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Forecasting demand for lodging properties at a resort : a comparison of methods Roth, Dylan 2003

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Forecasting Demand for Lodging Properties at a Resort: A Comparison of Methods by Dylan Roth B . C o m m . , Q u e e n ' s University, 1998 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S FOR THE DEGREE OF M A S T E R O F S C I E N C E IN B U S I N E S S A D M I N I S T R A T I O N IN T H E F A C U L T Y O F G R A D U A T E S T U D I E S FACULTY OF C O M M E R C E AND BUSINESS ADMINISTRATION DEPARTMENT OF MARKETING W e a c c e p t this the>is-as conforming to the/required standard  '(Df.J^i^FJbtler)  ( D r . / ^ h a ^ e Weinberg)  (riryJohp Claxton)  "(fjavid J e n k i n s )  T H E UNIVERSITY O F BRITISH C O L U M B I A March 2003 © Dylan R o t h , 2 0 0 3  In  presenting  degree freely  at  this  the  available  copying  of  department publication  in  partial  fulfilment  of  the  University  of  British  Columbia,  I  agree  for  this or of  thesis  reference  thesis by  this  for  his  and  study.  scholarly  or  her  thesis  for  financial  of  Cot^  W^Of(  <-  The University of British C o l u m b i a Vancouver, Canada  Date  DE-6  (2/88)  Mfrroh  2-1  j  Z.O°^h  gain  shall  that  agree  purposes may  representatives,  permission.  Department  I further  requirements  be  it not  is  that  the  for  Library  advanced  shall  permission for  granted  by  understood be  an  allowed  the  make  extensive  head  that  without  it  of  copying my  my or  written  Abstract D e m a n d f o r e c a s t s a r e the m o s t important p i e c e of information u s e d to m a k e r e v e n u e m a n a g e m e n t d e c i s i o n s for l o d g i n g p r o p e r t i e s . H i g h d e m a n d f o r e c a s t s m a y l e a d to i n c r e a s e s in r o o m rates a n d s t a y restrictions w h i l e low d e m a n d f o r e c a s t s m a y result in price d e c r e a s e s a n d e a s i n g of s t a y restrictions. A n u m b e r of d e m a n d f o r e c a s t i n g m e t h o d s , both l o n g - t e r m ( m o r e than 9 0 d a y s prior to a target date) a n d short-term (within 9 0 d a y s of a target d a t e ) w e r e m o d e l l e d a n d c o m p a r e d for the l o d g i n g p r o p e r t i e s at a m a j o r North A m e r i c a n s k i resort. L o n g - t e r m f o r e c a s t i n g m e t h o d s i n c l u d e d r a n d o m w a l k , multiplicative H o l t - W i n t e r s , A R l M A ( a u t o r e g r e s s i v e integrated m o v i n g a v e r a g e ) , linear r e g r e s s i o n , a n d n o n l i n e a r r e g r e s s i o n . S h o r t - t e r m m o d e l s i n c l u d e d the five l o n g - t e r m f o r e c a s t i n g m e t h o d s a s well a s additive p i c k u p a n d a r e g r e s s i o n - b a s e d b o o k i n g c u r v e m o d e l . In t e r m s of long-term f o r e c a s t s , the n o n l i n e a r r e g r e s s i o n m e t h o d w a s f o u n d to b e s u p e r i o r w h i l e c a p a c i t y w a s trending u p w a r d but after a c a p a c i t y s h o c k ( u n e x p e c t e d l o s s in c a p a c i t y ) the r a n d o m w a l k m e t h o d p r o v e d o p t i m a l . In t e r m s of short-term f o r e c a s t s , the r e g r e s s i o n - b a s e d b o o k i n g c u r v e m o d e l w a s o p t i m a l i n - s a m p l e a n d d a t a w a s not t e s t e d out of s a m p l e . Further, the l o n g - t e r m n o n l i n e a r r e g r e s s i o n m o d e l a n d short-term r e g r e s s i o n - b a s e d b o o k i n g c u r v e m o d e l explicitly d e f i n e d s e a s o n a l p e r i o d s . T h e s e statistically d e f i n e d s e a s o n a l p e r i o d s s h o u l d h e l p m a n a g e m e n t s e t s e a s o n a l rate targets a s w e l l a s better u n d e r s t a n d typical b o o k i n g patterns a m o n g p e r i o d s .  ii  Table of C o n t e n t s  ABSTRACT  II  TABLE OF CONTENTS  Ill  LIST O F T A B L E S  V  LIST O F F I G U R E S  VI  1  INTRODUCTION  1  2  FORECASTING APPROACH  3  3  LONG-TERM MODELS  6  3.1 3.2  RESORT OVERVIEW M O D E L EFFICACY CRITERIA  3.3 3.4 3.5 3.6 3.7 3.8  R A N DOM W A L K ( R W ) MULTIPLICATIVE HOLT-WINTERS ( H W ) AUTOREGRESSIVE INTEGRATED M O V I N G A V E R A G E ( A R I M A ) L I N E A R REGRESSION ( L R ) N O N - L I N E A R REGRESSION ( N L ) L O N G - T E R M M O D E L COMPARISON  4  6 8  SHORT-TERM MODELS 4.1 4.2  4.3 4.4 5  24  ADDITIVE PICKUP (AP) BOOKING C U R V E ESTIMATE ( B C )  4.2.1 4.2.2 4.2.3  24 25  Booking curve baseline model Booking curve adjustment Short-term weighting function  28 30 31  COMPARING SHORT-TERM FORECASTS B O O K I N G C U R V E D E C I S I O N SUPPORT S Y S T E M DISCUSSION  5.1  10 10 13 15 18 21  36 38 39  M O D E L E X T E N S I O N S IN A C A P A C I T Y C O N S T R A I N E D E N V I R O N M E N T  46  APPENDIX A - D A T A PREPARATION AND TRANSFORMATION  48  APPENDIX B - MULTIPLICATIVE HOLT-WINTERS  49  APPENDIX C - ARIMA  51  APPENDIX D - LINEAR R E G R E S S I O N  53  APPENDIX E - POISSON REGRESSION  57  APPENDIX F - NONLINEAR REGRESSION  59  APPENDIX G - BASELINE REGRESSION  63  APPENDIX H - A P P R O A C H E S T O BOOKING C U R V E A D J U S T M E N T  71  APPENDIX I - ARIMA MODELLING OF BOOKING C U R V E A D J U S T M E N T  72  iii  REFERENCES  74  iv  List of T a b l e s  T a b l e 1:  F o r e c a s t i n g c h o i c e s m a d e in resort l o d g i n g m o d e l s  T a b l e 2:  L o n g - t e r m H o l t - W i n t e r s multiplicative m o d e l results ( M a y 1 5 , 1 9 9 8 to April 2 9 , 2 0 0 2 )  13  L o n g - t e r m linear r e g r e s s i o n m o d e l results ( M a y 15, 1 9 9 8 to April 2 9 , 2 0 0 2 )  17  L o n g - t e r m n o n l i n e a r r e g r e s s i o n m o d e l results ( M a y 15, 1 9 9 8 to April 2 9 , 2 0 0 2 )  21  I n - s a m p l e long-term m o d e l c o m p a r i s o n s (April 2 8 , 2 0 0 1 to April 2 7 , 2 0 0 2 )  22  O u t of s a m p l e long-term m o d e l c o m p a r i s o n s (July 2 9 , 2 0 0 2 to November 30, 2002)  22  O u t of s a m p l e long-term m o d e l c o m p a r i s o n s ( D e c e m b e r 1, 2 0 0 2 to January 22, 2003)  23  B o o k i n g c u r v e b a s e l i n e r e g r e s s i o n m o d e l results ( M a y 1 5 , 1 9 9 8 to April 2 9 , 2 0 0 2 )  30  T a b l e 3:  Table 4:  T a b l e 5:  T a b l e 6:  T a b l e 7:  T a b l e 8:  T a b l e 9:  3  I n - s a m p l e short-term m o d e l c o m p a r i s o n s ( A u g u s t 6, 2 0 0 2 to April 2 7 , 2 0 0 2 )  37  T a b l e 10:  Booking curve estimate calculation approaches  71  Table 11:  A R I M A s p e c i f i c a t i o n s for b o o k i n g c u r v e error f o r e c a s t s  73  v  List of Figures  Figure 1 A:  Figure 1B:  N o r m a l i z e d a v e r a g e daily r o o m nights for the 0 1 / 0 2 s e a s o n by w e e k  7  N o r m a l i z e d a v e r a g e daily r o o m nights for the 0 1 / 0 2 s e a s o n by d a y of w e e k  7  Figure 2:  E s t i m a t e d m a x i m u m daily d e m a n d for o n e b e d r o o m units b y y e a r  F i g u r e 3:  P r e d i c t e d s h a r e of m a x i m u m daily d e m a n d for o n e b e d r o o m units  19  o n s e l e c t e d d a t e s in 0 2 / 0 3 ( 6 1 5 units = 1 0 0 % )  20  Figure 4:  T y p i c a l b o o k i n g c u r v e ( b o o k i n g c u r v e for target d a t e of J u l y 5, 2 0 0 1 )  25  Figure 5A:  L o w d e m a n d period - c o n v e x b o o k i n g c u r v e ( b o o k i n g c u r v e for target d a t e of D e c e m b e r 6, 2 0 0 1 ) H i g h d e m a n d period - c o n c a v e b o o k i n g c u r v e ( b o o k i n g c u r v e for  26  target d a t e of D e c e m b e r 2 8 , 2 0 0 1 )  26  Figure 6:  F l o w c h a r t of f o r e c a s t i n g p r o c e s s for b o o k i n g c u r v e s h o r t - t e r m e s t i m a t e  27  F i g u r e 7:  G e n e r i c logistic c u r v e  28  F i g u r e 8:  M e a n s q u a r e error of o n e b e d r o o m long-term N L e s t i m a t e s a n d  Figure 5B:  short-term b o o k i n g c u r v e projections at different l e a d t i m e s  32  Figure 9A:  O n e b e d r o o m w e i g h t i n g function for short-term e s t i m a t e  34  Figure 9 B :  T w o b e d r o o m w e i g h t i n g function for short-term e s t i m a t e  35  Figure 9 C :  T h r e e plus b e d r o o m w e i g h t i n g function for short-term e s t i m a t e  35  F i g u r e 10:  M e a n s q u a r e error of o n e b e d r o o m long-term N L e s t i m a t e s , s h o r t - t e r m b o o k i n g c u r v e projections, a n d short-term b o o k i n g c u r v e e s t i m a t e s at different l e a d t i m e s I n - s a m p l e m e d i a n a b s o l u t e p e r c e n t a g e error ( M d A P E ) for f o r e c a s t i n g methods a c r o s s lead times  36  Figure 11:  F i g u r e 12:  Figure 13A:  Figure 13B:  37  P o r t i o n of d e c i s i o n - s u p p o r t s y s t e m output p a g e (data a s at D e c e m b e r 9, 2 0 0 2 )  38  A v e r a g e FIT daily d e m a n d in 0 1 / 0 2 s u m m e r s e a s o n a n d c o r r e s p o n d i n g s e a s o n a l period c l a s s i f i c a t i o n by w e e k  41  A v e r a g e FIT daily d e m a n d in 0 1 / 0 2 winter s e a s o n a n d c o r r e s p o n d i n g s e a s o n a l p e r i o d c l a s s i f i c a t i o n by w e e k  42  vi  Figure 14A:  Figure 14B:  F i g u r e 15:  E x p e c t e d FIT o n e b e d r o o m b o o k i n g c u r v e s for a S a t u r d a y v s . T h u r s d a y in J u l y 2 0 0 3 ( T h u r s d a y = J u l y 17, 2 0 0 3 , S a t u r d a y = J u l y 1 9 , 2 0 0 3 )  43  E x p e c t e d FIT o n e b e d r o o m b o o k i n g c u r v e s for a W i n t e r D a t e v s . F a l l D a t e (Winter D a t e = M a r c h 17, 2 0 0 3 , Fall D a t e = S e p t e m b e r 19, 2 0 0 3 )  44  C o m p a r i s o n of o n e b e d r o o m n o r m a l i z e d a v e r a g e daily r o o m rates to n o r m a l i z e d a v e r a g e d a i l y d e m a n d in the 0 1 / 0 2 s e a s o n b y w e e k ( a v e r a g e = 1.00)  45  Vll  1  INTRODUCTION  F o r e c a s t i n g d e m a n d is a critical c o m p o n e n t of r e v e n u e m a n a g e m e n t for l o d g i n g o p e r a t o r s (hotel a n d rental p r o p e r t i e s ) . L o d g i n g units a r e p e r i s h a b l e inventory s i n c e r e v e n u e f r o m a l o d g i n g unit o n a certain d a t e is lost f o r e v e r if the unit is not filled. G i v e n that the majority of l o d g i n g c o s t s a r e f i x e d , l o d g i n g o p e r a t o r s m u s t t a k e a p p r o p r i a t e a c t i o n s to m a x i m i z e l o d g i n g r e v e n u e . T h e d e m a n d f o r e c a s t is the p i e c e of information u p o n w h i c h r e v e n u e m a n a g e m e n t d e c i s i o n s a r e m a d e . A d e m a n d f o r e c a s t that is h i g h e r t h a n e x p e c t a t i o n m a y l e a d to i n c r e a s e s in p r i c e , s t a y c o n t r o l s (i.e. m i n i m u m two night s t a y ) a n d other restrictions. O n the other h a n d , a d e m a n d f o r e c a s t l o w e r than e x p e c t a t i o n m a y trigger p r o m o t i o n s , price d i s c o u n t s , a n d a lifting of restrictions. In this p a p e r , d e m a n d is u s e d s y n o n y m o u s l y with b o o k i n g s a n d is d e f i n e d a s o c c u p i e d r o o m nights. F o r e x a m p l e , a r e s e r v a t i o n for 2 units at 7 nights is e q u i v a l e n t to 2 r o o m nights p e r d a y for a total of 14 r o o m nights.  L o d g i n g d e m a n d e s t i m a t e s w e r e c a l c u l a t e d for a m a j o r North A m e r i c a n s k i resort.  Long-term  e s t i m a t e s w e r e c a l c u l a t e d m o r e than 9 0 d a y s prior to a target d a t e , w h i l e s h o r t - t e r m e s t i m a t e s w e r e c a l c u l a t e d within 9 0 d a y s of a target d a t e . S t a n d a r d f o r e c a s t i n g m o d e l s i n c l u d i n g linear r e g r e s s i o n , multiplicative H o l t - W i n t e r s , A R I M A ( a u t o r e g r e s s i v e integrated m o v i n g a v e r a g e ) , a n d r a n d o m - w a l k w e r e u s e d to c r e a t e long-term f o r e c a s t s . T h e s e w e r e c o m p a r e d to a n o n l i n e a r r e g r e s s i o n m o d e l built s p e c i f i c a l l y to c a p t u r e the r e s o r t ' s y e a r l y d e m a n d trend a n d s e a s o n a l i t y (e.g. r e g i o n a l s c h o o l h o l i d a y s ) .  W i t h i n the four y e a r s a m p l e p e r i o d , the n o n l i n e a r r e g r e s s i o n  p r o v i d e d s u p e r i o r e s t i m a t e s to the other m o d e l s . H o w e v e r , out of s a m p l e , the n o n l i n e a r r e g r e s s i o n m o d e l only p r o v i d e d a m a r g i n a l i m p r o v e m e n t o v e r other m o d e l s d u r i n g the first four m o n t h s of the out of s a m p l e p e r i o d . A t this 4 - m o n t h point a n u n d e r l y i n g a s s u m p t i o n of the n o n l i n e a r m o d e l (linearly i n c r e a s i n g y e a r l y d e m a n d ) w a s v i o l a t e d a s the resort e x p e r i e n c e d a l a r g e d e c r e a s e in c a p a c i t y w h e n a major l o d g i n g property s w i t c h e d r e s e r v a t i o n m a n a g e m e n t p r o v i d e r s . After this c a p a c i t y s h o c k , r a n d o m w a l k p r o v i d e d the best e s t i m a t e s for the r e m a i n d e r of the out of s a m p l e period (and h a d p r o v i d e d the s e c o n d b e s t e s t i m a t e s a m o n g l o n g - t e r m f o r e c a s t i n g m e t h o d s prior to the a s s u m p t i o n violation). A s a result, a ' c u s t o m i z e d ' n o n l i n e a r r e g r e s s i o n m o d e l is a r e c o m m e n d e d long-term f o r e c a s t i n g m e t h o d for r e s o r t s w h i c h a r e e x p e r i e n c i n g p r e d i c t a b l e y e a r l y shifts in d e m a n d ( i n c r e a s i n g or d e c r e a s i n g ) w h i c h c a n b e a p p r o x i m a t e d b y a f u n c t i o n a l f o r m s u c h a s a linear trend or m o d i f i e d e x p o n e n t i a l c u r v e . If c h a n g e s in y e a r l y d e m a n d a r e s p o r a d i c or s m a l l the r a n d o m w a l k m e t h o d is r e c o m m e n d e d for l o n g - t e r m e s t i m a t e s s i n c e it is s i m p l e a n d robust.  S h o r t - t e r m e s t i m a t e s ( e s t i m a t e s within 9 0 d a y s of a target date) typically c o m e in two v a r i e t i e s . T h e first variety i n c l u d e s the s a m e m o d e l s u s e d for long-term f o r e c a s t s but with a s h o r t e r f o r e c a s t i n g h o r i z o n . In other w o r d s , t h e s e m o d e l s u s e p a s t c o m p l e t e s t a y information to f o r e c a s t  1  future c o m p l e t e s t a y s . T h e s e c o n d variety of short-term m o d e l s i n c l u d e s m o d e l s that i n c o r p o r a t e current b o o k i n g s for future d a t e s ( b o o k i n g s to date). A d d i t i v e p i c k u p ( A P ) m o d e l s a r e s i m i l a r to r a n d o m w a l k m e t h o d s ; they u s e current b o o k i n g s a n d a d d last y e a r ' s p i c k u p ( n u m b e r of r e s e r v a t i o n s m a d e in the prior y e a r f r o m Y d a y s out up until the target date) to c o m e up with a short-term f o r e c a s t . T h e other t e s t e d m o d e l i n c o r p o r a t i n g b o o k i n g s to d a t e is a ' c u s t o m i z e d ' b o o k i n g c u r v e m o d e l . T h i s m o d e l c r e a t e s a b a s e l i n e b o o k i n g c u r v e (pattern of b o o k i n g s o v e r t i m e for a particular target d a t e ) a n d then c o m p a r e s a c t u a l b o o k i n g s to the b a s e l i n e b o o k i n g s in o r d e r to project d e m a n d for the target d a t e .  T h i s projected d e m a n d is t h e n c o m b i n e d with a  l o n g - t e r m n o n - l i n e a r e s t i m a t e ; the w e i g h t b e t w e e n e s t i m a t e s d e t e r m i n e d b y the n u m b e r of d a y s (lead time) f r o m the target d a t e .  Within the four y e a r s a m p l e p e r i o d , the ' c u s t o m ' b o o k i n g c u r v e m o d e l p r o v i d e d s u p e r i o r e s t i m a t e s to all other m o d e l s ( A P a n d long-term m o d e l s ) . Further, the b o o k i n g c u r v e m o d e l a n d long-term n o n l i n e a r m o d e l explicitly d e f i n e s e a s o n a l p e r i o d s b a s e d o n d e m a n d . T h e s e statistically significant s e a s o n a l p e r i o d s p r o v i d e m a n a g e m e n t with v a l u a b l e information for setting r o o m rate targets s i n c e r o o m rates a r e s e t to c o r r e s p o n d to distinct d e m a n d l e v e l s . A s w e l l , the b o o k i n g c u r v e m o d e l p r o v i d e s m a n a g e m e n t with e x p e c t e d b o o k i n g c u r v e s ; the s y s t e m a t i c build-up in b o o k i n g s for a particular target d a t e . T h e s e e x p e c t e d b o o k i n g c u r v e s quantify t h e r e l a t i o n s h i p b e t w e e n l e a d - t i m e a n d d e m a n d for a certain p e r i o d , h e l p i n g m a n a g e m e n t to identify the likely extent of last m i n u t e b o o k i n g s v e r s u s r e s e r v a t i o n s in a d v a n c e . H o w e v e r , w h i l e the n o n l i n e a r m o d e l a n d b o o k i n g c u r v e m o d e l h a v e m a n y benefits a n d a r e likely to i n c r e a s e the a c c u r a c y of f o r e c a s t s , the benefit of this additional a c c u r a c y is directly related to the a m o u n t of e x c e s s c a p a c i t y . A l a r g e a m o u n t of e x c e s s c a p a c i t y , a s is the c a s e in the resort s t u d i e d , l e a d s to a low c o s t of d e m a n d i n a c c u r a c y s i n c e all r e s e r v a t i o n s c a n b e a c c o m m o d a t e d r e g a r d l e s s of final d e m a n d . C o n s t r a i n e d c a p a c i t y e n v i r o n m e n t s , o n the other h a n d , h a v e a large opportunity c o s t of d e m a n d f o r e c a s t i n a c c u r a c y s i n c e h i g h - v a l u e r e s e r v a t i o n s (e.g. r e s e r v a t i o n s with high daily r o o m rates a n d long length of stay) s h o u l d b e prioritized a b o v e l o w - v a l u e r e s e r v a t i o n s . If the f o r e c a s t s for h i g h - v a l u e r e s e r v a t i o n s a n d l o w - v a l u e r e s e r v a t i o n s a r e i n a c c u r a t e in a situation of c o n s t r a i n e d c a p a c i t y , then r e s e r v a t i o n m a n a g e m e n t will m a k e s u b - o p t i m a l d e c i s i o n s with r e s p e c t to pricing, s t a y c o n t r o l s , a n d a p p r o p r i a t e mix of m a r k e t s e g m e n t s .  2  2  FORECASTING APPROACH  W e a t h e r f o r d , K i m e s , & S c o t t (2001) p r o v i d e a u s e f u l f r a m e w o r k for f o r e c a s t i n g d e m a n d for hotel p r o p e r t i e s . T h e y c o n t e n d there a r e s e v e n d e c i s i o n f a c t o r s that m u s t b e d e t e r m i n e d prior to a l o d g i n g f o r e c a s t a n d t h e s e a r e outlined in T a b l e 1, a s w e l l a s the a p p r o a c h t a k e n in this p a p e r .  Table 1: F o r e c a s t i n g c h o i c e s m a d e in resort l o d g i n g m o d e l s Weatherford et al. forecast c h o i c e s 1) What to forecast a) A r r i v a l s b) R o o m nights 2) Level of aggregation a) F u l l y a g g r e g a t e d b) A g g r e g a t e d b y rate c a t e g o r y with lengtho f - s t a y probability distributions c) A g g r e g a t e d b y length of stay with ratec a t e g o r y probability distributions d) Fully d i s a g g r e g a t e d (by rate c a t e g o r y with length of stay) 3)  4)  5)  6)  7)  U n c o n s t r a i n i n g method a) N o n e b) D e n i a l s d a t a c) M a t h e m a t i c a l m o d e l s i) Pickup ii) B o o k i n g c u r v e iii) P r o j e c t i o n Number of periods to include in forecast a) A l l b) S e l e c t e d n u m b e r W h i c h data to use a) O n l y c o m p l e t e s t a y - n i g h t s b) A l l d a t a ( c o m p l e t e a n d i n c o m p l e t e s t a y nights) Outliers a) Included b) Not i n c l u d e d  F o r e c a s t c h o i c e for resort estimates 1. b) R o o m nights  2. T h e a p p r o a c h t a k e n d e v i a t e s slightly f r o m the c h o i c e s s t a t e d b y W e a t h e r f o r d et. a l . (2001). B o o k i n g d a t a w a s a g g r e g a t e d by m a r k e t s e g m e n t ( i n d e p e n d e n t traveler, g r o u p , a n d o w n e r ) a s w e l l a s by b e d r o o m (one b e d r o o m (including s u i t e s ) , two b e d r o o m , a n d three plus b e d r o o m s ) . F o r e c a s t s w e r e p r o v i d e d for the i n d e p e n d e n t traveller s e g m e n t b y b e d r o o m type. 3. c) T h e r e is n o u n c o n s t r a i n i n g m e t h o d for long-term m o d e l s . F o r short-term m o d e l s , both p i c k u p a n d b o o k i n g c u r v e m e t h o d s a r e used.  4. a) A l l  5. b) A l l d a t a ; o n l y c o m p l e t e s t a y - n i g h t s a r e u s e d for long-term f o r e c a s t s while short-term f o r e c a s t s u s e d all d a t a . 6. a) O u t l i e r s i n c l u d e d  Level of forecast a c c u r a c y a) A g g r e g a t e d f o r e c a s t s , errors reported a s a v e r a g e a c r o s s all r e a d i n g d a y s b) A g g r e g a t e d f o r e c a s t s , errors reported for e a c h individual r e a d i n g d a y c) D i s a g g r e g a t e d f o r e c a s t s , errors reported a s a v e r a g e a c r o s s all r e a d i n g d a y s d) D i s a g g r e g a t e d f o r e c a s t s , errors reported for e a c h individual r e a d i n g d a y  7. a) A g g r e g a t e d f o r e c a s t s . . . w h i l e m o d e l s a r e c a l c u l a t e d at a d i s a g g r e g a t e level b y r e a d i n g d a y (as in d), d e c i s i o n s a b o u t the m o d e l ' s e f f i c a c y a r e r e p o r t e d at a n a g g r e g a t e level.  A n effective r e v e n u e m a n a g e m e n t s y s t e m u s e s e s t i m a t e s for both g u e s t arrivals a n d r o o m nights in o r d e r to m a x i m i z e r e v e n u e . P r e d i c t e d arrival distributions a r e important s o that the resort c a n i m p l e m e n t effective s t r a t e g i e s for s p e c i f i c arrival d a y s (i.e. price c h a n g e s a n d s t a y c o n t r o l s ) .  3  H o w e v e r , if c a p a c i t y is not e x p e c t e d to b e s u r p a s s e d then s t a y c o n t r o l s a r e n e v e r u s e d .  In  situations of c a p a c i t y s l a c k , p r e d i c t e d r o o m nights a l o n e , rather t h a n p r e d i c t e d r o o m nights b y arrival s e g m e n t , a r e g e n e r a l l y a d e q u a t e for r e v e n u e m a n a g e m e n t . R o o m nights for the i n d e p e n d e n t traveller s e g m e n t w e r e d e t e r m i n e d to b e the m o s t important e s t i m a t e s for the s t u d i e d resort s i n c e i n d e p e n d e n t travellers p a y h i g h e r r o o m rates than g r o u p r e s e r v a t i o n s , a n d their b o o k i n g s a r e m a d e c l o s e r to the target d a t e than o w n e r or g r o u p r e s e r v a t i o n s . S i n c e the resort rarely s o l d out (4 d a y s in the m o s t r e c e n t y e a r ) , a n d therefore e s t i m a t i n g the n u m b e r of r o o m s likely to b e o c c u p i e d w a s r e v e n u e m a n a g e m e n t ' s p r i m a r y c o n c e r n .  W e a t h e r f o r d et a l . (2001) f o u n d that s u m m i n g d i s a g g r e g a t e d hotel d e m a n d f o r e c a s t s p r o d u c e d a m o r e a c c u r a t e f o r e c a s t than a s i n g l e a g g r e g a t e d e m a n d f o r e c a s t . A s a result, the r e s o r t s ' b o o k i n g d a t a w a s d i s a g g r e g a t e d a s m u c h a s p o s s i b l e . R o o m night f o r e c a s t s w e r e o n l y c r e a t e d for the i n d e p e n d e n t traveller s e g m e n t a s t h e s e w e r e the m o s t p r e d i c t a b l e b o o k i n g s a n d did not suffer f r o m d a t a i n c o n s i s t e n c y p r o b l e m s at t h e resort l e v e l . G r o u p b o o k i n g s w e r e often e x c l u d e d f r o m the r e s e r v a t i o n m a n a g e m e n t s y s t e m until shortly prior to a target d a t e m a k i n g it p r o b l e m a t i c to d e t e r m i n e w h e n r e s e r v a t i o n s / c a n c e l l a t i o n s w e r e a c t u a l l y m a d e . T h e o w n e r b o o k i n g s w e r e g e n e r a l l y flat (did not c h a n g e m u c h f r o m 9 0 d a y s out up until the target d a t e ) d u e to i n c e n t i v e s for o w n e r s to c l a i m v a c a t i o n d a t e s far in a d v a n c e . A s a result, d e m a n d f o r e c a s t s w e r e not c r e a t e d for g r o u p a n d o w n e r s e g m e n t s . F u r t h e r m o r e , d e n i a l s ( r e q u e s t s for u n a v a i l a b l e l o d g i n g units) a n d t u r n d o w n s ( c u s t o m e r s r e f u s i n g a r o o m type at a certain p r i c e or s t a y control) w e r e not t r a c k e d b y r e s e r v a t i o n a g e n t s for historical d a t a . A s a result, it w a s d e e m e d p r o b l e m a t i c to d i s a g g r e g a t e b y rate c l a s s in f o r e c a s t s s i n c e e s t i m a t i n g a p p r o p r i a t e probability distributions for different rate c l a s s e s w o u l d b e c o n t r i v e d . F u r t h e r m o r e , d u e to the t r e m e n d o u s s e a s o n a l i t y of the resort (nearly 1 0 0 % o c c u p a n c y d u r i n g C h r i s t m a s period a n d often l e s s t h a n 1 0 % d u r i n g s h o u l d e r p e r i o d s - e . g . e a r l y N o v e m b e r a n d e a r l y M a y ) it w a s h y p o t h e s i z e d that s e a s o n a l i t y a l o n e w o u l d e x p l a i n m o s t of the d e m a n d v a r i a t i o n .  T h e third d e c i s i o n f a c t o r cited b y W e a t h e r f o r d et a l . is u n c o n s t r a i n i n g m e t h o d . In other w o r d s , w h a t t e c h n i q u e is u s e d to s e p a r a t e d e m a n d f r o m c a p a c i t y ? Q u i t e s i m p l y , it is i m p o s s i b l e to o c c u p y m o r e than 1 0 0 % of the resort's l o d g i n g units, yet this d o e s not limit d e m a n d to 1 0 0 % of c a p a c i t y . F o r all the l o n g - t e r m f o r e c a s t i n g m o d e l s there is n o u n c o n s t r a i n i n g m e t h o d . C o m p l e t e s t a y night information, by definition, is c o n s t r a i n e d b y the r e s o r t ' s c a p a c i t y s o t h e s e m o d e l s d o not c a p t u r e d e m a n d a b o v e c a p a c i t y . H o w e v e r , a s m e n t i o n e d earlier, d u e to the infrequent nature of s e l l o u t s at the resort, this w a s not s e e n a s a m a j o r p r o b l e m . T h e short-term f o r e c a s t i n g m o d e l s d o p r o v i d e u n c o n s t r a i n e d f o r e c a s t s . T h e additive p i c k u p m e t h o d m a y f o r e c a s t d e m a n d a b o v e c a p a c i t y if last y e a r ' s p i c k u p plus current b o o k i n g s a r e a b o v e c a p a c i t y . H o w e v e r , this m e t h o d m a y u n d e r e s t i m a t e total u n c o n s t r a i n e d d e m a n d if either current b o o k i n g s h a v e b e e n  4  limited b y c a p a c i t y or if last y e a r ' s p i c k u p w a s limited by c a p a c i t y . T h e short-term n o n l i n e a r r e g r e s s i o n m o d e l u s e s b o o k i n g c u r v e s (pattern of b o o k i n g s o b s e r v e d in s i m i l a r d a y s past) a s a b a s e l i n e to g a u g e future d e m a n d . H o w e v e r , s i m i l a r to the additive p i c k u p m o d e l , there is potential to u n d e r e s t i m a t e u n c o n s t r a i n e d d e m a n d if either the b o o k i n g c u r v e d e v e l o p e d f r o m prior y e a r d a t a w a s c o n s t r a i n e d b y c a p a c i t y or if current b o o k i n g s a r e c o n s t r a i n e d b y c a p a c i t y . D u e to the infrequent nature of r e s o r t - w i d e s e l l o u t s , both the additive p i c k u p m o d e l a n d short-term n o n l i n e a r r e g r e s s i o n m o d e l w e r e e x p e c t e d to b e v e r y c l o s e a p p r o x i m a t i o n s of u n c o n s t r a i n e d demand.  In t e r m s of d a t a u s e d for f o r e c a s t i n g , the entire four y e a r s a m p l e w a s utilized in the m o d e l a n a l y s i s a n d this i n c l u d e s b o o k i n g s that n e v e r m a t e r i a l i z e d d u e to c a n c e l l a t i o n s a n d n o - s h o w s . B y i n c l u d i n g c a n c e l l a t i o n s a n d n o - s h o w s , the m o d e l s a r e better a b l e to project future d e m a n d g i v e n current b o o k i n g s . A s w e l l , the d a t a w a s not s c r u b b e d to e x c l u d e outliers s i n c e large variation in d e m a n d (due to w e a t h e r , p r o m o t i o n s , r a n d o m n e s s , etc.) is typical in l o d g i n g f o r e c a s t i n g . R e a d e r s interested in a m o r e d e t a i l e d d e s c r i p t i o n of the d a t a p r e p a r a t i o n a n d t r a n s f o r m a t i o n p r o c e s s u s e d in the m o d e l s a r e referred to A p p e n d i x A .  A r e a d i n g d a y is d e f i n e d by W e a t h e r f o r d et a l . (2001) a s the d a y w h e n the n u m b e r of r e s e r v a t i o n s o n h a n d for a particular arrival d a y is u p d a t e d . A t the resort s t u d i e d , r e a d i n g d a y s w e r e g e n e r a l l y u p d a t e d o n a w e e k l y b a s i s within 9 0 d a y s of a target d a t e a n d u p d a t e d daily in the w e e k prior to a target d a t e . S i n c e resort r e v e n u e m a n a g e m e n t ' s a p p r o a c h to f o r e c a s t i n g w a s r a n d o m w a l k (last y e a r ' s o c c u p a n c y figure for long-term f o r e c a s t s a n d additive p i c k u p for short-term f o r e c a s t s ) it w a s straightforward to c o m p a r e m o d e l f o r e c a s t s to likely m a n a g e m e n t f o r e c a s t s for a n y g i v e n d a y of historical d a t a . In o r d e r to p r o v i d e m a x i m u m a c c u r a c y in f o r e c a s t c o m p a r i s o n s , e a c h d a y in the 9 0 d a y w i n d o w w a s treated a s a r e a d i n g d a y .  5  3  LONG-TERM MODELS  3.1  Resort Overview  R e v e n u e m a n a g e m e n t at the resort s t u d i e d m a n a g e s r o u g h l y 5 0 % of the b e d b a s e o n - m o u n t a i n . T h e fraction of r o o m s u n d e r m a n a g e m e n t h a s r e m a i n e d r o u g h l y c o n s t a n t in the p a s t five y e a r s a s the d e v e l o p m e n t of o n - m o u n t a i n properties b y the resort's real e s t a t e d i v i s i o n h a s l a r g e l y m a t c h e d d e v e l o p m e n t b y e x t e r n a l hotel c h a i n s . F o r resort m a n a g e d p r o p e r t i e s , the r e v e n u e m a n a g e m e n t d i v i s i o n is r e s p o n s i b l e for m a n a g i n g r e s e r v a t i o n s a s w e l l a s setting p r i c e s a n d s t a y restrictions. T h e following long-term a n d short-term m o d e l s a r e d e m a n d f o r e c a s t i n g m e t h o d s for F I T (free i n d e p e n d e n t traveller) s e g m e n t s o n l y .  T h e long-term m o d e l s c o n s i s t entirely of p a s t d e m a n d information.  T h e r e a d e r is referred to  F i g u r e 1 A a n d F i g u r e 1 B for a s a m p l e of r o o m nights in the 0 1 / 0 2 s e a s o n . A s c a n b e s e e n f r o m F i g u r e 1 A , there is t r e m e n d o u s s e a s o n a l i t y t h r o u g h o u t the y e a r . T h e first s h o u l d e r period f r o m late April until mid J u n e is v e r y s l o w , a s the s k i hill is c l o s e d for d o w n h i l l s k i i n g a n d s c h o o l is not yet out for the s u m m e r . T h e s u m m e r period h a s v e r y high o c c u p a n c y , a s the resort h a s m a n y s u m m e r travelers a n d on-hill activities, with d e m a n d p e a k i n g in the first w e e k e n d of A u g u s t .  In  the s e c o n d s h o u l d e r s e a s o n , d e m a n d g r a d u a l l y d e c l i n e s f r o m e a r l y S e p t e m b e r until mid N o v e m b e r , until the hill o p e n s for downhill s k i i n g in late N o v e m b e r . T h e b o o k i n g s t h e n r a m p up until the C h r i s t m a s period p e a k i n g at N e w Y e a r ' s . F r o m J a n u a r y until late M a r c h the hill is a g a i n v e r y high o c c u p a n c y with p e a k s for regional h o l i d a y s s u c h a s s c h o o l b r e a k s , a s well a s the w e e k e n d s s u r r o u n d i n g Martin L u t h e r K i n g D a y a n d P r e s i d e n t ' s D a y . F i g u r e 1 B s h o w s the daily variation in d e m a n d , with F r i d a y a n d S a t u r d a y nights c o m m a n d i n g g r e a t e r d e m a n d than w e e k d a y s . H o w e v e r , the daily variation c h a n g e s d r a m a t i c a l l y by s e a s o n a l p e r i o d , with w e e k e n d nights m a k i n g up a g r e a t e r proportion of r o o m nights in s h o u l d e r s e a s o n s w h i l e daily variation is m o r e e v e n l y distributed during high o c c u p a n c y p e r i o d s . B o t h w e e k l y s e a s o n a l i t y a s well a s d a i l y v a r i a t i o n in d e m a n d a r e m u c h g r e a t e r at resort h o t e l s t h a n at b u s i n e s s - o r i e n t e d h o t e l s .  As a  result, the variation in pricing is a l s o m u c h m o r e c y c l i c a l at a resort hotel t h a n at a b u s i n e s s - h o t e l w h i c h t e n d s to h a v e h i g h e r a v e r a g e o c c u p a n c y l e v e l s .  6  3.00  2.00 z E o o  ?  1.50  g 5  1.00  0.50 4  0.00  ////////////////////////// Week  Figure 1 A : N o r m a l i z e d a v e r a g e daily r o o m nights for the 0 1 / 0 2 s e a s o n by w e e k  7  T h e r e a r e n u m e r o u s a p p r o a c h e s to long-term daily d e m a n d f o r e c a s t i n g a n d final f o r e c a s t s a r e often a c o m b i n a t i o n of statistical e s t i m a t e s a n d m a n a g e r i a l j u d g e m e n t . T h i s p a p e r f o c u s e s o n a statistical a p p r o a c h to f o r e c a s t i n g w h i l e r e a d e r s i n t e r e s t e d in f o r m a l a p p r o a c h e s to the integration of m a n a g e r i a l j u d g e m e n t a n d statistical f o r e c a s t s a r e referred to G h a l i a & W a n g (1999).  Before  the different long-term f o r e c a s t i n g m o d e l s a r e d e s c r i b e d a n d c o m p a r e d , h o w e v e r , a p p r o p r i a t e criteria for m o d e l e f f i c a c y m u s t b e c h o s e n .  3.2  M o d e l Efficacy Criteria  Traditionally, m e a n s q u a r e error ( M S E ) is a s t a n d a r d error m e a s u r e for statistical m o d e l s . S p e c i f i c a l l y , the o b j e c t i v e of m o s t p a r a m e t e r e s t i m a t i o n a l g o r i t h m s is to m i n i m i z e M S E (as is the c a s e of all m o d e l s to b e t e s t e d in this p a p e r e x c e p t for A R I M A m o d e l s ) . H o w e v e r , M S E is f o u n d b y m a n y r e s e a r c h e r s to b e a p o o r m e a s u r e of f o r e c a s t validity. A r m s t r o n g a n d C o l l o p y (1992) in their oft cited w o r k " E r r o r M e a s u r e s F o r G e n e r a l i z i n g A b o u t F o r e c a s t i n g M e t h o d s " t e s t e d error m e a s u r e s a g a i n s t a n u m b e r of criteria i n c l u d i n g reliability, c o n s t r u c t validity, sensitivity to s m a l l c h a n g e s , protection a g a i n s t outliers, a n d relationship to d e c i s i o n - m a k i n g . T h e y r e c o m m e n d u s i n g d e v i a n t s of two different error m e a s u r e s , the relative a b s o l u t e error ( R A E ) a n d a b s o l u t e p e r c e n t a g e error ( A P E ) , in o r d e r to c h o o s e a m o n g f o r e c a s t i n g m e t h o d s . T h e R A E ( E q u a t i o n 1) for a s i n g l e e s t i m a t e is the ratio of the a b s o l u t e error of a particular f o r e c a s t i n g m e t h o d ( e . g . HoltW i n t e r s ' m e t h o d ) d i v i d e d by the error of the r a n d o m - w a l k m e t h o d . T h e A P E for a s i n g l e e s t i m a t e ( E q u a t i o n 2) m e a s u r e s the a b s o l u t e error a s a p e r c e n t a g e of the a c t u a l o b s e r v a t i o n . F o r a s i n g l e h o r i z o n A r m s t r o n g & C o l l o p y r e c o m m e n d u s i n g the m e d i a n relative a b s o l u t e error ( M d R A E ) w h e n a s m a l l n u m b e r of t i m e s e r i e s a r e a v a i l a b l e a n d the m e d i a n a b s o l u t e p e r c e n t a g e error ( M d A P E ) w h e n there a r e a l a r g e n u m b e r of s e r i e s ( E q u a t i o n s 3-4). T o c o m p a r e s e r i e s o v e r a l o n g h o r i z o n , they r e c o m m e n d the c u m u l a t i v e relative a b s o l u t e error ( C u m R A E ) for a s i n g l e s e r i e s a n d m e d i a n c u m u l a t i v e relative a b s o l u t e error ( M d C u m R A E ) for multiple s e r i e s ( s e e E q u a t i o n s 5-6).  8  (1)  F„ F.m,h  (2)  MdRAE  h  = Median(RAE  m  h s  MdAPE  h  = Median(APE  m  h s  m  m  CumRAE.,, =  ) for all s e r i e s s  (3)  ) for all s e r i e s s  (4)  IK  (5)  h=\  MdCumRAE  m  = Median[CumRAE  m  ] for all series s  (6)  where:  m h s p  F o r e c a s t i n g m e t h o d (e.g. H o l t - W i n t e r s , A R I M A , etc.) H o r i z o n (lead time) b e i n g f o r e c a s t (h>Q0 for l o n g - t e r m f o r e c a s t s ) Forecast series M e t h o d m f o r e c a s t for h o r i z o n h  1  m,h  1  rw,h  R a n d o m w a l k f o r e c a s t for h o r i z o n h A c t u a l o b s e r v a t i o n at h o r i z o n h  A RAE m,h h  R e l a t i v e a b s o l u t e error of m e t h o d m at h o r i z o n h A b s o l u t e p e r c e n t a g e error of m e t h o d m at h o r i z o n ( l e a d time) h  MdRAE  M e d i a n relative a b s o l u t e error of m e t h o d m, h o r i z o n h for all s e r i e s s  MdAPE  M e d i a n a b s o l u t e p e r c e n t a g e error of m e t h o d m, h o r i z o n h for all s e r i e s s  mh  mlt  CumRAE  m  R e l a t i v e a b s o l u t e error ( R A E ) of m e t h o d m s u m m a r i z e d a c r o s s all n h o r i z o n s of a particular s e r i e s  MdCumRAE..  M e d i a n C u m R A E of m e t h o d m for all s e r i e s s  N o w that a p p r o p r i a t e error m e t r i c s h a v e b e e n c h o s e n the l o n g - t e r m m o d e l e s t i m a t e s c a n b e c o m p a r e d a n d e v a l u a t e d . T h e five different long-term e s t i m a t i o n m e t h o d s i n c l u d e r a n d o m w a l k ( R W ) , linear r e g r e s s i o n ( L R ) , multiplicative H o l t - W i n t e r s ( H W ) , a u t o r e g r e s s i v e i n t e g r a t e d m o v i n g a v e r a g e ( A R I M A ) , a n d n o n l i n e a r r e g r e s s i o n ( N L ) . T h e m o d e l s w e r e c a l i b r a t e d u s i n g the entire four y e a r s a m p l e p e r i o d f r o m M a y 15, 1 9 9 8 to April 2 9 , 2 0 0 2 a n d f o r e c a s t s c o m p a r e d in y e a r four  9  (April 2 8 , 2 0 0 1 to A p r i l 2 7 , 2 0 0 2 ) . In other w o r d s , the entire four y e a r s a m p l e w a s u s e d to d e t e r m i n e the f u n c t i o n a l f o r m of e a c h m o d e l ( n u m b e r a n d type of p a r a m e t e r s ) , but d e p e n d i n g o n the m o d e l , the entire four y e a r s m a y not h a v e b e e n u s e d to c a l c u l a t e the p a r a m e t e r e s t i m a t e s for the i n - s a m p l e p e r i o d . S p e c i f i c a l l y , the L R a n d N L m o d e l s u s e d the entire four y e a r s a m p l e to c a l c u l a t e p a r a m e t e r e s t i m a t e s a n d the s a m e p a r a m e t e r e s t i m a t e s w e r e u s e d in y e a r four.  In  co n tra st, the H W a n d A R I M A m o d e l s u s e d o n l y s a m p l e d a t a prior to the i n - s a m p l e f o r e c a s t to c a l c u l a t e p a r a m e t e r e s t i m a t e s ; s o d a t a f r o m y e a r s o n e to three w e r e u s e d to c a l c u l a t e p a r a m e t e r e s t i m a t e s for y e a r four f o r e c a s t s .  U s i n g the A R I M A m o d e l a s a n e x a m p l e , it w a s d e t e r m i n e d u s i n g the entire four y e a r s a m p l e that an ARIMA(2,0,2)(1,0,0) (1,1,0) 7  364  functional f o r m for o n e b e d r o o m s b e s t fit the entire s a m p l e  d a t a s e t , yet the a c t u a l p a r a m e t e r v a l u e s for the A R I M A ( 2 , 0 , 2 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) 4 w e r e different for 7  36  y e a r four. F u r t h e r , s i n c e a long-term f o r e c a s t is d e f i n e d in this p a p e r a s a n y f o r e c a s t m a d e m o r e t h a n 9 0 d a y s prior to a target d a t e , the H W a n d A R I M A m o d e l s b e g a n to f o r e c a s t f r o m J a n u a r y 2 8 , 2 0 0 1 in o r d e r p r o v i d e l o n g - t e r m e s t i m a t e s for the y e a r four f o r e c a s t p e r i o d (April 2 8 , 2 0 0 1 to A p r i l 2 7 , 2 0 0 2 ) . T h e out of s a m p l e p e r i o d w a s f r o m J u l y 2 9 , 2 0 0 2 to J a n u a r y 2 2 , 2 0 0 3 . T h e out of s a m p l e p e r i o d did not b e g i n until J u l y 2 9 , 2 0 0 2 to a l l o w for a 9 0 d a y p e r i o d f r o m the m o s t r e c e n t i n - s a m p l e d a t e (April 2 9 , 2 0 0 2 ) u s e d to p a r a m e t e r i z e the m o d e l s .  3.3  R a n d o m Walk (RW)  R a n d o m w a l k s i m p l y m e a n s to m a k e p r e d i c t i o n s of future d e m a n d u s i n g p a s t d e m a n d directly (without a n y m o d e l l i n g p r o c e s s ) . In this p a p e r , in o r d e r to obtain the s a m e s e a s o n a l p e r i o d a n d d a y of w e e k , the final d e m a n d f r o m 3 6 4 d a y s prior (52 w e e k s ) is u s e d a s a n e s t i m a t e for future d e m a n d . F o r e x a m p l e , the final long-term d e m a n d e s t i m a t e for J u l y 3 0 , 2 0 0 2 is t a k e n f r o m the final d e m a n d for J u l y 3 1 , 2 0 0 1 .  3.4  Multiplicative Holt-Winters (HW)  A s t a n d a r d statistical d e m a n d f o r e c a s t is a s i m p l e e x p o n e n t i a l m o v i n g a v e r a g e m o d e l . W h i l e a s i m p l e e x p o n e n t i a l s m o o t h i n g m o d e l is not a n a p p r o p r i a t e m e t h o d for daily d e m a n d f o r e c a s t i n g w h e n s e a s o n a l i t y is p r e s e n t , it is a g o o d b a s e u p o n w h i c h to u n d e r s t a n d m o r e c o m p l e x s m o o t h i n g m o d e l s s u c h a s H W a n d A R I M A . T h e b a s i c f o r m of a n e x p o n e n t i a l s m o o t h i n g m o d e l is s h o w n in E q u a t i o n 7 (as d e r i v e d f r o m s m o o t h i n g m o d e l p r e s e n t a t i o n s in S A S E T S U s e r ' s G u i d e , 1 9 9 9 a n d C h a t f i e l d , 1 9 8 9 ) . A s c a n b e s e e n , the w e i g h t s d e c r e a s e in a c o n s t a n t p r o p o r t i o n , t h e r e b y giving m o r e w e i g h t to r e c e n t o b s e r v a t i o n s a n d l e s s w e i g h t to p a s t o b s e r v a t i o n s . E x p o n e n t i a l s m o o t h i n g is the p r o c e s s b y w h i c h the w e i g h t s a r e c a l c u l a t e d  10  r e c u r s i v e l y in o r d e r to m i n i m i z e the s q u a r e d error. T h e error t e r m in e x p o n e n t i a l s m o o t h i n g is s h o w n in E q u a t i o n 8, a n d s o (7) c a n b e restated in e r r o r - c o r r e c t i o n f o r m a s E q u a t i o n 9 or alternatively a s E q u a t i o n 10. A R I M A m o d e l s , to b e e x p l a i n e d in the p r o c e e d i n g s e c t i o n , a r e a large c l a s s of m o d e l s e x p r e s s e d in error-correction f o r m . T h e s i m p l e e x p o n e n t i a l m o v i n g a v e r a g e m o d e l is e x p r e s s e d a s a n A R I M A ( 0 , 1 , 1 ) in E q u a t i o n 1 1 .  Y = aY _ + a(\ - a)Y _ + a(l - a) Y _ +. 2  t  t  x  t  e( =Yt e  2  t  (7)  3  -Yt  1  (8)  1  Y = ae _ + Y _ t  t  x  Y = ae,_,  t  +  t  (9)  x  ae,_ + ae,_ 2  3  +... =  e,_,  (10)  7=1  (11)  {\-B)Y ^e {l-eB) t  t  where: Y  t  O b s e r v a t i o n at t i m e t  Y  E s t i m a t e d o b s e r v a t i o n at t i m e t  a  S m o o t h i n g p a r a m e t e r for t i m e - v a r y i n g m e a n t e r m  e  Error ( d i s t u r b a n c e ) t e r m at time t  B  B a c k w a r d shift o p e r a t o r (e.g.  T  T o t a l n u m b e r of time p e r i o d s for w h i c h o b s e r v a t i o n s exist  t  t  (\-B)y  t  =y ~y -\ t  t  )  T h e multiplicative H o l t - W i n t e r s m o d e l is b a s e d on a n e x p o n e n t i a l s m o o t h i n g m o d e l but i n c l u d e s p a r a m e t e r s to a c c o u n t for trend a n d s e a s o n a l i t y . T h e multiplicative v e r s i o n w a s u s e d s i n c e the additive v e r s i o n c a n b e e x p r e s s e d a s a n A R I M A ( 0 , 1 , p + 1 ) ( 0 , 1 , 0 )  p  m o d e l , a n d a multiplicative  m o d e l s e e m e d m o r e a p p r o p r i a t e s i n c e variation in d e m a n d is likely to i n c r e a s e with a n i n c r e a s e in y e a r l y d e m a n d . In the hotel industry, the multiplicative H o l t - W i n t e r s three p a r a m e t e r e x p o n e n t i a l s m o o t h i n g m e t h o d is a n industry s t a n d a r d ( B a k e r & C o l l i e r (1999)).  The H W model  a c t u a l l y h a s m o r e t h a n t h r e e p a r a m e t e r s , but it is r e f e r r e d to a s a t h r e e p a r a m e t e r m o d e l a s it h a s three s m o o t h i n g p a r a m e t e r s ; a l p h a s m o o t h e s the t i m e v a r y i n g m e a n - t e r m , g a m m a s m o o t h e s the t i m e - v a r y i n g s l o p e , a n d delta s m o o t h e s the t i m e - v a r y i n g s e a s o n a l contribution. T h e e s t i m a t e of the H W m o d e l is s h o w n in E q u a t i o n 12, with the s e p a r a t e e l e m e n t s of (12) d e t a i l e d in E q u a t i o n s 1 3 - 1 5 . F o r c o m p a r i s o n , the s i m p l e e x p o n e n t i a l s m o o t h i n g m o d e l is d e s c r i b e d a s a n H W m o d e l in E q u a t i o n 16. It s h o u l d b e n o t e d that H W is multiplicative s i n c e the t i m e - v a r y i n g m e a n a n d s l o p e t e r m s a r e multiplied b y the s e a s o n a l t e r m . T h i s results in s e a s o n a l variation i n c r e a s i n g a s the  11  trend or s l o p e t e r m s i n c r e a s e , w h e r e a s the additive H W m o d e l m a i n t a i n s c o n s t a n t s e a s o n a l variation a r o u n d the trend a n d s l o p e t e r m s .  (12)  Y (h) = (L hT )S _ l  l+  !  t  L =a(Y IS _ ) t  t  t  p+h  (13)  + (\-a)(L _,+T _,)  p  t  t  T ^ y m - L ^ + il-r)^  (14)  S,=5{Y IL )  + (\-8)S _  Y (h) = L,  since  l  t  t  t  (15)  p  T =0, S,_ t  p+h  =1  (16)  (17)  Y (h) = L,+hT +S _ t  l  l  p+h  where: Y  O b s e r v a t i o n at time t  Y  E s t i m a t e d o b s e r v a t i o n at time t  t  t  h  Forecast horizon  a  S m o o t h i n g p a r a m e t e r for t i m e - v a r y i n g m e a n t e r m  y  S m o o t h i n g p a r a m e t e r for t i m e - v a r y i n g s l o p e  5  S m o o t h i n g p a r a m e t e r for t i m e - v a r y i n g s e a s o n a l contribution  L,  S m o o t h e d level that e s t i m a t e s the t i m e - v a r y i n g m e a n t e r m  T  S m o o t h e d trend that e s t i m a t e s the t i m e - v a r y i n g s l o p e  S,_j  S m o o t h e d trend that e s t i m a t e s the t i m e - v a r y i n g s e a s o n a l contribution for o n e of  t  the p s e a s o n s (j=0,...,p-1)  T h e additive H W m o d e l ( E q u a t i o n 17) c a n b e e x p r e s s e d a s a n A R I M A ( 0 , 1 ,p+1 )(0,1,0)  p  whereas  the multiplicative H W m o d e l c a n n o t b e e x p r e s s e d a s a n A R I M A m o d e l . N o t e that the c o m p o n e n t s of the multiplicative H W m o d e l ( E q u a t i o n s 13-15) a r e not the s a m e a s the c o m p o n e n t s of the additive H W m o d e l . Further, in the additive H W m o d e l the s u m of the s e a s o n a l t e r m s is z e r o w h i l e in the multiplicative H W m o d e l the a v e r a g e of the s e a s o n a l t e r m s is o n e . M o d e l s w e r e c r e a t e d for o n e b e d r o o m d e m a n d a n d two/three b e d r o o m d e m a n d . T w o a n d three b e d r o o m s w e r e c o m b i n e d into a s i n g l e m o d e l s i n c e the H W a l g o r i t h m s require n o n - z e r o e l e m e n t s a n d there w e r e m a n y z e r o v a l u e d a y s for the three b e d r o o m t i m e - s e r i e s .  T o d e t e r m i n e starting v a l u e s of the trend c o m p o n e n t , S A S s o f t w a r e a l l o w s either a c o n s t a n t e s t i m a t e , linear trend e s t i m a t e , or q u a d r a t i c trend e s t i m a t e of the starting v a l u e . A t the resort s t u d i e d , the c o n s t a n t trend e s t i m a t e p r o v i d e d the l o w e s t M S E for o n e - s t e p a h e a d f o r e c a s t s .  12  Further, in the H W m o d e l e s t i m a t e d for the resort, the s e a s o n a l t e r m is a c t u a l l y the p r o d u c t of two t e r m s . T h e first s e a s o n a l t e r m is a w e e k l y t e r m , thus there a r e 5 2 s e a s o n a l w e e k p a r a m e t e r s . T h e s e c o n d s e a s o n a l t e r m is a d a y of w e e k t e r m , thus there a r e s e v e n d a y of w e e k p a r a m e t e r s . A s a result, there a r e 3 6 4 ( 5 2 X 7 ) u n i q u e s e a s o n a l f a c t o r s d e r i v e d f r o m 5 9 (52+7) s e a s o n a l p a r a m e t e r s . T h e multiplicative H W p a r a m e t e r e s t i m a t e s for the entire four y e a r s a m p l e ( s e p a r a t e m o d e l s for o n e b e d r o o m a s well a s two/three b e d r o o m s ) a r e s h o w n in A p p e n d i x B w h i l e the s u m m a r y m o d e l results a r e s h o w n b e l o w in T a b l e 2. T h e H W p a r a m e t e r e s t i m a t e s c a n b e c o m p a r e d directly to the n o r m a l i z e d r o o m night v a l u e s of the p a s t s e a s o n ( F i g u r e 1 A a n d F i g u r e 1 B ) . T h e H W w e e k l y p a r a m e t e r s c a n b e s e e n to b e quite s i m i l a r to the n o r m a l i z e d w e e k l y v a l u e s a l t h o u g h not a s e x t r e m e in high p e r i o d s , w h i l e the H W daily p a r a m e t e r s h a v e m o r e variation than n o r m a l i z e d daily d e m a n d v a l u e s .  Table 2: L o n g - t e r m H o l t - W i n t e r s multiplicative m o d e l results ( M a y 15, 1 9 9 8 to April 2 9 , 2 0 0 2 ) Model  T y p e of model  1 bedroom  Holt-Winters multiplicative  #of parameters 62  C l a s s e s of parameters • • •  2/3 bedroom  Holt-Winters multiplicative  62  • • •  R*  Smoothing p a r a m e t e r s (3) D a y of w e e k p a r a m e t e r s (7) Weekly parameters (52)  .57*  #of observations 1,446  Smoothing p a r a m e t e r s (3) D a y of w e e k p a r a m e t e r s (7)  .57*  1,446  Weekly parameters (52)  T h e R is calculated from 1 step-ahead forecasts for the entire sample period. 2  3.5  A u t o r e g r e s s i v e Integrated M o v i n g A v e r a g e (ARIMA)  A slightly m o r e c o m p l e x t i m e - s e r i e s a p p r o a c h than either R W or H W for m o d e l l i n g daily d e m a n d is a n A R I M A ( a u t o r e g r e s s i v e integrated m o v i n g a v e r a g e ) m o d e l . A R I M A m o d e l s , a s d i s c u s s e d p r e v i o u s l y , a r e m o d e l s d e s c r i b e d in error-correction f o r m . G e n e r a l l y , d a t a is d i f f e r e n c e d (often b y y e a r or b y s o m e other s e a s o n a l period) to i n d u c e stationarity, a n d t h e n the pattern of m o v e m e n t a r o u n d the m e a n t e r m is e s t i m a t e d . T h e pattern of m o v e m e n t a b o u t the m e a n is e s t i m a t e d u s i n g p o l y n o m i a l b a s e d m o d e l s . P o l y n o m i a l b a s e d m o d e l s a r e effective s i n c e t h e y a l l o w a large a m o u n t of variation in the w e i g h t i n g of p a s t o b s e r v a t i o n s b y u s i n g a m i n i m u m n u m b e r of p a r a m e t e r s . F o r e x a m p l e , a s m a l l n u m b e r of p a r a m e t e r s in the n u m e r a t o r a n d d e n o m i n a t o r of  13  the error structure (right h a n d s i d e of a n A R I M A s p e c i f i c a t i o n ) c a n interact to f o r m a c o m p l e x w e i g h t i n g pattern that c a n b e a p p l i e d to a n infinite n u m b e r of o b s e r v a t i o n s .  T h e p r o c e s s of e s t i m a t i n g a n A R I M A m o d e l c a n b e d e s c r i b e d a s a n a l y z i n g the r e s i d u a l s (error t e r m s ) of a t i m e - s e r i e s p r o c e s s a n d a d d i n g a p p r o p r i a t e p a r a m e t e r s until there is no l o n g e r a s y s t e m a t i c c o m p o n e n t in the r e s i d u a l s . O n c e the s y s t e m a t i c c o m p o n e n t (or s i g n a l ) h a s b e e n sufficiently m o d e l l e d , the n e w r e s i d u a l s a r e s a i d to h a v e b e e n r e d u c e d to white n o i s e . W h i t e n o i s e m e a n s a s t o c h a s t i c p r o c e s s with m e a n z e r o . A R I M A m o d e l s a r e not a u t o m a t i c ( i n d e p e n d e n t of m o d e l l e r j u d g e m e n t a n d s p e c i f i c a t i o n ) a s t h e y require the m o d e l l e r to a n a l y z e a u t o c o r r e l a t i o n , i n v e r s e a u t o c o r r e l a t i o n , a n d partial c o r r e l a t i o n plots of the error t e r m s in o r d e r to d e t e r m i n e a p p r o p r i a t e A R I M A p a r a m e t e r s . A s m e n t i o n e d , p a r a m e t e r s a r e d e e m e d a p p r o p r i a t e if t h e y a r e statistically significant (i.e. significant t-statistics at a 9 5 % level of c o n f i d e n c e ) a n d g e n e r a l l y a d d e d until the r e s i d u a l s a r e d e e m e d to b e white n o i s e (as t e s t e d b y c h i - s q u a r e statistics at a 9 5 % level of c o n f i d e n c e ) . S i n c e the A R I M A p r o c e s s is s o flexible, the s a m e w e i g h t i n g f u n c t i o n s c a n b e a c h i e v e d b y a variety of A R I M A s p e c i f i c a t i o n s . T h e r e f o r e , p a r s i m o n y is e x t r e m e l y v a l u a b l e in A R I M A m o d e l s , a n d t h e A k a i k e Information C r i t e r i o n ( A I C ) is often u s e d to j u d g e the a p p r o p r i a t e n e s s of different A R I M A s p e c i f i c a t i o n s ( a n d w a s the o b j e c t i v e u s e d in m o d e l l i n g A R I M A m o d e l s of daily d e m a n d for the resort s t u d i e d ) .  T h e strength a n d w e a k n e s s of A R I M A m o d e l s is that t h e y a r e often a b l e to c a p t u r e patterns not i m m e d i a t e l y a p p a r e n t to the r e s e a r c h e r . In the b e s t i n s t a n c e s , t h e y a l l o w d i s c o v e r y of n e w d a t a patterns a n d h e n c e p r o v i d e better f o r e c a s t s of future o b s e r v a t i o n s . In the w o r s t i n s t a n c e s , t h e y result in a m o d e l that c a n n o t b e interpreted or a m o d e l that h a s s i m p l y overfit the s a m p l e d a t a . Overfit m o d e l s a r e o v e r l y c o m p l e x a n d d o not p r o v i d e better f o r e c a s t s than s i m p l e r m o r e interpretable m o d e l s . In spite of A R I M A m o d e l r e s e r v a t i o n s , t h e s e m o d e l s h a v e b e e n u s e d e x t e n s i v e l y in f i n a n c i a l a n a l y s i s (e.g. prediction of s t o c k m a r k e t data) a n d a r e often c o m b i n e d with e c o n o m e t r i c m o d e l s to further s p e c i f y the error t e r m s g e n e r a t e d by a r e g r e s s i o n - b a s e d a n a l y s i s .  T w o different A R I M A m o d e l s (one, two/three b e d r o o m s ) w e r e s p e c i f i e d for the resort.  Three  b e d r o o m d a t a w a s c o m b i n e d with two b e d r o o m d a t a a s a n A R I M A m o d e l built o n three b e d r o o m d a t a m o d e l a l o n e did not p r o v i d e g o o d e s t i m a t e s d u e to m a n y z e r o v a l u e s . In fact, a two/three b e d r o o m m o d e l p r o v i d e d better e s t i m a t e s than the s u m of a n i n d e p e n d e n t two b e d r o o m m o d e l a n d a three b e d r o o m m o d e l . T h e results of the final A R I M A m o d e l s w e r e f a v o u r a b l e in that f e w p a r a m e t e r s w e r e r e q u i r e d ; the o n e b e d r o o m m o d e l r e q u i r e d o n l y s e v e n p a r a m e t e r s (including m e a n term) a n d the two/three b e d r o o m m o d e l r e q u i r e d six p a r a m e t e r s . T h e two m o d e l s c a n b e described as ARIMA(2,0,2)(1,0,0) (1,1,0) 4 7  36  a n d A R I M A ( 3 , 0 , 1 )(1,0,0) (0,1,1 ) 7  364  respectively.  The  p a r a m e t e r e s t i m a t e s of the o n e b e d r o o m m o d e l a r e s h o w n in E q u a t i o n s 18 a n d the p a r a m e t e r  14  e s t i m a t e s of the two b e d r o o m m o d e l a r e s h o w n in E q u a t i o n 19. P a r a m e t e r e s t i m a t e detail a n d m o d e l fit statistics s h o w n in A p p e n d i x C . It s h o u l d b e n o t e d that n e g a t i v e f o r e c a s t s w e r e r e p l a c e d with z e r o for all A R I M A f o r e c a s t s .  1-.2095 +.U6B (1 - .8755 +. 1185 )(1 - . 175B )(1 + .38 IB ) 2  (1-5  364  ) F , =28.939 +  4  3  (18)  364  1  t  (1-.9215X1-.4685 ) 364  (\-B )Y 3M  t  =  (1 -1.8685 +1.0445 - . 1695 )(1 - .0945 ) 2  3  7  (19)  where:  y,  Observation at time t  B  B a c k w a r d shift operator (e.g. ( 1 - B ) y = y — y -2) 2  t  t  t  R a n d o m disturbance (error) at time t  3.6  Linear R e g r e s s i o n (LR)  L i n e a r r e g r e s s i o n is a c o m m o n c o r r e l a t i o n - b a s e d statistical t e c h n i q u e that h a s at l e a s t o n e input v a r i a b l e , a n d c a l c u l a t e s c o e f f i c i e n t s for e a c h input v a r i a b l e s o that t h e m o d e l e s t i m a t e ( r e s p o n s e v a r i a b l e ) is a linear c o m b i n a t i o n of the input v a r i a b l e s . If a linear c o m b i n a t i o n of d a t a inputs is not a p p r o p r i a t e , often the v a r i a b l e s c a n b e t r a n s f o r m e d s o that e s t i m a t e s a r e still p o s s i b l e within a linear r e g r e s s i o n f r a m e w o r k (e.g. taking l o g s of the d a t a or t a k i n g z - s c o r e s of the data). F o r univariate time s e r i e s d a t a , the m o d e l l e r often c r e a t e s s e p a r a t e b i n a r y input v a r i a b l e s to s p e c i f y m u t u a l l y e x c l u s i v e s e a s o n a l p e r i o d s . F o r e x a m p l e , if a m o d e l l e r w a n t e d to c a l c u l a t e r e g r e s s i o n c o e f f i c i e n t s for 12 p e r i o d s ( m o n t h s ) within a d a t a s e t , s h e m a y c r e a t e 11 n e w input v a r i a b l e s (one m o n t h b e i n g the default m o n t h to p r e v e n t perfect collinearity a m o n g input v a r i a b l e s ) . In this e x a m p l e , a s p e c i f i c m o n t h l y input v a r i a b l e ( s a y F e b r u a r y ) w o u l d b e o n e if the o b s e r v a t i o n w a s t a k e n from this m o n t h , a n d z e r o o t h e r w i s e . In this m a n n e r , e a c h o b s e r v a t i o n w o u l d h a v e at m o s t o n e m o n t h l y v a r i a b l e that w a s n o n - z e r o .  T o c o n t i n u e the e x a m p l e , a s s u m i n g positive o b s e r v a t i o n s , if J a n u a r y w a s t a k e n to b e the default m o n t h , the c a l c u l a t e d r e g r e s s i o n coefficients for the other 11 m o n t h s c a n b e interpreted a s the d i f f e r e n c e b e t w e e n the s p e c i f i e d m o n t h a n d J a n u a r y . If the coefficient for F e b r u a r y w a s p o s i t i v e , then the e x p e c t e d s e a s o n a l i m p a c t of F e b r u a r y o n o b s e r v e d d a t a v a l u e s w o u l d b e h i g h e r than that for J a n u a r y . C o n v e r s e l y , if the coefficient for F e b r u a r y w a s n e g a t i v e , o n e w o u l d e x p e c t l o w e r o b s e r v e d v a l u e s for F e b r u a r y than that of J a n u a r y . In this m a n n e r , b i n a r y v a r i a b l e s w e r e c r e a t e d to r e p r e s e n t s p e c i f i c s e a s o n a l p e r i o d s for the resort.  15  T h e resort m a n a g e r s p r o v i d e d 13 different p e r i o d s t h e y v i e w e d a s distinct. S a m p l e p e r i o d s i n c l u d e d s e v e n winter p e r i o d s (winter is d e f i n e d a s all d a t e s in w h i c h the s k i hill is o p e n for downhill skiing) a n d s i x s u m m e r p e r i o d s . A p e r i o d c o u l d b e d e f i n e d both b y h a r d d a t e s (e.g. D e c e m b e r 2 0 to J a n u a r y 4 - Holiday to the following S a t u r d a y - President's  Period) Week).  a n d soft d a t e s (e.g. F r i d a y prior to P r e s i d e n t ' s D a y B i n a r y v a r i a b l e s w e r e i n c l u d e d in the input  d a t a s e t to r e p r e s e n t t h e s e p e r i o d s (e.g. the binary v a r i a b l e for President's  Week is o n e if a n  o b s e r v a t i o n falls within that w e e k a n d z e r o o t h e r w i s e ) . T h e 13 p e r i o d s w e r e b r o k e n d o w n further by s p e c i f y i n g w e e k s within p e r i o d s a n d the r e g r e s s i o n w a s run to s e e if the additional p a r a m e t e r s w e r e significant at a p=.05 l e v e l . In this w a y p e r i o d s w e r e further s e g m e n t e d or c o m b i n e d until e a c h p a r a m e t e r w a s significant.  T h e final o n e b e d r o o m m o d e l c o n t a i n e d 2 6 s e a s o n a l p e r i o d p a r a m e t e r s w h i l e t h e final t w o b e d r o o m m o d e l i n c l u d e d 2 9 s e a s o n a l period p a r a m e t e r s . T h e d a t a w a s a l s o partitioned b y d a y of w e e k , with a s e p a r a t e p a r a m e t e r for e a c h d a y of the w e e k if significant at a p=.05 l e v e l .  The  final o n e b e d r o o m m o d e l c o n t a i n e d o n e d a y of w e e k p a r a m e t e r w h i l e the final t w o b e d r o o m m o d e l c o n t a i n e d two d a y of w e e k p a r a m e t e r s . N e x t , partitions for d a y of w e e k s e a s o n a l period interactions w e r e c r e a t e d . After s o m e testing, only a w e e k e n d - s e a s o n a l p e r i o d interaction ( w e e k e n d d e f i n e d a s a F r i d a y or S a t u r d a y night) w a s f o u n d to b e significant a n d for o n l y s o m e of the s e a s o n a l p e r i o d s . T h e final o n e b e d r o o m m o d e l c o n t a i n e d nine w e e k e n d p e r i o d interaction p a r a m e t e r s w h i l e the final two b e d r o o m m o d e l c o n t a i n e d o n l y o n e w e e k e n d p e r i o d interaction parameter.  F i n a l l y , the m o d e l i n c l u d e d a y e a r t e r m to c a p t u r e b r o a d - b a s e d y e a r l y t r e n d .  A linear r e g r e s s i o n w a s not a p p r o p r i a t e to m o d e l three b e d r o o m d e m a n d a s it w o u l d l e a d to h e t e r o s k e d a s t i c i t y s i n c e s m a l l c o u n t d a t a violate the a s s u m p t i o n of normality n e c e s s a r y for linear r e g r e s s i o n . O n e a n d t w o b e d r o o m s , o n t h e o t h e r h a n d , h a v e c o u n t d a t a that a r e l a r g e e n o u g h to a d e q u a t e l y a p p r o x i m a t e a n o r m a l distribution.  In o r d e r to o v e r c o m e the h e t e r o s k e d a s t i c i t y  p r o b l e m inherent in s m a l l c o u n t d a t a a P o i s s o n r e g r e s s i o n m o d e l w a s u s e d to m o d e l three b e d r o o m d e m a n d . P o i s s o n r e g r e s s i o n e m p l o y s a q u a s i - m a x i m u m likelihood t e c h n i q u e w h i c h finds c o n d i t i o n a l probabilities b a s e d o n v a l u e s of the e x p l a n a t o r y v a r i a b l e ( s e e W o o l r i d g e , 1 9 9 9 for a full d i s c u s s i o n of P o i s s o n r e g r e s s i o n a n a l y s i s ) . E s s e n t i a l l y , the benefit of u s i n g a P o i s s o n distribution is that it c a n b e fully d e s c r i b e d b y the m e a n t e r m a l o n e , a n d this is e x p l o i t e d to form a log-likelihood function in o r d e r to c a l c u l a t e p a r a m e t e r e s t i m a t e s . M a t h e m a t i c a l l y , the probability that d e m a n d e q u a l s a s p e c i f i c v a l u e (conditional o n input v a r i a b l e s is s h o w n in E q u a t i o n 2 0 ) . Interpretation of the p a r a m e t e r e s t i m a t e s t h e m s e l v e s is quite s i m i l a r to linear r e g r e s s i o n . H o w e v e r , rather than the xB t e r m s predicting y directly a s in linear r e g r e s s i o n , exp(x/?) p r e d i c t s y in a P o i s s o n r e g r e s s i o n .  16  P(y = k\x)  (20)  = exp[-exp(xfi)][exp(xfi)f/k\  where: y  three b e d r o o m d e m a n d  k  v a l u e for three b e d r o o m d e m a n d (k = 0,1,...)  P  input d a t a c o e f f i c i e n t s  x  d a t a input v a l u e s (i.e. s e a s o n a l b i n a r y v a r i a b l e s )  T h e input d a t a for the P o i s s o n r e g r e s s i o n w a s v e r y s i m i l a r to the input d a t a for the linear r e g r e s s i o n . T h e final m o d e l i n c l u d e d 2 0 s e a s o n a l p e r i o d p a r a m e t e r s , 1 d a y of w e e k p a r a m e t e r , a n d 8 w e e k e n d p e r i o d interaction p a r a m e t e r s . F u r t h e r m o r e , no y e a r l y trend in the n u m b e r of units b o o k e d w a s o b s e r v e d for the three b e d r o o m m o d e l s o n o y e a r l y trend c o m p o n e n t w a s i n c l u d e d in the m o d e l . T h e linear r e g r e s s i o n a n d P o i s s o n r e g r e s s i o n results for the i n - s a m p l e p e r i o d a r e s h o w n in T a b l e 3^ with d e t a i l e d p a r a m e t e r e s t i m a t e s a n d m o d e l fit statistics for the o n e a n d two b e d r o o m m o d e l s s h o w n in A p p e n d i x D, a n d the p a r a m e t e r e s t i m a t e s a n d m o d e l fit statistics for the three b e d r o o m m o d e l s h o w n in A p p e n d i x E .  T a b l e 3 : L o n g - t e r m linear r e g r e s s i o n m o d e l results ( M a y 1 5 , 1 9 9 8 to A p r i l 2 9 , 2 0 0 2 ) Model 1 bedroom  2 bedroom  3+ bedroom  T y p e of model Linear regression  Linear regression  Poisson regression  #of parameters 38  37  29  C l a s s e s of parameters • •  G e n e r a l intercept (1) P e r i o d intercepts (26)  •  D a y of w e e k intercepts (1)  •  D e m a n d trend (1)  •  W e e k e n d period interactions (9)  •  G e n e r a l intercept (1)  • •  P e r i o d intercepts (32) D a y of w e e k intercepts (2)  • •  D e m a n d trend (1) W e e k e n d period interactions (1)  • •  P e r i o d intercepts (20) D a y of w e e k intercepts (1)  •  W e e k e n d period interactions (8)  R  z  #of  .77  observations 1,446  .72  1,446  .30*  1,446  Minimizing S S E (sum of square errors) is not the objective function of a Poisson regression; however, a linear regression was run with the same parameters to get an approximate R . 2  17  Non-Linear R e g r e s s i o n (NL)  3.7  T h e m o s t important d e c i s i o n to b e m a d e with r e s p e c t to a c u s t o m i z e d l o n g - t e r m m o d e l w a s a n a p p r o p r i a t e f u n c t i o n a l f o r m that w o u l d s p e c i f i c a l l y c a p t u r e the resort's d e m a n d situation (rather t h a n s a y a m o r e a u t o m a t i c m o d e l s u c h a s a H W m o d e l ) . Initially it h a d b e e n t h o u g h t that d e m a n d a s a p e r c e n t a g e of c a p a c i t y m a y b e a g o o d m e a s u r e of d e m a n d . O v e r the four y e a r period for w h i c h d a t a h a d b e e n p r o v i d e d , the l o d g i n g c a p a c i t y h a d i n c r e a s e d e a c h y e a r in r o u g h l y a linear trend. H o w e v e r , the o b s e r v e d o c c u p a n c y rates for t h o s e p e r i o d s w e r e not c o n s t a n t . W h a t w a s h a p p e n i n g w a s that the n u m b e r of units o c c u p i e d i n c r e a s e d w h e n c a p a c i t y i n c r e a s e d , but not in the s a m e proportion a s the i n c r e a s e in c a p a c i t y . F o r i n s t a n c e , the h i g h e s t d e m a n d p e r i o d of the y e a r , the N e w Y e a r ' s h o l i d a y , w o u l d g e n e r a l l y b e c l o s e to c a p a c i t y r e g a r d l e s s of the a b s o l u t e i n c r e a s e in c a p a c i t y for the y e a r . O n the other h a n d , s l o w s h o u l d e r p e r i o d s (e.g. e a r l y M a y a n d e a r l y N o v e m b e r ) s h o w e d a l m o s t n o i n c r e a s e in d e m a n d y e a r o v e r y e a r r e g a r d l e s s of n e w l y a d d e d c a p a c i t y . O t h e r p e r i o d s , d e f i n e d a s m i d to high s e a s o n , s h o w e d a n i n c r e a s e in d e m a n d y e a r o v e r y e a r , but not in the s a m e proportion a s the i n c r e a s e in c a p a c i t y . A s a result, in a n attempt to c a p t u r e the i d i o s y n c r a t i c d e m a n d e l e m e n t s of the resort s t u d i e d , a n o n l i n e a r r e g r e s s i o n m o d e l w a s e s t i m a t e d with two c o m p o n e n t s ( E q u a t i o n 2 1 ) .  UNITS ,  = DEMAND  day year  DEMAND SHARE  d a y  year  = /?„ +  * SHARE  (21)  day  YEAR  (22)  = f(seasonal period, day o f week, seasonal period day o f week interaction)  SHARE = day  year  (23)  (24)  1  1 + exp(-xp)  where: /?  input d a t a c o e f f i c i e n t s  x  d a t a input v a l u e s (i.e. s e a s o n a l b i n a r y v a r i a b l e s )  T h e first c o m p o n e n t is a n e s t i m a t e of the m a x i m u m potential daily d e m a n d in a g i v e n y e a r ( E q u a t i o n 2 2 a n d s h o w n g r a p h i c a l l y for o n e b e d r o o m units in F i g u r e 2). T h e s e c o n d c o m p o n e n t is a logistic function that d e t e r m i n e s the s h a r e of m a x i m u m daily d e m a n d (up to 1 0 0 % ) b a s e d on s e a s o n a l f a c t o r s ( E q u a t i o n 2 3 a n d s h o w n g r a p h i c a l l y for o n e b e d r o o m units in F i g u r e 3). A logistic function is a p p r o p r i a t e a s a s h a r e of d e m a n d function s i n c e it is b o u n d e d b e t w e e n z e r o a n d o n e ( E q u a t i o n 2 3 is e x p r e s s e d m a t h e m a t i c a l l y in E q u a t i o n 24). T h e 0 c o e f f i c i e n t s a r e thus e s t i m a t e d s o that the linear xfi t e r m s a r e e x t r e m e l y positive in high d e m a n d p e r i o d s a n d e x t r e m e l y n e g a t i v e in l o w d e m a n d p e r i o d s .  18  E s t i m a t i n g daily d e m a n d a s a s h a r e of e s t i m a t e d m a x i m u m potential daily d e m a n d w a s e x p e c t e d to p r o v i d e a better e s t i m a t e of d e m a n d than d e m a n d a s a s h a r e of c a p a c i t y . T h i s h y p o t h e s i s w a s s u p p o r t e d b y a n a l y z i n g d a t a during the four y e a r s a m p l e p e r i o d ; i n c r e a s e s in y e a r l y c a p a c i t y often did not l e a d to a proportionate i n c r e a s e in y e a r l y d e m a n d . A logistic s h a r e of d e m a n d c o m p o n e n t c r e a t e s a multiplicative s e a s o n a l c o m p o n e n t rather than a n a d d i t i v e s e a s o n a l c o m p o n e n t a s in linear r e g r e s s i o n . A multiplicative m o d e l is m o r e intuitive s i n c e t h e variation in d e m a n d a m o n g s e a s o n a l p e r i o d s is likely to i n c r e a s e with o v e r a l l y e a r l y d e m a n d rather than s t a y i n g c o n s t a n t . S t a t e d differently, a n additive m o d e l is b a s e d o n t h e a s s u m p t i o n that t h e d i f f e r e n c e in units o c c u p i e d b e t w e e n high a n d l o w d e m a n d p e r i o d s r e m a i n s c o n s t a n t f r o m y e a r to y e a r . A multiplicative m o d e l , o n the other h a n d , is b a s e d o n t h e a s s u m p t i o n that the d i f f e r e n c e in units o c c u p i e d b e t w e e n high a n d l o w d e m a n d p e r i o d s is a proportion of o v e r a l l m a x i m u m d e m a n d . In a multiplicative m o d e l , a s overall y e a r l y d e m a n d i n c r e a s e s , t h e d i f f e r e n c e in units o c c u p i e d b e t w e e n high a n d l o w d e m a n d p e r i o d s i n c r e a s e s .  700 -,  615  600 A  500  1 Bedroom Unit Demand  A  400 -  300 -J  200 A  100 A  0 '98/99  '99/00  '00/01  '01/02  '02/03  Year Figure 2: E s t i m a t e d m a x i m u m daily d e m a n d for o n e b e d r o o m units b y y e a r  19  Figure 3 : P r e d i c t e d s h a r e of m a x i m u m daily d e m a n d for o n e b e d r o o m units o n s e l e c t e d d a t e s in 0 2 / 0 3 ( 6 1 5 units = 1 0 0 % )  T h r e e b e d r o o m d e m a n d , a s m e n t i o n e d earlier, h a d s m a l l c o u n t d a t a ( c a p a c i t y l e s s than 15 with a n a v e r a g e n u m b e r of o c c u p i e d units l e s s t h a n 3). S i m i l a r to linear r e g r e s s i o n , a n o n l i n e a r r e g r e s s i o n w o u l d h a v e h a d h e t e r o s k e d a s t i c i t y p r o b l e m s with s u c h c o u n t d a t a . A s a result, the s a m e three b e d r o o m P o i s s o n r e g r e s s i o n e s t i m a t e s that w e r e c o m b i n e d with l i n e a r r e g r e s s i o n e s t i m a t e s w e r e c o m b i n e d with n o n l i n e a r r e g r e s s i o n e s t i m a t e s for a g g r e g a t e d e m a n d e s t i m a t e s .  T h e input v a r i a b l e s for the n o n l i n e a r r e g r e s s i o n m o d e l b e l o n g to the s a m e c a t e g o r i e s of input v a r i a b l e s i n c l u d e d in t h e linear r e g r e s s i o n m o d e l : s e a s o n a l p e r i o d p a r a m e t e r s , d a y of w e e k p a r a m e t e r s , s e a s o n a l period w e e k e n d interaction p a r a m e t e r s , a n d a y e a r l y trend p a r a m e t e r . T h e d i f f e r e n c e is that all the inputs e x c l u d i n g the y e a r l y trend input w e r e i n c l u d e d in a logistic function ( E q u a t i o n 2 3 ) that w a s then c o m b i n e d with a y e a r l y d e m a n d function b a s e d o n y e a r . T h e final o n e b e d r o o m logistic function c o n t a i n e d 3 2 s e a s o n a l p e r i o d p a r a m e t e r s , 2 d a y of w e e k p a r a m e t e r s , a n d 9 w e e k e n d period interaction p a r a m e t e r s . T h e final two b e d r o o m logistic function i n c l u d e d 3 3 s e a s o n a l p e r i o d p a r a m e t e r s , 5 d a y of w e e k p a r a m e t e r s , a n d 3 w e e k e n d p e r i o d interaction p a r a m e t e r s . N o n l i n e a r m o d e l results a r e s h o w n in T a b l e 4 w h i l e d e t a i l e d  20  p a r a m e t e r e s t i m a t e s a n d m o d e l fit statistics a r e s h o w n in A p p e n d i x F. R e s u l t s of the three b e d r o o m P o i s s o n r e g r e s s i o n a r e s h o w n in A p p e n d i x E .  T a b l e 4: L o n g - t e r m n o n l i n e a r r e g r e s s i o n m o d e l results ( M a y 15, 1 9 9 8 to A p r i l 2 9 , 2 0 0 2 ) Model 1 bedroom  T y p e of model Nonlinear regression  #of parameters 45  • • •  2 bedroom  Nonlinear regression  43  29  P e r i o d intercepts (32) D a y of w e e k intercepts (2) D e m a n d trend (2)  .82  #of observations 1,446  W e e k e n d period interactions (9)  •  P e r i o d intercepts (33) D a y of w e e k intercepts (5) D e m a n d trend (2) W e e k e n d period interactions (3)  .78  1,446  P e r i o d intercepts (20) D a y of w e e k intercepts (1) W e e k e n d period interactions (8)  .30*  1,446  • • Poisson regression  R  •  •  3+ bedroom  C l a s s e s of parameters  • • •  *Minimizing SSE (sum of square errors) is not the objective function of a Poisson regression; however, a linear regression was run with the same parameters to get an approximate R . 2  3.8  Long-Term Model Comparison  T h e long-term m o d e l s w e r e c r e a t e d to f o r e c a s t d e m a n d m o r e than 9 0 d a y s prior to a target d a t e . A s a result, the five l o n g - t e r m m o d e l s ( R W , H W , L R , A R I M A , N L ) w e r e c o m p a r e d within a n ins a m p l e p e r i o d a s well a s within a n out of s a m p l e p e r i o d . A p p r o p r i a t e f u n c t i o n a l f o r m s for all l o n g t e r m m o d e l s w e r e c o n s t r u c t e d u s i n g the entire four y e a r s a m p l e . T h e m o d e l e s t i m a t e s w e r e then f o r e c a s t out for y e a r 4 within s a m p l e a n d the results c o m p a r e d . F o r the L R a n d N L m o d e l , the entire s a m p l e w a s u s e d to c a l c u l a t e p a r a m e t e r e s t i m a t e s (input c o e f f i c i e n t s ) a n d t h e s e s a m e c o e f f i c i e n t s w e r e u s e d for the i n - s a m p l e f o r e c a s t s . T h e H W a n d A R I M A m o d e l s a l s o u s e d the entire four y e a r s a m p l e to d e t e r m i n e m o d e l structure ( e . g . n u m b e r a n d t y p e of p a r a m e t e r s for the A R I M A m o d e l s ) . H o w e v e r , p a r a m e t e r e s t i m a t e s for t h e s e m o d e l s v a r y b y d a y , s o the e s t i m a t e s for y e a r 4 w e r e b a s e d o n d a t a up to y e a r 3 a n d then f o r e c a s t out for y e a r 4 . R W is not b a s e d o n a n y m o d e l , a n d d e m a n d e s t i m a t e s w e r e s i m p l y t a k e n f r o m 3 6 4 d a y s prior.  21  T h e results of the five m o d e l s in the i n - s a m p l e p e r i o d a r e s h o w n in T a b l e 5. T h r e e different error m e a s u r e s a r e s h o w n : M S E ( m e a n s q u a r e error), M d A P E ( m e d i a n a b s o l u t e p e r c e n t a g e error), a n d C u m R A E ( c u m u l a t i v e relative a b s o l u t e error). F o r the i n - s a m p l e p e r i o d , the N L m o d e l is s h o w n to b e s u p e r i o r o n all error m e a s u r e s a l t h o u g h A r m s t r o n g & C o l l o p y s u g g e s t C u m R A E is the m o s t robust error m e t r i c for m o d e l c o m p a r i s o n in this i n s t a n c e . A C u m R A E v a l u e of .738 i n d i c a t e s that the N L m o d e l c o n t a i n s 7 3 . 8 % of the c u m u l a t i v e error of the R W m e t h o d , t h e r e b y indicating a 2 6 . 2 % i m p r o v e m e n t o v e r R W . A C u m R A E v a l u e of 1.356 for the H W m o d e l i n d i c a t e s e s t i m a t e s that a r e 3 5 . 6 % m o r e i n a c c u r a t e than R W .  T a b l e 5: I n - s a m p l e long-term m o d e l c o m p a r i s o n s (April 2 8 , 2001 to A p r i l 2 7 , 2 0 0 2 ) Model R a n d o m walk ( R W ) Nonlinear regression (NL) ARIMA H o l t - W i n t e r s multiplicative ( H W ) Linear regression (LR)  MSE  CumRAE  12,711  1.000  6,333 10,431 22,175 9,113  .738 .993 1.356 .992  MdAPE 25.7% 20.4% 31.8% 37.5% 33.9%  T h e m o d e l s w e r e a l s o c o m p a r e d out of s a m p l e . S i n c e l o n g - t e r m e s t i m a t e s a r e f o r e c a s t s  more  than 9 0 d a y s prior to a target d a t e , the m o d e l s f o r e c a s t d e m a n d m o r e than 9 0 d a y s after the last d a t e of i n - s a m p l e d a t a (April 2 9 , 2 0 0 2 ) . A s a result the out of s a m p l e p e r i o d w a s J u l y 2 9 , 2 0 0 2 to N o v e m b e r 3 0 , 2 0 0 2 a n d the results a r e s h o w n in T a b l e 6. A s c a n b e s e e n in T a b l e 6, the N L m o d e l is still s u p e r i o r , but by a m u c h n a r r o w e r m a r g i n of i m p r o v e m e n t (3.3%) t h a n i n - s a m p l e ( 2 6 . 2 % ) . A s at D e c e m b e r 1, 2 0 0 2 the resort u n e x p e c t e d l y lost 1 0 5 units of c a p a c i t y d u e to a hotel property s w i t c h i n g r e s e r v a t i o n m a n a g e m e n t provider. A s a result, the a s s u m p t i o n u n d e r l y i n g the N L a n d L R m o d e l s w a s v i o l a t e d , a n d the quality of e s t i m a t e s significantly d e t e r i o r a t e d . T h e l o s s of c a p a c i t y a l s o i n c r e a s e d the error in the other l o n g - t e r m f o r e c a s t i n g m e t h o d s ( s e e M d A P E m e a s u r e s ) but s i n c e the other m e t h o d s w e r e not b a s e d o n a n i n c r e a s i n g y e a r l y trend in d e m a n d t h e y w e r e not a s a d v e r s e l y a f f e c t e d . T h e out of s a m p l e p e r i o d post D e c e m b e r 1, 2 0 0 2 is s h o w n in T a b l e 7. In this p e r i o d , the R W m e t h o d is far s u p e r i o r to other long-term m e t h o d s ; providing a m i n i m u m 3 8 % i m p r o v e m e n t o v e r all other l o n g - t e r m m o d e l s .  Table 6: O u t of s a m p l e long-term m o d e l c o m p a r i s o n s (July 2 9 , 2 0 0 2 to N o v e m b e r 3 0 , 2 0 0 2 ) Model R a n d o m walk ( R W )  MSE  CumRAE  Nonlinear regression (NL) ARIMA  10,987 7,187 9,932  1.000 .967 1.098  H o l t - W i n t e r s multiplicative ( H W ) Linear regression (LR)  24,273 10,475  1.610 1.398  22  MdAPE 23.1% 30.8% 32.0% 35.6% 72.0%  T a b l e 7: O u t of s a m p l e long-term m o d e l c o m p a r i s o n s ( D e c e m b e r 1, 2 0 0 2 to J a n u a r y 2 2 , 2 0 0 3 ) Model R a n d o m walk ( R W ) Nonlinear regression (NL) ARIMA H o l t - W i n t e r s multiplicative ( H W ) Linear regression (LR)  MSE  CumRAE  MdAPE  3,325 11,176 5,269 37,106 8,580  1.000 1.698 1.377  13.8% 28.5% 27.3% 35.6% 26.0%  3.033 1.667  23  4  SHORT-TERM MODELS  M o s t p a p e r s o n hotel f o r e c a s t i n g e m p l o y o n e of two a p p r o a c h e s : a l o n g - t e r m f o r e c a s t or a shortt e r m f o r e c a s t . T h e long-term f o r e c a s t u s e s past y e a r s ' d a t a o n d a i l y o c c u p a n c y to predict d a i l y o c c u p a n c y in the future.  L o n g - t e r m f o r e c a s t s ignore the b u i l d u p of b o o k i n g s for d a t e s in the  future (they a r e not a d j u s t e d for a c t u a l b o o k i n g s to date). S h o r t - t e r m f o r e c a s t s , o n the other h a n d , a n a l y z e the build-up of b o o k i n g s for a s i n g l e future d a t e (target d a t e ) , a n d t h e n project final d e m a n d for that target d a t e b a s e d o n a c t u a l b o o k i n g s to d a t e . S o m e short-term f o r e c a s t s treat target d a t e s in isolation, ignoring the final o c c u p a n c y f i g u r e s of y e a r s p a s t w h i l e o t h e r s integrate both a c t u a l b o o k i n g s to d a t e a s well a s final o c c u p a n c y f i g u r e s f r o m y e a r s p a s t . T h i s p a p e r a n a l y z e s integrative short-term f o r e c a s t s a s they u s e all a v a i l a b l e information a n d will b e s h o w n to p r o v i d e better e s t i m a t e s than either m o d e l s b a s e d entirely o n p a s t o c c u p a n c y d a t a or m o d e l s b a s e d entirely o n b o o k i n g s to d a t e . T h e two short-term m e t h o d s to b e s t u d i e d i n c l u d e additive p i c k u p ( A P ) a n d a c u s t o m i z e d b o o k i n g c u r v e ( B C ) m o d e l w h i c h is b a s e d o n a n o n - l i n e a r m o d e l n e a r l y identical to the long-term N L m o d e l .  4.1  Additive P i c k u p (AP)  A P is a s i m p l e yet robust short-term f o r e c a s t i n g m e t h o d w h i c h a u t o m a t i c a l l y i n t e g r a t e s prior y e a r o c c u p a n c y d a t a a s well a s a c t u a l b o o k i n g s to d a t e . It c a n b e thought of a s a d e t a i l e d r a n d o m w a l k . T h e A P e s t i m a t e for a target d a t e is b o o k i n g s to d a t e plus e x p e c t e d p i c k u p . A P is u s e d e x t e n s i v e l y in the airline industry for f o r e c a s t i n g p a s s e n g e r p i c k u p (short-term p a s s e n g e r d e m a n d ) ; for s p e c i f i c m o d e l s p e c i f i c a t i o n s s e e H a r r i s & M a r u c c i , 1 9 8 3 a n d L ' H e u r e u x , 1 9 8 6 . O f t e n a d e v i a n t of a direct A P m e t h o d is u s e d w h e r e a n e x p o n e n t i a l m o v i n g a v e r a g e of a s u b s e t of flights' p i c k u p is u s e d to predict p i c k u p for a current flight. T h e s u b s e t of a p p r o p r i a t e flights m a y b e b a s e d o n d a y of w e e k , s e a s o n a l p e r i o d , or o p e r a t i n g e n v i r o n m e n t s u c h a s a f a r e s a l e .  F o r the resort s t u d i e d , e x p e c t e d p i c k u p w a s d e f i n e d a s the p i c k u p that w a s e x p e r i e n c e d in the y e a r prior (364 d a y s prior s o that the p i c k u p is from the s a m e d a y of w e e k a n d s e a s o n a l period). A s a n e x a m p l e , s u p p o s e it is 15 d a y s prior to a target d a t e of D e c e m b e r 2 0 , 2 0 0 3 a n d there a r e 5 0 0 b o o k i n g s to d a t e . T o find the e x p e c t e d p i c k u p o n e w o u l d look at last y e a r ' s b o o k i n g s for the target d a t e D e c e m b e r 2 1 , 2 0 0 2 in the 15 d a y p e r i o d prior to the target d a t e ; let's s a y there w e r e 3 0 0 b o o k i n g s in that p e r i o d . In this c a s e , the A P e s t i m a t e for D e c e m b e r 2 0 , 2 0 0 3 is 8 0 0 (500 b o o k i n g s to d a t e + 3 0 0 e x p e c t e d p i c k u p ) . T h e o n e c o m p l i c a t i o n that s h o u l d b e m e n t i o n e d is that s o m e of the b o o k i n g s to d a t e will c a n c e l . T h e m e t h o d u s e d in this p a p e r w a s to i n c l u d e all c a n c e l l a t i o n s a s part of the e x p e c t e d p i c k u p . F o r e x a m p l e , s u p p o s e the b o o k i n g s to d a t e Y d a y s prior to a target d a t e w e r e 5 0 0 . F u r t h e r s u p p o s e that in the y e a r prior, Y d a y s b e f o r e the target d a t e , 4 0 0 n e w b o o k i n g s w e r e m a d e in the Y d a y interval a n d 100 c a n c e l l a t i o n s w e r e m a d e in the  24  Y d a y interval. T h e n there w o u l d b e a n e x p e c t e d p i c k u p of 3 0 0 units (400 n e w b o o k i n g s l e s s 1 0 0 cancellations).  4.2  B o o k i n g C u r v e Estimate (BC)  In o r d e r to u n d e r s t a n d t h e B C e s t i m a t e it is important to e x p l o r e t h e c o n c e p t of a b o o k i n g c u r v e . T h e typical b o o k i n g c u r v e (pattern of b o o k i n g s o v e r time) for a s p e c i f i c d a t e in t h e future (target date) is g e n e r a l l y a c o n v e x c u r v e , with t h e m o s t b o o k i n g s o c c u r r i n g in the w e e k i m m e d i a t e l y prior to a target d a t e ( s e e F i g u r e 4 ) .  250  Cumulative Bookings  0.00  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  1.00  Lead time (0=90 days, 1=Target Date)  Figure 4: T y p i c a l b o o k i n g c u r v e ( b o o k i n g c u r v e for target d a t e of J u l y 5, 2 0 0 1 )  H o w e v e r , t h e b o o k i n g c u r v e c h a n g e s in a n a b s o l u t e s e n s e (overall n u m b e r of b o o k i n g s ) a n d relative s e n s e ( s h a p e of b o o k i n g c u r v e ) d e p e n d i n g o n t h e t i m e of y e a r . H i g h d e m a n d p e r i o d s g e n e r a l l y yield b o o k i n g c u r v e s that a r e c o n c a v e with high o v e r a l l b o o k i n g s w h i l e l o w d e m a n d p e r i o d s p r o d u c e c u r v e s that a r e c o n v e x with l o w overall b o o k i n g s ( s e e F i g u r e 5 A a n d F i g u r e 5 B ) . R e s o r t hotels t e n d to d i s p l a y m o r e s e a s o n a l i t y than b u s i n e s s - o r i e n t e d hotels a n d a s s u c h the variation of b o o k i n g c u r v e s for a resort hotel t e n d to b e larger t h a n that of a b u s i n e s s hotel.  25  80  Cumulative Bookings  0J 0.00  , 0.10  r0.20  , 0.30  , 0.40  , 0.50  , 0.60  , 0.70  , 0.80  , 0.90  , 1.00  Lead time (0=90 days, 1=Target Date)  Figure 5A: L o w d e m a n d p e r i o d - c o n v e x b o o k i n g c u r v e ( b o o k i n g c u r v e for target d a t e of D e c e m b e r 6, 2 0 0 1 ) 800  -i  100  A  Cumulative Bookings  O-l 0.00  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  0.10  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  1.00  Lead time (0=90 days, 1=target date)  Figure 5B: H i g h d e m a n d p e r i o d - c o n c a v e b o o k i n g c u r v e ( b o o k i n g c u r v e for target d a t e of December 28, 2001)  26  T h i s p a p e r utilizes a n a p p r o a c h s i m i l a r to that u s e d by R a j o p a d h y e et a l . ( 1 9 9 9 ) in w h i c h l o n g t e r m e s t i m a t e s a r e u s e d to predict future d e m a n d , a n d t h e s e e s t i m a t e s a r e c o n t i n u a l l y a d j u s t e d b a s e d o n b o o k i n g s to d a t e . T h e p r o c e s s to a c h i e v e the resort's s h o r t - t e r m b o o k i n g c u r v e e s t i m a t e is outlined in F i g u r e 6.  Actual Bookings to Date  1r Prior Years' Data  •  Booking Curve Baseline Model  Expected Bookings to Date  Booking Curve Adjustment  Booking Curve Projection Estimate  Weighting Function  Prior Years' Data  Long-Term Model  Short-Term Demand Estimate  Long-Term Demand Estimate  Figure 6: F l o w c h a r t of f o r e c a s t i n g p r o c e s s for b o o k i n g c u r v e s h o r t - t e r m e s t i m a t e  T h e short-term d e m a n d e s t i m a t e is a w e i g h t e d a v e r a g e o f t h e l o n g - t e r m d e m a n d e s t i m a t e a n d the b o o k i n g c u r v e projection e s t i m a t e . T h e long-term e s t i m a t e is d e r i v e d f r o m a m o d e l b a s e d o n final d a i l y o c c u p a n c y f i g u r e s in y e a r s past. In this p a p e r t h e N L m o d e l is u s e d to p r o d u c e a l o n g term e s t i m a t e (although theoretically a n y of the long-term m o d e l s c o u l d b e u s e d ) . T h e b o o k i n g c u r v e projection e s t i m a t e , o n the other h a n d , is c o m p o s e d of two s t e p s . First, a b a s e l i n e b o o k i n g c u r v e m o d e l u s e s t h e pattern of b o o k i n g s to d a t e in y e a r ' s p a s t to c r e a t e e x p e c t a t i o n s of current b o o k i n g s to d a t e . S e c o n d , e x p e c t e d b o o k i n g s to d a t e a n d a c t u a l b o o k i n g s to d a t e a r e input into a n a d j u s t m e n t function that c r e a t e s a n e s t i m a t e of final d e m a n d ( b o o k i n g c u r v e projection e s t i m a t e ) . Finally, a w e i g h t i n g function ( b a s e d o n l e a d time) c o m b i n e s the l o n g - t e r m e s t i m a t e a n d b o o k i n g c u r v e projection e s t i m a t e to c o m e up with t h e s h o r t - t e r m d e m a n d e s t i m a t e .  In this  w a y , t h e short-term d e m a n d e s t i m a t e is c o n t i n u a l l y u p d a t e d a s n e w b o o k i n g s a r e m a d e , e x i s t i n g b o o k i n g s a r e c a n c e l l e d , a n d t h e target d a t e a p p r o a c h e s .  27  4.2.1 Booking curve baseline model T h e b o o k i n g c u r v e b a s e l i n e m o d e l is s i m i l a r to the long-term N L m o d e l in its structure (two c o m p o n e n t n o n l i n e a r r e g r e s s i o n for o n e a n d two b e d r o o m s a n d P o i s s o n r e g r e s s i o n for three p l u s b e d r o o m s ) . H o w e v e r , rather than p r o v i d e a s i n g l e point e s t i m a t e for a target d a t e , t h e m o d e l p r o v i d e s a n e s t i m a t e for e a c h of the 9 0 d a y s prior to a target d a t e a s well a s the target d a t e itself. T h e m a j o r d i f f e r e n c e b e t w e e n t h e short-term B C m o d e l a n d the long-term N L m o d e l is t h e i n c l u s i o n of a l e a d time e l e m e n t . A l e a d time e l e m e n t e n a b l e s t h e m o d e l to a c c o u n t for the i n c r e a s e in b o o k i n g s a s t h e target d a t e a p p r o a c h e s . T h e l e a d time e l e m e n t is a l s o interacted with s e a s o n a l p e r i o d b i n a r y v a r i a b l e s s o that t h e s h a p e of the b o o k i n g c u r v e c a n v a r y b y p e r i o d (i.e. c o n c a v e for high d e m a n d d a y s a n d c o n v e x for l o w d e m a n d d a y s ) . T h e l e a d time p a r a m e t e r s a s w e l l a s t h e l e a d time s e a s o n a l period interaction p a r a m e t e r s a r e c a p t u r e d in the logistic c o m p o n e n t of the m o d e l . T h e logistic function for s h a r e of d e m a n d w a s u s e d s i n c e it is a g o o d r e p r e s e n t a t i o n of the b o o k i n g c u r v e ( s e e F i g u r e 7). A n a p p r o a c h i n g target d a t e is e q u i v a l e n t to following the logistic c u r v e f r o m left to right for a s p e c i f i e d interval.  Share of Maximum Daily Demand (%)  -5  -4  -3  -2  -1  0  1  Transformed Date Values  Figure 7: G e n e r i c logistic c u r v e  28  2  3  4  5  T h e left h a n d s i d e of the logistic function c l o s e l y r e s e m b l e s the 9 0 - d a y b o o k i n g c u r v e for m o s t d a y s with a traditional c o n v e x build u p ( i m a g i n e F i g u r e 5 A s u p e r i m p o s e d o n the left h a n d s i d e of F i g u r e 7). T h e right h a n d s i d e of the logistic function c l o s e l y r e s e m b l e s a 9 0 - d a y b o o k i n g c u r v e for a high d e m a n d d a y ( i m a g i n e F i g u r e 5 B s u p e r i m p o s e d o n the right h a n d s i d e of F i g u r e 7). T h e r e f o r e , c h o o s i n g a n a p p r o p r i a t e intercept a l o n g the logistic function for a s p e c i f i c target d a t e (to m a r k the b e g i n n i n g of a s p e c i f i c time interval) a s well a s i n c l u d i n g l e a d time s e a s o n a l p e r i o d interactions p r o v i d e s a flexible functional form to a p p r o x i m a t e b o o k i n g c u r v e s for a s p e c i f i c target d a t e a n d l e a d time. T h e large a m o u n t of variation e x p l a i n e d by the b o o k i n g c u r v e b a s e l i n e r e g r e s s i o n m o d e l s is e v i d e n c e of the a p p r o p r i a t e n e s s of the logistic f u n c t i o n a l f o r m within the n o n l i n e a r r e g r e s s i o n m o d e l ( T a b l e 8). T h e b a s e l i n e r e g r e s s i o n p a r a m e t e r e s t i m a t e s a n d m o d e l fit statistics a r e s h o w n in A p p e n d i x G .  29  Table 8: B o o k i n g c u r v e b a s e l i n e r e g r e s s i o n m o d e l results ( M a y 15, 1 9 9 8 to A p r i l 2 9 , 2 0 0 2 ) Model 1 bedroom  T y p e of model Nonlinear regression  #of Parameters 112  C l a s s e s of parameters  R  •  P e r i o d intercepts (57)  .84  #of observations 131,586  •  .82  131,586  .41*  131,586  •  D a y of w e e k intercepts (5) D e m a n d trend (2)  •  L e a d - t i m e e l e m e n t s (2)  •  Period lead-time interactions (24) W e e k e n d period interactions (22)  •  2 bedroom  Nonlinear regression  85  • • • • • •  3+ bedroom  Poisson regression  80  P e r i o d intercepts (47) D a y of w e e k intercepts (5) D e m a n d trend (2) L e a d - t i m e e l e m e n t s (2) Period lead-time interactions (15) W e e k e n d period interactions (14)  •  G e n e r a l intercept (1)  • •  P e r i o d intercepts (50) D a y of w e e k intercepts (5) L e a d - t i m e e l e m e n t s (1) Period lead-time interactions (12) W e e k e n d period interactions (11)  • • •  2  Minimizing S S E (sum of square errors) is not the objective function of a Poisson regression; however, a linear regression was run with the same parameters to get an approximate R . 2  4.2.2  Booking curve adjustment  E x p e c t e d b o o k i n g s to d a t e for a s p e c i f i c target d a t e a n d l e a d time from the b o o k i n g c u r v e b a s e l i n e m o d e l is u s e d a s a b a s e l i n e figure to b e c o m p a r e d with a c t u a l b o o k i n g s to d a t e .  A  b o o k i n g c u r v e projection of final d e m a n d ( n u m b e r of units d e m a n d e d at the target d a t e w h e n l e a d time e q u a l s z e r o ) is thus the e x p e c t e d b o o k i n g s for the target d a t e a d j u s t e d b y a function of the a c t u a l b o o k i n g s to d a t e . F i v e different a p p r o a c h e s for a b o o k i n g c u r v e projection w e r e a t t e m p t e d . T h e i d e a w a s to a d j u s t t h e projection b y a n a m o u n t p r o p o r t i o n a l to t h e d e v i a t i o n (actual l e s s e x p e c t e d b o o k i n g s ) at a certain l e a d time ( s e e A p p e n d i x H for c a l c u l a t i o n s a n d notation for the first four a p p r o a c h e s ) . T h e fifth a p p r o a c h w a s s o m e w h a t different in that it e m p l o y e d a n A R I M A m o d e l to e s t i m a t e the pattern of b o o k i n g c u r v e errors to d a t e , a n d t h e n projected that pattern to the target d a t e . T h e results of the A R I M A m o d e l w e r e m i x e d , a n d the  30  a p p r o a c h w a s ultimately d i s c a r d e d d u e to additional c o m p l e x i t y in c o m p u t a t i o n a n d i m p l e m e n t a t i o n o n resort ( s e e A p p e n d i x I for results of the A R I M A a p p r o a c h ) . O f the five a p p r o a c h e s , the direct multiplicative a p p r o a c h w a s the m o s t straightforward a p p r o a c h a n d led to the g r e a t e s t r e d u c t i o n in s q u a r e d error a l t h o u g h all of the first four m e t h o d s p r o v i d e d v e r y s i m i l a r i m p r o v e m e n t s . In the direct multiplicative a p p r o a c h , the b o o k i n g c u r v e projection results from multiplying the b a s e l i n e e s t i m a t e by the ratio of a c t u a l to e x p e c t e d b o o k i n g s to d a t e ( s e e Equation 25).  EB  LT=0  = BCE  (25)  LT=y  where:  AB  LT=Y  A c t u a l bookings to date Y days prior to the target date (lead time = Y )  EB  LT=Y  Expected bookings to date Y days prior to the target date (from baseline booking curve)  EB  Expected bookings on the target date (from baseline booking curve)  LT=0  BCE ^ LT  4.2.3  Y  B o o k i n g curve estimate o f final demand at Y days prior to target date  Short-term weighting function  T h e short-term e s t i m a t e is a w e i g h t e d a v e r a g e of the b o o k i n g c u r v e projection a n d t h e l o n g - t e r m e s t i m a t e . E c o n o m e t r i c literature p r o v i d e s m a n y e x a m p l e s of situations w h e r e c o m b i n e d f o r e c a s t s p r o v i d e s u p e r i o r results to s i n g l e f o r e c a s t s . In fact, c o m b i n e d f o r e c a s t s will a l w a y s b e o p t i m a l a s l o n g a s f o r e c a s t s a r e u n b i a s e d (Min & Z e l l n e r , 1 9 9 3 ) . H o w e v e r , M i n & Z e l l n e r g o o n to p r o v e that c o m b i n i n g b i a s e d f o r e c a s t s d o e s not n e c e s s a r i l y p r o v i d e s u p e r i o r f o r e c a s t s . A s a result, a linear r e g r e s s i o n m o d e l (no intercept) w a s u s e d to c o m b i n e f o r e c a s t s at e a c h l e a d time a s this w a s a m e t h o d that w o u l d m i n i m i z e the s q u a r e d error r e g a r d l e s s of w h e t h e r or not b i a s w a s p r e s e n t . T h i s m o d e l a l l o w e d o n e e s t i m a t e to b e w e i g h t e d b e t w e e n 0 % a n d 1 0 0 % d e p e n d i n g o n its contribution to M S E . In fact, b i a s w a s likely for the s h o r t - t e r m m o d e l g i v e n the u n d e r l y i n g y e a r l y trend in d e m a n d w a s likely to either o v e r e s t i m a t e or u n d e r e s t i m a t e a c t u a l y e a r l y d e m a n d , w h i c h w o u l d then b i a s all daily e s t i m a t e s .  T h e short-term m o d e l w e i g h t c h a n g e s at different l e a d t i m e s s i n c e the error of the b o o k i n g c u r v e projection is not c o n s i s t e n t a c r o s s l e a d t i m e s . Instead, b o o k i n g c u r v e e s t i m a t e s at l o n g l e a d t i m e s (i.e. 9 0 d a y s prior to a target date) h a v e h i g h e r errors than b o o k i n g c u r v e e s t i m a t e s at short l e a d - t i m e s . T h i s is b e c a u s e b o o k i n g c u r v e projections at l o n g l e a d t i m e s h a v e f e w e r b o o k i n g s to d a t e in w h i c h to m a k e a f o r e c a s t a n d m u s t f o r e c a s t further out. L o n g - t e r m e s t i m a t e s , o n the other h a n d , d o not v a r y b a s e d o n b o o k i n g s to d a t e a s t h e y a r e c o n s t r u c t e d entirely f r o m prior  31  y e a r s ' d a t a . F i g u r e 8 c o m p a r e s the m e a n s q u a r e error of l o n g - t e r m N L e s t i m a t e s a n d s h o r t - t e r m b o o k i n g c u r v e projection e s t i m a t e s at different l e a d t i m e s o v e r a three y e a r i n - s a m p l e p e r i o d ( M a y 14, 1 9 9 9 to A p r i l 2 9 , 2 0 0 2 ) .  16,000  14,000  12,000  - Long-term NL estimate  MSE  Booking curve projection  8,000 4  4,000 J  2,000  0.00  0.10  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  1.00  Lead time (0 = 90 days, 1 • target date)  Figure 8:  M e a n s q u a r e error of o n e b e d r o o m long-term N L e s t i m a t e s a n d s h o r t - t e r m b o o k i n g c u r v e p r o j e c t i o n s at different l e a d t i m e s  In o r d e r to integrate the t i m e - d e p e n d e n t error of b o o k i n g c u r v e projections into better s h o r t - t e r m e s t i m a t e s , a w e i g h t i n g f u n c t i o n is u s e d to b a l a n c e the contribution of l o n g - t e r m e s t i m a t e s a n d b o o k i n g c u r v e p r o j e c t i o n s . T h e w e i g h t i n g function b e t w e e n t h e l o n g - t e r m e s t i m a t e a n d t h e b o o k i n g c u r v e projection in this p a p e r is s i m i l a r to t h e a p p r o a c h t a k e n t o p r o j e c t e d d e m a n d e s t i m a t e s in R a j o p a d h y e et a l . ( 1 9 9 9 ) . R a j o p a d h y e et a l . u p d a t e their w e i g h t i n g f u n c t i o n b a s e d on t h e m e a n s q u a r e error ( M S E ) of a short-term A R I M A f o r e c a s t a n d M S E of a l o n g - t e r m A R I M A f o r e c a s t . S i n c e short-term f o r e c a s t s typically h a v e s m a l l e r M S E t h a n d o l o n g - t e r m f o r e c a s t s a s the target d a t e n e a r s , t h e s h o r t - t e r m f o r e c a s t s a r e w e i g h t e d m o r e h e a v i l y c l o s e r to t h e target d a t e . S i m i l a r to t h e M S E ratio c a l c u l a t e d in R a j o p a d h y e et a l . , t h e w e i g h t i n g f u n c t i o n in this p a p e r is b a s e d o n a linear r e g r e s s i o n ( n o intercept) of the s a m p l e d a y s ( 1 , 4 4 6 d a y s ) at e a c h l e a d t i m e (from 9 0 d a y s out to 1 d a y out) to d e t e r m i n e the optimal w e i g h t i n g b e t w e e n l o n g - t e r m N L e s t i m a t e a n d b o o k i n g c u r v e projection ( s e e E q u a t i o n 2 6 a n d E q u a t i o n 2 7 ) .  32  STE  LT=Y  = a BCE LT=Y  LT=Y  AB _ IT  LT-Y  a  =  n  —  LT=Y  (26)  — LTE () 27  BCE ^ LT  + (1 - a )LTE  Y  — LTE  where:  AB  LT=0  STE  LT=Y  A c t u a l b o o k i n g s o n the target d a t e (LT=0) S h o r t - t e r m e s t i m a t e Y d a y s prior to target d a t e  BCE  B o o k i n g c u r v e projection Y d a y s prior to target d a t e  LTE  L o n g - t e r m N L e s t i m a t e ( d o e s not c h a n g e a c r o s s l e a d t i m e s )  cc  R e g r e s s i o n w e i g h t i n g p a r a m e t e r Y d a y s prior to target d a t e  LT=Y  LT=Y  O n c e t h e w e i g h t i n g p a r a m e t e r , a l p h a , w a s d e t e r m i n e d for e a c h l e a d t i m e , a l p h a w a s e s t i m a t e d a s a g e n e r a l f u n c t i o n of l e a d time (T) in o r d e r to r e m o v e a n y i d i o s y n c r a t i c effect that m a y h a v e o c c u r r e d at a s p e c i f i c l e a d t i m e . A l p h a a s a g e n e r a l function of l e a d time (T) e x p l a i n e d o v e r 9 9 % of the variation in t h e original a l p h a e s t i m a t e s a n d h e n c e t h e g e n e r a l a l p h a f u n c t i o n w a s u s e d a s the w e i g h t i n g for all s h o r t - t e r m e s t i m a t e s . S e e E q u a t i o n s 2 8 - 3 0 for t h e g e n e r a l a l p h a f u n c t i o n s a n d F i g u r e 9 A to 9 C for a g r a p h i c a l r e p r e s e n t a t i o n of the w e i g h t i n g f u n c t i o n . F i g u r e 10 is identical to the M S E reported in F i g u r e 8 but with the addition of t h e s h o r t - t e r m e s t i m a t e .  A s can  b e s e e n from F i g u r e 1 0 , t h e short-term e s t i m a t e p r o v i d e s m u c h - i m p r o v e d f o r e c a s t s a c r o s s all lead times.  33  a(l  bedroom) = .1805 + .6476T - .5900T + .7A35T  3  (28)  d(2 _bedroom) = .2377 + .8530T - .5088T + .39817/  3  (29)  2  2  d(3_bedroom) = .332\ + .7028T-.\320T  2  + .11027/  3  (30)  where: a  Short-term estimate weighting parameter  T  L e a d t i m e e x p r e s s e d b e t w e e n 0 a n d 1. Y d a y s prior to a target d a t e ; T = ( 9 1 - Y ) / 9 1 . T h e r e f o r e , at Y = 0 ; T = 1 , at Y = 9 0 ; T = . 0 1 1 .  34  100%  Long-Term Estimate 70% 4 60% Short-Term Estimate Weighting  50% 4 40%  Booking Curve Projection 20% 4  10% 4  0  0  0  0.10  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  1.00  Lead Time (0=90 days , 1 = target date) F i g u r e 9 B : T w o b e d r o o m w e i g h t i n g f u n c t i o n for s h o r t - t e r m e s t i m a t e  100% 90% Long-Term Estimate 80%  70% 4 60% Short-Term Estimate Weighting  30% Booking Curve Projection 20%  10%  0  0 0  0-  10  020  0.30  0.40  0.50  0.60  0.70  Lead Time (0=90 days , 1 = target date) Figure 9 C : T h r e e p l u s b e d r o o m w e i g h t i n g f u n c t i o n for s h o r t - t e r m e s t i m a t e  35  0.80  0.90  1.00  18,000 •  16,000  \  \  12,000 -t  10,000 MSE  •«.  \V  - Long-term NL estimate Booking curve projection  8,000  -Short-term estimate  6,000  ,  2,000  r  ~ — * —  4  -  0.00  0.10  0.20  0.30  0.40  ^  ^  ^  ^  ^  0.50  0.60  0.70  0.80  0.90  1.00  Lead time (0 = 90 days, 1 = target date)  Figure 10: M e a n s q u a r e error of o n e b e d r o o m long-term N L e s t i m a t e s , s h o r t - t e r m b o o k i n g c u r v e p r o j e c t i o n s , a n d s h o r t - t e r m b o o k i n g c u r v e e s t i m a t e s at different l e a d t i m e s  4.3  C o m p a r i n g Short-Term F o r e c a s t s  T a b l e 9 c o m p a r e s s h o r t - t e r m f o r e c a s t s for the two short-term f o r e c a s t i n g m e t h o d s a s w e l l a s t h e five l o n g - t e r m m e t h o d s d u r i n g a 2 y e a r i n - s a m p l e p e r i o d . E a c h m e t h o d w a s f o r e c a s t o u t 9 0 d a y s a n d t h e r e w e r e s e v e n different 9 0 d a y f o r e c a s t p e r i o d s ( 6 3 0 d a y s ) in t h e s a m p l e p e r i o d ( A u g u s t 6, 2 0 0 0 to A p r i l 2 7 , 2 0 0 2 ) . T h e r e f o r e , m e a n error m e a s u r e s a r e t h e m e a n e r r o r s a c r o s s 9 0 l e a d t i m e s a c r o s s s e v e n different f o r e c a s t p e r i o d s . C o n c e p t u a l l y o n e might think of the m e d i a n error m e a s u r e a s t h e error 4 5 d a y s prior to a target d a t e for a typical f o r e c a s t p e r i o d . S i n c e e a c h d a y is c o n s i d e r e d a s e p a r a t e d a t a s e r i e s (due to the u s e of b o o k i n g s to date), t h e M d A P E error m e a s u r e a s r e c o m m e n d e d b y A r m s t r o n g & C o l l o p y (1992) is t h e m o s t a p p r o p r i a t e ( s e e E q u a t i o n 2 a n d E q u a t i o n 4 for M d A P E c a l c u l a t i o n ) . A s c a n b e s e e n , t h e i m p r o v e m e n t of the B C m e t h o d o v e r t h e A P m e t h o d is significant (29.4%).  36  T a b l e 9: I n - s a m p l e s h o r t - t e r m m o d e l c o m p a r i s o n s ( A u g u s t 6, 2 0 0 0 to April 2 7 , 2 0 0 2 ) Model  MSE  MdCumRAE  MdAPE  1.00 .69  21.1%  Models using complete stay information only: R a n d o m walk (RW)  12,882 6,381  Nonlinear regression (NL) ARIMA  8,605 24,397  H o l t - W i n t e r s multiplicative ( H W ) Linear regression (LR)  31.2%  .84  27.0% 37.3%  1.41 .78  8,255  29.6%  Models using both complete stay information and bookings to date:  Additive pickup (AP)  4,781  .57  17.7%  Booking curve (BC)  3,180  .49  12.5%  T h e error of the s e v e n f o r e c a s t i n g m e t h o d s a l s o c h a n g e s a c r o s s l e a d t i m e s , with the t w o shortt e r m f o r e c a s t i n g m e t h o d s p r o v i d i n g c l e a r l y s u p e r i o r f o r e c a s t s c l o s e r to the target d a t e ( s e e F i g u r e 11). S h o r t - t e r m f o r e c a s t s w e r e not c o m p a r e d out of s a m p l e d u e to t h e t i m e effort r e q u i r e d a n d the m i n i m a l m a n a g e r i a l benefit g i v e n a c a p a c i t y - s l a c k e n v i r o n m e n t .  60%  0% B  C  AP  NL  ARIMA  LR  RW  HW  Forecasting Method  Figure 11: I n - s a m p l e m e d i a n a b s o l u t e p e r c e n t a g e error ( M d A P E ) for f o r e c a s t i n g m e t h o d s a c r o s s lead times  37  4.4  B o o k i n g C u r v e D e c i s i o n Support S y s t e m  A d e c i s i o n - s u p p o r t s y s t e m ( D S S ) b a s e d o n the B C m o d e l (which w a s c a l i b r a t e d u s i n g S A S statistical software) w a s built in M i c r o s o f t E x c e l s o that the resort c o u l d c a l c u l a t e short-term e s t i m a t e s in the current s e a s o n o n a n o n g o i n g b a s i s . T h e D S S c o n s i s t s of a n input w o r k s h e e t w h i c h is l i n k e d to the resort r e s e r v a t i o n m a n a g e m e n t s y s t e m that p r o v i d e s the n u m b e r of r o o m nights b o o k e d to d a t e ( e x c l u d i n g g r o u p a n d o w n e r b o o k i n g s ) for the next 9 0 d a y s by b e d r o o m type. T h e s e b o o k i n g f i g u r e s a r e a u t o m a t i c a l l y u s e d in B C m o d e l c a l c u l a t i o n s (with a p p r o p r i a t e v a l u e s b a s e d o n target d a t e : s e a s o n a l p e r i o d , d a y of w e e k , y e a r ) to f o r e c a s t short-term d e m a n d e s t i m a t e s e x p r e s s e d in the output w o r k s h e e t ( s e e F i g u r e 12). T h e output p a g e p r o v i d e s f o r e c a s t s by b e d r o o m a s well a s in a g g r e g a t e , a n d p r o v i d e s e x p e c t e d p i c k u p b e t w e e n b o o k i n g s to d a t e a n d final d e m a n d e s t i m a t e s s o that m a n a g e r s c a n s c r u t i n i z e a n d monitor a c t u a l v e r s u s e x p e c t e d p i c k u p . T h e D S S a l s o h a s c h a r t s of final d e m a n d e s t i m a t e s a n d e x p e c t e d b o o k i n g c u r v e s that a r e a u t o m a t i c a l l y u p d a t e d from the resort r e s e r v a t i o n s y s t e m .  90-DAY S H O R T - T E R M ESTIMATES Date of data extract 1 bedroom capacity 2 bedroom capacity 3 bedroom capacity Total capacity  09-Dec-02 836 147 . 27 1,010 TOTAL UNITS Day of Week Lead Long-Term Expected (1 = Time Estimate (before Bookings ; Short-Term Pickup . Forecast as % MONDAY) (Days) booking data) to Date Estimate (units) of capacity 1 0 75 92 92 o 9% 2 1 ; 75 79 84 5 8% 3 2 75 96 107 11 11% 4 3 77 121, 139 18 14% 5 4 230 339 403 64 40% 6 5 339 4.13 511 98 51% 7 6 138 430 510 80 50% 1 7 118 168 214 46 21% 2 8 itali' 128 118 173 • •' . "V45 17% .." 118 3 9 142 193 51 19% 1  Target Date 09-Dec-02 10-Dec-02 11-Dec-02 12-Dec-02 13-Dec-02 14-Dec-02 15-Dec-02 16-Dec-02 17-Dec-02 18-Dec-02  Figure  :  f  12: P o r t i o n of d e c i s i o n - s u p p o r t s y s t e m output p a g e (data a s at D e c e m b e r 9, 2 0 0 2 )  38  5  DISCUSSION  F o r a hotel with fixed c a p a c i t y , W e a t h e r f o r d , K i m e s & S c o t t (2001) f o u n d that four f o r e c a s t i n g m e t h o d s for hotel d e m a n d ( e x p o n e n t i a l s m o o t h i n g , m o v i n g a v e r a g e , linear r e g r e s s i o n , a n d additive p i c k u p ) p e r f o r m e d e q u a l l y w e l l . In the c a s e of the resort s t u d i e d , with i n c r e a s i n g y e a r l y c a p a c i t y , this w a s c e r t a i n l y not t h e c a s e . F o r l o n g - t e r m f o r e c a s t s , a s s u m i n g s t a b l e y e a r l y t r e n d , the n o n l i n e a r r e g r e s s i o n w a s slightly s u p e r i o r to the r a n d o m w a l k m e t h o d , a n d c l e a r l y s u p e r i o r to the A R I M A , linear r e g r e s s i o n , a n d multiplicative H o l t - W i n t e r s m o d e l s . F u r t h e r , in a situation of a d o w n w a r d c a p a c i t y s h o c k , a s w a s e x p e r i e n c e d at the resort o n D e c e m b e r 1, 2 0 0 1 , r a n d o m w a l k w a s c l e a r l y s u p e r i o r to all other long-term m o d e l s . T o g e n e r a l i z e to other resort l o d g i n g p r o p e r t i e s , g i v e n a p r e d i c t a b l e y e a r l y trend in d e m a n d , a n o n l i n e a r r e g r e s s i o n m o d e l is r e c o m m e n d e d . T h e p e r f o r m a n c e of a n A R I M A m o d e l w a s a l s o quite g o o d in both c a p a c i t y situations ( p r e d i c t a b l e a n d u n p r e d i c t a b l e c a p a c i t y ) w h i l e the p e r f o r m a n c e of a linear r e g r e s s i o n m o d e l a n d multiplicative H o l t - W i n t e r s m o d e l w e r e c l e a r l y i n a d e q u a t e in all c a p a c i t y s i t u a t i o n s .  In  t e r m s of short-term d e m a n d f o r e c a s t i n g , a b o o k i n g c u r v e m o d e l a s d e v e l o p e d in this p a p e r p e r f o r m e d very w e l l i n - s a m p l e a n d c a n only b e a s s u m e d to b e the c a s e in a n out of s a m p l e setting with p r e d i c a b l e c a p a c i t y .  In t e r m s of m a n a g e r i a l i m p l i c a t i o n s , this p a p e r h a s b a s i c a l l y g i v e n s u p p o r t to the resort m a n a g e m e n t ' s p r a c t i c e of r a n d o m w a l k for long-term f o r e c a s t s a n d additive p i c k u p for short-term f o r e c a s t s . N o n l i n e a r r e g r e s s i o n long-term m o d e l s a n d short-term b o o k i n g c u r v e m o d e l s p r o v i d e m a r g i n a l i m p r o v e m e n t s in the resort's d e m a n d f o r e c a s t i n g g i v e n a p r e d i c t a b l e c a p a c i t y e n v i r o n m e n t or a n u p w a r d d e m a n d s h o c k . H o w e v e r , g i v e n the resort h a s a large a m o u n t Of c a p a c i t y s l a c k , m o r e a c c u r a t e d e m a n d f o r e c a s t s will likely h a v e a s m a l l i m p a c t o n l o d g i n g o p e r a t i o n s . R a t h e r , the resort s h o u l d revisit t h e s e m o d e l s if c a p a c i t y b e c o m e s s t r a i n e d . If sell outs b e c o m e m o r e frequent then d e m a n d f o r e c a s t i n g a c c u r a c y b e c o m e s m u c h m o r e important. F u r t h e r m o r e , the b o o k i n g c u r v e m o d e l c a n b e a d j u s t e d slightly to p r o v i d e u n c o n s t r a i n e d d e m a n d e s t i m a t e s by arrival d a t e . U n c o n s t r a i n e d d e m a n d e s t i m a t e s by m a r k e t s e g m e n t , length of s t a y , a n d arrival d a t e a r e critical inputs into intelligent r e v e n u e m a n a g e m e n t d e c i s i o n s during p e r i o d s of c o n s t r a i n e d c a p a c i t y .  A c o m p a r i s o n of the m o d e l s b a s e d o n f o r e c a s t a c c u r a c y a l o n e is p r o b a b l y insufficient for a c o m p l e t e e v a l u a t i o n of m o d e l e f f e c t i v e n e s s . G i v e n that the m o d e l s a r e u s e d in a b u s i n e s s context, the insight that the m o d e l s m a y s h e d o n the l o d g i n g e n v i r o n m e n t is a n important m a n a g e m e n t consideration. T h e A R I M A models, while providing decent forecasts (especially short-term), a r e n e a r l y uninterpretable.  E v e n if the A R I M A e q u a t i o n s ( E q u a t i o n s 18-19) w e r e  e x p r e s s e d a s a w e i g h t i n g of p a s t o b s e r v a t i o n s , the differencing of the d a t a a n d long t i m e - s p a n r e q u i r e d m a k e m a n a g e m e n t insight f r o m t h e s e m o d e l s v e r y unlikely. T h e R W m o d e l s , o n the  39  other h a n d , a r e v e r y straightforward to u n d e r s t a n d a n d h a v e p r o v i d e d v e r y g o o d p r e d i c t i o n s . Unfortunately, other than p r o v i d i n g a g o o d e s t i m a t e of d e m a n d in the current p e r i o d , it is difficult to d e c i p h e r h o w m u c h of the R W e s t i m a t e is d u e to a s y s t e m a t i c s e a s o n a l c o m p o n e n t a n d h o w m u c h is d u e to r a n d o m flux. T h e H W m o d e l is v e r y g o o d in this r e g a r d a s it explicitly m o d e l s t h e s y s t e m a t i c p e r i o d c o m p o n e n t a n d interpretation of t h e s e p e r i o d s is s t r a i g h t f o r w a r d .  For example,  o n e n e e d o n l y to multiply the a p p r o p r i a t e w e e k of the y e a r p a r a m e t e r b y d a y of w e e k p a r a m e t e r to s e e h o w that d a y c o m p a r e s to the a v e r a g e d a y (1.00) or a n y other d a y of the y e a r . Unfortunately, the H W m o d e l likely s i m p l i f i e s too m u c h a s the d a y of the w e e k effect is not c o n s t a n t t h r o u g h o u t the y e a r a n d g r o u p i n g the y e a r by c h r o n o l o g i c a l w e e k m i s s e s important e v e n t s that s p a n l e s s than o n e w e e k s u c h a s P r e s i d e n t ' s d a y , N e w Y e a r s , a n d w e e k e n d f e s t i v a l s . T h e simplicity of the H W m o d e l , w h i l e readily interpretable, is likely r e s p o n s i b l e for its p o o r f o r e c a s t i n g p e r f o r m a n c e ( e s p e c i a l l y long-term f o r e c a s t s ) .  L R , N L , a n d B C m e t h o d s explicitly m o d e l all s e a s o n a l c o m p o n e n t s ( p e r i o d , d a y of w e e k , y e a r l y trend, interactions b e t w e e n p e r i o d a n d d a y of w e e k ) a n d likely strike the b e s t b a l a n c e b e t w e e n a straightforward interpretation a n d a level of s o p h i s t i c a t i o n that p r o v i d e s g o o d e s t i m a t e s . A l l three of t h e s e m o d e l s h a v e statistically t e s t e d the s i g n i f i c a n c e of s e a s o n a l p e r i o d s a n d thus p r o v i d e a reliable b a s e f r o m w h i c h m a n a g e m e n t c a n v i e w p e r i o d s a s b e i n g truly distinct.  In the  d e v e l o p m e n t of t h e s e m o d e l s , m a n a g e m e n t c l a i m e d to v i e w the l o d g i n g s e a s o n a s 13 distinct p e r i o d s (6 s u m m e r p e r i o d s a n d 7 winter p e r i o d s ) . T h e s e 13 p e r i o d s w e r e u s e d a s the starting point for t h e s e m o d e l s but the predictability of FIT d e m a n d h a s a l l o w e d further r e f i n e m e n t of t h e s e original 13 p e r i o d s into a s m a n y a s 5 7 distinct p e r i o d s in the c a s e of the B C o n e b e d r o o m m o d e l . F u r t h e r r e f i n e m e n t of d e m a n d p e r i o d s s h o u l d b e v e r y helpful for m a n a g e m e n t a s it s e t s rate targets a n d m a n a g e s e x p e c t a t i o n s of s e a s o n a l d e m a n d .  F i g u r e 1 3 A a n d F i g u r e 1 3 B s h o w the original 13 p e r i o d s a s d e f i n e d b y resort m a n a g e m e n t , a n d a further r e f i n e m e n t of t h e s e p e r i o d s a s d e f i n e d b y the N L o n e b e d r o o m m o d e l . B o t h f i g u r e s s h o w the a v e r a g e daily d e m a n d ( a v e r a g e d by w e e k or p e r i o d , w h i c h e v e r w a s s m a l l e r ) in the 0 1 / 0 2 s e a s o n . T h e b o l d red d a s h e d lines indicate the original 13 p e r i o d s d e f i n e d b y resort m a n a g e m e n t a n d t h e b l a c k d a s h e d l i n e s r e p r e s e n t s u b - p e r i o d s within t h e original 1 3 p e r i o d s . A s e v i d e n c e d by v i s u a l i n s p e c t i o n , the additional p e r i o d s d o s e e m to d i s c r i m i n a t e truly different d e m a n d l e v e l s within the original p e r i o d s . A s w e l l , the w e e k e n d - p e r i o d interaction p a r a m e t e r s a n d d a y of w e e k p a r a m e t e r s of the B C , N L , a n d L R m o d e l s s h o u l d further a i d m a n a g e m e n t in setting rates b y d a y of w e e k within larger s e a s o n a l p e r i o d s .  40  Figure 13A: A v e r a g e FIT daily d e m a n d in 0 1 / 0 2 s u m m e r s e a s o n a n d c o r r e s p o n d i n g s e a s o n a l p e r i o d c l a s s i f i c a t i o n by w e e k  41  600  Week  Figure 13B: A v e r a g e FIT daily d e m a n d in 0 1 / 0 2 winter s e a s o n a n d c o r r e s p o n d i n g s e a s o n a l period classification by w e e k  B e y o n d s e a s o n a l p e r i o d c l a s s i f i c a t i o n , the B C m o d e l p r o v i d e s m a n a g e m e n t with e x p e c t e d b o o k i n g c u r v e s for a n y target d a t e . T h i s s h o u l d p r o v e a u s e f u l c o m p l e m e n t to r a w p i c k u p n u m b e r s t a k e n f r o m A P m o d e l s . A chart of e x p e c t e d b o o k i n g s ( b a s e l i n e b o o k i n g c u r v e ) is a c o m p e l l i n g v i s u a l i z a t i o n of s y s t e m a t i c d e m a n d b u i l d - u p . F o r e x a m p l e , k n o w i n g that a c e r t a i n d a y of w e e k within a p e r i o d h a s c o n s i s t e n t l y s h o w n a large proportion of l a s t - m i n u t e b o o k i n g s s h o u l d r e a s s u r e m a n a g e m e n t of its current pricing if r o o m b o o k i n g s a r e short of b u d g e t e d r o o m nights c l o s e to the target d a t e . A t the v e r y least, e x p e c t e d b o o k i n g c u r v e s p r o v i d e a n o t h e r r e f e r e n c e point for d e t e r m i n i n g w h e t h e r last y e a r ' s p i c k u p ( A P m o d e l ) is r e p r e s e n t a t i v e of historical patterns or w h e t h e r it m a y h a v e b e e n a n a b e r r a t i o n . F i g u r e 1 4 A s h o w s h o w d a y of w e e k c a n h a v e a v e r y large i m p a c t o n the b o o k i n g c u r v e a s it c o m p a r e s the e x p e c t e d b u i l d u p in b o o k i n g s for a T h u r s d a y night a n d S a t u r d a y night within the s a m e w e e k in J u l y 2 0 0 3 . T h e c u r v e s a r e n e a r l y identical up to a b o u t 2 7 d a y s out f r o m the target d a t e (indicated by a d a s h e d line) at w h i c h point the S a t u r d a y night is e x p e c t e d to get a n a c c e l e r a t i o n of b o o k i n g s a b o v e a n d b e y o n d the T h u r s d a y night.  42  600  0  0.1  0.2  0.3  0.4  0.5  r  I  1  1  1  0.6  0.7  0.8  0.9  1  Lead Time (0 = 90 days lead time)  Figure 14A: E x p e c t e d F I T o n e b e d r o o m b o o k i n g c u r v e s for a S a t u r d a y v s . T h u r s d a y in J u l y 2 0 0 3 (Thursday = July 17, 2 0 0 3 , Saturday = July 19, 2003)  F i g u r e 1 4 B s h o w s a n e x a m p l e of h o w different p e r i o d s c a n a l s o result in v e r y different b o o k i n g c u r v e s . T h e winter s e a s o n d a t e ( M a r c h 1 7 , 2 0 0 3 ) h a s a n a l m o s t linear b u i l d u p in b o o k i n g s w h i l e the fall d a t e ( S e p t e m b e r 1 9 , 2 0 0 3 ) is e x p e c t e d to g e t a l a r g e proportion o f last m i n u t e b o o k i n g s . A t 31 d a y s out ( m a r k e d b y a d a s h e d line), there is a d i f f e r e n c e of 9 4 r o o m s b o o k e d w h i l e t h e final d e m a n d is e x p e c t e d to differ b e t w e e n t h e t w o d a t e s by o n l y 6 r o o m s .  43  Figure 14B: E x p e c t e d F I T o n e b e d r o o m b o o k i n g c u r v e s for a W i n t e r D a t e v s . F a l l D a t e (Winter D a t e = M a r c h 1 7 , 2 0 0 3 , Fall D a t e = S e p t e m b e r 19, 2 0 0 3 )  M e n t i o n h a s t h u s far b e e n m a d e that identifying distinct s e a s o n a l p e r i o d s m a y h e l p in setting r a t e s . W h i l e the g o a l of this p a p e r h a s b e e n to c o m p a r e m e t h o d s for e s t i m a t i n g d e m a n d , the r e l a t i o n s h i p b e t w e e n d e m a n d a n d rates is ultimately the m o s t i m p o r t a n t i s s u e for r e v e n u e management.  F i g u r e 15 s h o w s the a v e r a g e o n e b e d r o o m daily r o o m rate (by w e e k ) a n d a v e r a g e  daily d e m a n d (by w e e k ) with the a v e r a g e daily r o o m rate a n d a v e r a g e daily d e m a n d both n o r m a l i z e d to b e 1.00. T h e c o r r e l a t i o n b e t w e e n a v e r a g e daily rate a n d a v e r a g e d a i l y d e m a n d is . 8 3 . A s c a n b e s e e n f r o m t h e chart, a v e r a g e r o o m rates c l o s e l y m a t c h d e m a n d o v e r t h e winter s e a s o n w h i l e not m a t c h i n g h i g h d e m a n d p e r i o d s in the s u m m e r p e r i o d . F u r t h e r , the s h o u l d e r p e r i o d s d o not s e e a s u b s e q u e n t d e c r e a s e in r o o m rates w h e n d e m a n d t r o u g h s .  44  3.00  2.50  2.00  - 1.50  -Avg_Rate - AvgDemand  1.00  0.50  0.00  Week  Figure 15: C o m p a r i s o n  of o n e b e d r o o m n o r m a l i z e d a v e r a g e daily r o o m rates to n o r m a l i z e d a v e r a g e d a i l y d e m a n d in the 0 1 / 0 2 s e a s o n by w e e k ( a v e r a g e = 1.00)  F i g u r e 15 d o e s not n e c e s s a r i l y imply that rates a r e i n a p p r o p r i a t e , a s resort m a n a g e r s h a v e s t a t e d that the s u m m e r d e m a n d is c o m p r i s e d of r e g i o n a l g u e s t s w h o a r e m o r e p r i c e s e n s i t i v e , a n d a s s u c h m a n a g e m e n t h a s l e s s flexibility to i n c r e a s e r o o m rates w h e n s u m m e r d e m a n d i n c r e a s e s . H o w e v e r , the r e l a t i o n s h i p b e t w e e n d e m a n d a n d r o o m rates s h o u l d definitely b e e x p l o r e d further, a n d u s i n g the s e a s o n a l p e r i o d s d e f i n e d by the N L a n d B C m o d e l s is a g o o d starting point.  Resort  m a n a g e r s h a v e s t a t e d that m u c h of the a d j u s t m e n t of r o o m rates is d o n e o n a n o n g o i n g b a s i s in c o n j u n c t i o n with the r e s o r t ' s c a l l c e n t e r (which b o o k s c l o s e to 6 0 % of r o o m b o o k i n g s ) . S p e c i f i c a l l y , the r e v e n u e m a n a g e r s monitor c o n v e r s i o n rates (calls that e n d u p in b o o k i n g s ) a n d r e f u s a l rates ( p e r c e n t of c a l l s w h e r e a s p e c i f i e d r o o m is t u r n e d d o w n d u e to p r i c e ) . If c o n v e r s i o n rates d r o p too low (e.g. m u c h b e l o w 3 0 % ) or refusal rates c l i m b too high ( e . g . a b o v e 1 3 % ) t h e n this is a n indicator t h e c u r r e n t r o o m rates a r e too h i g h . C o n v e r s e l y , h i g h c o n v e r s i o n r a t e s a n d low r e f u s a l s m a y indicate p r i c e s a r e too low. F u r t h e r a n a l y s i s of r o o m r a t e s , e x p e c t e d d e m a n d , call v o l u m e , c a l l c l a s s i f i c a t i o n , a n d a c t u a l d e m a n d is b e y o n d the s c o p e of this p a p e r , but a p p e a r s to b e a fruitful a r e a for future a n a l y s i s . T h e current s e a s o n ( 0 2 / 0 3 ) is t h e first s e a s o n that call c e n t e r information r e g a r d i n g r o o m d e m a n d is b e i n g s y s t e m a t i c a l l y r e c o r d e d .  45  5.1  M o d e l E x t e n s i o n s in a C a p a c i t y C o n s t r a i n e d Environment  In a n e n v i r o n m e n t of frequently c o n s t r a i n e d c a p a c i t y , a d d i t i o n a l a c c u r a c y in d e m a n d f o r e c a s t s is v a l u a b l e a n d worthy of m o d e l l i n g effort. A s s u m i n g this c a p a c i t y c o n s t r a i n e d e n v i r o n m e n t a n u m b e r of potential e x t e n s i o n s in the short-term b o o k i n g c u r v e m o d e l a r e p r o p o s e d . First, the m o d e l c a n b e i m p r o v e d by u s i n g a larger n u m b e r of inputs a n d h e n c e p r o v i d e m o r e a c c u r a t e l o d g i n g d e m a n d e s t i m a t e s . S e c o n d , the m o d e l c a n b e a d a p t e d to integrate m o r e c l o s e l y with a r e v e n u e m a n a g e m e n t s y s t e m or other o p t i m i z a t i o n e n g i n e (although n o n e currently e x i s t s at the resort). T h i r d , the m o d e l c a n b e e x t e n d e d to i n c l u d e g r o u p a n d o w n e r b o o k i n g s . F o u r t h , the m o d e l c a n b e e x t e n d e d to i n c l u d e other o n - m o u n t a i n s o u r c e s of r e v e n u e in o r d e r to a c h i e v e a m o r e g l o b a l o b j e c t i v e of resort r e v e n u e m a x i m i z a t i o n .  It is well k n o w n (and c o n f i r m e d by the resort's i n - h o u s e r e s e a r c h ) that b e s i d e s d a y of w e e k a n d time of y e a r , w e a t h e r is the s i n g l e m o s t important factor in predicting d e m a n d at a s k i resort. Q u i t e s i m p l y , g o o d s n o w b r i n g s c r o w d s . Inputs into the r e g r e s s i o n m o d e l s c o u l d i n c l u d e s n o w b a s e (relative to a historical a v e r a g e ) , projected s n o w f a l l a n d past s n o w f a l l (e.g. in a w e e k prior to a target d a t e ) for winter s e a s o n s .  In w a r m e r s e a s o n s , w h i l e likely to h a v e l e s s of a n effect,  t e m p e r a t u r e a n d rainfall f o r e c a s t s m a y a l s o i m p r o v e l o d g i n g d e m a n d p r e d i c t i o n s .  B e s i d e s m o r e a c c u r a t e d e m a n d f o r e c a s t s , f o r e c a s t s that c a n b e e a s i l y integrated with a n o p t i m i z a t i o n e n g i n e w o u l d p r o v e u s e f u l . F o r e x a m p l e , c r e a t i n g c o m p l e m e n t a r y m o d e l s to predict arrival distributions (rather than o c c u p i e d r o o m nights) a n d d e m a n d by rate c l a s s a n d length of s t a y w o u l d further f o r m a l i z e the r e v e n u e m a n a g e m e n t p r o c e s s at the resort. B e g i n n i n g this y e a r (02/03) the resort is t r a c k i n g t u r n d o w n a n d d e n i a l information. T h i s information s h o u l d p r o v e i n v a l u a b l e in building m o r e d i s a g g r e g a t e d f o r e c a s t s a n d probability distributions that w o u l d b e c l a s s i f i e d b y r o o m type, m a r k e t s e g m e n t , rate c l a s s , a n d length of s t a y . O n l y b y p r o v i d i n g d i s a g g r e g a t e e s t i m a t e s in t e r m s of both length of s t a y , arrivals, a n d p r i c e probabilities c a n a l g o r i t h m s b e d e v e l o p e d to o p t i m i z e r e v e n u e .  D e m a n d e s t i m a t e s for i n d e p e n d e n t t r a v e l e r s s h o u l d b e integrated with d e m a n d e s t i m a t e s for g r o u p s a n d o w n e r s . O w n e r e s t i m a t e s a r e important a s far a s t h e y l o w e r a v a i l a b l e c a p a c i t y , w h i l e g r o u p d e m a n d e s t i m a t e s a r e important in t e r m s of p r i c e sensitivity, resort p r o m o t i o n , a n d a n a l y s i s of long-term c o n t r a c t s with w h o l e s a l e r s .  In the c a s e of the resort s t u d i e d , the resort r e c e i v e s r e v e n u e f r o m resort o p e r a t i o n s (ski tickets, rentals, f o o d a n d b e v e r a g e , retail) a s w e l l a s l o d g i n g . T h e r e f o r e , it m a k e s s e n s e to i n c l u d e t h e s e s o u r c e s of a n c i l l a r y r e v e n u e w h e n building r e v e n u e m a x i m i z a t i o n m o d e l s . In other w o r d s , s i n c e l o d g i n g g u e s t s will b e s p e n d i n g o n hill, the o b j e c t i v e s h o u l d b e to m a x i m i z e resort profit rather  46  than l o d g i n g profit a l o n e . F o r e x a m p l e , it m a y b e p r u d e n t to l o w e r l o d g i n g rates in o r d e r to b o o s t l o d g i n g o c c u p a n c y , with the a s s u m p t i o n that lost l o d g i n g r e v e n u e (due to l o w e r l o d g i n g p r i c e s ) w o u l d b e m o r e than offset by a n c i l l a r y r e v e n u e o n m o u n t a i n . In m a n y s i t u a t i o n s , the e f f i c a c y of rental p o o l m a n a g e r s a s j u d g e d b y c h a l e t o w n e r s is the o c c u p a n c y rate a c h i e v e d rather than r e v e n u e r e c e i v e d (in fact this w a s the p r i m a r y factor that led to the l o s s of units u n d e r m a n a g e m e n t in the current s e a s o n at the resort s t u d i e d ) . W h i l e r e v e n u e r e c e i v e d s h o u l d b e the rational e c o n o m i c o b j e c t i v e of c h a l e t o w n e r s , a f o c u s o n o c c u p a n c y rates m a y benefit resort m a n a g e m e n t in m a x i m i z i n g resort profit ( a s s u m i n g m a x i m u m resort profit c o m e s at the c o s t of l o w e r l o d g i n g profit a n d h i g h e r l o d g i n g o c c u p a n c y ) . It s h o u l d b e n o t e d that the a b o v e h y p o t h e s e s s h o u l d b e a n a l y z e d further, a n d that other c o n s i d e r a t i o n s / c o n s t r a i n t s to resort profit m a x i m i z a t i o n i n c l u d e s k i hill c a p a c i t y , d e s i r e d c l i e n t e l e / s n o b a p p e a l , a n d o v e r a l l g u e s t e x p e r i e n c e . A s a result, it m a y not m a k e s e n s e to offer rock bottom l o d g i n g rates to attract m o r e s k i e r s to the m o u n t a i n if it is at o d d s with the r e s o r t ' s s t r a t e g y in t e r m s of a p p r o p r i a t e target m a r k e t a n d atmosphere.  47  APPENDIX A - DATA PREPARATION AND TRANSFORMATION  D a t a r e c e i v e d f r o m the resort c o v e r e d the historical p e r i o d M a y 15, 1 9 9 8 to A p r i l 2 9 , 2 0 0 2 . T h e r e s e r v a t i o n d a t a w a s r e c e i v e d in a r a w table format with s e p a r a t e t a b l e s for r e s e r v a t i o n s , g u e s t information, a n d l o d g i n g unit information. T h i s r e s e r v a t i o n d a t a w a s c o n v e r t e d to r o o m night information u s i n g S A S statistical s o f t w a r e .  E s s e n t i a l l y , e a c h r e s e r v a t i o n w a s c l a s s i f i e d by m a r k e t  s e g m e n t (group, i n d e p e n d e n t traveler, o w n e r ) a n d b e d r o o m (one, two, three p l u s ) . O n c e c l a s s i f i e d , g r o u p a n d o w n e r b o o k i n g s w e r e e x c l u d e d . R o o m night information w a s then p r o c e s s e d u s i n g a l o o p i n g algorithm by c o u n t i n g the n u m b e r of distinct units to b e r e n t e d for a s p e c i f i c target d a t e at e a c h of 9 0 d a y s prior to a target d a t e . A r e s e r v a t i o n w a s i n c l u d e d in the r o o m night tally for a s long a s it r e m a i n e d o n the b o o k s up until the target d a t e . T h i s w a y , b o o k i n g s that e v e n t u a l l y b e c a m e c a n c e l l a t i o n s w o u l d b e i n c l u d e d in the s a m p l e d a t a for a s long a s they w e r e o n the b o o k s . If a b o o k i n g w a s c a n c e l l e d , the r e s e r v a t i o n w a s r e m o v e d f r o m the b o o k s u p o n the d a t e of c a n c e l l a t i o n .  C o n s i d e r a r e s e r v a t i o n for a t w o - b e d r o o m unit with a n arrival d a t e of F e b . 2, 2 0 0 2 , a d e p a r t u r e d a t e of F e b . 9, 2 0 0 2 , a r e s e r v a t i o n d a t e of J a n . 3, 2 0 0 2 , a n d a c a n c e l l a t i o n d a t e of J a n . 2 9 , 2 0 0 2 . T h i s r e s e r v a t i o n is a p p l i e d to s e v e n different target d a t e s (nights of F e b . 2, 2 0 0 2 to F e b . 8, 2 0 0 2 ) for o n e t w o - b e d r o o m unit. Further, the r e s e r v a t i o n is o n the b o o k s for 2 6 d a y s ( J a n . 3, 2 0 0 2 to J a n . 2 9 , 2 0 0 2 ) until the c a n c e l l a t i o n is m a d e . F o r the target d a t e of F e b . 2, 2 0 0 2 the b o o k i n g is i n c l u d e d for l e a d t i m e d a y s 3 0 to 4 ( F e b . 2, 2 0 0 2 l e s s J a n . 3, 2 0 0 2 e q u a l s l e a d time d a y 3 0 ; F e b . 2, 2 0 0 2 l e s s J a n . 2 9 , 2 0 0 2 e q u a l s l e a d time d a y 4). T h e b o o k i n g for the target d a t e of F e b . 8, 2 0 0 2 is i n c l u d e d for l e a d time d a y s 36 to 10 ( F e b . 8, 2 0 0 2 l e s s J a n . 3, 2 0 0 2 e q u a l s l e a d time d a y 3 6 ; F e b . 8, 2 0 0 2 l e s s J a n . 2 9 , 2 0 0 2 e q u a l s l e a d t i m e d a y 10).  48  A P P E N D I X B - MULTIPLICATIVE  HOLT-WINTERS  Multiplicative Holt-Winters O n e B e d r o o m Model  Fit  statistics:  N u m b e r of o b s e r v a t i o n s : 1,446 S u m of s q u a r e s total: 2 7 , 6 9 5 , 4 4 1 M e a n s q u a r e error: 8 , 5 8 4 M e a n a b s o l u t e p e r c e n t a g e error: 3 9 . 3 % M e a n a b s o l u t e error: 54.1 R : .57  D e g r e e s of f r e e d o m : 1,386 S u m of s q u a r e s error: 1 1 , 9 8 9 , 0 3 4 R o o t m e a n s q u a r e error: 9 2 . 7 M e a n p e r c e n t error: - 1 8 . 5 % M e a n error: - 6 . 5 Sigma: 92.65  2  Smoothing  parameters:  A l p h a ( m e a n - t e r m ) : .20 Day of week  G a m m a ( s l o p e - t e r m ) : .20  D e l t a ( s e a s o n a l - t e r m ) : .25  parameters:  M o n d a y : .58 F r i d a y : 1.58  T u e s d a y : .68 S a t u r d a y : 1.94  Weekly  parameters  18-MAY: 29-JUN: 10-AUG: 21-SEP: 02-NOV: 14-DEC: 25-JAN: 07-MAR: 18-APR:  .34 .81 1.94 .72 .25 .83 .86 1.31 .49  (approximate  2 5 - M A Y : .46 0 6 - J U L : 1.12 1 7 - A U G : 1.74 2 8 - S E P : .73 0 9 - N O V : .20 2 1 - D E C : 2.02 0 1 - F E B : 1.02 14-MAR: 2.12 2 5 - A P R : .28  beginning  W e d n e s d a y : .67 S u n d a y : .70  Thursday: .85  date):  0 1 - J U N : .44 1 3 - J U L : 1.24 2 4 - A U G : 1.45 0 5 - O C T : .71 1 6 - N O V : .21 2 8 - D E C : 2.77 0 8 - F E B : 1.48 2 1 - M A R : 1.45  08- J U N : .45 2 0 - J U L : 1.73 3 1 - A U G : .91 1 2 - O C T : .96 2 3 - N O V : .49 0 4 - J A N : 1.69 1 5 - F E B : 1.66 2 8 - M A R : 1.13  0 2 - M A Y : .19  0 9 - M A Y : .20  49  1 5 - J U N : .51 27-JUL: 2.09 0 7 - S E P : 1.05 1 9 - O C T : .43 3 0 - N O V : .36 1 1 - J A N : .85 2 2 - F E B : 1.87 0 4 - A P R : .90  22-JUN: 03-AUG: 14-SEP: 26-OCT: 07-DEC: 18-JAN: 29-FEB: 11-APR:  .57 2.11 .89 .33 .52 .89 1.58 .62"  MuItiplicative Holt-Winters Two/Three B e d r o o m Model  Fit  statistics:  N u m b e r of o b s e r v a t i o n s : 1,446 S u m of s q u a r e s total: 8 1 9 , 8 3 8 M e a n s q u a r e error: 2 5 2 M e a n a b s o l u t e p e r c e n t a g e error: 5 1 . 4 % M e a n a b s o l u t e error: 9.7 R : .57  D e g r e e s of f r e e d o m : 1,386 S u m of s q u a r e s error: 3 4 9 , 0 3 4 R o o t m e a n s q u a r e error: 15.9 M e a n p e r c e n t error: - 2 8 . 6 % M e a n error: -1.2 S i g m a : 15.87  2  Smoothing  parameters:  A l p h a ( m e a n - t e r m ) : .20 Day  of week  G a m m a ( s l o p e - t e r m ) : .20  D e l t a ( s e a s o n a l - t e r m ) : .25  parameters:  M o n d a y : .77 F r i d a y : 1.53  T u e s d a y : .78 S a t u r d a y : 1.77  Weekly  parameters  18-MAY: 29-JUN: 10-AUG: 21-SEP: 02-NOV: 14-DEC: 25-JAN: 07-MAR: 18-APR:  .30 .70 1.68 .70 .39 .89 .98 1.27 .60  (approximate  25-MAY: 06-JUL: 17-AUG: 28-SEP: 09-NOV: 21-DEC: 01-FEB: 14-MAR: 25-APR:  .54 1.36 1.52 .72 .34 1.98 1.23 1.81 .27  W e d n e s d a y : .68 S u n d a y : .68  beginning  0 1 -J U N : 13-JUL: 24-AUG: 05-OCT: 16-NOV: 28-DEC: 08-FEB: 21-MAR:  .50 1.34 1.54 .83 .36 2.44 1.50 1.60  0 2 - M A Y : .15  T h u r s d a y : .79  date): 08- J U N : 20-JUL: 31-AUG: 12-OCT: 23-NOV: 04-JAN: 15-FEB: 28-MAR:  .40 1.90 1.12 .88 .78 1.65 1.43 1.30  0 9 - M A Y : .13  50  15-JUN: 27-JUL: 07-SEP: 19-OCT: 30-NOV: 11-JAN: 22-FEB: 04-APR:  .50 1.95 .94 .41 .37 .87 1.72 .97  22-JUN: 03-AUG: 14-SEP: 26-OCT: 07-DEC: 18-JAN: 29-FEB: 11-APR:  .50 1.85 .72 .33 .48 .85 1.64 .76  APPENDIX C - ARIMA A R I M A O n e B e d r o o m Model Conditional  Parameter  Term MU Moving Average Terms MA1,1 MAI,2 A u t o r e g r e s s i v e Terms AR1,1 AR1,2 Weekly Autoregressive AR2,1 Yearly Autoregressive AR3,1  Least  Squares  Estimate  Standard Error  28.93936  5.06860  0.20889 -0.11588 0.87504 -0.11799 Term 0.17547 Term -0.38091  Estimation Approx > |t|  Lag  5.71  <.0001  0  0.04011 0.03713  5.21 -3.12  <.0001 0.0018  2 4  0.03003 0.02957  29.14 -3.99  <.0001 <.0001  1 3  0.03131  5.60  <.0001  7  0.03654  -10.42  <.0001  364  t  Value  Pr  Mean  Constant Variance Std E r r o r AIC SBC Number of * AIC and  Parameter MU MAI, 1 MAI, 2 AR1, 1 AR1, 2 AR2,1 AR3,1  To Lag  6 12 18 24 30 36 42 48  1 0 -0 0 -0 -0 -0  0 6 14 23 32 39 42 54  00 24 61 82 67 31 14 12  8.005583 2204.724 46.95449 11407.18 11442.09 Residuals 1082 SBC do not i n c l u d e l o g d e t e r m i n a n t .  C o r r e l a t i o n s of MAI, 1 MAI, 2 0 . 002 -0.002 1.000 -0.223 -0.223 1.000 0 . 655 -0.144 -0.444 0.518 0 . 052 -0.145 -0.023 0.003  Parameter AR1,1 0 . 001 0 .655 -0.144 1.000 -0.651 0.092 -0.031  Estimates AR1, 2 AR2,1 -0.004 -0.002 -0.444 0 . 052 0 . 518 -0 .145 -0 . 651 0 . 092 1. 000 -0.261 -0.261 1. 000 0 . 032 -0.037  Check  Residuals  DF  Autocorrelation Pr > ChiSq  0 6 12 18 24 30 36 42  .0001 0 .3965 0 .2632 0 . 1610 0 .1112 0 . 1190 0 . 2224 0 .0996  0 0 0 0 0 0 0 -0  MU 000 002 002 001 004 002 015  ChiSquare  Estimate Estimate Estimate  <  -0 -0 -0 -0 0 0 -0 0  Model  004 014 039 022 006 004 008 010 for  of  -Autocorrelations -  001 032 067 052 026 035 030 089  0 -0 -0 0 -0 -0 -0 -0  variable  E s t i m a t e d Mean P e r i o d ( s ) of D i f f e r e n c i n g Autoregressive  Factor 1 Factor 2 Factor 3  1:  015 016 015 064 039 057 024 035  0 0 0 0 0 -0 -0 0  001 015 018 013 061 006 008 014  -0 0 -0 0 -0 0 0 -0  FIT_1  28.93936 364  Factors  1 - 0.87504 B * * ( l ) + 0.11799 B**(3) 1 - 0 . 17547 B**(7) 1 + 0 . 3 8 0 9 1 B**(364) Moving  Factor  AR3,1 -0.015 -0.023 0 . 003 -0.031 0 . 032 -0.037 , 1.000  Average  Factors  1 - 0.20889 B**(2) + 0.11588 B**(4)  51  029 013 030 029 026 033 003 024  0 0 -0 -0 -0 0 0 -0  005 052 008 009 037 017 031 023  A R I M A Two/Three B e d r o o m Model Conditional  Parameter  Least  Squares  Standard Error  Estimate  Average Term MA1,1 0.92053 Yearly Moving Average Term MA2,1 0.46835 Autoregressive Terms AR1,1 1.86845 AR1,2 -1.04364 AR1,3 0.16905 Weekly Autoregressive Term AR2 , 1 0 .09361  Estimation  t  Value  Pr  Approx > |t|  Lag  Moving  0 . 03577  25 . 73  < . 0001  0 . 03620  12 . 94  < .0001  0 . 04501 0 . 06061 0 . 03101  41 .51 -17.22 5.45  0 . 03248  2 .88  1 364 1 2 3  < . 0001 < .0001 < .0001 0 .0040  Variance Estimate 62.67246 Std E r r o r Estimate 7.916594 AIC 7553.798 SBC 7583.717 Number o f R e s i d u a l s 1082 * AIC and SBC do not i n c l u d e l o g d e t e r m i n a n t .  Parameter  MAI, 1 MA2, 1 AR1, 1 AR1, 2 AR1, 3 AR2, 1  To Lag  6 12 18 24 30 36 42 48  1. 000 -0.003 0 . 742 -0.372 -0 . 231 0.296  ChiSquare  0 11 17 23 24 32 34 44  Correlations of MAI, 1 MA2,1  00 37 58 02 91 42 53 65  DF  0 6 12 18 24 30 36 42  -0 . 003 1 . 000 0 . 059 -0 . 089 0 . 089 0 . 010  Autocorrelation Pr > ChiSq < 0001 0 004  0 0 0 0 0 0 0  0776 1291 1897 4106 3481 5387 3612  -0 0 0 0 -0 -0 -0  Model  004 023 013 003 033 015 043  for  Parameter AR1,1  Estimates AR1, 2  0 . 742 0 . 059 1 .000 -0 . 858 0 .319 0 .205  -0 .372 -0.089 -0 . 858 1. 000 -0.759 -0.107  Check  of  AR2,  Autocorrelations  -0 0 0 0 -0 -0 -0 -0  021 004 034 015 000 007 019 076  variable  0 -0 0 0 0 0 -0 0  022 078 016 045 036 023 011 004  Autoregressive  0 009 0 012 -0 011 -0 015 0 018 -0 033 -0 009 ' 0 009  -0 -0 -0 0 -0 0 -0 -0  026 0 025 -0 053 -0 046 0 001 0 061 0 010 - 0 035 0  FIT_23  364  Factors  1 - 1.86845 B * * ( l ) + 1.04364 B**(2) - 0.16905 B**(3) 1 - 0.09361 B**(7)  Moving  F a c t o r 1: F a c t o r 2:  1  0 .296 0 . 010 0 .205 -0.107 -0.064 1. 000  Residuals  P e r i o d ( s ) of D i f f e r e n c i n g No mean term i n t h i s model.  F a c t o r 1: F a c t o r 2:  AR1,3  -0.231 0 .089 0 .319 -0.759 1 .000 -0.064  Average  Factors  1 - 0.92053 B * * ( l ) 1 - 0.46835 B**(364)  52  043 004 027 014 010 012 032 010  APPENDIX D - LINEAR REGRESSION Linear R e g r e s s i o n O n e B e d r o o m M o d e l  Analysis  Source  Model Error Corrected  Total  of  Variance  DF  Sum o f Squares  Mean Square  37 1408 1445  21268760 6426681 27695441  574831 4564.40381  Root MSE Dependent Mean C o e f f Var  67.56037 174.41286 38.73589  F  Value  125.94  R-Square Adj R-Sq  Pr  > F  < . 0001  0.7680 0.7619  Parameter E s t i m a t e s Dependent V a r i a b l e : FIT_1  Variable  Label  DF  Parameter Estimate  General InterceptIntercept Intercept 1 93 20633 Yearly Trend Parameter SN 1 29 67332 Seasonal Period Parameters (Summer. SI SI 1 -182 87644 S1_DG 1 60 75948 S2 S2 1 -109 12375 S2_D 1 33 89286 S2_EF 1 109 59592 S2_G 1 163 57477 S4 S4 1 105 84829 1 S4_BC 40 20238 S4_F 1 -58 54762 S4_G 1 -114 27018 S5_BE 1 -71 35990 S5_G 1 -103 52768 S6 S6 1 -153 78780 Seasonal Period Parameters (Winter Wl Wl 1 -121 68495 W1_D 1 39 26190 W1_E 1 115 62976 W2 W2 1 211 12617 W2_B 1 53 75000 W2_C 1 -79 89079 W3_FG 1 72 83232 W4 W4 1 229 36735 W5 W5 1 101 50453 W5_C 1 -122 18418 W6 W6 1 240 56038 W7_D 1 -81 75047 W7_EF 1 -12 7 13583 Day of Week Parameter SAT SAT 1 51 88946  53  Standard Error  t  Value  Pr  >  Itl  6 03273  15 45  <  .0001  1 61654  18 36  <  .0001  9 11 8 14 11 31 9 11 14 16 7 24 7  57005 36755 96432 74288 71884 24706 19576 65527 74288 88808 98835 34367 55306  -19 5 -12 2 9 5 11 3 -3 -6 -8 -4 -20  11 34 17 30 35 23 51 45 97 77 93 25 36  .0001 . 0001 < . 0001 0 . 0217 < .0001 < .0001 < .0001 0 .0006 < . 0001 < . 0001 < . 0001 < .0001 < .0001  9 14 18 13 18 27 14 15 10 31 15 13 13  39803 74288 65431 56494 05627 10031 15107 08652 75644 83587 75840 66522 79685  -12 2 6 15 2 -2 5 15 9 -3 15 -5 -9  95 66 20 56 98 95 15 20 44 84 27 98 21  .0001 0 .0078 < .0001 < .0001 0 .0030 0 . 0033 < . 0001 < . 0001 < . 0001 0 . 0001 < .0001 < .0001 < .0001  6 49948  7 98  < <  <  <  .0001  Linear R e g r e s s i o n O n e B e d r o o m Model (contd.) Parameter Variable  Label  Weekend Period S_12_WD S3_WD S4_WD S_56_WD W1_WD W3_WD W4_WD W6_WD W7 WD  DF  Interaction 1 1 1 1 1 1 1 1 1  Estimate  Parameters 53 37471 178 04065 59 01425 100 60750 63 54727 55 18529 -120 82670 -134 58223 76 31825  54  Standard Error  9 24 11 9 13 11 22 22 12  11061 52147 57597 00418 43712 46217 43503 89227 37653  t Value  5 7 5 11 4 4 -5 -5 6  86 26 10 17 73 81 39 88 17  Pr > |t| < < < < < < < < <  0001 0001 0001 0001 0001 0001 0001 0001 0001  Linear R e g r e s s i o n T w o B e d r o o m Model Analysis  Source  Model Error Corrected  Total  of  Variance  DF  Sum o f Squares  Mean Square  36 1409 1445  504306 202603 706909  14009 143 .79194  Root MSE Dependent Mean Coeff Var  11.99133 27.12379 44.20963  R-Square Adj R-Sq  Value  Pr  97 .42  < .0001  > F  0.7134 0.7061  Parameter E s t i m a t e s Dependent V a r i a b l e : FIT_2  Variable  Label  DF  Parameter Estimate  General Intercept Parameter Intercept Intercept 1 -7 .18376 General Intercept Parameter SN 1 3 .85587 Seasonal Period Parameters (Summer) S1_DE 1 4 .79338 SI FG 1 6 .94904 S2 S2 1 7 .91945 S2_E 1 16 .59821 S2 F 1 20 .40404 S2_G 1 24 .86609 S3 S3 1 41. 35331 S4 S4 1 41. 86588 S4_EF 1 -4 .86607 S4_G 1 -12 .80107 S5 S5 1 11. 89670 S5_BD 1 -4 .64286 S5_G 1 -8 .79774 Seasonal Period Parameters (Winter> W1_BC 1 • 4 48195 . W1_D 1 11. 30338 W1_E 1 26 .81794 W2 W2 1 53 .52199 W3 W3 1 19. 18433 W3_D 1 5 .76190 W3_E 1 13 .15476 W3_FG 1 13 .80060 W4 W4 1 53 .16052 W4_B 1 -13 .53906 W5 W5 1 42 .87725 W5_C 1 -20 .57188 W6 W6 1 49 .20994 W7 W7 1 44 .03552 W7_C 1 -15 .62500 W7_D 1 -28 .51786 W7_E 1 -35 .81509 W7_F 1 -45 .71576  55  Standard Error  t  Value  Pr  >  |t  1 13703  -6 32  < 0001  0 28504  13 53  < 0001  1 2 1 2 2 5 3 1 1 2 1 1 4  85556 74099 38184 53363 57096 49774 56366 38184 96254 91469 64566 85030 43843  2 2 5 6 7 4 11 30 -2 -4 7 -2 -1  58 54 73 55 94 52 60 30 48 39 23 51 98  0 0099 0 0113 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001 0 0133 < 0001 < 0001 0 0122 0 0477  1 2 3 1 1 2 2 2 2 4 1 5 2 1 2 2 3 5  78699 40022 10680 69482 52886 61672 61672 73335 40022 43765 90484 65149 15325 78699 77545 77545 01970 61428  2 4 8 31 12 2 5 5 22 -3 22 -3 22 24 -5 -10 -11 -8  51 71 63 58 55 20 03 05 15 05 51 64 85 64 63 28 86 14  0 0122 0001 0001 0001 0001 0 0278 < 0001 < 0001 < 0001 0 0023 < 0001 0 0003 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001 < < < <  Linear R e g r e s s i o n T w o B e d r o o m Model (contd.)  Variable  Label  Weekend Period Interaction S5_WD Day of Week Parameters FRI FRI SAT SAT SUN SUN  DF  Parameter Estimate  Standard Error  1  Parameter 18 50671  2 14253  8 64  < 0001  1 1 1  5 82153 11 50077 2 11260  0 97760 0 97770 0 93696  5 95 11 76 2 25  < 0001 < 0001  56  t  Value  Pr  >  |t|  0 0243  APPENDIX E - POISSON REGRESSION  P o i s s o n R e g r e s s i o n Three B e d r o o m Model  Model I n f o r m a t i o n Data S e t Distribution Link Function Dependent V a r i a b l e O b s e r v a t i o n s Used  MONTH.ALL_TR_3 Poisson Log FIT_3 1446  C r i t e r i a F o r A s s e s s i n g Goodness Of F i t Criterion Deviance S c a l e d Deviance Pearson C h i - S q u a r e S c a l e d Pearson X2 Log L i k e l i h o o d  DF 1417 1417 1417 1417  Value 2105.3556 2105.3556 2165.7431 2165.7431 1641.2989  Value/DF 1.4858 1.4858 1.5284 1.5284  The GENMOD Procedure A l g o r i t h m converged.  A n a l y s i s Of Parameter E s t i m a t e s  Parameter  Seasonal  DF  Period  S1_BC S1_DF S2 S2_B S2_EF S_34 S4_E S5 S5_BD S6  Seasonal  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  Day of Week  Standard Error  Parameters 0 0 0 -0 0 1 0 0 -0 0 0 l 0 1 0 1 1 2 1 -0  3747 5032 3294 0786 4361 9394 5509 0281 6372 7557  Wald 95% C o n f i d e n c e Limits  ChiSquare  Pr > ChiSq  (Summer)  2380 6223 7532 3886 5260 4741 1866 9180 2357 7819  Period Parameter  Wl W2 W2_C W3 W3_E W4 W5 W6 W7 W7_CF SAT  Estimate  0 0 0 0 0 0 0 0 0 0  1331 0941 0754 1591 0934 0367 0890 0694 0860 0687  0 0 0 0 0 0 0 0 0 0  0884 0626 1547 0558 0960 0808 0629 0811 0586 0909  s  -0 0 0 -0 0 1 0 0 -0 0  0230 4379 6054 7004 3429 4021 0122 7820 4042 6473  0 0 0 -0 0 1 0 1 -0 0  4989 8067 9009 0767 7091 5460 3610 0540 0671 9165  3 43 99 5 31 1611 4 175 7 129  19 77 81 96 71 04 40 08 51 60  0 1 0 0 0 1 1 1 1 -0  2015 3804 0262 9692 2479 7809 4275 8692 5223 9339  0 1 0 1 0 2 1 2 1 -0  5479 6260 6325 1879 6243 0978 6742 1871 7521 5775  17 575 4 373 20 575 607 625 779 69  97 79 53 82 63 47 60 23 61 07  0 0332 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001  14 07  0 0002  0 0739 < 0001 < 0001  0 0146 < 0001 < 0001 0 0360 < 0001 0 0061 < 0001  (Winter) < 0001 < 0001  Parameter  i  0 1595  0 0425  0 0762  57  0 2429  P o i s s o n R e g r e s s i o n Three B e d r o o m Model (contd.)  Parameter  Standard Error  DF  Estimate  Weekend  Period  Interaction  WKD_P1 WKD_P2 WKD_P5 WKD_P6 WKD_P7 WKD_P9 WKD_P10 WKD_P12 Scale  1 1 1 1 1 1 1 1 0  0 0 0 0 0 0 -0 -0 1  4396 4280 7953 5263 7438 2750 4642 5932 0000  Wald 95% C o n f i d e n c e Limits  Pr > ChiSq  Parameters 0 0 0 0 0 0 0 0 0  1247 0938 0880 1112 1259 0892 1422 1442 0000  NOTE: The s c a l e parameter was h e l d  0 0 0 0 0 0 -0 -0 1  1951 2442 6229 3084 4970 1002 7429 8759 0000  0 0 0 0 0 0 -0 -0 1  6840 6117 9678 7443 9906 4498 1855 3105 0000  fixed.  Lagrange M u l t i p l i e r S t a t i s t i c s Parameter Intercept  ChiSquare  '  Chi-Square 1.7128  58  Pr > C h i S q 0.1906  12 20 81 22 34 9 10 16  42 84 74 40 89 51 66 92  0 0004 < 0001 < 0001 < 0001 < 0001 0 0020 0 0011 < 0001  APPENDIX F - NONLINEAR REGRESSION  Nonlinear R e g r e s s i o n O n e B e d r o o m Model Dependent V a r i a b l e A_TERM Method: Gauss-Newton OTE: Convergence c r i t e r i o n met.  Estimation  Summary  Gauss-Newton Method Iterations 7 R 287E-6 PPC(D7) 000042 RPC(D7) 000121 Object 68E-10 Objective 4971593 O b s e r v a t i o n s Read 144S O b s e r v a t i o n s Used 1446 Observations Missing 0 NOTE: An i n t e r c e p t was not s p e c i f i e d f o r t h i s model,  Source  Regression Residual Uncorrected Total Corrected Total  Parameter  Yearly  DF  Sum o f Squares  Mean Square  44 1402 1446 1445  66710946 4971593 71682539 27695441  1516158 3546 .1  Estimate  Trend Parameters TREND1 23 5.0 TREND2 75.9307 Period Parameters (Summer) PI -3.2618 P1_DG 1.1976 P2 -1.8741 P2_D 0 .4532 P2_EF 1.3201 P2_G 2.0503 P4 0.5275 P4_BC 0.4574 P4_F -0.6225 P4_G -1.1311 P5 -0.7504 P5_BE -0.6508 P5_G -1.0100 P6 -2 . 2529 P6_BE -0 . 7657  Std  Approx Error  F  Value  427.56  < . 0001  A p p r o x i m a t e 95% C o n f i d e n c e Limits  9.2855 3 .2913  216 . 8 69 .4741  253 . 2 82 3872  0 .2502 0 .2663 0 .1180 0 . 1711 0 1320 0 3356 0 0983 0 1318 0 1358 0 1505 0 0979 0 1129 0 3068 0 .1976 0 2328  -3 7527 0 6752 -2 1056 0 1176 1 0612 1 3920 0 3346 0 1987 -0 8889 -1 4265 -0 9426 -0 8723 -1 6119 -2 .6405 -1 2223  -2 1 -1 0 1 2 0 0 -0 -0 -0 -0 -0 -1. -0  59  Approx Pr > F  7709 7199 6427 7888 5789 7086 7204 7160 3560 8358 5583 4293 4081 8652 3091  Nonlinear R e g r e s s i o n O n e B e d r o o m Model (contd.)  Parameter  Estimate  Period Parameters (Winter) P7 -2 . 1008 P7_D 0 . 5077 P7_E 1 .3932 P8 1 . 9622 P8_B 176 . 9 P8_C -1 . 0090 P9 -0 . 7077 P9_E 0 .3210 P9_FG 0 .9432 P10 2 . 9136 Pll 0 .5959 P11_C -1 .3452 P12 3 .1819 P13 0 .1974 P13_C -0 . 7718 P13_D -1 .4838 P13 EF -2 . 1677 Day of Week Parameters (l=Monday) D6 0.6524 D7 0.1638 Weekend Period Interaction Parameters WDS1S2 0 8070 WDS3 1 8641 WDS4 0 9408 WDS5S6 1 3709 WDW1 0 8586 WDW3 0 6771 WDW4 -2 1914 WDW6 -2 5790 WDW7 0 5690  Std  Approx Error  A p p r o x i m a t e 95% C o n f i d e n c e Limits  0 0 0 0  1465 1852 1811 3228  -2 .3883 0 . 1444 1 . 0379 1 .3290  -1 0 1 2  8134 8711 7485 5954  0 0 0 0 0 0 0 1 0 0 0 0  3714 0746 1238 1334 7672 1120 4011 0118 0996 1385 1623 2090  -1 . 7377 -0 . 8541 0 .0782 0 .6815 1 .4087 0 .3762 -2 . 1320 1 . 1970 0 .00199 -1 . 0434 -1 .8021 -2 .5777  -0 -0 0 1 4 0 -0 5 0 -0 -1 -1  2804 5612 5638 2050 4185 8156 5585 1667 3927 5001 1654 7577  0 .0736 0 . 0618 0 0 0 0 0 0 0 0 0  60  1131 5462 1629 1111 1573 1120 7458 9879 1322  0 .5081 0 . 0426 0 0 0 1 0 0 -3 -4 0  5851 7927 6213 1530 5500 4575 6544 5169 3096  0 . 7968 0 .2849 1 2 1 1 1 0 -0 -0 0  0289 9355 2603 5888 1673 8967 7284 6410 8284  Nonlinear Regression Two Bedroom Model Dependent V a r i a b l e FIT_2 Method: Gauss-Newton NOTE: Convergence c r i t e r i o n met.  Estimation  Summary  Method Iterations R PPC(D7) RPC(WB13) Object Obj e c t i v e O b s e r v a t i o n s Read O b s e r v a t i o n s Used Observations Missing  Gauss-Newton 7 2.647E-6 0.000012 0 .000044 1.27E-10 152506 . 8 1446 1446 0  The NLIN Procedure NOTE: An i n t e r c e p t was not s p e c i f i e d f o r t h i s model.  Source  Regression Residual Uncorrected Total Corrected Total  Parameter  Yearly  Trend Parameters TREND1 TREND2 P e r i o d Parameters P1_DE P1_FG P2 P2_E P2_F P2_G P3 P4 P4_EF P4_G P5 P5_BD P5_G P6_CD  DF  Sum o f Squares  Mean Square  40 1406 1446 1445  1618224 152507 1770731 706909  40455.6 108 . 5  Estimate  31 5383 10 4943 (Summer) 1 0387 1 2220 1 3089 1 4409 1 6528 2 4922 3 7753 4 1942 -0 5138 -1 1505 1 8613 -0 4661 -0 7547 -0 3947  Std  Approx Error  F  Value  Approx Pr > F  372.97  < .0001  A p p r o x i m a t e 95% C o n f i d e n c e Limits  1 2706 0 4590  29 0458 9 5940  34 0307 11 3946  0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 1 1 1 2 3 -0 -1 1 -0 -1 -1  1 1 1 1 2 3 4 4 -0 -0 2 -0 0 0  61  2442 3468 2102 1780 1840 4711 4029 2335 1635 1959 2241 1516 4000 3349  5598 5418 8966 0918 2919 5680 9850 7362 8345 5349 4217 7635 5394 0517  5177 9023 7212 7901 0137 4164 5655 6521 1931 7662 3009 1687 0301 2623  Nonlinear R e g r e s s i o n T w o B e d r o o m Model (contd.)  Parameter  Estimate  Period Parameters (Winter) 0.7919 P7_BC P7_D 1.4522 P7_E 2.6357 6.4129 P8 P8_B 812 . 0 P9 2 .2680 P9_D 0 . 5027 P9_E 0 . 9797 P9_FG 1.1693 P10 61 . 7079 P10_B 57 . 7100 Pll 4.6024 P11_C -1.7863 P12 368 . 8 P13 4.9684 -1.8392 P13_C P13_D -2.7805 P13_E -3.4934 P13_F -5.0951 Day of Week Parameters (1= Monday) Dl -3 .0880 D4 0.2193 D5 1. 0322 D6 1. 6507 D7 0.3696 Weekend Period Interaction Parameters WB5 0.7393 WB6 0.6485 WB13 -0.5984  Std  Approx Error  A p p r o x i m a t e 95% C o n f i d e n c e Limits  Seasonal  0 0 0 1  2568 2632 2626 2468  0 0 2 3  2882 9358 1206 9671  1 1 3 8  2957 9686 1508 8588  0 0 0 0 0  2078 1679 1751 1834 5122  1 0 0 0 60  8604 1734 6362 8096 7031  2 0 1 1 62  6757 8320 3233 5291 7128  0 2989 0 4989  4 0161 -2 7649  5 1887 -0 8077  0 0 0 0 1  3442 3021 3142 3516 6321  4 -2 -3 -4 -8  2932 4317 3968 1832 2969  5 -1 -2 -2 -1  6435 2466 1642 8037 8934  0 0 0 0 0  1958 0912 1034 1115 0891  -3 0 0 1 0  4722 0404 8294 4320 1949  -2 0 1 1 0  7038 3982 2350 8694 5443  0 1716 0 2787 0 2116  62  0 4026 0 1019 -1 0135  1 0760 1 1951 -0 1833  APPENDIX G - BASELINE  REGRESSION  B a s e l i n e R e g r e s s i o n (nonlinear) O n e B e d r o o m Model Dependent V a r i a b l e FIT_1 Method: Gauss-Newton NOTE: Convergence c r i t e r i o n met.  Estimation  Summary-  Method Iterations Subiterations Average S u b i t e r a t i o n s R PPC(WC5) RPC(WC7) Obj ect Obj e c t i v e O b s e r v a t i o n s Read O b s e r v a t i o n s Used Observations Missing  Gauss-Newton 20 2 0 .1 5 . 78E-6 0 .000467 0.049505 9 .346E-9 1.5702E8 131586 131586 0  NOTE: An i n t e r c e p t was not s p e c i f i e d f o r t h i s model.  Source  Regression Residual Uncorrected Total Corrected Total  Parameter  Yearly  DF  Sum o f Squares  Mean Square  112 131474 131586 131585  1.4115E9 1.5702E8 1.5685E9 9.8471E8  12602761 1194.3  Approx Estimate  Trend Parameters TREND1 299 . 6 TREND2 118 . 6 Period Parameters (Summer) P1_B 0 .4745 P1_C 0 .3188 P1_D 1 .4219 P1_E 1 . 8412 P1_F 1 .4939 P1_G 2 . 1718 P2 -0 . 0652 P2_B 0 . 1250 P2_C 0 . 1460 P2_D 0 .3472 P2_E 1 .0071 P2_F 1 . 1594 P2_G 1 .4427 P4 2 1906 P4_B 0 . 1718 P4_C 0 .0943 P4_E -0 .0805 P4_F -0 .3306 P4_G -0 . 7391  Std  Approximate Error  9 5%  F  Value  10552 . 5  Confidence Limits  4 6484 1 8370  290 . 5 115 . 0  308 . 7 122 . 2  0 1527 0 1699 0 1129 0 1077 0 1185 0 1583 0 1552 0 0347 0 0344 0 0323 0 0287 0 0284 0 0414 0 .1058 0 0100 0 0102 0 0109 0 0121 0 0166  0 . 1752 -0 .0143 1 .'2006 1 .6302 1 .2616 1 . 8616 -0 .3693 0 . 0570 0 . 0785 0 .2838 0 . 9509 1 . 1038 1 .3615 1 9833 0 . 1521 0 . 0742 -0 . 1017 -0 .3543 -0 . 7717  0 . 7738 0 . 6519 ' 1. 6433 2 .0523 1 . 7262 2 .4821 0 .2390 0 .1930 0 .2135 0 .4105 1 . 0632 1 .2151 1 . 5239 2 3979 0 . 1915 0 . 1144 -0 .0592 -0 .3069 -0 . 7065  63  Approx Pr > F  < .0001  B a s e l i n e R e g r e s s i o n (nonlinear) O n e B e d r o o m Model (contd.) Approx Parameter  Seasonal  Period Parameters P5 P5_B P5_C P5_D P5_E P5_G P6 P6_B P6_C P6_D P6_E Seasonal Period Parameters P7_B P7_C P7_D P7_E P8 P8_B P8_C P9 P9_B P9_C P9_D P9_E P9_F P9_G P10 P10_B Pll P11_B P11_C P12 P12_B P13 P13_B P13_C P13_D P13_E P13_F Day of Week Parameters (1 Dl D4 D5 D6 D7 Seasonal Period Lead Time B B2 B3 B4 B5 B6 B7 B8 B9 BIO  Estimate  Std  Approximate Error  (Summer) contd. 0 . 9046 0 . 1368 -0 .4706 0 .0198 -0 .4202 0 .0193 -0 .3880 0 .0190 -0 . 1042 0 .0166 -0 . 7148 0 . 0412 -0 .5575 0 .4620 -0 . 5323 0 . 0549 -0 . 9356 0 .0710 -0 .9741 0 0730 -0 . 9276 0 . 1244 (Winter) 0 . 1739 0 . 0373 -0 . 0360 0 . 0405 0 .6539 0 . 0329 1 .4913 0 . 0308 2 . 6874 0 . 1124 0 .4065 0 .00919 -0 . 7174 0 .0157 2 . 1928 0 . 1084 -0 . 0492 0 .0133 -0 .3038 0 .0146 -0 .2197 0 . 0141 0 . 1691 0 . 0125 0 .6026 0 . 0121 0 . 9630 0 . 0275 4 . 9812 0 . 1052 -0 . 8181 0 .0151 2 . 7607 0 . 1087 0 . 0435 0 . 0118 -0 . 7283 0 . 0386 3 . 9043 0 . 1057 0 . 0700 0 . 0201 2 .5969 0 . 1107 -0 . 1867 0 .0105 -0 . 7043 0 .0131 -1 .5075 0 . 0209 -2 . 1900 0 . 0368 -3 .3720 0 .2521 = Monday) -5 .2222 0 . 1042 0 . 0493 0 .00458 0 . 1632 0 . 0154 0 2966 0 . 0154 0 0176 0 .00465 i n t e r a c t i o n Parameters 0 3563 0 .0126 2 7764 0 .1108 4 0208 0 .1074 2 3603 0 . 0241 2 4883 0 .0922 2 5387 0 .4687 2 3528 0 . 1037 2 8170 0 . 0422 1 6793 0 . 0324 0 8467 0 . 0232  64  95%  Confidence Limits  0 . 6364 -0 . 5094 -0 .4580 -0 .4251 -0 . 1368 -0 .7956 -1 .4630 -0 .6399 -1 .0749 -1. 1172 -1 .1714  1 1729 -0 4319 -0 3825 -0 3508 -0 0716 -0 6340 0 3479 -0 4247 -0 7964 -0 .8310 -0 6838  0 .1007 -0 .1154 0 .5895 1 .4310 2 .4671 0 .3884 -0 . 7482 1 . 9804 -0 .0753 -0 .3325 -0 . 2474 0 . 1446 0 .5789 0 . 9091 4 . 7749 -0 . 8476 2 . 5477 0 . 0203 -0 .8039 3 .6971 0 . 0306 2 .3799 -0 .2074 -0 . 7300 -1 . 5485 -2 .2621 -3 . 8661  0 0 0 1 2 0 -0 2 -0 -0 -0 0 0 1 5 -0 2 0 -0 4 0 2 -0 -0 -1 -2 -2  2470 0435 7183 5516 9077 4245 6866 4053 0231 2751 1920 1936 6263 0169 1875 7886 9738 0667 6527 1116 1093 8140 1661 6786 4665 1180 8778  -5 .4264 0 . 0403 0 . 1331 0 .2665 0 . 00850  -5 0 0 0 0  0180 0583 1934 3268 0267  0 .3316 2 . 5593 3 8102 2 3130 2 3076 1 6200 2 1495 2 7343 1 6158 0 8012  0 2 4 2 2 3 2 2 1 0  3810 9936 2313 4075 6690 4573 5562 8997 7427 8922  B a s e l i n e R e g r e s s i o n (nonlinear) O n e B e d r o o m Model (contd.) Approx Estimate  Parameter  Std  Approximate Error  95%  Confidence Limits  Seasonal  Period Lead Time Interaction Parameters (contd.) Bll 1 9279 2 0567 1.9923 0. 0329 B12 1.6666 0 .0277 1 6123 1 7208 2.0104 B13 0 .0403 1 9314 2 0895 sriod Lead Time Interaction Para; meters WB 0 . 5295 0 .0175 0 4952 0 5638 WB3 0.7336 0 .0289 0 6769 0 7902 WB4 -0 .3546 0 .0157 -0 3854 -0 3237 WB5 0.7073 0 .0172 0 6737 0 7409 WB6 1 .0736 0 .0779 0 9209 1 2263 WB7 0 .2568 0 .0237 0 2105 0 3032 WB8 -0 .4050 0 .0170 -0 4383 -0 3717 WB10 -1 .1678 0 .0186 -1 2043 -1 1313 WB11 -0.3596 0 .0187 -0 3963 -0 3230 WB12 -1.4620 0 .0234 -1 5078 -1 4162 WB13 -0.2831 0 .0157 -0 3139 -0 2523 Exponent Parameters for Seasonal Period Lead Time Interactions (e.g. B2*T" where T=lead time between 0 and 1) 16 0914 0 8948 14 3376 17 8452 C2 0 8596 0 0560 0 7498 0 9693 C3 0 3351 0 0153 0 3051 0 3650 C4 1 4256 0 0308 1 3653 1 4860 C5 1 2074 0 0642 1 0816 1 3332 C6 1 5767 0 3297 0 9305 2 2228 C7 0 7442 0 0524 0 6416 0 8468 C8 0 6351 0 0199 0 5960 0 6742 C9 0 8465 0 0342 0 7794 0 9135 C10 2 2505 0 1032 2 0483 2 4528 Cll 1 2605 0 0459 1 1706 1 3504 C12 1 3545 0 0454 1 2655 1 4435 C13 0 9487 0 0368 0 8766 1 0209 Exponent Parameters for Weekend Seasonal Period Lead Time Interactions (e.g. WB3*WEEKEND*T where T=lead time between 0 and 1, WEEKEND=1 i f a or Saturday night) 2 7774 0 1658 2 .4525 3 . 1023 WC3 -0 2562 0 0167 -0 .2890 -0 .2234 WC4 1 0027 0 1612 0 .6868 1 .3186 WC5 -0 0232 0 0468 -0 . 1149 0 . 0685 WC6 0 1010 0 2468 -0 .3829 0 . 5848 WC7 0 1942 0 2051 -0 .2078 0 . 5963 WC8 0 3478 0 0695 0 .2116 0 .4841 WC10 -0 1091 0 0103 -0 . 1292 -0 . 0889 WC11 -0 1494 0 0583 -0 .2637 -0 . 0351 WC12 -0 0289 0 0143 -0 . 0568 -0 . 00094 WC13 0 1968 0 1010 -0 .30116 0 .3949 2  c  C3  wc  65  Friday  Baseline R e g r e s s i o n (nonlinear) T w o B e d r o o m Model  Dependent V a r i a b l e FIT_2 Method: Gauss-Newton NOTE: Convergence c r i t e r i o n met.  Estimation  Summary  Method Iterations R PPC(C_1011) RPC(C_1011) Object Objective O b s e r v a t i o n s Read O b s e r v a t i o n s Used Observations Missing  Gauss-Newton 45 8 .307E-6 0 .000196 0.000254 1.88E-10 7532086 131586 131586 0  NOTE: An i n t e r c e p t was not s p e c i f i e d f o r t h i s model.  Source  Regression Residual Uncorrected Total Corrected Total  Parameter  Yearly  DF  Sum o f Squares  Mean Square  85 131501 131586 131585  69107115 7532086 76639201 41937547  813025 57 .2778  Estimate  Trend Parameters TREND1 35 . 7075 TREND2 14 .4171 e r i od Parameters (Summer) P1_DE 1 . 9699 P1_FG 2 .2014 P2 1 .4666 P2_B -0 .3220 P2_D -0.1274 P2_E 0.9098 P2_F 1 . 0240 P2_G 1 .6122 P3 3 . 8499 P4 3 .6717 P4_B 0.0416 P4_C 0 .1129 P4_E -0.2848 P4_F -0 .1618 P4_G -0 . 5726 P5 1 .9732 P5_B -0 .3236 P5_C -0 . 5789 P5_D -0.3272 P5_G -1. 0436 P6 -1.1979 P6_C -0.5374 P6_D -0.7844  Std  Approx Error  Value  Approx Pr > F  14194.4  < .0001  F  A p p r o x i m a t e 95% C o n f i d e n c e Limits  0 3258 0 1286  35 0689 14 1650  36 3461 14 6691  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1 1 1 -0 -0 0 0 1 3 3 0 0 -0 -0 -0 1 -0 -0 -0 -1 -1 -0 -1  2 2 1 -0 -0 0 1 1 4 3 0 0 -0 -0 -0 2 -0 -0 -0 -0 -0 -0 -0  3  66  1060 1100 1485 0322 0290 0208 0207 0375 1075 1059 0121 0120 0130 0126 0155 1230 0231 0261 0232 0640 2356 0928 1114  7621 9858 1756 3851 1843 8691 9835 5387 6392 4642 0178 0894 3103 1865 6031 7321 3689 6301 3727 1690 6596 7193 0029  1776 4170 7577 2589 0705 9506 0646 6856 0605 8793 0653 1364 2593 1371 5422 2143 2782 5277 2818 9183 7362 3555 5660  B a s e l i n e R e g r e s s i o n (nonlinear) T w o B e d r o o m Model (contd.)  Parameter  Estimate  Period Parameters (Winter) P7_B 0.2209 P7_C -0.1911 P7_D 0.4585 P7_E 1.7226 P8 5 .2560 P8_B 0.43 98 P8_C -0 . 5051 P9 3.4555 P9_C -0.1193 P9_D 0.1611 P9_E 0 . 6330 P9_FG 0 . 8219 P10 6.3793 P10_B -0.4505 Pll 4.6634 P11_B 0.0391 P11_C -0.7306 P12 5.8774 P13 4.4725 P13_B -0.3237 P13_C -1.1948 P13_D -2.0204 P13_E -2.9338 P13_F -4 .2937 Day of Week Parameters (1 = Monday) Dl -5.5321 D4 1060 D5 5753 D6 6332 D7 0605 Period Lead Time Interactions B 1 1450 B2 1 2430 B4 0 9535 B5 0 8846 B6 2 2000 B7 2 0859 B8 0 4572 B_1011 0 3312 B12 0 2207 B13 0 8927 Briod Lead Time Interactions WB 0 3940 WB3 0 5351 WB4 -0 5179 WB5 0 8392 WB6 1 0675 WB7 0 1479 WB8 -0 9414 WB10 -2 3222 WB11 -0 4434 WB12 -1 9353 WB13 -0 7828  Std  A p p r o x i m a t e 9 5% C o n f i d e n c e Limits  Approx Error  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0415 0485 0390 0351 1057 0154 0196 1041 0149 0137 0127 0130 1105 0209 1049 0143 0367 1075 3508 0126 0159 0210 0346 2173  0 . 1047 0 .00608 0.0101 0.0100 0.00614  0 -0 0 1 5 0 -0 3 -0 0 0 0 6 -0 4 0 -0 5 3 -0 -1 -2 -3 -4  1396 2862 3821 6539 0488 4098 5434 2515 1485 1342 6080 7965 1627 49154578 0111 8025 6667 7850 3485_ 2261 0615 ' 0015 7195  -5.7373 0 .0940 0 .5556 0 .6136 0 . 0485  0 -0 0 1 5 0 -0 3 -0 0 0 0 6 -0 4 0 -0 6 5 -0 -1 -1 -2 -3  3022 0960 5348 7914 4632 4699 4668 6594 0900 1880 6579 8474 5958 4094 8691 0670 6588 0882 1600 2990 1636 9793 8660 8679  -5 . 3269 0 .1179 0 .5950 0.6529 0 . 0726  0 0 0 0 0 0 0 0 0 0  0173 1010 0213 0714 2192 1086 0370 0279 0374 3290  1 1 0 0 1 1 0 0 0 0  1111 0451 9118 7446 7703 8730 3846 2766 1474 2479  1 1 0 1 2 2 0 0 0 1  1790 4409 9951 0246 6297 2989 5298 3858 2941 5374  0 0 0 0 0 0 0 0 0 0 0  0159 0331 0117 0270 1054 0288 0260 0348 0193 0274 0138  0 0 -0 0 0 0 -0 -2 -0 -1 -0  3628 4703 5408 7863 .8609 0915 9922 3903 4814 9890 8097  0 0 -0 0 1 0 -0 -2 -0 -1 -0  4252 5999 4949 8921 2741 2043 8905 2541 4055 8817 7558  67  B a s e l i n e R e g r e s s i o n (nonlinear) T w o B e d r o o m Model (contd.)  Parameter  Estimate  Std  Approx Error  A p p r o x i m a t e 9 5% C o n f i d e n c e Limits  Exponent Parameters for Seasonal Period Lead Time Interactions B2*T° where T=lead time between 0 and 1) C 0 . 9418 0 . 0258 0.8912 0 9923 C2 0 .6642 0 . 0926 0 .4826 0 8458 C4 1.4767 0.0718 1.3360 1 6174 C5 1.4063 0 . 1725 1.0681 1 7444 C7 0.2152 0.0197 0.1767 0 2538 C_1011 4 .1863 0 . 6267 2.9580 5 4145 C13 0 . 1883 0 . 0928 0.00636 0 3703 Exponent Parameters for Weekend Seasonal Period Lead Time Interactions (e.g. WB5*WEEKEND*T where T=lead time between 0 and 1, WEEKEND=1 if a or Saturday night, 0 otherwise) WC 6.2312 0.4447 5.3596 7.1028 WC5 0.3805 0.0744 0.2347 0.5263 WC8 0.1220 0.0207 0.0815 0.1625 2  >,C5  68  Friday  B a s e l i n e R e g r e s s i o n (Poisson) Three B e d r o o m Model  Model  Information  Data S e t Distribution Link Function Dependent V a r i a b l e O b s e r v a t i o n s Used  Criteria Criterion Deviance S c a l e d Deviance Pearson C h i - S q u a r e S c a l e d Pearson X2 Log L i k e l i h o o d  MONTH.ALL_TR_3 Poisson Log FIT_3 131586  F o r A s s e s s i n g Goodness Of F i t DF 13E4 13E4 13E4 13E4  Value 154544.8740 154544.8740 147757.0998 147757.0998 14550.0430  Value/DF 1.1752 1. 1752 1.1236 1.1236  The GENMOD Procedure A l g o r i t h m converged.  Analysis  Of Parameter E s t i m a t e s  Standard Wald 95% C o n f i d e n c e Parameter DF Estimate Error Limits General Intercept Parameter 1 -1.2267 Intercept 0.0309 -1 2873 -1 1662 Period Seasonal (Summer) Parameters SI B l 0.2301 0.0350 0 1616 0 2987 S1_C l 0.6894 0.0317 0 6273 0 7516 S1_D l 0.9263 0.0298 0 8679 0 9846 S1_E l 0.6613 0.0302 0 6022 0 7205 S1_F l 0.7138 0.0324 0 6502 0 7774 S2 l 0.6046 0.0373 0 5315 0 6776 S2_B l -1.2675 0.0330 -1 3322 -1 2028 S2_CD l -0.3522 0.0197 -0 3909 -0 3136 S2_E l 0.3997 0.0195 0 3615 0 4379 S2_F l 0.6024 0.0188 0 5656 0 6393 S3 l 1.7301 0.0540 1 6243 1 8359 S4 l 1.4885 1 4237 0.0331 1 5534 S4_B l 0.2730 0.0132 0 2472 0 2989 S4_D 1 0.0595 0.0142 0 0316 0 0873 S4_E l 0.2020 0.0135 0 1756 0 2285 S4_F l 0.0926 0.0140 0 0651 0 1201 S5 l 1.3419 0.0340 1 2753 1 4085 S5_B l -0.3028 0.0168 -0 3358 -0 2698 S5_C 1 -0.2754 0.0166 -0 3080 -0 2428 S5_D l -0 .4254 0.0176 -0 4599 -0 3908 S5_E l -0.1264 0.0158 -0 1573 -0 0955 S6 l 1.0702 0.0352 1 0012 1 1392 SS_C l -0.1081 0.0178 -0 1430 -0 0732 Seasonal Pei'iod Parameters (Winter) Wl 1 0.3386 0.0404 0 2595 0 4178 W1_B 1 -0.1563 0.0255 -0 2063 -0 1063 W1_C 1 -0.2723 0.0264 -0 3239 -0 2206 W1_D 1 0.0777 0.0240 0 0305 0 1248 W1_E l 0.3322 0.0253 0 2826 0 3818 W2 1 2.3765 0.0337 2 3105 2 4425 W2_C 1 0.1324 0.0187 0 0957 0 1691 W3 1 1.6435 0.0347 1 5755 1 7115 W3 B l -0.3153 0.0186 -0 3518 -0 2787 W3_C 1 -0.2188 0.0181 -0 2544 -0 1833 W3_D 1 -0.1544 0.0178 -0 1893 -0 1194  69  ChiSquare  Pr > ChiSq  1577 68  <  0001  43 472 968 480 484 263 1473 318 420 1024 1026 2025 428 17 223 43 1558 324 273 582 64 924 36  27 35 54 27 16 08 06 44 80 93 51 74 48 52 57 50 97 18 77 41 35 57 86  <  0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001  70 37 106 10 172 4979 50 2245 286 145 75  25 52 77 43 49 85 04 51 07 56 03  < < < < < < < < < < < < < < < < < < < < < <  < 0001 < 0001  0001 0 0012 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001 <  W3_E W3_F W3_G W4 W4_B W5 W5_B W5_C W6 W6_B W7 W7_B W7_C W7_D W7_E W7_F  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  Day of Week  0 0 -0 2 -0 2 0 -0 2 -0 2 -0 -0 -1 -1 -1  3329 0382 3396 8293 1838 2089 0583 3978 9556 0808 5086 1997 4802 5010 3569 8073  0 0 -0 2 -0 2 0 -0 2 -0 2 -0 -0 -1 -1 -1  3018 0031 4501 7610 2254 1391 0271 4612 8875 1347 4431 2256 5083 5418 3993 9100  0 0 -0 2 -0 2 0 -0 3 -0 2 -0 -0 -1 -1 -1  3640 0734 2291 8976 1423 2788 0896 3345 0237 0269 5741 1737 4520 4602 3145 7046  440 4 36 6590 75 3836 13 151 7235 8 5628 227 1116 5202 3932 1189  38 55 29 30 28 97 38 44 79 64 91 40 46 78 15 47  <  0001 0 0329 < 0001 < 0001 < 0001 < 0001 0 0003 < 0001 < 0001 0 0033 < 0001 < 0001 < 0001 < 0001 < 0001 < 0001  0 0067 0 0066 0 0066 0 0059 0, 0066  0 0 0 0 0  0123 0369 0868 0518 0885  0 0 0 0 0  0385 0629 1125 0751 1142  14 56 230 114 238  45 33 23 60 64  0 0001 < 0001 < 0001 < 0001 < 0001  1 0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0  0475 4192 2866 1580 2419 2365 3305 8150 3428 8385 7164 9245 6571  1 0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0  1645 5680 0383 0249 1025 0838 4951 6684 2063 6832 5619 7691 5200  1372 169 6 7 23 16 96 393 62 368 263 456 283  37 12 58 25 47 89 65 03 19 68 22 21 20  < <  0 0 0 0 0 0 0 0 -0 0 -0  2872 3575 2651 2166 6488 3779 6531 2061 4619 2819 6057  0 0 0 0 0 0 0 0 -0 0 -0  3588 4128 4060 2593 6958 4391 7201 2531 3880 3546 5210  312 746 87 478 3140 682 1617 366 508 294 680  86 39 20 09 29 80 62 83 50 12 90  <  Parameters  TUE WED THR SAT SUN  l l l l l  Lead Time  Element Parameters  T2 T2_P2 T2_P3 T2_P4 T2_P5 T2_P6 T2_P7 T2_P8 T2_P9 T2_P10 T2_P11 T2_P12 T2_P13  l l l l l l l 1 l l l l l  Weekend  0159 0179 0564 0349 0212 0357 0159 0323 0347 0275 0334 0132 0144 0208 0216 0524  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  Seasonal  WKD_P1 WKD_P2 WKD_P3 WKD_P4 WKD_P5 WKD_P6 WKD_P7 WKD_P9 WKD_P10 WKD_P11 WKD_P12  l l l l l l 1 l l l 1  Scale  0  0 0 0 0 0 1 0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0  0254 0499 0996 0635 1013 1060 4936 1624 0914 1722 1601 4128 7417 2746 7608 6391 8468 5886  Period  0 3230  0 0 0 0 0 0 0 -0 0 -0  3852 3355 2380 6723 4085 6866 2296 4249 3183 5634  1 0000  0 0 0 0 0 0 0 0 0 0 0 0 0  0299 0380 0633 0340 0355 0390 0420 0374 0348 0396 0394 0396 0350  0 0 < < < < < < < < <  0001 0001 0103 0071 0001 0001 0001 0001 0001 0001 0001 0001 0001  Parameters 0 0 0 0 0 0 0 0 0 0 0  0183 0141 0359 0109 0120 0156 0171 0120 0188 0186 0216  0 0000  NOTE: The s c a l e parameter was h e l d  1 0000  fixed.  70  1 0000  < < < < < < < < < <  0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001  APPENDIX H - A P P R O A C H E S TO BOOKING CURVE ADJUSTMENT Table 10: B o o k i n g c u r v e e s t i m a t e c a l c u l a t i o n a p p r o a c h e s Method B o o k i n g C u r v e Adjustment Calculation  R e d u c t i o n in S q u a r e d Error from Baseline  Direct Multiplicative  44.98%  Mean Absolute Percentage Error ( M A P E )  MAPE _A  (  LT  MAPE  K  Geometric Mean Absolute P e r c e n t a g e Error ( G M A P E )  J  LT=y  f GMAPE ^  +1  J  44.96%  EB _ LT  0  44.95%  N  LT  ®LT=Y  1  +1  \GMAPE  LT=Y/  Median Absolute P e r c e n t a g e Error ( M d A P E )  'MdAPE Dlt=y y  A u t o r e g r e s s i v e integrated moving average (ARIMA) where:  A  LT=  MdAPE  LT=Y  44.91% +1  y  -  See Appendix I  L e a d t i m e ; n u m b e r of d a y s prior to a target d a t e . T a r g e t date o c c u r s w h e n L T = 0 . A c t u a l b o o k i n g s (room nights r e s e r v e d to b e o c c u p i e d o n the target date) Y d a y s prior to t h e target d a t e . E x p e c t e d b o o k i n g s ( r o o m nights r e s e r v e d to b e o c c u p i e d on the target date) Y d a y s prior to t h e target d a t e . E x p e c t e d bookings taken from booking curve baseline m o d e l for a s p e c i f i c target d a t e a n d l e a d time.  LT AB  LT=Y  FB  B o o k i n g d e v i a t i o n Y d a y s prior to target d a t e  ^LT=Y  where:  (AR  FR  ~  L T = Y  N  J7D \  ^  D  LT=Y  ;  A b s o l u t e p e r c e n t a g e error Y d a y s prior to a target d a t e . N o t e : A P E is g e n e r a l l y c a l c u l a t e d a s a p e r c e n t a g e of a c t u a l v a l u e ( A B ) , w h e r e a s in this c a s e it is c a l c u l a t e d a s a p e r c e n t a g e of e x p e c t e d v a l u e ( E B ) s o that it m a y b e u s e d a s a multiplier.  APE  LT=Y  where:  FR ^pj?  _  — AR LT=Y  LT=Y  FB  L T = Y  LT=Y  MAPE  M e a n a b s o l u t e p e r c e n t a g e error Y d a y s prior to a target d a t e , n is t h e n u m b e r of d a y s in t h e s a m p l e (n=1,446).  L T = Y  where:  MAPE  = -  LT=Y  GMAPE  ^APE __ LT  YJ  G e o m e t r i c m e a n a b s o l u t e p e r c e n t a g e error Y d a y s prior to a target d a t e , n is t h e n u m b e r of d a y s in t h e s a m p l e (n=1,446).  L T = Y  where:  GMAPE  LT=y  =  (TT  1 1 APE  V 1=1  MdAPE  LT=YJ  )  LT=Y  where:  MdAPE  ir=y  = Median{APE  )\fi  LT=Yi  eN  M e d i a n a b s o l u t e p e r c e n t a g e error Y d a y s prior to a target d a t e . N is the s a m p l e s e t of d a y s Y d a y s prior to a target d a t e (N=1,446).  71  APPENDIX I - ARIMA MODELLING OF BOOKING CURVE ADJUSTMENT T h e short-term b o o k i n g c u r v e m o d e l w a s d e v e l o p e d to p r o v i d e i m m e d i a t e e s t i m a t e s of d e m a n d at the resort u s i n g a n E x c e l s p r e a d s h e e t without a n y t e c h n i c a l a s s i s t a n c e . In o r d e r to i n v e s t i g a t e w h e t h e r a m o r e t h o r o u g h p r o c e d u r e c o u l d p r o v i d e better e s t i m a t e s (although requiring a m o r e s o p h i s t i c a t e d statistical a p p l i c a t i o n than M S E x c e l ) a n A R I M A a p p r o a c h w a s t e s t e d for the booking curve adjustment.  B o o k i n g c u r v e b a s e l i n e error is d e f i n e d a s the d i f f e r e n c e b e t w e e n the  b o o k i n g c u r v e b a s e l i n e e s t i m a t e of b o o k i n g s to d a t e a n d a c t u a l b o o k i n g s to d a t e for a s p e c i f i c target d a t e a n d l e a d t i m e . T h e test w a s to s e e w h e t h e r a n A R I M A a d j u s t e d projection w o u l d p r o v i d e m o r e a c c u r a t e results than the direct multiplicative a p p r o a c h . T h e m a j o r d i f f e r e n c e b e t w e e n a n A R I M A a p p r o a c h a n d the direct multiplicative ( D M ) a p p r o a c h is the n u m b e r of a c t u a l b o o k i n g to d a t e t e r m s u s e d in the b o o k i n g c u r v e projection. T h e D M e s t i m a t e u s e s a s i n g l e b o o k i n g to d a t e term at the m o s t r e c e n t l e a d time a n d multiplies that b y a b a s e l i n e d e r i v e d multiple. T h e A R I M A a p p r o a c h e s t i m a t e s the pattern of b a s e l i n e error t e r m s (all error t e r m s to d a t e for a s p e c i f i c target date) a n d projects that pattern out to the target d a t e . T h e p r o j e c t e d A R I M A error e s t i m a t e is then a d d e d to the original b a s e l i n e d e m a n d e s t i m a t e in o r d e r to a c h i e v e the b o o k i n g c u r v e projection e s t i m a t e .  T h e p r o c e d u r e w a s to a p p l y s e v e n different A R I M A s p e c i f i c a t i o n s to b o o k i n g c u r v e b a s e l i n e errors prior to a target d a t e , a n d t h e n u s e the b e s t fitting A R I M A s p e c i f i c a t i o n b a s e d o n A k a i k e Information C r i t e r i o n ( A I C ) to f o r e c a s t b o o k i n g c u r v e errors out to the target d a t e . In o r d e r to h a v e a n a d e q u a t e a m o u n t of d a t a u p o n w h i c h to c r e a t e a n A R I M A f o r e c a s t for e a c h target d a t e , a m i n i m u m of 4 0 d a t a points (lead time d a y s 9 0 to 50) w a s u s e d to c a l i b r a t e the A R I M A error f o r e c a s t . A s the target d a t e a p p r o a c h e d , m o r e d a t a w a s u s e d to c a l i b r a t e the A R I M A f o r e c a s t (for e x a m p l e , 10 d a y s prior to a target d a t e , 8 0 d a t a points w o u l d b e u s e d to c a l i b r a t e the A R I M A m o d e l ; l e a d time d a y s 9 0 to 10). In o r d e r to d e t e r m i n e w h i c h A R I M A s p e c i f i c a t i o n s w e r e a p p r o p r i a t e , a stratified s a m p l e of 4 5 d a t a s e t s w a s m o d e l l e d u s i n g s t a n d a r d A R I M A p r o c e d u r e s ( s e e B o x & J e n k i n s (1976)) in S A S E T S ( E c o n o m e t r i c a n d T i m e S e r i e s ) statistical s o f t w a r e .  The  d a t a s e t s r e p r e s e n t e d a n e q u a l mix of o c c u p a n c y (high, m e d i u m , a n d l o w d a y s ) , b e d r o o m s (one, two, three p l u s ) , a n d l e a d t i m e s ( r a n d o m l y c h o s e n b e t w e e n l e a d time d a y s 5 0 a n d 1). 3 0 of the 4 5 s a m p l e d a t a s e t s w e r e c o m p l e t e l y d e s c r i b e d (white n o i s e a c h i e v e d in m o d e l r e s i d u a l s ) b y 7 different A R I M A s p e c i f i c a t i o n s , w h i l e the r e m a i n i n g 15 s a m p l e d a t a s e t s w h i l e h a v i n g m o r e p a r a m e t e r s c o u l d b e a p p r o x i m a t e d quite well b y the 7 b a s i c s p e c i f i c a t i o n s ( s e e T a b l e 11 for 7 A R I M A specifications).  72  T a b l e 11: A R I M A s p e c i f i c a t i o n s for b o o k i n g c u r v e error f o r e c a s t s  Label AR(1)  ARIMA description ARIMA(1,1,0)  Deterministic drift No  Notation  A R ( 1 ) with drift  ARIMA(1,1,0)  Yes  MA(1)  ARIMA(0,1,1)  No  V>,  M A ( 1 ) w i t h drift  ARIMA(0,1,1)  Yes  V=  White noise  ARIMA(0,1,0)  No  Vy, = a,  L i n e a r trend  ARIMA(0,1,0)  Yes  Vy, = // + a  Q u a d r a t i c trend  ARIMA(0,2,0)  Yes  V y , =ju + a  Vy, =ju +  yi  —  =aXl-9,B) +  M  a (l-OiB) t  t  2  t  Where: B o o k i n g c u r v e error e s t i m a t e at time t  y  t  a  R a n d o m c o m p o n e n t at time t  ju  D e t e r m i n i s t i c drift  B  B a c k w a r d shift o p e r a t o r ( e . g . Ba,  V  B a c k w a r d difference operator = 1-5  t  =  a_ ) t  x  (e.g. Vy,  = (1 - B)y  t  = y  t  -  y,_ ) x  T h e algorithm written in S A S E T S m o d e l l e d all 7 A R I M A s p e c i f i c a t i o n s for a s p e c i f i c l e a d time a n d target d a t e . T h e s p e c i f i c a t i o n with the l o w e s t A I C w a s then u s e d to f o r e c a s t a h e a d h p e r i o d s (h = l e a d time) to p r o v i d e a n error e s t i m a t e for the target d a t e . T h i s e s t i m a t e d error term w a s then a d d e d to the original b o o k i n g c u r v e b a s e l i n e for a final b o o k i n g c u r v e projection e s t i m a t e . F i n a l A R I M A projections w e r e t h e n c o m p a r e d to D M b o o k i n g c u r v e p r o j e c t i o n s . A s a m p l e of 101 d a y s (with 5 0 l e a d t i m e e s t i m a t e s for e a c h d a y ) s p a n n i n g a s a m p l e p e r i o d f r o m M a y 15, 1 9 9 9 to J a n . 2 0 , 2 0 0 2 w a s u s e d to c o m p a r e e s t i m a t e s . T h e results of the A R I M A m e t h o d s w e r e m i x e d . B a s e d o n the traditional M S E metric, the A R I M A e s t i m a t e s w e r e 6 0 % w o r s e t h a n the D M e s t i m a t e s . H o w e v e r , b a s e d o n the m o r e robust m e d i a n a b s o l u t e p e r c e n t a g e error ( M d A P E ) metric ( s e e E q u a t i o n 2 a n d E q u a t i o n 4), the A R I M A results p r o v i d e d a 1 4 % i m p r o v e m e n t o v e r the D M m e t h o d . D u e to the a d d e d c o m p u t a t i o n a l c o s t (in both c o m p u t e r time a n d m o d e l configuration) a n d inability to i m p l e m e n t A R I M A m e t h o d s at t h e resort, the D M m e t h o d w a s u s e d for the b o o k i n g c u r v e a d j u s t m e n t in the b o o k i n g c u r v e short-term e s t i m a t e .  73  REFERENCES A r m s t r o n g , J . & C o l l o p y , F. (1992). E r r o r m e a s u r e s for g e n e r a l i z i n g a b o u t f o r e c a s t i n g m e t h o d s : E m p i r i c a l c o m p a r i s o n s . International J o u r n a l of F o r e c a s t i n g , 8, 6 9 - 8 0 . B a k e r , T. & C o l l i e r , D. (1999). A c o m p a r a t i v e r e v e n u e a n a l y s i s of hotel yield m a n a g e m e n t h e u r i s t i c s . D e c i s i o n S c i e n c e s , 30(1), 2 3 9 - 2 6 3 . B o x , G . & J e n k i n s , G . (1976). T i m e S e r i e s A n a l y s i s : F o r e c a s t i n g a n d C o n t r o l , R e v i s e d E d i t i o n . San Francisco: Holden-Day. C h a t f i e l d , C . (1989). C h a p m a n and Hall.  T h e A n a l y s i s of T i m e S e r i e s : A n Introduction, 4 e d i t i o n . th  New York:  G h a l i a , M . & W a n g P . (2000). Intelligent s y s t e m to s u p p o r t j u d g m e n t a l b u s i n e s s f o r e c a s t i n g : T h e c a s e of e s t i m a t i n g hotel r o o m d e m a n d . I E E E T r a n s a c t i o n s of F u z z y S y s t e m s . 8(4), 3 8 0 - 3 9 7 . H a r r i s , P . & M a r u c c i , G . (1983). A short term f o r e c a s t i n g m o d e l . A G I F O R S S y m p o s i u m P r o c . 23. Memphis, U S A . L ' H e u r e u x , E . (1986). A n e w twist in f o r e c a s t i n g s h o r t - t e r m p a s s e n g e r p i c k u p . S y m p o s i u m Proc. 26. Bowness-On-Windermere, England.  AGIFORS  M i n , C . & Z e l l n e r , A . (1993). B a y e s i a n a n d n o n - B a y e s i a n m e t h o d s for c o m b i n i n g m o d e l s a n d f o r e c a s t s with a p p l i c a t i o n s to f o r e c a s t i n g international g r o w t h r a t e s . J o u r n a l of E c o n o m e t r i c s . 5 6 , 89-118. R a j o p a d h y e , M . , G h a l i a , M . , W a n g , P . , B a k e r , T., & E s t e r , C . (1999). F o r e c a s t i n g u n c e r t a i n hotel room d e m a n d . Proc. A m e r i c a n Control Conference, S a n Diego, C A , pp.1925-1999. S A S Institute Inc. (1999). Carolina, U S A .  S A S E T S U s e r ' s G u i d e , V e r s i o n 8, [Electronic v e r s i o n ] .  North  W e a t h e r f o r d , L , K i m e s , S . , & S c o t t , D. (2001). F o r e c a s t i n g for hotel r e v e n u e m a n a g e m e n t : Testing aggregation against disaggregation. Cornell Hotel and Restaurant Administration Quaterly, August, 53-64. W o o l r i d g e , J . (2000). College Publishing.  Introductory E c o n o m e t r i c s : A M o d e r n A p p r o a c h . U S A : S o u t h - W e s t e r n  74  

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