Basic Business Statistics (9th Edition) Chapter 16 Time-Series Forecasting and Index Numbers © 2004 Prentice-Hall, Inc.

Chapter Topics The Importance of Forecasting Component Factors of the Time-Series Model Smoothing of Annual Time Series Moving averages Exponential smoothing Least-Squares Trend Fitting and Forecasting Linear, quadratic and exponential models © 2004 Prentice-Hall, Inc.

Chapter Topics Holt-Winters Method for Trend Fitting and Forecasting (continued) Holt-Winters Method for Trend Fitting and Forecasting Autoregressive Models Choosing Appropriate Forecasting Models Time-Series Forecasting of Monthly or Quarterly Data Pitfalls Concerning Time-Series Forecasting Index Numbers © 2004 Prentice-Hall, Inc.

The Importance of Forecasting Government Needs to Forecast Unemployment, Interest Rates, Expected Revenues from Income Taxes to Formulate Policies Marketing Executives Need to Forecast Demand, Sales, Consumer Preferences in Strategic Planning © 2004 Prentice-Hall, Inc.

The Importance of Forecasting (continued) College Administrators Need to Forecast Enrollments to Plan for Facilities, for Student and Faculty Recruitment Retail Stores Need to Forecast Demand to Control Inventory Levels, Hire Employees and Provide Training © 2004 Prentice-Hall, Inc.

What is a Time Series? Numerical Data Obtained at Regular Time Intervals The Time Intervals Can Be Annually, Quarterly, Monthly, Daily, Hourly, Etc. Example: Year: 1994 1995 1996 1997 1998 Sales: 75.3 74.2 78.5 79.7 80.2 © 2004 Prentice-Hall, Inc.

Time-Series Components Trend Cyclical Time-Series Seasonal Irregular © 2004 Prentice-Hall, Inc.

Trend Component Overall Upward or Downward Movement Data Taken Over a Period of Years Upward trend Sales Time © 2004 Prentice-Hall, Inc.

Cyclical Component Upward or Downward Swings May Vary in Length Usually Lasts 2 - 10 Years 1 Cycle Sales © 2004 Prentice-Hall, Inc.

Seasonal Component Upward or Downward Swings Regular Patterns Observed Within 1 Year Sales Summer Winter Spring Fall Time (Monthly or Quarterly) © 2004 Prentice-Hall, Inc.

Irregular or Random Component Erratic, Nonsystematic, Random, “Residual” Fluctuations Due to Random Variations of Nature Accidents Short Duration and Non-Repeating © 2004 Prentice-Hall, Inc.

Example: Quarterly Retail Sales with Seasonal Components © 2004 Prentice-Hall, Inc.

Example: Quarterly Retail Sales with Seasonal Components Removed © 2004 Prentice-Hall, Inc.

Multiplicative Time-Series Model Used Primarily for Forecasting Observed Value in Time Series is the Product of Components For Annual Data: For Quarterly or Monthly Data: Ti = Trend Ci = Cyclical Ii = Irregular Si = Seasonal © 2004 Prentice-Hall, Inc.

Moving Averages Used for Smoothing Series of Arithmetic Means Over Time Result Dependent Upon Choice of L (Length of Period for Computing Means) To Smooth Out Cyclical Component, L Should Be Multiple of the Estimated Average Length of the Cycle For Annual Time Series, L Should Be Odd © 2004 Prentice-Hall, Inc.

Moving Averages Example: 3-Year Moving Average First average: (continued) Example: 3-Year Moving Average First average: Second average: © 2004 Prentice-Hall, Inc.

Moving Average Example John is a building contractor with a record of a total of 24 single family homes constructed over a 6-year period. Provide John with a 3-year moving average graph. Year Units Moving Ave 1994 2 NA 1995 5 3 1996 2 3 1997 2 3.67 1998 7 5 1999 6 NA © 2004 Prentice-Hall, Inc.

Moving Average Example Solution Year Response Moving Ave 1994 2 NA 1995 5 3 1996 2 3 1997 2 3.67 1998 7 5 1999 6 NA Sales L = 3 8 6 4 2 94 95 96 97 98 99 No MA for the first and last (L-1)/2 years © 2004 Prentice-Hall, Inc.

Moving Average Example Solution in Excel Use Excel formula “=average(cell range containing the data for the years to average)” Excel Spreadsheet for the Single Family Home Sales Example © 2004 Prentice-Hall, Inc.

Example: 5-Period Moving Averages of Quarterly Retail Sales © 2004 Prentice-Hall, Inc.

Exponential Smoothing Weighted Moving Average Weights decline exponentially Most recent observation weighted most Used for Smoothing and Short-Term Forecasting Weights are: Subjectively chosen Range from 0 to 1 Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting © 2004 Prentice-Hall, Inc.

Exponential Weight: Example Year Response Smoothing Value Forecast (W = .2, (1-W)=.8) 1994 2 2 NA 1995 5 (.2)(5) + (.8)(2) = 2.6 2 1996 2 (.2)(2) + (.8)(2.6) = 2.48 2.6 1997 2 (.2)(2) + (.8)(2.48) = 2.384 2.48 1998 7 (.2)(7) + (.8)(2.384) = 3.307 2.384 1999 6 (.2)(6) + (.8)(3.307) = 3.846 3.307 © 2004 Prentice-Hall, Inc.

Exponential Weight: Example Graph Sales 8 6 4 2 Data Smoothed 94 95 96 97 98 99 Year © 2004 Prentice-Hall, Inc.

Exponential Smoothing in Excel Use Tools | Data Analysis | Exponential Smoothing The damping factor is (1-W ) Excel Spreadsheet for the Single Family Home Sales Example © 2004 Prentice-Hall, Inc.

Example: Exponential Smoothing of Real GNP The Excel Spreadsheet with the Real GDP Data and the Exponentially Smoothed Series © 2004 Prentice-Hall, Inc.

Linear Trend Model Use the method of least squares to obtain the linear trend forecasting equation: Year Coded X Sales (Y) 94 0 2 95 1 5 96 2 2 97 3 2 98 4 7 99 5 6 © 2004 Prentice-Hall, Inc.

Linear trend forecasting equation: Linear Trend Model (continued) Linear trend forecasting equation: Excel Output Projected to year 2000 © 2004 Prentice-Hall, Inc.

The Quadratic Trend Model Use the method of least squares to obtain the quadratic trend forecasting equation: Year Coded X Sales (Y) 94 0 2 95 1 5 96 2 2 97 3 2 98 4 7 99 5 6 © 2004 Prentice-Hall, Inc.

The Quadratic Trend Model (continued) Excel Output Projected to year 2000 © 2004 Prentice-Hall, Inc.

The Exponential Trend Model After taking the logarithms, use the method of least squares to get the forecasting equation: or Year Coded X Sales (Y) 94 0 2 95 1 5 96 2 2 97 3 2 98 4 7 99 5 6 Excel Output of Values in Logs © 2004 Prentice-Hall, Inc.

The Least-Squares Trend Models in PHStat Use PHStat | Simple Linear Regression for Linear Trend and Exponential Trend Models and PHStat | Multiple Regression for Quadratic Trend Model Excel Spreadsheet for the Single Family Home Sales Example © 2004 Prentice-Hall, Inc.

Model Selection Using Differences Use a Linear Trend Model If the First Differences are More or Less Constant Use a Quadratic Trend Model If the Second Differences are More or Less Constant © 2004 Prentice-Hall, Inc.

Model Selection Using Differences (continued) Use an Exponential Trend Model If the Percentage Differences are More or Less Constant © 2004 Prentice-Hall, Inc.

The Holt-Winters Method Similar to Exponential Smoothing Advantages Over Exponential Smoothing Can detect future trend and overall movement Can provide intermediate and/or long-term forecasting Two Weights 0<U<1 and 0<V<1 are to Be Chosen Smaller values of U give more weight to the more recent levels and less weight to earlier levels Smaller values of V give more weight to the current trends and less weight to past trends © 2004 Prentice-Hall, Inc.

The Holt-Winters Method © 2004 Prentice-Hall, Inc.

The Holt-Winters Method: Example Year Sales (Yi ) Level (Ei ) U =.2 Trend (Ti ) V = .2 94 2 NA 95 5 5-2=3 96 .2(5+3)+.8(2)=3.2 .2(3)+.8(3.2-5)=-.84 97 .2(3.2-.84)+.8(2)=2.07 .2(-.84)+.8(2.07-3.2)=-1.07 98 7 .2(2.07-1.07)+.8(7)=5.8 .2(-1.07)+.8(5.8-2.07)=2.77 99 6 .2(5.8+2.77)+.8(6)=6.51 .2(2.77)+.8(6.51-5.8)=1.12 © 2004 Prentice-Hall, Inc.

The Holt-Winters Method: Forecasting © 2004 Prentice-Hall, Inc.

Holt-Winters Method: Plot of Series and Forecasts Excel Spreadsheet with the Computation 1994 Forecasts for 2000 to 2005 © 2004 Prentice-Hall, Inc.

Autoregressive Modeling Used for Forecasting Takes Advantage of Autocorrelation 1st order - correlation between consecutive values 2nd order - correlation between values 2 periods apart Autoregressive Model for p-th Order: Random Error © 2004 Prentice-Hall, Inc.

Autoregressive Model: Example The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last 8 years. Develop the 2nd order autoregressive model. Year Units 93 4 94 3 95 2 96 3 97 2 98 2 99 4 00 6 © 2004 Prentice-Hall, Inc.

Autoregressive Model: Example Solution Develop the 2nd order table Use Excel to estimate a regression model Year Yi Yi-1 Yi-2 93 4 --- --- 94 3 4 --- 95 2 3 4 96 3 2 3 97 2 3 2 98 2 2 3 99 4 2 2 00 6 4 2 Excel Output © 2004 Prentice-Hall, Inc.

Autoregressive Model Example: Forecasting Use the 2nd order model to forecast number of units for 2001: © 2004 Prentice-Hall, Inc.

Autoregressive Model in PHStat PHStat | Multiple Regression Excel Spreadsheet for the Office Units Example © 2004 Prentice-Hall, Inc.

Autoregressive Modeling Steps 1. Choose p : Note that df = n - 2p - 1 2. Form a Series of “Lag Predictor” Variables Yi-1 , Yi-2 , … ,Yi-p 3. Use Excel to Run Regression Model Using All p Variables 4. Test Significance of Ap If null hypothesis rejected, this model is selected If null hypothesis not rejected, decrease p by 1 and repeat © 2004 Prentice-Hall, Inc.

Selecting a Forecasting Model Perform a Residual Analysis Look for pattern or direction Measure Residual Error Using SSE (Sum of Square Error) Measure Residual Error Using MAD (Mean Absolute Deviation) Use Simplest Model Principle of parsimony © 2004 Prentice-Hall, Inc.

Cyclical effects not accounted for Residual Analysis e e Time Time Random errors Cyclical effects not accounted for e e Time Time Trend not accounted for Seasonal effects not accounted for © 2004 Prentice-Hall, Inc.

Measuring Errors Choose a Model that Gives the Smallest Measuring Errors Sum Square Error (SSE) Sensitive to outliers © 2004 Prentice-Hall, Inc.

Measuring Errors Mean Absolute Deviation (MAD) (continued) Mean Absolute Deviation (MAD) Not sensitive to extreme observations © 2004 Prentice-Hall, Inc.

Principle of Parsimony Suppose 2 or More Models Provide Good Fit to Data Select the Simplest Model Simplest model types: Least-squares linear Least-squares quadratic 1st order autoregressive More complex types: 2nd and 3rd order autoregressive Least-squares exponential Holt-Winters Model © 2004 Prentice-Hall, Inc.

Forecasting with Seasonal Data Use Categorical Predictor Variables with Least-Squares Trend Fitting Forecasting Equation (Exponential Model with Quarterly Data): The bj provides the multiplier for the j -th quarter relative to the 4th quarter Qj = 1 if j -th quarter and 0 if not Xi = the coded variable denoting the time period i © 2004 Prentice-Hall, Inc.

Forecasting with Quarterly Data: Example Standards and Poor’s Composite Stock Price Index: Quarter 1994 1995 1996 1997 Excel Output r2 is .98 Appears to be an excellent fit. © 2004 Prentice-Hall, Inc.

Forecasting with Quarterly Data: Example (continued) Excel Output Regression equation for the first quarters: © 2004 Prentice-Hall, Inc.

Forecasting with Quarterly Data: Example (continued) 1st quarter of 1994: 1st quarter of 1998: © 2004 Prentice-Hall, Inc.

Forecasting with Quarterly Data in PHStat Use PHStat | Multiple Regression Excel Spreadsheet for the Stock Price Index Example © 2004 Prentice-Hall, Inc.

Index Numbers Measure the Value of an Item (Group of Items) at a Particular Point in Time as a Percentage of the Item’s (Group of Items’) Value at Another Point in Time A price index measures the percentage change in the price of an item (group of items) in a given period of time over the price paid for the item (group of items) at a particular point of time in the past Commonly Used in Business and Economics as Indicators of Changing Business or Economic Activity © 2004 Prentice-Hall, Inc.

Simple Price Index Selection of the Base Period Should be a period of economic stability rather than one at or near the peak of an expanding economy or declining economy Should be recent so that comparisons are not greatly affected by changing technology and consumer attitudes or habits © 2004 Prentice-Hall, Inc.

Simple Price Index: Example Given the prices (in dollars per pound) for apples, construct the simple price index using 1980 as the base year. Base Year © 2004 Prentice-Hall, Inc.

Shifting the Base © 2004 Prentice-Hall, Inc.

Shifting the Base: Example Change the base year of the simple price index of apples from 1980 to 2000: New Base Year © 2004 Prentice-Hall, Inc.

Aggregate Price Index Reflects the Percentage Change in Price of a Group of Commodities (Market Basket) in a Given Period of Time Over the Price Paid for that Group of Commodities at a Particular Point of Time in the Past Affects the Cost of Living and/or the Quality of Life for a Large Number of Consumers Two Basic Types Unweighted aggregate price index Weighted aggregate price index © 2004 Prentice-Hall, Inc.

Unweighted Aggregate Price Index © 2004 Prentice-Hall, Inc.

Unweighted Aggregate Price Index (continued) Easy to Compute Two Distinct Shortcomings Each commodity in the group is treated as equally important so that the most expensive commodities per unit can overly influence the index Not all commodities are consumed at the same rate, but they are treated the same by the index © 2004 Prentice-Hall, Inc.

Unweighted Aggregate Price Index: Example Given the prices (in dollars per pound) for apples, bananas and oranges, compute the unweighted aggregate price index using 1980 as the base year: Base Year © 2004 Prentice-Hall, Inc.

Weighted Aggregate Price Indexes Allow for Differences in the Consumption Levels Associated with the Different Items Comprising the Market Basket by Attaching a Weight to Each Item to Reflect the Consumption Quantity of that Item Account for Differences in the Magnitude of Prices Per Unit and Differences in the Consumption Levels of the Items Two Types that are Commonly Used The Laspeyres price index The Paasche price index © 2004 Prentice-Hall, Inc.

Laspeyres Price Index Uses the Consumption Quantities Associated with the Base Year © 2004 Prentice-Hall, Inc.

Laspeyres Price Index: Example Given the prices (in dollars per pound) and per capita consumption (in pounds) for apples, bananas, and oranges, compute the Laspeyres price index using 1980 as the base year: © 2004 Prentice-Hall, Inc.

Paasche Price Index Uses the Consumption Quantities Experienced in the Year of Interest Instead of Using the Initial Quantities © 2004 Prentice-Hall, Inc.

Paasche Price Index Advantage Disadvantages (continued) Advantage A more accurate reflection of total consumption costs at the point of interest in time Disadvantages Accurate consumption values for current purchases are often hard to obtain If a particular product increases greatly in price compared to other items in the market basket, consumers will avoid the high-priced item out of necessity, not because of changes in preferences © 2004 Prentice-Hall, Inc.

Paasche Price Index: Example Given the prices (in dollars per pound) and per capita consumption (in pounds) for apples, bananas, and oranges, compute the Paasche price index using 1980 as the base year: © 2004 Prentice-Hall, Inc.

Pitfalls Concerning Time-Series Forecasting Taking for Granted the Mechanism that Governs the Time Series Behavior in the Past Will Still Hold in the Future Using Mechanical Extrapolation of the Trend to Forecast the Future Without Considering Personal Judgments, Business Experiences, Changing Technologies, Habits, Etc. © 2004 Prentice-Hall, Inc.

Chapter Summary Discussed the Importance of Forecasting Addressed Component Factors of the Time-Series Model Performed Smoothing of Data Series Moving averages Exponential smoothing Described Least-Squares Trend Fitting and Forecasting Linear, quadratic and exponential models © 2004 Prentice-Hall, Inc.

Chapter Summary (continued) Discussed Holt-Winters Method of Trend Fitting and Forecasting Addressed Autoregressive Models Described Procedure for Choosing Appropriate Models Addressed Time-Series Forecasting of Monthly or Quarterly Data (Use of Dummy-Variables) Discussed Pitfalls Concerning Time-Series Forecasting Described Index Numbers © 2004 Prentice-Hall, Inc.