© 2000 Prentice-Hall, Inc. Chap The Least Squares Linear Trend Model Year Coded X Sales
© 2000 Prentice-Hall, Inc. Chap The Least Squares Linear Trend Model (Continued) Excel Output Projected to year 2001
© 2000 Prentice-Hall, Inc. Chap Year Coded X Sales The Quadratic Trend Model
© 2000 Prentice-Hall, Inc. Chap The Quadratic Trend Model (Continued) Excel Output
© 2000 Prentice-Hall, Inc. Chap The Exponential Trend Model or Excel Output of Values in logs Year Coded Sales
© 2000 Prentice-Hall, Inc. Chap 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
© 2000 Prentice-Hall, Inc. Chap Model Selection Using Differences Use an Exponential Trend Model if the Percentage Differences Are More or Less Constant (continued)
© 2000 Prentice-Hall, Inc. Chap 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 pth order: Random Error
© 2000 Prentice-Hall, Inc. Chap 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
© 2000 Prentice-Hall, Inc. Chap Autoregressive Model: Example Solution Year Y i Y i-1 Y i Excel Output Develop the 2nd order table Use Excel to run a regression model
© 2000 Prentice-Hall, Inc. Chap Autoregressive Model Example: Forecasting Use the 2nd order model to forecast number of units for 2001:
© 2000 Prentice-Hall, Inc. Chap Autoregressive Modeling Steps 1. Choose p: note that df = n - p Form a series of “lag predictor” variables Y i-1, y i-2, … y i-p 3. Use excel to run regression model using all p variables 4. Test significance of a p If null hypothesis rejected, this model is selected If null hypothesis not rejected, decrease p by 1 and repeat
© 2000 Prentice-Hall, Inc. Chap 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
© 2000 Prentice-Hall, Inc. Chap Residual Analysis Random errors Trend not accounted for Cyclical effects not accounted for Seasonal effects not accounted for T T T T ee e e 00 00
© 2000 Prentice-Hall, Inc. Chap Measuring Errors Choose a model that gives the smallest measuring errors Sum square error (SSE) Sensitive to outliers
© 2000 Prentice-Hall, Inc. Chap Measuring Errors Mean absolute deviation (MAD) Not sensitive to extreme observations (continued)
© 2000 Prentice-Hall, Inc. Chap Principal of Parsimony Suppose 2 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
© 2000 Prentice-Hall, Inc. Chap Forecasting With Seasonal Data Use categorical predictor variables with least- square trending fitting Exponential model with quarterly data: The b i provides the multiplier for the ith quarter relative to the 4th quarter Q i = 1 if ith quarter and 0 if not X j = the coded variable denoting the time period
© 2000 Prentice-Hall, Inc. Chap Forecasting With Quarterly Data: Example Quarter Standards and Poor’s Composite Stock Price Index: Excel Output Appears to be an excellent fit. r 2 is.98
© 2000 Prentice-Hall, Inc. Chap Quarterly Data: Example Excel Output Regression Equation for the first quarter:
© 2000 Prentice-Hall, Inc. Chap Chapter Summary Discussed the importance of forecasting Addressed component factors of the time- series model Performed smoothing of data series Moving averages Exponential smoothing
© 2000 Prentice-Hall, Inc. Chap Chapter Summary Described least square trend fitting and forecasting Linear, quadratic and exponential models Addressed autoregressive models Described procedure for choosing appropriate models Discussed seasonal data (use of dummy variables) (continued)