Analysis with Time Dependent Costs

Practical Applications The independent variables we have used so far-machine-hours, labor-hours-have been continuous variables, that is, variables that can take on a continuous set of values. Explanatory or independent variables can also be discrete or categorical, taking on only a finite number of different values. For example, in estimating utilities expense, the cost of heating will be higher in winter months than in summer months, To incorporate this effect, a dummy variable can be included in the regression model. The variable will be set equal to 1 during the winter months, and to 0 during all the other months. The coefficient of this variable will then estimate the increased cost during the winter months when the variable is defined to be 1) relative to all other months. A dummy variable can also be useful to control for jumps or shifts in fixed costs. Supervision or manning levels may have increased at some point, or increased charges due to new machines or expansion of floor space may have been occurred sometime during the estimation period. A simple linear regression of overhead cost (OH) versus direct labor-hours (DLH) yields: OH, = 16,056 + 5.13 DLH, R° = 0.16, DW = 0.97 (5.46) (1.77) The explanatory power of the regression is low, the coefficient of DLH is barely statistically significant (using a one-sided test at the 5% levell, and the Durbin-Watson statistic indicates a high degree of positive au-tocorrelation. A plot of residuals versus time (see Exhibit 5-7) shows that the first six observations have negative residuals while five of the next six have positive residuals. This evidence suggests that around the end of month 6 an increase in fixed costs occurred. To test this hypothesis, we define a dummy variable, JUMP, that equals 0 for observations 1, 2,.6 and equals 1 for observations 7, 8, ..., 12. Rerunning the regression with the dummy variable, JUMP, leads to the following output: OH, = 12,679 + 7.66 DLH, + 1,646 JUMP, R' = 0.77; DW = 2.4 (7.55) (4.78) (5.20) The explanatory power of the regression is now quite good. The estimated coefficients of the DLH and JUMP variables are highly statistically significant. If we wanted to predict costs beyond period 12, we would add the coefficients of the intercept and dummy variable together to obtain OH = 14,325 + 7.66 DLH. In another example, perhaps fixed costs increased more gradually because of a steady increase in the price level.