June 1982 Division of Research Graduate School of Business Administration The University of Michigan MEASURING THE PRECISION OF STATISTICAL COST ALLOCATIONS Working Paper No. 305 Roger L. Wright The University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the expressed permission of the Division of Research.

ABSTRACT This paper examines the use of regression analysis and other statistical methods for allocating indirect costs. When multiple regression is used to estimate the weights of several allocation factors, conventional standard errors and correlation coefficients can be misleading with respect to the statistical precision of the cost allocations. Alternative measures of precision which use linear prediction theory and Bayesian inference are developed. The proposed methods are illustrated in the context of a university indirect cost study. KEY WORDS: Bayesian inference, Generalized linear model, Joint costs, BLU Prediction -ii

Author' s Footnote: Roger L. Wright is Associate Professor of Statistics, Graduate School of Business Administration, The University of Michigan, Ann Arbor, MI 48109. The author wishes to thank M. Bryant, J. S. Demski, and W. E. Wecker for helpful suggestions on earlier versions of this paper. -iii

1. INTRODUCTION This paper examines the use of statistical methods, particularly regression analysis, in cost allocation studies. Regression analysis is widely used by economists and managerial accountants to estimate cost relationships, but there seems to be no satisfactory technique for determining the statistical precision of the allocations that result from these estimates. Conventional measures of precision can be misleading because they fail to take into account certain aspects of the allocation process. A successful technique must also separate the statistical issues from the difficult policy decisions that arise in connection with the allocation of fixed and joint costs. This paper introduces a new approach that is based on the most relevant features of the cost allocation process, and seems to yield a more appropriate measure of the precision of a regression-based cost allocation. Section 2 of the paper discusses several alternative statistical procedures that can be appropriate when costs are separable, i.e., free of fixed or joint components. This discussion includes an approach based on Goldberger's (1962) theory of prediction under the generalized linear regression model. Section 3 develops the main approach of the paper, which uses Bayesian inference with a class of multiple regression procedures for allocating both separable and joint costs. The Bayesian approach isolates the statistical precision achieved when estimates are used to implement a prespecified allocation policy formulated in terms of the parameters of the underlying cost function. Section 4 illustrates these methods in the context of a university indirect cost study. The rest of the present section gives a brief introduction to the cost allocation problem.

-2 -Although the need for cost allocations arises in a variety of settings, a generic example is a firm composed of a home office and several divisions. Each division produces a single product, using its own resources together with some home office services. In this situation the firm will usually allocate part of its home office costs to each division. Cost allocations generate substantial revenues for firms working under cost-plus contracts or having significant indirect cost charges built into their fees. Cost allocations also play a central role in public utility rate setting. Even unregulated firms with no opportunities for external cost reimbursement usually have elaborate accounting procedures for allocating costs. A lively and rather analytical literature in accounting and economics has sprung up around the cost allocation problem. Most of the early accounting texts dealt at length with rules for allocating costs within divisions of the firm or over time (for a review, see Solomons (1968)). Eventually many accounting theoreticians and economists such as Thomas (1974) began to argue that "any allocation of joint costs is arbitrary and serves no useful purpose" (Kaplan 1977, p. 52). Others, though conceding the essential arbitrariness of many cost allocations, have recognized that they are often unavoidable. Using mostly game-theoretic approaches, they have tried to develop allocation procedures that are equitable and consistent with beneficial cooperation among managers (e.g., Shubik 1962 and Hamlen, Hamlen, and Tschirhart 1980). Quite recently, Zimmerman (1979) and Demski (1981) have suggested that cost allocations can help coordinate managers' individual utility functions with the interests of the firm. Most of the recent literatureent it is concerned with the problem of allocating joint costs, or common costs, which are often somewhat circuitously described

-3 -as "costs which cannot be readily identified with individual products [or divisions]" (Billera, Heath, and Verrecchia 1981, p. 185). Joint costs comprise the fixed costs of a shared facility and any other costs linked to the activities of two or more divisions. Many accountants feel that "when the costs of producing multiple products are separable, they are readily identified with individual products, so no particular problem arises..." (Billera, Heath, and Verrecchia 1981, p. 185). However, whenever costs are indirect, whether they are separable or joint, statistical methods are usually needed to implement cost allocation procedures (see for example, Benston (1966), Johnston (1960), or chapters 3 and 14 of Dopuch, Birnberg, and Demski (1974)). These statistical techniques use available cost data, collected either administratively or through sampling, to estimate the cost relationships that form the basis for allocations. While most statistical procedures yield estimates that are substantially free of bias, they often differ markedly in terms of their statistical precision. Methods with poor precision will usually be rejected, especially when cost negotiations are involved. 2. ALLOCATING SEPARABLE COSTS 2.1 Proportional Allocation Home office costs are said to be separable if we can define the costs incurred by each division. This means that the costs incurred by each division are technically observable, although often sufficiently difficult to trace to make measurement impractical. We assume throughout Section 2 that costs are separable, although only total home office costs are observed. In this case, our problem might be called disaggregation; but to preserve continuity with Section 3 we will continue to speak of allocation.

-4 -The most common procedure is to distribute home office costs among several divisions in proportion to an observed variable which is regarded as an index of the service provided by the home office. The allocation variable usually measures the activity within each division, often reflecting a relevant factor of production such as labor hours or wages, or sometimes measuring the division's output or cash flow. Special studies may be undertaken to justify the choice of the allocation variable (e.g., Verrecchia 1981), but once this variable is determined the approach is usually straightforward. A careful formulation of the proportional allocation procedure will facilitate the discussion of more complex approaches. Consider a particular period i; let yi denote the costs of the home office to be allocated among several divisions, let x. denote the total value of the allocation variable across all of these divisions (xi > 0), and let xOi denote the value of the allocation variable within a particular division of interest —say, division 0. Under proportional allocation, the portion of the home office cost allocated to division 0 is aiYi, where ai = xOi/x. The statistical properties of proportional allocation depend on the nature of the underlying cost relationships. Suppose that yi is related to x. in the manner defined by a simple regression model with zero intercept: yi = xi + (1) E(ui) = 0, V(ui) = c2 Assume also that the actual unobserved cost incurred by the home office in serving division 0 in period i is separable, is denoted yo, and is similarly related to Xoi:

-5 - Yoi = XOi + uOi' (2) E(UOi) = 0, V(UOi) = yo, 0 < Y < 1. Finally, assume that u0i and u. - uOi are uncorrelated so that C(u0i, ui) = E(u0iui) 2 = ya Then aii - Yi = aiui - i' (3) E(aiYi - Yoi) = 0, ~~2 2 V(aiYi - Yoi) = (ai 2aiY + y)a Thus, given (1) and (2), the allocated cost is an unbiased predictor of the 2 incurred cost with variance proportional to a. Three points might be made about proportional allocation. (a) The 2 relationship of the prediction variance to a in (3) helps to explain why the correlation between yi and x. is often considered in selecting an allocation variable. (b) The allocation proportion a. can be written as BXoi/3xi, where f is any estimator of C. Here the choice of 8 and its precision is irrelevant, but this suggests a natural extension of proportional allocation to situations involving multiple allocation variables. (c) The unbiasedness of proportional allocation depends critically on the validity of (2). For example, if the 0 are actually unequal in (1) and (2), the allocation will be biased. Proportional allocation can also result in bias if the costs are not separable. Suppose, for example, that (1) and (2) are replaced by

-6 - y = 0 + x.i + ui Yoi = l1XOi + Uoi' Then aiYi - Yoi = ai8o + aiui - Ugi' Thus, in this case the allocated cost aiYi as a predictor of the incurred cost yoi is biased by the allocated fixed cost ai.%. In order for an allocation procedure to be accepted as equitable, it should be generally regarded as free of bias. In some applications the possibility of bias can be reduced substantially by utilizing multiple allocation variables, or by using statistical sampling to improve the data base. 2.2 Finite Population Sampling When proportional allocation is suspected of bias, statistical sampling is sometimes used to directly estimate the cost Yoi incurred by the home office in serving division 0. For example, many firms allocate the cost of their capital to divisions. As part of this allocation, statistical sampling is often used to estimate the current value of the inventories held by each division (e.g., Newman 1976). Another example is the use of work sampling to study the activities of employees (e.g., Mandel 1971). In other circumstances, there may be a particular activity variable that would provide an acceptable basis for proportional allocation but which is observable only through sampling. An example occurs in electric utility load research. tbst electric utilities would like to allocate certain capacityrelated costs to rate classes in proportion to each class's consumption of electricity during certain hours of peak system-wide demand. A common procedure is to estimate each class's consumption during these peak hours by using

-7 - special metering equipment on a sample of customers. This application is more fully described in Argonne National Laboratory's Load Research Manual (1980). Statistical sampling is often regarded as the most objective approach to the allocation of separable costs, but it can be expensive. However, sampling procedures can often be made more efficient by taking advantage of the association between the target variable and other auxiliary variables that are known throughout the population. The model-based survey sampling techniques of Cassel, Sarndal, and Wretman (1976), Kaplan (1973), and Royall (1976) offer a useful link to other regression-based allocation methods (Wright 1981). 2.3 A Prediction Theory Approach Goldberger's (1962) theory of prediction under the generalized linear model yields some useful insights for the use of several allocation variables. Suppose that we observe an N-dimensional vector y = (y1...YN )' which gives the home office costs in N periods. We also observe two N x k matrices X and X0. X describes the activities of all divisions and X0 describes the particular activities of division 0. We continue to assume that costs are separable, and we let yO be the N-dimensional vector of unobserved costs incurred by the home office in serving division O. We generalize (1) and (2) by assuming that 2 there exists an unknown k-dimensional vector 0, an unknown scalar a > 0, and known N x N matrices 00' 0,' and Ql such that (X'Ql X) exists and y = Xg + u, (4) YO = X0 + UO' E(u) = 0, E(u0) = 0, V(u) = E(uu') 21

-8 - V(uo) = oo 2 C(uo,U) = E(uoU') 01 Here 201 will usually be nonzero, since u0 is a component of u. Imagine momentarily that f is known. Given 13, the best linear unbiased (BLU) predictor of yO is Y01, where YO = E(yolU) (5) -l = Xo0 + ol u In the spirit of Shillinglaw (1963), we call y01 the costs attributable to division 0, and we call yO - y01 the unidentifiable costs of division 0. The unidentifiable costs are uncorrelated with y and have variance V(y0 - Y01) = (00 - 501l ll )a (6) When 3 is unknown, we consider the class of unbiased linear predictors of YO or Yo1, i.e., predictors of the form YO= Ay for some fixed N x N matrix A, such that E(y0) = E(yo) (7) = (Yo1) We call V(y0 - yo) the prediction variance of yO, and we call V(yO - Y01) the allocation variance. Several useful results follow from Goldberger (1962) or Fuller (1980). (a) The unbiasedness condition (7) is equivalent to the algebraic identity

-9 - AX = XO. (8) Thus, if b is any k-dimensional vector, y0 = Xb +A(y - Xb), (9) so that yO is the sum of a linear prediction of yo based on X0 and b plus an allocation of the associated total residual cost associated with b. (b) The prediction variance of y0 is V(yO - y0) = V(Au - uO) (10) = (AS211 A' - 0A' - AQ0 + 0)2, and the allocation variance of yO is V(y0 - Y01) = V(y - yO) - V(yO - Y01)' (11) where V(y0 - Y01) is given by (6) and does not depend on A. (c) V(yO - y0) and V(y0 - Y01) are both minimized subject to (8) by choosing A equal to A*, where A* = (X - X01)C + n01 ~1, (12) C = (X'1 1X) -1X -1 Xol 1 011 11x This gives the best linear unbiased allocation X^ 01A -1 ^ YO = x + o1211 u, (13) B = Cy, u = y - Xf. Also, Yo - Yo01 = (Xo - X01)( - ), V(yo - y01) = (Xo - XO0)V(1)(XO - Xol)

-10 - where A -1 -1 2 V(g) = (X'Qi1 X)1 r2 (d) Let yO be defined by (13) and let X = X - X, C 2 C(u - uO, u) = ac A A -1 A y =X + S 21 u. c c cl 11 u Then the BLU allocation is complete in the sense that y0 + c = y. Thus, prediction theory yields a seemingly optimal procedure for allocating separable costs under the criteria of unbiasedness and minimum variance. As an example, consider (4) with k = 1, xi > 0, 2ll = diag(xi), and Q00 = Q01 = diag(x0i). Then the BLU predictor gives proportional allocation A* = diag(x0i/xi), and the allocation variance V(YO - Y01) is zero. Nevertheless, the prediction approach has serious practical limitations. It is unlikely that Q01 will be known or satisfactorily estimable when yO is unobserved. Moreover, it is not even clear how to define yO and hence (4) when the cost function for y includes joint costs. 3. ALLOCATING JOINT COSTS THROUGH REGRESSION 3.1 Bayesian Prediction When joint costs are involved, nonstatistical considerations dominate the choice of an allocation policy. However, as pointed out by Dopuch, Birnberg, and Demski (1974, p. 581), "even though allocations of fixed costs

-11 -are necessarily arbitrary, they may produce reasonably equitable results if the allocation method is based on the relative amount of services consumed by the receiving departments." As an example, suppose that the home office provides two services in the quantities xli and x2i, and has the cost function yi = 0 + lXli + 2X2i' Also assume that division 0 consumes all of the first service and none of the second service. Then, as a matter of policy the firm might define the costs attributable to division 0 to be ai(Gl, 02)Yi. where ai(31, 02) = g1Xli/(/1Xli + 82x2i ) In this case the costs allocated to division 0 could be calculated as ai.(1, 32)yi,where P1 and f2 are suitable estimates of,1 and 02. To generalize this example, we assume throughout this section that an allocation policy has been specified in terms of the parameters of a regression model for the cost function of the home office. The firm is assumed to implement the desired allocation by estimating these parameters. Our interest is not in the choice of allocation policy but only in the allocation variance induced by these estimates. Specifically, we assume a model for the total home office cost y as in (4): y = X g + u, (14) E(u) = 0, V(u) = 1O. Although we no longer assume that yO is defined, we do assume an allocation policy that defines the costs attributable to division 0 —say, yol —in terms of a prespecified N x N matrix A(B):

-12 - O = A()y. (15) Here each entry of A(P) —say, a. (ij() —is assumed to be a continuously differentiable function of 0 in the region of interest. The Bayesian approach to statistical inference (e.g., Zellner 1971) is convenient in discussing inference about y01 conditional on the observed y. (A more conventional approach might be developed along the lines of McDonald (1981).) In Bayesian inference, 1 is regarded as a random variable. Assume for convenience that the prior information is relatively vague, that X'Q11 X is not poorly conditioned, and that N is reasonably large. Then the posterior distribution of 1 is approximately normal, with mean 3 and variance V(B), where i is as in (13) and v(-) = (x' -1 ^2 V() =) (X 211 X)1 a, (16) ^2 -1 ^ -1 ^ a = (N - k) u'11 u. The posterior distribution of B induces the posterior distribution of Yo1 through (15). For large N, the posterior distribution of y01 can be easily approximated through a Taylor's series expansion of A(O3) around 3. Define D(6) to be the N x k matrix with entry d. ij() given by -17 N dij() = Yh(aaih(3)/a3) (17) h=l Here 0j is the jth entry of the k-dimensional vector 1, and yh is the hth entry of y. Then, in the neighborhood of 3, aih() aih() + Y (3a h()/30 )(8 - j) (18) so that Y1 = A(8)y + D(1)(g - 0). Thus, the posterior distribution of y01 is approximately normal, with mean YO1 and variance V(y01), where

-13 - A A YO = A(8)y, V(y01) = D(3)V(O)D(3)'. (19) Here V(y01) provides a measure of the precision associated with the allocation yo.1 In many applications, A(B) may take a fairly simple form. For each period i, let Qi and QOi be two known k-dimensional vectors such that Qi' > 0, and define A(6) by a.. (13) = (20) aii (o) Q0li B/Qii ' (20) aij () =, i * j. Then the cost attributable to division 0 in period i can be written as aii(8)yi = Qoi'B + aii(B)[(Xi - Qi)' + u], which generalizes (5). Here yi and ui are the ith entries of y and u, and X. ' is the ith row of X. Also, if we let Di()' denote the ith row of D(1), (17) implies that -1 Di(B) = Yi(Qi') [Qi - aii()Q ] (21) Now suppose that we are interested in the total costs attributable to division 0 throughout all N periods —say, T = e'y01 —where e' = (1...1). Then (19), (20), and (21) imply that the posterior distribution of T is approximately normal, with mean T and standard deviation sT, where N T = (Qoi '/Qi' )yi (22) i=l N N -= Qoi t + % (QOi0, /Qi ')[(Xi - Qi)' + u], and i=l i=l N N sT = Di(3) V() D ()1, a i= i=l and where V(1) is defined as in (16).

-14 -Note that if Q0i is proportional to Qi —say, QOi = ciQi —then T is N simply ) ciYi and sT = 0. Proportional allocation is a special case of i=l this. 3.2 Examples (a) Assume the separable cost framework (4), and let Qi' and QOi' be the ith rows of X and XO, respectively. Then N T = I [E(Yi )/E(yi)]yi i=l N N i= E(Y o) + [E(yO )/E(yi)]ui i=l i=l 1 N A= A T (X0OilXi ) i=l N N Z X0i t +. (X1oit/Xi'3)u. iA 11 Thus, the allocation T is proportional with respect to Xi 'f, and can be written as the sum of a simple regression-based predicted cost and a proportional allocation of the residual. (b) Suppose that (14) is a simple linear cost function yi = 1 + 2xi + Ui. Perhaps x. is the total wages paid in all divisions, and xgi is the total wages paid in division 0. If the firm wished to allocate the fixed cost f1 in proportion to each division's share of the variable cost, we would take Q = (0 xi), QOi' = (0 X0i). Alternatively, the firm might decide to allocate one-half of the fixed costs to division 0, using

-15 - Qi' = (1 xi), Qoi = (.5 x0i) (c) Suppose that home office costs are assumed to be a quadratic function of the wages xli and x2i in each of two divisions, 1 and 2: i = o0 + lx1li + 22x2i + 12Xix + 2 + 1iX2i.. Suppose further that we want to allocate home office costs in each period i 2 in proportion to the cost components j.x.. + 0..x.. uniquely associated with each division j = 1, 2. Then, to allocate costs to division 1, use Qi = (0 li x2i Xli 2i 0) QOi= (0 Xli 0 xli 0 0). In these and similar situations, the statistical precision of the allocated costs can be evaluated using (19) or (22). This Bayesian approach is simpler to implement than the predictive approach of Section 2.3, since there is no need to specify the covariance matrix 01. The Bayesian approach can be adapted to a wide variety of models and allocating policies, and seems to capture the spirit of accounting practice. Although superficially similar, the predictive and Bayesian approaches are conceptually quite distinct. The predictive approach is built around a causal model for the costs actually incurred by each division, so that it is not applicable to the allocation of joint costs. The Bayesian approach is based on an allocation policy expressed in terms of the parameters of the home office cost function; this policy may be more or less arbitrary, but nevertheless it is taken to be prespecified. The Bayesian approach moves the focus from the conventional emphasis on the precision of the estimated regression

-16 -coefficients to the more relevant precision of the allocations determined by these estimates and the given policy. 4. AN APPLICATION TO INDIRECT COST ACCOUNTING FOR UNIVERSITIES 4.1 Background In determining costs for federal grants and contracts, educational institutions are currently governed by the cost accounting principles established by the Office of Management and Budget's Circular A-21 (1979). These standards prescribe how indirect costs should be distributed among major functions which usually include instruction, departmental research, and organized (contract) research. The principal indirect cost categories are depreciation and use allowances, general administrative expenses, and operation and maintenance expenses. According to Circular A-21 (1979, p. 12371), "the overall objective of the indirect cost allocation is to distribute these indirect costs to the major functions in proportions reasonably consistent with the nature and extent of their use of the institution's resources." Consider the case of centrally supplied heating and other utilities. The standard approach is to allocate utility expenses to the individual functions performed in each building in proportion to the square feet of space used by each function. However, the cost distribution methodology is not restricted to proportional allocation. Circular A-21 allows cost determinations to be based on special cost analysis studies which take into account multiple allocation variables together with suitable weighting factors. These cost analysis studies must (a) provide adequate documentation for review, (b) give cost distributions in proportion to benefits, (c) be statistically sound, (d) be performed at the institution itself, and (e) be reviewed periodically. The methodology developed in the previous section seems to comply with the provisions of Circular A-21 for special cost studies.

-17 - 4.2 A Specific Example In considering the allocation of its utility expenses, one large university felt that proportional allocation was biased because various types of space incurred utility costs at different rates. A special study was undertaken to determine the feasibility of a cost allocation that differentiated four types of space: nonlaboratory space (XN), including offices, classrooms, and dormitories; laboratory space (xL); automobile parking space (xp); and common space (xc), including entranceways and halls. The study involved N = 187 buildings in regular use at the university. A data base which described these 187 buildings was assembled. These data included the total number of square feet of each of the four types of space for each building as well as the amount of each type of space used by the function of interest —say, the organized research function. The study data base also included the utility cost of each building during the 1979/80 fiscal year. The total utility cost of these 187 buildings was $19,175,000. Excluding structural space, these buildings had a total size of 15,045,000 square feet, of which 54% was nonlaboratory, 11% was laboratory, 11% was parking, and 24% was common. The organized research function utilized 1,755,000 square feet, of which 24% was nonlaboratory, 42% was laboratory, and 34% was common. Virtually no parking was assigned to organized research. Conventional allocation on a total square foot basis is straightforward. Since organized research utilized 11.7% of the total space, the research function might be assigned $2,243,000 of the total utility cost. Alternatively, the utility cost of each building might be allocated on a total square foot basis. Under this procedure, the total organized research allocation would be increased to $3,890,000. This method still utilizes proportional allocation,

-18 -and no statistical estimation is involved. The substantial increase in the allocation to organized research in going from a total to a building-bybuilding procedure was caused by the high utility costs in buildings with research activity. This strengthened the conviction that the various types of space were not homogeneous in their use of utilities. On the basis of preliminary analysis of the data base, the following regression model seemed to be appropriate: yi = SNXNi Lx + iLi + pXi + cXi + i (23) E(ui) = 0, 2 2 V(u.) = oE(y)2, C(ui, uj) = 0, i * j. Because of the heteroscedastic specification, an iterative weighted-leastsquares algorithm was used to estimate the parameters of this model (Maddala 1977, p. 261). This yielded the estimated regression equation (with standard errors): i = 0.8223 xi + 3.1961 xL + 0.1253 xp + 2.0050 x, (24) (0.1066) (0.5428) (0.0413) (0.3395) a =.5987. The estimated coefficients were consistent with the expectation that utility costs varied by type of space. The results showed that utility costs averaged about $0.82 per square foot of nonlaboratory space and about $3.20 per square foot of laboratory space. The rather high cost of common space, about $2.00 per square foot, was surprising but seemed reasonable when the nature of the space was considered. The cost of parking space ($0.13 per square foot) reflected the cost of lighting.

-19 - The university felt that it was equitable to allocate utility costs to each function in proportion to the expected costs incurred by its space based on (23). Since no joint costs were involved, the procedure followed example (a) of Section 3.2. Using (22) and (24), the allocation to organized research was $4,302,000. However, this approach was questioned because of the poor standard errors in (24), all of which exceed 12% of their respective coefficients. In contrast, (22) gave the Bayesian standard deviation of the allocation as $66,000, i.e., only 1.5% of the allocation. This was regarded as much more satisfactory. A small simulation experiment was carried out to check this analysis. For simplicity, a sampling orientation was adopted rather than a strict Bayesian view. The space variables were considered to be fixed for the 187 buildings, but the utility costs were randomly generated. For this purpose, (24) was taken to be the true model with the disturbances assumed to be normally distributed. The simulated utility costs were used with the assumed coefficients to calculate T. Then the simulated data were analyzed using the preceding estimation procedure, and T and sT were calculated using (22). This entire procedure was repeated 100 times to get some indication of the sampling relationship between T and T, and the validity of sT. The distribution of the relative allocation error was examined in the 100 trials. The coverage of Bayesian confidence intervals for the allocation was also investigated. The results of this simulation experiment supported the methodology in every respect. The allocation errors were immaterial, as expected. In the 100 trials, all of the relative allocation errors, (T - T)/T, were less than 4%,

-20 -and 97% of them were less than 3%. Moreover, there was no indication of any bias in the estimated allocation. In fact, the average relative allocation error was less than 0.3% and was not statistically significant. The simulation experiment also demonstrated that the standard error sT gives a realistic estimate of the allocation error. This was determined by calculating the observed coverage of various Bayesian confidence intervals, i.e., the proportion of simulation trials for which T was in the interval T + z sT. The observed coverage was compared to the nominal coverage expected under the standard normal distribution. For T + 1 sT, the observed coverage was 70%, slightly exceeding the nominal coverage of 68%. For T + 2 s, the observed coverage was 97%, again slightly above the nominal coverage of 95%. The same pattern occurred for the interval T + 3 sT where the observed coverage was 100%. This suggests that, if anything, sT may be slightly conservative. 5. SUMMARY This paper has examined the precision of a cost allocation based on a statistically estimated multiple regression model. Our central model characterizes the total costs incurred by a home office in serving several divisions. When the central model can be expanded to characterize separable divisional costs, linear prediction theory gives a specific allocation procedure that is unbiased and has minimum variance among all linear allocations. Alternatively, if costs are joint or separable with unknown covariances, and if a linear allocation policy can be specified in terms of the parameters of the central model, then Bayesian inference yields measures of the precision of the resulting cost allocations.

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