1 FORECASTING Regression Analysis Aslı Sencer Graduate Program in Business Information Systems

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 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 Interpretation of Coefficients

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, S Y,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, S Y,X is the same for each x. Errors are independent of each other. Errors are normally distributed with mean=0 and variance= S Y,X. for each x.

10 Answers: ‘how strong is the linear relationship between the variables?’ Correlation coefficient, r  Values range from -1 to +1  Measures degree of association Used mainly for understanding Correlation

11 Sample Coefficient of Correlation

12 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 r 2 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 + b 1 X 1 + b 2 X 2  Floorspace X 1 and advertising expense X 2 make forecasts of hardware outlet sales Y: Ŷ =  22, X X 2  The above was obtained in a computer run using 10 data points.  Forecast with X 1 =2,500 sq.ft. and X 2 =$750: Ŷ =  22, (2,500)+23.41(750)=$23,129

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

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