Statistics for Managers using Microsoft Excel 3rd Edition Chapter 13 Time Series Analysis © 2002 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 square trend fitting and forecasting Linear, quadratic and exponential models © 2002 Prentice-Hall, Inc.

Chapter Topics Autoregressive models (continued) Autoregressive models Choosing appropriate forecasting models Time series forecasting of monthly or quarterly data Pitfalls concerning time-series analysis © 2002 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 © 2002 Prentice-Hall, Inc.

The Importance of Forecasting (continued) College administrators need to forecast enrollments to plan for facilities and for faculty recruitment Retail stores need to forecast demand to control inventory levels, hire employees and provide training © 2002 Prentice-Hall, Inc.

Time-Series Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc. Example: Year: 1994 1995 1996 1997 1998 Sales: 75.3 74.2 78.5 79.7 80.2 © 2002 Prentice-Hall, Inc.

Time-Series Components Trend Cyclical Time-Series Seasonal Random © 2002 Prentice-Hall, Inc.

Trend Component Overall upward or downward movement Data taken over a period of years Upward trend Sales Time © 2002 Prentice-Hall, Inc.

Cyclical Component Upward or downward swings May vary in length Usually lasts 2 - 10 years 1 Cycle Sales © 2002 Prentice-Hall, Inc.

Seasonal Component Upward or downward swings Regular patterns Observed within 1 year Sales Summer Winter Spring Fall Time (Monthly or Quarterly) © 2002 Prentice-Hall, Inc.

Random or Irregular Component Erratic, nonsystematic, random, “residual” fluctuations Due to random variations of Nature Accidents Short duration and non-repeating © 2002 Prentice-Hall, Inc.

e.g.: Quarterly Retail Sales with Seasonal Components © 2002 Prentice-Hall, Inc.

e.g.: Quarterly Retail Sales with Seasonal Components Removed © 2002 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 © 2002 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 © 2002 Prentice-Hall, Inc.

Moving Averages Example: Three-year moving average First average: (continued) Example: Three-year moving average First average: Second average: © 2002 Prentice-Hall, Inc.

Moving Average Example John is a building contractor who has constructed 24 single-family homes over a six-year period. Provide John with a three-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 © 2002 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 © 2002 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 © 2002 Prentice-Hall, Inc.

e.g.: 5-point Moving Averages of Quarterly Retail Sales © 2002 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 Ranges from 0 to 1 Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting © 2002 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 © 2002 Prentice-Hall, Inc.

Exponential Weight: Example Graph Sales 8 6 4 2 Data Smoothed 94 95 96 97 98 99 Year © 2002 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 © 2002 Prentice-Hall, Inc.

Example: Exponential Smoothing of Real GNP The EXCEL spreadsheet with the real GDP data and the exponentially smoothed series © 2002 Prentice-Hall, Inc.

The Least Squares Linear Trend Model Year Coded X Sales (Y) 95 0 2 96 1 5 97 2 2 98 3 2 99 4 7 00 5 6 © 2002 Prentice-Hall, Inc.

The Least Squares Linear Trend Model (continued) Excel Output Projected to year 2001 © 2002 Prentice-Hall, Inc.

The Quadratic Trend Model Year Coded X Sales (Y) 95 0 2 96 1 5 97 2 2 98 3 2 99 4 7 00 5 6 © 2002 Prentice-Hall, Inc.

The Quadratic Trend Model (continued) Excel Output Projected to year 2001 © 2002 Prentice-Hall, Inc.

The Exponential Trend Model or Year Coded X Sales (Y) 95 0 2 96 1 5 97 2 2 98 3 2 99 4 7 00 5 6 Excel Output of Values in logs © 2002 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 © 2002 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 © 2002 Prentice-Hall, Inc.

Model Selection Using Differences (continued) Use an exponential trend model if the percentage differences are more or less constant © 2002 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 © 2002 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 eight years. Develop the second order Autoregressive model. Year Units 93 4 94 3 95 2 96 3 97 2 98 2 99 4 00 6 © 2002 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 © 2002 Prentice-Hall, Inc.

Autoregressive Model Example: Forecasting Use the second order model to forecast number of units for 200x: © 2002 Prentice-Hall, Inc.

Autoregressive Model in PHStat PHStat | multiple regression Excel spreadsheet for the office units example © 2002 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 © 2002 Prentice-Hall, Inc.

Selecting A Forecasting Model Perform a residual analysis Look for pattern or direction Measure sum of square error - SSE (residual errors) Measure residual error using MAD Use simplest model Principle of parsimony © 2002 Prentice-Hall, Inc.

Cyclical effects not accounted for Residual Analysis e e T T Random errors Cyclical effects not accounted for e e T T Trend not accounted for Seasonal effects not accounted for © 2002 Prentice-Hall, Inc.

Measuring Errors Choose a model that gives the smallest measuring errors Sum square error (SSE) Sensitive to outliers © 2002 Prentice-Hall, Inc.

Measuring Errors Mean Absolute Deviation (MAD) (continued) Mean Absolute Deviation (MAD) Not sensitive to extreme observations © 2002 Prentice-Hall, Inc.

Principal of Parsimony Suppose two or more models provide good fit for data Select the simplest model Simplest model types: Least-squares linear Least-square quadratic 1st order autoregressive More complex types: 2nd and 3rd order autoregressive Least-squares exponential © 2002 Prentice-Hall, Inc.

Forecasting With Seasonal Data Use categorical predictor variables with least-square trending fitting Exponential model with quarterly data: The bi provides the multiplier for the i-th quarter relative to the 4th quarter. Qi = 1 if i-th quarter and 0 if not Xj = the coded variable denoting the time period © 2002 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. © 2002 Prentice-Hall, Inc.

Forecasting With Quarterly Data: Example (continued) Excel Output Regression Equation for the first quarter: © 2002 Prentice-Hall, Inc.

Forecasting with Quarterly Data in PHStat Use PHStat | multiple regression Excel spreadsheet for the stock price index example © 2002 Prentice-Hall, Inc.

Pitfalls Regarding Time-Series Analysis Assuming 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, and habits, etc. © 2002 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 square trend fitting and forecasting Linear, quadratic and exponential models © 2002 Prentice-Hall, Inc.

Chapter Summary Addressed autoregressive models (continued) 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 analysis © 2002 Prentice-Hall, Inc.