report – development of variabilities

REPORT – DEVELOPMENT OF VARIABILITIES June 1, 2015

DETERMINATION by the Canadian Transportation Agency of the variable portions of railway company cost accounts for the Canadian National Railway Company and the Canadian Pacific Railway Company.

File No. OP-014-375-003/2014

This is the Canadian Transportation Agency (Agency) staff's report on the review of the factors used in determining the variable portions of the prescribed railway companies’ cost accounts under the Agency’s costing model. BACKGROUND The Agency Rail Regulatory Costing System The Agency has been maintaining for close to half a century a rail regulatory costing system (Rail Costing System) to meet its evolving responsibilities in respect of regulatory cost and rate determinations. The Rail Costing System was established in 1969 following a Cost Inquiry (Cost Inquiry) conducted by the Canadian Transport Commission (CTC). Railway companies, shippers and shipper associations, the provincial governments, academics, and other interested parties participated in the Cost Inquiry. This Inquiry was aimed at defining the meaning of “actual loss” incurred by railway companies in meeting their imposed public burdens, including from the continued operation of uneconomical branch lines or passenger transportation services. The Rail Costing System was subsequently used for the payment of subsidies to the railway companies for actual losses incurred in branch line and passenger service operations, and for other regulatory purposes defined under the Railway Act. The Rail Costing System continues to be a critical tool in support of the Agency’s current responsibilities, including the setting of interswitching rates, the provision of costing information needed by parties involved in a Final Offer Arbitration process, the determination of cost adjustments for the maximum revenue entitlement program, the resolution of disputes between railway companies and public passenger service providers. The Agency also annually publishes a guide on the railway charges for crossing maintenance and construction. The Rail Costing System is, in its most basic form, an activity-based system that decomposes all rail movements and services into defined activities. The cost incurred by the railway company in performing each activity is collected by the railway company as prescribed in the Uniform Classification of Accounts. For each of these activities, the Rail Costing System provides a

- 2 -

REPORT – DEVELOPMENT OF VARIABILITIES

rigorous and empirical method for determining the associated costs that are fixed (that do not vary with proxies of traffic level) and variable (that do vary with proxies of traffic level). The need to differentiate between the variable and fixed component of each railway cost accounts stems from the long-established principle that rates established by the Agency must at least reflect the long-run variable costs of the railway activity considered. To determine the variable portions of the railway cost accounts, the Agency determines the cost to perform each unit of each activity by dividing the cost in each activity account by a quantity representing a traffic volume proxy thought to have caused that cost. A factor, called the variability factor or the variability, is then applied to this estimated cost per unit of that traffic volume proxy to determine the portion that was caused strictly by changes in traffic volume. This variable portion of the cost per unit of traffic volume is called a unit cost. The rest of the cost in that account is determined to be unrelated to changes in that specific traffic volume proxy, and is considered fixed with respect to that proxy. The traffic volume proxies which drive costs fall in two broad categories, workload units and cost subtotals. Workload units are physical measurements of railway activities (e.g. gross ton-miles) whereas the cost subtotals are grouping of costs (e.g. railway car maintenance costs) which drive other cost complexes (e.g. car maintenance supervision costs). For any specific railway movement or service for which a cost is to be determined, it is notionally possible to identify all of the associated workload units. When that is established, the Railway Costing Model enables one to determine the variable costs for the associated operational cost categories, and then in turn the cost subtotals which drive the overhead-type costs. When all these costs are summed up, the total variable costs of a given railway movement or service can be established. The variability factors for a number of cost accounts were assumed to be self-evident during the 1969 Cost Inquiry. For example, the cost of fuel was assumed to be fully related to changes in the miles performed by locomotives, i.e., variability is 100 percent, and the cost of maintaining tunnels and bridges was assumed to be completely unrelated to the traffic volume, and variability was set to 0 percent). The variability factors for most other cost accounts were first determined in 1969 and then updated annually using statistical methods, i.e., regression analysis. Regression analysis allows one to determine empirically and efficiently by how much one factor (say a given workload unit) causes another (say a given railway operational expense account) to vary. The central role of regression analysis to determine variabilities The Rail Costing System in Canada is grounded in the concept of variable costs, and regression analysis plays a central role in determining the component of the costs that are variable. In Reasons for Order No. R-6313 Concerning Cost Regulations, pages 329-331, issued in August

- 3 -


1969 following the Cost Inquiry, the CTC stated the following with respect to the role of regression analysis:

Before considering the issues raised in this Inquiry, there is one matter that the Committee believes is deserving of special mention. Throughout these reasons, reference will be made to the technique known as regression analysis. In explaining its findings relative to items and factors of cost, the Committee considers it to be useful to include a brief and general description of this technique. Regression analysis, as used in Canadian railway costing, is a mathematical technique whereby variations in expense items are related to corresponding variations in railway traffic. When there are good economic or engineering explanations for these relationships, the degree of correlation, or variability, shown by the regression analysis is presumed to reflect causation, that is, the extent to which the expense item is caused by traffic. Regression analysis derives these relationships by comparing a number of independent observations of the same expense item at different levels of traffic. The Canadian railways use “cross-section” analysis which involves comparing the traffic and expense levels of the various geographical divisions or areas of the railway during the same time period. This comparison of the divisional relationships between expense and traffic reveals an average relationship which is presumed to apply throughout the railway. As an example, yard costs could be developed for the Canadian Pacific by comparing yard expense on its 28 operating divisions with the corresponding yard switching minutes on the same divisions. This comparison might indicate that on the average a difference of one yard switching minute results in a difference of 40 cents of yard expense. In costing any specific traffic movement, each yard switching minute would then be presumed to have a variable cost of 40 cents. Simple regression is used when there is only one causative influence, or independent variable, being analyzed in relation to a given expense item. Multiple regression is used when two or more independent variables are studied to determine the simultaneous relationship of each to an expense category. The most significant example of multiple regression is the analysis used by the railway companies for track and roadway maintenance expense. These expenses in each of the railway companies’ operating divisions are compared with the corresponding divisional totals of the following items:

- 4 -


1. Gross ton-miles, freight and passenger; 2. Yard-switching miles; 3. Roadway miles; 4. An index of track grade and curvature.

Mathematical techniques make it possible to calculate the proportion of total roadway maintenance expenses which correlate with each of these four independent variables. The expenses are thus divided between those which vary with traffic (gross ton miles and yard switching miles) and those which vary with the characteristics of the roadway network (road miles and curvature). The independent variables (presumed to be causative factors) used in regression analysis may be other expense items as well as measures of traffic or network characteristics. For example, superintendence is compared with the expense involved in the functions which are supervised. For each regression calculation, there are a number of statistical tests which provide quantitative expressions of the “fit” or degree of correlation, between individual independent variables and the expense item under study (dependent variable). In the case of multiple regression, tests can indicate the extent to which the particular combination of independent variables explains total variation in the dependent expense category.

Data needed to update the variabilities The regression models used to determine the variabilities that have been in use since 1969 require regionally-based information to obtain enough observations for calibration. The UCA directs that most of the financial and operations data submitted by the railway to the Agency be collected according to geographic cost centre. Before 1987, both CN and CP operated more decentralised rail networks, with the ability to match costs and operating statistics at the District level. The Districts were semi-autonomous operating unites based on a geographic partition of the overall network. In 1987, CN had 26 and CP had 23 operating Districts. This provided sufficient data to calibrate the variabilities. However, with industry deregulation in 1987 and the significant rationalization of the rail networks that followed, by 1997, both prescribed railway companies had become more centralized and had moved away from the District structure. Except in selected cases, the centralization left both railway companies with limited ability to match cost and operating statistics at less than system level. As a result, for the most part both prescribed railway

- 5 -


companies lost the capability to capture the regionally-based information needed to calibrate the regression models. In 2000, the Agency adopted fixed variabilities that would be updated on a periodic basis. Three main factors for this were:

• the previous costing model was dependent on regionally-based data that was no longer available

• there was then limited awareness of the statistical techniques that could be used to take advantage of national data

• the need to reduce the regulatory burden on the railway companies with respect to maintaining the Rail Costing System.

Accordingly, the variability factors of a number of unit costs were fixed to their average values between 1992 and 1997. The variability factors that were fixed in 2000 were intended to be updated annually. However, they have not been updated since 2000 and as a result the variabilities used to date still reflect their average values determined empirically between 1992 and 1997. This was primarily due to the lack of suitable regionally-based data, as indicated above, given the statistical techniques then being utilized. With respect to the calibration of the variabilities, the disappearance of regionally-based data is not necessarily problematic, as the relationship between the cost and its causal factors can only be observed in a meaningful way for many accounts at the system level, as opposed to the regional level. An example is freight car maintenance, whose cost is related to the miles of service performed by the car. As freight cars travel freely across Canada and the U.S., the cost of maintenance performed in a given facility can only be matched to the system total miles performed by the car, not to the miles performed in specific geographic areas. Statistical techniques The variabilities that have been in use since 1969 are derived from linear regression analysis. When using this technique, a workload unit is presumed to cause a given expense category to change at a fixed rate; for example, a gross ton-mile could cause a fixed 10 percent change in a specific cost account. In non-linear regression analysis, in addition to allowing for a fixed rate of change per unit of workload, the technique can also test whether the rate of change increases or decreases according to the value of the workload unit. Thus, at a certain time, a gross ton-mile could cause a 5 percent change in a specific cost account and at another point a10 percent change, and vice versa. Non-linear regression does not presume a linear relationship and allows the “evidence to speak for itself” more naturally as it supports a broader range of relationships.

- 6 -


When using national level data, time trends are embedded in the time series and must be removed in order to assess the impact of a given workload unit on costs independently of time trends. The use of national level data to calibrate the variabilities requires the use of data filtering techniques to achieve that objective. REVIEW OF VARIABILITIES BY AGENCY STAFF In 2009, the Agency initiated a review of the variability factors which had been fixed based on regression analyses performed between 1992 and 1997 and in some cases even earlier. This review was due to concerns that, as railway operations have changed significantly since that time, the fixed variabilities had clearly become outdated and may no longer reflect the current relationships between cost and traffic volume. Furthermore, new statistical techniques were found that could be applied to the available railway national data. The Agency staff review process involved five major steps as follows: Step 1: Assessment of current model Step 2: First preliminary results Step 3: Exchange of views on first preliminary results Step 4: Second preliminary results Step 5: Exchange of views on second preliminary results Step 1: Assessment of current model In June 2010, a first review of the past regression analysis was completed by Agency staff. This review involved an extensive assessment of the statistical properties of the existing linear model. It also involved a comparison with the results obtained when applying a more powerful non-linear model to the original data. Staff concluded that the fixed variability factors currently in use, comprising averages of the annual variabilities determined between 1992 and 1997, were out of date and were based on linear regression models that tended to overstate, sometimes significantly, the variability factors for some accounts. It was concluded that all the variability factors needed to be updated. This conclusion was shared with CN and CP staff who then acknowledged the necessity to update the variability factors. CN and CP agreed with Agency staff on the data and analysis needed to update the variabilities. They also agreed with Agency staff that linear and non-linear regression analysis should be used to test the causal relationships between costs and traffic volume.

- 7 -


Step 2: First preliminary results Data The accounting and management systems accounting systems used by CN and CP did not have the capability to match cost and operating statistics for geographic partitions of the network in respect of all of CP’s cost accounts and most of CN’s cost accounts. The data used for each cost account for both CN and CP comprised annual observations of cost and corresponding traffic volume causal variables from 1992 to 2011. However, CN provided for four cost accounts – Track & Roadway Investment, Track & Roadway Maintenance, Signals Investment, and Signals Maintenance - data that matched cost and operating statistics for 14 geographic cost centres. For each of these four accounts CN also provided five years of data comprising 14 geographic-level observations for each year, a total of 70 observations each. Therefore, for these four accounts both the annual time series data and the cross-sectional data were analyzed. For all cost accounts except the four named above, the regression analysis was calibrated only on annual time series data, while for the four accounts the regression analysis was calibrated on both time series and cross-sectional data. When performing time series analysis, the advanced statistical techniques adopted and which will be further explained in this Report requires a number of treatments to the raw data to remove the effect of inflation and time-trends and to express the observations relative to the best possible reference point. The net effect of these three changes ensures that the data used in the calibration:

- reflect real quantities, e.g. changes in the price of fuel is removed from the fuel costs time series, so that it becomes in effect an index of the quantity of fuel actually consumed;

- remove time trends in the time series so that it does become “stationary”, for instance the efficiency gains over time would be removed from both the railway costs and workload units so the transformed data reflect changes as compared to a stationary state;

- are all positive and be measured from an appropriate reference point. In the vast majority of cases, this was achieved by adding a positive quantity to each of the filtered observation so that the lowest value of each times series is just positive. This ensures that the maximum log likelihood technique can be used (it requires positive values) while calibrating the equations with observations that are measured from a reference point that is the closest possible to the transformed observations, thus allowing variations to be more easily detected.

By using transformed data measured from the specified reference point that are expressed in real terms and that are free of time trends, it is possible to observe more clearly the causal link that exists between the variation of a given workload unit on the variation of a given cost. In contrast to simply calibrating the model on the raw data which incorporate both inflationary and time

- 8 -


trend effect, this should reveal the extent to which a specific cost varies in relation to changes in a specific workload unit. Data treatments, filtering and transformation All raw financial data are treated to remove the effects of inflation in input prices and were expressed in constant 2011 dollars. As mentioned, this ensures that the relationship that is established between certain factors and railway costs do not incorporate any inflationary component, i.e., that costs are expressed in real terms. Wherever appropriate for a given unit cost account, multi-year averaging of the data is employed as dictated by the CN and CP costing manuals. The main issue with using time series data for analysis of variabilities is that the data contains trends related to time, which could lead to misleading results. Any relationships derived using such data would capture not only the relationship between a cost and its traffic volume cause at a point in time, but also the effects of technological, network, and process changes over time. To avoid such biased regressions the time effect – or the trend – is removed from each variable. The most common de-trending approach is to apply a filter to the variables. Several de-trending filters are available; the most commonly used being the Hodrick-Prescott (HP) filter which is used in this analysis. The HP filter includes a parameter λ, set by the analyst, which controls the shape of the Trend component in the data. If λ=0, then the Trend component comprises the entire series, and a de-trended series would yield 0 at each observation point. In other words, the variation of the entire times series would be in this case entirely explained by the time trend, leaving no other reasons for the transformed time series to vary from its stationary level. As λ increases, the Trend component is assumed to be less and less of the original raw observations. After some preliminary tests, a filter parameter λ=1600 was used to de-trend all observations. Later in this Report, there will be further discussion of the importance of this choice. After applying a filter to remove time-related trends and ensure that the remaining data is “stationary”, the data contains both positive and negative values, as it measures at that stage variation around the de-trended level. It is necessary to re-scale the filtered numbers so that the minimum value of the series is positive, as the regression analysis techniques cannot handle negative values. This has the effect of moving the referential system to a different location and to measure the filtered observations from that new reference point. While there is in theory an infinite number of potential reference points, only two are in practice adopted. One, the most reasonable, is a small transformation, by means of the addition of a constant that is just large enough to turn the smallest observation positive. As all the transformed observations are relatively close to the origin, the transformation is said to be to the origin. This transformation is the smallest one that can turn all values into positive quantities while being the vantage point that is at the shortest possible distance from the data clouds.

- 9 -


Stated another way, a small transformation enables the chosen regression technique to be applied to values that are observed from the closest vantage point. The further away you move from that referential point, the more distant the data points are and the less relative variability they display. Taken to an extreme, observing the data points from an extremely distant origin would turn the cloud of data into indistinguishable quantities, from which variabilities could not be detected. Furthermore, it can be mathematically proven that the transformation to the origin is the one that results in the least impact to the estimates of the original parameters as a result of the need to shift the cloud of data into the positive quadrant. Appendix I presents the mathematical demonstration of this statement. The other is a much larger transformation, by adding a constant that is large enough to shift the de-trended observations to the approximate space of the original sample, so that the mean of the de-trended observations is equal to the mean of the original sample. The transformation is said to be to the sample mean. Staff have statistically measured both approaches and concluded that in the vast majority of cases, the transformation to the origin leads to both more statistically robust and more intuitively reasonable results. Nevertheless, in a very few instances, the results from the transformation to the mean may be accepted for the moment. Regression Model Specifications For each cost account, both a linear and a non-linear relationship between cost and traffic volume variables were tested using Box-Cox model specifications. The Box-Cox transformation is an extensively-studied statistical technique in which data is transformed into a new form which preserves important aspects of the original data and allows the standard statistical assumptions, that the population being studied is normally distributed with a common variance and additive error structure, to be satisfied. The Box-Cox transformation reduces anomalies such as non-additivity, non-normality, and heteroscedasticity in the data. The models specified were of the form: Linear Specification: 𝑦𝑦 = 𝛼𝛼0 + ∑ 𝛼𝛼𝑖𝑖𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 + 𝜀𝜀

Non-Linear Box-Cox Specification: 𝑦𝑦𝜃𝜃−1𝜃𝜃

= 𝛼𝛼0 + ∑ 𝛼𝛼𝑖𝑖𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 + 𝜀𝜀 Where: y is the regulatory cost for the account; xi represent i causal traffic volume-related variables; λ, α0 and αi are parameters determined in the regressions; and, ε is random noise. For each cost account, eight alternative analysis models reflecting the two different types of data transformation, two different HP filter parameters, and two different model specifications, were prepared for the analysis. The alternatives were as follows:

- 10 -


Model 1: HP filter λ parameter = 100; small post-filter transformation (to the origin); linear model specification

Model 2: HP filter λ parameter = 100; small post-filter transformation (to the origin); non-

linear model specification Model 3: HP filter λ parameter = 100; large post-filter transformation (to the sample mean);

linear model specification Model 4: HP filter λ parameter = 100; large post-filter transformation (to the sample mean);

non-linear model specification Model 5: HP filter λ parameter = 1600; small post-filter transformation (to the origin);

linear model specification Model 6: HP filter λ parameter = 1600; small post-filter transformation (to the origin); non-

linear model specification

Model 7: HP filter parameter = 1600; large post-filter transformation (to the sample mean); linear model specification

Model 8: HP filter λ parameter = 1600; large post-filter transformation (to the sample

mean); non-linear model specification The coefficients of each regression models were estimated using the maximum likelihood approach as implemented in the STATA software package. Standard statistical tests were then used to assess the statistical validity (well-behaved equation with positive parameter estimates) goodness-of-fit (linear and non-linear R2 values, t and F values, log-likelihood ratios) and overall quality of each estimated regression equation. Calculation of the variabilities For each cost account, the regression equation that best met the statistical tests was selected as the best estimate of the relationship between the regulatory cost and its causal traffic volume-related variables. The results suggested that for most cost accounts Model 6, the regression model based on an HP filter λ parameter of 1600, with only a small post-filter transformation, and using a non-linear model specification, produced the best statistical fits. Therefore, for all cost accounts, the variability factor associated with a specific traffic volume-related variable, or the proportion of the regulatory cost that varies with changes in each causal variable, were estimated from the regression parameters derived from the Model 6 regression equation. Appendix II presents the mathematical derivations used to estimate variabilities from the estimated equations.

- 11 -


Results for CN The analysis showed that for most cost accounts the relationship between cost and traffic volume is non-linear, as the non-linear models showed superior statistical properties (positive signs for all coefficients and higher log-likelihood ratios) compared to the linear models. This is a significant improvement over regression analyses performed between 1992 and 1997, where only linear relationships were analyzed and regression models of questionable validity were used to estimate the existing variabilities. A total of 74 cost accounts were analyzed. Several of the estimated non-linear regression models (35) showed high goodness of fit (non-linear R2 values greater than 70 percent) and most of these allowed intuitively plausible variability estimates to be derived. Other regression models (22) showed only moderate goodness of fit (non-linear R2 values between 50 percent and 70 percent), but most also produced intuitively plausible variability estimates which may be deemed acceptable considering the limitations of the data used in the analysis. A number of accounts (17) showed low goodness-of-fit (non-linear R2 values below 50 percent), but even some these revealed variability estimates that seemed intuitively implausible. However, the regression equations for 16 accounts resulted in variability estimates that seemed implausible (too high or too low) based on staff’s knowledge of railway operations. Staff identified the following reasons for the unexpected variability estimates for these accounts:

• The cost activity may have been too broadly defined in 1969 and may contain too many currently unrelated sub-activities which are affected differently by changes in the causal traffic volume-related variables;

• The traffic volume-related variables used in the analysis may no longer appreciably influence cost changes in the account due to technological or operational changes since 1969, and different or additional causal variables need to be identified for the account; and

• Possible errors in the data. On September 11, 2013, Agency staff sent CN the regression analysis results and variability estimates, along with a detailed discussion of each account for which the variability was considered to be too high or too low and the possible reasons for that result. The report included a recommendation to CN for acceptance of the estimated variabilities for 58 accounts, and an invitation to meet with Agency staff to find solutions for the 16 cost accounts for which the variabilities seemed too high or too low. Results for CP The analysis showed that, similar to the results for CN, for most CP cost accounts the relationship between cost and traffic volume is non-linear. This is a significant improvement over all previous regression analyses performed to date, where only linear relationships were

- 12 -


analyzed and regression models of questionable validity were used to estimate the existing variabilities. A total of 85 cost accounts were analyzed. The best-fit regression models (Model 6) for 26 cost accounts showed high goodness of fit, from most of which could be derived intuitively plausible variability estimates; the regression models for 36 accounts showed only moderate goodness of fit, but still produced for most accounts intuitively plausible variability estimates, which may be deemed acceptable considering the limitations of the data used in the analysis; and, the regression equations for 30 accounts showed low goodness of fit, some of which, nevertheless produced intuitively plausible reliability estimates. The variability estimates for 27 of the 85 cost accounts seemed too high or too low based on staff’s knowledge of railway operations, for similar reasons as described for CN. On December 3, 2013, Agency staff sent CP the regression analysis results and variability estimates, along with a detailed discussion of each account for which the variability was considered to be too high or too low and the possible reasons for that result. The report included a recommendation to CP for acceptance of the estimated variabilities for 58 accounts, and an invitation to meet with Agency staff to find solutions for the 27 cost accounts for which the variabilities seemed too high or too low. Step 3: Exchange of views on first preliminary results CN’s comments On February 10, 2014, CN submitted its comments on Agency staff’s analysis and proposed new variabilities. CN first expressed its agreement that the current variabilities employed by the Agency cannot be relied on for a variety of reasons. However, CN indicates that it sees significant challenges in attempting to update them with statistical analysis because of the long life span of rail assets, the lack of available data points (only two railway companies) and the numerous other factors that have changed in the rail industry over the last 20 years. CN adds that, although it would agree that the overall outcome of the proposed new variabilities makes logical sense from a directional perspective (i.e., that more of the railway companies’ costs are fixed vs. only about 20 percent using the current rates), its review and analysis indicates the statistical methods employed have serious flaws. In a separate document, CN discusses in greater detail the specific issues it has with the technical analysis. Primarily, CN is of the opinion that the post-filter shifting or transformation of data points arbitrarily shifts the intercept (i.e., the amount of fixed costs). CN points out that, although the analysis employed may be the best regression fit, and while it may be the best indicator of the slope, it yields an intercept which is completely dependent on the choice of the size of the

- 13 -


transformation. CN states that unfortunately this is the exact thing the analysis is trying to measure (i.e., the variability or fixed portion). CN states its opinion that the shift is arbitrary, and is only needed to satisfy the requirements of the particular statistical methodology employed. According to CN, if a different methodology had been chosen (for example, one with no filtering, or a filter with a different parameter), a different shift would have been applied to arrive at a different variability. CN states that it cannot accept a variability determined by an arbitrary factor imposed by the statistical method chosen. CN has other objections to the statistical analysis. First, according to CN, it has a fundamental objection to using filters. The objective of the analysis is to study costs and their supposed drivers, and the drivers were chosen because of a belief in their cost driving ability. The fact that both a cost and its driver may be going up or down in the same time frame is in fact the relationship that is the focus of the study. If the trend is filtered out so that only variations from the trend are observed, which may leave essentially what may be only noise in the data, the variability of these “variations from trend” is not the same thing as the variability of the whole expense. CN states that the analysis is therefore ignoring all the amounts in the eliminated trend, amounts that have fixed and variable portions and that should legitimately influence the variability of the expense. CN notes that there are three recommended values for the HP filter λ parameter: 100 for yearly data series, 1600 for quarterly data series, and 14400 for monthly data series. CN suggests that the default λ value of 1600 only means that the most commonly expected data is quarterly, not that 1600 should by default be used on all data series. CN states that it prefers using the smaller recommended parameter of 100 to smooth out the noise, so that the data analyzed is the portion where the noise has been removed, not the portion where the base data is eliminated and the noise is analyzed. CN also notes that, while a smaller shift or transformation generally produces a better fit than a large transformation, it does not overcome the fact that there are still many cases where the larger transformation yields better results, and suggests that this highlights the arbitrary nature of the shift and associated results. Furthermore, CN suggests that there are many results which are completely illogical based on its experience and knowledge of railway operations. CN listed examples of accounts whose estimated variabilities it considered to be illogical. CN concludes its comments with the following:

These are only a few examples (there are more) but the fact that the results seem so obviously illogical calls into question the validity of the entire method. Due to the apparently extreme difference in opinion on the validity of the methods employed, CN urges the Agency to have a third party statistical expert review both party’s [sic] positions and provide an opinion. CN continues to believe that it

- 14 -


may not be possible to statistically determine the variability with any reliability due to the lack of data and preponderance of other variables affecting the relationships. CN believes this exercise and the illogical results are extremely strong evidence to support this claim. Our position on the issue of variability is that it is better suited to a conceptual approach whereby each account is reviewed and determined to be either fixed or variable based on logic, common sense and knowledge of railroad operations. Admittedly, this involves some subjective judgment, however in our experience this has produced better results than a pure statistical analysis. We would be happy to discuss any of our preceding comments further with Agency staff if need be. [Emphasis added]

CP’s comments CP filed no formal written response to the proposed new variability estimates. Instead, it chose to engage with Agency staff, both over the phone and in a day-long conference meeting, to review those cost accounts where unexpectedly high or low variability estimates were obtained, and to discuss potential solutions for updating the variability estimates for those accounts. Agency staff response to CN’s and CP’s comments The need for a time trend filter Staff explained to CN that its objection to the use of a time filter ignores the fact that variability is intended to measure only how changes in the traffic volume variable affect costs at a given point in time. However, observations of the traffic volume and cost changes over time would include major changes, in the rail network, in applications of technology, and in production processes, among others. In the absence of a filter to remove some of these effects, any observed changes in cost cannot be attributed solely to changes in traffic volume, and the observed relationship would not represent the variability. The HP Filter’s λ parameter Staff explained that contrary to CN’s assertion, the lower the λ parameter, the greater the portion of the observed relationship that is attributed to time trend effects. When λ = 0, the entire relationship is attributed to the trend effect, and as λ increases the time trend component of the observations decreases. Therefore, using a λ parameter of 100 instead of 1600, as advocated by CN, would reduce, not increase, the portion of each observations available to identify the unbiased relationship between cost and traffic volume.

- 15 -


The post-filter shift or transformation Staff agrees with CN’s position that the choice of transformation would influence the intercept term in the regression equation and hence the estimated variability. However, despite CN’s suggestion that an infinite number of transformations can be performed on the sample, staff explained that in reality only two transformations are widely accepted to make sense: a small transformation to the origin, and a larger transformation to the original sample space, as previously explained. As indicated previously, it can be mathematically demonstrated that the transformation to the origin is the one that results in the least impact to the estimates of the original parameters as a result of the need to shift the cloud of data into the positive quadrant. Nevertheless, staff accepted that it was reasonable to test both transformations and to choose one based on not just the statistical measures of goodness-of-fit, but in addition on assessment of the two alternative variability estimates based on common sense, experience, and knowledge of railway operations. Step 4: Second preliminary results In part due to CN’s comments, and partly in an attempt to find solutions for those accounts for which the estimated variabilities appeared to be too high or too low based on staff’s experience and knowledge of railway operations, Agency staff undertook an expansion of the analysis to consider additional alternative statistical analysis for each cost account. The major features of the revised analysis are discussed below. Data For each unit cost account, data used in the analysis comprise the time series of dependent and independent variables, over the period 1992 – 2011, as used previously. In addition, CN also provided, for four unit costs accounts, 102-Track and Roadway Investment, 149-Signals Investment, 401-Track & Roadway Maintenance, and 441-Signals Maintenance, panel data sets, each comprising five years of observations of cost and operating statistics for 14 geographic cost centres. The panel data was analyzed separately from the time-series data for these four accounts. For CP, the same time series data was used as in the previous analyses. Data treatments and transformation Two alternative sets of time series data treated to remove inflations effects and to conform with the costing manuals were used:

A. Filtered Data – both dependent and independent variables are further treated to remove the time trend component from the data; and

B. Unfiltered Data – no further treatment of the data The HP filter is applied to de-trend the data in the Alternative A scenario. The HP filter includes a parameter λ, set by the analyst, which controls the shape of the Trend component in the data. If

- 16 -


λ=0, then the Trend component comprises the entire series, and a de-trended series would yield 0 at each observation point. As λ increases, the Trend component is assumed to be less and less of the original observations. Two alternative values of the HP λ parameter were investigated:

(a) λ=100, which assumes that a very large portion of each observation is due to time Trend; and;

(b) λ=1600, which assumes a much smaller time trend component. At λ=1600, the Trend component is very close to a straight line, and further increases in λ have very little effect on the Trend component or the estimated variabilities. Further tests were conducted on selected cost accounts with λ=10,000 and λ=50,000 to assess the influence of the choice of the λ value on the resulting variabilities. The results indicate that λ values greater than 1600 have no meaningful effect on the estimated variabilities; the final regression equations and estimated variabilities with those two extreme values of λ remained almost identical to the results with λ=1600. After the trend component is removed from the dependent or independent variable, the remaining de-trended observations may include both positive and negative values. As negative values are not possible in a log-likelihood estimation methodology, a simple transformation (that is, a single constant added to each observation) is applied to the de-trended observations to eliminate the negative values. Such transformation is necessary only when the data is filtered. Two types of transformations were investigated:

(a) small transformation – a constant large enough to convert the highest negative value to a positive value is added to each de-trended observation; this has the effect of shifting the entire set of observations above and close to the origin; and

(b) large transformation – a constant equal to the mean value of the original sample is added to each de-trended observation; this has the effect of shifting the entire set of observations to around the mean of the raw observations.

Functional relationships For both Alternative A (filtered data) and Alternative B (unfiltered data) analysis scenarios, the following functional relationship was examined for each unit cost account:

y = f(x) where: y is the dependent variable (cost) and x is the set of explanatory variables. In addition, also for Alternative B analysis scenarios, the following functional relationship was examined for each unit cost account:

- 17 -


y = f(x,T)

where: y is the dependent variable (cost), x is the set of explanatory variables, and T is a simple time trend variable. That is, time is an endogenous variable in the modelled relationship. The following linear and non-linear regression specifications were developed to test the relationship between dependent and independent variables. The independent variables may represent single or multiple causal variables, and may also include the time trend variable. Linear Specification: 𝑦𝑦 = 𝛼𝛼0 + 𝛼𝛼1 ∗ 𝑥𝑥 + 𝜀𝜀

Left Box-Cox Specification: 𝑦𝑦𝜆𝜆−1𝜆𝜆

= 𝛼𝛼0 + 𝛼𝛼1 ∗ 𝑥𝑥 + 𝜀𝜀

Right Box-Cox Specification: 𝑦𝑦 = 𝛼𝛼0 + 𝛼𝛼1 ∗ �𝑥𝑥𝜆𝜆−1𝜆𝜆�+ 𝜀𝜀

Lambda Box-Cox Specification: 𝑦𝑦𝜆𝜆−1𝜆𝜆

= 𝛼𝛼0 + 𝛼𝛼1 ∗ �𝑥𝑥𝜆𝜆−1𝜆𝜆�+ 𝜀𝜀

Full (Theta) Box-Cox Specification: 𝑦𝑦𝜃𝜃−1𝜃𝜃

= 𝛼𝛼0 + 𝛼𝛼1 ∗ �𝑥𝑥𝜆𝜆−1𝜆𝜆� + 𝜀𝜀

Supplementary analysis A number of cost accounts revealed 0 percent or even negative variabilities with most or all of the 30 regression models tested, or revealed variabilities that were considered to be highly improbable. In many of these cases convergence of the likelihood function was not achieved. This may reflect quality problems in the data, or improper specification of the dependent and independent variables. Staff knowledge of railway operations suggest that many of these accounts should have non-zero variabilities, as there is reason to expect that over the long term the costs arising from the activities represented in these accounts would vary with changes in traffic levels. For these accounts another set of 30 regression models were investigated using total cost (as opposed to separate labour and material+other costs) as the dependent variable. In some cases, specifically for Accounts 441, 501, and 537, the total cost models revealed more reasonable variabilities than the separate cost models, and those variables are recommended for use in unit cost development.

- 18 -


For four accounts where it was possible to match operating statistics and cost over geographic areas less than the total system, specifically Accounts 102, 149, 401, and 441, CN provided cross-sectional data comprising five years of observations of costs and operating statistics over 15 regions, a total of 75 observations for each account. No filtering of time trend or resulting data transformation was required for this cross-sectional data. The five regression specifications were investigated for each account using this data. Variability results considered to be more consistent with expectations were obtained for Accounts, 102 and 149, and are recommended for use in unit cost development. Summary of Regression Analysis Models For all unit costs accounts except for the four accounts for which panel data was provided, a total of 30 regression analysis models were investigated for each account. For the four accounts with panel data, an additional five models representing use of the panel data set, and resulting in a total of 35 regression models each, were investigated. A summary of the 30 regression analysis models investigated for each account is presented below.

Model 01: Filtered time series data; HP filter λ parameter = 100; small post-filter transformation (to the origin); linear model specification

Model 02: Filtered time series data; HP filter λ parameter = 100; small post-filter

transformation (to the origin); left Box-Cox model specification Model 03: Filtered time series data; HP filter λ parameter = 100; small post-filter

transformation (to the origin); right Box-Cox model specification Model 04: Filtered time series data; HP filter λ parameter = 100; small post-filter

transformation (to the origin); lambda Box-Cox model specification Model 05: Filtered time series data; HP filter λ parameter = 100; small post-filter

transformation (to the origin); full Box-Cox model specification Model 06: Filtered time series data; HP filter λ parameter = 100; large post-filter

transformation (to the sample mean); linear model specification Model 07: Filtered time series data; HP filter λ parameter = 100; large post-filter

transformation (to the sample mean); left Box-Cox model specification Model 08: Filtered time series data; HP filter λ parameter = 100; large post-filter

transformation (to the sample mean); right Box-Cox model specification Model 09: Filtered time series data; HP filter λ parameter = 100; large post-filter

transformation (to the sample mean); lambda Box-Cox model specification

- 19 -


Model 10: Filtered time series data; HP filter λ parameter = 100; large post-filter

transformation (to the sample mean); full Box-Cox model specification Model 11: Filtered time series data; HP filter λ parameter = 1600; small post-filter

transformation (to the origin); linear model specification

Model 12: Filtered time series data; HP filter λ parameter = 1600; small post-filter transformation (to the origin); left Box-Cox model specification

Model 13: Filtered time series data; HP filter λ parameter = 1600; small post-filter

transformation (to the origin); right Box-Cox model specification Model 14: Filtered time series data; HP filter λ parameter = 1600; small post-filter

transformation (to the origin); lambda Box-Cox model specification Model 15: Filtered time series data; HP filter λ parameter = 1600; small post-filter

transformation (to the origin); full Box-Cox model specification Model 16: Filtered time series data; HP filter λ parameter = 1600; large post-filter

transformation (to the sample mean); linear model specification Model 17: Filtered time series data; HP filter λ parameter = 1600; large post-filter

transformation (to the sample mean); left Box-Cox model specification Model 18: Filtered time series data; HP filter λ parameter = 1600; large post-filter

transformation (to the sample mean); right Box-Cox model specification Model 19: Filtered time series data; HP filter λ parameter = 1600; large post-filter

transformation (to the sample mean); lambda Box-Cox model specification Model 20: Filtered time series data; HP filter λ parameter = 1600; large post-filter

transformation (to the sample mean); full Box-Cox model specification

Model 21: Unfiltered time series data; no data transformation; endogenous time trend variable in equation; linear model specification

Model 22: Unfiltered time series data; no data transformation; endogenous time trend

variable in equation; left Box-Cox model specification

Model 23: Unfiltered time series data; no data transformation; endogenous time trend variable in equation; right Box-Cox model specification

- 20 -


Model 24: Unfiltered time series data; no data transformation; endogenous time trend variable in equation; lambda Box-Cox model specification

Model 25: Unfiltered time series data; no data transformation; endogenous time trend

variable in equation; full Box-Cox model specification

Model 26: Unfiltered time series data; no data transformation; no endogenous time trend variable in equation; linear model specification

Model 27: Unfiltered time series data; no data transformation; no endogenous time trend

variable in equation; left Box-Cox model specification

Model 28: Unfiltered time series data; no data transformation; no endogenous time trend variable in equation; right Box-Cox model specification

Model 29: Unfiltered time series data; no data transformation; no endogenous time trend

variable in equation; lambda Box-Cox model specification

Model 30: Unfiltered time series data; no data transformation; no endogenous time trend variable in equation; full Box-Cox model specification

For the four accounts for which panel data was available, the following models were analyzed:

Model 31: Panel data set; no time trend filters; linear model specification Model 32: Panel data set; no time trend filters; left Box-Cox model specification Model 33: Panel data set; no time trend filters; right Box-Cox model specification Model 34: Panel data set; no time trend filters; lambda Box-Cox model specification Model 35: Panel data set; no time trend filters; full Box-Cox model specification

The coefficients of each regression models were estimated using the maximum likelihood approach as implemented in the STATA software package. Standard statistical tests were then used to assess the statistical validity (well-behaved equation with positive parameter estimates) goodness-of-fit (linear and non-linear R2 values, t and F values, log-likelihood ratios) and overall quality of each estimated regression equation. Calculation of the variabilities For each cost account, the variability factors associated with a specific traffic volume-related variable, or the proportion of the regulatory cost that varies with changes in each causal variable, were estimated from the regression parameters derived from each of 30 regression models. Appendix II presents the mathematical derivations used to estimate variabilities from the estimated equations.

- 21 -


Analysis results In this revised approach, no attempt was made to pick a single model that gave the best results for all accounts so that the estimated variabilities for all accounts would be derived from the regression equations based on that single model. Instead, each cost account was reviewed independently, and all 30 or more models were examined to identify the model that best represented the relationship, based on statistical goodness of fit measures in the underlying equation as well as the reasonableness of the derived variability based on common sense, experience and knowledge of railway operations. Of the 84 CN cost accounts analyzed, only five accounts did not provide reasonable variability estimates from at least one of the modelled relationships. It was concluded that these five accounts provided unexpected variability estimates because of either an over-broad specification of the cost activity or an imperfect selection of causal traffic volume-related variables. For four of these accounts, the variability estimated using CP’s data, as CP also had the same cost activities with the same causal variables, were considered the most reasonable empirically determined estimates to use for those CN cost accounts. For the remaining one of the five CN accounts with unacceptable variability estimates, the estimated variability of a cost activity judged to be similar to that account was considered the best empirically-determined estimate of the possible relationship. Of the 85 CP cost accounts analyzed, 34 accounts did not provide reasonable variability estimates from at least one of the modelled relationships. Staff identified as a possible cause for the relatively larger number of accounts that showed unexpected variability, in addition to either an over-broad specification of the cost activity or an imperfect selection of causal traffic volume-related variables, data collection problems experienced by CP. In 2008, CP discovered some irregularities in the assignment of costs to the specified UCA accounts, and undertook a re-mapping exercise to correct the anomalies it could identify in its financial and operational data in the period 2006-2010. However, the re-mapping did not extend to historical data earlier than 2006. There is a possibility that data for years 1992-2005 still contains anomalies that may distort the true relationships between CP’s cost and traffic volume-related variables. As CN also has in many cases the same cost activities with the same causal variables, the relationships developed using CN data were considered the most reasonable empirically determined estimates to use for those CP cost accounts. CP agreed to the questionable variabilities for the 25 accounts being replaced with more reasonable variabilities estimated using CN data. Step 5: Exchange of views on second preliminary results On November 6, 2014, Agency staff sent CN its recommended variabilities for each CN cost account and on November 7, 2014, Agency staff sent CP its recommended variabilities for each CP cost account.

- 22 -


The package sent to both CN and CP included a document explaining the analysis methodologies employed to derive the recommended variabilities, and another document showing, for each account, the data treatment, data transformation, and regression model specification that produced the recommended variability estimate. It was explained that, for most accounts, the variability estimate derived from the model with the best statistical properties and goodness-of-fit (highest R2 value) was selected, though in some cases judgement was applied to select the “most reasonable” variability estimate and not necessarily the one with the highest R2. The letter included in the package pointed out that though there may be some specific accounts for which CN or CP may consider the estimated variabilities to be counter-intuitive, Agency staff is of the opinion that in the aggregate the recommended variabilities are demonstrably superior to the existing fixed variabilities, as they are derived from properly specified regression equations with well-behaved statistical properties. The letter noted, however, that the recommendations represented staff’s view of an interim approach to developing, in the long term, a methodology for estimating variabilities that is statistically sound and results in intuitively plausible variability estimates for all accounts, and that such a methodology would be expected to include changes in the dependent and independent variables specified for some cost accounts, refinements in the statistical analysis methodologies employed, and use of more appropriate data. The letter also noted that, in the short term, staff’s view is that the analysis presented had exhausted all possible analytical options to identify reasonable variabilities for the cost accounts, given the existing structure of cost accounts and available data. Finally, the letter advised CN and CP that staff intended to present, by November 28, 2014, the recommended set of variabilities to the Agency for approval for use in the development of the CN 2012 and 2013 unit costs. CN and CP were invited to review the attached set of variabilities and identify any specific accounts for which they consider the recommended variability to be unrepresentative of the expected relationship, and to provide objective reasons for that consideration. Staff offered to discuss alternative options for the accounts to which CN or CP object, and to agree on solutions for such accounts in the set of variabilities to be presented to the Agency for approval. CN’s comments on the recommended set of variabilities On November 21, 2014, CN provided its comments on the revised variabilities recommended by Agency staff. CN indicates that its concerns with Agency staff’s recommended variabilities have already been communicated in its comments on the initial variability estimates, and that without reiterating the entirety of that communication, the following recaps some of the salient points made at the time.

- 23 -


CN sees, in general, very significant challenges in attempting to determine variabilities with statistical analysis because of the extremely long life span of rail assets, the lack of available data points (only 2 railway companies) and the numerous other factors that have changed in the rail industry over the last 20 years. For example, CN is of the opinion that, more specifically, the statistical methods employed by Agency staff are invalid for the purposes of determining the variability level, and CN had recommended that Agency staff consult a third party statistical expert if needed to confirm CN’s conclusions. CN adds that, in addition to its own statistical analysis which points to fundamental flaws in the methods employed, it is of the opinion that the results speak for themselves in that they produce conclusions that are clearly wrong for a significant number of accounts. In its view, this is obvious evidence that supports its conclusions that the methods are flawed, and the fact that Agency staff had to employ different methods to different accounts or arbitrarily choose different constants to avoid illogical results is further evidence of the validity of its opinion. CN reiterates that its view on the issue of variability is that it is better suited to a conceptual approach whereby each account is reviewed and determined to be either fixed or variable based on logic, common sense and general knowledge of railroad operations. CN suggests that it uses this approach internally, and in its experience this has produced better results than a pure statistical analysis. CP’s comments on the recommended set of variabilities In a letter dated November 21, 2014, CP indicates its concerns with the final recommended variabilities, which relate to both the approach and the results. CP acknowledges that there have been several challenges to the process of updating the estimates of the variable portions of activity costs, driven primarily by changes in the way CP collects operational and financial data. However, CP suggests that there are several fundamental issues with the methodologies employed in calculating the proposed variabilities. First, CP states that the review of variabilities was undertaken before a review of the unit cost models themselves, and therefore the empirical work was done on models that were mis-specified. The grouping of railway activities into cost accounts and the choice of traffic volume related variables to explain the cost changes should all have been reviewed prior to the calculation of variabilities. In CP’s opinion without this review, empirical relationships may be impossible to establish as the causal relationship would be looked for in the wrong place, or the cost accounts may involve aggregations of activities with dissimilar relationships to the selected causal variables.

- 24 -


Second, CP states that the variabilities were calculated based only on empirical data, which all parties agree has significant shortcomings. CP contends that Agency staff’s strategy for overcoming these shortcomings was, instead of introducing logic or redesigning the cost accounts and the causal variables, rather to employ an increasingly complex set of mathematical techniques, which served only to obfuscate empirical challenges and not address them. CP’s position is that the variabilities should not be amended until there has been a proper review of the aggregation of activities into cost accounts and of the traffic volume-related variables thought to influence each cost account. CP suggests proceeding in any other way is counterproductive and will result in variabilities being calculated off of cost models that are fundamentally mis-specified. CP notes that the importance of an accurate estimate of variability should not be understated, because issuing an interim solution could have a long-term impact on the railway companies. It points out that the roughly 20 percent contribution to fixed costs has been echoed repeatedly for over two decades in the industry, by both railway companies and shippers alike, during political discussions and proceedings such as Final Offer Arbitration under the CTA. Hence, creating a new cost demarcation between variable and fixed costs, which drastically departs from the traditional average of 80/20 to 60/40, as the proposed new variabilities would do, would set a new long-term precedent in the industry. CP points out that the CTA specifies in multiple instances that cost determinations must compensate railway companies for the variable costs incurred, which leaves the contribution to fixed cost to be assessed based on unknown considerations. CP is concerned that the proposed variabilities could lead to railway companies being mandated to charge non-compensatory rates, causing irreparable harm to the railway companies and their infrastructure. After a conference call with Agency staff to discuss its concerns with the recommended variabilities for a number of cost accounts, CP followed up with a formal letter to the Agency dated December 10, 2014, reinforcing its primary concerns. CP states in the letter that the proposed new variabilities, which would change the demarcation between variable and fixed costs from 80/20 to around 60/40 could be used as an unjustifiable mechanism to reduce railway rates since a full contribution to fixed costs is not always applied. CP notes that examples of this were seen when the Agency for several years applied a 7.5% contribution to fixed costs for regulated interswitching rates and a 3% contribution on the hopper car maintenance revision in 2008, while it had always been accepted that the ongoing contribution was close to 20 percent.

- 25 -


CP suggests that this demonstrates to what extent contribution determinations can be used inconsistently, for various underlying reasons, where the exercise itself should be straightforward, as an evidence-based recognition of the railway’s right to recover system total fixed costs. In summary, CP requested that the proposed new variabilities not be implemented until the cost relationships are fully reviewed, and pledged to continue to work with the Agency on such a review. Agency staff provide this report to the Agency with the recommendation that these findings be accepted.

Prepared by Industry Determinations and Analysis Directorate Industry Regulation and Determinations Branch Principal Authors: Koby Kobia, PhD, Transportation Economics, Chief Economist Hakan Andic, PhD, Econometrics, Economist Ryan Dallaway, M.A., Economics, Senior Economist Tom O'Hearn, M.A., Economics, Senior Economist Rakesh Manhas, M.A., Economics, Senior Economist Wilfred Wong, M.A., Economics, Senior Economist Omar Abdi, M.A., Economics, Economist

- 26 -


APPENDIX I: THE POST-FILTER TRANSFORMATION As a result of removing trends from time series data, the filtered database included negative numbers. A non-linear model does not allow negative numbers powered to a small factor because a negative number cannot be powered with a factor that is less than one. For this reason, the reference frame where all the data are expressed must be rescaled to the positive quadrant. The purpose of this Appendix is to show that a small transformation, one where the filtered observations are shifted so that the smallest observation is just above the origin, point O(0,0), is indeed the one that will have the least impact on the estimate of the original parameters that would have prevailed if no such rescaling was made. In other words, the proof will show that the shift to the origin is the most efficient rescaling in this context. The proof is provided for the most general case, which is the full Box-Cox model. Monotonic Transform of a Random Variable

The following property makes it much easier to write the log Likelihood of a Full Box-Cox.

Let Y be a continuous random variable with a probability density function f(y).

Let Z = g(Y) be a monotonic differentiable transform of Y as is represented by the following graphic:

- 27 -


The points (z0<Z<z1) and (y0<Y<y1) represent the same event viewed on the Z and Y axes. Hence, their probabilities are equal so that P(z0<Z<z1) = P(y0<Y<y1).

Bearing this in mind, let’s develop the probability of the Z event:

𝑃𝑃(𝑍𝑍0 < 𝑍𝑍 < 𝑍𝑍1) = � 𝑓𝑓𝑍𝑍(𝑧𝑧)𝑑𝑑𝑧𝑧

𝑍𝑍1

𝑍𝑍0

= � 𝑓𝑓𝑍𝑍(𝑧𝑧)𝑑𝑑𝑧𝑧

𝑔𝑔(𝑌𝑌1)

𝑔𝑔(𝑌𝑌0)

= � �𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑑𝑑𝑧𝑧𝑑𝑑𝑌𝑌�

𝑌𝑌1

𝑌𝑌0

𝑑𝑑𝑌𝑌

However, 𝑃𝑃(𝑍𝑍0 < 𝑍𝑍 < 𝑍𝑍1) = 𝑃𝑃(𝑌𝑌0 < 𝑌𝑌 < 𝑌𝑌1) = ∫ 𝑓𝑓𝑌𝑌(𝑌𝑌)𝑑𝑑𝑌𝑌𝑌𝑌1𝑌𝑌0

Hence we have:

� �𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑑𝑑𝑧𝑧𝑑𝑑𝑌𝑌�

𝑌𝑌1

𝑌𝑌0

𝑑𝑑𝑌𝑌 = � 𝑓𝑓𝑌𝑌(𝑌𝑌)𝑑𝑑𝑌𝑌

𝑌𝑌1

𝑌𝑌0

Therefore, by identification the density of the untransformed variable Y is given by:

𝑓𝑓𝑌𝑌(𝑌𝑌) = 𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑑𝑑𝑧𝑧𝑑𝑑𝑌𝑌

= 𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑔𝑔′(𝑌𝑌)

Now let’s we apply this property to the box-cox transformation of Y given by:

𝑍𝑍 = 𝑔𝑔(𝑌𝑌) =�𝑌𝑌𝜃𝜃∗ − 1�

𝜃𝜃∗

The density of Y can be written directly as a function of the density of Z. Since Z follows a normal distribution we thus have:

𝑓𝑓𝑌𝑌(𝑌𝑌) = 𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑑𝑑𝑧𝑧𝑑𝑑𝑌𝑌

= 𝑓𝑓𝑍𝑍(𝑔𝑔(𝑌𝑌))𝑔𝑔′(𝑌𝑌) =1

𝜎𝜎√2𝜋𝜋𝑒𝑒𝑥𝑥𝑒𝑒 �

−12𝜎𝜎2

��𝑌𝑌𝜃𝜃 − 1�

𝜃𝜃− ℎ(𝑋𝑋)�

2

� 𝑌𝑌𝜃𝜃−1

- 28 -


Where: ℎ(𝑋𝑋) = 𝑏𝑏0 + 𝑏𝑏1�𝑥𝑥1𝜆𝜆−1�

𝜆𝜆+ 𝑏𝑏2

�𝑥𝑥2𝜆𝜆−1�𝜆𝜆

+ ⋯+ 𝑏𝑏𝑝𝑝�𝑥𝑥𝑝𝑝𝜆𝜆−1�

𝜆𝜆= 𝑏𝑏0 + 𝐾𝐾(𝑋𝑋)

The Log Likelihood of a Box-Cox Model

As we know the density of the untransformed variable Y, we can write the log likelihood of the model:

ln 𝐿𝐿(𝜃𝜃, 𝜆𝜆, 𝑏𝑏0, 𝑏𝑏1, … , 𝑏𝑏𝑝𝑝)

= � ln�𝑓𝑓𝑌𝑌(𝑦𝑦𝑖𝑖)�𝑛𝑛

𝑖𝑖=1

= � ln�1

𝜎𝜎√2𝜋𝜋𝑒𝑒𝑥𝑥𝑒𝑒 �

−12𝜎𝜎2

��𝑦𝑦𝑖𝑖𝜃𝜃 − 1�

𝜃𝜃−�𝑏𝑏𝑗𝑗

�𝑥𝑥𝑖𝑖𝑗𝑗𝜆𝜆 − 1�𝜆𝜆

𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0�

2

�𝑦𝑦𝑖𝑖𝜃𝜃−1�𝑛𝑛

𝑖𝑖=1

= 𝑛𝑛 𝑙𝑙𝑛𝑛 �1

𝜎𝜎√2𝜋𝜋� −

12𝜎𝜎2

��𝑦𝑦𝑖𝑖𝜃𝜃 − 1�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0�

2

+ �𝑙𝑙𝑛𝑛�𝑦𝑦𝑖𝑖𝜃𝜃−1�𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

The estimated values of parameters 𝜃𝜃, 𝜆𝜆, 𝑏𝑏0, 𝑏𝑏1, … , 𝑏𝑏𝑝𝑝 will be at such a point where ln L will be maximized.

The first order conditions for a global maximum are as follows:

⎩⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎧𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝜃𝜃

= 0 =−12𝜎𝜎2

�2��𝑦𝑦𝑖𝑖𝜃𝜃 − 1�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0�𝑦𝑦𝑖𝑖𝜃𝜃𝑙𝑙𝑛𝑛(𝑦𝑦𝑖𝑖)𝜃𝜃 − �𝑦𝑦𝑖𝑖𝜃𝜃 − 1�

𝜃𝜃2+ �𝑙𝑙𝑛𝑛(𝑦𝑦𝑖𝑖)

𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝜆𝜆

= 0 =−12𝜎𝜎2

� 2��𝑦𝑦𝑖𝑖𝜃𝜃 − 1�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)�𝑏𝑏𝑗𝑗𝑥𝑥𝑖𝑖𝑗𝑗𝜆𝜆 𝑙𝑙𝑛𝑛�𝑥𝑥𝑖𝑖𝑗𝑗�𝜆𝜆 − �𝑥𝑥𝑖𝑖𝑗𝑗𝜆𝜆 − 1�

𝜆𝜆2

𝑝𝑝

𝑗𝑗=1

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝑏𝑏𝑗𝑗

= 0 =−12𝜎𝜎2




𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)�𝑥𝑥𝑖𝑖𝑗𝑗𝜆𝜆 − 1�

𝜆𝜆

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝑏𝑏0

= 0 =−12𝜎𝜎2




𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)𝑛𝑛

𝑖𝑖=1

- 29 -


The last equation tells us that the constant can be expressed as a function of all other parameters:

𝑏𝑏0 =1𝑛𝑛��

�𝑦𝑦𝑖𝑖𝜃𝜃 − 1�𝜃𝜃

−�𝑏𝑏𝑗𝑗�𝑥𝑥𝑖𝑖𝑗𝑗𝜆𝜆 − 1�

𝜆𝜆

𝑝𝑝

𝑗𝑗=1

� =𝑛𝑛

𝑖𝑖=1

1𝑛𝑛��

�𝑦𝑦𝑖𝑖𝜃𝜃 − 1�𝜃𝜃

� −��1𝑛𝑛�𝑏𝑏𝑗𝑗


𝑛𝑛

𝑖𝑖=1

�𝑝𝑝

𝑗𝑗=1

𝑛𝑛

𝑖𝑖=1

The Impact of a Change in the Reference Frame Now we can analyze the effect of a change in the reference frame. Such a change happens when all Y are increased by the same number s≥0 and all X are increased by the same number t≥0.

Formally, this shift in the reference frame is noted:

(𝑦𝑦 + 𝑠𝑠 ; 𝑥𝑥 + 𝑡𝑡)

To see the impact of such a shift, we must look at its effect on the log likelihood function that is solely and directly used to estimate the parameters. The shifted log likelihood will be given by the following expression:

ln 𝐿𝐿(𝜃𝜃, 𝜆𝜆, 𝑏𝑏0, 𝑏𝑏1, … , 𝑏𝑏𝑝𝑝)

= 𝑛𝑛 𝑙𝑙𝑛𝑛 �1

𝜎𝜎√2𝜋𝜋�

−1

2𝜎𝜎2��

�(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃 − 1�𝜃𝜃

−�𝑏𝑏𝑗𝑗��𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�

𝜆𝜆− 1�

𝜆𝜆

𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0�

2𝑛𝑛

𝑖𝑖=1

+ �𝑙𝑙𝑛𝑛�(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃−1�𝑛𝑛

𝑖𝑖=1

This shift will result in a deviation of estimated parameters since the first order condition will now be:

⎩⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎧𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝜃𝜃 = 0 =

−12𝜎𝜎2� 2�

�(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃 − 1�𝜃𝜃 −�𝑏𝑏𝑗𝑗

��𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�𝜆𝜆 − 1�𝜆𝜆

𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0�(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃𝑙𝑙𝑛𝑛(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃 − �(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃 − 1�

𝜃𝜃2 + �𝑙𝑙𝑛𝑛(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝜆𝜆 = 0 =

−12𝜎𝜎2� 2�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)�𝑏𝑏𝑗𝑗�𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�𝜆𝜆𝑙𝑙𝑛𝑛�𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�𝜆𝜆 − ��𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�𝜆𝜆 − 1�

𝜆𝜆2

𝑝𝑝

𝑗𝑗=1

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝑏𝑏𝑗𝑗

= 0 =−12𝜎𝜎2� 2�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)��𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�𝜆𝜆 − 1�

𝜆𝜆

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝑙𝑙𝑛𝑛𝐿𝐿𝜕𝜕𝑏𝑏0

= 0 =−12𝜎𝜎2�2�



𝑝𝑝

𝑗𝑗=1

− 𝑏𝑏0� (−1)𝑛𝑛

𝑖𝑖=1

- 30 -


The last equation tells us that the constant can be expressed as a function of all other parameters:

𝑏𝑏0(𝑠𝑠, 𝑡𝑡) = 1𝑛𝑛∑ ��(𝑦𝑦𝑖𝑖+𝑠𝑠)𝜃𝜃−1�

𝜃𝜃− ∑ 𝑏𝑏𝑗𝑗

��𝑥𝑥𝑖𝑖𝑖𝑖+𝑡𝑡�𝜆𝜆−1�

𝜆𝜆𝑝𝑝𝑗𝑗=1 � = 𝑛𝑛

𝑖𝑖=11𝑛𝑛∑ ��(𝑦𝑦𝑖𝑖+𝑠𝑠)𝜃𝜃−1�

𝜃𝜃� − ∑ �1

𝑛𝑛∑ 𝑏𝑏𝑗𝑗

��𝑥𝑥𝑖𝑖𝑖𝑖+𝑡𝑡�𝜆𝜆−1�

𝜆𝜆𝑛𝑛𝑖𝑖=1 �𝑝𝑝

𝑗𝑗=1𝑛𝑛𝑖𝑖=1

This demonstrates that the impact of the change in the reference frame is to change the optimal values of our parameters, including b0. Therefore, the size of the change in reference frame will have a direct impact on the calculation of our variabilities.

The Optimal Shift in the Frame of Reference

Because the constant depends on all other parameters (including s and t) and is paramount in calculating the level of fixed cost, we must choose s≥0 and t≥0 so that it has a minimum impact on the estimated parameters.

The impact on the constant can be expressed by the following distance function:

𝑑𝑑(𝑠𝑠, 𝑡𝑡) = 𝐷𝐷(𝑏𝑏0(𝑠𝑠, 𝑡𝑡), 𝑏𝑏0) = |𝑏𝑏0(𝑠𝑠, 𝑡𝑡) − 𝑏𝑏0|

Our objective is to choose s and t that minimize the above distance function.

Now, it is obvious that d(0,0)=0. So s*=0 and t*=0 are optimal solutions to the problem. However, since we cannot compute our model when s and t are equal to zero, we need s>0 and t>0. (This is called an interior solution). The following property is well known in maths and provides the necessary conditions for a local interior solution:

If we note ▼d(s,t) the vector of partial derivatives of d(s,t) (also called the gradient vector);

(s*>0;t*>0) is a solution to {min(d(s,t))} ==> ▼d(s,t)=0 at (s*,t*)

Corollary: If ▼d(s*,t*) ≠ 0 then it is impossible that (s*,t*) is a local solution.

Now, let’s check if there could be any (s*,t*) so that ▼d(s*,t*)=0

∇𝑑𝑑(𝑠𝑠, 𝑡𝑡) = �𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝑠𝑠

,𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝑡𝑡

,𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝜃𝜃

,𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝜆𝜆

,𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝑏𝑏𝑗𝑗

�

For ▼d(s*,t*)=0 to be true, all these elements have to be equal to zero at (s*>0,t*>0)

However,

- 31 -


�𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝑠𝑠


� = �1𝑛𝑛�(𝑦𝑦𝑖𝑖 + 𝑠𝑠)𝜃𝜃−1 ,−��

1𝑛𝑛�𝑏𝑏𝑗𝑗�𝑥𝑥𝑖𝑖𝑗𝑗 + 𝑡𝑡�

𝜆𝜆−1𝑛𝑛

𝑖𝑖=1

�𝑝𝑝

𝑗𝑗=1

𝑛𝑛

𝑖𝑖=1

�

Hence,

�𝑦𝑦𝑖𝑖 > 0, 𝑥𝑥𝑖𝑖𝑗𝑗 > 0, 𝑠𝑠 > 0, 𝑡𝑡 > 0� => �𝜕𝜕𝑑𝑑(𝑠𝑠, 𝑡𝑡)𝜕𝜕𝑠𝑠


� ≠ (0,0) => ∇𝑑𝑑(𝑠𝑠, 𝑡𝑡) ≠ (0,0,0,0,0)

Therefore, we have shown that there is no possible interior solution to the minimization problem. Moreover, the gradient vector is never null, and we have d(s,t)>0 if s>0 and t>0,and d(0,0)=0. Hence, any increase in s and t from (0,0) will cause d(s,t) to increase.

We can illustrate this general result with a particular example. If we look closer to the formula that describes the constant b0, we can see that it is the difference between the sums of two kind monotonic transformations:

𝑔𝑔(𝑥𝑥) = �𝑥𝑥𝜃𝜃−1�𝜃𝜃

and 𝐾𝐾(𝑥𝑥) = �𝑥𝑥𝜆𝜆−1�𝜆𝜆

Let’s say b is a constant that is the difference between g(x) and K(x). Now let’s consider an increase from a given point x to a shifted point (x+t):

- 32 -


The initial non-shifted point noted x is shifted by different increasing values (t1<t2<t3). We can see that the difference between g(x) and K(x) will always decrease and will even become negative after a certain level of shifting as it is the case for x+t3. That means the absolute value of the difference between the initial constant and the shifted constant will keep increasing.

As a result, the objective must be to make the values of s and t as close as possible to zero.

Hence, our objective can be expressed in the following optimization problem:

�min𝑠𝑠,𝑡𝑡

(𝑠𝑠, 𝑡𝑡)

𝑦𝑦 + 𝑠𝑠 ≫ 0𝑥𝑥 + 𝑡𝑡 ≫ 0

This optimization problem says that s and t should be minimized under the constraint that the resulting shifted vector of observation y+s and x+t must have positive elements.

- 33 -


The solution to the program is trivial:

(i) For y:

𝑠𝑠∗ = �−𝑚𝑚𝑚𝑚𝑛𝑛{𝑦𝑦𝑖𝑖}𝑖𝑖=1𝑛𝑛 ,𝑚𝑚𝑚𝑚𝑛𝑛{𝑦𝑦𝑖𝑖}𝑖𝑖=1𝑛𝑛 < 0

0,𝑚𝑚𝑚𝑚𝑛𝑛{𝑦𝑦𝑖𝑖}𝑖𝑖=1𝑛𝑛 ≥ 0

(ii) For x:

𝑡𝑡∗ = �−𝑚𝑚𝑚𝑚𝑛𝑛{𝑥𝑥𝑖𝑖}𝑖𝑖=1𝑛𝑛 ,𝑚𝑚𝑚𝑚𝑛𝑛{𝑥𝑥𝑖𝑖}𝑖𝑖=1𝑛𝑛 < 0

0,𝑚𝑚𝑚𝑚𝑛𝑛{𝑥𝑥𝑖𝑖}𝑖𝑖=1𝑛𝑛 ≥ 0

- 34 -


APPENDIX II: CALCULATION OF VARIABILITIES Linear Model For a linear model, the variability for the kth independent variable is estimated as:

𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) = 𝛼𝛼𝑘𝑘∗𝑥𝑥𝑘𝑘��𝑌𝑌�

Where: bk

* is the estimated parameter for the kth causal variable; xk is the kth causal variable evaluated at the mean of the sample of observations; and Y is the regulatory cost evaluated at the mean of the sample of observations. Left Box-Cox Model For the non-linear equation represented by Model 6, the variability for the kth independent variable is estimated as follows: For both the linear and non-linear models, evaluating the variability at the sample means of the observations is based on the statistical property that in a regression model (both linear and non-linear), when a constant term is included the equation the random noise term in the regression equation cancels out and is zero at the sample means. The portion of the cost in that account which is not attributable to any of the explanatory variables is considered to be fixed. Note: Once they have been estimated, the parameters are written with a star— such as 𝑏𝑏0∗. The average residuals still cancels out once the coefficients are estimated so that:

�(𝑌𝑌𝜃𝜃∗ − 1)

𝜃𝜃∗�

��= 𝑏𝑏0∗ + 𝑏𝑏1∗(𝑋𝑋1)�� + ⋯+ 𝑏𝑏𝑝𝑝∗�𝑋𝑋𝑝𝑝��

However, to be more precise, we need to find a way to isolate the expenses (Y). First, for any average, there exists a number (z) so that:

�𝑧𝑧𝜃𝜃∗ − 1�𝜃𝜃∗

= ��𝑍𝑍𝑗𝑗𝜃𝜃

∗− 1�

𝜃𝜃∗

𝑛𝑛

𝑗𝑗=1

= �(𝑍𝑍𝜃𝜃∗ − 1)

𝜃𝜃∗�

��

And of course,

𝑧𝑧 = �𝑍𝑍𝑗𝑗

𝑛𝑛

𝑗𝑗=1

= �̅�𝑍

- 35 -


Therefore, there is a unique combination of observed expense (y) and independent variables (x1, x2,...xp) so that the equation holds without any residuals:

�𝑦𝑦𝜃𝜃∗ − 1�𝜃𝜃∗

= 𝑏𝑏0∗ + 𝑏𝑏1∗𝑥𝑥1 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝 It is then possible to isolate the expenses:

𝑦𝑦 = ��𝑏𝑏0∗ + 𝑏𝑏1∗𝑥𝑥1 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝�𝜃𝜃∗ + 1�1𝜃𝜃∗

This equation can then be simplified such as:

𝑦𝑦 = �𝐶𝐶𝜃𝜃∗ + �𝑏𝑏1∗𝑥𝑥1 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝�𝜃𝜃∗��𝑃𝑃𝑢𝑢𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣𝑢𝑢𝑖𝑖𝑣𝑣𝑣𝑣𝑣𝑣𝑢𝑢 𝑢𝑢𝑒𝑒𝑒𝑒𝑢𝑢𝑒𝑒𝑡𝑡

�

1𝜃𝜃∗

Where 𝐶𝐶𝜃𝜃∗ = (𝑏𝑏0∗)𝜃𝜃∗ + 1 Therefore, as it was the case for the simple non-linear version, there is still a unique combination of observed expense and independent variable where the theoretical equation fits perfectly with observations, and the variability factors have to be evaluated at that perfect combination. Following this property, the contribution for a given kth independent variable to the variable costs will be as follows:

𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣𝑚𝑚𝑏𝑏𝐶𝐶𝑡𝑡𝑚𝑚𝐶𝐶𝑛𝑛(𝑥𝑥𝑘𝑘) = 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘

𝑏𝑏1∗𝑥𝑥1 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝

It can be shown that this contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:

𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣𝑚𝑚𝑏𝑏𝐶𝐶𝑡𝑡𝑚𝑚𝐶𝐶𝑛𝑛(𝑥𝑥𝑘𝑘) = ∈ (𝑥𝑥𝑘𝑘)

∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)

Where ∈ (𝑥𝑥𝑘𝑘) = 𝜕𝜕𝑦𝑦/𝑦𝑦

𝜕𝜕𝑥𝑥𝑘𝑘/𝑥𝑥𝑘𝑘 is the elasticity of expense subject to an increase of 𝑥𝑥𝑘𝑘

This property is exactly the same in a linear model. On the other hand, the fixed costs components will be given by:

𝐶𝐶 = ��𝑏𝑏0∗ + 𝑏𝑏1∗0 + 𝑏𝑏2∗0 + ⋯+ 𝑏𝑏𝑝𝑝∗0�𝜃𝜃∗ + 1�1/𝜃𝜃∗

- 36 -


Therefore, the total variability factor (V) will be given by the following ratio:

𝑉𝑉 = 1 − 𝐶𝐶𝑦𝑦

The variability factor 𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) related to the independent variable 𝑥𝑥𝑘𝑘 will then be the product of the total variability factor and the contribution of that independent variable 𝑥𝑥𝑘𝑘 to variable costs:

𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) = 𝑉𝑉 .𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣𝑚𝑚𝑏𝑏𝐶𝐶𝑡𝑡𝑚𝑚𝐶𝐶𝑛𝑛(𝑥𝑥𝑘𝑘) In case there is only one explanatory variable, V will give the variability related to that variable. Full Box-Cox Model As all the models mentioned above can be expressed as a sub-case of a Full Box-Cox Model, the following calculations can be applied to all of them: The average residuals still cancels out once the coefficients are estimated so that:

�(𝑌𝑌𝜃𝜃∗ − 1)

𝜃𝜃∗�

��= 𝑏𝑏0∗ + 𝑏𝑏1∗ �

�𝑋𝑋1𝜆𝜆∗ − 1�𝜆𝜆∗

��

+ ⋯+ 𝑏𝑏𝑝𝑝∗ ��𝑋𝑋𝑝𝑝𝜆𝜆

∗ − 1�𝜆𝜆∗

��

However, to me more precise, we need to find a way to isolate the expenses (Y). First, for any average, there exists a number (z) so that:

�𝑧𝑧𝜃𝜃∗ − 1�𝜃𝜃∗

= ��𝑍𝑍𝑗𝑗𝜃𝜃

∗− 1�

𝜃𝜃∗

𝑛𝑛

𝑗𝑗=1

= �(𝑍𝑍𝜃𝜃∗ − 1)

𝜃𝜃∗�

��

Therefore, there is a unique combination of observed expense (y) and independent variables (x1, x2,...xp) so that the equation holds without any residuals:

�𝑦𝑦𝜃𝜃∗ − 1�𝜃𝜃∗

= 𝑏𝑏0∗ + 𝑏𝑏1∗�𝑥𝑥1𝜆𝜆

∗ − 1�𝜆𝜆∗

+ 𝑏𝑏2∗�𝑥𝑥2𝜆𝜆

∗ − 1�𝜆𝜆∗

+ ⋯+ 𝑏𝑏𝑝𝑝∗�𝑥𝑥𝑝𝑝𝜆𝜆

∗ − 1�𝜆𝜆∗

It is then possible to isolate the expenses:

𝑦𝑦 = ��𝑏𝑏0∗ + 𝑏𝑏1∗�𝑥𝑥1𝜆𝜆

∗ − 1�𝜆𝜆∗

+ 𝑏𝑏2∗�𝑥𝑥2𝜆𝜆

∗ − 1�𝜆𝜆∗

+ ⋯+ 𝑏𝑏𝑝𝑝∗�𝑥𝑥𝑝𝑝𝜆𝜆

∗ − 1�𝜆𝜆∗

� 𝜃𝜃∗ + 1�

1𝜃𝜃∗

- 37 -


This equation can then be simplified such as:

𝑦𝑦 = �𝐶𝐶𝜃𝜃∗ + �𝑏𝑏1∗𝑥𝑥1𝜆𝜆∗ + 𝑏𝑏2∗𝑥𝑥2𝜆𝜆

∗ + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗�𝜃𝜃∗

𝜆𝜆∗��𝑃𝑃𝑢𝑢𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣𝑢𝑢𝑖𝑖𝑣𝑣𝑣𝑣𝑣𝑣𝑢𝑢 𝑢𝑢𝑒𝑒𝑒𝑒𝑢𝑢𝑒𝑒𝑡𝑡

�

1𝜃𝜃∗

Where 𝐶𝐶𝜃𝜃∗ = �𝑏𝑏0∗ −1𝜆𝜆∗∑ 𝑏𝑏𝑘𝑘∗𝑝𝑝𝑘𝑘=1 � 𝜃𝜃∗ + 1

Therefore, as it was the case for the simple non-linear version, there is still a unique combination of observed expense and independent variable where the theoretical equation fits perfectly with observations, and the variability factors have to be evaluated at that perfect combination. Following this property, the contribution for a given kth independent variable to the variable costs will be as follows:

𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣𝑚𝑚𝑏𝑏𝐶𝐶𝑡𝑡𝑚𝑚𝐶𝐶𝑛𝑛(𝑥𝑥𝑘𝑘) = 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆

∗

𝑏𝑏1∗𝑥𝑥1𝜆𝜆∗ + 𝑏𝑏2∗𝑥𝑥2𝜆𝜆

∗ + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗

It can be shown that this contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:


∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)



This property is exactly the same in a linear model. On the other hand, the fixed costs components will be given by:

𝐶𝐶 = ��𝑏𝑏0∗ + 𝑏𝑏1∗�0𝜆𝜆∗ − 1�

𝜆𝜆∗+ 𝑏𝑏2∗

�0𝜆𝜆∗ − 1�𝜆𝜆∗

+ ⋯+ 𝑏𝑏𝑝𝑝∗�0𝜆𝜆∗ − 1�

𝜆𝜆∗� 𝜃𝜃∗ + 1�

1/𝜃𝜃∗


𝑉𝑉 = 1 − 𝐶𝐶𝑦𝑦

- 38 -


The variability factor 𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) related to the independent variable 𝑥𝑥𝑘𝑘 will then be the product of the total variability factor and the contribution of that independent variable 𝑥𝑥𝑘𝑘 to variable costs:

𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) = 𝑉𝑉 .𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣𝑚𝑚𝑏𝑏𝐶𝐶𝑡𝑡𝑚𝑚𝐶𝐶𝑛𝑛(𝑥𝑥𝑘𝑘) In case there is only one explanatory variable, V will give the variability related to that variable. PROOFS The linear model and the Traffic-Elasticities of Total Cost: The Linear Specification is given by: 𝑦𝑦 = 𝛼𝛼0 + ∑ 𝛼𝛼𝑖𝑖𝑖𝑖 ∗𝑥𝑥𝑖𝑖 + 𝜀𝜀 Traditionally, the variability of a given dependent variable is given by:

𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) = 𝛼𝛼𝑘𝑘∗𝑥𝑥𝑘𝑘��𝑦𝑦�

As we have at the optimum where residual cancels out that: 𝑦𝑦� = 𝛼𝛼0∗ + ∑ 𝛼𝛼𝑖𝑖∗𝑖𝑖 ∗ 𝑥𝑥𝚤𝚤� Where αk

* is the estimated parameter for the kth causal variable Now, let’s show that this result can be expressed in terms of elasticity. At the optimum, the elasticity of expense subject to an increase of 𝑥𝑥𝑘𝑘 is given by:

∈ (𝑥𝑥𝑘𝑘) =

𝜕𝜕𝑦𝑦�𝑦𝑦�𝜕𝜕�̅�𝑥𝑘𝑘�̅�𝑥𝑘𝑘

=𝜕𝜕𝑦𝑦�𝜕𝜕�̅�𝑥𝑘𝑘

�̅�𝑥𝑘𝑘𝑦𝑦�

= 𝛼𝛼𝑘𝑘∗�̅�𝑥𝑘𝑘𝑦𝑦�

It is now easy to establish a connection between the contribution to the total variable cost of a given independent variable and its elasticity:

𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣(𝑥𝑥𝑘𝑘) =𝛼𝛼𝑘𝑘∗�̅�𝑥𝑘𝑘

𝛼𝛼1∗�̅�𝑥1 + ⋯+ 𝛼𝛼𝑝𝑝∗ �̅�𝑥𝑝𝑝=

(𝛼𝛼𝑘𝑘∗�̅�𝑥𝑘𝑘)/𝑦𝑦�𝛼𝛼1∗�̅�𝑥1𝑦𝑦� + ⋯+

𝛼𝛼𝑝𝑝∗ �̅�𝑥𝑝𝑝𝑦𝑦�

= ∈ (𝑥𝑥𝑘𝑘)

∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)

Now let’s call C the fixed component in this linear model; we thus have:

𝐶𝐶 = 𝛼𝛼0∗ + � 0𝑖𝑖

∗ 𝑥𝑥𝚤𝚤� = 𝛼𝛼0∗


- 39 -


𝑉𝑉 = 1 − 𝐶𝐶𝑦𝑦�

= 1 − 𝛼𝛼0∗

𝑦𝑦�

Hence,

𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) = 𝑉𝑉 .𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣(𝑥𝑥𝑘𝑘) = �1 − 𝛼𝛼0∗

𝑦𝑦��

𝛼𝛼𝑘𝑘∗�̅�𝑥𝑘𝑘𝛼𝛼1∗�̅�𝑥1 + ⋯+ 𝛼𝛼𝑝𝑝∗ �̅�𝑥𝑝𝑝

� = �𝑦𝑦� − 𝛼𝛼0∗

𝑦𝑦��

𝛼𝛼𝑘𝑘∗�̅�𝑥𝑘𝑘𝑦𝑦� − 𝛼𝛼0∗

� = 𝛼𝛼𝑘𝑘∗𝑥𝑥𝑘𝑘��𝑦𝑦�

We have shown that the procedure we have followed for a multiple variable Box-Cox Model is consistent with a traditional linear-model procedure.

Full Box-Cox Models and Traffic-Elasticities: Let’s prove that the contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:


∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)=

𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆∗





First, we have shown that the optimum where residual cancels out can be written as:

𝑦𝑦 = �𝐶𝐶𝜃𝜃∗ + �𝑏𝑏1∗𝑥𝑥1𝜆𝜆∗ + 𝑏𝑏2∗𝑥𝑥2𝜆𝜆

∗ + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗�𝜃𝜃∗

𝜆𝜆∗�1𝜃𝜃∗

Where 𝐶𝐶𝜃𝜃∗ = �𝑏𝑏0∗ −1𝜆𝜆∗∑ 𝑏𝑏𝑘𝑘∗𝑝𝑝𝑘𝑘=1 � 𝜃𝜃∗ + 1

Following this expression, the elasticity of a given independent variable can be easily computed:


𝜕𝜕𝑦𝑦𝑦𝑦𝜕𝜕𝑥𝑥𝑘𝑘𝑥𝑥𝑘𝑘

=1𝜃𝜃∗�𝐶𝐶𝜃𝜃∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗

𝑝𝑝

𝑗𝑗=1

�𝜃𝜃∗

𝜆𝜆∗�

1𝜃𝜃∗−1

𝜃𝜃∗

𝜆𝜆∗𝑏𝑏𝑘𝑘∗𝜆𝜆∗𝑥𝑥𝑘𝑘

𝜆𝜆∗−1𝑥𝑥𝑘𝑘 �𝐶𝐶𝜃𝜃∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗

𝑝𝑝

𝑗𝑗=1

�𝜃𝜃∗

𝜆𝜆∗�

−1𝜃𝜃∗

Many terms cancel out in this expression which is equivalent to:

- 40 -


∈ (𝑥𝑥𝑘𝑘) =𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘

𝜆𝜆∗

𝐶𝐶𝜃𝜃∗ + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆∗𝑝𝑝

𝑗𝑗=1 � 𝜃𝜃∗

𝜆𝜆∗

Therefore, contribution for a given kth independent variable to the variable costs is given by:

𝐶𝐶𝐶𝐶𝑛𝑛𝑡𝑡𝑣𝑣(𝑥𝑥𝑘𝑘) =∈ (𝑥𝑥𝑘𝑘)

∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ �𝑥𝑥𝑝𝑝�

=




𝜆𝜆∗

𝑏𝑏1∗𝑥𝑥1𝜆𝜆∗



𝜆𝜆∗+ ⋯+

𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗



𝜆𝜆∗

=𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆

∗



Left Box-Cox Models and Traffic-Elasticities: Let’s prove that the contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:


∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)=

𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝑏𝑏1∗𝑥𝑥1 + 𝑏𝑏2∗𝑥𝑥2 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝




𝑦𝑦 = �𝐶𝐶𝜃𝜃∗ + �𝑏𝑏1∗𝑥𝑥1 + 𝑏𝑏2∗𝑥𝑥2 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝�𝜃𝜃∗�1𝜃𝜃∗

Where 𝐶𝐶𝜃𝜃∗ = (𝑏𝑏0∗)𝜃𝜃∗ + 1 Following this expression, the elasticity of a given independent variable can be easily computed:



=1𝜃𝜃∗�𝐶𝐶𝜃𝜃∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗

𝑝𝑝

𝑗𝑗=1

� 𝜃𝜃∗�

1𝜃𝜃∗−1

𝜃𝜃∗𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘 �𝐶𝐶𝜃𝜃∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗

𝑝𝑝

𝑗𝑗=1

� 𝜃𝜃∗�

−1𝜃𝜃∗


- 41 -



𝐶𝐶𝜃𝜃∗ + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝑝𝑝𝑗𝑗=1 � 𝜃𝜃∗



∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ �𝑥𝑥𝑝𝑝�

=

𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝐶𝐶𝜃𝜃∗ + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗

𝑝𝑝𝑗𝑗=1 � 𝜃𝜃∗

𝑏𝑏1∗𝑥𝑥1𝐶𝐶𝜃𝜃∗ + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗

𝑝𝑝𝑗𝑗=1 � 𝜃𝜃∗

+ ⋯+𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝


=𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘

𝑏𝑏1∗𝑥𝑥1 + 𝑏𝑏2∗𝑥𝑥2 + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝

Right Box-Cox Models and Traffic-Elasticities: Let’s prove that the contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:


∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)=







𝑦𝑦 = 𝐶𝐶 +1𝜆𝜆∗�𝑏𝑏1∗𝑥𝑥1𝜆𝜆

∗ + 𝑏𝑏2∗𝑥𝑥2𝜆𝜆∗ + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆

∗�

Where 𝐶𝐶 = 𝑏𝑏0∗ −1𝜆𝜆∗∑ 𝑏𝑏𝑘𝑘∗𝑝𝑝𝑘𝑘=1


- 42 -




=1𝜆𝜆∗𝜆𝜆∗𝑏𝑏𝑘𝑘∗�𝑥𝑥𝑗𝑗𝜆𝜆

∗−1�𝑥𝑥𝑘𝑘 �𝐶𝐶 +1𝜆𝜆∗��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗

𝑝𝑝

𝑗𝑗=1

��

−1


∈ (𝑥𝑥𝑘𝑘) =𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆

∗

𝐶𝐶 + 1𝜆𝜆∗ �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗𝑝𝑝𝑗𝑗=1 �



∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ �𝑥𝑥𝑝𝑝�

=







+ ⋯+𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆

∗




∗



Lambda Box-Cox Models and Traffic-Elasticities: Let’s prove that the contribution for a given kth independent variable to the variable costs is exactly equal to its contribution to the elasticity of total expenses subject to an increase in the all the independent variable so that:


∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ (𝑥𝑥𝑝𝑝)=







𝑦𝑦 = �𝐶𝐶𝜆𝜆∗ + �𝑏𝑏1∗𝑥𝑥1𝜆𝜆∗ + 𝑏𝑏2∗𝑥𝑥2𝜆𝜆

∗ + ⋯+ 𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗��

1𝜆𝜆∗

Where 𝐶𝐶𝜆𝜆∗ = �𝑏𝑏0∗ −1𝜆𝜆∗∑ 𝑏𝑏𝑘𝑘∗𝑝𝑝𝑘𝑘=1 � 𝜆𝜆∗ + 1

- 43 -





=1𝜆𝜆∗�𝐶𝐶𝜆𝜆∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗

𝑝𝑝

𝑗𝑗=1

��

1𝜆𝜆∗−1

𝑏𝑏𝑘𝑘∗𝜆𝜆∗𝑥𝑥𝑘𝑘𝜆𝜆∗−1𝑥𝑥𝑘𝑘 �𝐶𝐶𝜆𝜆

∗ + ��𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆∗

𝑝𝑝

𝑗𝑗=1

��

−1𝜆𝜆∗



𝜆𝜆∗

𝐶𝐶𝜆𝜆∗ + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆∗𝑝𝑝

𝑗𝑗=1 �



∈ (𝑥𝑥1)+∈ (𝑥𝑥2) + ⋯+∈ �𝑥𝑥𝑝𝑝�

=



𝑗𝑗=1 �



𝑗𝑗=1 �+ ⋯+

𝑏𝑏𝑝𝑝∗𝑥𝑥𝑝𝑝𝜆𝜆∗


𝑗𝑗=1 �


∗



Traffic-Elasticity of cost and the non-linearity of cost functions: We have shown in section (i) that in the case of a linear model, the traffic-elasticity of Cost subject to an increase in a given kth independent variable was exactly equal to its variability. Hence, we had the following property:

� ∈ (𝑥𝑥𝑘𝑘)𝑝𝑝

𝑘𝑘=1

+𝐶𝐶𝑦𝑦

= �


𝑝𝑝

𝑘𝑘=1

+

𝜕𝜕𝑦𝑦𝑦𝑦𝜕𝜕𝐶𝐶𝐶𝐶

= 1

Therefore, as all quantities are positive, we have for all k, the following inequality:

∀𝑘𝑘,∈ (𝑥𝑥𝑘𝑘) ≤ 1

- 44 -


Hence, the linear model imposes any traffic-elasticity to be very moderate. For instance, if Gross Ton Miles increases by 10 percent, the linear model makes it impossible to have an impact of Total Track and Roadway Maintenance Cost greater than 10%. This is a very rigid costing structure that may be released in some non-linear environments. In a Full Box-Cox Model, the sum of elasticities will turn to the following expression:

� ∈ (𝑥𝑥𝑘𝑘)𝑝𝑝

𝑘𝑘=1

+∈ (𝐶𝐶) = �


𝑝𝑝

𝑘𝑘=1

+


= �𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘

𝜆𝜆∗



𝜆𝜆∗+

𝐶𝐶𝜃𝜃∗



𝜆𝜆∗

𝑝𝑝

𝑘𝑘=1

= ∑ 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘

𝜆𝜆∗𝑘𝑘 + 𝐶𝐶𝜃𝜃∗



𝜆𝜆∗=�∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆

∗𝑝𝑝𝑗𝑗=1 � 𝜃𝜃

∗

𝜆𝜆∗ + 𝐶𝐶𝜃𝜃∗



𝜆𝜆∗+

∑ 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆∗ �1 − 𝜃𝜃∗

𝜆𝜆∗�𝑘𝑘



𝜆𝜆∗

= 1 + �1 −𝜃𝜃∗

𝜆𝜆∗�

∑ 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝜆𝜆∗

𝑘𝑘



𝜆𝜆∗

⎩⎪⎨

⎪⎧= 1 𝑚𝑚𝑓𝑓

𝜃𝜃∗

𝜆𝜆∗= 1 ↔ 𝜃𝜃∗ = 𝜆𝜆∗

< 1 𝑚𝑚𝑓𝑓 𝜃𝜃∗

𝜆𝜆∗> 1 ↔ 𝜃𝜃∗ > 𝜆𝜆∗

> 1 𝑚𝑚𝑓𝑓 𝜃𝜃∗

𝜆𝜆∗< 1 ↔ 𝜃𝜃∗ < 𝜆𝜆∗

Therefore it is the data that will dictate the capacity of individual elasticities to exceed 1. However, that doesn’t prevent variabilities to add to 1 since we have:

�𝑣𝑣𝑣𝑣𝑣𝑣(𝑥𝑥𝑘𝑘) +𝑝𝑝

𝑘𝑘=1

𝐶𝐶𝑦𝑦

= �1 −𝐶𝐶𝑦𝑦��


∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆∗𝑝𝑝

𝑗𝑗=1+𝐶𝐶𝑦𝑦

= 1𝑝𝑝

𝑘𝑘=1

Now, we will do the above calculation for the 3 other Box-Cox versions that are less flexible. For the Left Box-Cox,

- 45 -


�


𝑝𝑝

𝑘𝑘=1

+




+𝐶𝐶𝜃𝜃∗


𝑝𝑝

𝑘𝑘=1

= 1 + (1 − 𝜃𝜃∗)∑ 𝑏𝑏𝑘𝑘∗𝑥𝑥𝑘𝑘𝑘𝑘


�= 1 𝑚𝑚𝑓𝑓 𝜃𝜃∗ = 1

< 1 𝑚𝑚𝑓𝑓 𝑚𝑚𝑓𝑓 𝜃𝜃∗ > 1 > 1 𝑚𝑚𝑓𝑓 𝑚𝑚𝑓𝑓 𝜃𝜃∗ < 1

For the Right Box-Cox,

�


𝑝𝑝

𝑘𝑘=1

+



𝜆𝜆∗

𝐶𝐶 + �∑ 𝑏𝑏𝑗𝑗∗𝑥𝑥𝑗𝑗𝜆𝜆∗𝑝𝑝

𝑗𝑗=1 � 1𝜆𝜆∗

+𝐶𝐶


𝑗𝑗=1 � 1𝜆𝜆∗

𝑝𝑝

𝑘𝑘=1

= 1 + �1 −1𝜆𝜆∗�



𝑗𝑗=1 � 1𝜆𝜆∗

�= 1 𝑚𝑚𝑓𝑓 𝜆𝜆∗ = 1

< 1 𝑚𝑚𝑓𝑓 𝑚𝑚𝑓𝑓 𝜆𝜆∗ < 1 > 1 𝑚𝑚𝑓𝑓 𝑚𝑚𝑓𝑓 𝜆𝜆∗ > 1

For the Lambda Box-Cox,

∀ 𝜆𝜆∗ , �


𝑝𝑝

𝑘𝑘=1

+



𝜆𝜆∗


𝑗𝑗=1 �+

𝐶𝐶𝜆𝜆∗


𝑗𝑗=1 �

𝑝𝑝

𝑘𝑘=1

= 1

Which is consistent with the first scenario of a Full Box-Cox Model when 𝜃𝜃∗ = 𝜆𝜆∗ And of course, in all these three other versions, as it was the case for the Full Box-Cox Model, variabilities add to 1 independently from their properties in terms of elasticity.

report – development of variabilities

Documents