An important application of regression models is to predict or forecast values of the de- pendent variable, given values for the independent variables. Forecasts can be computed directly from the estimated regression model using the coefficient estimates in that model, as shown in Equation 12.25. Predictions from the Multiple Regression Models Given that the population regression model y = Bo + B1X1 + BxX+---+ BxXx: + / holds and that the standard regression assumptions are valid, let bo, by, ...,bx be the least squares estimates of the model coefficients, B., where j = 1,...,K, based on the x1, Ky,...,xx (i = 1,...,n) data points. Then, given a new obser- vation of a data point, 41, +1, X2,0 +1,..., XXX1+1 the best linear unbiased forecast of yn+ is 9 = bo + bxu + b2x1; + ... + bx i = 1 + 1 (12.25) It is very risky to obtain forecasts that are based on X values outside the range of the data used to estimate the model coefficients because we do not have data evidence to support the linear model at those points. In addition to the predicted value of Y for a particular set of x, terms, we are often interested in a confidence interval or a prediction interval associated with the prediction. As we discussed in Section 11.6, the confidence interval includes the expected value of Y with probability 1 a. In contrast, the prediction interval includes individual predicted values-expected values of Y plus the random error term. To obtain these intervals, we need to compute estimates of the standard deviations for the expected value of Y and for the individual points. These computations are similar in form to those used in simple regression, but the estimator equations are much more complicated. The standard devia- tions for predicted values, so, are a function of the standard error of the estimate, s; the standard deviation of the predictor variables; the correlations between the predictor vari- ables; and the square of the distance between the mean of the independent variables and 12.6 Prediction 491 the X terms for the prediction. This standard deviation is similar to the standard devia- tion for simple regression predictions in Chapter 11. However, the equations for multiple regression are very complex and are not presented hereinstead, we compute the values using Minitab. The standard deviations for the prediction interval, the confidence inter- val, and the corresponding intervals are computed by most good statistics packages. Excel does not have the capability to compute the standard deviation of the predicted variables. k46 Example 12.10 Forecast of Savings and Loan Profit Margin (Regression Model Forecasts) You have been asked to forecast the savings and loan profit margin for a year in which the percentage net revenue is 4.50 and there are 9,000 offices, using the savings and loan regression model. Data are stored in the file Savings and Loan