An integrated model of oil production.

1. INTRODUCTION

What determines oil production in a specific region? Physical scientists such as the late M. King Hubbert (1956, 1962, 1967, 1971, 1974, 1982) stress that current production is governed mainly by cumulative production and technological conditions, but downplay the role

of explicit economic variables.(1) Hubbert's view is that cumulative production in a region is a proxy for geological knowledge gained through exploration and development. In mature regions that have been produced for many years, it eventually comes to reflect the inescapable fact of reserve depletion. And since reserves ultimately govern production, the eventual decline in reserves causes an eventual decline in the rate of production.

By contrast, economists stress that production is governed mainly by expected profits. They emphasize the importance of real prices, costs, taxes, and other economic variables, but pay secondary attention to geophysical constraints [Adelman (1993, 1995), Adelman and Lynch (1997), Porter (1995, 1997)]. Thus in modeling production, physical scientists and economists focus on different explanatory variables. Cleveland (1991), Kaufmann (1991), Cleveland and Kaufmann (1991), and Pesaran and Samiei (1995) have successfully incorporated both physical and economic variables in Hubbert-type models, as we discuss below.

The goals of this paper are three. First, we show how physical reserves and economic variables can be combined to yield a more reliable model of oil production than models based either on reserves alone or on economic variables alone. Second, we specify and estimate four alternative models of annual oil production in the U.S. lower 48 states. The sample period is 1950 to 1996.(2) Third, we use the four models to forecast production for the years 1994-1996. A partial adjustment model based on reserves, lagged production, and the real price of oil yields much smaller forecast errors than models based on reserves alone or on real price alone.

Model I stipulates that current production is determined by last year's reserves and nothing else. Model II stipulates that current lower 48 production depends on the current real price of oil and prorationing of production in Texas until 1973, but not explicitly on reserves. Model III combines the reserves in Model I with the real price and prorationing of Model II. Finally, Model IV extends Model III by incorporating a simple distributed lag in the adjustment of actual production to the desired production rate. We find that oil reserves are a key determinant of production. We also find that the real price of oil and prorationing are important in explaining production. After accounting for the non-price influences on production, the one-year elasticity of supply with respect to the real price of oil is approximately 0.03; the long-run equilibrium elasticity is about 0.12. The long-run elasticity of production with respect to last year's reserves is approximately one or somewhat larger, depending on the model specification.

2. HUBBERT'S MODEL AND ECONOMIC EXTENSIONS

In 1956 Hubbert predicted that oil production in the lower 48 would reach a peak in 1970. Hubbert was correct. Figure 1 shows that annual oil production in the lower 48 increased from about 2600 million barrels in 1956 to a peak in 1970 of slightly more than 3200 million barrels. Since 1970, production has declined almost monotonically (a slight increase occurred between 1982 and 1984) to a rate of about 1663 million barrels per year in 1996.(3) So by 1996, production in the lower 48 had fallen to roughly one-half of the all-time high reached in 1996.(4)

Hubbert forecasted the growth, the peak, and the decline in lower 48 oil production using a life-cycle model he thought generally applicable to exhaustible resources. His life-cycle model (1962, 1967, 1982) is based on a logistic curve to describe the time path of cumulative production.

Hubbert's model of cumulative oil production, [Q.sub.t], can be written as

[Q.sub.t] = [Q.sub.[infinity]]/[1 + [N.sub.0][e.sup.-a(t - [t.sub.0])]], (1)

where t is time, [t.sub.0] is an arbitrary starting date for observations, [Q.sub.[infinity]] is ultimate cumulative production, and [N.sub.0] = ([Q.sub.[infinity]] - [Q.sub.0])/[Q.sub.0]. [N.sub.0] has important geophysical meaning: since [Q.sub.0] is cumulative production up through year [t.sub.0], ([Q.sub.[infinity]] - [Q.sub.0])/[Q.sub.0] is the volume remaining to be produced expressed as a percentage of cumulative production up through time to.

To express the annual production rate, differentiate equation (1) with respect to time to obtain

d[Q.sub.t]/dt = [q.sub.t] = [Q.sub.[infinity]] a[N.sub.0][e.sup.a(t - [t.sub.0])] / [1 + [N.sub.0][e.sup.-a(t-[t.sub.0)].sup.2] (2)

Equation (2) is a symmetrical, bell-shaped curve for the annual rate of production, [q.sub.t]. By making an appropriate choice of t and [t.sub.0], one may use a stochastic version of equation (2) to estimate the three parameters [Q.sub.[infinity]], a, and [N.sub.0].

But equation (2) is highly nonlinear in these parameters. So researchers have estimated them in one of two ways. Pesaran and Samiei (1995) estimated three parameters in one step using maximum likelihood non-linear least squares.(5)

Alternatively, [Q.sub.[infinity]], a, and [N.sub.0] can be estimated from equation (1) by choosing initial values for [Q.sub.t] and [t.sub.0], then performing an iterative grid search over different values of a and [N.sub.0] to estimate [Q.sub.t].(6) The first differences in the successive estimates of cumulative production then yield estimates of the annual rate of production, conditional on [Q.sub.[infinity]], [t.sub.0], [Mathematical Expression Omitted], and [Mathematical Expression Omitted]:

[Mathematical Expression Omitted] (3)

The estimated values of [Mathematical Expression Omitted] trace a symmetric, bell-shaped curve of annual oil production. Kaufmann (1991) and Cleveland and Kaufmann (1991) quite naturally call this a production curve (or we could think of it as an estimated production curve derived from the estimated parameters of a Hubbert-type logistic equation). Kaufmann (1991), Cleveland and Kaufmann (1991), and Pesaran and Samiei (1995) attempted to use economic and political variables to explain the differences between actual production rates [q.sub.t] and Hubbert-type estimates of production rates [Mathematical Expression Omitted]. We presently discuss this in more detail.

Despite its success in predicting annual oil production, Hubbert's lifecycle model has been widely criticized. One shortcoming is that it includes no explicit economic variables. Ryan (1965) stresses that the ultimate recovery of oil or natural gas changes with economic and technological conditions. As the price of oil increases, secondary recovery efforts may become profitable.

Technological advances may also lower the cost of secondary recovery, and make such efforts profitable. Ryan dismisses Hubbert's method of forecasting as unreliable, and believes Hubbert's equation for cumulative production cannot distinguish reliably among a wide range of values for ultimate recovery.

Wiorkowski (1981) reviews much of the literature on statistical methods that have been used to estimate the volume of oil and natural gas ultimately recoverable. Fletcher (1974) criticizes the use of the logistic function because of the symmetry it imposes on the annual production curve: the right and left sides of the curve must be mirror images. To avoid the restriction of symmetry, Moore (1970) uses the Gompertz curve. Wiorkowski estimated the Weibull function and generalized Richards functions, since they are less restrictive than the logistic equation.

Cleveland and Kaufmann (1991) compare production curves estimated by each of the above techniques to an estimate using Hubbert's logistic model. They find that, "Despite its lack of a theoretical basis and its alleged misspecification, Hubbert's symmetric parabola predicts production in the lower 48 states more accurately than asymmetric curves, economic models, or delphi techniques."(7)

Kaufmann (1991) uses a two stage process to combine economic and regulatory variables with Hubbert's method of forecasting production. The first stage is to take the difference between actual production [q.sub.t] and a production curve estimate [Mathematical Expression Omitted], and then to divide this difference by the estimated value.

[Mathematical Expression Omitted] (4)

[R.sub.t] represents errors of estimate expressed in percentages. In the second stage, [R.sub.t] is regressed against a set of economic and regulatory variables. [Mathematical Expression Omitted] is used to construct a new forecast by multiplying the production curve forecast by [Mathematical Expression Omitted]. Kaufmann found the result to be a curve that fits the actual data quite closely. Pesaran and Samiei (1995) extend in several ways the suggestions of Kaufmann (1991) and Cleveland and Kaufmann (1991) to combine economic variables with Hubbert-type models.

Is there not a more direct method of estimating production? Why does the first difference in cumulative production derived from a logistic time trend alone fit the annual oil production rates in the U.S. lower 48 states so closely?(8) A time trend alone explains nothing. It is merely a proxy, or instrument, for geophysical and economic variables that themselves evolve over time.(9) To understand the time path of production one must identify these variables.

The Importance of Reserves

Compare the time path of reserves in Figure 2 to the path of annual production in Figure 1. The shape of the reserve curve is similar to the actual production curve, although reserves reach their peak in 1961 while production peaks in 1970. Because reserves are the physical inventory that feeds production, the rise and fall of production must roughly follow the rise and fall of reserves. The two would coincide, of course, if the ratio of reserves to production were constant. In fact, the reserve/production ratio varied from about 12.5 during the years in 1950-1955 to about 10.0 during the years 19901996.

In the following section we show how economic variables can be combined with reserves to obtain a more reliable model than models based either on reserves alone or on economic variables alone.

3. THE MODELS:

Combining Reserves with Economics and Regulation

Before oil production can occur, an inventory of developed reserves must already be on hand. Successful exploration leads to new oil field discoveries. Development of these discoveries increases the stock of reserves.

Today's stock of reserves results from investment and production decisions made in the past. If investors determine that the expected profit from oil exploration is higher than that from competing investment opportunities, then exploratory investment occurs. The expected profit calculation involves the likelihood of discovery, the expected wellhead price, the expected price of related goods and the expected cost of exploratory drilling. Successful exploration leads to development of reserves which can then be produced.

Pindyck (1978) developed an elegant model of a competitive firm that produces and explores for an exhaustible resource. By allowing the firm to devote resources to exploration as well as to production, Pindyck extends Hotelling's (1931) well-known model in several ways.

Pindyck's model has two critical implications for modeling oil production. The first concerns the firm's decision to produce or not produce. If the real wellhead price of oil exceeds the marginal cost of production from current reserves, then it produces. Second, because the discounted present value of the firm is linear (not concave) in current production, whenever the firm produces, it produces at the maximum rate permitted by reserves and geophysical conditions (downhole pressure, intrusion of water into the volumes of produced liquids, and so on). Thus reserves are an essential geophysical constraint on production. But the decisions to produce or not to produce from current reserves, and to invest in finding new reserves, are based fundamentally on economics.

In years past, oil production was subject to considerable government regulation. The Texas Railroad Commission began regulating oil production in the 1930's. At the time, Texas accounted for 76% of total U.S. production. By restricting Texas' production, the Texas Railroad Commission was able to boost prices to levels higher than they otherwise would have been. The Commission computed the so-called allowable production by a formula specifying the number of days in the year that oil wells were allowed to produce. By limiting supply, higher wellhead prices were supported. Since 1973, Texas oil production has been unrestricted.

The percentage of days in a year when production was allowed is designated here as TRC. From an economic viewpoint, TRC is a government-mandated capacity constraint. 1962 was the year of tightest constraint, when production was limited to 0.27 of the year. Since 1973 production has been unrestricted, meaning that TRC = 1.0 from 1973 onward. Our use of the TRC capacity constraint is quite similar to the methodology originally suggested by Kaufmann (1991, especially pp. 117-119).

Geophysical and Econometric Models

3.1 Model I (Pure Geophysical Model)

In modeling production, petroleum engineers often focus upon the geophysical variables such as reserves, water intrusion, permeability of the deposits, and so on. Pindyck's model, as noted above, implies that known reserves are a partial constraint on production. The model can be specified as

ln([q.sub.t]) = ln ([[Alpha].sub.0]) + [[Alpha].sub.1] ln ([RESRV.sub.t-1]) + [[Alpha].sub.4] dum1 + [e.sub.1t], (5)

where [RESRV.sub.t-1] is the stock of reserves measured at the end of the previous year, dum 1 is a dummy variable to measure the significance of the change in the data gathering agency (see the discussion in the data section) and [e.sub.1t] is an error term. If [[Alpha].sub.1] [greater than] 0, an increase in the stock of known reserves increases production. The logarithm of lagged reserves is a predetermined variable: Its value influences current-year production, but cannot be codetermined with it. Model I omits real price as an explicit economic variable and prorationing as a production constraint.

3.2 Model II (Price/Regulatory Model)

The price/regulatory model, Model II, specifies that the real price of oil and the TRC variable determine current production. Thus, Model II omits known reserves, the key geophysical variable in Model I:

[Mathematical Expression Omitted]. (6)

[[Alpha].sub.2] shows the elasticity of supply in response to changes in real price, given last year's reserves and the pre-1973 prorationing of production in Texas. And [[Alpha].sub.3] [greater than] 0 shows the elasticity of production in the lower 48 in response to changes in prorationing by the Texas Railroad Commission. Model II can be interpreted as a conditional supply equation. On this interpretation, [[Alpha].sub.2] is the long-run elasticity of supply. To view equation (6) rigorously as a conditional supply equation requires that we treat real oil prices and the TRC prorationing constraint as predetermined variables.

3.3 Model III (Integrated Model)

One may easily combine Models I and II as follows:

[Mathematical Expression Omitted], (7)

In natural logs this is:

ln ([q.sub.t]) = ln([Alpha]) + [[Alpha].sub.1] ln([RESRV.sub.t-1]) + [[Alpha].sub.2] ln([RP.sub.t]) + [[Alpha].sub.3] [TRC.sub.t] + [[Alpha].sub.4] dum1 + [[Epsilon].sub.3t]. (8)

Model I and Model II are nested in Model III. If [[Alpha].sub.2] and [[Alpha].sub.3] are restricted to equal zero, we obtain Model I. If [[Alpha].sub.1] is restricted to equal zero, we obtain Model II. Model III is the unrestricted integrated model against which the restrictions of Models I and II can be tested. The interesting econometric question is this: How much more information about oil production do we gain from the integrated Model III by comparison with the information in Model I alone or the information in Model II alone? As we show in section 5, the answer is quite a lot.

3.4 Model IV (Integrated Partial Adjustment Model)

One can specify Model III as a partial adjustment model and there are sound economic reasons to do so.(10) Once an oil well is in production, lifting costs may be low relative to the costs of adjusting the production rate. If so, producers react gradually to changes in the economic environment. Rather than react instantaneously to every change in price, producers adjust production in stages as trends in the movement of price become clear. The desired level of production is:

[Mathematical Expression Omitted], (9)

and taking logarithms, we obtain

[Mathematical Expression Omitted]. (10)

[Mathematical Expression Omitted] is the logarithm of the desired level of oil production in period t. This target level of output is determined using reserves on hand at the beginning of the year, the current real price of oil, and the TRC prorationing variable.

Logarithmic adjustment of actual production to the desired output takes place according to:

[Mathematical Expression Omitted] (11)

[Delta] represents the rate at which producers adjust ln([q.sub.t]) to the desired level [Mathematical Expression Omitted]. If [Delta] = 0, there is no adjustment, and if [Delta] = 1, adjustment is complete within the year. The smaller is [Delta] the slower the adjustment. The error term [[Omega].sub.t], represents the disturbance in the adjustment process.

Substituting equation (10) into equation (11) and simplifying gives the partial adjustment equation.

[Mathematical Expression Omitted] (12)

or

[Mathematical Expression Omitted] (13)

The coefficients [[Gamma].sub.1], ..., [[Gamma].sub.5] are short-run (one-year) coefficients of production. The long-run coefficients can be obtained from the coefficients of equation (13) as follows:

[Delta] = 1 - [[Gamma].sub.5]

ln([[Beta].sub.0]) = [[Gamma].sub.0]/1 - [[Gamma].sub.5]

[[Beta].sub.1] = [[Gamma].sub.1]/1 - [[Gamma].sub.5]

[[Beta].sub.2] = [[Gamma].sub.2]/1 - [[Gamma].sub.5]

[[Beta].sub.3] = [[Gamma].sub.3]/1 - [[Gamma].sub.5]

[[Beta].sub.4] = [[Gamma].sub.4]/1 - [[Gamma].sub.5] (14)

[[Beta].sub.1] is the long-run elasticity of production with respect to reserves, and [[Beta].sub.2] is the long-run elasticity of production with respect to the real price of oil. If the error term [[Omega].sub.t], is serially independent, the estimated coefficients of equation (13) are asymptotically unbiased and consistent. If [[Omega].sub.t] is normally distributed, then the estimators [Mathematical Expression Omitted], are also normally distributed, and [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted] are asymptotically normal.

4. DATA

Twentieth Century Petroleum Statistics, 1997 (TCPS) published by DeGolyer and MacNaughton (1997) contains most of the data used here. Production and reserve data were collected by the American Petroleum Institute (API) for years before 1977. But these data were collected by the Energy Information Agency (EIA) from 1977 onward. Because the two agencies used different methods of measuring and reporting these series, a dummy variable, labeled dum1, is used to measure the significance of this change. From 1950 to 1976, dum1 has a value of zero. From 1977 to 1991, dum1 has a value of one. The series we label RESRV consists of year-end reserves for the lower 48 states measured in millions of barrels. This series was constructed in accordance with the U.S. Department of Energy, Energy Information Administration (1984) report which describes a procedure for linking the API series with the EIA series.(11) For years 1949 to 1964, RESRV was calculated by subtracting Alaska's reserves from those of the entire U.S.

The series for annual oil production in the lower 48 states is taken from TCPS (1997). Production is measured in millions of barrels. For years prior to 1966, but after production began in Alaska, total U.S. production is adjusted downward each year by subtracting Alaska's annual production to give production in the lower 48. This series is labeled q and contains observations from 1950 to 1996.

The U.S. average wellhead price of oil per barrel in current dollars is located on page 104 of TCPS (1997). To convert these to 1982 dollars we deflate current-dollar prices by the producer's price index located on page 103. The resulting series of constant dollar wellhead prices is labeled RP.

We obtain the percentage of each year Texas oil wells were allowed to produce from the 1989 Oil and Gas Annual Report, Railroad Commission of Texas, page I-6. After 1972, Texas oil production was not prorationed. The final series, labeled TRC, is stated in decimals rather than percentages and ranges from 1950 to 1996. The value of TRC is 1.00 after 1972, signifying that no production was prorationed.

5. ESTIMATION

Results from estimating Models I through IV are collected in Table 1. The very low Durbin-Watson statistics from Models I, II, and III show that each exhibits highly autocorrelated residuals. For the moment, ignore this obviously serious problem and the fact that the reported standard errors and t statistics (for testing hypotheses that the parameters are different from zero) are incorrect.

Focusing on the estimated coefficients, we find several consistent patterns:

1. Models I, III, and IV yield long-run elasticities of production with respect to lagged reserves of one or slightly higher. The estimate of 1.291 from Model IV is statistically different from zero at P [less than or equal to] 0.01, but is not significantly different from one. (If the (reserve/production) ratio were constant, this elasticity would be exactly one. In fact, the reserve/production ratio exhibited some variation in the sample.)

2. Models II, III, and IV all show positive but very low elasticities of production with respect to the real price of oil. The estimated elasticities range from 0.057 to 0.190, but only in Model II is the estimate statistically different from zero.(12) The estimated long-run elasticity from Model IV is 0.123.

3. As the Texas Railroad Commission relaxed its prorationing constant, oil production in the lower 48 increased. Our estimate of 0.531 obtained from Model III is most nearly comparable to the estimate of 0.36 obtained by Kaufmann (1991, page 119).

[TABULAR DATA FOR TABLE 1 OMITTED]

The estimated coefficient of adjustment in Model IV is significantly different form zero and significantly less than one. Adjustment is incomplete within one year, so the time-series on production contains quite a bit of inertia: each year's production depends a great deal on last year's production, other things equal.

Model IV appears to be well specified. The Breusch-Godfrey test(13) for serially independent disturbances shows that the hypothesis of non-autocorrelated disturbances is acceptable. We regard this integrated model with partial adjustment to be the most reliable of the four alternatives.

A Further Comparison of the Models

Even though Models I, II, and III display strongly autocorrelated errors, it is useful to compare them in three ways:

5.1 Comparisons of R-squares

The percentage of variation in the logarithm of production around its mean explained by Geophysical Model I is [R.sup.2](ModelI) = 0.650. And the percentage of variation explained by Price/Regulatory Model II is [[R.sup.2].sub.(ModelII)] = 0.517. But combining Models I and II into the Integrated Model III yields an [[R.sup.2].sub.(ModelIII)] of 0.867. Thus the Integrated Model III is a substantive improvement over either Model I or Model II. That [[R.sup.2].sub.(ModelIII)] is so much larger than [[R.sup.2].sub.(ModelI)] or [[R.sup.2].sub.(ModelII)] means that the explanatory information in Model I (contained in lagged reserves) is different from the explanatory information in Model II (contained in real oil prices and the TRC prorationing constant). By adding lagged reserves from Model I we greatly increase the explanatory power of Model II. By adding real oil prices and prorationing from Model II we greatly increase the explanatory power of Model I. And to restrict ourselves either to Model I or to Model II is to commit model specification error.

Yet even the integrated Model III contains specification error of some sort, as shown by its strongly autocorrelated residuals. We believe that the considerable inertia in annual production can be adequately dealt with by the simple partial adjustment incorporated in Model IV.

5.2 Illustrative Tests of Model Restrictions

The integrated model is unrestricted because it includes all of the variables in Models I and II. We test the Model II restriction that [[Alpha].sub.1] = 0 by means of an F-test. The calculated value for F is 111.01. At the 1% level of significance, we reject the restriction [[Alpha].sub.1] = 0 in favor of the alternative that ln([RESRV.sub.t-1]) is significant.

To test the Model I restrictions that [[Alpha].sub.2] = 0 and [[Alpha].sub.3] = 0 the computed F-statistic is 68.99. The critical value for [F.sub.(2,42)] is 5.16 at the 1% significance level. The null hypotheses [[Alpha].sub.2] = [[Alpha].sub.3] = 0 is rejected, indicating that both ln([RP.sub.t]) and [TRC.sub.t] belong in the model. We stress that these tests of restrictions are illustrative, not precise, because they are conducted with residuals that are highly autocorrelated. The consequence of autocorrelation is that our F-tests do not have the usual properties as when the disturbances are normally and independently distributed.

5.3 Time Paths of Actual and Estimated Production

The antilogarithms of actual and estimated production obtained from Model I are shown in Figure 3. They show two things quite clearly. First, the (antilogarithmic) errors of estimate are strikingly autocorrelated. Second, the Model's estimated production begins a sharp decline in 1968, when reserves began their downward march. But actual production commenced a strong decline four years later. Model I shows with stark clarity that production follows reserves.

Model II is a very different story. The antilogarithms of actual and estimated production are shown in Figure 4. The antilogarithms of estimated production are practically constant from 1950 through 1973 because the real price of oil was nearly constant: during these 24 years the real price ranged from a minimum of $8.32 a barrel in 1951 to a maximum of $9.90 a barrel in 1957. Actual production increased from 1950 through 1970 chiefly because of larger reserves, which are omitted from Model II. Only after 1975 does Model II begin to track (with large errors!) the downturn in production.

Antilogarithms of estimated production obtained from Model III are much more on target. Figure 5 shows that Model III sort of mimics the production increase until 1970 because it includes the information on increases in reserves. Then it closely tracks the protracted production decline from 1972 through 1996. Model III displays an obviously stronger ability than Model I or Model II to explain the time path of production.

Antilogarithms of actual and estimated production using Model IV are shown in Figure 6. This model, which is Model III supplemented by the logarithm of last year's production, is a major improvement over any of the other models. Its in-sample errors of estimate are quite small. And as we see in the next section, Model IV produces much more accurate forecasts than the other models.

6. ALTERNATIVE MODEL FORECASTS

To test the forecasting properties of the four models, we re-estimate each model using the subsample years 1950-1993. We then make forecasts of oil production for the years 1994-1996, conditional on the known values of the explanatory variables in each model. Table 2 presents these forecasts from the four models for the years 1994-1996. One is struck by how poorly the price/regulatory Model II forecasts production. Its forecasts are always too high with a large forecast error. Its mean absolute percentage forecast error is a whopping 25.9 percent.

Table 2. Results of 3-Year Ahead Forecast

                               1994       1995        1996

Actual                        1697       1673       1663
Model I Geophysical           1910.2     1866.7     1873.9
Model II Price/Regulatory     2083.6     2099.6     2150.3
Model III Integrated          1494.4     1418.8     1457.3
Model IV Long Run Adj.        1679.8     1607.1     1566.5

                               Mean Absolute      Mean Absolute
                                   Error             % Error

Model I Geophysical                205.9              12.3%
Model II Price/Regulatory          433.5              25.9%
Model III Integrated               220.8              13.2%
Model IV Long Run Adj.              59.8               3.6%

The pure geophysical Model I forecasts much better than the price/regulatory Model II, implying that the stock of oil reserves is a better predictor of production than the real price of oil. Nonetheless, Model I always yields production forecasts that are too high. Its mean absolute percentage forecast error of 12.3 percent is far from reassuring. By contrast, integrated Model III always produces forecasts that are too low. Its mean absolute error and mean absolute percentage error are comparable to those of Model I - uncomfortably large.

Model IV easily wins the forecasting contest. Its forecast errors are obviously much smaller than those of Models I, II, and III. The main reason, of course, is that Model IV has lagged production as a right-hand variable that works to put its forecasts somewhat on track. Its one-year ahead forecast error in 1994 is very small, but its three-year ahead forecast error in 1996 is 96 million barrels, or about 5.8 percent of actual 1996 production. The lesson is clear: lagged production greatly improves forecasts of future production. Because Model IV includes lagged production, its forecasts are much more accurate than those obtained from models that exclude it.

7. SUMMARY AND CONCLUSIONS

1. Predicting production by reserves alone (Model I) or by the real price of oil alone (Model II) yields needlessly large forecast errors. A simple model that combines reserves, the real price of oil, and lagged production (Model IV) yields much more accurate production forecasts.

2. The partial adjustment Model IV yields the smallest in-sample sum of squared residuals, and much smaller forecast errors than the other three models.

3. Models III and IV combine the physical constraint on production imposed by reserves with economic conditions embedded in real oil prices and, until 1973, the political constraint of prorationing. The obvious advantage of Models III and IV is that changes in reserves and economic conditions can be analyzed simultaneously.

4. The purpose of a model is important. Hubbert's use of the first derivative of a logistic function to predict lower 48 peak annual oil production in 1970 was on target. His model also predicts correctly the continuing decline in lower 48 annual oil production. But we believe that an integrated, partial adjustment model along the lines proposed here is preferable for assessing the impact of regulation or a change in economic conditions on oil production.

We are pleased to acknowledge the suggestions of participants in the Energy/Resources Workshop at Texas A&M University, Dr. Robert A. Wattenbarger, Department of Petroleum Engineering, and of three anonymous referees. This research was supported by an Interdisciplinary Research Grant awarded by the Associate Provost for Research, Texas A&M University.

1. Cumulative production is of course the sum of past years' annual production. Cumulative production thus embodies the entire history of reserve discovery, production, and eventual reserve depletion.

2. In an earlier (October, 1997) version, we used the years 1965-1991 as the sample period. A referee suggested that we extend the sample, which now covers 47 rather than 27 years. Results from estimating the four models are not sensitive to the choice of sample years.

3. The oil production numbers reported in Figure 1 and used for statistical estimation cover crude oil only, and exclude natural gas liquids.

4. Lower 48 production in 1996 relative to that in 1970 was (1663 million barrels/3236 million barrels) = 51.4 percent of production in 1970.

5. See Pesaran and Samiei (1995, equation (18), pp. 548-549). Their observations cover the U.S. lower 48 states for the years 1926-1990.

6. See Kaufmann (1991, pp. 113-114).

7. See Figure 4 in Cleveland and Kaufmann (1991), and their discussion on page 35.

8. The U.S. lower 48 states was the sample studied by Cleveland (1991), Kaufmann (1991), and Pesaran and Samiei (1995).

9. Cleveland (1991) and Kaufmann (1991, pp. 122-124) reason that the rise in the lower 48 production curve until 1970, then its subsequent decline, may be an inverted image of the long-run average cost of production. They argue that economies of scale attributable to early discoveries of giant fields combined with improved production technology to cause long-run average and marginal costs to decline until around 1970. Thereafter, the cost-increasing effects of reserve depletion overshadowed the cost-reducing effects of better production technology, thus causing long-run average and marginal costs to increase. Their reasoning is certainly consistent with the facts that oil reserves in the lower 48 increased until 1961, remained nearly constant from 1961 until 1968, then declined almost continuously from 1968 through 1996. Reserve depletion in the lower 48 is a salient fact: reserves in 1968 were 30,334 million barrels, but declined to 16,743 million barrels in 1996.

10. This partial adjustment model is developed in greater detail in Moroney (1997).

11. The authors would like to thank the anonymous referee who made us aware of this report.

12. Kaufmann (1991) used as the real price variable running averages of real oil prices (1982 dollars) lagged one and two years, and lagged three, four, and five years. His use of lagged real oil prices is more likely to show long-run production responses than our current real oil price. Kaufmann (1991, page 119) found these lagged oil prices to be highly significant. Pesaran and Samiei (1995, Table 4, page 553) estimated a variant of Kaufmann's model, and found a running average of real oil prices lagged three, four, and five years to be marginally significant - the t-statistics were 1.86 or lower in two of three variants of the model.

13. See Godfrey (1978).

REFERENCES

Adelman, M. (1993). The Economics of Petroleum Supply. Cambridge, MA: MIT Press.

Adelman, M. (1995). "Sustainable Growth and Valuation of Mineral Reserves." In J. R. Moroney (Ed.), Sustainable Economic Growth, pp. 45-68. Greenwich, Connecticut: JAI Press, Inc.

Adelman, M. and M. Lynch (1997). "Fixed View of Resource Limits Creates Undue Pessimism." Oil and Gas Journal (April 7): 56-60.

Cleveland, C. (1991). "Physical and Economic Aspects of Resource Quality: The Cost of Oil Supply in the Lower 48 United States, 1938-1988." Resources and Energy 13: 163-188.

Cleveland, C. J. and R. Kaufmann (1991). "Forecasting Ultimate Oil Recovery and Its Rate of Production: Incorporating Economic Forces into the Models of M. King Hubbert." The Energy Journal 12(2): 17-46.

DeGolyer and MacNaughton (1997). Twentieth Century Petroleum Statistics. Dallas, TX: DeGolyer and MacNaughton.

Fletcher, R. (1974). "The Quadratic Law of Damped Exponential Growth." Biometrics 30: 111-124.

Godfrey, L. (1978). "Testing Against General Autoregressive and Moving Average Error Models When the Regressors Include Lagged Dependent Variables." Econometrica 46: 1293-1302.

Hotelling, H. (1931). "The Economics of Exhaustible Resources." Journal of Political Economy 39 (April): 137-175.

Hubbert, M. (1962). "Energy Resources. A Report to the Committee on Natural Resources," Publication 1000-D. Technical report, National Academy of Sciences, Washington, D.C.

Hubbert, M. (1967). "Degree of Advancement of Petroleum Exploration in the United States." American Association of Petroleum Geologists Bulletin 51: 2207-2227.

Hubbert, M. (1971). "The Energy Resources of the Earth." Scientific American (September): 61-70.

Hubbert, M. (1974). U.S. Energy Resources, a Review as of 1972. Washington, D.C.: U.S. Government Printing Office.

Hubbert, M. (1982). "Techniques of Prediction as Applied to the Production of Oil and Gas." In S. Glass (Ed.), Oil and Gas Supply Modeling, pp. 16-141. Washington, D.C.: National Bureau of Standards, Special Publication 631.

Hubbert, M. K. (1956). "Nuclear Energy and the Fossil Fuels." In Drilling and Production Practices, pp. 7-25. Washington, DC: American Petroleum Institute (March).

Kaufmann, R. K. (1991). "Oil Production in the Lower 48 States, Reconciling Curve Fitting and Econometric Models." Resources and Energy 31: 111-127.

Moore, C. (1970). "Future Petroleum Provinces of the United States-A Summary." Technical report, National Petroleum Council, Washington, DC.

Moroney, J. R. (1997). Exploration, Development, and Production: Texas Oil and Gas, 1970-1995. Greenwich, Conn.: JAI Press.

Pesaran, M. and H. Samiei (1995). "Forecasting Ultimate Resource Recovery." International Journal of Forecasting 11: 543-555.

Pindyck, R. S. (1978). "The Optimal Exploration and Production of Nonrenewable Resources." Journal of Political Economy 86(5): 841-861.

Porter, E. (1995). "U.S. Petroleum Supply: History and Prospects." In J. Moroney (Ed.), Advances in the Economics of Energy and Resources, Volume 9. Greenwich, CT: JAI Press Inc.

Porter, E. (1997). "Are We Running Out of Oil?" In J. R. Moroney (Ed.), Energy Supply and Demand, pp. 185-251. Greenwich, Connecticut: JAI Press.

Ryan, J. M. (1965). "Discussion." Bulletin of the American Association of Petroleum Geologists 49(10): 1713-1727.

U.S. Department of Energy, Energy Information Administration (1984). "Two Approaches to the Linkage of U.S. Oil and Gas Reserve Estimates." Technical Report DOE/EIA-0452, Washington, D.C. (July).

Wiorkowski, J. J. (1981). "Estimating Volumes of Remaining Fossil Fuel Resources: A Critical Review." Journal of the American Statistical Association 76(375): 534-559.