forecasting marginal costs of a multiple-output production technology

Journal of Forecasting, Vol. 12, 421-436 (1993)

Forecasting Marginal Costs of a Multiple- output Production Technology

GEORGE M. LADY Temple University, Philadelphia, PA, U. S. A.

and

CARLISLE E. MOODY College of William and Mary, Willamsburg, VA, U.S.A.

ABSTRACT This paper presents the results of fitting a scaled translog restricted profit function to ‘pseudo’-data formed by repeated runs of a large linear programming model of domestic and international refining. The translog approximation is designed to estimate the marginal cost of producing eight petroleum products given the amounts of each product demanded and the price of crude oil. We test the model against out-of-sample data from the refinery model and historical data. The model is used in the US Department of Energy’s Annual Energy Outlook forecasting system.

KEY WORDS Pseudo-data Forecasting Petroleum prices Translog

The Energy Information Administration (EIA) prepares forecasts of domestic energy production, consumption, distribution, and prices which are published in the Annual Energy Outlook (AEO-EIA, 1991a). The forecasting methodology entails projecting a general equilibrium for domestic energy markets given the characteristics of the international energy markets and the domestic macroeconomy. An important component of the forecasting system is the Oil Marketing Module (OMM). This model forecasts the supply prices or ‘refinery gate’ prices of domestically produced petroleum products, conditional upon the world price of crude oil and the amount of each product supplied. These prices are the marginal costs of production for each product which are then marked up to retail by the costs of distribution and marketing. In use, the OMM is iterated to convergence with product-demand models for each forecast year. Because of the amount of detail required, large linear programming models are often used to estimate the marginal costs of refining. Although the EIA has several refinery LP models available to it, they are too large and slow to be used with the other models in the AEO forecasting system.

In this paper we present the version of OMM being used by the Energy Information Administration. EIA specified that the model be small enough to incorporate into EIA’s PC- based AEO forecasting system, retain a relationship to the technical detail of refinery 0277-6693/93/0SO421-16$13 .OO Received July 1991 0 1993 by John Wiley & Sons, Ltd. Revised November i992

422 Journal of Forecasting Vol. 12, Zss. No. 5

modelling, conform to the constraints implied by neoclassical production theory, and yield accurate forecasts of the refinery gate prices. Using the pioneering work of Griffin (1977a,b, 1978, 1979), we developed a model based on a scaled restricted translog profit function and its first partial derivatives. The equations are estimated on ‘pseudo’-data generated by repeated runs of a large LP refinery model known as the Oil Trade Model (OTM-DAC, 1989). The resulting model is much smaller and faster than the OTM model. The OTM model occupies 1.4 megabytes and solves in 10 minutes on a 386/25 PC. The OMM model based on pseudo- data occupies 74 kilobytes and solves in 3-4 seconds on the same computer. We test the model’s forecasting accuracy below against both historical data and out-of-sample pseudo- data. To our knowledge, this is the only translog approximation to a linear programming model in use as a forecasting model, although Finan and Griffin (1978) used a similar approach to estimate input-output coefficients for a macroeconometric model.

THE SCALED RESTRICTED TRANSLOG PROFIT FUNCTION

Let x be a vector of n outputs, z a vector of m inputs, and f (x, z) represent the multiple output production technology such that f (x, z) = 0 for feasible production points. For p the vector of output prices and w the vector of input prices, solutions to the linear programming representations of refining approximate solutions to the problem:

Select non-negative values for (x, z) such that, ?r = px - wz is a maximum, subject to f(x, z) = 0

Dual to the multiproduct production function f (x, z ) is the profit function, ?r = g(p, w), which gives the value of (maximum) profit found in solution to problem (1). Solutions to (1) are invariant with respect to absolute, but not relative, changes in input and output prices. As a result, the profit function is homogeneous of degree one in input and output prices. Given this, it can be shown that ax/ap; = X i and ar /aw , = - z,, respectively, for the ith output and the gth input; and further, a In r / d In p ; = p i x ; / r and a In r/a In w, = - w,z,/?r.

The linear programming representation of refining technology is homogeneous of degree one in all inputs since, for a given solution, if all constraints were changed by a given proportionate amount the solution values for outputs and variable inputs would change by the same proportionate amount. As a result, all model solutions must be constrained by bounds on facilities. In the short run such bounds are due to the assumption that existing capital is fixed. For the long-run solutions, facilities can be added, but only if it is economic to do so. Thus, for any long-run solution, capital is fixed at the margin in that no facilities can be added if product prices are too low. Since some factors are fixed, g ( p , w) is termed the ‘restricted profit function’ and the technology represented by f (x, z) exhibits (almost) homogeneity of degree k < 1. In particular, if f(x, z) = 0, then f (Xkx, Xz) = 0. For the multiproduct production function almost homogeneous of degree 1 and k, it can be shown that

Revenue = px = (1/(1 - k ) ) r

As a result, the restricted profit function is homogeneous of degree l / ( l - k ) in p and homogeneous of degree - k/(l - k) in w.

and Cost = wz = (k/(l - k ) ) x

‘ The derivations in this and the next section are based upon Lau (1978).

G. M. Lady and C. E. Moody Forecasting Marginal Costs 423

APPROXIMATING THE SCALED RESTRICTED PROFIT FUNCTION

Griffin (1977a) first approximated the profit function from pseudo-data generated by a linear programming refining model using the translog functional form. However, this earlier effort differs in several important particulars from the problem being considered here. Griffin formulated the problem as one of revenue maximization constrained by fixed total costs and the multiproduct technology. Given solutions to this problem, Griffin scaled output prices such that profits equalled zero. Under these circumstances, the profit function corresponding to the multiproduct production function identifies the price possibility frontier for which profits are constant and equal to zero. The profit function was then submitted to a linear transformation by adding (constant) total costs. The first partial derivatives of the log of the transformed profit function to the logs of output and input prices are, respectively, the revenue share of each output and the negative cost share of each input. These share equations as approximated by the translog functional form are the basis for the coefficient estimation from the pseudo- data. Significantly, Griffin did not estimate the transformed profit function itself.

In the OMM model we must forecast the marginal costs in both absolute and relative terms. This made it impossible to scale output prices such that profits are zero. However, the share equations remain desirable and convenient as a basis for the estimation. As a result, we selected the restricted profit function scaled down to its (1 - k)th root as the function to be approximated by the translog functional form. For v = T ' - ~ , it can be shown that a In v/d In pi = pixi/Revenue and a In v / a In wg = - wgzg/Revenue.

The OMM model is also required to forecast the amount of petroleum products imported into the USA. Therefore, we must also make product imports endogenous to the model used to forecast product marginal costs. We did this by treating product imports as 'inputs' to the enterprise of satisfying domestic product demand. In solution, the refinery gate price and the (landed) import price are equal for each product category. Therefore, in the approximation of the restricted profit function with imports treated as inputs, the prices of the seven products that are imported appear in two roles, first as an output price and then as an input price corresponding to imports of the product. Because the activity of importing petroleum products is part of the process of satisfying domestic demand, but not part of the technology of refining crude oil, we assume that imports are separable from outputs in the multiproduct production function. As a result, import prices do not appear as arguments in the output share equations and output prices do not appear as arguments in the import share equations. The prices of all other inputs to refining except crude oil were held constant for all the OTM runs made to generate the pseudo-data. Accordingly, the restricted profit function to be estimated has sixteen arguments: eight supply prices (i.e. marginal costs) for output product categories (gasoline (GSH), LPG (LPG), jet fuel (JET), petrochemicals (PCH), distillate (DST), low- sulphur residual fuel (LSR), high-sulphur residual fuel (HSR), and other products (OTH)), seven prices for product imports as inputs, and the refinery's acquisition price for crude oil. For w , the price of crude oil, the translog approximation of the scaled restricted profit function is given as

15 l 8 8 ln(7r'-k)=ao+ C a i l n p i + - C C c ; j l n p i l n p j + C d i ,16 lnp i ln w

i = l 2 i = l j = 1 i = 9


The equations to be estimated are the first partial derivatives of equation (2). There are eight output share equations:

8

j= 1 (1 - k) a In ~ / a In pi = piXi/R = ai + C C i j In pj + di,16 In W , i = 1,2, ..., 8 (3)

where R is revenue, and seven input share equations,

There is a sixteenth equation for crude oil input to refining: 8

j=1 (1 - k) a In r/a In w = - w z l 6 / ~ = 616 + C dl6j In Ph-8 + d16.16 In w . ( 5 )

The characteristics of the restricted profit function constrain the values of the coefficients. The constraints are:

Symmetry:

c. . IJ - - Cji, e8h = ehg, dk.16 = dl6.k.

T l - k is homogeneous of degree one in output prices:

T l - k is homogeneous in input prices: I5 15 16

g=9 h=9 k = 9 C ezh + C (dk.i6+ d i a , k ) = O .

The sum of output shares (= I ) is invariant to a change in any one price: 8 8 C C i j = 0 for each J , C di.16 = 0. i = l i=l

Also, the degree of homogeneity is 16

g = 9 k = - C bg.

The coefficients of equations (3)-(5) were estimated from the pseudo-data subject to the constraints (6)-(9). As discussed in the next section, equation (2) was not included in the estimation. Thus, the coefficient uo of equation (2) was not estimated directly. Instead, as discussed in the fifth section, this coefficient was set when calibrating the forecasting equations to a base case.

GENERATION OF PSEUDO-DATA

OTM is a highly detailed linear programming model (with some non-linear features) that simulates the domestic refining sector and world markets for petroleum products for ten discrete geographic regions. As such it contains endogenous supply functions for crude oil and petroleum product imports. The pseudo-data are to be used to estimate the coefficients in


equations (3)-(5). Accordingly, we specified the OTM model for variations in each output price holding all other prices constant. Two data sets were created representing a short-run and long-run version of OTM. The short-run version fixed capacities and other operating features of refineries at the most recently available historical values (1988). In the long-run version of the model we removed all capacity constraints and allowed OTM to add capacity using the model’s own optimization criteria. For each of the short- and long-run data sets we considered three scenarios corresponding to alternative assumptions about the world oil price which spanned the range of crude oil price variation typically assumed for the annual AEO forecasts.

For each world oil price scenario, we designed a base or reference case. To do this, we ran OTM such that it satisfied US product demand quantities set equal to 1988 actual levels. To get a fully consistent set of prices and quantities several additional model runs were necessary. The point of the additional runs was to adjust the estimates of prices and quantities such that if one of them (e.g. prices) were assumed as model inputs, then the other (e.g. quantities) would be found in the model’s solution. Given a consistent base case, we made ‘scoping’ runs by varying each price above and below its base case price. The purpose of these runs was to find ‘corner solutions’ for each price; that is, the smallest price variation, both above and below, from the base case price (all else equal to the base case values) such that further price variation would produce no significant changes in domestic product output. The corner solution prices were the maximum and minimum prices used for pseudo-data generation. Sixteen price-variation runs were conducted for each product price by dividing the range of variation into 15 equal segments. We changed each price individually with all other prices set at base case values. This design provided a base case and 128 additional price-variation cases for each world oil price scenario. As a result, the short- and long-run pseudo-data sets each consist of the results of 387 OTM runs.

MODEL ESTIMATION

The model to be estimated consists of the eight output share equations (3), the seven input import share equations (4), and the crude oil input share equation (3, to which we append a vector of independently and identically normally distributed random errors e = [Ei, cg, &16].

The estimation is done subject to the constraints (6)-(9). The use of pseudo-data allows us to avoid many problems that are common in non-

experimental data. In generating the pseudo-data, each model run is independent so that there is no autocorrelation among the data. Further, the model is the same from run to run so that there is no heteroskedasticity. We generated the pseudo-data by varying relative prices to assure sufficient sampling variability. We chose the prices to be orthogonal to avoid collinearity. Finally, the exogenous variables are truly exogenous.

The estimation technique is iterated Zellner Efficient Least Squares (IZELS), also known as Iterated Seemingly Unrelated Regressions (ITSUR), which minimizes the trace of the residual covariance matrix. The error terms are correlated across equations because the revenue shares sum to one. To restore non-singularity to the residual covariance matrix, one equation must be deleted. Here, the share of other products (S,) equation is removed. In the absence of heteroskedasticity and autocorrelation, IZELS is numerically equivalent to full information maximum likelihood (Oberhofer and Kmenta, 1974). The estimates are therefore invariant to the choice of equation to be eliminated (Barton, 1969).

The translog model is only a local approximation to a given production surface. It may not fit all observations adequately. We therefore subject the model to several tests of adequacy:

426 Journal of Forecasting Vol. 12, Iss. No. 5

sample fit (coefficients of determination and t-ratios), sign conditions on the predicted shares, and convexity. The signs of the revenue shares must be positive. If the predicted revenue share is negative, the model does not fit the data well at that point. However, value shares for imported products can be positive or negative. Finally, to be well behaved, the price possibility frontier must be convex. Convexity is assured if the first seven principal minors of the Hessian matrix are positive.

The program used to produce the long- and short-run estimates are identical except for the input data sets. The long-run model uses all 387 observations while the short-run data set excludes those observations where the model ran up against one or more constraints. The short-run pseudo-data were generated with capacities and other capital facilities fixed. For the long-run pseudo-data, the model was allowed to change capacities if it was economic to do so. Although the corner solutions found with the ‘scoping runs’ were for own-price variations, a few cases of cross-price corner solutions were later found and removed from the short-run data. As a result, the final short-run data set has 369 observations.

SHORT-RUN RESULTS

Table 1 contains the summary statistics for the 16 share equations. The minimum and maximum values of the product import shares (S9-Sl5) are of opposite signs. This means that OTM exports some of the products, changing the negative value share into a positive revenue share, for some of the observations.2

The sixteen equations require the estimation of 85 coefficients. The coefficient values are given in the Appendix. Of the 85 coefficients, 70 were significant at the 0.05 level, two-tailed.

Table I . Summary statistics for fuel shares: short run

Share Name Fuel N Mean Min M U RSQ

s1 GSH s2 LPG s3 JET s4 PCH s5 DST S6 LSR s7 HSR S8 OTH s9 GSH s10 LPG s11 JET s12 PCH S13 DST S14 LSR S15 HSR S16 CRD

Gasoline Liquified petroleum gas Jet fuel Petrochemicals Distillate fuel Low-sulphur residual fuel High-sulphur residual fuel Other Imported gsh Imported lpg Imported jet Imported pch Imported dst Imported Isr Imported hsr Crude oil

369 369 369 369 369 369 369 369 369 369 369 369 369 369 369 369

0.4770 0.0697 0.1122 0.0470 0.1663 0.0453 0.0262 0.0559

- 0.0258 - 0.0096 - 0.0244 - 0.0064

0.0055 -0.0179 - 0.0047 - 0.6627

0.2912 0.0303 0.0006 O.oo00 0.0333 0.0076 O.oo00 O.oo00

- 0.0543 - 0.0288 - 0.0528 - 0.0383 - 0.0250 - 0.0895 - 0.0647 -0.7595 -

0.5734 0.741 0.1667 0.579 0.1989 0.626 0.1343 0.709 0.2874 0.604 0.1803 0.890 0.1249 0.515 0.1318 0.015 0.0013 0.589 0.0007 0.656 O.oo00 0.486 O.oo00 0.518 0.0727 0.531 0.0137 0.594 0.0175 0.242

-0.4881 0.011

’Interestingly, OTM and OMM both forecast net exports for distillate in many scenarios. Since this did not have historical precedent at the time (third quarter, 1990). the model’s reviewers expressed some consternation over these results. As it happens, the US has been a net exporter of distillate for the months of December 1990 to March 1 9 9 1 (EIA, 1991b, p. 19).


Since only asymptotic standard errors can be calculated, the appropriate distribution is normal. The squares of the simple correlation coefficients between the predicted and actual shares are presented in the column headed RSQ in Table I. Most of the R-squares appear adequate, especially for the revenue shares where the highest is 0.89 for low-sulphur resid and, except for other product (SS), the lowest is 0.515 for high-sulphur resid.

The R-square for gasoline share is a respectable 0.741. The R-squares for the product imports are also respectable except for high-sulphur resid (S15, 0.24) and crude oil (S16, 0.01 1). The particularly anaemic R-square for crude oil may be a result of the small amount of variation in the crude price.

The first seven principal minors of the Hessian matrix, estimated at each data point, are positive, showing that the translog model estimates a convex price possibility frontier. Finally, we find that the model predicts a negative revenue share for 17 observations out of 369, indicating that the model is not monotonic at those points. We found 53 sign changes in the cost share equations. Thus the translog model violates monotonicity for 17 observations and disagrees with the OTM model about importing versus exporting product in 53 cases.

In general, the estimation statistics and informal adequacy tests suggest that the translog model yields an adequate approximation of the price possibility frontier generated by the OTM model for the short-run data set. We test its performance later in the paper.

LONG-RUN RESULTS

Table I1 shows summary statistics for the long-run fuel shares. The minimum and maximum values for the import cost shares are opposite signs showing that all products except jet fuel and petrochemicals are exported at one time or another in the data set. The squares of the simple correlation coefficients between the predicted and actual shares are also reported in

Table 11. Summary statistics for fuel shares: long-run

Share Name

S1 GSH s2 LPG s3 JET s4 PCH s5 DST S6 LSR s7 HSR S8 OTH s9 GSH S10 LPG s11 JET S12 PCH S13 DST S14 HSR S15 LSR S16 CRD

~~~ ~

Fuel

Gasoline Liquified petroleum gas Fuel Petrochemicals Distillate fuel Low-sulphur residual fuel High-sulphur residual fuel Other Improted gsh Imported lpg Improted jet Imported pch Imported dst Imported hsr Imported Isr Crude oil

N Mean Min

385 385 385 385 385 385 385 385 385 385 385 385 385 385 385 385

0.4622 0.0775 0.0993 0.0456 0.1792 0.0473 0.0298 0.0587

- 0.0766 - 0.0102 - 0.0178 - 0.0002 - 0.0029 -0.0187 - 0.0045 - 0.6805

0.2621 0.0384 0.0024 O.oo00 0.0887 O.oo00 O.oo00 0.0031

- 0.21 12 - 0.0362 - 0.0937 - 0.0226 - 0.0237 - 0.0701 -0.0417 - 0.7501

0.6624 0.1645 0.2264 0.1130 0.2795 0.1130 0.0979 0.1357

- 0.0055 - 0.0037 O.oo00 O.oo00 0.0225 0.0053 0.0086

- 0.5663

0.546 0.505 0.628 0.585 0.557 0.431 0.370 0.056 0.420 0.262 0.537 0.255 0.457 0.279 0.354 0.660

Alternative ways to project the supply price of the other product category and crude runs to still are presented later. Complete results are available from the authors on request.


Table 11. Except for 'others', the R-squares are adequate for the most part, although somewhat lower than the corresponding R-squares for the short-run data set. However, the greatest difference is in the crude import share equation where the R-square is an impressive 0.66, due to the fact that the price of crude oil is allowed to vary much more widely in the long-run data set. Out of 85 coefficient estimates 13 are not significant at the 0.05 level.

The first seven principal minors of the Hessian matrix, evaluated at each data point, are all positive, signifying a convex price possibility surface for the long-run version of the translog model. Finally, the number of negative revenue shares predicted by the long-run model is only nine out of 387 observations, showing that monotonicity is violated in very few cases. The number of sign changes in the cost share equations is also small. Cost shares with predicted signs different from the actual occur in only 30 cases.

The model seems to fit the data very well. Monotonicity is violated in only a few cases, the estimated price possibility frontier is convex, the t-ratios are good, and the R-squares appear to be adequate.

MODEL ADJUSTMENTS

Solving the OMM model requires finding prices, product imports, and crude runs to still as a solution to the system of equations (2)-(5) given the quantity of each product category to be supplied and the price of crude oil. Before using the model in a given application, the model is calibrated to a base case, usually the last observed historical period. Calibration requires setting the shift coefficients ao, ai, and b,, at those values such that the solution values from the model replicate the base case exactly. Because of the separability assumption, the system of equations is decomposable. Equations (2) and (3) can be solved for the total value of products and the individual product prices independent of equations (4) and (5 ) . Given this solution, equations (4) can be solved for individual product imports and equation ( 5 ) solved for crude runs. Although equations (4) and ( 5 ) can be solved analytically, equations (2) and (3) cannot.

We initially attempted to solve equations (2) and (3) using Newton's iteration method (Shoup, 1984, pp. 64-9). Preliminary trials found this system of equations to be unstable and the initial price vector assumed was often not within the zone of convergence for the algorithm. In contrast, the eight share equations (3) were very stable. Given product quantities and an assumed total value of products, equations (3) could be solved readily (e.g. often in three or four iterations). Accordingly, the solution algorithm was revised to solve for total revenue = px independent of the individual prices. Given an assumed value for px, we can solve equations (3) numerically. Given this solution, the simulated value for px was found:

In(simulated(px)) = In(l/(l - k ) ) + ln(7r1-')/(l - k )

where k = - Cb, and In(n'-') was computed as given in equation (2). The simulated value for revenue was then compared to the value initially assumed. If the difference was sufficiently large, then the midpoint of the two values was chosen as the new assumed value and the algorithm was iterated. Using this approach, the algorithm was generally successful, but often required tens of iterations.

Further testing revealed that solutions to the nine equations (2) and (3) often involved considerable errors in the level of the prices found. In contrast, if the correct value for total revenue was specified exogenously, the prices found in solution to equation (3) were usually very accurate. The source of the error stems from the absence of equation (2) from the system


of equations for which the coefficients were estimated. Moreover, the value of k used in computing In(simulated(px)) is an artifact of the way in which the model is calibrated to a base case and the coefficient values from which it is derived are not chosen in a fashion constrained by the error associated with equation (2). As a result, we decided to amend the OMM model so that the total value of products would be estimated separately from the solution to equations (3).

The new function chosen reflected the hypothesis that the value of total refined products amounts to a markup of the price of crude oil with an adjustment for the general level of liquid throughput. The functional form estimated was

PX = Aw"QP

where Q is the sum of the amounts of all products to be produced and w is the price of crude oil. The pseudo-data set used to estimate the coefficients A, a, and p was formed by varying the level of all prices f 10% around each of the crude price cases used for the original pseudo- data. The results of the regression were good and are given in Table I11 (with t-values in parentheses).

Given the amount of each product category to be supplied and the price of crude oil, the OMM model is solved as follows. px is estimated using the new revenue equation in Table 111. Then, given px, equations (3) are solved for prices using Newton's iteration method with an error tolerance of 0.005 for each share equation. Then, given px and prices, equations (4) are solved analytically for product category imports and, finally, equation ( 5 ) is solved for crude runs.

The regression results and the results of testing the estimated model revealed that the accuracy in projecting the price for the 'other' product category was particularly poor. Inspection of the pseudo-data revealed that the variation of solution quantities for the 'other' product category was not always monotonic with respect to variations in the price of other product categories. This circumstance is due to the composite price computed for this product category being the weighted average of a group of product types related only by their small yields. These products include aviation gasoline, naphtha solvents, kerosene, lubricants, and asphalt. As a result, the statistical estimate of this particular price cannot be made with sufficient accuracy. In the same spirit as the new revenue equation, the price of the 'other' product category was computed as a markup of the crude price. The equation for this product category is

p s = B + C w

The values for B and C, given in Table IV, were derived from the OTM solutions for the base case runs for each of the crude price levels considered.

Table 111. Estimated revenue equation

Short run Long run

In ( A ) - 0.864 0.438 a 0.750 0.890

P 1.645 1.013

RSQ 0.982 0.999

(19.9) (128)

(18.4) (79.9)

Table 1V. Adjustments for other price category

Coefficient Short run Long run

B 1.95 4.00 C 0.90 0.87


Product imports are estimated within OTM by an explicit comparison of the economics of domestic versus foreign refining to the exclusion of any other consideration. In fact, there is considerable inertia to changes in petroleum trading supply patterns due to long-standing contractual relationships. Further, changes in infrastructure and other measures necessary to large changes in import and export patterns are not modelled explicitly within OTM. As a result, we decided that the model’s solution should be constrained to reporting changes in product imports consistent with historical patterns.

The adjustments called for were applied in two steps. First, the proportion of all product imports as a fraction of total products supplied was given an upper and lower bound. If the model’s solution fell outside of the bound, then total product imports were reset equal to the bound violated. Second, if the total volume of product imports was reset to a bound, then individual product imports were reset to historical proportions of the total. The short-run proportions are based upon 1989 levels. The long-run proportions reflect small adjustments to the short-run proportions to reflect increased shares consistent with current secular trends. The bounds and individual product import proportions are given in Table V.

Although an estimate of crude runs is provided by equation (9, the amount of crude runs is constrained in its relationship to other magnitudes in the model’s solution by material balance conditions. In particular, the following relationship must hold:

Total product supplied = crude runs to still + other liquid input (i.e. NGLs) + total product imports + refinery gain - liquids consumed in refining (10)

Projections of NGL are provided OMM as input from other supply modules of the AEO forecasting system. Further, refinery gain and liquids consumed as refinery fuel are small and reasonably fixed proportions of total refinery output. As a result, we decided to use the above material balance equation to compute crude runs using the model’s projection of total imports (as adjusted above), an exogenous value for NGL inputs, and assuming that (refinery gain - liquids consumed) = - 2% of total domestic production, the historical average. Table VI shows the solution to OMM with all adjustments.

The short-run version of OMM is used for forecasts one year ahead. The long-run version is used for forecasts of five or more years ahead. For intermediate projections of two, three,

Table V. Bounds and proportions for product imports ~~ ~

Product category Short run Long run ~~~

GSH LPG JET PCH DST LSR HSR Upper bound on total Lower bound on total

0.26 0.11 0.08 0.10 0.14 0.18 0.13 0.12 0.08

0.26 0.10 0.10 0.11 0.16 0.17 0.10 0.19 0.08

G. M. Lady and C. E. Moody Forecasting Marginal Costs 43 1

Table VI. Model solution algorithm

Summary of model inputs Summary of model outputs

Product supplied for eight products The price of crude oil NGL production

Eight product supply prices Seven product imports Crude runs to still

Steps of solution algorithm

(1) Calculate total revenue: px = A waQB. (2) Calculate supply prices, pi, i = 1,2, ..., 8, from equations (3) using Newton’s iteration method. (3) Recalculate the price of ‘other’ products: ps = B + Cw. (4) Solve for product imports and test to see if total imports within bound. If not, set total at bound

( 5 ) Solve for crude runs to still using the material balance equation (10). violated and recalculate product imports using the proportions in Table V.

and four years ahead, both versions of the model are solved and the solutions combined as a weighted average: Z = W S S + WLL where Z is the reported solution, ws is the short-run weight, WL is the long-run weight, and S and L are the short- and long-run solutions, respectively. The weights are as follows: two years, ws = 2/3, WL = 1/3; three years, ws = 1/2, WL = 1/2; four years, ws = 1/3, WL = 2/3.

MODEL PERFORMANCE

A test of the robustness of the model’s representation of the refining technology is the model’s performance for an out-of-sample test consisting of a case in which the price of crude oil is varied while the prices of the products are held constant. If OMM is an adequate representation of the refining technology, its solution should satisfy the materials balance conditions. A summary of short-run model solutions for liquids input and output is given in Table VII for a base, high, and low crude price case.

The material balance conditions for model solutions require that crude inputs plus other liquids inputs and refinery gain equal total refinery output. These conditions are satisfied for all solutions for both OTM and OMM. The OTM solutions are constrained to satisfy these conditions; so that, the issue at stake is whether or not the OTM can find a feasible solution. The OMM is constrained to satisfy these conditions, but not necessarily with all non-negative

Table VII. Model solutions: varying crude price with product prices fixed (millions of barrelslday)

Crude runs to

still

Low price OTM 14.79 $13.77/bbl OMM 15.74 Base case OTM 13.23 $14.1O/bbl OMM 13.26 High price OTM 12.26 $15.13/bbl OMM 10.29

Total

liquids + gain output

1.26 16.05 1.31 17.04 1.31 14.55 1.36 14.62 1.23 13.49 1.41 11.71

Other domestic Total

product imports

Total supply

Runs as To of

base runs

2.48 1.48 1.35 1.27

-0.19 1.60

18.53 18.53 15.89 15.89 13.30 13.30

112 119 nla n/a 93 78


values. In fact, both models solved for all of the cases with entirely plausible projections of refinery liquids throughput.

An analysis of these cases provides insight into differences between OTM and OMM as finally configured with the adjustments described in the previous section. Compared to the base case, the OTM projections of refinery crude input varied in the range of +12% for the low crude price to -7% for the high crude price. The corresponding variation for OMM was +19% to - 22%. These differences arise because of treatment of product imports in the two models. Since the price of imported product is the same as product supplied domestically, the objective function of OTM attempts to satisfy demand through domestic production to the highest degree possible. As a result, the OTM solutions for the cases at issue here will tend to keep the level of domestic production as high as possible with the swing in total supply reflected (to a proportionately greater degree) by changes in the level of product imports. In fact, for the high crude price case the OTM solution shows a net export of products (the volume of distillate exports exceeds imports of all other products (see footnote 2)). The OTM solutions in Table VII are another example of import projections that fail to account for the many institutional factors that would inhibit large swings in trade patterns.

As discussed in the previous section, the OMM projections are adjusted to keep product imports within the bounds of historical patterns. Consequently, for the cases in Table VII, the level of product imports in the OMM solution will change far less as total supply changes than the level of imports in the OTM solution (the OMM import solution is at the lower bound for the low crude price and base case, and at the upper bound for the high crude price case). Since OMM is constrained to satisfy the product supply quantities in the OTM solutions, the residual variation must come in variation of the projected crude input to refineries. For this reason the variation in crude input for OMM will be larger than for OTM for these cases. In terms of OMM's estimate of OTM marginal costs, the mean absolute percentage error is 5.1% for the base case, 9.6% for the low crude price case, and 16.6% for the high crude price case.

Since OMM is designed to become part of a forecasting model and will have the responsibility of forecasting petroleum product prices, we conclude with a comparison OMM and OTM for their out of sample forecasts of historical data for the years 1986, 1987, 1988,

Table VIII. Backcasting test: OMM versus OTM

Variable\ year 1986 1987 1988 1989

OMM versus actual Prices: RMSVoE Prices: RMSVoE (w/o LPG) Total imports: Vo error Crude runs: Vo error OTM versus actual Prices: RMSVoE Prices: RMS%E (w/o LPG) Total imports: % error Crude runs: Vo error OMM versus OTM Prices: RMSVoE Prices: RMSVoE (w/o LPG) Total imports: To error Crude runs: 9'0 error

9.4 9.5

-1.3 3.2

11.8 11.3 5.7 0.0

10.6 3.5

- 6.6 3.3

26.1 4.3 1.7 4.9

19.0 6.1

-12.3 3.5

5.9 3.1

16.0 1.4

17.5 9.5

- 9.9 5.9

14.3 9.4

- 8.3 2.6

3.3 1.9

-1.8 3.3

29.5 11.0 - 2.8

5.7

22.4 10.6 -1.0

0.7

5.2 1.8

- 3.8 4.9


and 1989. The capital bounds and other features of the short run version of OTM are not realistic for years further back than 1986. The results of the backcast comparison are given in Table VIII.

The error given for prices is the root mean square percentage error (RMSVoE). Since actual data for refinery gate prices do not exist for the ‘other’ price category, this price was omitted when comparing each model’s results to actual data. Generally, the largest error in backcast price projections was for the LPG product category. In fact, EIA does not collect refinery gate prices for this product category. The price used for the historical data was that of propane. This is because propane is quantitatively the largest component of LPG sales. Further, the propane price is intermediate to other LPG prices (e.g. ethane sells for lower prices while butanes sell for higher prices). Nevertheless, it is not surprising that the apparent error in the backcasts is the largest for the LPG product category. The error without LPG shows how well the models perform apart from this troublesome product category. Generally, both models perform surprisingly well, given that neither is estimated from historical data. Significantly, OMM forecasts historical data about as well as OTM does.

SUMMARY AND CONCLUSIONS

The form of OMM as ultimately designed differed in several respects from the model as it was originally conceived. The limits of the translog approximation of the scaled-down restricted profit function are a particular handicap when attempting to project price levels as compared to relative prices. Further, the use of equation (2) as an estimate of price level when the shift coefficients of the share equations are set when calibrating the model to a base case does not appear valid in any event. The stand-alone revenue equation that was used proved to be very accurate and no greater source of error than the share equations themselves. Further, the version of the model with the stand-alone revenue equation and the numerical solution to the share equations (3), even if no more accurate (and it was considerably more accurate), solved between one and two orders of magnitude faster than the numerical solution to systems (2) and (3) apart from its occasional failure to converge at all. As a practical matter, use of the stand- alone revenue equation was a successful compromise.

Since the ‘other’ product category is the most subject to aggregation error, the failure to derive good estimates for this category is not surprising. Inspection of the pseudo-data revealed very non-systematic responses in model solutions to the systematic price-variation scenarios. The adjusted forecast for this price was considerably more accurate than that provided from the solution to the share equations.

Product imports are quantitatively small and highly volatile as solution variables. Further, the OTM model itself may not adequately constrain its estimate given historical trade patterns. Although the method for bounding and apportioning product import quantities based upon historical patterns is the least satisfying aspect of the adjustments for OMM, we concluded that the cost of seeking improvements exceeds the potential for a significantly better model. The advantage of estimating crude runs from the material balance equation follows in part from the potential for adjusting imports, and, more significantly, from the fact that the NGL input to refining is projected outside of OMM. Given that the total volume of products is determined through iteration and NGL inputs are given, there is no reason to submit the crude input variable to the potential errors of the estimate of the price possibility frontier. The model forecasts the remaining seven product prices with no adjustments.

Altogether, the attempt to approximate OTM with OMM was a success. In the comparisons

434 Journal of Forecasting Vol. rz, Iss. No. 5

made, OMM provided very accurate estimates of OTM. Yet OMM solves in a second or two compared to many minutes for OTM. Moreover, the size of the model itself when installed as software is vanishingly small compared to a large, array-driven linear (and, in some respects, non-linear) program such as OTM. The speed of OMM is an important consideration in that the model must be iterated with a demand module in the many grand iterations of the AEO modelling system in seeking multi-year equilibria. The size of the model is important if the entire modelling system is to fit on a microcomputer. If changes in technology, environmental constraints, the characteristics of petroleum products, or other circumstances cause the current pseudo-data base to go out of date, then the re-estimation of OMM could be achieved easily. Finally, OMM forecasts historical data as well as the much larger OTM model, at a far lower cost. Forecasts of petroleum product prices based on OMM are available in EIA (1991), Tables A3, B3, C3, and D3.

ACKNOWLEDGEMENTS

The development of the version of the OMM model reported here was sponsored by the US Department of Energy’s Energy Information Administration (EIA) and conducted as an account of work by Decision Analysis Corporation of Virginia (DAC). William Johnson of DAC was responsible for the many technical decisions necessary to using the Oil Trade Model (OTM) as the source of pseudo-data. EIA also sponsored a continuous technical review of the model’s development. Helpful comments and suggestions were received from Dr William Robinson of EIA and Professors Timothy Considine of Pennsylvania State University and Alan Manne of Stanford University. The present paper was also substantially improved by the suggestions of an anonymous referee. Of course, responsibiiity for any remaining errors resides with the authors.

REFERENCES

Barton, A. P., ‘Maximum likelihood estimation of a complete system of demand equations’, European

Christensen, L. R., ‘Conjugate duality and the transcendental logarithmic production function’,

DAC, The Oil Trade Model (OTM) Documentation, User’s Manual, and Data Base, Vienna, VA:

EIA, Annual Energy Outlook 1990: Long Term Projections, DOEIEIA-0383(90), Washington, DC:

EIA, 199I Annual Energy Outlook, DOE/EIA-0383(91), Washington, DC: Energy Information

EIA, Petroleum Supply Monthly, DOE/EIA-0109(91/04), Washington, DC: Energy Information

Finan, W. F. and Griffin, J. M., ‘A post Arab oil embargo comparison of alternative 1-0 coefficient

Griffin, J . M., ‘Long-run production modeling with pseudo data: electric power generation’, Bell Journal

Griffin, J . M., ‘The econometrics of joint production: another approach’, Review of Economics and

Griffin, J . M., ‘Joint production technology: the case of petrochemicals’, Economerrica, 46 (1978),

Griffin, J . M., ‘Statistical cost analysis revisited’, Quarterly Journal of Economics, 93 (1979), 107-129.

Economic Review, 1 (1969), 7-73.

Econometrica, 39 (1971), 255-256.

Decision Analysis Corporation of Virginia, f989.

Energy Information Administration, US Department of Energy, 1990.

Administration, US Department of Energy, 1991a.

Administration, US Department of Energy, 1991b.

modelling techniques’, Resources and Energy, 1 (1978), 315-324.

of Economics, 8 (1977a), 112-127.

Statistics, 59 (1977b), 389-397.

379-396.


Lau, L., ‘Applications of profit functions’, in Fuss and McFadden (eds), Production Economics: A Dual

Oberhofer, W. and Kmenta, J . , ‘A general procedure for obtaining maximum likelihood estimates in

Shoup, T. E., Applied Numerical Methods for the Microcomputer, Englewood Cliffs, NJ, Prentice-Hall,

Approach to Theory and Applications, Vol. 1, Amsterdam: North-Holland, 1978, pp. 133-216.

generalized regression models’, Econometricu, 42 (1974), 579-590.

1984.

APPENDIX

Table AI. Short-run revenue share translog coefficients

COL# 0 1 2 3 4 5 6 7 8 16 EQ# Shift GSH LPG JET PCH DST LSR HSR OTH CRD

0.632 0.231

- 0.120 -0.917

1.197 0.394 0.293

- 0.710

1.849 - 0.223 -0.818 - 0.952

0.170 0.130 0.041

-0.197

- 0.223 0.146 0.115 0.082

-0.117 0.009

- 0.001 - 0.010

- 0.818 0.115 3.076

- 0.366 -1.895 -0.124 - 0.043

0.056

- 0.952

- 0.366 0.082

2.423 - 0.928 - 0.409 - 0.105

0.256

0.170 -0.117 -1.895 - 0.928

- 0.030

- 0.333

3.114

0.019

0.130 0.009

-0.124 - 0.409 - 0.030

-0.119 - 0.065

0.609

0.041 - 0.001 - 0.043 - 0.105

0.019

0.245 -0.119

- 0.037

- 0.197 - 0.010

0.056 0.256

- 0.333 - 0.065 - 0.037

0.331

- 0.144 - 0.033

0.105 0.347

-0.390 -0.111 - 0.070

0.296

Table AII. Short-run import and crude value share translog coefficients

COL# 0 9 10 11 12 13 14 15 16 EQ# Shift GSH LPG JET PCH DST LSR HSR CRD

9 - 0.001 10 0.083 11 -0.000 12 -0.065 13 0.005 14 0.146 15 0.021 16 0.567

0.309 - 0.058 - 0.097 - 0.190 - 0.055 - 0.037

0.042 0.087

- 0.058 0.039 0.032

- 0.004 - 0.046

0.016 0.001 0.004

- 0.097 0.032 0.510

- 0.043 - 0.354 - 0.036 - 0.005

0.004

- 0.190 - 0.004 - 0.043

- 0.294 - 0.097 - 0.002

0.061

0.591

- 0.055 - 0.046 - 0.354 - 0.294

0.823 - 0.162 - 0.024

0.106

- 0.037

- 0.036 - 0.097 - 0.162

0.016

0.306

0.086 - 0.108

0.042 0.001

- 0.005 - 0.002 - 0.024 - 0.108

0.097 - 0.000

0.087 0.004 0.004 0.061 0.106 0.086

- 0.000 - 0.330

~~~~~ ~

Table AIII. Long-run revenue share translog coefficients

COL# 0 1 2 3 4 5 6 7 8 16 EQ# Shift GSH LPG JET PCH DST LSR HSR OTH CRD

1 - 0.737 2 0.378 3 0.620 4 0.098 5 0.253 6 0.236 7 0.750 8 - 0.597

4.719 - 0.630 -1.995 -1.182 -0.084 - 0.303 - 0.404 -0.123

- 0.630 0.577 0.069 0.323

- 0.235 -0.173

0.034 0.035

-1.995 0.069 8.340

-1.535 - 4.457 - 0.305 - 0.003 -0.114

-1.182 0.323

-1.535 3.199

- 0.447 -0.154 - 0.260

0.056

- 0.084 - 0.235 - 4.457 - 0.447

5.675 - 0.235 - 0.181 - 0.036

- 0.303 -0.173 - 0.305 -0.154 - 0.235

1.507 - 0.358

0.021

- 0.404

- 0.003 - 0.260 -0.181 -0.358

1.284 -0.112

0.034 - 0.123

0.035

0.056 - 0.036

0.021 -0.112

0.274

-0.114

0.146 - 0.018 - 0.142

0.034 - 0.058 - 0.057 -0.150

0.246


Table AIV. Long-run import and crude value share translog coefficients

COL# 0 9 10 11 12 13 14 15 16 EQ# Shift GSH LPG JET PCH DST LSR HSR CRD

9 - 0.574 10 0.162 1 1 0.290 12 -0.005 13 0.048 14 0.270 15 0.220 16 0.402

1.868 - 0.27 1 -1.265 -0.123 - 0.03 1 - 0.287 - 0.145

0.373

- 0.27 1 0.166 0.161 0.001

- 0.041 -0.114

0.019 0.057

-1.265 0.161 2.456

- 0.237 - 0.740 - 0.384 - 0.236

0.193

- 0.123 0.001

- 0.237 0.493

- 0.080 - 0.012 - 0.066

0.025

-0.031 -0.041 -0.740 - 0.080

0.876 - 0.063

0.001 0.064

- 0.287 -0.114 - 0.384 -0.012 - 0.063

0.990

0.026 - 0.228

- 0.145

- 0.236 - 0.066

0.001 - 0.228

0.670

0.019

- 0.037

0.373 0.057 0. I93 0.025 0.064 0.026

- 0.037 -0.641

Authors ’ biographies: George M. Lady is Associate Professor of Economics at Temple University. His research areas are mathemetical enconomics, methodology, and energy economics, and his publications include articles in the journals Econometrica, Journal of Mathematical Social Sciences, and Energy Economics. Carlisle E. Moody is Professor of Economics at the College of William and Mary. His research areas are applied econometrics and forecasting and his publications include articles in the journals Land Economics, Rand Journal of Economics and Energy Economics.

Authors’ addresses: George M. Lady, Economics Department, Temple University, Philadelphia, PA 19122, USA. Carlisle E. Moody, Economics Department, College of William and Mary, Willamsburg, VA 23185, USA.

forecasting marginal costs of a multiple-output production technology

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