forecasting

INFORMATION TO USERS

This manuscript has been reproduced from the microfilm master. UMI films

the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer.

The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations

and photographs, print bleedthrough, substandard margins, and improper

alignment can adversely affect reproduction

In the unlikely event that the author did not send UMI a complete manuscript

and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.

Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing

from left to nght in equal sections with small overlaps

ProQuest Information and Learning 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA

800-521-0600

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

NOTE TO USERS

This reproduction is the best copy available.

UMI


ABSTRACT

Title of Thesis: Forecasting Hotel Occupancy Rates in EgyptUsing Time Series Models

Degree Candidate: Mohamcd A. Ibrahim

Degree and Year: Master of Science. 1995

Thesis directed by: Dr. Emmanuel T. AcquahAssociate Professor. Department of Agriculture,University of Maryland Eastern Shore

Key words: Forecasting. Time Series, Occupancy Rate, Market Segments,Hospitality. Accuracy

Forecasting occupancy rates is important for effective hotel management. This

study examines empirically the use of five time series models (moving average. Brown’s.

Holt’s, Winter’s and AR1MA) in forecasting hotel occupancy rates in Egypt. These

models were fitted and tested using actual monthly occupancy rates for two types of

hotels (city center and resort). For the city center hotel. ARIMA model forecasted more

accurately than the other investigated models. Results were different for the resort hotel

where Brown’s model outperformed all ether models in prediction.

Results of this study showed generally that simple time scries models (Brown’s,

Holt’s & moving average) performed as well as or better than sophisticated models

(Winter’s & ARIMA) in forecasting hotel occupancy rates in Egypt. Since simple time

scries models arc easy to develop and economical in terms of time and skill levels of

users, results of this study have important implications for the use of time series

forecasting techniques in the hotel industry.


FORECASTING HOTEL OCCUPANCY RATES IN EGYPT USING TIME SERIES MODELS

by

MOHAMED A. IBRAHIM

Thesis Submitted to the Graduate Faculty of the Department of Agriculture at the University

of Maryland Eastern Shore in partial fulfillment of the requirement

for the degree MASTER OF SCIENCE

1995

Advisory Committee:Emmanuel Acquah, Ph.D, Advisor Imtiaz Ahmad, Ph.D John Dienhart. Ph.D Frank Lin, Ph.D


UMI Number EP13939

UMIUMI Microform EP 13939

Copyright 2005 by ProOuest information and Learning Company All rights reserved This microform edition is protected against

unauthorized copying under Title 17. United States Code

ProOuest Information and Learning Company 300 North Zeeb Road

P O Boi 1346 Ann Artxx Ml 48106-1346


DedicatedTo my father Professor Abdalla Fathy Ibrahim

who did not live to read this work.

ii


ACKNOWLEDGEMENT

I would like to start by thanking Allah for giving me the knowledge, effort and

patience to complete this study. I would also like to express my gratitude to Dr.

Emmanuel Acquah my advisor and the other members of graduate committee, Drs.

Imtiaz Ahmad, John Dienhart and Frank Lin for their support and advice during the

preparation of this manuscript.

My deep appreciations are also extended to Drs. William Hytche, Mortimer

Neufville, Yousef Hafez and George Shorter for all their support during my study at the

University of Maryland Eastern Shore.

Special acknowledgement to my mother Nagat and my family Omneya, Dalia,

Sarah and Hakam for their prayers, love and support during the preparation of this work.

My gratitudes are extended to my colleagues Greg Early and Richard Coe for

their friendship and help in preparing this manuscript.

Mohamed A. Ibrahim

iii


TABLE OF CONTENTSPage

List of Tables viList of Figures vii

CHAPTER 1. Introduction 1Statement of the Problem 3Objectives 3

CHAPTER 2. Review of Literature 4Judgmental forecasting 4Econometric forecasting 5Time series forecasting 9

CHAPTER 3. Methodology 17Development of Questionnaire 17Sample 17Data Analyses 19

Measuring the Accuracy of Forecasting Models 19Time Series Models Formulation 20Moving Average Model 20Exponential Smoothing Models 21Autoregressive Integrated Moving Average Models 28

CHAPTER 4. Results and Discussion 36Time Scries Models Fit 37Time Scries Models Forecasts 40

CHAPTER 5. Conclusions and Recommendations 50

REFERENCES 53

APPENDIXES 57A. Questionnaire 58B. Time series plots of models fitting for Cairo hotel 66C. Time series plots of models fitting for Hurgada hotel 72

v


LIST OF TABLES

Tables Page

1. Unconditional Least Squares ARIMA Model for the Cairo Hotel 38

2. Unconditional Least Squares ARIMA Model for the Hurgada Hotel 39

3. Six Months Actual Occupancy versus the Forecast of 41 Time Series Models for the Cairo Hotel

4. Six Months Actual Occupancy versus the Forecast of 42 Time Series Models for the Hurgada Hotel

5. Sum of Squared Errors of Forecast (SEE) for the Cairo Hotel 49

6. Sum of Squared Errors of Forecast (SEE) for the Hurgada Hotel 49

VI


LIST OF FIGURES

Figures Page

1. Time Series Plot for the Cairo Hotel 26

2. Time Series Plot for the Hurgada Hotel 27

3. Autocorrelation for the Cairo Hotel 32

4. Partial Autocorrelation for the Cairo Hotel 33

5. Autocorrelation for the Hurgada Hotel 34

6. Partial Autocorrelation for the Hurgada Hotel 35

7. The Forecast of the Best Models to Forecast for the Cairo Hotel 44

8. The Forecast of the Best Models to Forecast for the Hurgada Hotel 45

9. SSE of Time Series Model Forecasts for the Cairo Hotel 46

10. SSE of Time Series Model Forecasts for the Hurgada Hotel 47

11. Moving Average Model Fit for the Cairo Hotel 67

12. Brown’s Model Fit for the Cairo Hotel 68

13. Holt’s Model Fit for the Cairo Hotel 69

14. Winter’s Model Fit for the Cairo Hotel 70

15. ARIMA Model Fit for the Cairo Hotel 71

vii


LIST OF FIGURES (continued)

Figures Page

16. Moving Average Model Fit for the Hurgada Hotel 73

17. Brown’s Model Fit for the Hurgada Hotel 74

18. Holt’s Model Fit for the Hurgada Hotel 75

19. Winter’s Model Fit for the Hurgada Hotel 76

20. ARIMA Model Fit for the Hurgada Hotel 77

viii


CHAPTER 1

INTRODUCTION

Hold management companies that want to meet future challenges of the

hospitality industry must take positive actions toward adopting accurate forecasting

methods. Accurate forecasting of hotd occupancy rates is very important in all areas of

hotd operations. Short term occupancy rates are needed for daily and weekly employees

scheduling; especially where hotds are highly dependent on part time labor. To

maximize revenues, hotd managers must adopt yidd management strategies which

depend on accurate forecasting of slow and peak occupancy seasons and plan to operate

on maximum capacity in all seasons. Marketing strategies, menu devdopment decisions,

employee hiring, training programs and capital investment decisions must be based on

predicting future hotel occupancies accurately.

To address the critical need for forecasting in the hospitality industry, the majority

of hotd operators devdop their own forecasting methods depending on simple averages

of historical business observations, judgements based on experience, or sometimes a mix

of both methods. Hospitality managers use their own judgment based on their experience

of the economic and business activity in estimating future sales (Crange <fc Andrew

1992). The studies that have compared judgmental with quantitative forecasting indicated

that judgmental forecasting is less accurate than quantitative forecasting. Geurts and Kelly

(1986) conducted a study to forecast retail sales, they concluded that judgmental

forecasting is inferior to time series or econometric short term sales forecasts.

1


Makridakis (1986) indicated that people’s judgement is one of the obstacles for

successful implementation of forecasting. He reported that individuals prefer making

forecasts judgementally. People believe that their knowledge of the product, market and

customers as well as their insight and inside information give them a unique ability to

forecast judgementally. The author indicated that individuals do not usually consider that

quantitative methods can do an adequate job. He concluded in his study that when a

number of frequently required forecasts are needed, the judgmental method is not a good

practice due to its inaccuracy compared with quantitative methods.

Andrew, Crange and Lee (1990) reported that the application of quantitative

methods in forecasting has been ignored by the hotels managers and educators. This may

be due to the lack of experience in the use of quantitative methods in forecasting or the

lack of evidence of its accuracy in hotels applications.

The cost of judgmental forecasting errors in hotels can be high due to many

reasons such as the loss of food and beverage because of over production, high inventory

costs due to over stocking, loss of customers to competition due to over estimation of

prices in slow occupancy periods, and high labor costs due to over staffing and

scheduling of employees.

Quantitative methods in forecasting takes management from the uncertain

environment into the probability of what might happen in the future by looking at

different variables that affect a dependent variable, or by looking at the historical patterns

and trends over a period of time in the dependent variable in order to capture these

patterns and trends to project the future.


Problem Statement

The problem is the inability of hotels to forecast occupancy rates accurately.

Purpose

This study will provide evidence of the efficiency and accuracy of quantitative

methods, namely time series models, in forecasting hotels occupancy rates in Egypt.

Objectives

The objectives of this study were to:

(i) Forecast hotel occupancy rates in Egypt using moving average, exponential

smoothing, and autoregressive integrated moving average (ARIMA) time series

forecasting models.

(ii) Determine which of the time series models is more accurate in forecasting hotel

occupancy rates in selected hotels.

(iii) Determine the effect of type of hotel operation, location and market segmentation

on forecasting errors of the models.

(iv) Determine which of the time series models would be most appropriate for use by

hospitality managers based on accuracy and simplicity.

3


CHAPTER 2

LITERATURE REVIEW

Judgmental Forecasting

Crange and Andrew (1992) reported that the judgmental forecasting is the

technique most often used by the majority of hotels and restaurants. The judgmental

technique consists of an intuitive forecast based on the manager's collective experience

regarding the variable in question (for example occupancy rate).

Makridakis and Wheelwright (1989) reported the disadvantages of judgmental

forecasting. The authors reported that humans tend to be optimistic and underestimate

future uncertainty. They indicated that the reasons for the poor performance of

judgmental forecasters are because humans are not capable of separating wishful

thinking, politics, personal consideration, and their own emotional state from an objective

evaluation of the past, present or future.

Simon and Newell (1971) discussed the reasons for the poor performance of

judgmental forecasting methods. They pointed out that because people have limited

information processing capacity, they can not deal directly with large problems. Instead

they use a simplifying heuristic approach in their problem solving efforts. In the end,

they look for satisfying, rather than optimizing, solutions since in the complex

environment in which we live, optimizing is well beyond present human intellectual

abilities.

4


Mabert (1975) reported a direct comparison between judgmental and quantitative

methods of forecasting. He found that forecasts based on opinions of the sales force and

corporate executives in different industries gave less accurate results over the five year

period covered by the study than did the quantitative forecasts of three methods

(exponential smoothing, harmonic smoothing and ARIMA). In a comparison between the

performance of judgmental and quantitative methods in forecasting, Adam and Ebert

(1976) found that exponential smoothing method produced forecasts that were statistically

more accurate than those of human forecasters.

Many studies have been conducted on the accuracy of judgmental versus statistical

models. For example, Armstomg (1983), Lorek and Patz (1976), Cleveland and Tiao

(1976), Hogarth (1975) and Dalrymple (1975), all concluded that quantitative methods

provided better forecasts than judgmental methods.

Econometric Forecasting

In econometric forecasting models, the value of a certain dependent variable is

a function of one or more other independent variables. Makridakis & Wheelwright (1989)

indicated that the strength of an econometric model used as a forecasting method is that

a manager can develop a range of forecasts corresponding to a range of values for the

different independent variables. They also reported on three main disadvantages of

econometric models. First, it requires large amounts of data and information about all

the variables used in the model. Information on several independent variables and the

dependent variable must be collected to build the model, the extensive data collection

makes econometric models expensive and time consuming. Secondly, since explanatory

5


models generally relate several factors, they usually take a long time to develop and are

more sensitive to changes in the underlying relationships. As a result, they must be

constantly revised and redeveloped according to changes in the relationship. Finally, they

require an estimation of future values of the independent variables before the output can

be forecasted. All these factors make the econometric models time consuming in

collecting the data and difficult to develop, especially the model identification and

estimation which requires a full understanding of economic theory, statistics and

econometrics.

Steckler (1968) conducted a study to predict economic activities namely inflation,

using econometric models. The author reported that the results suggest that econometric

modes have not been entirely successful in forecasting economic activity. Similarly,

Cooper and Jorgenson (1969) conducted a study to predict the components of the national

income and product accounts by using econometric and exponential smooching models.

They indicated that the econometric models are generally not superior to purely

smooching methods of forecasting.

Naylor and Seaks (1972) conducted a study to forecast investment using

econometric and ARIMA models. The authors reported that few forecasting quarterly

investment from 1963 to 1967 the ARIMA forecasts were superior to econometric models

in forecasting quarterly investment rates based on average absolute error.

Cleary and Fryk (1974) conducted a study to forecast the telephone demand using

econometric model versus time series models namely ARIMA. They concluded that

ARIMA models outperformed the econometric model in forecasting telephone demand.

6


Binroth, Burshtein, Haboush and Hartz (1979) conducted a study to forecast

monthly rubber commodity prices using time series versus econometric models. They

concluded that the ARIMA time series model used was superior to the econometric model

in forecasting monthly rubber commodity prices. The authors reported that the advantage

of ARIMA model over the econometric model comes from the ability of ARIMA models

to elucidate the various components of time series such as non-stationarities (trends) and

cyclic, seasonal and random effects better than econometric models. They indicated the

disadvantages of econometric models are because they require the user to determine

which predictor variables to use, which transformations of the input data to perform, and

also require the forecasts of predictor variables that are not leading indicators of the

independent variable. The human estimation of the model could lead to identification

errors which increase the forecast errors. The automatic methods namely time series

eliminate human intervention in the model identification and estimation which eliminate

human errors and lead to accurate forecasts.

Schmidt (1979) conducted a study to compare the performance of econometric

versus time series models, namely ARIMA, in forecasting monthly retail sales. The

author concluded that the ARIMA model performed better than the econometric model

in forecasting monthly retail sales.

Geurts and Kelly (1986) conducted a study to forecast sales in retail stores. The

authors used two time series models versus an econometric model to forecast monthly

sales. They reported that there were many disadvantages in using econometric models.

First, the future values of the causal variables had to be predicted. Second, in the case

7


of using a lagged causal variable to predict sales one period ahead, the reported causal

variable value is often a preliminary figure that is later revised. Both of these factors can

cause data in an econometric model to be inaccurate and the model to be weak in its

ability to forecast. Third, the continual need to gather data can make these models time

consuming and expensive to use. And fourth, causal relationships can change over time,

making it necessary to constantly update or totally redesign the model. They concluded

in their study that time series models namely exponential smoothing and ARIMA are

superior to econometric models in retail sales forecasting.

Crange & Andrew (1992) conducted a study to forecast restaurant monthly sales

using econometric versus time series models. They reported that econometric models

utilize a regression equation or equations to establish a causal relationship between the

dependent variable (hotel occupancy) and independent variable such as disposable

income, consumer price index and unemployment. They indicated that the advantage of

econometric models over time series models is that the econometric model may respond

more quickly in its predictions to the changed situation than the time series models,

especially when there is a turning point in the economy due to unexpected economic

events. They concluded in their study that the time series models outperformed the

econometric models in forecasting monthly restaurant sales. The findings are significant

in a fact that restaurant sales are similar to hotel sales (occupancy rates), both are

influenced by the same economic variables and business activities.

The majority of research that compared the performance of econometric with time

series models in forecasting has shown that time series forecasts were superior to

8


econometric models in forecasting different dependent variables in different industries.

Other studies that compared the accuracy of econometric models with time series models

Groff (1973). Christ (1975). Narashimhan (1975), Nelson (1972). Mahmoud (1984) and

Makridakis (1986) all concluded that time series methods are generally superior to

econometric models in short term sales forecasting. The significance of the previously

investigated research is that in all the studies conducted in forecasting sales in different

industries concluded that time series models outperformed the econometric models in

sales forecasting.

Time Series Forecasting

The time series model looks for time patterns (trends, cycles, and seasonal

influences) in a single series of data and captures them in mathematical equations. These

mathematical relationships are then used to project into the future the historical time

patterns in the data. Time series models assumes explicitly that the underlying pattern can

be identified solely on the basis of historical data from that series (Crange Sc Andrew

1992). Three of time series models techniques will be used in this study; moving

average, exponential smoothing and ARIMA.

Moving Average Model

The moving average model uses a process of averaging different time series

observations to smooth (average) the random fluctuations in different points of historical

data. In the averaging process the model gives equal weights to all observations in the

time series data. This moving average model is obtained by finding the mean of the most

recent observations of the time series data and using it to forecast the next period. The

9


future forecast is simply the average of the most recent observations.

Exponential Smoothing Models

In the exponential smoothing method, the historical data (monthly occupancy

rates), are used to obtain a smoothed value for the series. That smoothed value is then

extrapolated to become the forecast for the future value of the series. Exponential

smoothing operates in a manner similar to that of moving averages by smoothing

historical observations to eliminate randomness.

Makridakis & Wheelwright (1989) reported that the exponential smoothing models

apply an unequal set of weights to past data. These weights decay in an exponential

manner from the most recent data value to the most distant value. The goal of

exponential smoothing is to distinguish between the random fluctuations and the basic

underlying pattern by "smoothing* the historical values. The estimated weights are used

to eliminate the randomness found in the historical sequence and basing a forecast on the

smoothed pattern of the data. One of the advantages of the exponential smoothing over

the moving average is that it gives more importance to recent observations by assigning

high value weights to them than those of earlier observations. This technique makes the

exponential smoothing models require less amount of historical data to forecast the next

data point than the moving average because it assumes that the next point of the series

is a function of the last point multiplied by a certain weight, in addition to a residual. In

forecasting hotel occupancy rates this will be an advantage because the most recent

occupancy rates could be used to prepare the future forecast without the need to have all

the historical time series data.

10


Crange & Andrew (1992) conducted a study to forecast restaurant sales using time

series and econometric models. They reported that the exponential smoothing method has

many advantages over the econometric models. First, the data required to build the

model is less than that of the econometric models. Time series models requires only one

variable consisting of historical data observations over a period of time such as, historical

occupancy rates. This makes time series models less expensive and less time consuming

to develop. Secondly, time series models are easy to develop. They require minimum

training in the area of quantitative methods to prepare accurate forecast. Third, the

forecasting method could be constantly repeated by minimum data entry of the latest

historical observation to obtain a new forecast without the need to revise the model as

in the case of causal econometric models if the relationship between variables changes,

the model will need to be revised and extra data collection might be needed. Last,

Andrew et al. (1990), Crange et. al (1992) and Geurts et. al (1986) support the

superiority of the time series models (Exponential smoothing and ARIMA) to other

forecasting models, namely, econometric models in forecasting sales in retail business

and the hospitality industry.

Many studies have been conducted on comparing the performance of three

different forecasting models such as moving averages, exponential smoothing and

econometric. Kirby (1966) found that in terms of month to month forecasting accuracy,

the exponential smoothing methods did the best forecast. In a study conducted by Levine

(1967), the same three forecasting methods examined by Kirby were compared. Levine

concluded that although the moving average method had the advantage of simplicity,

1 1


exponential smoothing offered the best accuracy for short-term forecasting. Other studies

reported by Gross and Ray (1965), Raine (1971) and Krampf (1972) have arrived at

similar conclusions.

Autoregressive Integrated Moving Average

The Autoregressive Integrated Moving Average (ARIMA) models, are a

specialized class of linear filtering techniques that completely ignore causal variables in

making forecasts. The ARIMA is a highly refined curve fitting device that uses current

and past values of the dependent variable to produce accurate forecasts. The ARIMA

methodology is appropriate if the observations of time series are statistically dependent

on or related to each other (Makridakis & Wheelwright 1989).

Makridakis & Wheelwright (1989) reported, the ARIMA method of forecasting

is different from most methods because it does not assume any partial pattern in the

historical data of the series being forecasted. The ARIMA uses an iterative approach of

identifying a possibly useful model from a general class of models. The chosen model

is then checked against the historical data to see whether it accurately describes the

series. The model fits well if the residuals between the forecasting model and the

historical data points are small, randomly distributed, and independent. If the specified

model is not satisfactory, the process is repeated by using another model designed to

improve on the original one. This process is repeated until a satisfactory model is found.

There are three types of ARIMA models. The selection of the proper model

depends on the patterns of time series data being used. The first type is the ARIMA

technique for stationary data in which its average value does not change over time is

12


considered auto regressive (AR). This is because the AR model depends on capturing the

relationships among different time series points and uses the weights of different

correlations over a period of time to predict the future. The second type is the moving

average (MA) model which could be applied if the data is not stationary. The MA is

used to smooth, the recent observations and predict the future. The third model is the

integration between both AR and MA to form ARIMA model. If the time series data has

a mix of stationary and non stationary data the ARIMA model is used with both

autoregressive and moving average terms.

In a study conducted by Nelson (1972), a comparison was made between ARIMA

models and structural econometric models. It was found that the ARIMA models were

superior to econometric models in fitting the data and also in forecasting. The output of

ARIMA models had less forecasting errors than econometric models. Cooper (1972)

concluded that ARIMA forecasting models can be constructed to predict economic

variables about as well as econometric models. Naylor and Seaks (1972) conducted a

study to forecast investment. They reported that the ARIMA results were significantly

better than econometric models in forecasting the future in all cases under investigation;

ARIMA provided better forecasts by a factor of almost two to one.

Other studies by Christ (1951), Steckler (1968), Cleary and Fryk (1974), Cooper

and Nelson (1975) and McWhorther (1975) concluded that ARIMA were superior to

econometric models in forecasting future business or economic activities. The suggested

reason for the better performance of ARIMA models was the inability of econometric

models to accommodate structural changes in the economy (Christ, 1951; Steckler, 1968;

13


Cooper, 1972; Nelson, 1972).

There is considerable disagreement concerning comparisons between ARIMA and

smoothing methods. Newbold and Granger (1974) indicated that the ARIMA forecasts

did better than two fully automatic forecasting models, the Holt, Winters models and one

model based on stepwise regression. A similar conclusion was reached by Reid (1969)

who reported that the ARIMA method was clearly better than exponential smoothing

model.

Groff (1973) conducted a study to forecast retail sales using time series models.

The author, arrived at a rather different conclusion. He reported that the forecasting

errors of the best of the ARIMA models that were tested were either approximately equal

to or greater than the errors of the corresponding exponentially smoothed models for

most series. Similarly, Geurts and Ibrahim (1975) conducted a study to forecast tourism

demand; they found that the exponentially smoothed models patterned on Brown's model

and the ARIMA approach seemed to perform equally well.

Crange and Andrew (1992), conducted a study to investigate the appropriateness

of various classes of forecasting models in forecasting restaurant sales. The authors used

two time series models; exponential smoothing and ARIMA versus one econometric

model. They concluded that the fitted time senes models provided a degree of accuracy

better than the fitted econometric model. In forecasting seven months of the sales ahead,

the time series models generally performed better than the econometric model. The

researchers reported that the ARIMA model was superior to the exponential smoothing

model in fitting and forecasting seven months ahead. They added that the time senes

14


models were easy to develop and use, required minimum data collection, and less

experience to build which made it ideal for the use in the hospitality industry for

forecasting.

Andrew, Crange and Lee (1990), conducted a study in which they forecasted hotel

occupancy rates in a full service city center hotel. They used time series analyses,

particularly the ARIMA model and exponential smoothing. They concluded that the

ARIMA model performed better than the exponential smoothing model in fitting and

forecasting future six months occupancy rate for a full service city center hotel. They

reported that the findings of their study could vary for different types of hotel operations,

and they suggested that this should be investigated.

Based on the findings of Andrew, Crange and Lee (1990) and Crange and Andrew

(1992) the accuracy of time series models (exponential smoothing and ARIMA) is

comparable or better than econometric models in forecasting hotel and restaurant monthly

sales. They are also more economical to develop and implement than econometric models

based on the data requirement and skills required to develop. This will make the time

series models more applicable for use in the short term of daily, weekly and monthly

occupancy rate forecasts (which is the main purpose of this study).

The significance of previously investigated studies support that the time series

models (exponential smoothing and ARIMA), are superior to econometric models in sales

forecasting. However, there is a disagreement on the superiority of exponential

smoothing or ARIMA models in forecasting sales.

The literature supports that the time series models are more accurate in

15


forecasting monthly variables in different industries like commodity prices, retail stores

sales and hotels occupancy rates. This study will use time series models (Brown’s, Holt,

Winter’s, ARIMA and moving average), because they have proven to be superior to

econometric models in fit and in forecasting of future hotel and restaurant sales based on

the findings and recommendations of Andrew, Crange and Lee (1990) and Crange and

Andrew (1992).

This study will not include econometric models because the review of literature

revealed that the majority of research that compared the performance of econometric with

time series models in forecasting proved that time series forecasts were superior to

econometric models. Steckler (1968), Groff (1973), Cleary and Fryk (1974), Christ

(1975), Cooper, Nelson (1975) Geurts and Kelly (1986) and Crange and Andrew (1992)

all concluded that time series methods are superior to econometric in sales forecasting.

The second reason behind not including the econometric models in the study is that the

main purpose of the study was to provide a simple accurate forecasting method for the

hotel managers to use in all areas of hotel operations. The econometric models need

special experience and training in the area of statistics, econometrics and economic

theory (not acquired by most hotel managers) which makes it beyond the ability of hotel

managers to use as a management tool for daily, weekly and monthly short term

forecasting.

16


CHAPTER 3

METHODOLOGY

The purpose of the study was to provide evidence of the efficiency and accuracy

of quantitative methods, specifically time series models, in forecasting hotels occupancy

rates in Egypt. Time series data of historical occupancy rates was collected from two

hotels in Egypt to fit the time series models and forecast six months beyond the fitting

horizon.

Development of Questionnaire

A questionnaire was developed to collect historical time series data of monthly

occupancy rates. The questionnaire included two sections. The first section included

questions to collect information about the hotel such as size, number of rooms, number

of restaurants, management company and forecasting techniques currently applied. The

second section consisted of tables to collect time series data of monthly occupancy rates

starting January 1991 until June 1994. Appendix A presents a sample of the questionnaire

used for data collection.

Sample

The subjects in this study were two hotels in Egypt. The first was the Cairo

Sheraton hotel and the second was the Hurgada Sheraton hotel. They were selected to

accomplish the objectives of the study according to the following criteria:

(1) Same management company to control for management techniques used in

forecasting; and to address the first objective which was to forecast monthly

occupancy rates for both hotels. Also to achieve the second objective which was to

17


find which time series model was more accurate in forecasting the occupancy in the

selected hotels.

(2) The two different locations to address the third objective of the study which was to

fmd-out if the forecast errors will be affected by changing the location of the hotel.

(3) Two different type of hotels (city center, resort) to address the third objective which

was to check if the forecasting accuracy of different models will be affected by

changing the type of hotel. This characteristic also was to accomplish objective four

which was to recommend which was the most appropriate model to be used by the

hotel managers in Egypt based on accuracy and simplicity.

18


Data Analyses

Measuring the Accuracy of Time Series Forecasting Models

The word accuracy refers to "goodness of fit," which in turn refers to how well

the forecasting model is able to reproduce the data that is already known. In econometric

models, goodness of fit measures predominate. In time series models, it is possible to use

a sub-set of the known data to forecast the rest of the known data, enabling one to study

the accuracy of forecasts more directly (Makridakis and Wheelwright, 1989).

The sum of squared errors (SSE), statistical measure of forecasting accuracy, was

used in this study to determine which model would best fit the data and forecast the

future. The most accurate model would be the one which had the lowest SSE value.

Sum of squared errors

The SSE is found simply by squaring every forecasted error and totaling all of

the squared errors. The following is the equation to obtain SSE

SSE =E [eJ3

e, = forecast error

The SSE test was used to test for goodness of fit during the process of fitting the

time series models to the data. The best model that fitted the data was the one that had

minimum SEE value. Similarly the same test was used to measure the accuracy of

forecast, and the model that forecasted six months ahead with lowest SSE was the most

accurate.

19


Alternative Time Series Models Formulation

Makridakis and Wheelwright (1989) explained the model formulation of different time

series models as follows:


Ft*i= S, X i - : (1-1)N

Where

F,*, * forecast for time t + 1

S, = Smoothed value at time t

N = The number of observations included in average

This means that once the forecast for the period t (that is F), was obtained could

obtain the forecast for period t+1 by adding X/N and subtracting X ^ N -

The value of equation (1.1) was called the moving average model and could be modified

to obtain

Ft.,= 2 L * 2 U + F. (1.2)N N

Written in this form, each new forecast based on a moving average was an adjustment

of the preceding moving average forecast. It is also easy to see why the smoothing effect

increased as N became larger, a much smaller adjustment was being made between each

forecast.

In this study, a double moving average was used to smooth (average) the random

fluctuation, trend and seasonality in the data. The first 12 observations were added and

20


divided by 12 to get the the second year values, Then the twelve obtained observations

of the second year were averaged again to get the next values.

Exponential Smoothing Models

The technique of exponential smoothing could be developed using Equation 1.2

If one only had the most recent observed value and the forecast was made for the same

period, equation 1.2 could be modified so that in place of the observed value in period

t-N one could employ an approximate value. A reasonable test would be to forecast the

value from the preceding period. Thus equation 1.2 could be modified to give

F«*,=& + E. + F, (1.3)N N

This equation can be written as

Ft. l =_LXt + ( l - J J F t (1-3.a)N N

What is presented in Equation 1.3.a is a forecast that weighed the most recent

observation with a weight of value 1 / S and the most recent forecast with a weighted

value of (1JJ.N

If we substitute the symbol alpha in place of 1/N, the following is obtained

F ,^ * o X, + (1 • o) F, (1.4)

This equation is the general form used in computing a forecast by the method of

exponential smoothing. This equation corrects one of the problems associated with

moving average in that extensive historical data is no longer needed to be stored. Rather,

only the most recent observation, the most recent forecast, and a value of a are required

to prepare a new forecast. If Equation 1.4 is expanded by substituting the value for Ft

21


which is equal to F,= a X,., + (1 - a) F,., (the value of one iag behind), we get

Ft«.j= a X, + (1 - a) [a X*, + (1 - a) Fwl]= a X, + o ( l - a)X,., + (1 - a)2 F„ (1.5)

However,F,*,= a Xt.2 + (1 - a) F,.j

If this substitution process is carried out even further, we obtain the relationship

F„ , = o Xt + a (l - or) + (1 - a)2 F„2 + (1 • o)J F„2

Brown’s One parameter model

F,«.j = a X, + a(l • oOX,., + (1 * a)2 X,.2 +

(1 - a )J X,, + (1 - a)4 X ^ (1.6)

From this equation it can be seen that Brown's exponential smoothing model gives

decreasing weights to older observed values; that is , since a is a number between 0 and

1; thus (1-a) is also a number between 0 and 1. the weights a , a (l-a ), a (l-a )2, etc. have

exponentially decreasing values. Hence, the name exponential smoothing is used because

it gives less importance for the old data observations to forecast the next data point.

The method of single parameter exponential smoothing is appropriate when the

data series contains a horizontal pattern and does not have a trend. If Brown's model is

used with data series that contain a trend, the forecast will trail behind (lag) that trend.

To solve this problem two parameters model are used to take into account the presence

of a trend.

22


Holt’s two parameters model

T,= B(S, - S,.|)+ (l-B)Tt., (1.7)

where

S, = Equivalent of single exponential smoothed value

B = Smoothing coefficient, analogous to a

T, = Smoothed trend in data series

The principle underlying Equation 1.7 is the same as that of single exponential

smoothing represented by Equation 1.4. The most recent trend, (St-Swl) is weighted by

B and the last smoothed trend T,.,, is weighted by (1-B). The sum of these weighted

values is the new smoothed trend value. Holt’s two parameter model uses equation (1.7)

to obtain a smoothed value of the trend and combines this trend with the standard

smoothing equation to obtain

S,= a X, + (1-a) (Sv,+ T wl) (1.8)

The difference between Equation 1.8 and equation 1.7, is the additional term TM

that is added to S,., to adjust the smoothed values for the trend pattern in the data series.

The model will give weights to the data series that has a trend to remove the impact of

that trend. The best value of a and B are found by trying various combinations of values

between 0 and 1 and then selecting the set values that minimizes the SEE. Holt’s model

is superior to Brown's model only if the time series data has a trend. However, if the

data has both trend and seasonal variation Brown’s model will not be able to smooth the

impact of seasonality in the data and this will result in high forecasting errors due to the

seasonality in the data.

23


The data was fitted to Holt’s two parameters model in order to get the best

fit.(Recall that the best fitted model was the one that had minimum SSE). The value of

or and 6 coefficient was repetitively changed until minimum SEE was reached.

W inter’s three parameters model

Winter’s three parameter exponential smoothing model had an extra advantage of

being capable of dealing with seasonal data that has a trend. This model was based on

three equations, each equation smoothed one of the three components: randomness, trend,

or seasonality. The following are equations representing Winter's three parameters model

Equation 1.9 which attempts to smooth the randomness in the data. Equation 1.10

attempts to smooth the trend of the time series data and Equation 1.11 attempts to smooth

the seasonal factor in the time series data.

S ,- <O L+ ( l - a X ^ + T ^ ) (1.9)

T,« B(S, - SVI)+ (i-8)TM (1.10)

it* e _&_+ (i-0) (i.ii)s,

whereS ■ smoothed value of deseasonalized seriesT ■ Smoothed value of trendI ■ Smoothed value of seasonal factorL ■ length of seasonality

Winter's model smoothed each component of the time series separately. This

model was ideal when analyzing data that contained trends and seasonality. This model

used three different coefficients to smooth the data that has a trend and seasonality. First,

24


it gave exponential values between 0 and 1 to every observation of the data series a to

smooth the randomness of the data, then it used B to remove the trend effect from the

data and 0 to smooth the seasonality. The integrated three equations (1.9,1.10,1.11)

treated the randomness, trend and seasonality in the data to transform the data into a

stationary data in order to give the best fit. The data was fitted to Winter’s three

parameters model in order to get the best fit. The best fit is the one with the minimum

SEE.

Exponential Smoothing Model Identification

To identify the exponential smoothing models, the data was plotted to detect

different patterns (trend, cycles or seasonality). The data of 36 months (January 91 to

December 93) for both subjects were entered in to the computer and plotted to get a time

series plot. Six months of the data (January 94 to June 94) were held in reserve to be

forecasted and to be used as a test of the accuracy of different time series models. Figure

1 presents a time series plot for Cairo hotel and Figure 2 presents time series plot of

Hurgada data.

25


CAIR

O (X

CIJI

'ANC

Y UA

TKS

70-

3 0 -

8 15 22 29 36

JANUARY 9 1 TO DECEMBER 9 3

Figure 1. Time Series Plot for the Cairo Hotel.

Figure 1 indicated that the occupancy for the Cairo hotel is characterized with

increasing trend in the first 12 months until the occupancy reached over 90 percent The

second 12 months are characterized with stability of the occupancy between the range of

90 and 70 percent. The third 12 months are characterized with increasing and decreasing

random shocks (random fluctuations). The change of the behavior of the time senes data

in the three years is due to the different types of market segments in the Cairo hotel

operation. As was indicated in the questionnaire, it is a full service city center hotel,

which gives it the advantage of targeting different market segments. The hotel uses

26


different marketing techniques to approach different markets, this cause the occupancy

not to vary in different seasons. This is accomplished by replacing one market segment

with another market segment (for example, travel with corporate) to keep the hotel

occupied at a stable rate.

8 0 -

20

29 36228 151JANUARY 9 1 TO DECEMBER 9 3

Figure 2. Time Series Plot for the Hurgada Hotel.

Figure 2 presented the a time series plot for the Hurgada hotel. The occupancy

is characterized with constant peaks and valleys every 12 months in the occupancy rate

except the last 12 months have many random shocks. These peaks and valleys are

existent almost in every year due to changing seasons because as it was reported in the

questionnaire the hotel depended only on the travel market which is fluctuating according

to different seasons.

27


Autoregressive Integrated Moving Average Models Formulation

Autoregressive Model

An autoregressive (AR) model takes the form

Y t = 8, Y t _, + Bj Y t2 + . . . + 6P Y tp + e, (1.12)

Where:

YT — dependent variable (future occupancy rate)

Y t i . Y T 2, Y T.p = independent variables that are dependent variables lagged at specific time periods.

8, , 6;, 8p= regression coefficients

eT = residual term that represents random events not explained by the model

Equation 1.12 introduced autoregressive models. In this equation the regression

coefficients were found by using the nonlinear least squares method which used an

iterative solution technique to calculate the parameters rather than using direct

computation. Preliminary estimates were used as starting points: then these estimates

were systematically improved until optimal values were found. The optimal value is the

one that has the best fit. (minimum SSE). Furthermore, the variance for equation 1.12

was calculated in a manner that takes into account the fact that the independent variables

were correlated with each other. Finally, equation 1.12 does not contain a constant term.

This approach was used because the dependent variable values (Ys) are expressed as

deviations from their mean.

Seasonal Autoregressive Model

A seasonal autoregressive (SAR) model takes the form

Yt = 8, YT1J + Bj YT lJ + . . .+ Bp Yt t + Cr (1.12.a)

28


The model in Equation (1.12.a) is similar to AR model but is adjusted according

to the season. In this case the data was considered monthly seasonal data.

Unfortunately, not all data series can be handled with autoregressive models. For

this reason the Box-Jenkins (1976) approach also uses the moving average model.


A moving average model (MA) takes the form

Yt = Ct - WUT., - - ....... - W ^ (1.13)

Where:

Yt = dependent variable (future occupancy rate)

W „W 2,W, = weights decaying exponentially

Cj = residual term that represents random events not explained by the model

Equation 1.13 is similar to Equation 1.12 except that the dependent variable YT depends

on previous values of the residuals rather than on the variable itself.

Moving average models provide forecasts of YT based on a linear combination of

past errors, whereas autoregressive models express YT as a linear function of some

number of actual past values of YT. It is customary to show the weights with negative

coefficients, even though the weights can be either positive or negative. The sum of W,

+ W2 + .......+ W, does not need to equal 1 and the values of W, are independent

weights estimated by the model not depending on moving average of the previous

observations to obtain a new observation as they are with the moving average

computation.

29


Seasonal Moving Average Model

A seasonal moving average model (SMA) lakes the form

Yt= Cj - W UT.l2 - - .......- (1.13.a)

The model in Equation 1.13.a is similar to the AR but adjusted for the monthly seasonal

data.

Autoregressive Integrated Moving Average Model

In addition to AR and MA models, the two can be mixed, providing a third class

of general models called ARIMA. Equation 1.12 and 1.13 were combined to form:

Yt = 6, Yt ., + Bj Yt_2 + . . .+ BP YT.P + tx

- W UT., - W * „ - .......-W „T„. (1.14)

Seasonal Autoregressive Integrated Moving Average Model

In addition to SAR and SMA models, the two can be mixed, providing a third

class of general models called SARIMA. Equation 1.12.a and Equation 1.13.a were

combined again to form:

YT= B, Yj.jj + B} Yt.u + . . .+ 8P YTP +

- W lcT1J - W ^ , , - ........... - W ^ . (1.14.a)

ARIMA (1,1,1,1) means that this model has AR, SAR, MA and SMA terms. In

ocher words the SARIMA model is the seasonal auto regressive integrated moving

average model. Similarly the ARIMA (1,1) model uses combinations of past values and

past errors and offer a potential for fitting models that could not be adequately fitted by

using an AR or an MA model separately.

30


Autoregressive Integrated Moving Average Models Identification

Selection of an appropriate model of autoregressive integrated moving average

(ARIMA) could be made by comparing the distributions of autocorrelation coefficients

of the time series being fitted with the theoretical distributions for the various models,

this was done to decide if the model is a moving average, seasonal moving average, auto

regressive, seasonal autoregressive or, ARIMA. Before the selection of the model, the

data was transformed by seasonally differencing to remove the seasonal variation and

non-seasonally differencing to remove the trend.

Autocorrelation

Autocorrelations and partial autocorrelations were used to heip identify an

appropriate ARIMA model for forecasting. They allowed the analyst to identify the

degree of relationship between current values of a variable and earlier values of the same

variable while holding the effects of all other variables (time lags) constant.

The degree of the relationship between variables was measured by the correlation

coefficient which varies between +1 and -1. A value close to +1 implied a strong

positive relationship between the two variables. This meant that when the value of one

variable increased, the value of the other tended to increase also. Similarly, a correlation

coefficient close to • 1 indicated the opposite; an increase in one variable was associated

with a decrease in the other. A coefficient of 0 indicated that the two variables were

unrelated. An auto correlation coefficient is similar to a correlation coefficient except that

it describes the association (mutual relationship) among values of the same variable but

at different time periods (Makridakis Sc Wheelwright 1989).

31


The data was plotted to get autocorrelation plot. A seasonal fluctuation and trends

were detected in the data which needed a transformation of seasonally and non-seasonally

differrtncing to remove the seasonality and trend. Partial autocorrelation then was

obtained to verify that the data became stationary before fitting it to one of the ARIMA

models.

- 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAG

12345678 9

1011

CORK. ♦ -------- «•-------- ♦ -------- ----------0 . S 8 8 >0 . 3 0 5 >0 . 1 3 6 >0 . 0 8 0 >0 . 0 2 5 >0 . 0 7 8 >

- 0 . 0 4 1 >- 0 . 0 9 1 >- 0 . 0 5 1 >- 0 . 1 0 2 >

- 0 . 0 9 5 >

• ft ft ftft**f t *

• ft* ftft

ftftft • ft

• •ft* ftftft

MEAN OF THE S E R I E S 7 3 . 6 5 0 6STD . DEV. OF S E R I E S 1 7 . 8 4 2 6NUMBER OP CAS ES 36

Figure 3. Autocorrelation Plot For The Cairo Hotel.

Figure 3 presented autocorrelation plot of the Cairo hoed. It indicated that the

values of correlations were decreasing exponentially until a point where they reverted.

This indicated that the data fit one of the seasonal moving average modds of ARIMA.

32


- 1 . 0 - 0 . 3 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAC CORR. --------- -- ------«.-------- ---------- ♦ _ -------------- ------------+ -------- +

1 0 . 5 3 8 >2 - 0 . 0 6 2 > * * * <3 - 0 . 0 2 8 > • * <4 0 . 0 3 9 > • * <5 - 0 . 0 4 1 > • • <6 0 . 1 1 9 > • • • • <7 - 0 . 2 0 1 <8 - 0 . 0 0 3 > • <9 0 . 0 7 4 > * * * <

10 - 0 . 1 6 3 > * <11 0 . 0 6 2 > • • • <

MEAN OF THE S E R I E S 7 3 . 6 5 0 6STD. DEV. OF S E R I E S 1 7 . 8 4 2 6NUMBER OF CASES 36

Figure 4. Partial Autocorrelation for the Cairo Hotel

Figure 4 presented partial autocorrelation plot for the Cairo hotel, it showed the

relations between specific lags holding all others constant; it indicated that after seasonal

and non-seasonal differencing transformation the relationship among data points became

random and valid to be fitted using the ARIMA model.

33


- 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAG CORR. ———f

1 0 . 3 5 9 > * * • * • • * * > *2 0 . 2 5 2 > * * * * * * * <3 0 . 0 6 5 > * * * <4 - 0 . 0 3 6 > * * <5 - 0 . 3 0 1 > * * * * * * * * * <6 - 0 . 3 5 7 > * * * * * * * * * * <7 - 0 . 1 7 7 > * * * * * <8 - 0 . 2 1 5 > * * * * * * <9 0 . 0 2 5 > * * <

1 0 0 . 0 0 7 > * <1 1 0 . 1 7 3 > * * * * * <

KEAN OF THE S E R I E S 6 2 . 1 3 8 9STD . DEV. OF S E R I E S 1 7 . 4 8 3 6NUMBER OF CASES 3 6

Figure 5. Autocorrelation Plot for the Hurgada Hotel.

Figure 5 presented the autocorrelation plot for the Hurgada hotel. The correlation

for different time lags indicated that the first three observations are related and the

correlation decrease exponentially. Then the relation reverts into a negative correlation

with random increases and decreases in the coefficient, this indicated that the best

ARIMA model to fit the data was the integrated autoregressive moving average ARIMA

model.

34


LAC 1 2345678 9

10 11

KEAN OF THE S E R I E S 6 2 . 1 3 8 9STD . DEV. OF S E R I E S 1 7 . 4 8 3 6NUMBER OF CASES 36

Figure 6. Partial Autocorrelation for the Hurgada Hotel.

Figure 6 presented partial autocorrelation plot; it showed the relationship among

specific lags holding all others constant. It indicated that the relationship between

different lags was a random one and the data could be analyzed using ARIMA models.

The autocorrelation test was performed to identify which ARIMA model best fit the data

for both hotels. The ARIMA models were estimated using unconditional ordinary least

square regression. The t-test and the goodness of t test (p-value) were used to test the

validity of the selected coefficients of ARIMA models.

35


CORR. 0 . 3 5 9 0 . 1 4 1

- 0 . 0 7 4 - 0 . 0 8 3 ■ 0 . 3 0 7 ■ 0 . 2 0 7 0 . 1 1 9

- 0 . 0 9 1 0 . 1 6 9

- 0 . 0 8 0 0 . 0 1 6

• 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 u . o w . o i . . w

* * * * * * * *

• •* * * *

* • • *

• * *

CHAPTER 4

RESULTS AND DISCUSSION

The purpose of this study was to help provide the evidence of the efficiency and

accuracy of time series methodologies in forecasting hotels occupancy rates in Egypt.

Five time series models were used to fit time series data starting January 1991 until

December 1993 and to forecast six months ahead.

Characteristics of the Sample

A questionnaire was developed to collect monthly occupancy rate form two hotels

in Egypt. The results of the data collected reported that the forecasting process was

conducted by the sales, front office or reservation departments. The two hotels reported

that they did not use any quantitative methods in forecasting and that the forecasts were

prepared based on actual reservation figures adjusted by judgement. Both hotels were

different in size, type of operation and major market segmentation. The Cairo hotel was

a large city center hotel with 688 rooms, nine restaurants and three bars. The Hurgada

hotel was a resort hotel with 119 rooms, two restaurants and two bars. The major market

segment for Cairo was a mix of corporate, government and travel business. In the

Hurgada hotel travel was the major market segment which was due to its location in a

remote resort area.

36


Time Series Models Fit

Brown’s model

The data was first fitted to Brown’s model in order to get the best fit. The best

fit is the one that had minimum sum of squared errors SSE. The value of a coefficient

was changed repeatedly until minimum SSE was reached. The model that had the best

fit was(cr = 0.67) for the Cairo hotel and (a = 0.48) for the Hurgada hotel. These were

the models used to forecast six months ahead.

Holt’s model

The data was fitted to Holt’s model in order to get the best fit. The best fit is the

one that had minimum sum of squared errors SSE. The value of a , B and parameters

was repetitively changed until minimum SSE was reached. The Holt's model that had the

best fit was (a = .67, 8 = 0.01) for the Cairo hotel and (a = 0.48, 8 = 0.01) for the

Hurgada hotel. The fitted models were used to forecast six months ahead.

W inter’s model

The data was fitted to Winter’s model in order to get the best fit. The best fit is

the one that had minimum sum of squared errors SSE. The value of a , 8 and 0

coefficient was repetitively changed until minimum SSE was reached. The model that had

the best fit was (o = .67, 8 = 01, 0 = .99) for the Cairo hotel and (a = 0.48, 8 =

0 .01, 0 * 0.99) for the Hurgada hotel. These model were used to forecast six months

ahead.

37


ARIMA Model

The ARIMA model estimation was performed using ordinary least squares. The

coefficients were tested for significance using t-test and p-test. As presented in Table 1,

the ARIMA model for the data collected from the Cairo hotel proved to be ARIMA

(0,0,0,1). This means that only seasonal moving average (SMA) coefficient was used in

the model with a t-value of 39.56 and a p-value of 0.0000. The autoregressive (AR),

moving average (MA) and seasonal autoregressive (SAR) were not used in the model

because they proved not to be significantly different from zero.

Table 1. Unconditional Least Squares ARIMA Model for the Cairo Hotel

TERM COEFFICIENT STD ERROR T VALUE P VALUE

SMA 1 1.00985 0.02553 39.56 0.0000

Table 2 presents the result of ARIMA model for Hurgada hotel time series data,

the ARIMA model proved to be ARIMA (1,1.0.0). This means that only AR and MA

coefficients were used in this model. The parameters SMA and SAR were not used in

the model because they proved not to be significantly different form zero. The t-value

for AR was -3.29 and the p-value was 0.0010. For the MA t-value was 27.04 and p~

value was 0.0000.

38


Table 2. Unconditional Least Squares ARIMA Model for the Hurgada Hotel

TERM COEFFICIENT STD ERROR T VALUE P VALUE

AR 1 - 0.47159 0.14320 -3.29 0.0010

MA 1 0.94184 0.03484 27.04 0.0000

Appendix B presents time series plots of actual occupancy rate versus the best fit

of different time series models used to forecast the occupancy for the Cairo hotel.

Appendix C presents a lime series plot of actual occupancy rate versus the best fit of

different time series models used to forecast the occupancy for the Hurgada hotel.

39


Time Series Models Forecast

Six months of the collected data (January 1994 to June 1994), were reserved to

be compared with the forecast of the different models. The six months were forecasted

by different time series models. The difference between the actual occupancy and the

forecasted was measured using SSE test to measure the accuracy of forecasts. The most

accurate model in forecasting was the one that had minimum SSE for forecast.

Table 3 presents the results of the Cairo hotel and Table 4 presents the results of

the Hurgada hotel. Both tables presents actual six months occupancy rates from January

1994 untill June 1994 and the forecast of the five different time series models for the

same period.

The first objective of the study was successfully achieved as indicated in the

results presented in Table 3 and Table 4, the results are very close to the actual

occupancy rates. Most of the values were within 10 points deviation from the actual

occupancy rate. These deviations were within the acceptable range for the hotel industry

and considered very accurate compared to the existing judgmental forecasting errors in

hotels. The results presented in Table 3 and Table 4 indicated that the forecast of the

time series models was accurate and could be used to forecast hotels occupancy rates in

Egypt The time series models were able to smooth the random variations, trends, and

seasonality in the data to forecast six months occupancy rates with minimum deviations

from the actual occupancy figures.

40


Reproduced

with perm

ission of the

copyright ow

ner. Further

reproduction prohibited

without

permission.

Table 3 Six Months Actual Occupancy versus the Forecast ofTiroe Series Models for the Cairo Hotel.

Month Actual

Occupancy

Models

ARIMA

F R

BROWN

F R

HOLT

F R

MVAV

F li

WINTRR

F R

Jan 72 50 59 50 13 00 58 40 14 10 61 93 10 57 54 35 18 15 58 55 13 95

Feb 45 70 58 00 -12 30 56 14 -1044 62 04 -16 34 52 88 -7.18 63.12 -17 42

Mar 52 00 56 50 i o 53 89 -1 89 62 41 -1041 5141 0 59 36.21 15.79

Apr 57 20 55 00 2 20 51 63 5.57 62 25 -5 05 49 94 7 26 64 44 -7 24

May. 61.40 53 50 7 90 49 38 1202 62.35 -095 48 47 12 93 60.90 0 50

Jun. 48 20 52 00 -3 80 47 13 1 07 62 16 -14 26 47 00 1 20 57 37 -9 17

F~ Forecast

F-a F.rror

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Table 4 Six Months Actual Occupancy versus the Forecast o f Time Series Models for the Hurgada Hotel

Month Actual Models

Occupancy ARIMA UROWN HOLT MVAV W IN! RR

F H F I F R F R F R

Jan 70 00 74.25 -4.25 64 29 5 71 65 68 4 32 52 30 17.70 43 03 26 97

Feb. 72 00 67 75 4 25 63 63 8 37 65 69 6 31 51 48 20 52 56.21 15.79

Mar 67 00 71.97 -4.97 62 98 4 02 65 71 1 29 5065 16 35 37.20 29.80

Apr 66 00 71 14 -5.14 62 33 3 67 65 72 0 28 49 83 16.17 61 44 4 56

May. 52 00 72 70 -20 70 61.69 -9 69 65.74 -13.74 49 00 3.00 43.73 8 27

Jun 470 0 73 12 -26 12 61 0 3 ■1403 65 75 -18 75 48 18 -1 18 54.74 -7.74

Fra Forecast

H“ Hrror

To address the second objective of the study, the accuracy of forecasts was

measured for each model using the SSE test. The results for the Cairo hotel are presented

in Table 5 and the results for the Hurgada hotel presented in Table 6. The most accurate

time series model to forecast the occupancy for the Cairo hotel was the ARIMA model.

It had the minimum SSE for forecasts which was (422.20). This result is consistent with

the findings of Andrew, Crange and Lee (1990) who conducted a similar study to

forecast hotel occupancy rate in a full service city center hotel using different time series

models. The authors reported that the ARIMA model was the best time series model to

forecast hotel occupancy rate.

The results were different for the Hurgada hotel. The most accurate model to

forecast the occupancy rates was Brown's exponential smoothing which had the minimum

SSE for forecasts equal to (423.00). This indicates that when changing the type of hotels

from a city center to resort hotel the ability of different time series models to forecast

the occupancy changed. This is because each hotel data is characterized with a certain

pattern that can be best captured using the specific model that can handle this specific

type of data.

43


73-

ocU 66' p A N

59 •Y

RAT 5 2 iES

4 5 -

21 3 54 6

- A C T U A L

- A R IM A

- B R O W N

- M V A V

JANUARY 94 TO JUNE 94

Figure 7. The Forecast of the Best Models to Forecast for the Cairo Hotel

Figure 7 summarizes the forecasts of the best three models used to forecast the

occupancy for the Cairo hotel. Although the actual occupancy had random shocks

(unexpected fluctuations), the ARIMA, Brown and moving average time senes models

were able to forecast accurately. The most accurate model to forecast was the ARIMA

(SEE - 422.20), Brown’s exponential smoothing model came next and very close to

ARIMA (SEE=488.00). The third most accurate model was the moving average

(SEE=602.70).

44


occupANCY

RATES

75

68

61

54

47

cw421 J

- a c t u a l

-• BR OW N

- HOLT

- ARIMA

JANUARY 94 TO JUNE 94

Figure 8. The Forecast of the Best Models to Forecast for the Hurgada Hotel

Figure 8 summarizes the forecast of the best three models used to forecast the

occupancy for the Hurgada hotel. The actual occupancy was decreasing exponentially.

The most accurate model to forecast for six months ahead was the Brown's exponential

smoothing (SEE=423.00). Holt’s exponential smoothing model was next (SEE=600.60),

which was followed by the ARIMA (SEE= 1198.00). Brown's model was the only model

to respond to the exponential decrease in the actual occupancy.

The results of this study indicate that the accuracy of time series models

(forecasting errors) change by changing the location, type of hotel operation and market

45


segmentation. There is no single time series model that can be generalized to be used

based on its accuracy in forecasting hotel occupancy rates. The results suggest that

different models must be investigated for different types of hotels until the most accurate

model that fits a specific hotel is found. Then this model can be used to forecast the

occupancy rates for that particular hotel. This is because the patterns in the data for

different types hotel operations are different. The patterns can be best smoothed only by

using the specific model that was built to capture these patterns and use it to forecast the

future. This is supported by the findings of Makridakis (1986) who indicated that no

single method is superior for all accuracy measures and forecasting horizons.

HVAV BROWN BOLT W INTO ARIMA

Figure 9. SSE of Time Series Model Forecasts for the Cairo Hotel.

46


Figure 9 indicated that the ARIMA model was the most accurate model to forecast

the occupancy rates for the Cairo hotel. Brown’s model was next which was followed by

the moving average model. The plot showed that although Brown’s model came second

in accuracy, it was very close in its prediction to the ARIMA model.

2500

2000 J

1500-i'

SSE

1000

500

MVAV BROWN HOLT WINTER ARIMA

Figure 10. SSE of Time Scries Model Forecasts for the Hurgada Hotel.

Figure 10 indicated that Brown’s model was the most accurate model to forecast

the occupancy rates for the Cairo hotel. Holt’s model was next which was followed by

the ARIMA model. The plot showed that although the ARIMA model came third in its

accuracy, the simple moving average model was very close to the performance of the

47


ARIMA model in forecasting occupancy rates.

The results presented in Table 5, Table 6, Figure 7 Figure 8 Figure 9 and Figure

10 indicated that the accuracy of the most simple methods such as simple exponential

smoothing (Brown’s model), is comparable to or better than the most sophisticated time

series methods (Winter’s & ARIMA models). This is good news for the hospitality

industry because the simple methods used in this study are accurate and easy to develop.

This advantage makes it applicable in the area of forecasting for hotel occupancy rates

in Egypt. Hotels of any size can implement the simple time series models by using a

personal computer, simple statistical analyses software and minimum training for the

person preparing the forecast.

48


Table 5. Sum of Squared Errors of Forecast (SEE) for the Cairo Hotel.

Moving average Brown Holt Winter ARIMA

602.70 488.00 716.80 884.14 422.20

Table 6. Sum of Squared Errors of Forecast (SEE) for the Hurgada Hotel.

Moving average Brown Holt Winter ARIMA

1274.00 423.00 600.60 2013.80 1198.00

49


CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

For forecasting monthly occupancy rates in a selected sample of two hotels in

Egypt, the ARIMA model proved to be the most accurate model in forecasting six

months ahead in a full service city center hotel. In forecasting the same period for the

resort hotel, Brown’s exponential smoothing model outperformed all other time series

models.

In general, the fitted (moving average. Brown’s and Holt) simple time series

models performed close to or better than most sophisticated time series (ARIMA and

Winter’s) models. These results are significant in the fact that simple time series models

are easier to implement using very limited resources such as a personal computer and a

time series software to produce accurate forecasts for hotels.

The study indicates that the accuracy of forecasts varied for different types of

hotels. The more sophisticated method ARIMA. was more accurate in forecasting the

occupancy rate in a full service city center hotel, but did not perform as accurately for

a resort hotel. The time series simple models (Brown's and Holt’s) outperformed the

more sophisticated models (ARIMA) to forecast for the resort hotel. The findings of this

study indicate that the accuracy of different time series models vary by changing the hotel

type, location and market segment. This is due to changing patterns in the data of every

hotel. This will raise the question “which of the investigated models is the most accurate

in forecasting hotel occupancy rates ?’ . To answer to this question, the model that can

best smooth the variability and capture the trends of the time series data will be the one

50


that most accurately forecasts the future. There is no specific model that can be used in

all forecasting situations. Different time series models must be investigated first to find

which is the most accurate model to forecast the data of a specific type of hotels. Once

the model is found it could be implemented to regularly forecast occupancy for that type

of hotels. Monitoring the model is very important to make sure that the patterns in the

data being forecasted are consistently repeated over the time and do not turn around in

a reverse trend because this will affect the ability of the model being used to forecast.

This study is significant because the simple time series models can be prepared

with a minimum data entry, minimum data collection (historical data of one variable) and

minimum training to prepare accurate forecasts and use it as a management tool for better

planing in all ares of the hospitality industry. The empirical evidence provided in this

study on the accuracy of time series applications in the hotel industry should be given a

serious consideration by the hospitality educators. It might be useful to adopt this method

of forecasting due to its importance and cost effectiveness as a management tool in the

hospitality industry.

Future research should be focused on investigating the monthly occupancy rates

when disaggregated into its components by weekly occupancy. Weekly occupancy rates

should then be forecasted and aggregated to re-establish the monthly occupancy rates.

This technique could be applied to determine weather or not there will be an increase in

the accuracy of time series models. In another area for future research, it would be

valuable to investigate the occupancy time series data when disaggregated into its

components by market segments (conference, walk in, day use, travel). Each segment

51


should be forecasted using time series models and then aggregated to form the forecasted

occupancy rates. This technique could enhance the accuracy of the time series forecasting

models.

The time series models used in this study were only investigated in the area of

occupancy forecasting. For future research, one might apply the time series models in

different areas of the hotel to investigate their accuracy in forecasting other variables like

inventory levels, table turnover in the food and beverage outlets, material requirement

planning for food production, and forecasting power needs such as energy usage in

hotels.

52


References

Andrew, W. P., Crange, D., & Lee, K. (1990). Forecasting hotel occupancy rates with time series models: An empirical analysis. Hospitality Research Journal. 14,(2), 93-127.

Adam, E. E. and Ebert, R. J. (1976). A comparison of human and statistical forecasting.A l H t l a n s . , S. 120-127.

Armstrong, J. (1983). Relative Accuracy of judgmental and extrapolation methods in forecasting annual earnings. Journal of Forecasting. 2, 437-447.

Box, G. E. P. and Jenkins G. M. (1976). Time series analysis forecasting and control (rev. ed.). San Francisco, CA: Holden-Day.

Binroth, W., Burshtein I., Haboush, R. Sc Hartz J. (1979). A comparison of commodity price forecasting by Box-Jenkins and Regression - based techniques Technological Forecasting and Sociai Change. ]4 , 169-180.

Cerullo, M. J. and Avila, A. (1975). Sales forecasting practices-a survey. Managerial Planning. September/October, 33-39.

Christ, C. F. (1951). A test of an Econometric model of the United States. 1921-1974. In Conference on Business Cycles. New York: National Bureau of Economic Research.

Christ, C. F. (1975). Judging the performance of econometric models of the U.S. economy. International Economic Review. 1£,(1), 57-81.

Clearly, J. P. and Fryk, D. A. (1974). A comparison of ARIMA and econometric models for telephone demand. Proceedings of American Statistical Association. Business and Economic Section, 448-450.

Cleveland, W. P. and Tiao, G. C., (1976). Decomposition of seasonal time series: A model for the census X-H program. Journal of American Statistical Association. 71. 581-587.

Cooper, R. L. (1972). The predictive performance of quarterly econometric models of the U.S. Conference on Econometric Models of Cyclical Behavior. New York: National Bureau of Economic Research.

53


Cooper, J. P. and Jergenson, D. W. (1969). The predictive performance of Quarterly econometric models of the U.S. Conference on Econometric Models of Cyclical Behavior, Harvard University, November, 14-15.

Cooper, J. P. and Nelson, C. R. (1975). The ex ante performance of the St. Louis and FRB-MIT-PENN econometric models and some results on composite predictors. Journal of Money. Credit and Banking. February, 1-32.

Crange, D. and Andrew, P. (1992). A comparison of time series and economic models for forecasting restaurant sales. International Journal of Hospitality Management. 11.(2), 129-143.

Dalrymple, D. J. (1975). Sales forecasting methods and accuracy. Business Horizons. 5, 69-74.

Geurts, M. D. and Kelley, J. P. (1986). Forecasting retail sales using alternative models. International Journal of Forecasting. 2. 261-272.

Geurts. M. D. and Ibrahim, I. B. (1975). Comparing the Box-Jenkins approach with the exponentially smoothed forecasting model application to Hawan tourists. Journal of Marketing Research. 12. 182-188.

Granger. C. W. J. (1969). Investigating causal relations by econometric models and cross spectral methods. Economctrica. 32 , 424-438.

Groff, G. K. (1973). Empirical comparison of models for short range forecasting. Management Science. 200). 22-31.

Gross, D. and Ray, J. L. (1965). A general purpose forecasting simulator.Management Science. fiU(6). 119-135.

Hollier, R. H., Khir, M. & Storey, R. R. (1981). A comparison of short-term adaptive forecasting methods. OMEGEA International Journal of Management Science. 2. 96-98.

Hogarth, R. M., (1975). Cognitive processes and the assessment of subjective probability distributions. Journal of The American Statistical Association. 7Q, 271-290.

Kirby, R. M. (1966). A comparison of short and medium range statistical forecasting methods. Management Science. B15(4), 202-210.

Krampf. R. F. (1972). The turning point problem in smoothing models. Unpublished Ph.D. thesis. University of Cincinnati.

54


Lorek, K. S., McDonald, C. L. & Patz, D. H. (1976). A comparative examination of management forecasts and Box-jenldns forecasts of earnings. Accounting Review. 2, 321-330.

Levine, A. H. (1967). Forecasting techniques. Management Accounting. January, 86-95.

Libby, R. (1976). Man versus model of man: some conflicting evidence. Organizational Behavior and Human Performance. 4, 1-12 .

Mahmoud, E. (1984). Accuracy in forecasting: A survey. Journal of Forecasting. 2,139- 159.

McWhorter, A. Jr. (1975). Time series forecasting using the kalman filter: An empirical study. Proceedings of the American Statistical Association. Business and Economic Section, 436-446.

Makridakis, S. and Hibon, M. (1979). Accuracy of forecasting: An empirical investigation. The Journal of the Roval Statistical Society. 2, 97-145.

Makridakis, S. (1986). The art and science of forecasting: an assessment and future directions techniques. International Journal of Forecasting. 2. 15-39.

Makridakis, S. and Wheelwright S. (1989). Forecasting Methods for Management. New York, John Wiley & Sons.

Maibert, V. A. (1975). Statistical versus sales force-executive opinion short-range forecasts: a time series analysis case study. Krannert Graduate School, Perdue University (working paper).

Narashimhan, G. V. L. (1975). A comparison of predictive performance of alternative forecasting techniques: time series models vs. an econometric model. Proceedings of the American Statistical Association. 1, 459-464.

Nelson, C. R., (1972). The prediction performance of the FRR-MIT PENN model of the U.S. Economy, American Economic Review. 62. December, 902-917.

Newbold, P. and Greanger, C.W.J. (1974). Experience with forecasting univariate time series and the combination of forecasts. Journal of Roval statistical society. 122(2). 131-165.

Nylor, T. H. and Seaks, T.G. (1972). Box-Jenldns methods: an alternative to econometric models. International Statistical Review. 40(2). 113-137.

55


Poulos, L., Kvanli, A. and Pavur, R. (1987). A comparison of the Box-Jenkins method with that of automated forecasting methods. International Journal of Forecasting. 2, 261-267.

Raine, J. E. (1971). Self-adaptive forecasting considered., Decision Science. 2(2), 264- 279.

Reid, D. J. (1969). A comparative study of time series prediction techniques on economic data. Unpublished Ph.D. thesis, University of Nottingham.

Reid, D. J. (1971). Forecasting in action; comparison of forecasting techniques in economic time series. Proceedings of the Joint Conference of Operation Research Society.

Reid, D. J. (1975). A review of short term projection technique. Practical Aspects of Forecasting, Operational Research Society. London. 6, 8-25.

Schmidt, J. (1979). Forecasting state retail sales: Econometric versus time series models. The Annuals of Regional Science. 8. 91-101.

Simon, H. and Newell A., (1971). Human problem solving: The state of the theory in 1970. American Psychologist.28. 1-39.

Steckler, H. O. (1968). Forecasting with econometric models: an evaluation. Econometnca. 26. 437-463.

56


a p p e n d ix e s

57


APPENDIX A

Sample of the questionnaire used for data collection

S8


Predicting Future occupancy Rates in Hotels Using

Forecasting Models

1) Name ..................................................

Title..................................................

A d d r e s s .......................................... C i t y .........

Country ......................................................

2) Hotel Name...........................................................

Address.........................................

C i t y .........................................................

Country.........................

3) Number of Rooms...................

4) Number of Restaurants..........................

5) Number of Bars....................................

6) How is your hotel managed ? Please circle one only.

a) Owner b) Management Company c) Franchise

d) Other, please s p e c i f y ...........................................

7) If the hotel was not managed by the owner, then what is the

name of the firm managing the hotel ?

59


8) What is the major market segment of the hotel ?

Please circle one segment only.

a) Corporate b) Resort c) Travel d) Government

e) Conference f) Transient

g) Other, please specify

9) Do you use computer software in occupancy forecasting

Yes No

10) If yes, please list the name of the software used.

11) Please circle the type of computer hardware used in

forecasting.

12) If a personal computer is used for forecasting, is it:

a) a Stand alone system (just for forecasting) or,

b) an integrated system with hotel software applications such

as front office, yield management, etc.

13) If you do not use computers, please circle one cr more of the

following methods used for occupancy forecasting.

a) Quantitative methods b) Judgment based on experience

c) Sales people reports

d) Other please specify ...................................

a) Personal computer b) Main frame

60Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

14) If quantitative methods are used in forecasting what are the

methods used. (Please circle one or more).

a) Simple average b) time series

c) Econometric methods

d) Other, please specify....................................

15) How often do you forecast the occupancy ? (Please circle one or more of the following).

a) Daily b) Weekly c) Monthly

d) Quarterly e) every 6 months f) yearly

g) Other, please specify...........................................

16) The forecasting process is done by: (Please circle one or more

of the following)

a) Sales person b) Sales manager

c) Marketing manager d) General Managere) Other, please specify .........................................

17) Please complete the attached tables for the historical monthly

occupancy percentages; starting January 1989 thru June 1994,

and list vour forecast from July,1994 till December 1994 in

the attached tables.

61


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Occupancy R ates (1991)M k ( Sgmnt JJ*1! D t f c t e

R ack Rale

T rav e l A tcn l .

Conference

T rans ien t

C roups

C ovenm cnl

C orpo ra teHi

oo muF§lit R n w P i n SHI mhm M ' k ' mWM

D e c : : ‘!

1

Total M onthly Occupancy

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

O ccupancy Rates (1992)M kt Sgmnt 5 y 6 8 ^ Nov'«> h c c ; f e ;

R ack R ale

T ra v e l A fen l.

Conference

T ra m le n l.

C r o u p i

Covcnm enl

C o rp o ra te m Ml m md& T,*Nov.' i> 'te =

1

—— ■■ 1 \

Tnlal Monllily Occupancy

Occ

upan

cy

Rat

es

(199

3)

•k

uV.

JC

64


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

O ccupancy Hales In Percent (1994)Forecast Foreca st Forecast Forecast Forecast Forecast

M arket Segm ent 1 Jan. ' Feb: WAr. Apr.’ May.’ \ ' w "■June '■Sept"; i ' O t U - Nov'. . Dec.;Rack RaleT ra w l Accnl.ConferenceT ra n i ic n lG ro u p iC o w n m e n fC o ipo ra lc

other V fl|aH U»tJJfl*W' Occupancy/Mk| Sfmal '

4* v . Jan. Feb. f>Ur.

<

Apr. May.***■ Forecast Forecast Forecast Forecast Forecast Forecast

’June Ju ly Aue. Sept. Oct. N ov . Dec.

Tolal M onthly Occupancy

APPENDIX B

Time scries plots of models fitting for Cairo Hotel

66


CAllt

O (K

.'CU

PAN

CY

KA

'IIO

S

APPENDIX B.l

5 0 -

3 0 -

4 32 9 3 68 22l 1 5


Figure II. Moving Average Model Fit for the Cairo Hotel.

• A c tu a l

• Forecast

67


CAIR

O OC

CUPA

NCY

RA'I»

>

APPENDIX B.2

90 4

70i

433629228 15Case Number

A lp h a “ 0 6 7

Figure 12. Brown’s Model Fit for the Cairo Hotel.

• A c tu a l

• Forecast

68


APPENDIX B.3

9 0<

>

7 0 -

SO

3 0

2 9 3 6 4 3228 151Case Number

A lp h a - 0 . 6 7 B a ta ■ 0 0 1

Figure 13. Holt’s Model Fit for the Cairo Hotel.

69


• A c tu a l

• P o ra c a s t

CAIK

O (X

X’U

I’ANC

Y ItA

'IKS

APPENDIX B.4

9 0 -

70

F o ra c a s t

22 29 368 431 15Case Number

Alpha « 0 67 B a ta - 0.01 G a m m a - 0 .99

Figure 14. Winter’s Model Fit for the Cairo Hotel.

70


APPENDIX B.5

<

>z<

5<9 50-<* '

3081 22 29 3615 4 3

Case Number

Figure 15. ARIMA Model Fit for the Cairo Hotel.

71


A c tu a l

F o r a c a s t

APPENDIX C

Time scries plots of models fitting for Hurgada Hotel

72


APPENDIX C .l

£<

%<<*

100 -

6 0 -

4 0 -

20 ■ —

8 IS 22 29


36 43

Figure 16. Moving Average Model Fit for the Hurgada Hotel.

73


Actual

Forecast

APPENDIX C.2

too •

>

6 0 t

X<<

29 361 8 1 5 22 43

Case Number Alpha - 0 .48

Figure 17. Brown’s Model Fit for the Hurgada Hotel.

74

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

A c tu a l

For«c«t

APPENDIX C.3

£< 80-1>Z<

p 60-P

<<

298 362215 431Case Number

Alpha - 0 .4 8 Beta - 0 01

Figure 18. Holt’s Model Fit for the Hurgada Hotel.

75


Actual

F o r e c a s t

IIUI

iGAD

A (X

X.'U

PANC

Y R

A'I

KS

APPENDIX C.4

100 ■

60 ■

4 0 -

36 4 3298 22151Case Number

Alpha - 0 .4 8 Bata - 0.01 Gamm a - 0 .99

Figure 19. Winter’s Model Fit for the Hurgada Hotel.

• Actual

• Forecast

76


APPENDIX C.5

100 ■

<

>5<

>rx<<

36 4329228 151Case Number

Figure 20. ARIMA Model Fit for the Hurgada Hotel.

77


Actual

F o r e c a s t

forecasting

Documents

Transcript of forecasting