Business Forecasting

1. Business Forecasting Chapter 9 Advanced Regression Methodologies in Forecasting

2. Chapter Topics • Proxy and dummy variables • Selection of Independent variables • All possible regression • Stepwise regression • Lagged variables • Distributed lag models • Adaptive expectation models • Partial adjustment model • Chapter Summary

3. Dummy Variable Models (with two Levels) Given: Y = Assessed Value of House X1 = Square footage of House X2 = Desirability of Neighborhood = Desirable (X2 = 1) Undesirable (X2 = 0) 0 if undesirable 1 if desirable Same slopes

4. Dummy Variable Models (with two Levels) (continued) Y (Assessed Value) Same slopes Desirable Location b0 + b2 Undesirable Intercepts different b0 X1(Square footage)

5. Interpretation of the Dummy Variable Coefficient (with two Levels) Example: : Annual salary of college graduate in thousand $ 0 nonbusiness degree : : GPA 1 business degree With the same GPA, college graduates with a business degree are making an estimated 6 thousand dollars more than graduates with a nonbusiness degree on average.

6. Selection of Independent Variables • Theory should be the guiding principle. • Follow the principle of “Parsimony.” • Less is better than more. • A number of approaches can be adopted: • All possible Regression • Stepwise Regression • Forward stepwise selection • Backward stepwise elimination

7. Stepwise Regression • Forward Stepwise Selection • A simple regression model is developed with one independent variable, and other variables are added to the equation without deleting any variable as the process continues. • May lead to a model that includes too many variables without satisfying the theoretical framework or the statistical criterion for selection.

8. Stepwise Regression • Backward stepwise elimination • Begins with a model that considers all variables of interest and subsequently reduces the number until the “best” model that satisfies theory and is statistically significant is found. • The steps in using this model are: • Choose a value of and .

9. Stepwise Regression • The stepwise approach considers the k possible one-independent variable regression model of the form: • where is a different potential independent variable in a different model. • If the t Statistic shows that values for are not significant, the stepwise procedure terminates otherwise the variable included.

10. Stepwise Regression • The stepwise approach considers the k possible one-independent variable regression model of the form: • where is a different potential independent variable in a different model. • If the t Statistic shows that values for are not significant, the stepwise procedure terminates otherwise the variable is included.

11. Stepwise Regression • After each addition or deletion of an independent variable to the model, the stepwise procedure checks all the independent variables included and removes those independent variables that have the smallest (in absolute value) t Statistic from the model and that are considered not significant at the level.

12. Lagged Variables • In a dynamic model the variable of interest may be influenced by the value of the same variable in the previous time period. • In these cases the explanatory variable determines the dependent variable with a lag.

13. Lagged Variables (continued) • A regression model where the dependent variable depends not only on the current value, but also on a one-period lag, may be written as: where is the value of X one period before i, and is the error term.

14. Distributed Lagged Models • In this model we will consider the fact that the recent past may have more impact on the variable of interest than a more distant time period. • Hence in specifying a distributed lag model we assign weights ( ) to each variable that is lagged.

15. Adaptive Expectation Models • In this model, the dependent variable is impacted by the “expectation” of the causal variable . • Suppose capital investment is the dependent variable and is influenced by corporate earnings. • Since decisions to invest might be tied to the expected future earnings, the independent variable is unobservable.

16. Adaptive Expectation Models • How then do we capture the notion of “expectation” in a model? where is the independent variable representing the expectation.

17. Adaptive Expectation Models • Since is unobservable, we must make some assumptions about the way in which expectations are formed:

18. Adaptive Expectation Models where is the actual earnings at time t, and the expectation at time t and the expectations are adjusted proportionally by the factor to the difference between expected values in period t –1 and the actual value of X in t.

19. Adaptive Expectation Models • To link the observable process of and we put the two equations together. With simple manipulations of the equations, we are able to create a model for prediction. • Note that we substituted t−1 for t. The result is:

20. Adaptive Expectation Models • We multiply both sides of the previous two equations by , and subtract it from the first equation to get:

21. Adaptive Expectation Models • This last equation can be rewritten as: Where: • We now have a multiple regression equation that can be used for prediction.

22. Partial Adjustment Model • The partial adjustment model presumes that habit plays a critical role in how consumers and producers do not move completely from one equilibrium point to another. • The theory behind the partial adjustment model is that the behaviorally desired level of Y in period t is an unobservable variable Y* that can be written as:

23. Partial Adjustment Model • For reasons such as inertia, habit, cost, or other constraints, actual behavior in each period closes only part of the gap between last period’s actual and this period’s desired Y *. This can be stated as:

24. Partial Adjustment Model • Similarly to the adaptive expectation model, we combine equations and rearrange the terms to arrive at: where: • The coefficient on identifies the adjustment factor .

25. Chapter Summary • Proxy and dummy variables • Selection of independent variables • All possible regression • Stepwise regression • Lagged variables • Distributed lag models • Adaptive expectation models • Partial adjustment model