regions. Models Region 1 RT*RS — Wayne & Oakland 2 RT*SS 3 ST*RS 4 ST*SS Mean Absolute Error, Number of Jobs — 2,623 3,956 4,707 5,207 738 1,026 6,252 6,251 1,826 1,976 2,521 2,411 Northwest Michigan 186 987 530 955 Northeast Michigan 64 475 153 478 131 1,814 725 1,840 Eastern Upper Peninsula 91 1,038 1,086 1,356 Western Upper Peninsula 36 168 838 838 Out-State Urban Counties South Rural Mackinac Straits Note: RT: Regional trend component; RS: Regional seasonal component. ST: Statewide trend component; SS: Statewide seasoanl component. * The number of tourism-related jobs for the base month (July 1974) is 83,086 for Wayne and Oakland counties, 48,067 for other urban counties, 45,438 for south rural counties, 8,862 for northwest Michigan, 5,392 for northeast Michigan, 2,312 for Mackinac Straits region, 4,210 for the eastern upper peninsula, and 4,440 for the western upper peninsula. 140 SUMMARY Structural and tine series forecasting models for tourismrelated employment were developed for the state of Michigan and 8 sub-regions. Each forecasting model consists of a trend model and a seasonal model which were estimated separately and then multiplied together. Statewide and regional models differ in their structures, accuracy, and sources of forecast errors. All or some of these four structural variables are included in the individual struc­ tural trend models if they have the signs of coefficients as hypothesized. All three time variables are included in each time series model. Error models can capture the part of the trend component not explained by the economic variables or the time variables. The statewide seasonal model includes the first, second and fourth harmonics. The state of Michigan has a strong summer sea­ son and smaller semiannual and quarterly patterns of seasonal fluctuations in tourism-related employment across the year. In each regional seasonal model, the first and second harmonics can explain 94 percent or more variance in the regional seasonality series. Some of the other four harmonics are also included in the individual seasonal models. Differences in the structures between seasonal models show how regions differ from each other in seasonal patterns of tourism-related employment. All Structural forecasting models accurately fit the data and most of them can forecast within 3.25% error in average, except for the Mackinac Straits region (7.75%), for the next 12 141 months from July 1983 to June 1984. deteriorate slowly over time. The transfer function models Forecast accuracy of the trend models is improved using adaptive forecasting. Seasonal models for those tourism-dependent regions did not perform well because these regions experienced large changes in their seasonal factors over the period examined. Generally, seasonal (trend) models contribute most forecast errors of structural forecasting models for northern (southern) regions. All time series models can forecast within 2.20% error, except the models for northwest Michigan, the Mackinac Straits region and the eastern upper peninsula. The structural models perform better than time series models for northwest Michigan, the Mackinac Straits region and the eastern upper peninsula. The time series models perform bet­ ter for the other five sub-regions. Finally, the statewide structural models do not generalize well to regional levels over the period examined. The state of Michigan differs from its individual sub-regions in trend, sea­ sonal, or both patterns of tourism-related employment. This result again suggests that sub-regions are distinguishable from each other in the patterns of tourism activity, and each region needs to have its own forecasting model to guide tourism planning and management in the region. CHAPTER VI CONCLUSIONS This study was designed to (1) identify and quantify temporal and spatial patterns of tourism activity in Michigan, (2) test alternative methods associated with the development of tourism forecasting models, and (3) examine thegeneralizability of statewide models to regional levels. Methods employed to select the decomposition approach, to identify the seasonal patterns of tourism activity and the rela­ tionships underlying fluctuations in the trends of tourism acti­ vity over time, and to select techniques for forecast modelbuilding were introduced and discussed in Chapter III. Statewide and regional trend and seasonal patterns of tourism-related employment were empirically analyzed and reported in Chapter IV. In Chapter V, patterns of tourism-related employment in each study region were specified as a structural forecasting models. Each model and a time series consists of a transfer function model of the trend and a harmonic model of seasonality. The mod­ els were subsequently used as forecasting equations to predict monthly tourism-related employment in each of the study regions. The forecasting performance of each model was evaluated using four measures of forecast accuracy. Finally, the ability of the statewide models to forecast at the regional levels was tested. 142 143 Chapter III, IV, and V report results relevant to these study objectives. All five study objectives were successfully achieved. This concluding chapter summarizes the study, addresses major study limitations, provides recommendations for future research, discusses important findings of the study and suggest possible applications of the study results. SUMMARY OF THE STUDY The first objective of this study was to identify and quan­ tify trend and seasonal patterns of tourism activity in the state of Michigan and its individual sub-regions. Data used here are monthly tourism-related employment series (1974-84) systemati­ cally collected and reported by Michigan Employment Security Com­ mission. The application of multiplicative decomposition to the data series for each study region produced a trend and a season­ ality series. Trend and seasonal patterns were identified from these series for each study region. The second objective of this analysis was to compare trend and seasonal patterns across several regions of the state. Analysis of the behavior of these trend and seasonality series for the state and individual sub-regions revealed that patterns of tourism-related employment in individual study regions differ with each other in trend, seasonality, or both. The third and fourth objectives of this study were to develop, estimate and evaluate alternative models for forecasting tourism activity in the state as a whole and individual 144 sub-regions. A structural forecasting model composed of a struc­ tural transfer function model of the trend and a harmonic model of seasonality were developed for each of the study regions. Similar models using pure time series specifications were also estimated and compared with the statewide models. Both types of trend models performed quite well and had no serial auto­ correlation problem. The harmonic model accurately fit the sea­ sonal patterns over the period examined and predicted the future seasonal patterns within 3% of error, except for the Straits region. Statewide and regional forecasting models differ with each other in structure, accuracy and sources of forecasting errors. For each study region, the trend and seasonal models were combined to produce monthly forecasts for tourism-related employ­ ment in each region. Forecast errors of structural models aver­ age 3.30% or less for the next 12 monthly observations (July 1983-June 1984), except the model for the Straits region. Fore­ cast eroors of time series models average 2.20% or less, except the models for the three most heavily tourism-dependent regions. Forecast errors increase with increases in the lead time for forecast but can be decreased by using adaptive forecsting. Rel­ ative forecast errors of models for northern Michigan are larger than those for southern regions. For the structural forecasting models, most of forecast errors are contributed by forecast errors of seasonal models for northern Michigan, while they are evenly shared by forecast errors of both trend and seasonal mod­ els for southern regions. This is because the percentage changes in seasonal factors for tourism-related employment are larger in 145 northern Michigan than in southern Michigan over the period fore­ casted . The final objective of this study was to test the generalizability of statewide models to the regional level. tural models were used for these tests. The struc­ Four alternative models were obtained by substituting statewide trend, seasonal, or both components for the corresponding component(s) of the individual regional structural models. They were then used to forecast .future levels of tourism-related employment in individual subregions of Michigan. Regional models based on statewide trend, seasonal or both components have larger errors than the estimated regional models. It is concluded that patterns of statewide tourism-related employment do not generalize well to the regional level. To reduce the costs of developing regional models the model consisting of the statewide trend component and the regional seasonal component is recommended for Wayne and Oakland counties and southern rural counties. The regional models should be used for the other six sub-regions of Michigan. STUDY LIMITATIONS AND FUTURE RESEARCH Several limitations to the data utilized, the isolation of tourism component from the data, the stability of the models, and the applications of the study results should be recognized. Research approaches to overcome these limitations are recom­ mended . 146 First, the study period covered a time-span of only eleven years (1974-84). With only eleven years of data the cyclical component of the series could not be identified, since it takes about 5 to 7 years to complete a business cycle. Thus, cyclical and trend components were combined and interpreted as a trend component in the present study. Study of the characteristics and underlying forces of the cycle could not be performed. The ana­ lysis of the business cycle may provide some important informa­ tion, such as the length of a cycle, fluctuation patterns of the cycle, and the relationships of the cycle to other components of the series. The second limitation concerns the definition of tourismrelated jobs. The patterns of tourism-related employment can only be applied to the employment of the private tourism-related sectors, since employees of state and local government and family business related to tourism activity are not included in the employment series. Also, the number of tourism-related jobs do not represent the number of full-time jobs solely created by tourism activity, since a half-time or a full-time job is counted as a job and tourism related employees serve not only tourists but also other travelers, and local residents. The tourism com­ ponent and its importance to overall tourism-related employment in a region was not clearly understood. It is recommended that the tourism component of tourismrelated employment should be clearly isolated and identified. Two approaches to extract the tourism component from tourismrelated employment data series are recommended here. One is to take the lowest level of tourism-related employment over the year 147 as serving local residents. The difference between the lowest level of employment and the levels for other months could be treated as the tourism component of tourism-related employment. The assumption made here is that tourism-related services demanded by local permanent populations only create a certain number of jobs which are stable over the year. this approach was proved by O'Donnell (1970). The usefulness of Another approach proposed by Stynes (1986) involves developing a general model for the employment serving local populations. The tourism component of tourism-related employment can be derived by subtracting the prediction of the model from the reported level of employment. The third limitation concerns the definition of tourism. composition of tourism industry has not been well defined. The The tourism-related employment statistics used in this study do not encompass all of the jobs in tourism-related businesses. Whether the results for the 9 tourism-related series would also hold for other tourism series is an open question. It is recommended that the trend and seasonal patterns of other tourism-related data series (e.g. hotel/motel and eating and drinking places tax revenues, and gasoline sales) should be examined and compared with those of tourism-related employment data series (or hotel/motel employment data series if available). Differences in patterns of these data series can show the ranges of fluctuations in trend and seasonality of tourism activity. If such differences are significant, how to incorporate two or more tourism-related data series into one model in order to produce better tourism forecasts becomes an interesting research topic. 148 Fourth, trend and seasonal patterns of aggregate tourismrelated employment may not be consistent with those of the indi­ vidual sectors. The lack of data disaggregated by region for individual sectors restricts attempts to examine the differences in the behavior between the aggregate employment series and employment series for individual sectors. Information, such as which sector(s) contribute most to the fluctuations in the series and changes in the relative importance of the individual sectors to the overall tourism-related employment levels, may be helpful in interpreting patterns of tourism activity in each region. It is recommended that the trend and seasonal patterns of the individual tourism-related sectors, especially hotel/motel and eating and drinking places should be identified region and compared. such comparisons. for each Valuable information may be obtained from Regional planners can determine which sectors have the potential in creating new jobs in the region based upon their grwoth in employment together with other economic develop­ ment factors. Business managers can better develop marketing and management plans to compete with their competitors in other regions if they know the trend and seasonal patterns of tourismrelated businesses in their own region and the other regions. Information on differences in the trend and seasonal patterns between the individual sectors and their aggregate enable researchers to determine whether the present aggregate trend and seasonal patterns can represent these patterns of tourism acti­ vity in the region, and, if not, which sector(s) can do better. 149 The fifth limitation concerns the predictive ability of the seasonal model. The nine-year monthly averages of seasonal fac­ tors were used to represent the seasonal patterns over time, since generally these seasonal factors have been relatively stable for the past nine years (1974:07-1983:06). During this period some changes in seasonal factors for the individual regions were observed. However, no effort was devoted to under­ standing and capturing these changes. The seasonal model devel­ oped here provided reasonably accurate fit to past data but may not be able to capture changes in the future. Models which permit the twelve seasonal factors to change over time should be developed and compared with the fixed models here. Three approaches for developing seasonal models are recom­ mended. One is to develop a transfer function model using all or some of six harmonics as the independent variables in an additive form. The error model should predict part of seasonal variations not captured by these harmonics. seasonal Box-Jenkins models. Another approach is to develop Finally, a simple moving average or exponential smoothing model for the seasonal factors may suffice. Comparing these alternative models with the seasonal model in this study may enable researchers to better understand the chang­ ing patterns of seasonality. Sixth, regional trend and seasonal patterns are not always consistent with those of each county within the region. County planners and managers should be careful in taking and interpret­ ing these regional patterns as the county's patterns to develop tourism plans for the county. The only regional variable used in the study is population size. Variations in the regional 150 employment series may be better captured using more variables at the regional or local levels. A research effort should be directed toward collecting data for a number of regional or local variables in addition to popu­ lation size. Variables, such as weather index for travel, dis­ tance to and economic environment of the major market areas, and a tourism attraction index for the area are candidates to improve regional models. These variables can be helpful in not only dif­ ferentiating counties to form more homogeneous sub-regions of the state but also in explaining trends and seasonal patterns of tourism activity. They are also useful in examining differences in these two patterns between the individual counties and the region as a whole and between regions. Continuing research is therefore necessary to monitor patterns of tourism activity in the region and its counties. The last study limitation concerns the stability of the forecasting models over time. The economic structure of a tourism destination changes over time, the economic environment of the market areas at the national, state and regional levels change, the taste and preference of visitors changes, and weather conditions affecting the seasonality change. Therefore the trend and seasonal patterns of tourism activity in an area are likely to change over time. There is no assurance that the underlying forces of the trends and seasonality are stable over time. If changes in the trend and seasonal patterns of tourism activity are not acknowledged, forecasts may lead planners and managers to develop inappropriate tourism development and manage­ ment plans and related policy. Therefore, cautious 151 interpretation and application of the findings and the forecast­ ing models of the study to some time beyond a certain time limit (like two years) are highly recommended. The usefulness of the models for forecasting should be carefully monitored over time, and the forecasting models should be updated and re-tested peri­ odically. CONCLUSIONS AND DISCUSSIONS Five major conclusions can be drawn from the study. They are: (1) the relative magnitudes of fluctuations in trends and seasonality of tourism-related employment in Michigan do vary across regions; (2) both the economic variables and the time variables perform well in capturing the fluctuations in the trends of tourism-related employment? (3) the statewide forecast­ ing models do not generalize well to the regional level; (4) decomposing a time series enables us to better identify and model the trend and seasonal patterns of tourism activity; and (5) transfer function techniques, harmonic analysis and the combined forecasting methods should receive more attentions in tourism forecasting. First, spatial and temporal variations in tourism-related employment do exist in Michigan. regions change over time. Patterns in the individual This suggests that tourism activity should be carefully monitored and tracked to better understand past tourism patterns. Understanding historical patterns is essential to projectthem into the future. Different patterns of tourism-related employment between study regions are identified, 152 showing that regions are distinguishable by trend, seasonal, or both patterns. This finding suggests that statewide tourism plans for Michigan must accommodate regional variations. Each study region should have its own plans for tourism development. This study provides forecasting models to forecast tourismrelated employment in individual study regions. Such forecasts can be useful inputs to guide statewide and regional tourism development. Second, time series factors and economic variables can suc­ cessfully track the trends of tourism activity. Generally, eco­ nomic variables perform slightly better than time variables in capturing fluctuations in tourism activity in the most tourismdependent regions. Economic variables may capture the fluctua­ tions of economic phenomena which in turn impact tourism busi­ nesses better than time series variables. It is hypothesized that tourism businesses are quite sensitive to the fluctuations in the economic enviroment of the nation, the state, and the region. If possible, economic variables at the national, state and region levels should be included in the model for tourism activity. Third, predictions from statewide forecasting model should not be generalized to the local level without first checking the consistency of the respective patterns of tourism activity in the state and its sub-regions. If tourism patterns of the state and the sub-region are sigificantly different, using statewide fore­ casts to guide local tourism planning and management may result in inaccurate expectations on the levels of tourism activity, inappropriate and inefficient uses of local tourism resources, 153 and finally instability of local economic situations. This sug­ gests that tourism patterns of an area should be carefully iden­ tified and monitored. Regional planners and managers should understand the patterns of tourism businesses in their own regions and know the differences in the patterns between regions to enable themselves to make better tourism development and man­ agement plans. On the other hand, local variations in the pat­ terns of tourism activity in the region should be considered in developing the regional tourism plans. If local variations are sigificant, it may call for a closer examination of tourism pat­ terns across the region to group those counties with highly homogeneous tourism patterns into a region for further study. The fourth conclusion is decomposing a time series is useful in better identifying and modeling the behavior of the time series. Multiplicative decomposition models outperform additive decomposition models in fitting the data especially for tourismdependent regions. £his provides strong empirical support for their use recommended by BarOn (1972;1975) and Wheelwright and Makridakis (1983;1985). Researchers can clearly identify the trend and seasonal patterns and separate their respective effects on the overall behavior of the series. The underlying forces of the trends and seasonality can be carefully examined in order to better quantify and model these forces to capture the trends and seasonality of the series. From the forecasts of the trend and seasonal models planners and managers can know the future changes in the volume of tourism activity are due to fluctuations in trend, seasonality, or both. For each case, appropriate plans can be developed to deal with the foreseen situations. 154 The last conclusion is the transfer function techniques, harmonic analysis and methods of combining structural and time series models are applicable in tourism forecasting. The trans­ fer function techniques employed in the study have a strong theo­ retical basis (Pindyck and Rubinfeld, 1981). Such techniques did perform well in not only identifying the relationships of inde­ pendent variables and dependent variable but also correcting for the autocorrelation problems encountered in the pure regression models to increase the "goodness-of-fit" and predictive ability of the model. updated. Also, the transfer function model is easily These findings provide strong empirical support for their use recommended by Wander and Erden (1980) and Weller and Kurre (1987). The harmonic model of seasonality performs well, particu­ larly if no other variables are available to fully explain the variations in the seasonality series and predict future seasonal patterns. Variables like temperature, precipitation and snowfall for a given month can be used to explain the monthly seasonal fluctuations but are hard to accurately predict for the forecast­ ing purposes. In this situation, descriptive measures of the periodic behavior of the seasonal patterns produced by harmonic analysis can be modeled and used to accurately predict future sesonal patterns. These measures enable planners and managers to determine the existence of a single or multiple tourism seasons, the peak timing of the season, and the relative importance of individual seasonal fluctuations in a year. 155 This study has demonstrated how the methods of combining structural and time series models can be used to develop fore­ casting models to successfully track and forecast the behavior of the series. The trend and seaonal patterns of the series were successfully modeled using transfer function techniques and har­ monic analysis respectively. Through the processes of developing these models forces underlying the fluctuations in the trends and periodic patterns of the seasonality were better identified and understood. The excellent performance of the combined forecasting meth­ ods observed in the present study provide further support for the potential uses of such methods on the other types of data series to help betteridentify and quantify the It is believed behavior of the series. that information provided by the combined fore­ casting model can help managers and planners better understand the patterns of tourism activity in an area, and consequently improve tourism planning and management decisions. APPLICATIONS OF RESULTS How can these results be directly applied to decision-making or to the management and development of tourism businesses in the state or a region? The first obvious use would be to employ the forecasting models to produce short-term predictions of tourismrelated employment. These short-term predictions would, for example, help tourism planners of the state foresee future changes in the volume of tourism activity in the state as a whole and its individual sub-regions and then to efficiently allocate 156 available resources to promote tourism businesses in regions with high potential growth in tourism-related employment. Also, with these predictions regional planners and managers can better sup­ ply tourism opportunities and services to meet the demand of tourism activity in the region. If a downward trend of tourism activity in a region is predicted, planners and managers can act together to develop appropriate plans to sustain and/or stimulate tourism businesses in the region. Knowing the seasonal patterns of tourism-related employment across the regions help business managers in better coordinating advertising, inventory and staff programs to compete with competitors in the region and in the other competitive regions. Although the forecasting models look complicated and sophis­ ticated, they are relatively easy to manipulate to produce fore­ casts. For example, the forecast equations can be programmed onto a LOTUS 1-2-3 spreadsheet file. In this form users only need to enter the predicted values of independent variables to produce forecasts. Forecast accuracy depends on the lead time of forecasts, i.e. the lead time equals 3 when forecasting for the next 3 months without using new observed values of the dependent variable. The larger the lead time of a forecast the larger the forecast error. model. Forecast errors can be reduced by updating the Users plug the new observed values of dependent and inde­ pendent variables into the spreadsheet file and then the whole model is updated immediately without additional costs of reestimating the whole equation. This simple procedure for updat­ ing the model is one of the characteristics of the transfer func­ tion model. A transfer function model can automatically 157 re-calculate the values for the autoregressive and moving average components of the error model once the new values of dependent and independent variables are entered. It should be noted that the whole model needs to be re-estimated after a certain period of time or whenever there occur unusual situations of tourism activity. Another use of the models is to perform the sensitivity ana­ lysis. From this analysis how much impact of changes in the individual independent variables on the dependent variable can be understood, when other variables are held constant. The other use of the models is to track and monitor the impact of a (statewide or regional) tourism promotion program on tourism activity in an area, such as "Say Yes to Michigan" cam­ paign. The predictions of the modls can be used as the control data because they are produced assuming that all underlying forces of the dependent variable are constant over time. By com­ paring the observations and predictions the magnitude of the impact of a campaign can be better understood and justified. The study is part of efforts of the Travel, Tourism, Recreation and Resource Center at Michigan State University to collect, analyze, and disseminate tourism-related information to help the tourism industry in the state. This analysis provides useful information that will help in better understanding the patterns of tourism activity in the state. The analysis and modeling procedures employed in this study provide a useful framework to track, monitor and forecast tourism activity at the state, regional and local levels. With the identified tourism patterns and tourism forecasting models for tourism activity in 158 the state, the center can better assist planners and business managers in developing appropriate tourism development and management plans for their areas. The study provides useful information on temporal and spa­ tial variations in tourism related employment in Michigan. These information enable planners and managers to better understand the patterns of tourism activity in various regions of the state. Predictions of the models are helpful in designing future tourism development plans. It is further hoped that these procedure and methods for analyzing and forecasting tourism activity can stimu­ late more research in the procedure and methods employed to bet­ ter monitor and forecast tourism activity. BIBLIOGRAPHY 159 160 BIBLIOGRAPHY Archer, Brian H. (1976). Demand Forecasting in Tourism. University of Wales Press. Archer, Brian H. (1980). "Forecasting Demand: Quantitative and and Intuitive Techniques", International Journal of Tourism Management(March!. 5-12. Artus, J. R. (1972) . "An Econometric Analysis of International Travel", International Money Fund Staff Papers (19). 579-614. Askari, H. (1971). 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Travel Data Center's 1985 Travel Outllok Forum. Wander, A. S. and D. J. Van Erden. (1980). "Estimating the Demand for International Tourism Using Time Series Analysis", Tourism Planning and DevelopmentIssues, ed. by Hawkins et al., Washington D. C.: George Washington University. Weller, Barry B. and James A. Kurre. (1987). "Applicability of the Transfer Function Approach to Forecasting Employment Levels in Small Regions", The Annals of Regional Science. Vol. XXI, No. 1. Wheelwright, S. C. and Makridakis, S. (1985). Methods for Management. 4th ed. New York: Sons. Forecasting John Wiley and APPENDIX A DEFINITIONS OF HARMONIC STATISTICS MEASURES 165 The formula for the Fourier series commonly used in harmonic analysis, defined by Rayner (1971), is: Xk = [AkSlN(k9+k ] =A0+A1SIN(9+2)+ •••+ AkSIN(kO+<|>k)+ .. .+ANSIN(N9+(j)k) (A.1) Xk is the time series, Ak is the amplitude for harmonic k. Ak measures the magnitude of the fluctuation of sine curve k around the arithmetic mean of the data series (A0) . k)cos(kQ)+Akcos(k)sin(ke) ] (A.3) When ak and bk are defined as ak « Aksin(k), and (A.4) bk = Akcos(k), (A.5) 166 Equation (A.3) becomes Xk - [akcos(k0)+bksin(ke). (A.6) Then, Ak and 0k can be computed from ak and bk , since cos2 (w) + sin2 (w) = 1. (A.7) The amplitude, Ak , is defined by, Ak - (ak2+bk2)V2. (A.8) The relative amplitude, Rk , is defined by, Rk * (Ajc/Aq )*100.0. (A.9) The phase angle, <{)k , is defined by, The percentage of the total variance explained by harmonic k is, v. - cr 2 . ^ k cr2 (A. 14) APPENDIX B SUMMARY TABLES AND FIGURES FOR CHAPTER V 168 Table Bl.l: Harmonic Measures of Tourism-Related Employment in Wayne and Oakland Counties, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Ai 1.00 .04 .02 .01 .01 .00 .00 Phase Angle Relative Variance ^ (degrees) Amplitude Explained 259 198 155 230 260 270 4.1 1.5 .5 .9 .3 .3 82.6 11.5 1.3 3.7 .5 .4 Peak Timing Aug. 5 Table Bl. 2: Harmonic Measures of Tourism-Related Employment in Out-State Urban Counties, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Ai 1.00 .03 .02 .01 .01 .00 .00 Phase Angle Relative Variance i (degrees) Amplitude Explained 264 207 100 238 262 270 3.3 1.8 .7 .7 .2 .2 71.9 21.1 3.1 3.4 .3 .2 Peak Timing Aug. 9 169 Table B1.3: Harmonic Measures of Tourism-Related Employment in South Rural Counties, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Ai 1.00 .09 .01 .01 .01 .00 .00 Phase Angle Relative Variance (degrees) Amplitude Explained 256 170 152 240 244 270 9.1 1.0 1.0 .7 .4 .2 96.8 1.3 1.1 .6 .2 .0 Peak Timing Aug. 1 Table Bl. 4: Harmonic Measures of Tourism-Related Employment in Northwest Michigan, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Ai 1.00 .18 .09 .00 .00 .02 .00 Phase Angle Relative Variance (degrees) Amplitude Explained 250 62 205 299 170 270 17.9 8.9 .3 .3 1.7 .1 79.7 19.5 .0 .0 .6 .0 Peak Timing Aug. 6 170 Table B1.5: Harmonic Measures of Tourism-Related Employment in Northeast Michigan, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude A^ 1.00 .17 .03 .00 .01 .00 .00 Phase Angle Relative Variance (degrees) Amplitude Explained 247 81 212 299 160 90 17.0 3.4 .5 1.0 95.7 3.9 .1 .3 .2 .2 .0 .0 Peak Timing Aug. 8 Table B1.6: Harmonic Measures of Tourism-Related Employment in the Mackinac Straits Region, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Phase Angle Relative Variance A^ <)>£ (degrees) Amplitude Explained 1.00 .81 .18 .04 .02 .04 .00 253 68 48 276 202 90 80.9 17.9 4.3 1.6 4.4 .2 94.8 4.6 .3 .0 .3 .0 Peak Timing 171 Table B1.7: Harmonic Measures of Tourism-Related Employment in the Eastern Upper Peninsula, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Phase Angle Relative Variance Ai ()>^ (degrees) Amplitude Explained 1.00 .44 .08 .01 .01 .02 .00 251 67 82 272 215 90 43.8 8.2 1.5 1.3 2.1 .0 Peak Timing 96.2 3.4 .1 .1 .2 .0 Table B1.8: Harmonic Measures of Tourism-Related Employment in the Western Upper Peninsula, 1974-84. Harmonic i 0 1 2 3 4 5 6 Amplitude Aj[ 1.00 .06 .03 .01 .01 .01 .00 Phase Angle Relative Variance (j>£ (degrees) Amplitude Explained 236 72 179 234 206 270 5.9 3.3 .9 1.1 1.0 .2 71.5 22.5 1.5 2.4 2.0 .1 Peak Timing 172 Table B2.1: Forecasts for Tourism-Related Employment In Wayne and Oakland Counties, 1983.07-1984.06. ACTUAL REGIONAL REGIONAL ACTUAL ACTUAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR MOUTH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS (X) 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 93,582 93,797 94,026 94,405 94,955 95,518 96,044 96,552 97,011 97,413 97,762 98,089 93,612 93,847 94,061 94,254 94,431 94,587 94,732 94,860 94,969 95,062 95,146 95,228 1.03 1.04 1.04 1.01 0.99 0.99 0.94 0.94 0.95 1.00 1.04 1.06 1.03 1.02 1.02 1.01 1.00 0.99 0.95 0.94 0.97 0.99 1.03 1.05 96,562 95,992*1 97,355 95,397 97,353 96,154 94,678 94,834 93,919 94,885 94,098 93,886 90,398 89,701 90,609 89,252 91,848 92,098 97,559 94,196 101,517 98,081 103,576 100,272 570b 1958 1199 -156 •966 212 697 1357 -250 3363 3436 3304 0.59c 2.01 1.23 0.16 1.03 0.23 0.77 1.50 0.27 3.45 3.38 3.19 a. Colum 3 multiplied to column 5. b. Column 6 minus column 7. c. Column 8 divided by column 6. Table B2.2: Forecasts for Tourism-Related Employment in Out-State Urban Counties, 1983.07-1984.06. ACTUAL ACTUAL ACTUAL REGIONAL REGIONAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR HONtH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS (X) 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 a. b. c. 64,999 65,220 65,431 65,661 65,914 66,202 66,488 66,734 66,924 66,980 66,957 66,954 64,939 65,063 65,193 65,323 65,444 65,573 65,640 65,684 65,727 65,770 65,816 65,862 1.02 1.02 1.04 1.02 1.01 1.00 0.96 0.95 0.96 0.99 1.03 1.05 Column3 multiplied to column S. Column6 minus colum 7. Column8 divided by column 6. 1.00 1.01 1.03 1.00 1.00 1.00 0.96 0.95 0.97 1.00 1.04 1.04 65,942 66,205 67,625 67,119 66,219 65,875 63,579 63,263 64,226 66,499 69,209 70,601 65,097® 65,637 66,850 65,454 65,285 65,389 63,175 62,435 63,785 65,717 68,559 68,645 845b 568 775 1,665 934 486 404 828 441 782 650 1,956 1.28° 0.86 1.15 2.48 1.41 0.74 0.63 1.31 0.69 1.18 0.94 2.77 173 Table B2.3: Forecasts for Tourism-Related Employment in South Rural Counties, 1983.07-1984.06. ACTUAL REGIONAL REGIONAL ACTUAL ACTUAL REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR REGIONAL TREND (X) MONTH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 49,418 49,733 50,057 50,342 50,550 50,738 50,889 50,986 51,085 51,123 51,109 51,112 49,333 49,353 49,402 49,430 49,546 49,602 49,655 49,713 49,749 49,789 49,792 49,844 1.10 1.10 1.07 1.01 0.98 0.97 0.92 0.91 0.94 0.98 1.03 1.07 1.09 1.08 1.05 1.02 0.98 0.94 0.91 0.91 0.93 0.98 1.03 1.07 54,147 54,615 53,604 50,805 49,476 48,993 46,850 46,601 47,932 50,022 52,482 54,573 53,727*1 53,245 52,026 50,420 48,696 46,857 45,405 45,031 46,171 48,618 51,409 53,539 420b 1,370 1,578 385 780 2,136 1,445 1,570 1,761 1,404 1,073 1,034 0.78c 2.51 2.94 0.76 1.58 4.36 3.08 3.37 3.67 2.81 2.05 1.89 a. Column 3 multiplied to column*5. b. Column 6 minus colum 7. c. Column 8 divided by column 6. Table B2.4: Forecasts for Tourism-Related Employment in Northwest Michigan, 1983.07-1984.06. ACTUAL ACTUAL ACTUAL REGIONAL REGIONAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR MONTH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS (X) 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 a. b. c. 9,148 9,182 9,221 9,263 9,295 9,323 9,321 9,276 9,232 9,183 9,136 9,106 9,166 9,169 9,182 9,189 9,221 9,236 9,258 9,277 9,287 9,293 9,294 9,316 1.29 1.33 ’ 1.18 1.00 0.36 0.92 0.89 0.88 0.86 0.85 0.97 1.13 Column3 multiplied to column S. Column6 minus column 7. Column8 divided by column 6. 1.25 1.25 1.13 0.98 0.89 0.89 0.91 0.90 0.86 0.86 0.95 1.12 11,766 12,233 10,830 9,249 8,024 8,579 8,305 8,174 7,922 7,822 8,855 10,293 11,445® 11,494 10,419 9,025 8,229 8,196 8,428 8,341 7,977 7,982 8,875 10,423 321b 739 411 224 -205 383 -123 -167 -55 -160 -20 -130 2.73c 6.04 3.79 2.43 2.55 4.46 1.48 2.04 0.70 2.05 0.23 1.26 174 Table 82.5: Forecasts for Tourism-Related Employment in Northeast Michigan, 1983.07-1984.06. ACTUAL REGIONAL REGIONAL ACTUAL ACTUAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR <%) FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS MONTH TREND 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 5,870 5,882 5,895 5,908 5,926 5,940 5,939 5,923 5,906 5,895 5,887 5,875 5,862 5,866 5,868 5,872 5,874 5,877 5,878 5,880 5,881 5,882 5,883 5,884 1.14 1.17 1.10 1.00 0.96 0.92 0.91 0.90 0.88 0.93 1.05 1.14 1.19 1.19 1.13 1.04 0.96 0.91 0.87 0.85 0.85 0.90 1.00 1.11 6,694 6,844 6,453 5,887 5,668 5,456 5,393 5,321 5,179 5,510 6,188 6,733 6,953a 6,967 6,602 6,088 5,641 5,339 5,132 4,999 5,019 5,317 5,894 6,549 -259bb •123 •149 •201 27 117 261 322 160 193 294 184 3.87° 1.79 2.30 3.42 0.47 2.15 4.83 6.05 3.08 3.50 4.75 2.73 a. Column 3 multiplied to column 5. b. Column 6 minus column 7. c. CoIurn 8 divided by column 6. Table B2.6: Forecasts for Tourism-Related Employment in the Mackinac Straits Region, 1983.07-1984.06. ACTUAL ACTUAL ACTUAL REGIONAL REGIONAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR MONTH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS 1.84 1.84 1.59 1.19 0.63 0.51 0.44 0.41 0.42 0.52 1.08 1.51 CoIu m 3 multiplied to column 5. Column6 minus column 7. Column8 divided by column 6. 1.94 1.93 1.57 1.07 0.68 0.47 0.39 0.35 0.38 0.60 1.04 1.58 3,264 3,272 2,842 2,134 1,133 908 775 736 751 920 1,905 2,655 3,444“ 3,432 2,790 1,909 1,208 846 704 634 688 1,073 1,874 2,844 o 1,774 1,777 1,780 1,784 1,787 1,791 1,795 1,797 1,798 1,799 1,801 1,803 CO a. b. c. 1,774 1,781 1,786 1,790 1,785 1,778 1,775 1,773 1,767 1,762 1,757 1,754 JO 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 -160 52 225 •75 62 71 102 63 •153 31 •189 5.52c 4.88 1.83 10.56 6.61 6.77 9.14 13.91 8.41 16.60 1.64 7.13 175 Table B2.7: Forecasts for Tourism-Related Employment in the Eastern Upper Peninsula, 1983.07-1984.06. ACTUAL REGIONAL REGIONAL ACTUAL ACTUAL REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR REGIONAL TREND FORECAST ERRORS (%) FORECAST SEASONAL FORECAST SERIES MONTH TREND 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 2,943 2,944 2,945 2,949 2,951 2,951 2,950 2,945 2,934 2,917 2,897 2,883 2,963 2,966 2,971 2,977 2,982 2,987 2,992 2,996 2,997 2,995 2,996 3,002 1.47 1.51 1.36 1.15 0.83 0.74 0.63 0.63 0.65 0.76 1.06 1.29 1.49 1.50 1.32 1.07 0.85 0.72 0.66 0.63 0.66 0.78 1.02 1.30 4,328 4,441 4,007 3,399 2,443 2,180 1,852 1,851 1,902 2,211 3,073 3,726 4,414*1 4,437 3,929 3,184 2,541 2,154 1,968 1,893 1,973 2,345 3,056 3,898 -86b 4 78 215 •98 26 •116 -42 •71 -134 17 •172 1.98c 0.10 1.96 6.32 4.02 1.20 6.24 2.29 3.71 6.08 0.55 4.61 a. Column 3 multiplied to column 5. b. Colum 6 minus column 7. c. Column 8 divided by column 6. Table B2.8: Forecasts for Tourism-Relsted Employment in the Western Upper Peninsula, 1983.07-1984.06. ACTUAL ACTUAL ACTUAL REGIONAL REGIONAL REGIONAL TREND REGIONAL SEASONAL REGIONAL REGIONAL FORECAST ABS ERROR MONTH TREND FORECAST SEASONAL FORECAST SERIES FORECAST ERRORS (X) 1983.07 1983.08 1983.09 1983.10 1983.11 1983.12 1984.01 1984.02 1984.03 1984.04 1984.05 1984.06 a. b. c. 5,950 5,976 6,006 6,038 6,062 6,077 6,092 6,108 6,121 6,122 6,120 6,120 5,952 5,981 6,006 6,026 6,043 6,057 6,068 6,076 6,080 6,083 6,085 6,086 1.08 1.10 1.06 1.01 0.97 1.02 0.97 0.96 0.96 0.94 0.96 1.04 Column3 multiplied to column 5. Column6 minus colum 7. Column8 divided by column 6. 1.07 1.08 1.05 1.01 0.97 1.00 0.97 0.96 0.95 0.92 0.98 1.03 6,427 6,545 6,323 6,057 5,872 6,190 5,915 5,885 5,846 5,747 5,861 6,353 6,396° 6,470 6,314 6,062 5,867 6,041 5,864 5,837 5,804 5,601 5,978 6,289 31b 75 9 -5 5 149 51 48 42 146 -117 64 0.49° 1.14 0.15 0.08 0.09 2.41 0.87 0.81 0.72 2.53 2.00 1.00 WAYNE AND O A K L A N D COUNTIES 104 1 02 - 100 - 96 94 92 90 88 - 86 - 176 TO U R ISM —RELATED EM PLOY M ENT (T h o u sa n d s) 98 - 84 82 80 78 76 7 4 --- 1984.07 1974.08 MONTH ACTUAL Figure Bl.l: PREDICTED Actual and Predicted Tourism-Related Employment in Wayne and Oakland Counties, 1974.08-1984.06. OUT-STATE URBAN COUNTIES 72 68 - 66 - 64 62 60 58 177 T O U R ISM —RELATED EMPLOYMENT (T h o u sa n d s) 70 - 56 54 52 50 48 46 4 4 ---- 1984.07 1980.07 1974.08 MONTH ACTUAL Figure B1.2: PREDICTED Actual and Predicted Tourism-Belated Employment in Out-State Urban Counties, 1974.08-1984.06. T O U R ISM —RELATED EM PLOYM ENT (T h o u sa n d s) SOUTH R U R A L COUNTIES 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 1974.08 1980.07 ACTUAL Figure B1.3: MONTH + 1984.07 PREDICTED Actual and Predicted Tourism-Belated Employment in Southern Rural Counties, 1974.08-1984.06. 10 - 179 TOURISM -RELATED EMPLOYM ENT (T h o u sa n d s) N O R T H W E S T MICHIGAN ACTUAL Figure B1.4: 1984.07 1980.07 1974.08 MONTH + PREDICTED Actual and Predicted Tourism-Related Employment in Northwest Michigan, 1974.08-1984.06. N O R T H E A S T MICHIGAN 7.2 TO U R ISM —RELATED EM PLOYMENT (T h o u sa n d s ) 7.06.8 - 6.6 - 6.4 6.2 - 6.05.8 5.6 5.4 5.2 - 5.0" 4.8 4.6 4.4 4.2 - 4.0H— * 1974.08 1980.07 ACTUAL Figure B1.5: MONTH + 1984.07 PREDICTED Actual and Predicted Tourism-Related Employment in Northeast Michigan, 1974.08-1984.06. M A C K I N A C STRAITS R E G I O N 3.6 3.4 - 2.6 - 2.4 2.2 - 2.0" 181 T O U R ISM —RELATED EMPLOYM ENT (T h o u sa n d s) 3.2 - 1.4 - 1.0- 0.8 - 0.6 - 0.4 H— 1 ACTUAL Figure B1.6: 1984.07 1980.07 1974.08 MONTH + PREDICTED Actual and Predicted Tourism-Related Employment in the Mackinac Straits Region, 1974.08-1984.06. THE EASTERN UPPER PENINSULA 4.0- 3.5 182 T O U R ISM —RELATED EMPLOYM ENT (T h o u sa n d s ) 4.5 - 2.5 - 2.0“ 1.5 ACTUAL Figure Bl.7: 1984.07 1980.07 1974.08 MONTH + PREDICTED Actual and Predicted Tourism-Related Employment in the Eastern Upper Peninsula, 1974.08-1984.06. THE WESTERN UPPER PENINSULA 6.8 6.6 - T O U R ISM —RELATED EM PLOYM ENT (T h o u san d s) 6.4 6.2 - 6 .0 - 5.8 5.6 5.4 5.2 - 5 .0 4.8 4.6 4.4 4.2 - 4 .0 - f i— 1974.08 ACTUAL Figure Bl.8: 1984.07 1980.07 MONTH + PREDICTED Actual and Predicted Tourism-Belated Employment in the Western Upper Peninsula, 1974.08-1984.06.