Systematic Variations in Costs

Cross Sectional Cost Dummy variables also arise when estimating a cross-sectional cost relationship. Suppose we had a company with a large number of retail chains around the country, and we wished to estimate the variable selling expenses as a function of the gross sales of each store. Initially, we might estimate a simple model using the observations in the same period for each store, i: Selling expensei = Bo + B1 salesi A more careful study would reveal that wages and expenses are highly variable across the country. Northeastern urban stores may have higher hourly wages than rural southwestern stores. Such variations would cause the estimated relationship to have low explanatory power because of the omitted variables (region, urban/rural). To control for this systematic variation in wage expense, we could split the stores into mutually exclusive and collectively exhaustive categories in which stores considered to be reasonably homogeneous would be grouped together within a category. For example, we could define the following 0-1 dummy variables: D1 = (1, if northeast - 0, elsewhere) D2 = (1, if southwest - 0, elsewhere etc.) S1 = (1, if urban store - 0, elsewhere) S2= (1, if suburban store - 0, elsewhere) S3 = (1, if rural store - 0, elsewhere) The coefficients of these dummy variables will measure the shift in fixed costs associated with having a store in a given region. When splitting observations into these mutually exclusive categories, we must remember to either omit one category from the regression or estimate the relationship without the constant term. Otherwise we will have a severe problem, since the independent variables will be perfectly collinear. For example, suppose we classify stores into three categories, as above—urban, suburban, and rural-where each store is in one (and only one) of the three categories. The constant term always equals 1, so that for every observation we would have constant = S1 + S2 + S3 With this perfect collinearity, we will be unable to obtain regression es-timates. The regression program either will not work or will give nonsense results. To avoid this problem, simply omit one category, say rural stores, and estimate coefficients for the constant and the remaining categories. We then interpret the coefficient of the categorical dummy variables as the increase (or decrease) in fixed cost relative to the omitted category; that is, the coefficient of S, would represent the difference in fixed costs between an urban store and a rural store. Dummy variables are aptly named. People occasionally use these categorical variables when they have not thought enough about how to measure the relevant effect. It would be preferable to identify the specific factor that causes costs to shift across categories. For example, instead of using a dummy variable for each season to explain monthly heating expense, we could use the continuous variable of median monthly temperature or median degree days a degree day measures the number of degrees below a standard temperature, such as 68 degrees Fahrenheit each day). Then if we had an unusually warm or cold month, the regression model would be better able to explain or predict actual heating expense. In the cross-sectional retail store example, we could use median wage levels in each region, a continuous variable available from governmental statistics, rather than a dummy variable for each region. Again, such a variable would do better in tracking costs if there were relative shifts in wages among the various regions in the future. Dummy variables are best used when we do not believe that the dependent variable will vary linearly with the independent variable. In such cases, we can define separate dummy variables for subsets of the independent variable and not impose a linearity assumption. We can let the data suggest how the dependent variable varies across the range of values in the independent variable.