Business Statistics A DecisionMaking Approach 6 th Edition
Business Statistics: A Decision-Making Approach 6 th Edition Chapter 15 Analyzing and Forecasting Time-Series Data Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Chap 15 - 1
Chapter Goals After completing this chapter, you should be able to: n Develop and implement basic forecasting models n Identify the components present in a time series n Compute and interpret basic index numbers n n Use smoothing-based forecasting models, including single and double exponential smoothing Apply trend-based forecasting models, including linear trend, nonlinear trend, and seasonally adjusted trend Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 2
The Importance of Forecasting n n Governments forecast unemployment, interest rates, and expected revenues from income taxes for policy purposes Marketing executives forecast demand, sales, and consumer preferences for strategic planning College administrators forecast enrollments to plan for facilities and for faculty recruitment Retail stores forecast demand to control inventory levels, hire employees and provide training Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 3
Time-Series Data n n n Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc. Example: Year: 1999 2000 2001 2002 2003 Sales: 75. 3 74. 2 78. 5 79. 7 80. 2 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 4
Time Series Plot A time-series plot is a two-dimensional plot of time series data n n the vertical axis measures the variable of interest the horizontal axis corresponds to the time periods Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 5
Time-Series Components Time-Series Trend Component Seasonal Component Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Cyclical Component Random Component 6
Trend Component n Long-run increase or decrease over time (overall upward or downward movement) n Data taken over a long period of time Sales Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. d ren t d r a pw U Time 7
Trend Component (continued) n n Trend can be upward or downward Trend can be linear or non-linear Sales Time Downward linear trend Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Time Upward nonlinear trend 8
Seasonal Component n n n Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Sales Summer Winter Spring Fall Time (Quarterly) Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 9
Cyclical Component n n n Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to trough 1 Cycle Sales Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Year 10
Random Component n n Unpredictable, random, “residual” fluctuations Due to random variations of n n n Nature Accidents or unusual events “Noise” in the time series Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 11
Index Numbers n n Index numbers allow relative comparisons over time Index numbers are reported relative to a Base Period Index Base period index = 100 by definition Used for an individual item or measurement Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 12
Index Numbers (continued) n Simple Index number formula: where It = index number at time period t yt = value of the time series at time t y 0 = value of the time series in the base period Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 13
Index Numbers: Example n Company orders from 1995 to 2003: Index Year Number of Orders (base year = 2000) 1995 272 85. 0 1996 288 90. 0 1997 295 92. 2 1998 311 97. 2 1999 322 100. 6 2000 320 100. 0 2001 348 108. 8 2002 366 114. 4 2003 384 120. 0 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Base Year: 14
Index Numbers: Interpretation n Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Orders in 1996 were 90% of base year orders Orders in 2000 were 100% of base year orders (by definition, since 2000 is the base year) Orders in 2003 were 120% of base year orders 15
Aggregate Price Indexes n An aggregate index is used to measure the rate of change from a base period for a group of items Aggregate Price Indexes Unweighted aggregate price index Weighted aggregate price indexes Paasche Index Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Laspeyres Index 16
Unweighted n Aggregate Price Index Unweighted aggregate price index formula: where It = unweighted aggregate price index at time t pt = sum of the prices for the group of items at time t p 0 = sum of the prices for the group of items in the base period Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 17
Unweighted Aggregate Price Index Example Automobile Expenses: Monthly Amounts ($): Index Year Lease payment Fuel Repair Total (2001=100) 2001 260 45 40 345 100. 0 2002 280 60 40 380 110. 1 2003 305 55 45 405 117. 4 2004 310 50 50 410 118. 8 n Combined expenses in 2004 were 18. 8% higher in 2004 than in 2001 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 18
Weighted Aggregate Price Indexes n Paasche index qt = weighting percentage at time t n Laspeyres index q 0 = weighting percentage at base period pt = price in time period t p 0 = price in the base period Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 19
Commonly Used Index Numbers n Consumer Price Index n Producer Price Index n Stock Market Indexes n Dow Jones Industrial Average n S&P 500 Index n NASDAQ Index Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 20
Deflating a Time Series n n n Observed values can be adjusted to base year equivalent Allows uniform comparison over time Deflation formula: where = adjusted time series value at time t yt = value of the time series at time t It = index (such as CPI) at time t Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 21
Deflating a Time Series: Example n Which movie made more money (in real terms)? Movie Title Total Gross $ 1939 Gone With the Wind 199 1977 Star Wars 461 1997 Titanic 601 Year (Total Gross $ = Total domestic gross ticket receipts in $millions) Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 22
Deflating a Time Series: Example (continued) Movie Title Total Gross (base year = 1984) Gross adjusted to 1984 dollars 1939 Gone With the Wind 199 13. 9 1431. 7 1977 Star Wars 461 60. 6 760. 7 1997 Titanic 601 160. 5 374. 5 Year CPI n Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. GWTW made about twice as much as Star Wars, and about 4 times as much as Titanic when measured in equivalent dollars 23
Trend-Based Forecasting n Estimate a trend line using regression analysis Year 1999 2000 2001 2002 2003 2004 Time Period (t) 1 2 3 4 5 6 n Sales (y) Use time (t) as the independent variable: 20 40 30 50 70 65 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 24
Trend-Based Forecasting (continued) n Time Year Period (t) 1999 2000 2001 2002 2003 2004 1 2 3 4 5 6 The linear trend model is: Sales (y) 20 40 30 50 70 65 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 25
Trend-Based Forecasting (continued) n Year Time Period (t) Sales (y) 1999 2000 2001 2002 2003 2004 2005 1 2 3 4 5 6 7 20 40 30 50 70 65 ? ? Forecast for time period 7: Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 26
Comparing Forecast Values to Actual Data n n The forecast error or residual is the difference between the actual value in time t and the forecast value in time t: Error in time t: Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 27
Two common Measures of Fit n Measures of fit are used to gauge how well the forecasts match the actual values MSE (mean squared error) n Average squared difference between yt and Ft MAD (mean absolute deviation) n n Average absolute value of difference between yt and Ft Less sensitive to extreme values Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 28
MSE vs. MAD Mean Square Error Mean Absolute Deviation where: yt = Actual value at time t Ft = Predicted value at time t n = Number of time periods Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 29
Autocorrelation (continued) n n Autocorrelation is correlation of the error terms (residuals) over time Here, residuals show a cyclic pattern, not random n Violates the regression assumption that residuals are random and independent Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 30
Testing for Autocorrelation n The Durbin-Watson Statistic is used to test for autocorrelation H 0: ρ = 0 HA : ρ ≠ 0 (residuals are not correlated) (autocorrelation is present) Durbin-Watson test statistic: Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 31
Testing for Positive Autocorrelation H 0: ρ = 0 (positive autocorrelation does not exist) HA : ρ > 0 (positive autocorrelation is present) § Calculate the Durbin-Watson test statistic = d (The Durbin-Watson Statistic can be found using PHStat or Minitab) § Find the values d. L and d. U from the Durbin-Watson table (for sample size n and number of independent variables p) Decision rule: reject H 0 if d < d. L Reject H 0 0 Inconclusive d. L Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Do not reject H 0 d. U 2 32
Testing for Positive Autocorrelation (continued) n Example with n = 25: Excel/PHStat output: Durbin-Watson Calculations Sum of Squared Difference of Residuals 3296. 18 Sum of Squared Residuals 3279. 98 Durbin-Watson Statistic 1. 00494 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 33
Testing for Positive Autocorrelation (continued) n Here, n = 25 and there is one independent variable n Using the Durbin-Watson table, d. L = 1. 29 and d. U = 1. 45 n n d = 1. 00494 < d. L = 1. 29, so reject H 0 and conclude that significant positive autocorrelation exists Therefore the linear model is not the appropriate model to forecast sales Decision: reject H 0 since d = 1. 00494 < d. L Reject H 0 0 Inconclusive d. L=1. 29 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Do not reject H 0 d. U=1. 45 2 34
Nonlinear Trend Forecasting n n A nonlinear regression model can be used when the time series exhibits a nonlinear trend One form of a nonlinear model: Compare R 2 and sε to that of linear model to see if this is an improvement Can try other functional forms to get best fit Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 35
Multiplicative Time-Series Model n n n Used primarily forecasting Allows consideration of seasonal variation Observed value in time series is the product of components where Tt = Trend value at time t St = Seasonal value at time t Ct = Cyclical value at time t It = Irregular (random) value at time t Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 36
Finding Seasonal Indexes Ratio-to-moving average method: n n Begin by removing the seasonal and irregular components (St and It), leaving the trend and cyclical components (Tt and Ct) To do this, we need moving averages Moving Average: averages of consecutive time series values Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 37
Moving Averages n n Used for smoothing Series of arithmetic means over time Result dependent upon choice of L (length of period for computing means) To smooth out seasonal variation, L should be equal to the number of seasons n n For quarterly data, L = 4 For monthly data, L = 12 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 38
Moving Averages (continued) n Example: Four-quarter moving average n First average: n Second average: n etc… Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 39
Seasonal Data Quarter Sales 1 2 3 4 5 6 7 8 9 10 11 etc… 23 40 25 27 32 48 33 37 37 50 40 etc… Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. … … 40
Calculating Moving Averages Average Period 4 -Quarter Moving Average 23 2. 5 28. 75 2 40 3. 5 31. 00 3 25 4. 5 33. 00 4 27 5. 5 35. 00 5 32 6. 5 37. 50 6 48 7. 5 38. 75 7 33 8. 5 39. 25 8 37 9. 5 41. 00 9 37 10 50 11 40 Quarter Sales 1 etc… n Each moving average is for a consecutive block of 4 quarters Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 41
Centered Moving Averages n Average periods of 2. 5 or 3. 5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages Average Period 4 -Quarter Moving Average Centered Period Centered Moving Average 2. 5 28. 75 3 29. 88 3. 5 31. 00 4 32. 00 4. 5 33. 00 5 34. 00 5. 5 6 36. 25 6. 5 35. 00 etc… 37. 50 7 38. 13 7. 5 38. 75 8 39. 00 8. 5 39. 25 9 40. 13 9. 5 41. 00 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 42
Calculating the Ratio-to-Moving Average n Now estimate the St x It value Divide the actual sales value by the centered moving average for that quarter n Ratio-to-Moving Average formula: n Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 43
Calculating Seasonal Indexes Quarter Sales Centered Moving Average 1 23 2 40 3 25 29. 88 4 27 32. 00 5 32 34. 00 6 48 36. 25 7 33 38. 13 8 37 39. 00 9 37 40. 13 10 50 etc… 11 40 … … A Decision-Making … Approach, 6 e ©… Business Statistics: 2005 Prentice. Hall, Inc. Ratio-to. Moving Average 0. 837 0. 844 0. 941 1. 324 0. 865 0. 949 0. 922 etc… … … 44
Calculating Seasonal Indexes (continued) Quarter Sales Centered Moving Average 1 23 2 40 3 25 29. 88 Fall 4 27 32. 00 5 32 34. 00 6 48 36. 25 Fall 7 33 38. 13 8 37 39. 00 9 37 40. 13 10 50 etc… Fall 11 40 … … A Decision-Making … Approach, 6 e ©… Business Statistics: 2005 Prentice. Hall, Inc. Ratio-to. Moving Average 0. 837 0. 844 0. 941 1. 324 0. 865 0. 949 0. 922 etc… … … Average all of the Fall values to get Fall’s seasonal index Do the same for the other three seasons to get the other seasonal indexes 45
Interpreting Seasonal Indexes n Suppose we get these seasonal indexes: Seasonal Index n Interpretation: Spring 0. 825 Spring sales average 82. 5% of the annual average sales Summer 1. 310 Summer sales are 31. 0% higher than the annual average sales Fall 0. 920 etc… Winter 0. 945 = 4. 000 -- four seasons, so must sum to 4 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 46
Deseasonalizing n n The data is deseasonalized by dividing the observed value by its seasonal index This smooths the data by removing seasonal variation Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 47
Deseasonalizing (continued) Quarter Sales 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 37 50 40 Seasonal Index Deseasonalized Sales 0. 825 1. 310 0. 920 0. 945 0. 825 1. 310 0. 920 … 27. 88 30. 53 27. 17 28. 57 38. 79 36. 64 35. 87 39. 15 44. 85 38. 17 43. 48 … Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. etc… 48
Unseasonalized vs. Seasonalized Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 49
Forecasting Using Smoothing Methods Exponential Smoothing Methods Single Exponential Smoothing Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. Double Exponential Smoothing 50
Single Exponential Smoothing n n A weighted moving average n Weights decline exponentially n Most recent observation weighted most Used for smoothing and short term forecasting Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 51
Single Exponential Smoothing (continued) n The weighting factor is n n Subjectively chosen Range from 0 to 1 Smaller gives more smoothing, larger gives less smoothing The weight is: n n Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 forecasting Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 52
Exponential Smoothing Model n Single exponential smoothing model or: where: Ft+1= forecast value for period t + 1 yt = actual value for period t Ft = forecast value for period t = alpha (smoothing constant) Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 53
Exponential Smoothing Example n Suppose we use weight =. 2 Quarter (t) Sales (yt) Forecast from prior period 1 23 NA 2 40 23 3 25 26. 4 4 27 26. 12 5 32 26. 296 6 48 27. 437 7 33 31. 549 8 37 31. 840 9 37 32. 872 10 50 33. 697 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prenticeetc… Hall, Inc. Forecast for next period (Ft+1) 23 (. 2)(40)+(. 8)(23)=26. 4 (. 2)(25)+(. 8)(26. 4)=26. 12 (. 2)(27)+(. 8)(26. 12)=26. 296 (. 2)(32)+(. 8)(26. 296)=27. 437 (. 2)(48)+(. 8)(27. 437)=31. 549 (. 2)(48)+(. 8)(31. 549)=31. 840 (. 2)(33)+(. 8)(31. 840)=32. 872 (. 2)(37)+(. 8)(32. 872)=33. 697 (. 2)(50)+(. 8)(33. 697)=36. 958 etc… F 1 = y 1 since no prior information exists 54
Sales vs. Smoothed Sales n n Seasonal fluctuations have been smoothed NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only. 2 Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 55
Double Exponential Smoothing n n n Double exponential smoothing is sometimes called exponential smoothing with trend If trend exists, single exponential smoothing may need adjustment Add a second smoothing constant to account for trend Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 56
Double Exponential Smoothing Model where: yt = actual value in time t = constant-process smoothing constant = trend-smoothing constant Ct = smoothed constant-process value for period t Tt = smoothed trend value for period t Ft+1= forecast value for period t + 1 t = current time period Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 57
Double Exponential Smoothing n n n Double exponential smoothing is generally done by computer Use larger smoothing constants and β when less smoothing is desired Use smaller smoothing constants and β when more smoothing is desired Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 58
Exponential Smoothing in Excel n Use tools / data analysis / exponential smoothing n The “damping factor” is (1 - ) Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 59
Chapter Summary n n Discussed the importance of forecasting Addressed component factors present in the time-series model Computed and interpreted index numbers Described least square trend fitting and forecasting n n linear and nonlinear models Performed smoothing of data series n n moving averages single and double exponential smoothing Business Statistics: A Decision-Making Approach, 6 e © 2005 Prentice. Hall, Inc. 60
- Slides: 60