Chapter 16 Student Lecture Notes 16-1

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1 Chapter 16 Student Lecture Notes 16-1 Basc Busness Statstcs (9 th Edton) Chapter 16 Tme-Seres Forecastng and Index Numbers 2004 Prentce-Hall, Inc. Chap 16-1 Chapter Topcs The Importance of Forecastng Component Factors of the Tme-Seres Model Smoothng of Annual Tme Seres Movng averages Exponental smoothng Least-Squares Trend Fttng and Forecastng Lnear, quadratc and exponental models 2004 Prentce-Hall, Inc. Chap 16-2 Chapter Topcs (contnued) Holt-Wnters Method for Trend Fttng and Forecastng Autoregressve Models Choosng Approprate Forecastng Models Tme-Seres Forecastng of Monthly or Quarterly Data Ptfalls Concernng Tme-Seres Forecastng Index Numbers 2004 Prentce-Hall, Inc. Chap 16-3

2 Chapter 16 Student Lecture Notes 16-2 The Importance of Forecastng Government Needs to Forecast Unemployment, Interest Rates, Expected Revenues from Income Taxes to Formulate Polces Marketng Executves Need to Forecast Demand, Sales, Consumer Preferences n Strategc Plannng 2004 Prentce-Hall, Inc. Chap 16-4 The Importance of Forecastng (contnued) College Admnstrators Need to Forecast Enrollments to Plan for Facltes, for Student and Faculty Recrutment Retal Stores Need to Forecast Demand to Control Inventory Levels, Hre Employees and Provde Tranng 2004 Prentce-Hall, Inc. Chap 16-5 What s a Tme Seres? Numercal Data Obtaned at Regular Tme Intervals The Tme Intervals Can Be Annually, Quarterly, Monthly, Daly, Hourly, Etc. Example: Year: Sales: Prentce-Hall, Inc. Chap 16-6

3 Chapter 16 Student Lecture Notes 16-3 Tme-Seres Components Trend Cyclcal Tme-Seres Seasonal Irregular 2004 Prentce-Hall, Inc. Chap 16-7 Trend Component Overall Upward or Downward Movement Data Taken Over a Perod of Years Sales Upward trend Tme 2004 Prentce-Hall, Inc. Chap 16-8 Cyclcal Component Upward or Downward Swngs May Vary n Length Usually Lasts 2-10 Years Sales 1 Cycle 2004 Prentce-Hall, Inc. Chap 16-9

4 Chapter 16 Student Lecture Notes 16-4 Seasonal Component Upward or Downward Swngs Regular Patterns Observed Wthn 1 Year Sales Summer Wnter Sprng Fall Tme (Monthly or Quarterly) 2004 Prentce-Hall, Inc. Chap Irregular or Random Component Erratc, Nonsystematc, Random, Resdual Fluctuatons Due to Random Varatons of Nature Accdents Short Duraton and Non-Repeatng 2004 Prentce-Hall, Inc. Chap Example: Quarterly Retal Sales wth Seasonal Components Quarterly wth Seasonal Components Sales Tme 2004 Prentce-Hall, Inc. Chap 16-12

5 Chapter 16 Student Lecture Notes 16-5 Example: Quarterly Retal Sales wth Seasonal Components Removed Quarterly wthout Seasonal Components Sales Y(t) Tme 2004 Prentce-Hall, Inc. Chap Multplcatve Tme-Seres Model Used Prmarly for Forecastng Observed Value n Tme Seres s the Product of Components For Annual Data: Y = TC I For Quarterly or Monthly Data: T = Trend C = Cyclcal Y = T S C I I = Irregular S = Seasonal 2004 Prentce-Hall, Inc. Chap Used for Smoothng Movng Averages Seres of Arthmetc Means Over Tme Result Dependent Upon Choce of L (Length of Perod for Computng Means) To Smooth Out Cyclcal Component, L Should Be Multple of the Estmated Average Length of the Cycle For Annual Tme Seres, L Should Be Odd 2004 Prentce-Hall, Inc. Chap 16-15

6 Chapter 16 Student Lecture Notes 16-6 Movng Averages (contnued) Example: 3-Year Movng Average Y1+ Y2 + Y3 Frst average: MA(3) = 3 Y2 + Y3 + Y4 Second average: MA(3) = Prentce-Hall, Inc. Chap Movng Average Example John s a buldng contractor wth a record of a total of 24 sngle famly homes constructed over a 6-year perod. Provde John wth a 3-year movng average graph. Year Unts Movng Ave NA NA 2004 Prentce-Hall, Inc. Chap Movng Average Example Soluton Year Response Movng Ave NA NA Sales L = No MA for the frst and last (L-1)/2 years 2004 Prentce-Hall, Inc. Chap 16-18

7 Chapter 16 Student Lecture Notes 16-7 Movng Average Example Soluton n Excel Use Excel formula =average(cell range contanng the data for the years to average) Excel Spreadsheet for the Sngle Famly Home Sales Example Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap Example: 5-Perod Movng Averages of Quarterly Retal Sales Quarterly 5-Perod Movng Averages Sales MA(5) Y(t) Tme 2004 Prentce-Hall, Inc. Chap Exponental Smoothng Weghted Movng Average Weghts declne exponentally Most recent observaton weghted most Used for Smoothng and Short-Term Forecastng Weghts are: Subjectvely chosen Range from 0 to 1 Close to 0 for smoothng out unwanted cyclcal and rregular components Close to 1 for forecastng 2004 Prentce-Hall, Inc. Chap 16-21

8 Chapter 16 Student Lecture Notes 16-8 Exponental Weght: Example E = WY + (1 W ) E 1 Year Response Smoothng Value Forecast (W =.2, (1-W)=.8) NA (.2)(5) + (.8)(2) = (.2)(2) + (.8)(2.6) = (.2)(2) + (.8)(2.48) = (.2)(7) + (.8)(2.384) = (.2)(6) + (.8)(3.307) = Prentce-Hall, Inc. Chap Exponental Weght: Example Graph Sales 8 6 Data 4 2 Smoothed Year 2004 Prentce-Hall, Inc. Chap Exponental Smoothng n Excel Use Tools Data Analyss Exponental Smoothng The dampng factor s (1-W ) Excel Spreadsheet for the Sngle Famly Home Sales Example Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap 16-24

9 Chapter 16 Student Lecture Notes 16-9 Example: Exponental Smoothng of Real GNP The Excel Spreadsheet wth the Real GDP Data and the Exponentally Smoothed Seres Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap Lnear Trend Model Use the method of least squares to obtan the lnear trend forecastng equaton: Year Coded X Sales (Y) Y = b + b X ˆ Prentce-Hall, Inc. Chap Lnear Trend Model Lnear trend forecastng equaton: Yˆ = b + b X = X 0 1 (contnued) Excel Output Coeffcents Intercept X Varable Sales Projected to year X 2004 Prentce-Hall, Inc. Chap 16-27

10 Chapter 16 Student Lecture Notes The Quadratc Trend Model Use the method of least squares to obtan the quadratc trend forecastng equaton: Year Coded X Sales (Y) ˆ = Y b b X b X 2004 Prentce-Hall, Inc. Chap The Quadratc Trend Model (contnued) Yˆ b bx bx X.214X 2 2 = = + Excel Output Coeffcents Intercept X Varable X Varable Sales Projected to year X 2004 Prentce-Hall, Inc. Chap The Exponental Trend Model After takng the logarthms, use the method of least squares to get the forecastng equaton: ˆ X Y = b0b or log Yˆ log 1 = b0 + X1log b1 Year Coded X Sales (Y) Coeffcents Intercept X Varable Excel Output of Values n Logs antlog( ) = 2.17 antlog( ) = 1.2 Y ˆ = X (2.17)(1.2) 2004 Prentce-Hall, Inc. Chap 16-30

11 Chapter 16 Student Lecture Notes The Least-Squares Trend Models n PHStat Use PHStat Smple Lnear Regresson for Lnear Trend and Exponental Trend Models and PHStat Multple Regresson for Quadratc Trend Model Excel Spreadsheet for the Sngle Famly Home Sales Example Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap Model Selecton Usng Dfferences Use a Lnear Trend Model If the Frst Dfferences are More or Less Constant Y Y = Y Y = L = Y Y n n 1 Use a Quadratc Trend Model If the Second Dfferences are More or Less Constant ( Y Y ) ( Y Y) = L= ( Y Y ) ( Y Y ) n n 1 n 1 n Prentce-Hall, Inc. Chap Model Selecton Usng Dfferences (contnued) Use an Exponental Trend Model If the Percentage Dfferences are More or Less Constant Y2 Y 1 Y3 Y 2 Yn Y n 1 100% = 100% = L = 100% Y1 Y2 Yn Prentce-Hall, Inc. Chap 16-33

12 Chapter 16 Student Lecture Notes The Holt-Wnters Method Smlar to Exponental Smoothng Advantages Over Exponental Smoothng Can detect future trend and overall movement Can provde ntermedate and/or long-term forecastng Two Weghts 0<U<1 and 0<V<1 are to Be Chosen Smaller values of U gve more weght to the more recent levels and less weght to earler levels Smaller values of V gve more weght to the current trends and less weght to past trends 2004 Prentce-Hall, Inc. Chap The Holt-Wnters Method ( 1 1) ( ) ( )( ) Level: E = U E + T + 1 U Y Trend: T = VT + 1 V E E 1 1 E : level of smoothed seres n tme perod E : level of smoothed seres n tme perod 1 T : value of trend component n tme perod T : value of trend component n tme perod 1 Y : observed value of the tme seres n perod U : smoothng constant (where 0 < U < 1) V : smoothng constant (where 0 < V < 1) E = Y and T = Y Y Prentce-Hall, Inc. Chap The Holt-Wnters Method: Example E = U ( E 1 + T 1) + ( U ) Y T = VT + ( V )( E E ) Level: 1 Trend: Year Sales (Y ) Level (E ) U =.2 Trend (T ) V = NA NA = (5+3)+.8(2)=3.2.2(3)+.8(3.2-5)= ( )+.8(2)=2.07.2(-.84)+.8( )= ( )+.8(7)=5.8.2(-1.07)+.8( )= ( )+.8(6)=6.51.2(2.77)+.8( )= Prentce-Hall, Inc. Chap 16-36

13 Chapter 16 Student Lecture Notes The Holt-Wnters Method: Forecastng = + ( ) Yˆ n+ j En j Tn where Yˆ n+ j: forecasted value j years nto the future En : level of smoothed seres n perod n Tn : value of trend component n perod n j : number of years nto the future Year 00: j = 1 Yˆ 00 = E99 + 1( T99 ) = ( 1.12) = Year 05: j = 6 Yˆ = E + 6 T = = ( ) ( ) Prentce-Hall, Inc. Chap Holt-Wnters Method: Plot of Seres and Forecasts Excel Spreadsheet wth the Computaton 14 Mcrosoft Excel Worksheet Forecasts for 2000 to Level (E) Seres (Y) 2004 Prentce-Hall, Inc. Chap Autoregressve Modelng Used for Forecastng Takes Advantage of Autocorrelaton 1st order - correlaton between consecutve values 2nd order - correlaton between values 2 perods apart Autoregressve Model for p-th Order: Y = A + AY + A Y + L + A Y + δ p p Random Error 2004 Prentce-Hall, Inc. Chap 16-39

14 Chapter 16 Student Lecture Notes Autoregressve Model: Example The Offce Concept Corp. has acqured a number of offce unts (n thousands of square feet) over the last 8 years. Develop the 2nd order autoregressve model. Year Unts Prentce-Hall, Inc. Chap Autoregressve Model: Example Soluton Develop the 2nd order table Use Excel to estmate a regresson model Excel Output Coeffcents Intercept 3.5 X Varable X Varable Year Y Y -1 Y Yˆ = Y.9375Y Prentce-Hall, Inc. Chap Autoregressve Model Example: Forecastng Use the 2nd order model to forecast number of unts for 2001: Yˆ = Y Y 2 Yˆ = Y.9375Y = = Prentce-Hall, Inc. Chap 16-42

15 Chapter 16 Student Lecture Notes Autoregressve Model n PHStat PHStat Multple Regresson Excel Spreadsheet for the Offce Unts Example Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap Autoregressve Modelng Steps 1. Choose p : Note that df = n -2p Form a Seres of Lag Predctor Varables Y -1, Y -2,,Y -p 3. Use Excel to Run Regresson Model Usng All p Varables 4. Test Sgnfcance of A p If null hypothess rejected, ths model s selected If null hypothess not rejected, decrease p by 1 and repeat 2004 Prentce-Hall, Inc. Chap Selectng a Forecastng Model Perform a Resdual Analyss Look for pattern or drecton Measure Resdual Error Usng SSE (Sum of Square Error) Measure Resdual Error Usng MAD (Mean Absolute Devaton) Use Smplest Model Prncple of parsmony 2004 Prentce-Hall, Inc. Chap 16-45

16 Chapter 16 Student Lecture Notes e Resdual Analyss e 0 0 e Random errors Tme Tme Cyclcal effects not accounted for e 0 0 Tme Tme Trend not accounted for Seasonal effects not accounted for 2004 Prentce-Hall, Inc. Chap Measurng Errors Choose a Model that Gves the Smallest Measurng Errors Sum Square Error (SSE) n SSE = Y Yˆ ( ) 2 = 1 Senstve to outlers 2004 Prentce-Hall, Inc. Chap Measurng Errors (contnued) Mean Absolute Devaton (MAD) n Y ˆ Y = 1 MAD = n Not senstve to extreme observatons 2004 Prentce-Hall, Inc. Chap 16-48

17 Chapter 16 Student Lecture Notes Prncple of Parsmony Suppose 2 or More Models Provde Good Ft to Data Select the Smplest Model Smplest model types: Least-squares lnear Least-squares quadratc 1st order autoregressve More complex types: 2nd and 3rd order autoregressve Least-squares exponental Holt-Wnters Model 2004 Prentce-Hall, Inc. Chap Forecastng wth Seasonal Data Use Categorcal Predctor Varables wth Least- Squares Trend Fttng Forecastng Equaton (Exponental Model wth Quarterly Data): Yˆ = b b b b b X Q1 Q2 Q The b j provdes the multpler for the j -th quarter relatve to the 4th quarter Q j = 1 f j -th quarter and 0 f not X = the coded varable denotng the tme perod 2004 Prentce-Hall, Inc. Chap Forecastng wth Quarterly Data: Example Standards and Poor s Composte Stock Prce Index: I Excel Output Quarter Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons r 2 s.98 Appears to be an excellent ft Prentce-Hall, Inc. Chap 16-51

18 Chapter 16 Student Lecture Notes Forecastng wth Quarterly Data: Example (contnued) Excel Output Coeffcents Standard Error t Stat P-value Intercept E-21 Coded X E-10 Q Q Q Regresson equaton for the frst quarters: log ˆ 10 Y = log10 b0 + Xlog10 b1 + Q1 log10 b2 = X Q X ( ) ( ) Y ˆ = Q Prentce-Hall, Inc. Chap Forecastng wth Quarterly Data: Example (contnued) 1 st quarter of 1994: log10 Yˆ 1994,1 = ( 0) ( 1) = Yˆ 1994,1 = 10 = or Yˆ = = ,1 1 st quarter of 1998: log10 Yˆ 1998,1 = ( 16) ( 1) = Yˆ 1998,1 = 10 = or Yˆ = = ,1 ( ) ( ) ( ) ( ) 2004 Prentce-Hall, Inc. Chap Forecastng wth Quarterly Data n PHStat Use PHStat Multple Regresson Excel Spreadsheet for the Stock Prce Index Example Mcrosoft Excel Worksheet 2004 Prentce-Hall, Inc. Chap 16-54

19 Chapter 16 Student Lecture Notes Index Numbers Measure the Value of an Item (Group of Items) at a Partcular Pont n Tme as a Percentage of the Item s (Group of Items ) Value at Another Pont n Tme A prce ndex measures the percentage change n the prce of an tem (group of tems) n a gven perod of tme over the prce pad for the tem (group of tems) at a partcular pont of tme n the past Commonly Used n Busness and Economcs as Indcators of Changng Busness or Economc Actvty 2004 Prentce-Hall, Inc. Chap base Smple Prce Index P I = 100 Pbase where I = smple prce ndex for year P = prce for year P = prce for the base year Selecton of the Base Perod Should be a perod of economc stablty rather than one at or near the peak of an expandng economy or declnng economy Should be recent so that comparsons are not greatly affected by changng technology and consumer atttudes or habts 2004 Prentce-Hall, Inc. Chap Smple Prce Index: Example Gven the prces (n dollars per pound) for apples, construct the smple prce ndex usng 1980 as the base year. Base Year Year ( ) P base P ( ) Prce Smple Prce Index (0.692/0.692)100 = (0.684/0.692)100 = (0.719/0.692)100 = (0.835/0.692)100 = (0.896/0.692)100 = Prentce-Hall, Inc. Chap I

20 Chapter 16 Student Lecture Notes old new Inew base new old Shftng the Base I I = 100 where I = new prce ndex I I new base = old prce ndex = value of the old prce ndex for the new base year 2004 Prentce-Hall, Inc. Chap Shftng the Base: Example Change the base year of the smple prce ndex of apples from 1980 to 2000: Year Prce Smple Prce Index Smple Prce Index (base = 1980) (base = 2000) (100.00/129.48)100 = (98.84/129.48)100 = (103.90/129.48)100 = (120.66/129.48)100 = (129.48/129.48)100 = New Base Year I new base I old new 2004 Prentce-Hall, Inc. Chap I Aggregate Prce Index Reflects the Percentage Change n Prce of a Group of Commodtes (Market Basket) n a Gven Perod of Tme Over the Prce Pad for that Group of Commodtes at a Partcular Pont of Tme n the Past Affects the Cost of Lvng and/or the Qualty of Lfe for a Large Number of Consumers Two Basc Types Unweghted aggregate prce ndex Weghted aggregate prce ndex 2004 Prentce-Hall, Inc. Chap 16-60

21 Chapter 16 Student Lecture Notes Unweghted Aggregate Prce Index n () t () t Σ 1P = IU = 100 n ( 0) Σ= 1P where t = tme perod (0, 1, 2, L) n = total number of tems under consderaton n () t Σ= 1P = sum of the prces pad for each of the n commodtes at tme perod t n ( 0) Σ = 1P = sum of the prces pad for each of the n commodtes at tme perod 0 () t IU = value of the unweghted prce ndex at tme t 2004 Prentce-Hall, Inc. Chap Unweghted Aggregate Prce Index (contnued) Easy to Compute Two Dstnct Shortcomngs Each commodty n the group s treated as equally mportant so that the most expensve commodtes per unt can overly nfluence the ndex Not all commodtes are consumed at the same rate, but they are treated the same by the ndex 2004 Prentce-Hall, Inc. Chap Unweghted Aggregate Prce Index: Example Gven the prces (n dollars per pound) for apples, bananas and oranges, compute the unweghted aggregate prce ndex usng 1980 as the base year: n ( 0) Base Year Σ = 1P = = Year t Prce Unweghted Aggregate ( t) Apples Bananas Oranges Prce Index I U ( 4) ( 4) ( 4) P n = = 1P IU = = 3 ( 0) 100 = P = Prentce-Hall, Inc. Chap Σ = + + =

22 Chapter 16 Student Lecture Notes Weghted Aggregate Prce Indexes Allow for Dfferences n the Consumpton Levels Assocated wth the Dfferent Items Comprsng the Market Basket by Attachng a Weght to Each Item to Reflect the Consumpton Quantty of that Item Account for Dfferences n the Magntude of Prces Per Unt and Dfferences n the Consumpton Levels of the Items Two Types that are Commonly Used The Laspeyres prce ndex The Paasche prce ndex 2004 Prentce-Hall, Inc. Chap Laspeyres Prce Index n () t ( 0) () t Σ= 1P Q I L n ( 0) ( 0) Σ= 1P Q = 100 where t = tme perod (0, 1, 2, L) n = total number of tems under consderaton Q = quantty of tem at tme perod 0 I = value of the Laspeyres prce ndex at tme t ( 0) () t L Uses the Consumpton Quanttes Assocated wth the Base Year 2004 Prentce-Hall, Inc. Chap Laspeyres Prce Index: Example Gven the prces (n dollars per pound) and per capta consumpton (n pounds) for apples, bananas, and oranges, compute the Laspeyres prce ndex usng 1980 as the base year: 3 ( 0) ( 0) P Q = ( 0.692)( 19.2) + ( 0.342)( 20.2) + ( 0.365)( 14.3) = = 1 Year t Apple Bananas Oranges Laspeyres P Q P Q P Q Prce Index ( )( ) ( )( ) ( )( ) 3 ( 4) ( 0) P 1 Q = = = ( ) I L = 100 = Prentce-Hall, Inc. Chap 16-66

23 Chapter 16 Student Lecture Notes n () t () t () t Σ= 1P Q I P n ( 0) ( 0) Σ= 1P Q () t () t P Paasche Prce Index = 100 where t = tme perod (0, 1, 2, L) n = total number of tems under consderaton Q = quantty of tem at tme perod t I = value of the Paasche prce ndex at tme t Uses the Consumpton Quanttes Experenced n the Year of Interest Instead of Usng the Intal Quanttes 2004 Prentce-Hall, Inc. Chap Paasche Prce Index (contnued) Advantage A more accurate reflecton of total consumpton costs at the pont of nterest n tme Dsadvantages Accurate consumpton values for current purchases are often hard to obtan If a partcular product ncreases greatly n prce compared to other tems n the market basket, consumers wll avod the hgh-prced tem out of necessty, not because of changes n preferences 2004 Prentce-Hall, Inc. Chap Paasche Prce Index: Example Gven the prces (n dollars per pound) and per capta consumpton (n pounds) for apples, bananas, and oranges, compute the Paasche prce ndex usng 1980 as the base year: 3 ( 0) ( 0) P Q = ( 0.692)( 19.2) + ( 0.342)( 20.2) + ( 0.365)( 14.3) = = 1 Year t Apple Bananas Oranges Laspeyres P Q P Q P Q Prce Index ( 4) ( 4) P ( 0.896)( 18.8) ( 0.491)( 31.4) ( 0.843)( 8.6) Q = = + + = ( ) I P = 100 = Prentce-Hall, Inc. Chap 16-69

24 Chapter 16 Student Lecture Notes Ptfalls Concernng Tme-Seres Forecastng Takng for Granted the Mechansm that Governs the Tme Seres Behavor n the Past Wll Stll Hold n the Future Usng Mechancal Extrapolaton of the Trend to Forecast the Future Wthout Consderng Personal Judgments, Busness Experences, Changng Technologes, Habts, Etc Prentce-Hall, Inc. Chap Chapter Summary Dscussed the Importance of Forecastng Addressed Component Factors of the Tme- Seres Model Performed Smoothng of Data Seres Movng averages Exponental smoothng Descrbed Least-Squares Trend Fttng and Forecastng Lnear, quadratc and exponental models 2004 Prentce-Hall, Inc. Chap Chapter Summary (contnued) Dscussed Holt-Wnters Method of Trend Fttng and Forecastng Addressed Autoregressve Models Descrbed Procedure for Choosng Approprate Models Addressed Tme-Seres Forecastng of Monthly or Quarterly Data (Use of Dummy-Varables) Dscussed Ptfalls Concernng Tme-Seres Forecastng Descrbed Index Numbers 2004 Prentce-Hall, Inc. Chap 16-72