Secular Cycles

Turchin and Nefedov (2009) empirically identified recurring secular or economic cycles in agrarian societies. Similar phenomena apply to oil and resource production, so we recall their findings.

The cycle begins with a period of growth, in population and living standards lasting on the order of a hundred of years, and comes a period of stagflation in which population density approaches the carrying capacity of the land (one says increased population pressure) lasting on the order of half a century. During the stagflation period, peasants leave the countryside for cities, the difference between the elite and the commoners increases, and the price of food rises relative to wages. Population ceases to grow, because food production ceases to grow. Initially, the elite is somewhat better off in the stagflation period, because wages are low and they can employ a larger number of former peasants who have left the countryside. As the stagflation period progresses, the ratio of elite population to working class population rises (the working class has a lower birthrate and a higher mortality rate due to malnutrition and cramped living conditions in cities) creating competition among the elite. Social mobility increases, mostly downward as elites lose their status. The inter-elite competition creates fissures which lead to civil war and the final crisis stage lasting a few decades in which population decreases and the state breaks down. There follows an inter-cycle lasting several decades before a new growth period ensues.

Secular cycles have been linked to unsustainable agricultural production systems schindlerar and schindlerar, Fraser and Rimas (2011).

An Empirical Study of Oil Prices

In this section, we study empirical data to understand what the price dynamics of the contraction phase of oil extraction might be.

We worked with data from BP’s 2016 Statistical Review. We used prices and extraction data from the BP’s data set:

Annual crude oil prices in 2015 US dollars per barrel (deflated using the Consumer Price Index for the US) available from 1861 to 2015.

Annual world oil production expressed as a daily mean in millions barrels per day (MMbbl per day) from 1965 to 2015. These data include crude oil, shale oil, oil sands, and NGLs (natural gas liquids—the liquid content of natural gas where this is recovered separately). However, these data exclude liquid fuels from other sources, such as biomass and derivatives of coal and natural gas.

We used data from Etemad et al. (1998) to obtain annual world crude oil extraction in thousand metric tons from 1937 to 1970. We used the world average conversion factor of 7.6 barrels per ton to convert this data to barrels per year.

Remark 3.1

All data we use are very approximate. Laherrère (2014) has exhaustively documented incoherence in extraction data from all standard sources. Laherrère states (correctly) that since the data are accurate to at most 3%, it is both illogical and misleading to provide the data beyond two decimal points. We repeat the error of BP of including extra decimals, because we did not want to alter the data. We use a single price for the price of oil provided by BP, but there is a large spectrum of prices for oil of different densities, chemistry, and provenance Laherrère (2015). BP groups extraction data for crude oil, condensate, and NGL’s, a large spectrum of products not all used for the same purpose and of course with different prices. The fact that our regression analysis works suggests that there are correlations within the data and averaging going on.

In light of Remark 3.1, our aim is not to get the best fit possible, but to understand factors that influence price.

Let \(\left( P_t\right) _{t}\) denote the time series of oil prices (in 2015 dollars adjusted for inflation) from year 1861 to year 2015 and \(\left( Q_t\right) _{t}\) the time series of quantities of oil extracted (in million barrels daily) from year 1965 to year 2015 for BP data and from 1937 to 1970 for Etemad and Luciani data from Etemad et al. (1998).

A possible first idea to study the series \(\left( P_t\right) _{t=1861 \dots 2015}\) may be to fit a classical time series model to the data. As the series of prices is clearly not stationary, one can apply a preliminary Box–Cox transformation to stabilize the variance. We thus define the log prices:

$$\begin{aligned} p_t=\log P_t. \end{aligned}$$ (3.1)

This transformation is consistent with the assumption of log normality of the prices often encountered in the literature. One can try to eliminate the trend by differentiating. The lag-1 difference operator \(

abla\) is defined for a series \(\left( x_t\right) _t\) by the following:

$$\begin{aligned}

abla x_t \mathop {=}\limits ^{\mathrm{{def}}}x_t-x_{t-1}. \end{aligned}$$

We are then led to study the so-called log return at year t:

$$\begin{aligned} r_t =

abla p_t=p_t-p_{t-1}=\log P_t-\log P_{t-1}. \end{aligned}$$ (3.2)

It is well known that the log returns data \((

abla p_t)_t\) may, at first glance, look like a white noise. But, by looking closer, one would observe that the magnitude of the fluctuations still depends on time. In fact, the series \((|r_t|)_t\) and \((r_t^2)_t\) are often strongly autocorrelated. Moreover, the distribution of the log returns may also present heavy tails. To consider these two stylized facts, the log returns are often modeled by non-linear models, such as ARCH/GARCH models or derivative models (Cheong 2009; Hou and Suardy 2012).

Many other techniques to forecast oil prices have been considered by researchers: financial models, structural models, and computational methods using artificial neural networks, see Behmiri and Manso (2013) and references therein. In structural models, oil prices are modeled as a function of explanatory variables, such as oil consumption, oil extraction, and even non-oil variables, such as interest rates. However, the use of such models is limited by the need of future values of the explanatory variables to derive predictions of oil prices. Furthermore, the number of possible explanatory variables may be very large.

Price Explained by Oil Extraction

The approach we consider here is structural. We try to derive information on the price from a single explanatory time series: oil extraction. From Fig. 1, one sees that \(Q_t\) cannot explain \(P_t,\) because the price P is not uniquely determined by the extracted quantity Q; in other words, several prices correspond to the same produced quantity. For this reason, we attempt to use, in addition to \(Q_t\), the lag-1 difference and the lag-2 difference of the series \((Q_t)_t\) at year t:

$$\begin{aligned}

abla Q_t&\mathop {=}\limits ^{\mathrm{{def}}}&Q_t-Q_{t-1},\end{aligned}$$ (3.3)

$$\begin{aligned}

abla ^2 Q_t&\mathop {=}\limits ^{\mathrm{{def}}}&

abla (Q_t-Q_{t-1})=Q_t-2\,Q_{t-1}+Q_{t-2}. \end{aligned}$$ (3.4)

Fig. 1 Oil price as a function of oil production from 1965 to 2015 Full size image

Note that \(Q_t\), \(

abla Q_t,\) and \(

abla ^2 Q_t\) are linearly independent, so their span is the same as that of \(Q_t\), \(Q_{t-1}\) and \(Q_{t-2}\). We prefer the former variables to the latter, because \(

abla Q_t\) and \(

abla ^2 Q_t\) are the discreet first and second derivatives of \(Q_t\) with step time \(h=1\) making them easier to interpret. From an economic point of view, it is quite natural to postulate that the market for oil a given year is influenced by production in the previous years.

We consider the following model:

$$\begin{aligned} p_t=a+b\,Q_t +c\,

abla Q_t+d\,

abla ^2 Q_t +\epsilon _t \end{aligned}$$ (3.5)

where a, b, c, and d are coefficients determined by the linear regression and \((\epsilon _t)_t\) is a centered second-order stationary process. Equation (3.5) is equivalent to

$$\begin{aligned} P_t=\exp (a+b\,Q_t +c\,

abla Q_t+d\,

abla ^2 Q_t+\epsilon _t). \end{aligned}$$ (3.6)

The dependency of price \(P_t\) on these variables is non-linear. As the logarithm function flattens large values, the model considers the inelasticity of oil prices. That is, small changes in the supply provoke large changes in price.

Data From 1965 to 2015

The R output for the linear regression with the data starting at year 1965 is given in Appendix Fig. 9. First, note that we have lost two years because of the lag-2 differences \((

abla ^2 Q_t)_t\) that are only available form year 1967 with the data set starting in 1965. Adjusted R-squared being 0.5544 means that the model explain \(55.44\%\) of the variance from the mean considering the number of explanatory variables. From the stars in the R output, we obtain four significant coefficients for the model (3.5) that are:

$$\begin{aligned} \begin{array}{lll} a &{} \approx &{} 2.1\\ b &{} \approx &{} 0.029\\ c &{} \approx &{} -0.20\\ d &{} \approx &{} 0.081.\\ \end{array} \end{aligned}$$

Remark 3.2

We considered several other models. The regression with the 3-lag difference \(

abla ^3Q_t\) in addition to the other variables gave a coefficient which was not significant. Using \(\tilde{p}_t \mathop {=}\limits ^{\mathrm{{def}}}\log (\log P(t))\) marginally improved the fit of the model (with the same variables). In fact, taking further logs also marginally improved the fit. We use the above model for simplicity and because, as we will see, the model is not robust over different time periods.

Fig. 2 Fitted price from 1967 to 2015 Full size image

We have plotted the adjusted prices and the real prices in Fig. 2. Moreover, the model allows us to derive a prediction of the price \(P_{t+1}\) at year t. Namely, if the residuals are independent, as they are centered, the prediction of \(\epsilon _{t+1}\) is zero. Then, from the extracted quantity at year \(t+1\), we obtain, at year, t a prediction of the price:

$$\begin{aligned} \hat{P}_{t+1}= & {} \exp \left( a+b\,Q_{t+1}+c\,

abla Q_{t+1}+d\,

abla ^2 Q_{t+1}\right) \\= & {} \exp \left( a+b\,Q_{t+1}+c\,\left( Q_{t+1}-Q_t\right) +d\,\left( Q_{t+1}-2\,Q_t+Q_{t-1}\right) \right) . \end{aligned}$$ (3.7)

The conclusions we have made above are, in fact, not justified here, because the residuals of the regression are correlated. One can observe the dependence of the residuals while plotting the autocorrelation function and the partial autocorrelation function with the R commands acf and pacf. One can also perform the Box–Pierce test, the turning point test, and the difference sign test with the R commands Box.test, turning.point.test, and difference.sign.test. All these tests reject the hypothesis of a white noise at the level \(\alpha =5\%\). We attempt then to fit an ARMA model to the residuals. As the partial autocorrelation at lag 1 was very high, we have chosen the AR(1) model. In fact, among the models ARMA(p, q) with \(p=0,1,2\) and \(q=0,1,2\), the one that minimizes the AIC and the BIC criteria is the AR(1) model. Finally, we perform a linear regression with a covariance structure of an AR(1) with the R command gls. The model is fitted by maximizing the log-likelihood. The R output is given in Appendix Fig. 11. The coefficients of the model (3.5) are as follows:

$$\begin{aligned} \begin{array}{lll} a &{} \approx &{} 1.6\\ b &{} \approx &{} 0.032\\ c &{} \approx &{} -0.062\\ d &{} \approx &{} 0.018 \end{array} \end{aligned}$$ (3.8)

with an AR(1)-noise \((\epsilon _t)_t\) such that:

$$\begin{aligned} \epsilon _t-\phi \,\epsilon _{t-1}=\eta _t \end{aligned}$$

with \(\phi\) estimated by 0.816 and \((\eta _t)_t\) an actual centered white noise with estimated standard deviation 0.474. The intervals R command allows us to obtain \(95\%\) CI for the model coefficients (see Fig. 10). The plot of price and fitted price is given in Fig. 3.

Fig. 3 Fitted price from 1967 to 2015 Full size image

Fig. 4 Percent increases in quantity from 1971 to 2015 Full size image

We tested the stability of the coefficients by trying the regression in different sub-intervals and found that the coefficients were not stable. This means that the model cannot be used directly to predict future prices; there is some sort of averaging going on. One must understand why the coefficients differ in different time frames and then analyze how future conditions might effect the coefficients. To this end, we split the data into four periods with different growth rate and price characteristics (see Fig. 4) and computed the coefficients of the classical linear regression for each period (see Figs. 5, 6).

Fig. 5 Adjusted price and linear regression coefficients for the first period Full size image

Fig. 6 Adjusted price and linear regression coefficients for the last three periods Full size image

Data From 1937 to 1970

From Fig. 7, one sees that, for this period, the price is a function of the extracted quantity. A linear regression of the log price with explanatory variables \(Q_t\) and \(

abla Q_t\) shows that the coefficient of \(

abla Q_t\) is not significant in the R output of Appendix Fig. 12. The R output for the sub-model of (3.5 with \(c=d=0\) is given in Appendix Fig. 13. Again, the residuals are not i.i.d. Therefore, we use the linear model with covariance structure ARMA(1, 1) chosen among the ARMA(p, q) models with \(p=0,1,2\) and \(q=0,1,2\) with the Akaike AIC criterion. The R output is given in Appendix Fig. 14. The model coefficients are as follows:

$$\begin{aligned} \begin{array}{lll} a &{} = &{} 2.943\\ b &{} = &{} -0.01\\ \end{array} \end{aligned}$$

with an ARMA(1, 1)-noise \((\epsilon _t)_t\) such that:

$$\begin{aligned} \epsilon _t-\phi \,\epsilon _{t-1}=\eta _t +\theta \,\eta _{t-1} \end{aligned}$$

with \((\phi ,\theta )\) estimated by (0.04, 0.69) and \((\eta _t)_t\) an actual centered white noise with estimated standard deviation 0.08. The intervals R command allows us to obtain \(95\%\) confidence intervals for the model coefficients (see Appendix Fig. 15). The plot of price and fitted price is given in Fig. 8.

Fig. 7 Oil price as a function of oil production from 1937 to 1970 Full size image

Fig. 8 Fitted price from 1937 to 1970 Full size image

Interpretation of the Results

It is well known that oil prices are correlated with economic growth. Indeed, some use estimates of future Gross World Product (GWP) to obtain estimates of future oil prices. It is not surprising then that the model (3.5) gives a reasonable fit of GWP if one replaces \(p_t\) by \(y_t \mathop {=}\limits ^{\mathrm{{def}}}\log Y_t\) where \(Y_t\) denotes GWP. We will make some conjectures about this relationship presently.

We begin by examining what the predictive model (3.7) for the period 1967–2015 using the coefficients (3.8).

If extraction is constant for 2 years, then \(

abla Q_t=

abla ^2 Q_t=0\). In that case, the model reduces to

$$\begin{aligned} p_{\mathrm {basic}}(t) = \exp ( 1.591 + 0.032 \times Q_t +\epsilon _t) \end{aligned}$$ (3.9)

with 0 the expected value of \(\epsilon _t\). We call this the basic price formula that predicts the price if extraction is constant. Note that the price increases with the quantity, consistent with a positive derivative in (5.11) as \(\dfrac{\partial {P_t}}{\partial {Q_t}} /P_t = .032>0\).

Suppose \(

abla Q_t=\) constant \(

e 0\). Then, \(

abla ^2 Q_t=0\) and the model becomes

$$\begin{aligned} p(t) = \exp (1.591 + 0.032 \times Q_t - 0.062 \times

abla Q_t). \end{aligned}$$ (3.10)

Note that the coefficient of \(

abla Q_t\) is about twice as large as the coefficient of \(Q_t\) and of opposite sign. Thus, \(

abla Q_t\) gives a larger price signal than \(Q_t\). The signal goes in the opposite direction of the change, but it only lasts for a year. One might understand this as follows: a rise (fall) in extraction causes growth (contraction) in the economy much more next year than this year (as in the example in “An Example” section).

Example 3.1

If extraction is constant at 88 MMbll per day, the basic price is \(\hat{P}_{\mathrm basic}(t) \approx \$82/barrel\). If \(Q_t= 88,\) \(Q_{t-1}=86,\) and \(Q_{t-2}=84,\) then \(

abla Q_t=

abla Q_{t-1}=2\) and \(

abla ^2 Q_t=0\). Then, \(\hat{P}_t \approx \$72\)/barrel less than the basic price, because extraction is rising. If \(Q_{t-1}=90\) and \(Q_{t-2}= 92,\) then \(

abla Q_t=

abla Q_{t-1}=-2\) and \(

abla ^2 Q_t= 0\). Then \(\hat{P}(t) \approx \$93\)/barrel greater than the basic price because extraction is falling.

The second derivative is negative at local maxima and positive at local minima, so that the second derivative will mollify the price change caused by the first derivative. This explains why peak extraction is frequently associated with low prices. A minimum in extraction will thus be associated with relatively high prices. Economically, this factor can be interpreted as follows: it takes 2 years for the economic growth (contraction) produced by an increase (decrease) in extraction to take hold. The first year it is rather fragile and easily reduced (increased) by a drop (rise) in extraction.

Example 3.2

1. If \(Q_t = 88\), \(Q_{t-1} =86\), and \(Q_{t-2} =88,\) then extraction reaches a local minimum at \(Q_{t-1}\). We compute \(\hat{P}(t) \approx \$78\) rather than $72 as in Example 3.1 with the same increase in extraction from 86 MMbll per day. 2. If \(Q_t = 88\), \(Q_{t-1}=90\), and \(Q_{t-2} =88\), then extraction reaches a local maximum at \(Q_{t-1}\). We compute \(\hat{P}(t) \approx \$86\) rather than $93 as in Example 3.1 with the same decrease in extraction from 90 MMbll per day. 3. It is interesting to note that if \(Q_t= q_0\rho ^t\) with \(\rho = (1+r)\) and the growth rate r in a reasonable range (\(0<r<.12\)), then \(\hat{p}(t)\) is an increasing function of t. For example, if \(Q_t = 80 (1.02)^t\), then \(\hat{p}(3) \approx\) $58.3 < $60.8 \(\approx \hat{p}(4)\), thus increasing extraction at a constant rate produces increasing prices. However, \(\hat{p}(4)\) is much lower than the basic price at the same extraction quantity which is $87.6.

We believe that the differences in the different periods we studied stem from the importance of oil in the economy, that is the size of the partial derivative of economic production with respect to oil varied during the different periods. From 1937 to 1970, oil extraction increased regularly and so was never an impediment to economic growth. In the 1970’s, oil shocks created irregular supplies and constraints in oil supplies reduced economic growth. Mathematically, this is expressed by a larger partial derivative of economic production with respect to oil extraction levels. this period, as in the period from 2005 to 2014. This leads us to the following conjectures in accord with (5.10) and our empirical results:

Conjecture 3.1

The principle signal of scarcity of oil production is that \(\dfrac{\partial {p}}{\partial {q}}> 0\).

Conjecture 3.2

Volatility in the supply of oil leads to increased price dependence on the discreet first and second derivatives of supply.

The standard scarcity rent view that lower quantities leads to higher prices can be explained by temporary dependence on the first- and second-discreet derivatives of extraction levels. After the short-term spikes, the price stabilizes with economic production adjusting to the new level of extraction. In other words, a shortage of important items in economic production ultimately leads to lower prices, because economic production contracts reducing demand. This can be read from Eq. (5.8).

In Alba et al. (1994), the authors note that \(\dfrac{\partial {p}}{\partial {q}}> 0\) is unstable. When extraction levels are increasing, \(\dfrac{\partial {p}}{\partial {q}}> 0\) encourages many actors to enter the extraction business which commonly leads to sharp increases in extraction levels and a price collapse. This occurred in the late 1970’s and early 1980’s and again from 2000 to 2013 when capital expenses increased at roughly 11% per year (Kopits 2014; Mushalnik 2016). If extraction levels are decreasing, then \(\dfrac{\partial {p}}{\partial {q}}> 0\) leads to bankruptcies of higher priced extractors which leads to sharply decreasing extraction levels.