FORECASTING Regression Analysis Aslı Sencer

1. Graduate Program in Business Information Systems FORECASTINGRegression AnalysisAslı Sencer

2. Regression in Causal Models • Regression analysis can make forecasts with with a non-time independent variable. • A simple regression employs a straight line. Ŷ(X) = a + bX • The dependent variable is not time periods, such as: • store size • order amount • weight • For 10 rail shipments, the transportation time Y was forecast for specific distance X.

3. Forecasting Transportation Time from Distance

4. Linear Regression Equations Equation: Slope: Y-Intercept:

5. Interpretation of Coefficients • Slope (b) • Estimated Y changes by b for each 1 unit increase in X • If b = 2, then transportation time (Y) is expected to increase by 2 for each 1 unit increase in distance (X) • Y-intercept (a) • Average value of Y when X = 0 • If a = 4, then transportation time (Y) is expected to be 4 when the distance (X) is 0

6. Least Squares Assumptions • Relationship is assumed to be linear. • Relationship is assumed to hold only within or slightly outside data range. • Do not attempt to predict time periods far beyond the range of the data base.

7. Random Error Variation • Variation of actual Y from predicted • Measured by standard error of estimate, SY,X • Affects several factors • Parameter significance • Prediction accuracy

8. Standard Error of the Estimate

9. Assumptions on Error Terms • The mean of errors for each x is zero. • Standard deviation of error terms , SY,X • is the same for each x. • Errors are independent of each other. • Errors are normally distributed with mean=0 and • variance= SY,X. for each x.

10. Correlation • Answers: ‘how strongis the linearrelationship between the variables?’ • Correlation coefficient, r • Values range from -1 to +1 • Measures degree of association • Used mainly for understanding

11. Sample Coefficient of Correlation

12. r = 1 r = -1 Y Y ^ Y = a + b X i i ^ Y = a + b X i i X X r = .89 r = 0 Y Y ^ ^ Y = a + b X Y = a + b X X X i i i i Coefficient of Correlation and Regression Model

13. Coefficient of Determination If we do not use any regression model, total sum of square of errors, SST If we use a regression model, sum of squares of errors Then sum of squares of errors due to regression We define coef. of determination

14. Coefficient of determination r2 is the variation in y • that is explained and hence recovered/eliminated • by the regression equation ! • Correlation coeficient r can also be found by using

15. Multiple Regression in Forecasting • Regression fits data employing a multiple regression equation with several predictors: Ŷ = a + b1X1 + b2X2 • Floorspace X1 and advertising expense X2 make forecasts of hardware outlet sales Y: Ŷ = -22,979 + 11.42X1 + 23.41X2 • The above was obtained in a computer run using 10 data points. • Forecast with X1 =2,500 sq.ft. and X2=$750: Ŷ = -22,979+11.42(2,500)+23.41(750)=$23,129

16. Guidelines for Selecting Forecasting Model • You want to achieve: • No pattern or direction in forecast error • Error = (Yi - Yi) = (Actual - Forecast) • Seen in plots of errors over time • Smallest forecast error • Mean square error (MSE) • Mean absolute deviation (MAD)

17. Trend Not Fully Accounted for Desired Pattern Error Error 0 0 Time (Years) Time (Years) Pattern of Forecast Error