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Accounting Educators’ Journal 2003 Volume XV
Should High-Low Go: An Analysis Using the
Bootstrap
Jon Woodroof
University of Tennessee
Terry Ward
Middle Tennessee State University
Bill Burg
University of Alabama - Birmingham
Abstract In order for managers to be able to estimate break-even numbers for budgeting purposes, historical total costs must be able to be separated into their fixed and variable cost components. There are generally two methods for teaching this: the high-low method, and the method of least squares regression. The high-low method is considered theoretically inferior to the method of least squares, yet it continues to be taught in accounting courses. The argument for its continuation has been that it is “quick and easy”. However, with the proliferation of electronic spreadsheets, this advantage can also be attributed to the method of least squares. The high-low method’s continued coverage in accounting textbooks would seem to indicate that educators feel that the results generated by each method are not significantly different. This paper compares these methods by using a bootstrapping technique. Bootstrapping facilitates the simulated generation of entire distributions from a sample and allows statistical comparisons to be made between the distributions. The results of this study indicate that the high-low method, while easy to use, may be giving results that are significantly different from results obtained from regression. Because students now have the ability to do regression easily and inexpensively using a spreadsheet, and because of the theoretical shortcomings of the high-low method, it may be that educators should discontinue using and teaching the high-low method altogether.
Introduction
In order for managers to be able to estimate break-even numbers for budgeting purposes, historical total costs (which are mixed in nature) must be able to be separated into their fixed and variable cost components. There are generally two methods for teaching this: the high-low method, and the method of least squares regression. Using the high-low method, one determines the slope of the variable cost line (and ultimately the amount of fixed costs) by identifying high and low total cost data points and high and low activity data points and dividing the change in total cost at these high and low levels by the change in activity. Alternatively using the method of least squares regression, the
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slope line is mathematically fitted to the data. The point where this slope line crosses the Y-axis indicates the amount of the total costs that are fixed.
Advocates of the high-low method have historically argued that it is "quick and easy" and does not require sophisticated analysis tools that may not be readily available to accounting practitioners and students. Advocates of the method of least squares regression insist that relying on only two data points to estimate the fixed and variable components of total costs (and discarding the rest of the data) is too simplistic an approach and may lead managers to make bad decisions.
Problem
The high-low method has been known to have serious theoretical and practical flaws for several decades (Johnson and Harrell, 1999). Conversely, the method of least squares regression has long been recognized as the superior technique (Nurnberg, 1977). Regression is a much more sophisticated mathematical technique, and accounting researchers more than 30 years ago showed that regression can be a valuable tool in accounting (Comiskey, 1966; Benston, 1966). Today, regression can be performed easily using a common electronic spreadsheet.
Because electronic spreadsheets have been readily available to both accounting practitioners and students for several years now, utilization of the least squares regression method should be on the increase, and teaching and promoting the simplistic (and perhaps flawed) high-low method should be on the decrease. However, there is no evidence that this is the trend. On the contrary, both methods continue to be taught in accounting principles, cost, and managerial accounting textbooks. In fact, texts published within the past few years continue to give a significant amount of coverage to the high- low method and include several high-low problems and exercises (Johnson and Harrell, 1999).
There can be only one good reason for this (reluctance to change is not considered here as a good reason). Evidently, there is a perception that the two methods do not produce results that materially differ from each other. The source of this perception is not entirely clear -- accounting research literature describing empirical studies on this topic is unusually sparse. One reason for this scarcity is that it is very difficult to get companies to release real cost data. A second reason is that the very exercise of comparing methods for separating total costs into their fixed and variable components is innately problematic. How do you statistically determine which method is "better"? And how can this determination for a given set of data in a particular industry be made easily and inexpensively? One approach for doing this is bootstrapping.
Bootstrapping
Bootstrapping is a powerful technique for making statistical inferences about a population characteristic (Efron, 1982). However, it differs from the traditional approach to statistical inferencing in that,
Bootstrapping relies on an analogy between the sample and the population from which the sample was drawn. The central idea is that it may sometimes be better to draw conclusions about the characteristics of a population strictly from the sample at hand, rather than by making perhaps unrealistic assumptions about that population (Mooney and Duval, 1993, p.1).
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The Research Question
For the current study, the high-low implementation based on total cost was dropped from the analysis due to Nurnberg's (as well as others) apriori conclusions concerning the nature of actual high-low pairs based upon highest and lowest cost levels. He states:
Implicit throughout most discussions of cost behaviour in the managerial accounting literature is the assumption of fewer errors in the measurement of activity levels than cost levels. This in turn is due to the fact that activity levels are often measured in physical terms and, as such, are devoid of the ambiguities inherent in accounting accruals, deferrals, and allocations, to which measurements of cost levels are subject. Accordingly, extreme activity levels are less likely to reflect the abnormal than extreme cost levels (Nurnberg, 1977, p. 433). Therefore, this paper uses the same Horngren data set (Appendix A) and uses bootstrapping to empirically compare the following three methods of separating total costs into their fixed and variable components: 1) least squares regression; 2) actual high-low pairs based upon highest and lowest activity levels; and 3) hypothetical high- low pairs based on absolute highs and lows regardless of actual pairing. The calculations for the three methods are shown in Table 2.
Table 2: Comparison of Methods to Separate Total Costs
Least Squares Regression
Actual Pairing Based on Activity
Hypothetical Pairing High Total Cost 11,000 11, Low Total Cost 6,400 6, High Activity 384,000^ 384, Low Activity 180,000 180, Variable cost per unit 0.02155 0.02255 0. Fixed cost component 2,918^ 2,341^ 2,
Thus, the research question to be investigated is, "Is there a significant difference among these three methods?" Stated in the null,
There is no difference among least squares regression, actual pairing high- low, and hypothetical pairing high-low in their abilities to separate total costs into their fixed and variable components.
Bootstrapping is used to address this question. The following section provides a discussion of a spreadsheet template that performs this powerful statistical technique.
The Spreadsheet Template
This template can be build using any electronic spreadsheet. For our example, we chose Corel’s Quattro Pro spreadsheet. First, the data (Appendix A) is entered into the Data page of the spreadsheet - shown in Figure 1. In column A is an index (1 through
- indicating the number of the particular data point. This will be used later to randomly select a data point. Of course, columns C and D are needed to calculate least squares regression (details of this calculation will not be presented here, since that is not the purpose of this paper). This page calculates and displays the variable and fixed components of total cost for the original data using the three methods.
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Figure 1: Data Page
The next spreadsheet page is the Sample page, shown in Figure 2. Each time the spreadsheet is recalculated, formulas on this page randomly sample the data on the Data page, and then calculate and display the variable and fixed components of total cost for the sampled data using the three methods.
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Figure 3: Bootstrap Macro
Here is what this macro is doing. First, a FOR loop is defined. The logic is: “For as long as the counter is less than or equal to the number of iterations, execute the code beginning in the start location”. B11 is the start location for the code that gets executed. So, for each increment, the spreadsheet is CALCulated (a new sample is drawn and the calculations using each of the three methods are performed). The results of these calculations (only the fixed component was used in the comparison) are displayed on the Pairings page in row 2, shown in Figure 4. This row is selected and copied to the row associated with the current iteration number. Figure 4 shows the results after six iterations. Five thousand iterations were run in order to give more data points for the resulting distributions.
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Figure 4: Pairings Page
The second macro is found on the Test Page shown in Figure 5. This macro uses the output of the 5000 iterations that were previously written to the Pairings Page. It takes the fixed cost components for the two models being compared and copies them into the range on the Test page beginning with B10. The vector of fixed cost points from one model is subtracted from the vector of fixed cost points from the second model, creating a vector of differences beginning in D10. The macro then copies the values found in the differences vector into column G and sorts this column in ascending order.
Column F contains percentage bins increments of .02%, from 0.00% to 100.00%. This assigns each of the 5000 ranked data points in the differences vector to a percentile place in the distribution. Then, this column is used to look up the difference associated with the particular level of confidence entered into cell G5. The respective lower and upper limits of the confidence interval are found by using the following two formulas:
Formula 5 (F3): @VLOOKUP(@VLOOKUP(J6,F11 through G5010,1)) Formula 6 (G3): @VLOOKUP(@VLOOKUP(J5,F11 through G5010,1)).
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Table 3: Results of Bootstrapping
Comparison Significance LSR vs. APA 95 % LSR vs. HP 99 % APA vs. HP not significant
In the case of the data set used in this study, the results indicate that using the least squares regression method of separating total cost into its fixed and variable components does produce results that are significantly different from the results produced by both implementations of the high-low method (while the two implementations of the high-low method were not found to produce significantly different results). Therefore, the null is rejected. Graphs showing the generated estimated distributions for the significant comparisons are shown in Figures 6 and 7. As can be readily observed, both graphs show distributions that are not normal. Bootstrapping is one of the few techniques that can be used to compare such distributions, and, as shown, it can be performed easily in a spreadsheet.
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500
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Frequency
900 2,000 3,100 4, Calculated Fixed Cost
Regression
Act High-Low
Comparing Act High-Low to Regression @ 95% confidence level
Figure 6: Regression vs. Actual High-Low Pairing Based on Activity
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500
1000
1500
2000
2500
Frequency
900 2,000 3,100 4, Calculated Fixed Costs
Regression
Hypo High-Low
Comparing Hypo High-Low to Regression @ 99% confidence level
Figure 7: Regression vs. Hypothetical High-Low Pairing
Conclusion and Future Direction
The technique of bootstrapping in a spreadsheet provides practitioners and educators with a powerful, yet simple to use, tool to compare methods for separating total costs into their fixed and variable components. The results of this study indicate that the high-low method, while easy to use, may be giving results that are significantly different from results obtained from regression. Because students now have the ability to do regression easily and inexpensively in a spreadsheet, and because of the theoretical shortcomings of the high-low method, it may be that educators should discontinue using and teaching the high-low method altogether.
Much more analysis needs to be done using real cost data from various industries. Researchers and accounting educators are encouraged to use this bootstrapping technique on various total cost data sets from various industries to compare the high-low method with the regression method in order to determine which method(s) of separating total costs into their fixed and variable components should continue to be used and taught. By doing this, a body of empirical evidence about these methods can be collected and analyzed.
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Appendix A: Horngren Data
Total Cost Trucking Labor Activity Direct
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