Chapter 8 Student Lecture Notes 8-1

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1 Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop and explain basic forecasing models Apply rend-based forecasing models, including linear rend, nonlinear rend, and seasonally adjused rend Use smoohing-based forecasing models, including single and double exponenial smoohing Examples of Forecasing Caegories of Forecasing Governmens forecas unemploymen, ineres raes, and expeced ax revenues for policy purposes Markeing execuives forecas demand, sales, and consumer preferences for sraegic planning College adminisraors forecas enrollmens o plan for faciliies and for faculy recruimen Reail sores forecas demand o conrol invenory levels, hire employees and provide raining Qualiaive forecasing echniques Based on saisical mehods for analyzing hisorical daa Qualiaive forecasing echniques Based on exper opinion and judgmen NOT a gu feel or an unsubsaniaed opinions Developing a Forecasing Model Forecasing Horizon Seps in forecas modeling (see Chaper ): model specificaion model fiing model diagnosis Goal: use he simples available model ha mees forecasing needs o provide good forecass for fuure performance Forecasing horizon is he lead ime necessary (or available) o develop he forecasing model Inermediae erm less han one monh Shor erm one o hree monhs Medium erm hree monhs o wo years Long erm wo years or more Forecasing period: he uni of ime for which forecass are o be made Forecasing inerval: he frequency wih which new forecass are prepared

2 Chaper Suden Lecure Noes - -Series Analysis Series Plo The process for using pas measuremens o generae forecass for he fuure Numerical daa obained a regular ime inervals The ime inervals can be annually, quarerly, daily, hourly, ec. Example: : :..... A ime-series plo is a wo-dimensional plo of ime series daa he verical axis measures he variable of ineres he horizonal axis corresponds o he ime periods Inflaion Rae (%) U.S. Inflaion Rae -Series Componens Trend Componen Trend Componen Seasonal Componen -Series Cyclical Componen Random Componen Long-run increase or decrease over ime (overall upward or downward movemen) Daa aken over a long period of ime Trend Componen (coninued) Trend can be upward or downward (recall Chapers -) Trend can be linear or non-linear (recall Chaper ) Seasonal Componen Shor-erm regular wave-like paerns (repeaing) Observed wihin year Ofen monhly or quarerly Winer Summer Downward linear rend Upward nonlinear rend Spring (Quarerly) Fall

3 Chaper Suden Lecure Noes - The paern iself repeas hroughou he ime series The shores period of repeiion is he recurrence period Examples: The recurrence period will be year a MOST Increase in visis o he docor in he Fall and Winer, decrease in he Spring and Summer Seasonal flucuaion in reails sales around various holidays (Chrismas, Moher s Day, ec.) Seasonal Componen (coninued) Cyclical Componen Long-erm wave-like paerns Regularly occur bu may vary in lengh Ofen measured peak o peak or rough o rough Cycle Recurrence period is longer han year Susained periods of highs and lows Cycles vary in lengh and inensiy Examples: Cyclical Componen Unemploymen raes Sock marke indexes New home sales (coninued) Random Componen Unpredicable, random, residual flucuaions Will be presen in virually all siuaions Due o random variaions of Naure Devasaing ornado his a manufacuring faciliy Accidens or unusual evens Unexpeced closing of a large employer in a communiy Noise in he ime series No discernable paern Trend-Based Forecasing Esimae a rend line using regression analysis () (y) Use ime () as he independen variable: ŷ b b () Trend-Based Forecasing (y) sales (coninued) The linear rend model is: ŷ.. rend

4 Chaper Suden Lecure Noes - () Trend-Based Forecasing (y)?? Forecas for ime period : sales. (coninued) ŷ.. () Comparing Forecas Values o Acual Daa The forecas error or residual is he difference beween he acual value in ime and he forecas value in ime : Error in ime : e y F Two Common Measures of Fi MSE vs. MAD Measures of fi are used o gauge how well he forecass mach he acual values (model diagnosis) MSE (mean squared error) squared difference beween y and F Mean Square Error MSE (y F ) n Mean Absolue Deviaion MAD y F n MAD (mean absolue deviaion) absolue value of difference beween y and F Less sensiive o exreme values RMSE (roo mean square error) Square roo of MSE where: y = Acual value a ime F = Prediced value a ime n = Number of ime periods Auocorrelaion True Forecass Auocorrelaion is correlaion of he error erms (residuals) over ime Here, residuals show a cyclic paern, no random Also called serial correlaion Residuals () Residual Plo () Violaes he regression assumpion ha residuals are random and independen True forecass are gauged by how well i forecass fuure values no how well i fis hisorical daa To deermine if he rend model produced a rue forecas, you have o wai unil he fuure ime acually arrives Can use spli samples Forecas bias Posiive underforecas Negaive overforecas

5 Chaper Suden Lecure Noes - Nonlinear Trend Forecasing Finding Seasonal Indexes A nonlinear regression model can be used when he ime series exhibis a nonlinear rend One form of a nonlinear model: y β β β ε Compare R and s ε o ha of linear model o see if his is an improvemen Can ry oher funcional forms o ge bes fi Raio-o-moving average mehod: Begin by removing he seasonal and irregular componens (S and I ), leaving he rend and cyclical componens (T and C ) To do his, we need moving averages : averages of consecuive ime series values Muliplicaive -Series Model Used primarily for forecasing Allows consideraion of seasonal variaion Observed value in ime series is he produc of componens where y T S T = Trend value a ime S = Seasonal value a ime C = Cyclical value a ime I C = Irregular (random) value a ime I s Used for smoohing Series of arihmeic means over ime Resul dependen upon lengh of period chosen for compuing means To smooh ou seasonal variaion, he number of periods should be equal o he number of seasons For quarerly daa, number of periods = For monhly daa, number of periods = s Seasonal Daa Example: Four-quarer moving average Firs average: Second average: ec Q Q Q Q average Q Q Q Q average (coninued) Quarer ec ec Quarerly Quarer

6 Chaper Suden Lecure Noes - Calculaing s Cenered s Quarer ec -Quarer Each moving average is for a consecuive block of quarers periods of. or. don mach he original quarers, so we average wo consecuive moving averages o ge cenered moving averages -Quarer ec Cenered Cenered Calculaing he Raio-o- Calculaing Seasonal Indexes Now esimae he S x I value Divide he acual sales value by he cenered moving average for ha quarer Raio-o- formula: S I y T C Quarer Cenered ec Raio-o ec Example:.. Fall Fall Fall Quarer Calculaing Seasonal Indexes Cenered ec Raio-o ec (coninued) all of he Fall values o ge Fall s seasonal index Do he same for he oher hree seasons o ge he oher seasonal indexes Inerpreing Seasonal Indexes Suppose we ge hese seasonal indexes: Season Seasonal Index Spring. Summer. Fall. Winer. Inerpreaion: Spring sales average.% of he annual average sales Summer sales are.% higher han he annual average sales ec =. -- four seasons, so mus sum o

7 Chaper Suden Lecure Noes - Deseasonalizing Deseasonalizing The daa is deseasonalized by dividing he observed value by is seasonal index T C I y S This smoohs he daa by removing seasonal variaion Quarer Seasonal Index Deseasonalized Example:.. ec (coninued) Unseasonalized vs. Seasonalized Seasonal Adjusmen Summarized : Unseasonalized vs. Seasonalized Quarer Deseasonalized. Compue each moving average. Compue he cenered moving averages. Isolae he seasonal componen by deermining he raio-o-moving average values. Deermine seasonal indexes and normalize if necessary. Deseasonalize he ime series. Develop rend line using deseasonalized daa. Develop unadjused forecass using rend projecion. Seasonally adjus he forecass Using Dummy Variables for Seasonaliy Forecasing Using Smoohing Mehods Can incorporae he seasonal componen using dummy variables in a regression model Example for seasons: Le x = if winer, if no winer Le x = if spring, if no spring Le x = if summer, if no summer (Fall is he defaul season) Model: F β β βx βx βx ε Single Exponenial Smoohing Exponenial Smoohing Mehods Double Exponenial Smoohing Used when here is no pronounced rend in he daa The goal is o smooh ou he irregular componen

8 Chaper Suden Lecure Noes - Exponenial Smoohing Single Exponenial Smoohing Assumes mos recen daa is more indicaive of possible fuure values Curren observaions can be weighed more heavily han older observaions The forecas developed reflec he curren daa more Good for shor erm forecasing and for ime series ha are no seasonal A weighed moving average Weighs decline exponenially Mos recen observaion weighed mos Used for smoohing and shor erm forecasing Easy o updae Single Exponenial Smoohing The weighing facor is Subjecively chosen Range from o Smaller gives more smoohing, larger gives less smoohing (coninued) The weigh is: Close o for smoohing ou unwaned cyclical and irregular componens Close o for forecasing Exponenial Smoohing Model or: Single exponenial smoohing model where: F F F (y y F ) ( ) F F + = forecas value for period + y = acual value for period F = forecas value for period = alpha (smoohing consan) Exponenial Smoohing Example vs. Smoohed Quarer () ec Suppose we use weigh =. (y ) ec Forecas from prior period NA ec Forecas for nex period (F + ) (.)()+(.)()=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. (.)()+(.)(.)=. ec F F = y since no prior informaion exiss y ( )F Seasonal flucuaions have been smoohed NOTE: he smoohed value in his case is generally a lile low, since he rend is upward sloping and he weighing facor is only. Quarer Smoohed

9 Chaper Suden Lecure Noes - Exponenial Smoohing in Excel Mean Absolue Percen Error Use: Daa / daa analysis / exponenial smoohing The damping facor is ( - ) where: y = Value of ime series in ime F = Forecas values for ime period n = Number of periods of available daa Chaper Summary Discussed he imporance of forecasing Addressed componen facors presen in he ime-series model Described leas square rend fiing and forecasing linear and nonlinear models Performed smoohing of daa series moving averages single and double exponenial smoohing

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