ESTIMATION OF DYNAMIC INVESTMENT

FUNCTIONS IN OIL EXPLORATION

Documento 89-05

Kenneth Hendricks

Alfonso Novales

Estimation of Dynamic Investment Functions in Oil Exploration

by

Kenneth Hendricks University of British Columbia

Alfonso Novales Universidad Complutense de Madrid

Revised

September 1988

Abstract

This paper studies the dynamics of the discovery process for oil and gas pools in the province of Alberta, Canada from 1950 to 1979. A costs of adjustment, rational expectations model is specified in which current and future returns from exploration are random, and the main source of the uncertainty is the amount of discoveries. The parameters are estimated from the stochastic Euler equation using the Generalized Methods of Moments estimator proposed by Hansen (1982). We find that, at the appropriate level of aggregation, there is a stable, quantifiable relationship between exploratory investment and returns which is consistent with optimizing behavior.

*We are indebted to R. Uhler for giving us access to the data. We would also like to thank Harry Paarsch and John Kennan for helpful discussions. Financial support from NSF Grant No. SES-8409133 and the Resources for the Future, Inc. is gratefully acknowledged. This paper was written while Hendricks was visiting the Hoover Institution.

1. Introduction

Empirical models of exploration can be grouped into two categories.

The first is supply models which focus on estimating the probability law

governing the sequence of discoveries . Such models usually ignore the

economic decisions which help determine the sample of drilling outcomes. The

second is econometric models which purport to estimate the effects of economic

incentives on the flows of drilling inputs and discoveries. These usually

involve an ad hoc specification of the forces determining supply in which

measures of drilling inputs and discoveries are regressed on one-period lags

of these variables, and proxies for prices and costs. An exception is the

work by Epple (1984), who develops an optimizing, dynamic model of investment

behavior with rational expectations.

Both categories have their problems. The supply models can predict

the probability of finding at least z more reserves if k more wells are

drilled, but they cannot address the issue of whether the k wells should be

drilled and if so, at what rate. Furthermore, their failure to account for

the behavioral process which generates the data is likely to bias the

forecasts. The econometric models have yielded notoriously unreliable

estimates of supply elasticities. In his evaluation of three of these

studies, Pindyck (1975) found that estimates of supply elasticities were

highly unstable with respect to both model specification and time period. He

attributed the instability to two factors: the absence of an optimizing

framework and an inadequate specification of the dynamic structure of the

discovery process. We would add a third factor, aggregation of investment

paths and returns from diverse geographical regions.

The work by Epple (1984) addresses some of these criticisms. He

derives the firm's decision rule from a dynamic, stochastic optimization model

in which the firm chooses a contingent production plan to maximize the

expected present value of resources. The distinguishing feature of

nonrenewable resources like oil and gas is, of course, that they are

depletable. This aspect is incorporated into Epple's model through the

assumption that the costs of finding and developing new deposits in future

periods depend upon cumulative production. Thus, in choosing its current

output, the firm must take into account the effect of its decision on expected

future extraction costs. Forecasts of ultimate recovery are calculated from

the condition that expected marginal extraction costs of the last unit

produced must be equal to the resource price.

Our modelling is in the same spirit as Epple's, but the approach is

quite different. We focus on the decision of the individual firm to drill

exploratory wells in a specific area, Alberta, Canada. The optimal drilling

plan is characterized as a solution to a stochastic Euler equation which

equates the expected returns of a exploratory well to expected marginal

drilling costs. Uncertainty about returns is caused primarily by randomness

in discoveries rather than in drilling costs. The effect of depletion is

incorporated into the model through the stochastic process determining

returns: average returns are assumed to decline over time. Our objective is

to establish whether a stable, quantifiable relationship between exploratory

investment and returns exists which is consistent with optimizing behavior.

In particular, we are interested in determining whether a dynamic model is

necessary to explain the data (i.e., are there any costs of adjustment), and

how information flows from drilling activity influence expectations of current

and future returns.

The presence of nontrivial investment dynamics has important

implications for measuring the impact of changes in the economic environment.

Previous studies have generally focused on short-run measures. But this is

valid only if exploratory investment is not a dynamic process. Otherwise, the

appropriate comparison is between the time paths of investment and discoveries

that are generated by alternative economic environments. In particular, one

needs to recognize the possibility that a once-and-for-all increase in, say,

the price of oil, is likely to "tilt" the drilling profile in a region to the

present. As a result, much of the initial response in discoveries may be a

transfer from the near future to the present, and does not represent finds

that would not have occurred otherwise.

It is also important to determine the role of information in the

exploration process. If information spillovers are substantial, the

competitive allocation of resources is probably not efficient. Too much

drilling may occur in areas where discoveries have been made, and too little

in areas which are relatively unexplored. Learning may also help explain why

estimates obtained from data which aggregate returns and investments from

diverse regions are often so imprecise and unstable. The response of

investment and discoveries in a region to changes in the economic environment

may depend upon its stage of development. In that case, there is no reason to

suppose that the relationship between aggregate investment and aggregate

returns is a stable one if the vintage distribution of areas is not

stationary.

Finally, we should note that, in contrast to the previous literature,

we do not try to forecast reserves. The reason is that the model is partial

equilibrium: it predicts the supply path of drilling inputs of a

representative firm when that firm faces a fixed probability law for drilling

returns. At the industry level, this probability law is almost certain to be

endogenous, since depletion effects implies that current drilling returns

should depend upon the past history of industry drilling inputs. This

endogeneity of the returns process needs to be taken into account when making

forecasts about a region's reserves and stocks of wells.

The paper is organized as follows. In the next section, we introduce

our model of the firm's investment problem. The estimation procedure is

outlined in Section 3. Although it is possible to solve explicitly for the

decision rule under appropriate assumptions on the stochastic processes, we

have chosen instead to estimate the parameters of the objective function from

the Euler equation using the Generalized Method of Moments estimator

introduced by Hansen (1982). The main reasons for doing so are the relatively

small number of observations, and our reluctance to impose the requisite

restrictions on the stochastic process for returns. We test the model using

the restrictions implied by the theory. Previous applications of this method

include studies by Hansen and Singleton (1982) on stock returns data and by

Pindyck and Rotemberg (1983, 1984) to energy demand.

The data is described in Section 4. A virtue of the data set is that

exploratory wells are distinguished from development wells, and discoveries

from revisions to initial estimates. This is important since the evidence

strongly suggests that discoveries from exploratory wells generate information

which affect drilling decisions on other prospects, but that revisions to

reserves due to development wells do not. The estimation results are

presented in Section 5, and conclusions follow in Section 6.

2. The Investment Model

We begin with a brief description of the exploration process in the

province of Alberta, Canada.

The ownership of natural resources in Canada is, for the most part,

invested in the provinces. In developing its hydrocarbon reserves, the

Alberta government has generally followed a policy of retaining public

ownership, and contracting with firms in the private sector to find and

produce the deposits. For example, in 1971, less than 12% of the total oil

and gas acreage in the province of Alberta was privately owned. The rest of

the acreage was leased by the government to oil and gas firms in a variety of

tenure arrangements.

Three types of licenses control exploration for oil and gas in

Alberta. Geophysical licenses give non-exclusive rights to conduct private

geological, seismic, magnetometer, and other types of surveys almost anywhere

in the province, including those areas subject to existing leases. The

license does not permit the drilling of test wells. Anyone can obtain a

geophysical lease, and it is renewable each year for a fee. Firms use the

surveys to identify promising areas, or prospects, over which it desires to

obtain exclusive rights to drill exploratory wells and to produce oil and gas.

Drilling and production rights are usually sold via a first price,

sealed bid auction. Initially, firms bid for a petroleum and natural gas

reservation, which gives the winner an exclusive right to drill in a specified

area. The size of the area cannot exceed 156 square miles. The term of a

reservation is four months, but it can be extended for several years with a

pledge to conduct an exploration program that meets the approval of the

government. If oil or gas is discovered, or prospects look promising, the

reservation holder would normally apply for petroleum and natural gas leases,

which confer long-term (at least 10 years) rights to produce oil and gas on

designated parts of the reservation. A lease cannot exceed eight or nine

square miles, or be less than a quarter-section. Furthermore, the reservation

holder cannot apply for more than 50% of any township subject to the

reservation, or for leases that are adjacent. This gives rise to a

"checkerboard" pattern of lease ownership. Leases not claimed by the

reservation holder are returned to the government as Crown reserves, and

subsequently sold in first-price, sealed bid auctions. The reservation holder

is required to submit cores and logs from any wells drilled, and this

information is made public one year after the application date.

The markets for land and drilling inputs into the exploration process

were competitive. For any given year in the sample period from 1946 to 1979,

the fraction of exploratory wells drilled by the top sixteen oil companies

ranged from a low of 22 per cent to a high of 40 per cent. Thus, the

individual firm's annual investment in exploratory wells was small relative to

the total number of wells drilled by the exploration industry. Wellhead

prices for the Alberta oil and gas products were clearly exogenous to the

firms, since these prices were determined by the international market, and

total output of oil and gas from Alberta was only a tiny fraction of the

volumes transacted in this market.

Wellhead prices for oil were roughly constant in nominal terms until

1973, after which they rose according to a formula imposed by the federal

government. Wellhead prices for gas increased steadily throughout the sample

period, but were generally too low prior to the 1970's to make exploration and

development of gas pools a profitable investment. Nominal drilling costs per

well were also basically constant throughout the sample period, ranging

between $70,000 to $130,000, depending upon the depth of the well. Thus, for

our sample, the main source of uncertainty in returns was the sizes of the

discoveries. Many prospects did not contain sufficient quantities of oil and

gas to make production profitable and, as a result, the cost of drilling

exploratory wells was not recovered.

We shall consider a somewhat stylized model of firm behavior. At the

beginning of each period, each firm must decide on the number of prospects it

should acquire and drill in that period, taking as given market prices of

prospects and wellhead prices of oil and gas products. Prospects not drilled

by the end of the period are returned to the government. We shall further

assume that prospects and drilling rigs are combined in fixed proportion to

produce wells. Given these assumptions, the planning problem of the

representative firm can be formulated in terms of choosing a contingent

drilling program to maximize the expected present value of discoveries :

\[E _ {t} \sum_ {j = 0} ^ {\infty} \delta^ {t + j} \left\{\left(v _ {t + j} a _ {t + j} - b _ {t + j}\right) x _ {t + j} - \alpha_ {1} x _ {t + j} - (1 / 2) \alpha_ {2} x _ {t + j} ^ {2} - (1 / 2) \alpha_ {3} \left(x _ {t + j} - x _ {t - 1 + j}\right) ^ {2} \right\}.\tag{2.1}\]

In the above notation: is the number of exploratory wells drilled

during period t, is the amount of reserves discovered per well during

period t, is the after-tax, present value price of a unit of reserves

discovered in period t, and is the cost of period t drilling inputs. The

vector of cost parameters, , is nonnegative, and the discount

parameter satisfies . The operator denotes the conditional

expectation: where y is a random variable, and is the

firm's information set at the beginning of period t. It includes, in

particular, and , for all , current and lagged prices of oil and

gas, and for i > 0. It does not include , which is random at the time

the firm drills the wells.

Define to be the dollar return per well drilled net of

drilling costs. We shall assume that the individual firm views the stochastic

process governing as lying outside of its control, and that it is of

exponential order less than . Then, differentiating (2.1) with respect to

yields the Euler equation:

\[\mathrm{E} _ {t} \left[ \pi_ {t} - \alpha_ {1} - \alpha_ {2} x _ {t} - \alpha_ {3} \left(x _ {t} - x _ {t - 1}\right) + \delta \alpha_ {3} \left(x _ {t + 1} - x _ {t}\right) \right] = 0.\tag{2.2}\]

The transversality condition is

\[\lim _ {t \rightarrow \infty} E _ {t} \delta^ {T} \left[ \pi_ {t} - \alpha_ {1} - \alpha_ {2} x _ {t} - \alpha_ {3} \left(x _ {t} - x _ {t - 1}\right) + \delta \alpha_ {3} \left(x _ {t + 1} - x _ {t}\right)\right] = 0.\tag{2.3}\]

Our assumption on ensures that equation (2.3) is satisfied. Also,

given our representative firm assumption, we shall assume that the solution to

the Euler equation is interior (i.e., for all t).

The strength of our model is that it captures two features which we

believe are essential to understanding the exploration process. First,

current and future returns are stochastic because of the randomness in

discoveries. If learning is present, and depletion effects are anticipated,

then expectations about these returns depend upon past discoveries and

drilling efforts. Second, fluctuations in drilling activity are costly. One

interpretation of this property is that there is a fixed factor, such as the

stock of drilling rigs, which makes it costly for a firm to change its level

of drilling activity from period to period. Note that if there are no costs

of adjustment (i.e., is equal to zero), then there are no dynamics to

explain, since drilling effort in each period would depend only upon the

expected returns in that period.

The restrictive aspect of the model lies in the assumption that firms

cannot hold inventories of prospects. Each firm acquires only as many

prospects as it intends to drill in that period. This assumption has several

important implications. It means that a firm cannot delay exploration of a

prospect until after the outcomes of wells being drilled in the current period

are known. Furthermore, information from current drilling efforts about

prospects sold in subsequent periods is incorporated into the prices of these

prospects since drilling outcomes are observable. Hence, a prospect has no

informational value to its owner, and private drilling returns are determined

solely by the amount of hydrocarbons which can be produced.

The purpose of the above assumption is to rule out any incentive firms

might have to hold prospects for speculative purposes. Its necessity follows

from limitations in the available data, which identifies the location of an

exploratory well and the number of wells drilled each year, but not which firms drilled them or their lease holdings. The empirical significance of the assumption is not clear. As mentioned previously, many tenure arrangements are in fact relatively short-lived, although in most instances, they seem to be longer than the period required to drill and evaluate core samples. Consequently, it is a restrictive assumption, and we shall have to bear it in mind when interpreting the estimation results.

Finally, we should point out that the model is partial equilibrium. The probability law governing drilling returns is assumed to be exogenous to the individual firm. In a more complete model, the amount of discoveries in each period would be determined simultaneously with investment in wildcat wells, and the supply paths of both drilling inputs and discoveries would be expressed as functions of the underlying cost technology and the probability law for discoveries.

3. Estimation Strategy

The literature suggests several procedures for estimating the model. The simplest method is the approach followed by McCallum (1982) and Kennan (1979). They use realizations of future variables in the Euler equation as proxies for their conditional expectations, and treat the estimation of the resulting model as an errors in variables problem. As Hansen and Sargent (1980, 1982) point out, this method ignores a number of restrictions implied the model. A major benefit of the linear-quadratic structure is that, given suitable restrictions on the stochastic environment, one can obtain closed form solutions for the decision rule. Hansen and Sargent exploit this property in developing full information estimators for models such as ours.

The methods of Hansen and Sargent involve estimation of a relatively

large set of parameters. In addition to those in the decision rule, one must

estimate the parameters of the vector autoregressive representation of the

instrumental variables, and the parameters of the linear projections of the

forcing variable, , and the decision variable, , on the set of

instruments. We have at most 33 observations, of which we lose at least two

due to lags in the decision rule and instruments. This number seems too few

to implement with any degree of confidence the Hansen-Sargent methods. In

addition, we were somewhat reluctant to impose the requisite restrictions on

the stochastic structure.

Hansen and Singleton (1982) suggest an alternative estimation strategy

based upon the General Methods of Moments estimator proposed by Hansen (1982).

This method interprets the Euler equation as implying a set of orthogonality

conditions, which can be used to obtain estimates for the parameters, and to

test the model. As the authors note, an important benefit of this approach is

that it does not require strong a priori assumptions about the properties of

the stochastic process of the forcing variable . In particular, it permits

conditional variances of the disturbances to depend on variables in the

information set.

In describing the estimation procedure, it will be useful to express the Euler equation in the form:

\[E _ {t} \left[ x _ {t} - \beta_ {0} - \beta_ {1} x _ {t - 1} - \beta_ {2} x _ {t + 1} - \beta_ {3} \pi_ {t} \right] = 0,\tag{3.1}\]

where:

\[\beta_ {0} = - \alpha_ {1} / [ \alpha_ {2} + \alpha_ {3} (1 - \delta) ]\]

\[\beta_ {1} = \alpha_ {3} / [ \alpha_ {2} + \alpha_ {3} (1 - \delta) ]\]

\[\beta_ {2} = \delta \alpha_ {3} / [ \alpha_ {2} + \alpha_ {3} (1 - \delta) ]\]

\[\beta_ {3} = 1 / \left[ \alpha_ {2} + \alpha_ {3} (1 - \delta) \right].\]

Our strategy will be to derive an estimator for , and then

obtain estimates for from the four equations given in (3.1). Standard

errors will then be obtained using the method.

Following Hansen and Singleton (1982), we define the function

\[h \left(Y _ {t + 1}, \beta\right) = x _ {t} - \beta_ {0} - \beta_ {1} x _ {t - 1} - \beta_ {2} x _ {t + 1} - \beta_ {3} \pi_ {t},\]

where is the vector of variables in the decision rule. Deviations of from zero can be interpreted as arising from errors in expectations. Letting denote this error, the Euler condition can then be written as

\[\mathrm{h} \left(\mathrm{Y} _ {\mathrm{t+1}}, \beta\right) + \mathrm{v} _ {\mathrm{t+1}} = 0\]

or:

\[\mathrm{x} _ {\mathrm{t}} = \beta_ {0} + \beta_ {1} \mathrm{x} _ {\mathrm{t} - 1} + \beta_ {2} \mathrm{x} _ {\mathrm{t} + 1} - \beta_ {3} \pi_ {\mathrm{t}} + \mathrm{u} _ {\mathrm{t} + 1},\tag{3.2}\]

with .

If expectations are rational, the residual in equation (3.2) should be unpredictable, given the available information: . Furthermore, residuals are serially uncorrelated, since, letting E denote the unconditional expectations operator, for all . Finally, although not required by the estimation procedure, we shall make the simplifying assumption that the errors are conditionally homoscedastic.

The Hansen-Singleton estimator is an instrumental variables procedure which minimizes the correlation between variables known at time and the residuals of equation (3.2). Ideally, the vector of instruments is the expectation of the gradient vector conditional on the information set at time . Hansen and Singleton note that, in general, choosing the optimal instruments requires a detailed specification of the economic environment. The task is greatly simplified if one is willing to

assume that conditional expectations coincide with linear projects. We make

this assumption. Combined with the conditional homoscedsticity assumption,

the estimation procedure then reduces to two step least squares.

The estimation method uses only four of the available orthogonality

conditions to identify the parameter vector . The remaining othogonality

conditions can then be used as a specification test since, if the model and

its behavioral restrictions are correct, the sample analogues of the cross-

products of the residuals and the instruments should be close to zero. The

appropriate test statistic for the hypothesis that these restrictions are

satisfied has been derived by Hansen (1982). He has shown that the asymptotic

distribution of T times the minimized value of the objective function is chi-

square with p-r degrees of freedom, where p is the number of instruments and r

is number of orthogonality conditions actually used for identifying .

To summarize: we propose to estimate the structural parameters of our

model directly from the stochastic Euler equation using the Generalized

Instrumental Variables estimator of Hansen and Singleton. The specification

of the model will be tested by examining whether or not (1) the residuals are

serially uncorrelated, (2) the set of overidentifying restrictions are

satisfied, and (3) the coefficients and satisfy the linear restriction

for reasonable values of . We also propose to test the hypothesis

that there are no costs of adjustment (i.e., ) by estimating the

static model in which wildcat drilling is a linear function of .

4. Description of Data

The data were obtained from the study submitted by Uhler (with

Eglington) (1983) to the Economic Council of Canada on the exploration for oil

and gas deposits in the province of Alberta, Canada. An important feature of

the data set is the geographical detail of the drilling and discovery

information.

Oil and gas pools in this study were indexed by areal location and

geological horizon (depth). Horizons refer to the substrata associated with

different geological periods. The province was divided into ten areas and ten

horizons, generating one hundred possible (horizon, area) pairs. For each

(horizon, area) pair, Uhler gives annual data on changes in oil and gas

reserves separately, and on drilling activity. The sample period is basically

from 1950 to 1979. Discoveries are reported separately from revisions, and

the latter are reported by vintage. That is, annual revisions of pools

discovered in year t are listed separately from those of pools discovered in

year s, s ≠ t.

The drilling data for each (horizon, area) pair reports the number of

wells drilled annually, and the type of well: wildcat wells, which are drilled

in search of new pools, and development wells, which are drilled in pools that

have already been discovered. Wells were allocated to each category under the

assumption that the deepest horizon penetrated is the target horizon.

Since the amount of drilling in many of the (horizon, area) pairs was

quite small, Uhler (1983) reduced the number of categories to eight, mainly by

aggregating across areas within an horizon. We combined the two Lower

Devonian categories, and dropped one category from the sample due to

incomplete drilling data. (It contained virtually no oil pools, and the

number of wells drilled prior to the 1970's was quite small.) The remaining

six categories are listed in Table 1. They are ordered by depth, with the

Lower Devonian horizon representing the deepest horizon. Gas reserves are

measured in billions of cubic meters, and oil reserves are measured in

millions of cubic meters. Gas reserves in pools which contain both oil and

gas are called associated gas (AG). Reserves in gas pools are called

nonassociated gas (NAG). The Upper Cretaceous horizon includes only oil and

gas pools discovered in the central part of Alberta. The other horizons

include all pools discovered in these horizons throughout the province.

The pools in the shallower horizons tend to have a higher gas content

than oil. The exception is the Upper Cretaceous horizon, where 170 million

cubic meters of oil were discovered. This figure is somewhat misleading,

however, since most of the oil reserves, 130 million cubic meters to be

precise, are in one pool, the gigantic Pembina pool. It was discovered in

1953, and accounts for approximately 11% of Alberta's oil reserves. Most of

the remaining 89% of Alberta's oil reserves lie in the deeper Devonian

horizons.

The differences in the ratio of the number of wildcat wells to the

number of development wells across horizons reflect differences in the size

distributions of pools. When a wildcat well discovers a pool, it is usually

completed and turned into a producing well. Development wells are drilled

only if the pool is sufficiently large that more than the one well is needed

for optimal production. In the Lower Devonian and Mannville horizons, the oil

and gas pools were relatively small, and frequently did not require any

development wells. The gigantic Pembina pool, on the other hand, required a

large number of development wells and, as a result, the ratio of wildcat to

development wells in this horizon was relatively low. Many of the pools in

the Upper Devonian horizon were also quite large.

What is the appropriate geographical and geological context for the

firm's investment decision? One possibility is the horizon, on the grounds

that the presence (or absence) of hydrocarbon deposits at one location in a

horizon changes the probability of finding deposits elsewhere in that horizon.

But, even if the formation of pools in different horizons are independent

geological events, when a well is drilled to an horizon, it necessarily

produces information about every horizon that lies above the target horizon.

Therefore, all horizons in a specific area is likely to be a better

approximation to the decision environment than the individual horizon.

Unfortunately, the raw data that would permit this type of aggregation

were not available. A crude alternative is to aggregate drilling investment

in horizons where drilling occurred in approximately the same areas at the

same time. Prior to the 1970's, most of the drilling activity in the Upper

Cretaceous, Mannville, Viking, Mississippi, and Upper Devonian horizons took

place in central Alberta. In the 1970's, drilling in the Mannville, Viking,

and Mississippi horizons shifted to the southwestern part of Alberta, near the

foothills, where most of the gas pools are located. Drilling in the Lower

Devonian horizon was concentrated almost exclusively in northwestern Alberta.

Moreover, almost all of the drilling in this area was targeted to the Lower

Devonian horizon, since other horizons in this area contained few oil or gas

pools. On the basis of these facts, we separated the Lower Devonian horizon

from all other horizons, and aggregated the latter into one category, called

Agg\LoDev. This classification represents our base case.

Our measure of exploratory drilling effort is the number of wildcat

wells. We regard expenditures on development wells as part of the cost of

developing the pool. Accordingly, the amount of oil and gas discovered by a

wildcat well drilled in year t is given by the average size of pools

discovered in that year, where size is measured by cumulative production.

Revisions to initial estimates of discoveries are assumed to reflect reporting

lags, and to add no new information about the prospect. We examine the

validity of these assumptions in the appendix, and find that they are strongly

supported by the data.

Annual prices for developed and undeveloped reserves of an "average"

pool in Alberta were computed using annual data on wellhead prices for oil and

gas, development costs, and taxes. The formula which Uhler used to calculate

the expected price of reserves in each year is based upon the following

assumptions: a fifteen year production life, a constant rate of production,

and price expectations which were static up to 1973 , and consistent with the

government-controlled growth rate in prices after 1973. The price shock of

1973 was assumed to be unanticipated. After deducting various operating

costs, taxes, and royalties from the wellhead price (measured in nominal

dollars), the flow of expected net returns were discounted back to the date of

discovery at the nominal rate of interest to yield a price for developed

reserves. The price of undeveloped reserves was obtained by deducting

development costs, which were assumed to occur in the first year of the

production period.

Drilling costs were obtained from the Annual Report on exploration

expenditures by Alberta Department of Energy and Natural Resources.

The empirical definitions of the variables introduced in Section 2 are

listed in the top panel of Table 2. All dollar figures are expressed in

millions of 1965 dollars. The gross return from a wildcat well is defined as

the sum of oil and gas revenues, although in most horizons, and for most of

the sample period, oil revenues was the main component. The net return is

calculated by deducting exploratory drilling costs. It does not net out

geophysical and land acquisition costs, since these data were available only

for Alberta as a whole, and not for the individual horizons. Means and

standard deviations for annual wildcat drilling, net returns, and profits are

reported in the bottom panel of Table 2 for each horizon, and for each of the

aggregates.

5. Estimation Results

The model was estimated by two step least squares. Instrumental

variables for and were constructed using as instruments one lag each of

discoveries, discoveries per well, and wildcat drilling, and contemporaneous

values of wildcat drilling, oil prices, and gas prices. The standard errors

for were calculated using a consistent estimate of the asymptotic

covariance matrix under heteroskedasticity, proposed by White (1982).

Standard errors for were derived using the covariance matrix A·Σ·A, where A

is the Jacobian of the mapping between and .

Table 3 gives the estimation results for the base case. The estimates

in column U correspond to the model with no restriction on the coefficients.

The estimates in column R are obtained under the restriction and a

value for . T, gives the chi-squared statistic which tests the

validity of this hypothesis. gives the chi-squared statistic for the

overidentifying restrictions of the dynamic model, and gives this statistic

for the restriction , which yields the static model. The numbers

which appear in parentheses after the chi-squared statistics are their

marginal significance levels.

The specification tests indicate that our model is consistent with the

data. The overidentifying restrictions are not rejected in either of the two

equations, although the marginal significance levels are not very high. The

linear restriction on the coefficients (i.e., ) was satisfied at

conventional significance levels. The first order autoregressions for the

residuals in each of the two equations revealed no evidence of serial

correlation. We checked the simple and partial autocorrelation functions of

the residuals, and did not find any contrary evidence. Of course, one should

remember that the sample size is small, so the confidence intervals for the

autocorrelations are quite wide (i.e., the tests surely have low power).

Most of the coefficients of the restricted model are significant at

conventional confidence levels, have the correct sign, and are stable across

the two classifications. Furthermore, the magnitudes of the associated cost

parameters do not appear to be implausible. The parameter can be

interpreted as average geophysical costs and land acquisition costs per well

over the sample period. For the province as a whole, these costs ranged

between $400,000 to $1 million per well during the sample period, which does

not differ too greatly from the point estimates of $1.2 and $1.5 million in

the Agg/LoDev and Lower Devonian horizons, respectively. The cost of

adjustment parameter is estimated to be about $15,000 to $28,000 per well,

which also seems plausible. However, the estimates for are consistently

negative, although not significantly different from zero.

The statistics for the set of overidentifying restrictions in the

static version of the model indicate that the null hypothesis of no costs of

adjustment (i.e., ) is overwhelmingly rejected in both the Lower

Devonian horizons and the unit called Agg/LoDev. Thus, the instantaneous

response in exploratory drilling to the discovery of a large pool, or an

unexpected rise in oil prices, can differ substantially from the long-term

response.

A striking result that deserves emphasis is the similarity of the

estimates across the two regions. In Figures 2 and 3, we have depicted the

actual drilling paths for the Agg/LoDev horizons and the Lower Devonian

horizons, respectively. The pattern of drilling activity in the latter case

was clearly quite different from that in the non-Lower Devonian horizons, even

after one accounts for differences in scale. Nevertheless, the same model is

capable of explaining both patterns.

Furthermore, the model performs well. The dashed lines in Figures 2

and 3 depict the predicted drilling paths for the two regions. The overall

fit in both cases is quite good. In the Agg/LoDev case, the model just misses

the spike in 1973, but it adjusts to it in a short period of time. In the

Lower Devonian case, the model captures quite well the turning points in 1960,

1964, and 1974.

Alternative Classifications of Pools

Table 4 reports the results of the restricted model when it is

estimated by horizon. As expected, the model does quite poorly in horizons

other than the Lower Devonian. The specification tests reveal inconsistencies

between the model and the data. The overidentifying restrictions are rejected

at the 5% significance level in the Upper Cretaceous and Mississippi horizons.

Support for the linear constraint is relatively weak in the Upper Cretaceous,

Viking and Upper Devonian horizons. Also, there is evidence of serial

correlation in the residuals for the Mississippi and Upper Devonian horizons.

Additional evidence for rejecting the model as a plausible explanation

of drilling behavior in non-Lower Devonian horizons is provided by the

parameter estimates themselves. The estimates for and are generally not

significantly different from zero, and in several instances, they have the

wrong sign. The implied estimates for are implausibly large in the Upper

Cretaceous and Upper Devonian horizons, and have the opposite signs to that

expected in the other three non-Lower Devonian horizons. The estimates for ,

and frequently have the wrong sign, although in most instances, they are

not significantly different from zero.

These results indicate that firms did not choose their drilling

programs to maximize the present value of discoveries within an horizon. Our

interpretation of this negative conclusion, in conjunction with the positive

conclusion obtained for the base case, is that the horizon was not the

geographical context in which firms were making their investment decisions.

Rather, the appropriate geographical context is one which aggregates different

horizons in a region.

6. Concluding Remarks

In this paper we have formulated and estimated a dynamic model of

exploratory drilling with rational expectations. The results provide some

important insights about the nature of the exploration process, which are

likely to be useful for policy-makers. In particular, we would emphasise the

following implications.

First, the data strongly rejects the hypothesis of a static model of

drilling behavior. This puts into question short-run measures of the impact

of changes in the economic environment on drilling and discovery rates. Most

of the initial response in discoveries may simply represent transfers from

future periods to the current period as the drilling profile is "tilted"

toward the present. Furthermore, they do not provide a measure of discoveries

that would not have been made if prices had not increased. The correct

approach to evaluating the impact of price and other policy changes is to

recognize that alternative economic environments generate different paths of

investment and discoveries, and to compare these paths.

Second, the data provides evidence that firms do not regard drilling

returns as a sequence of independent random variables. Instead, they use the

information from past realizations of returns and drilling efforts to forecast

current and future values of returns. Furthermore, their forecasts appear to

be consistent with the underlying probability law. The structure of this

probability law and the firm's response to drilling outcomes indicates that

information spillovers are likely to be an important determinant of drilling

behavior. This raises several issues, which could not be addressed at the

current level of aggregation, but could be studied if firm data were

available. How important are "bandwagon" effects in determining the

geographical pattern of drilling? What is the impact of different tenure

arrangements on rates of exploration?

Third, classifying pools and wells according to the informational

context in which drilling decisions are taken is important for obtaining

precise and stable estimates of the decision rule. In the Alberta basin, this

implied a classification that was based more on geographical than on

geological criteria. It is also apparent that aggregating investments from

regions which are in different stages of development, or have different

probability laws for returns, is in general not a good idea, since the

underlying probability laws and the expectations mechanism determining

investment behavior are region-specific.

Our costs-of-adjustment interpretation of the dynamics is subject to

an important caveat. It was derived under the assumption that the tenure of

prospects was equal to the drilling period. The presence of information

spillovers indicates that this assumption is crucial. Otherwise, firms have

an incentive to behave strategically, and delay drilling marginal prospects

until after drilling outcomes on prospects that are currently being drilled

are observed. As a descriptive matter, there is some reason to believe that

this assumption is not correct. Consequently, some of the dynamics detected

in the data may reflect the desire of firms to delay in order to learn more.

If so, the disaggregation of pools and wells must be extended to firms as well

as location. This is clearly the next step in the research.

Kaufman et. al. (1975, 1981), Meisner and Demirmen (1981) are good examples.

Three examples of this literature are Erickson and Spann (1971), Khazzoom (1971), and MacAvoy and Pindyck (1973).

See Hendricks and Kovenock (1989) for a theoretical analysis of the efficiency issue.

These restrictions did not apply for those parts of the province designated as Block 'A' areas. In these areas, a firm did not have to surrender any of the leases covered by the initial permit to the government.

It can be shown that this maximization problem is equivalent to maximizing the present value of production if output in period t is assumed to be proportional to stock of reserves in period t.

Nominal prices were essentially constant during this period.

In most cases, a comparison of to the covariance matrix obtained under the assumption of conditional homosckedasticity indicated that some heterosckedasticity was present.

References

1. Epple, D.N. (1985), "The Econometrics of Exhaustible Resource Supply: A Theory and an Application," in Energy, Foresight, and Strategy, ed. by T. Sargent, Resources For the Future, Washington, D.C..

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1. Erickson, E.W. and Spann, W. (1971), "Supply Response in a Regulated Industry: The Case of Natural Gas," Bell Journal of Economics and Management Science, 2(1), pp. 94-121.

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1. Fisher, F. M. (1964), Supply and Costs in the U.S. Petroleum Industry: Two Econometric Studies, John Hopkins Press, Baltimore.

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1. Hansen, L.P. and Sargent, T.J. (1980), "Formulating and Estimating Dynamic Linear Rational Expectations Models," Journal of Economic Dynamics and Control, 2, p.7-46.

References

1. Hansen, L.P. and Sargent, T.J. (1981), "Linear Rational Expectations Models for Dynamically Interrelated Variables," in: R.E. Lucas, Jr. and T.J. Sargent, eds., Rational Expectations and Econometric Practice, University of Minnesota Press, Minneapolis, MN.

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1. Hansen, L.P. and Sargent, T.J. (1982), "Instrumental Variables Procedures for Estimating Linear Rational Expectations Models," Journal of Monetary Economics, 9, pp. 263–296.

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1. Hansen, L.P. and Singleton, K.J. (1982), "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, 50 (5), pp. 1269–1286.

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1. Hendricks, K. and Kovenock, D. (1989), "Information Externalities, Asymmetric Information, and Efficiency: The Case of Oil Exploration", forthcoming, The Rand Journal, Summer, 1989.

References

1. Kaufman, G., Balcer, Y. and Kruyt, D. (1975), "A Probabilistic Model of Oil and Gas Discovery," in: Methods of Estimating the Volume of Undiscovered Oil and Gas Reserves, Studies in Geology, Vol. I, J.D. Hann (ed.), Tulsa, OK: American Association of Petroleum Geologists.

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1. Kaufman, G., Runggaldier, W., and Livne, Z. (1981), "Predicting the Time Rate of Supply from a Petroleum Play," in The Economics of Exploration for Energy Resources, edited by J.B. Ramsey, JAI Press, Inc.

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1. Kennan, J. (1979), "The Estimation of Partial Adjustment Models with Rational Expectations," Econometrica, 47.

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1. Khazzoom, J.D. (1971), "The FPC Staff's Econometric Model of Natural Gas Supply in the United States," Bell Journal of Economics and Management Science, (Spring), 2(1), pp. 51–93.

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1. MacAvoy, P.W. and R.S. Pindyck (1973), "Alternative Regulatory Policies for

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1. Dealing with the Natural Gas Shortage," Bell Journal of Economics and Management Science, (Autumn), 4(2), pp.454–598.

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1. McCallum, B.T. (1977) "Rational Expectations and the Natural Rate Hypothesis: Some Consistent Conjectures", Econometrica, 44, pp. 43–53.

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1. Meisner, J. and Demirmen, F. (1981), "The Creaming Method: A Bayesian Procedure to Forecast Future Oil and Gas Discoveries in Mature Exploration Provinces," Journal of the Royal Statistical Society, Series A, p. 1-31.

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1. Pindyck, R.S. (1974), "The Regulatory Implications of Three Alternative Econometric Models of National Gas Supply," Bell Journal of Economics and Management Science, 5(2), (Autumn), pp. 636–645.

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1. Pindyck, R.S. and J.J. Rotemberg (1983), "Dynamic Factor Demands and the Effects of Energy Price Shocks", The American Economic Review, 73, 5, (December), pp. 1066–1079.

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1. Pindyck, R.S. and J.J. Rotemberg (1983), "Dynamic Factor Demands Under Rational Expectations", Scandinavian Journal of Economics, 85, (July).

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APPENDIX

In this appendix, we examine the hypothesis that development wells are primarily investments in production, and that revisions to initial estimates do not represent new information about reserves. In our evaluation, we ignore gas pools because, for most of the sample period, gas prices were too low to make exploration for gas a profitable investment. Gas pools tended to be produced only if it was discovered as a byproduct of the search for oil pools, after the costs of drilling had been sunk.

Let denote the initial estimate of the amount of oil reserves discovered in period t, and let represent the period s revision of oil reserve estimates for these pools, s > t. Final reserves of vintage t oil pools are given by:

\[Z _ {t} = \Sigma_ {s \geq t} ^ {T} Z _ {s, t},\]

where T is equal to 1981, the last year for which reserve data are available. Total changes in oil reserves in period t due to development drilling are given by:

\[R _ {t} = \sum_ {s = 0} ^ {t - 1} z _ {t, s}\]

The Viking horizon contained virtually no oil pools. In the Upper Cretaceous, Mannville, and Mississippi horizons, oil pools were generally quite small, and the values of were frequently zero. (The correlation coefficients between initial and final reserve estimates in these pools are 0.75, 0.97, and 1.00 respectively.) Yet, in many of these years, substantial numbers of development wells were drilled. In the Devonian horizons, the oil pools were much larger, and final reserve estimates were often four to eight times as large as initial reserve estimates. However, even in these cases, the revisions exhibited a predictable pattern. The correlation coefficients between initial and final reserves for the Upper Devonian horizon and the two Lower Devonian categories are 0.94, 0.97, and 0.95 respectively. This evidence supports our hypothesis.

If discoveries from wildcat drilling represent new information, and revisions of initial reserve estimates represent anticipated information, then the former should be an important determinant of wildcat drilling, and the

latter should not. To test this implication, we regressed the number of wildcat wells drilled in period t, , on two lags of itself and one lag of each of the following variables: annual discoveries, , annual discoveries per wildcat well, , annual revisions in reserves, , and annual revisions per development well, . If our hypothesis is correct, the set of coefficients on the lagged discovery variables should be significantly different from zero, and the coefficients on the lagged revision variables should not.

The estimation results for the base case, which are reported in Table A1, support the hypothesis. The F-statistics which tests the null hypothesis that the intertemporal correlations between wildcat drilling and the reserve revision variables are not significantly different from zero are equal to 0.689 for Lower Devonian and 0.162 for Agg\LoD. These figures indicate that the hypothesis can be maintained at the 99% confidence level. By contrast, the intertemporal correlations between wildcat drilling and the discovery variables are significantly different from zero at conventional significance levels.

We also estimated the equations on horizon data to verify that the results did not depend on our form of aggregation. The estimates for the Upper Cretaceous, Mannville, Mississippi and Upper Devonian horizons are reported in Table A2. (The Viking horizon was omitted since it contained virtually no oil pools.) Once again, there is no evidence that the reserve revision variables have any marginal explanatory power. Note that the evidence for a feedback effect of discoveries onto wildcat drilling is much weaker at this level of aggregation. The intertemporal correlations between the lagged discovery variables and wildcat drilling are generally not significantly different from zero at the 95% confidence level. One explanation of this result is that the horizon is not the appropriate decision environment.

TABLE 1 Cumulative Stocks of Wells and Reserves By Horizon

* Exploration in the Viking, Mississippi, and Mannville horizons began in the 1940's. However, 1950 is the first year for which annual data are available.

Means and Standard Deviations of Wildcat Drilling and Returns*

*Standard deviations are displayed in parantheses.

Instrumental Variable Estimates (2SLS) of Cost Parameters Base Case

* t-statistics for the coefficients are displayed in parantheses. Q denotes the Box-Pierce statistic for serial correlation of any order. The number in parenthesis after the value of Q is its marginal significance level. T₁ is the chi-squared statistic for the hypothesis that β₂ = δ·β₁. T₂ is the chi-squared statistic for the overidentifying restrictions of the dynamic model, and T₃ is the statistic for the restriction that β₁ = β₂ = 0 (i.e., the static model). The numbers which appear in parenthesis after the chi-squared statistics are their marginal significance levels.

Instrumental Variable Estimates (2SLS) of Cost Parameters Classification of Pools by Geological Horizon

t-statistics for the coefficients are reported in parantheses. Q denotes the Box-Pierce statistic for serial correlation of any order. The number in paranthesis after the value of Q is its marginal significance level. is the chi-squared statistic for the hypothesis . is the chi-squared statistic for the overidentifying restrictions of the dynamic model, and is the statistic for the restriction . The numbers which appear in paranthesis after the chi-squared statistics are their marginal significance levels.

\[\underline {{A 1}} ^ {*}\]

Feedback Effects of Discoveries and Reserve Revisions on Wildcat Drilling Base Case

t-statistics for the coefficients are displayed in parantheses. Q denotes the Box-Pierce statistic for serial correlation of any order. The number of lags used in the computations is . The number in parenthesis below the value of Q is its marginal significance level. Marginal levels below .05 means that the null hypothesis of no serial correlation is rejected at the 95% confidence level.

TABLE A2* Feedback Effects of Discoveries and Reserve Revisions on Wildcat Drilling Individual Horizons

t-statistics for the coefficients are displayed in parantheses. Q denotes the Box-Pierce statistic for serial correlation of any order. The number of lags used in the computations is . The number in paranthesis below the value of Q is its marginal significance level.

FIGURE 2