An automatic procedure for maximizing R² in multiple regression. There are several approaches to stepwise regression including forward (step-up) and backward (step-down) versions. The forward version first enters the causal variable with the highest correlation to the dependent variable, then enters the one with the highest partial correlation (given the variable already included in the model), then enters the variable with the highest partial correlation (given the two variables already included), and so on, until certain stopping rules are encountered. One common rule is to include all those and only those variables that have a t-statistic equal to or greater than 1. According to Haitovsky (1969), this rule maximizes the adjusted R². The step-down version puts all of the variables in initially, then removes the one that contributes least to R², next removes from the remaining variables the one that contributes least, and so on. Stepwise regression does not use much prior knowledge, other than to propose a possible set of variables and a functional form. As a result, stepwise regression should not be used for forecasting. In addition, empirical evidence does not support the use of stepwise regression for forecasting. For example, Armstrong (1985, pp. 54) developed two models to forecast camera sales per capita in each of 11 countries. Each of these models was developed using data from 19 other counties. An exploratory model used stepwise regression, drawing from a set of 15 variables, and the model with the highest R² was selected as the forecasting model. A theory-based model was also developed by selecting seven variables, by putting a priori constraints on the signs, and by incorporating prior estimates of magnitudes. Although the exploratory model provided the best fit to the 19-country calibration data ( of 99.8% vs. 99.6%), its performance in forecasting for an 11-country validation sample was inferior; its mean absolute percentage error was 52% vs. 31% for the theory-based model. The average percentage error (using the signs) of the theory-based model was also lower at 5% vs. 38%. If despite this advice, you insist on using stepwise regression and associated measures of statistical significance, use the tables provided by McIntyre et al. (1983).

• Haitovsky, Y. (1969), “A note on the maximization of R²,” American Statistician, 23, (Feb.), 20-21.
• Armstrong, J. S. (1985), Long-Range Forecasting. New York: John Wiley. (Full text)
• McIntyre, S. H., D. B. Montgomery, V. Srinivasan & B. A. Weitz (1983), “Evaluating the statistical significance of models developed by stepwise regression,” Journal of Marketing Research, 20, 1-11.