Evidence of speculation in world oil prices

Abstract

It has recently been suggested that financial speculation is now playing an important role in daily price movements of global oil prices. This raises the question: what are important drivers of price changes given this new speculative regime? We identify new factors of the oil market related to speculation by fitting subset vector autoregression models with exogenous variables (SubVARX) and rank them by importance. Further, to account for model uncertainty and to obtain robust parameter estimation in this study, we apply a bootstrap model selection procedure. We find that certain speculative factors explain a large portion of the variation in oil price for the given data set.

Keywords

Model uncertainty oil bootstrap vector autoregression

1. Introduction

Increased volatility and price surges in the global oil market over the last 10 years have sparked a renewed interest in understanding the contributing factors to oil price dynamics. In particular, it has been noted that financial institutions have had a greater influence on the price of oil due to the allocation of assets to commodity index trading strategies increasing from 13 billion dollars in 2004 to 260 billion dollars as of March 2008. Commodities behave differently to stocks and bonds (Roache, 2008) providing investors with portfolio diversification opportunities (Chen and Pinsky, 2003; Jennings, 2012) and a hedge against inflation (Froot, 1995). More importantly, it has also been suggested that price speculation is now playing an important role in daily price movements (Juvenal and Petrella, 2011; Kilian and Murphy, 2011). This raises a number of questions. First, oil price models have traditionally relied on oil inventories as a factor of global oil prices due to the way they reflect the demand and supply imbalances that drive fluctuations in the oil price (He et al., 2010; Kilian and Murphy, 2011; Manera et al., 2007; Merino and Ortiz, 2005; Pang et al., 2011; Ye et al., 2002, 2005, 2006). However, given this new ‘speculative regime’ are such models still appropriate? Second, if we are in a new regime, are there new and relevant factors that one should incorporate into a model?

Answering these questions is important due to their numerous policy implications. It is now well understood that periods of high oil prices have a number of significant effects on the global economy. As most industrialised countries are net oil importers, periods of high oil prices can have significant negative effects on their economic growth and may even cause recessions (Gisser and Goodwin, 1986; Hamilton, 1983). In essence, higher prices transfer wealth from oil-importing economies to oil-exporting economies (Sachs et al., 1981). Higher oil prices have economy-wide implications including increasing the cost of production, increasing the cost of services, and increasing the cost of agriculture. Persisting high oil prices may lead to inflation (Cunado and Perez de Gracia, 2005) resulting in reduced consumer confidence as higher oil prices suppress consumption (Uri and Boyd, 1997). Oil prices can also have an impact on financial markets. Historically, the oil price shocks in 1973, 1979, and 1990 were followed by stock market downturns and high oil prices can also affect the financial performance and cash flows of firms. In turn, this affects shareholders due to reduced dividend payments and equity prices (Huang et al., 1996). Given the significant role the world oil market has in almost all aspects of the global economy, it is imperative that economic planners understand the factors that drive this market.

Our study identifies new factors related to financial speculation that are not yet well recognised in the oil markets. Our data-driven approach starts with the application of an unrestricted vector autoregression with exogenous variables (VARX). This approach is well known to be useful for determining relationships between economic variables and for conducting evidence-based policy. We have chosen a set of economic and financial variables through consultation with commodities strategists involved in the asset allocation decision process. While it has been identified that over-fitting can be an issue with such studies (Anderson and Burnham, 2003; Cox and Reid, 2000), market participants seeking to identify unrecognised investment opportunities do require such analysis. By only attempting to fit models with already well understood and well recognised parameters, market participants are unable to identify changes to market structure and any new relationships that may have emerged.

Most studies of this type face the problem of data-dredging or data-snooping bias. One way to circumvent this is to apply a statistical methodology that takes model uncertainty into account. Model uncertainty typically occurs as one only has a single realisation of the data generating process to analyse, thus any model fitted may only be capturing characteristics specific to the single sample path and may not generalise to the population. This can lead to a number of problems such as noise variables being frequently identified as true factors, true factors being excluded from the model, small p-values, and coefficients biased away from zero; see Anderson et al. (2001), Austin (2008), Bernanke et al. (2004), Clyde (2000), and Clyde and George (2004). This can lead to overestimation of the importance of the variables themselves.

Our study is the first to address the problem of model selection uncertainty in the context of commodity modelling. We apply non-parametric bootstrap techniques in conjunction with an automated general-to-specific (GetS) model selection algorithm to estimate coefficients for VARX models via ordinary least squares. The bootstrap provides the relative importance of variables as well as the relative importance of combinations of variables. This allows us to more accurately assess the importance of factors under study when compared to traditional methods. Whilst recent studies have investigated the role speculation as a whole has played on the oil market (e.g. Juvenal and Petrella, 2011), our study specifically investigates the importance of each of the factors under study. That is, previous studies have asked the question ‘How much do speculative factors influence the oil market?’ whereas we ask the question ‘which speculative factors affect the oil market and which ones are the most important?’ Further, combined with the application of VARX, our study is also the first to explicitly identify the time it takes for changes to these factors to flow through to the oil market.

2. Data

We base our study on a seven-dimensional vector time series $x_{t} = (x_{1} (t), x_{2} (t), \dots, x_{7} (t))'$ for days $t = 1, \dots, T$ , composed of two endogenous and five exogenous variables that are hypothesised speculative factors that drive the global oil market; see Table 1. Whilst it is tempting to throw as many financial market variables into the study as possible, when one attempts to model a large number of variables in combination with a large set of initial candidate models, model uncertainty is exacerbated (Anderson and Burnham, 2003; Chatfield, 1995; Miller, 1990; Zucchini, 2000). Attempting to model all possible factors at all possible lags is data dredging.

Table 1.

Description and sources of data.

	Data	Type	Description	Units	Source	Frequency
y ₁	x ₁	Endo.	S&P GSCI crude oil spot	US$ – price index	Thomson Reuters Datastream	Daily
y ₂	x ₂	Endo.	World oil supply – production	1000 B/Day	US Department of Energy	Monthly
z ₁	x ₃	Exo.	Industrial production	Index	OECD Statistics	Monthly
z ₂	x ₄	Exo.	US fed funds target rate	% p.a.	Thomson Reuters Datastream	Daily
z ₃	x ₅	Exo.	Inflation	CPI index	OECD Statistics	Monthly
z ₄	x ₆	Exo.	10 year government bond yield	% p.a.	Trading Economics	Daily
z ₅	x ₇	Exo.	US business (manufacturing) confidence indicator	Index	Thomson Reuters Datastream	Monthly

The endogenous (Endo.) and exogenous (Exo.) variables considered in our study. Business confidence is considered a leading indicator of economic activity and industrial production. To investors, a possible increase in economic activity would lead to greater demand for oil, therefore representing possible growth in the asset class and a good investment (Aastveity et al., 2012). Bonds and cash are seen as ‘safe’ alternative asset classes and provide a haven during volatile periods and therefore are considered as part of any asset allocation decision. Oil is often argued to provide inflation hedging properties and so the change to CPI has also been included (Chen and Pinsky, 2003; Frankel, 1986, 2008; Frankel and Rose, 2010).

While some series are only reported at low frequencies (e.g. monthly), our model must allow for signals from data series that are high frequency. Our model must also allow for signals that are low frequency but may not be released/reported at regular intervals. Given the critical time value of information, it is important to reflect changes to these series as they happen in our model. Therefore, we implemented our model based on a daily frequency and converted any data that is reported on a monthly basis to a daily frequency by carrying forward the last observation.

Over the sample period of March 1995 to August 2010 (consisting of 4022 data points), we consider daily changes $R_{t}^{i}$ calculated as

R_{t}^{i} = \log (\frac{x_{i} (t)}{x_{i} (t - 1)})

where $x_{i} (t)$ is the i th time series on day t. After constructing the daily changes vector time series $R (t) = (R_{1} (t), R_{2} (t), \dots, R_{7} (t))'$ we separate the series in the endogenous series $y_{t} = (y_{1} (t), y_{2} (t))$ and the exogenous series $z_{t} = (z_{1} (t), z_{2} (t), z_{3} (t), z_{4} (t), z_{5} (t))$ , thus mapping the data set x_t to y_t and z_t as described in Table 1.

Given the daily changes vector z_t, we also introduce the lags $z_{t - j}$ , for $j = 1, \dots, q$ where q represents the maximum significant lag. After consultation with market practitioners who are primarily focussed on short-term movements in the oil market, we choose a maximum lag length of $q = 21$ days (approximately four weeks of trading days). It was generally agreed that information contained in variables with lags any longer than 21 days would not be considered as part of their decision-making process.

3. Model and methodology

To robustly identify if speculation plays a role in the determination of world oil prices we have applied a methodology that we shall now describe.

3.1 The vector autoregression model

We begin the construction of the oil price model with a full-order unrestricted VARX of the form

y_{t} = \sum_{i = 1}^{m} B_{i} y_{t - i} + \sum_{j = 0}^{r} B_{m + 1 + j} z_{t - j} + ε_{t}

(1)

where $E [ε_{t}] = 0$ and $E [ε_{t} {ε^{'}}_{t}] = Q$ . Here, y_t and z_t are assumed to be zero-mean, wide-sense stationary time series of dimensions two and five respectively where $y_{t} = (y_{1} (t), y_{2} (t))'$ models the endogenous variables at time t, $z_{t} = (z_{1} (t), z_{2} (t), \dots, z_{5} (t))'$ models the exogenous variables at time t, and $ε (t)$ is a two-dimensional stationary vector stochastic process. For some choice of m and r, the matrices $B_{τ}$ for $τ = 1, 2, \dots, m$ and $B_{τ}$ for $τ = m + 1, m + 2, \dots, m + r + 1$ denote 2×2- and 2×5-dimensional matrices, respectively. A full-order vector autoregression (VAR) arises when we exclude the exogenous variables z_t.

3.2 Capturing dynamic relationships

In (1), all coefficients of the matrices $B_{τ}$ are non-zero. This assumption means that no subsets are considered and therefore the possibility of zero coefficients in (1) is also not considered. This requirement is restrictive if one wishes to exclude certain variables and their lags. Therefore, to capture the dynamic behaviour present in the data, we subsequently employed a model-fitting technique known as subset vector autoregression with exogenous variables (SubVARX).

Starting with a model of the form (1) and changing notation slightly to reflect that quantities defined previously may also depend on the model order and some lags, the multivariate SubVARX $(m, r)$ with included lags $I_{s}$ is given by

y_{t} = \sum_{i \in I_{s}}^{m} B_{i} y_{t - i} + \sum_{j \in I_{s}}^{r} B_{m + 1 + j} z_{t - j} + ε_{t}

(2)

This approach of first identifying the full model in the VARX avoids over-identifying restrictions and accidentally excluding important variables especially given the lack of strong economic theory on how restrictions should be imposed. Although economic theory may help to determine what the restrictions should be, we take a data-driven approach given our limited a priori knowledge.

3.3 Model selection

The structure of (2) gives a large number of possible models that could be fitted. In fact, $2^{m r}$ models are possible. Given a realisation of our data, we identified a SubVARX model of the form (2) using an automatic GetS econometric model selection procedure called Autometrics (Doornik, 2007).

Autometrics is a multiple search path strategy and revolves around a tree search that considers every possible model given the initial unrestricted model known as the general unrestricted model (GUM), that is, the VARX model (1). The true data generating process is assumed to be a subset of the GUM, that is, a particular choice of (2). The Autometrics algorithm estimates single equations using ordinary least squares as opposed to system estimation. While it could be argued that a system approach would be a better approach in cases where the variance covariance matrix of the system is non-diagonal, studies such as Krolzig (2000) and Hendry and Krolzig (2004) demonstrate that the parameters selected by Autometrics can successfully describe the true underlying data generating process, provide good size to power trade-offs, and are computationally more efficient when compared to full system estimation methods. Additionally, the GetS procedure can be applied equation-by-equation without loss of asymptotic efficiency (Krolzig, 2003).

To identify an optimal model, we selected a maximum lag q that ensures the true order (maximum lag of the endogenous and exogenous variables) of the model $(m, r)$ is captured (i.e. we need max(m, r) ≤ q). By setting the maximum lag for both exogenous and endogenous variables initially somewhat larger than the data indicate via analysing autocorrelations, then allowing Autometrics to determine the best full-order model, we are ensuring that the model selection process can capture the appropriate model. To begin we used maximum lag q of 21 days. Following determination of m and r, we obtained estimates of the $b_{τ}^{i, j}$ by assuming that every $b_{τ}^{i, j}$ is non-zero. This helps us model economic relationships appropriately by allowing for the possibility of entire lags or coefficient matrices being zero. It also avoids possible overparameterisation which can result in poor ex ante forecasting performance (Gooijer and Hyndman, 2006).

3.4 Model selection uncertainty

Identifying factors that influence world oil prices based on a single time series sample path is not robust against the problem of model selection uncertainty. This can lead to a number of problems such as certain irrelevant variables being identified as true factors, true factors being excluded from the model, small p-values, and coefficients biased away from zero; see Anderson et al. (2001), Austin (2008), Bernanke et al. (2004), Clyde (2000), and Clyde and George (2004). This can lead to overestimation of the importance of the factors themselves.

A number of approaches have been developed that attempt to deal with the model selection problem. Among them are bootstrap model averaging (Buckland et al., 1997), Bayesian model averaging (Clyde, 2000), bootstrap estimation of coefficients and confidence intervals (Austin, 2008), and multi-model inference (Anderson and Burnham, 2003). Each approach has its strengths and weaknesses. For example, some of these methods rely on distributional assumptions and a well-defined set of candidate models (Alfaro and Huelsenbeck, 2006). For an excellent outline of the model selection problem, see Anderson and Burnham (2004).

We applied a model-based resampling method called the non-parametric bootstrap (Efron and Tibshirani, 1994; Srinivas and Srinivasan, 2001). This approach builds a sampling distribution for the coefficient estimates by generating a large number of replicates of the original time series using a model-based resampling procedure. The only assumption required about the underlying data generating process is that it is based on i.i.d. innovations. This allows us to assess model uncertainty and identify possible model structures without the need to make distributional assumptions (Hinkley and Davison, 2006) and is particularly relevant in our setting as the residuals are not normally distributed; see Figure 1.

Figure 1.

Normality test for the residuals $ν_{t}$ .

The residuals $ν_{t}$ for $t = 1, \dots, T$ obtained by fitting the full model (1) to the original time series are not normally distributed. This guided our choice to use a non-parametric bootstrap procedure to generate replicates of the time series data.

3.5 Robustly identifying factors

To robustly identify factors, we combine the steps explained above to obtain L possible coefficient estimates $B^{* k}$ for $k = 1, \dots, L$ using the following methodology:

Generate the first set of approximately i.i.d. residuals $v_{t}$ for $t = 1, \dots, T$ , by fitting the full model (1) to the original time series. These residuals estimate the i.i.d. innovations $ε_{t}$ . This step also provides us with the coefficient estimates $\hat{B}$ from the full model fit.

Bootstrap the residuals by sampling with replacement from the components of $v_{t}$ and generate L resamples:

v_{t}^{* 1}, v_{t}^{* 2}, \dots, v_{t}^{* L} for t = 1, \dots, T

(3)

Generate L bootstrapped replicate time series using the model

y_{t}^{* k} = \sum_{i = 1}^{m} {\hat{B}}_{i} y_{t - i} + \sum_{j = 0}^{r} {\hat{B}}_{m + j + 1} z_{t - j} + v_{t}^{* k}

(4)

for $t = 1, \dots, T$ and $k = 1, \dots, L$ . This procedure creates a set of L replicate time series that resemble the original data series and allows us to (as far as possible) capture the main features of the underlying data generating process.

For each bootstrapped replicate time series $y_{t}^{* k}$ for $k = 1, \dots, L$ , apply the Autometrics model selection algorithm to identify a ‘winning model’ of the form (2) with coefficients $B^{* k}$ .

Store bootstrapped coefficient estimates $B^{* k}$ for $k = 1, \dots, L$ .

Although this methodology is highly computational, the main advantages of this approach are that an approximate sampling distribution for each of the coefficient estimates is established and the non-parametric bootstrap procedure (steps 2 and 3) allows us to estimate the sampling distribution of a statistic empirically without making assumptions about the form of the population. There are two sources of error when implementing the bootstrap: first, the error resulting from using a particular sample to represent the population, and second, the sampling error. We can mitigate sampling error by generating a large number of samples L.

In step 1, we fitted the model (1) that includes all available factors with all lags. By doing so, we are as far as possible removing any structure or dependence from the resultant observed residuals. Ideally, we would like to initially fit ‘all possible’ factors, so that we can be sure that the real model (assuming one exists) is nested within the full model fit. Of course, fitting all possible factors is not feasible in reality. Nevertheless, this is a necessary assumption that the model-based bootstrap procedure rests on.

We whitened the residuals before bootstrapping and then recolouring them to form the bootstrap. Our approach involved first fitting a very large number of predictors to help ensure as much as possible that the residuals did not contain any dependence structure. Further, analysis of the pre-whitened residuals indicated that there was no dependence remaining that required additional fitting (e.g. block bootstrap).

3.6 Relative importance of factors

As previously discussed, simply identifying important factors from the application of a model selection routine to the original time series may not give the researcher a deep enough appreciation of the importance of particular factors. For example, particular variables or combinations of variables may be selected, but not necessarily selected as part of the most frequently occurring models. That is, one might falsely conclude that a factor has no relationship to the response variable, simply because the most frequently occurring models excluded this variable as part of the model selection process. However, by tallying the number of times that any particular model is selected and dividing through by the number of bootstrap simulations L, we are able to calculate model selection probabilities or weights. In turn, this allows us to identify the relative importance of factors and combinations of factors.

To assess the relative importance of any particular factor we follow the approach of Anderson and Burnham (2003) and find the sum of the bootstrapped model selection weights for all the models that include the factor of interest. The higher the sum of the weights, the higher the relative importance of the factor. By following the same approach with pairs, triplets, and quadruplets, we are able to gain a better understanding of the importance of factor groupings. Care is needed in interpreting the weights. They should not be interpreted in absolute terms and should not be compared to the importance weights derived from studies where different model structures and/or variables are considered.

4. Empirical results

4.1 Initial model selected

We begin by examining the SubVARX model (2) identified by the automatic model selection algorithm applied to the original (non-bootstrapped) time series data with a significance level of 0.01, see Table 5. The factors for the oil supply equation S_t and the oil price equation O_t with the associated coefficient determined are given in Table 2. Also given are the standard errors, the t-values, and the t-probabilities for the coefficients. The results suggest that the important factors in the oil spot price equation O_t are the US 10 year government bond yield at lags 0 and 13 (US10GB and US10GB_13), US business confidence at lags 8 and 11 (USBC_8 and USBC_11), and the US inflation rate at lag 0 (USInfl). The important factor in the oil supply equation S_t is the US inflation rate at lag 11 (USInfl_11).

Table 2.

Initial models selected by the automatic model selection algorithm.

Oil Price Equation					Oil Supply Equation
Factor	Coefficient	Std error	t-value	t-prob	Factor	Coefficient	Std error	t-value	t-prob
OIL_2	−0.04	0.02	−2.82	*0.00	OIL_2	−0.00	0.00	−0.06	0.95
OIL_14	0.04	0.02	2.47	*0.01	OIL_14	−0.00	0.00	−0.30	0.77
OIL_15	0.04	0.02	2.64	*0.01	OIL_15	−0.00	0.00	−0.76	0.45
OIL_16	0.03	0.02	1.60	0.11	OIL_16	0.00	0.00	1.76	0.08
OilSup_3	0.51	0.20	2.60	*0.01	OilSup_3	0.00	0.02	0.07	0.94
US10GB	0.18	0.02	7.70	*0.00	US10GB	0.00	0.00	0.92	0.36
US10GB_4	0.06	0.02	2.58	*0.01	US10GB_4	0.00	0.00	1.62	0.11
US10GB_13	0.08	0.02	3.30	*0.00	US10GB_13	0.00	0.00	0.01	0.99
US10GB_19	−0.01	0.02	−0.24	0.81	US10GB_19	0.00	0.00	2.22	0.03
USCash_7	−0.01	0.01	−1.48	0.14	USCash_7	0.00	0.00	2.76	*0.01
USCash_14	0.01	0.01	2.70	*0.01	USCash_14	0.00	0.00	1.24	0.22
USCash_19	−0.01	0.01	−2.32	0.02	USCash_19	−0.00	0.00	−1.57	0.12
USBC_1	0.10	0.04	2.61	*0.01	USBC_1	−0.00	0.00	−0.05	0.96
USBC_8	0.11	0.04	3.03	*0.00	USBC_8	−0.00	0.00	−0.18	0.86
USBC_11	0.11	0.04	2.94	*0.00	USBC_11	0.00	0.00	0.60	0.55
USIP_8	−0.00	0.00	−2.19	0.03	USIP_8	0.00	0.00	0.43	0.67
USInfl	−1.22	0.32	−3.76	*0.00	USInfl	0.00	0.03	0.02	0.98
USInfl_7	0.75	0.32	2.31	0.02	USInfl_7	−0.00	0.03	−0.16	0.88
USInfl_11	0.02	0.32	0.07	0.94	USInfl_11	0.21	0.03	7.86	*0.00

The initial models for the oil price O_t and oil supply S_t selected by the automatic model selection algorithm Autometrics using a significance level of 0.01. The oil equation model O_t is shown on the left and the oil supply equation S_t is shown on the right. Conditional coefficient estimates (Coefficient) and conditional standard errors (Std error) are given and are calculated directly from the fitted model. The standard error reported estimates the variability conditional on the chosen model being selected and is determined as in (7). Also shown are the test statistics (t-value) and the p-values associated with each test statistic (t-prob). Significant t-probabilities (i.e. factors) are indicated with a star (*) and are calculated directly from the fitted model, see Appendix.

Some economists have previously suggested that a relationship between the government bond yields and oil prices exists whereby higher oil prices increase world net savings and then, subsequently, saved petrodollars are used to purchase securities. They suggest that higher oil prices would produce downward pressure on US government security yields. Some evidence of this relationship was found in the 12-month period through May 2005; see International Monetary Fund (2006) and Warnock and Warnock (2009). However, when industrial production was included in their regressions, this link became statistically insignificant. Interestingly, our initial model points to a new relationship: increasing government bond yields leads to increasing oil prices. One could argue that this relationship may be linked to government securities being sold from portfolios (hence, increasing bond yields) to purchase oil securities.

Identification of US business confidence at lags 8 and 11 as a significant factor indicates that it takes around one and a half to two trading weeks for shocks to business confidence to flow through to world oil prices. Business confidence is an important leading indicator of economic performance and an increase in economic activity would lead to greater demand for oil. One would typically argue that business confidence is a proxy for demand and higher demand would drive oil prices higher if it is not matched by sufficient oil supply. However, this does not rule out speculative play. Traders may use business confidence as an indicator of possible growth in the asset class and a sudden increase could signal a good time to invest (Aastveity et al., 2012).

The relationship between oil prices and inflation has been extensively discussed in the literature. The standard view is that higher oil prices increase inflation. However, much of the research on the relationship between oil prices and inflation focuses only on the relationship between either oil price levels (rather than returns) and inflation or the relationship between oil price changes and changes to CPI over a comparable period (e.g. daily correlations, monthly correlations, year-on-year changes). Our study however allows for the analysis of inter-temporal relationships.

Surprisingly, there are no significant lags for the oil supply factor and oil price factor appearing in either the oil equation or the oil supply equation. It would seem reasonable to expect that oil supply should impact on the price of oil to at least some degree. So we re-ran the algorithm with different settings for the p-values for the diagnostics and factor significance levels. We decreased the p-value to 0.10 from 0.01 with no significant change. These results appear to suggest that the supply levels themselves are not important. Relative inventories have historically been the important driver of oil markets rather than actual supply levels.

One should be careful in interpreting these results as factors are sometimes included simply because the corresponding variable in the other equation is significant according to its value. As a result, an insignificant factor will sometimes appear in an equation even if it is deemed to have no explanatory power according to its p-value. For example, the US inflation rate at lag 11 (USInfl_11) is the only one of two factors deemed important in the oil supply equation S_t and, although it is not significant in the oil equation O_t, it still appears as part of that equation. When we calculate the bootstrap coefficient and standard error estimates, it would have been more accurate to set these insignificant coefficients to zero. However, there will only be a small component of variation contributed by these variables, which will not materially affect our results.

4.2 Ranking models and factors

For each of the L replicate time series $y_{t}^{*}$ obtained from the bootstrap procedure, we applied the Autometrics model selection algorithm. This results in L potential models. Disregarding the model coefficients and simply tallying the models based on their SubVARX structure (the factors and the lags chosen), we tallied and then ranked the number of times the particular factor and lag appeared within the L models. In Table 3, we show the top 20 model structures ranked by the number of times they appear within the L models. The tally for each model structure is low due to the large number of potential SubVARX models. For example, the model structure with the highest tally (Model 1) was only selected 26 times out of a possible 10,000. Model 2, selected 25 times, is quite dissimilar to Model 1. Given the large number of candidate models, this is not a surprising result.

Table 3.

Ranking models and factors that explain price changes.

FactorAImportance:		SpeculativeAFactors										ClassicAFactors
		1.0000	1.0000	0.8426	0.6869	0.5517	0.4214	0.3827	0.3750	0.3143	0.2470	0.6904	0.6292	0.4741	0.3414	0.3017	0.2697	0.2508	0.2222	0.2174	0.2021
Model	Tally	US10GB	USInfl_11	USInfl	US10GB_13	US10GB_4	USInfl_7	USIP_8	US10GB_19 US10GB_2 US10GB_12			USBC_11	USBC_8	USBC_1	USBC_18	USIP	USIP_5	USIP_12	USBC_4	USBC_5	USBC_17
1	26	1	1	1	0	1	0	1	1	0	0	1	0	0	0	1	1	0	0	1	0
2	25	1	1	1	1	1	1	0	0	0	0	0	1	1	0	0	0	0	0	0	0
3	25	1	1	1	1	0	1	1	0	0	0	1	1	1	1	0	0	1	0	0	0
4	25	1	1	1	1	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0
5	24	1	1	1	1	1	0	1	0	0	0	1	0	0	0	0	1	0	0	0	0
6	23	1	1	1	1	0	1	1	0	0	0	1	0	0	0	0	0	0	0	0	0
7	22	1	1	1	1	0	0	0	1	1	1	1	0	1	0	0	0	1	0	0	0
8	22	1	1	0	1	1	1	0	0	0	1	0	1	0	0	0	0	0	0	0	0
9	22	1	1	1	0	1	0	0	0	0	0	0	1	0	0	1	0	0	1	0	1
10	22	1	1	0	0	1	0	0	1	0	0	1	0	0	0	0	0	0	0	1	0
11	22	1	1	1	1	0	1	0	0	0	1	0	1	0	0	0	0	0	0	0	0
12	21	1	1	1	0	1	1	0	0	0	0	1	1	0	0	1	1	0	1	0	0
13	20	1	1	0	0	1	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0
14	20	1	1	0	1	0	1	0	1	0	0	0	1	0	0	1	0	0	0	0	0
15	20	1	1	1	1	1	0	0	1	1	0	1	0	1	0	1	0	0	0	0	0
16	20	1	1	1	1	0	1	1	1	0	1	1	1	0	1	0	0	1	0	0	0
17	20	1	1	1	1	0	1	1	0	1	0	1	1	0	0	0	0	1	0	0	0
18	19	1	1	1	1	0	1	1	0	0	0	0	0	1	1	0	0	0	0	1	0
19	19	1	1	1	0	1	0	1	1	0	0	1	1	1	0	0	0	0	0	0	0
20	19	1	1	1	1	1	0	0	0	1	0	1	0	0	1	0	0	0	1	0	0

The top 20 models based on model tally (column 2) against the top 10 factor/lags related to financial speculation and the top 10 ‘classic’ factor/lags (i.e. related to supply and demand). The table ranks the models (model importance weight) and the factor/lags based on their frequency among the L models. We colour the table cell grey if the factor/lag is included in the specific model and white if it is not. For each factor/lag we give its importance (1.0 means the factor is included in 100% of all models) in the top row of the table. A standard error estimate for each factor can be found in Table 6.

Table 4.

Relative importance of factor groups and their lags.

SINGLES	RelativeImportance	PAIRS		RelativeImportance		TRIPLETS		RelativeImportance		QUADRUPLETS			RelativeImportance
SINGLES	RelativeImportance	PAIRS		RelativeImportance		TRIPLETS		RelativeImportance					RelativeImportance
US10GB	1.000	US10GB	USInfl_11	1.000	US10GB	USInfl_11	USInfl	0.843	US10GB	USInfl_11	USInfl	USBC_11	0.590
USInfl_11	1.000	US10GB/USInfl_11	USInfl	0.843	US10GB/USInfl_11	USInfl	USBC_11	0.590	US10GB/USInfl_11	USInfl	USBC_11	US10GB_13	0.404
USInfl	0.843	US10GB/USInfl_11	USBC_11	0.690	US10GB/USInfl_11	USBC_11	US10GB_13	0.474	US10GB/USInfl_11	USBC_11	US10GB_13	USBC_8	0.283
USBC_11	0.690	US10GB/USInfl_11	US10GB_13	0.687	US10GB/USInfl_11	US10GB_13	USBC_8	0.428	US10GB/USInfl_11	US10GB_13	USBC_8	US10GB_4	0.204
US10GB_13	0.687	US10GB/USInfl_11	USBC_8	0.629	US10GB/USInfl_11	USBC_8	US10GB_4	0.331	US10GB/USInfl_11	USBC_8	US10GB_4	USBC_1	0.160
USBC_8	0.629	US10GB/USInfl_11	US10GB_4	0.552	US10GB/USInfl_11	US10GB_4	USBC_1	0.268	US10GB/USInfl_11	US10GB_4	USBC_1	USInfl_7	0.111
US10GB_4	0.552	US10GB/USInfl_11	USBC_1	0.474	US10GB/USInfl_11	USBC_1	USInfl_7	0.202	US10GB/USInfl_11	USBC_1	USInfl_7	USIP_8	0.084
USBC_1	0.474	US10GB/USInfl_11	USInfl_7	0.421	US10GB/USInfl_11	USInfl_7	USIP_8	0.167	US10GB/USInfl_11	USInfl_7	USIP_8	US10GB_19	0.069
USInfl_7	0.421	US10GB/USInfl_11	USIP_8	0.383	US10GB/USInfl_11	USIP_8	US10GB_19	0.158	US10GB/USInfl_11	USIP_8	US10GB_19	USIP_11	0.062
USIP_8	0.383	US10GB/USInfl_11	US10GB_19	0.375	US10GB/USInfl_11	US10GB_19	USIP_11	0.122	US10GB/USInfl_11	US10GB_19	USIP_11	USBC_18	0.043
US10GB_19	0.375	US10GB/USInfl_11	USIP_11	0.355	US10GB/USInfl_11	USIP_11	USBC_18	0.133	US10GB/USInfl_11	USIP_11	USBC_18	US10GB_2	0.039
USIP_11	0.355	US10GB/USInfl_11	USBC_18	0.341	US10GB/USInfl_11	USBC_18	US10GB_2	0.102	US10GB/USInfl_11	USBC_18	US10GB_2	USIP	0.041
USBC_18	0.341	US10GB/USInfl_11	US10GB_2	0.314	US10GB/USInfl_11	US10GB_2	USIP	0.104	US10GB/USInfl_11	US10GB_2	USIP	USInfl_17	0.035
US10GB_2	0.314	US10GB/USInfl_11	USIP	0.302	US10GB/USInfl_11	USIP	USInfl_17	0.096	US10GB/USInfl_11	USIP	USInfl_17	USIP_5	0.036
USIP	0.302	US10GB/USInfl_11	USInfl_17	0.293	US10GB/USInfl_11	USInfl_17	USIP_5	0.084	US10GB/USInfl_11	USInfl_17	USIP_5	USInfl_5	0.044
USInfl_17	0.293	US10GB/USInfl_11	USIP_5	0.270	US10GB/USInfl_11	USIP_5	USInfl_5	0.118	US10GB/USInfl_11	USIP_5	USInfl_5	USIP_12	0.041
USIP_5	0.270	US10GB/USInfl_11	USInfl_5	0.266	US10GB/USInfl_11	USInfl_5	USIP_12	0.087	US10GB/USInfl_11	USInfl_5	USIP_12	US10GB_12	0.044
USInfl_5	0.266	US10GB/USInfl_11	USIP_12	0.251	US10GB/USInfl_11	USIP_12	US10GB_12	0.093	US10GB/USInfl_11	USIP_12	US10GB_12	US10GB_17	0.025
USIP_12	0.251	US10GB/USInfl_11	US10GB_12	0.247	US10GB/USInfl_11	US10GB_12	US10GB_17	0.068	US10GB/USInfl_11	US10GB_12	US10GB_17	US10GB_15	0.018
US10GB_12	0.247	US10GB/USInfl_11	US10GB_17	0.241	US10GB/USInfl_11	US10GB_17	US10GB_15	0.054	US10GB/USInfl_11	US10GB_17	US10GB_15	USBC_4	0.015
US10GB_17	0.241	US10GB/USInfl_11	US10GB_15	0.228	US10GB/USInfl_11	US10GB_15	USBC_4	0.043	US10GB/USInfl_11	US10GB_15	USBC_4	USBC_5	0.017
US10GB_15	0.228	US10GB/USInfl_11	USBC_4	0.222	US10GB/USInfl_11	USBC_4	USBC_5	0.063	US10GB/USInfl_11	USBC_4	USBC_5	USInfl_2	0.027
USBC_4	0.222	US10GB/USInfl_11	USBC_5	0.217	US10GB/USInfl_11	USBC_5	USInfl_2	0.054	US10GB/USInfl_11	USBC_5	USInfl_2	USBC_17	0.017
USBC_5	0.217	US10GB/USInfl_11	USInfl_2	0.206	US10GB/USInfl_11	USInfl_2	USBC_17	0.040	US10GB/USInfl_11	USInfl_2	USBC_17	USBC_9	0.016
USInfl_2	0.206	US10GB/USInfl_11	USBC_17	0.202	US10GB/USInfl_11	USBC_17	USBC_9	0.051	US10GB/USInfl_11	USBC_17	USBC_9	US10GB_8	0.016
USBC_17	0.202	US10GB/USInfl_11	USBC_9	0.199	US10GB/USInfl_11	USBC_9	US10GB_8	0.048	US10GB/USInfl_11	USBC_9	US10GB_8	USInfl_8	0.017
USBC_9	0.199	US10GB/USInfl_11	US10GB_8	0.198	US10GB/USInfl_11	US10GB_8	USInfl_8	0.054	US10GB/USInfl_11	US10GB_8	USInfl_8	USInfl_19	0.014
US10GB_8	0.198	US10GB/USInfl_11	USInfl_8	0.195	US10GB/USInfl_11	USInfl_8	USInfl_19	0.047	US10GB/USInfl_11	USInfl_8	USInfl_19	US10GB_5	0.012
USInfl_8	0.195	US10GB/USInfl_11	USInfl_19	0.177	US10GB/USInfl_11	USInfl_19	US10GB_5	0.041	US10GB/USInfl_11	USInfl_19	US10GB_5	USIP_1	0.016
USInfl_19	0.177	US10GB/USInfl_11	US10GB_5	0.171	US10GB/USInfl_11	US10GB_5	USIP_1	0.041	US10GB/USInfl_11	US10GB_5	USIP_1	US10GB_18	0.006
US10GB_5	0.171	US10GB/USInfl_11	USIP_1	0.166	US10GB/USInfl_11	USIP_1	US10GB_18	0.028	US10GB/USInfl_11	USIP_1	US10GB_18	USBC_10	0.007
USIP_1	0.166	US10GB/USInfl_11	US10GB_18	0.161	US10GB/USInfl_11	US10GB_18	USBC_10	0.026	US10GB/USInfl_11	US10GB_18	USBC_10	US10GB_9	0.009
US10GB_18	0.161	US10GB/USInfl_11	USBC_10	0.149	US10GB/USInfl_11	USBC_10	US10GB_9	0.027	US10GB/USInfl_11	USBC_10	US10GB_9	USIP_18	0.007
USBC_10	0.149	US10GB/USInfl_11	US10GB_9	0.147	US10GB/USInfl_11	US10GB_9	USIP_18	0.026	US10GB/USInfl_11	US10GB_9	USIP_18	US10GB_10	0.004
US10GB_9	0.147	US10GB/USInfl_11	USIP_18	0.140	US10GB/USInfl_11	USIP_18	US10GB_10	0.021	US10GB/USInfl_11	USIP_18	US10GB_10	USInfl_9	0.004
USIP_18	0.140	US10GB/USInfl_11	US10GB_10	0.137	US10GB/USInfl_11	US10GB_10	USInfl_9	0.019	US10GB/USInfl_11	US10GB_10	USInfl_9	US10GB_1	0.003
US10GB_10	0.137	US10GB/USInfl_11	USInfl_9	0.130	US10GB/USInfl_11	USInfl_9	US10GB_1	0.020	US10GB/USInfl_11	USInfl_9	US10GB_1	USInfl_6	0.004
USInfl_9	0.130	US10GB/USInfl_11	US10GB_1	0.128	US10GB/USInfl_11	US10GB_1	USInfl_6	0.022	US10GB/USInfl_11	US10GB_1	USInfl_6	USIP_13	0.003
US10GB_1	0.128	US10GB/USInfl_11	USInfl_6	0.123	US10GB/USInfl_11	USInfl_6	USIP_13	0.017	US10GB/USInfl_11	USInfl_6	USIP_13	USIP_17	0.005
USInfl_6	0.123	US10GB/USInfl_11	USIP_13	0.117	US10GB/USInfl_11	USIP_13	USIP_17	0.030	US10GB/USInfl_11	USIP_13	USIP_17	USBC	0.011
USIP_13	0.117	US10GB/USInfl_11	USIP_17	0.116	US10GB/USInfl_11	USIP_17	USBC	0.025	US10GB/USInfl_11	USIP_17	USBC	USIP_2	0.004
USIP_17	0.116	US10GB/USInfl_11	USBC	0.112	US10GB/USInfl_11	USBC	USIP_2	0.028	US10GB/USInfl_11	USBC	USIP_2	US10GB_16	0.005
USBC	0.112	US10GB/USInfl_11	USIP_2	0.110	US10GB/USInfl_11	USIP_2	US10GB_16	0.014	US10GB/USInfl_11	USIP_2	US10GB_16	US10GB_7	0.001
USIP_2	0.110	US10GB/USInfl_11	US10GB_16	0.108	US10GB/USInfl_11	US10GB_16	US10GB_7	0.021	US10GB/USInfl_11	US10GB_16	US10GB_7	USInfl_13	0.006
US10GB_16	0.108	US10GB/USInfl_11	US10GB_7	0.107	US10GB/USInfl_11	US10GB_7	USInfl_13	0.021	US10GB/USInfl_11	US10GB_7	USInfl_13	USBC_14	0.004
US10GB_7	0.107	US10GB/USInfl_11	USInfl_13	0.107	US10GB/USInfl_11	USInfl_13	USBC_14	0.012	US10GB/USInfl_11	USInfl_13	USBC_14	USInfl_4	0.004
USInfl_13	0.107	US10GB/USInfl_11	USBC_14	0.107	US10GB/USInfl_11	USBC_14	USInfl_4	0.016	US10GB/USInfl_11	USBC_14	USInfl_4	bsOilSup_8	0.003
USBC_14	0.107	US10GB/USInfl_11	USInfl_4	0.106	US10GB/USInfl_11	USInfl_4	bsOilSup_8	0.020	US10GB/USInfl_11	USInfl_4	bsOilSup_8	USBC_16	0.006
USInfl_4	0.106	US10GB/USInfl_11	bsOilSup_8	0.104	US10GB/USInfl_11	bsOilSup_8	USBC_16	0.015	US10GB/USInfl_11	bsOilSup_8	USBC_16	USIP_4	0.005
bsOilSup_8	0.104	US10GB/USInfl_11	USBC_16	0.095	US10GB/USInfl_11	USBC_16	USIP_4	0.011	US10GB/USInfl_11	USBC_16	USIP_4	USCash	0.004

The relative importance of not only single factors, but pairs, triplets, and quadruplets. As US10GB and Infl_11 are factors present in 100% of all L models, in this table we treat them as one factor.

Table 5.

Automated model selection algorithm versus the bootstrap.

Oil Equation-Automated Model Selection						Oil Equation-Bootstrapped Estimates
Factor	Coefficient	Std error	t-value	t-prob	ConditionalCoefficientEstimate	UnconditionalCoefficientEstimate	ModelUncertaintyContribution to Variability	ConditionalStandard Error	UnconditionalStandard Error	ModelUncertainty Contribution to Variability
US10GB	0.1832	0.0233	7.8500	−	0.1793	0.1793	0%	0.0247	0.0247	0%
US10GB_13	0.0760	0.0234	3.2600	0.0011	0.0895	0.0615	31%	0.0165	0.0437	164%
USBC_8	0.1158	0.0378	3.0600	0.0022	0.1319	0.0830	37%	0.0291	0.0678	133%
USBC_11	0.1127	0.0378	2.9800	0.0029	0.1291	0.0891	31%	0.0300	0.0647	115%
USInfl	−1.2428	0.3257	−3.8200	0.0001	−1.2574	−1.0595	16%	0.2695	0.5205	93%
USInfl_11	0.0657	0.3260	0.2010	0.8404	0.1127	0.1127	0%	0.3316	0.3316	0%

Oil Supply Equation-Automated Model Selection						Oil Supply Equation-Bootstrapped Estimates
Factor	Coefficient	Std error	t-value	t-prob	ConditionalCoefficientEstimate	UnconditionalCoefficientEstimate	ModelUncertaintyContribution to Variability	ConditionalStandard Error	UnconditionalStandard Error	ModelUncertaintyContribution to Variability
US10GB	0.0018	0.0019	0.9580	0.3383	0.0018	0.0018	0%	0.0018	0.0018	0%
US10GB_13	0.0000	0.0019	0.0008	0.9994	0.0001	0.0001	31%	0.0019	0.0016	17%
USBC_8	0.0000	0.0030	0.0049	0.9961	−0.0002	−0.0001	37%	0.0032	0.0025	21%
USBC_11	0.0018	0.0030	0.6080	0.5433	0.0027	0.0019	31%	0.0036	0.0032	10%
USInfl	0.0018	0.0260	0.0672	0.9464	0.0061	0.0051	16%	0.0257	0.0237	8%
USInfl_11	0.2098	0.0261	8.0500	−	0.2135	0.2135	0%	0.0259	0.0259	0%

Once we have our L bootstrapped replications, we can estimate the standard error of the parameter estimates in our model as the standard deviation of the coefficient estimates found over each of the bootstrapped replications. We can also estimate the population coefficients as the average of the bootstrapped fitted values. Conditional estimates therefore bias coefficient estimates away from zero and standard error estimates downwards. Unconditional estimates set parameter estimates to zero wherever the model selected (for a particular bootstrap replication) does not contain that factor. Conditional estimates do not account for model selection uncertainty. If we do not take model selection uncertainty into account then we introduce bias into our estimates. The difference between the unconditional and conditional estimates quantifies the model uncertainty component as described previously. According to the results of the automatic model selection algorithm the important factors in the oil spot price equation y₁ (t) are the US 10 year government bond yield (lags 0 and 13), the US business confidence (lags 8 and 11), and the US inflation rate (lag 0). The important factor in the oil supply equation y₂ (t) is the US inflation rate (lag 11). There are no significant lags for the oil supply and oil price variables appearing in either the oil equation or the oil supply equation.

Each model structure is also shown against 20 factor/lags. After calculating relative importance of each factor, we classified these factors into two types: the factors related to financial decision-making and the ‘classic’ supply-and-demand factors used for oil modelling. Within each of these types, our table shows the top 10 factor/lags based on the relative importance of that factor/lag. The relative importance is determined by the percentage of times the factor/lag appeared within the L models. An importance weight of 1.00 means that the particular factor/lag was included in 100% $y_{1}$ of the L models in our procedure.

Comparison of Model 1 in Table 3 to the initial model selected for O_t (see Table 2) shows that the top three factors (US10GB, USInfl_11, USInfl) were also selected by the initial model selection. However, US business confidence at lag 11 (USBC_11) with an importance weighting of 0.6904 and US 10 year government bond yield at lag 4 (USGB10_4) with an importance weighting of 0.5517 were not selected by the initial model selection. Comparing with Model 2, which includes the speculative factor/lags (US10GB, USInfl, US10GB_13, US10GB_4, USInfl_7) and the classic factor/lags (USBC_8, USBC_1), we see that USInfl_7 with an importance weight of 0.4214 is not selected by the initial model selection. Interestingly, Model 2 is more similar to the initial model selected than Model 1. However, overall, there is a lot of variability between the top 20 models so comparing models on a case-by-case basis is futile. Therefore, to better understand our results, in Table 3 we decided to split the factors into two categories: ‘speculative’ and ‘classic’. This categorisation is based on the view that the speculative factors could be considered variables that a trader may consider as a way to pre-empt increases in oil prices whilst classic factors could be considered as proxies for the supply-and-demand dynamics of oil prices. By making this categorisation and by colouring cells grey if the factor is included in the model and white otherwise, we observe that speculative factors have a higher presence in the top 20 models compared to classic factors.

The 10 year government bond yield at lag 0 was selected in 100% of all models and at lags 4 and 13 at least 50% of the time. From an asset-allocation perspective bonds are seen as a ‘quality’ safe asset and in times of financial market turmoil a flight to quality out of assets declining in value is often observed. If one asset class as part of your portfolio allocation strategy is experiencing significant volatility, the safe haven of bonds may present as attractive. In such cases yields are bid lower and prices higher. This can explain the positive and important relationship between decreasing oil prices and decreasing yields on bonds.

The 10 year government bond yield at lag 0 and the inflation rate at lag 11 (around the two-trading-week mark) are the most important factors given that they were chosen 100% of the time. Next, inflation at lag 0 was selected 84% of the time, followed by business confidence at lag 14, business confidence at lag 8 and then the 10 year government bond yield at lag 4.

Inflation was also selected in 100% of all models. Interestingly, when considering the coefficient estimates (see Table 6) we find that news of a positive shock to inflation causes the market to react negatively on a (very) short-term basis to oil, but this relationship is reversed over the longer period (lag 11 approximately two trading weeks). The reversal at the two-week mark (lag 11) is however the most important variable (selected 100% of the time) and lends support to the idea that oil is often considered an investable asset class that acts as an inflation hedge (Chen and Pinsky, 2003; Frankel, 1986, 2008; Frankel and Rose, 2010) and is therefore considered a speculative factor rather than a supply/demand factor.

Table 6.

Most important factors by bootstrap weight.

Factor	RelativeImportance	Oil equation		Oil supply equation
	RelativeImportance	UnconditionalCoefficientEstimate	UnconditionalStandard ErrorEstimate	UnconditionalCoefficientEstimate	UnconditionalStandard ErrorEstimate
US10GB	1.0000	0.1793	0.0247	0.0018	0.0018
USInfl_11	1.0000	0.1127	0.3316	0.2135	0.0259
USInfl	0.8430	−1.0595	0.5205	0.0051	0.0237
USBC_11	0.6900	0.0891	0.0647	0.0019	0.0032
US10GB_13	0.6870	0.0615	0.0437	0.0001	0.0016
USBC_8	0.6290	0.0830	0.0678	−0.0001	0.0025
US10GB_4	0.5520	0.0393	0.0386	0.0020	0.0023
USBC_1	0.4740	0.0600	0.0664	0.0000	0.0025
USInfl_7	0.4210	0.4486	0.5480	0.0011	0.0197
USIP_8	0.3830	−0.0018	0.0024	0.0000	0.0001
US10GB_19	0.3750	−0.0019	0.0166	0.0023	0.0030
USIP_11	0.3550	0.0004	0.0012	0.0001	0.0002
USBC_18	0.3410	0.0400	0.0582	−0.0002	0.0024
US10GB_2	0.3140	−0.0122	0.0231	0.0014	0.0023
USIP	0.3020	0.0012	0.0020	0.0000	0.0001
USInfl_17	0.2930	−0.2715	0.4462	0.0018	0.0198
USIP_5	0.2700	0.0012	0.0020	0.0000	0.0001
USInfl_5	0.2660	0.2327	0.4140	0.0001	0.0197
USIP_12	0.2510	−0.0012	0.0021	0.0000	0.0001
US10GB_12	0.2470	0.0015	0.0141	−0.0013	0.0024

This table summarises the top 20 factors by bootstrap weight for the oil equation y₁ and the oil supply equation y₂. These results suggest that the 10 year government bond yield at lag 0 and the inflation rate at lag 11 (around the two-trading-week mark) are the most important factors given that they were chosen 100% of the time in 10,000 simulations. Next, inflation at lag 0 was selected 84% of the time, followed by business confidence at lag 14, business confidence at lag 8, and then the 10 year government bond yield at lag 4.

By incorporating model uncertainty, we have obtained a more robust view of the driving factors of world oil prices. If we took the approach of only viewing factors as important if they were chosen as part of the initial model selection, we may have mistakenly concluded that certain factors are unimportant.

Due to the low tally for each model structure, it seems pointless to argue that one of the top 20 model structures is the ‘true model’. Apart from the speculative factors (US10GB, USInfl_11, USInfl), and to some extent the classic factor USBC_11, there is quite a high variability as to whether a factor is included or not. Given the high degree of variability, one should apply a grouping method to capture the basic structure (if any) that occurs most frequently. This could be achieved using clustering methods. However, we found that the high degree of variability limited the usefulness of clustering analysis in our study. Another method of identifying important groups of variables (and therefore possible model structures) is to calculate the relative importance weights of factor/lag groups.

4.3 Factor groupings

To understand the importance of certain factor/lag groups found by our procedure, we calculated the frequency of pairs, triplets, and quadruplets appearing in the L models found by our procedure, see Table 4.

These results indicate that, at a minimum, any model must include the factors US 10 year government bond yield (lag 0) and US inflation rate (lag 11). We see from the ‘singles’ and ‘pairs’ results that both these variables were selected 100% of the time and are important when it comes to explaining the global oil market.

Obviously correlation is something to consider here and we need to be aware that correlation can impact the interpretation of the relative importance. Multicollinearity is a situation when two or more variables are factors for the same variable and are correlated to one another. Multicollinearity makes it difficult to determine the relative importance of each factor on the dependent variable (Smith et al., 2009).

If we turn our attention to the ‘quadruplets’ rankings (which incidentally are by default quintuplets), we see that the second row contains the exact combination of these top five individually ranked factors. The ranking for this combination of factors is only 0.404. This tells us that this particular combination of factors was selected less than half the time. If the results of this investigation were intended to be used to build a model intended for predictive purposes, simply constructing a model based on single factor frequency alone is possibly not the best strategy.

Now consider the relative importance of the highest ranked triplets, see Table 7. We see that the combination of US 10 year government bond yield (lag 0), US inflation rate (lag 11) and US inflation rate (lag 0) yield a relative importance weighting of 0.843. Almost 85% of the time, these three variables turned up together as a result of the model selection process. Obviously, we would recommend that any further analysis with respect to building models for the oil market should take into consideration this combination. We also recommend examining subsets of sizes two, three, and four from this set of variables for use in multi-model inferences, in particular model averaging.

Table 7.

Relative importance of different factors and lags for the oil equation y₁.

SINGLES		Relative Importance		PAIRS			Relative Importance
US10GB		1.000		US10GB	USInfl_11		1.000
USInfl_11		1.000		US10GB/USInfl_11	USInfl		0.843
USInfl		0.843		US10GB/USInfl_11	USBC_11		0.690
USBC_11		0.690		US10GB/USInfl_11	US10GB_13		0.687
US10GB_13		0.687		US10GB/USInfl_11	USBC_8		0.629
USBC_8		0.629		US10GB/USInfl_11	US10GB_4		0.552
US10GB_4		0.552		US10GB/USInfl_11	USBC_1		0.474

TRIPLETS			Relative Importance	QUADRUPLETS				Relative Importance
US10GB	USInfl_11	USInfl	0.843	US10GB	USInfl_11	USInfl	USBC_11	0.590
US10GB/USInfl_11	USInfl	USBC_11	0.590	US10GB/USInfl_11	USInfl	USBC_11	US10GB_13	0.404
US10GB/USInfl_11	USBC_11	US10GB_13	0.474	US10GB/USInfl_11	USBC_11	US10GB_13	USBC_8	0.283
US10GB/USInfl_11	US10GB_13	USBC_8	0.428	US10GB/USInfl_11	US10GB_13	USBC_8	US10GB_4	0.204
US10GB/USInfl_11	USBC_8	US10GB_4	0.331	US10GB/USInfl_11	USBC_8	US10GB_4	USBC_1	0.160
US10GB/USInfl_11	US10GB_4	USBC_1	0.268	US10GB/USInfl_11	US10GB_4	USBC_1	USInfl_7	0.111
US10GB/USInfl_11	USBC_1	USInfl_7	0.202	US10GB/USInfl_11	USBC_1	USInfl_7	USIP_8	0.084

This table shows the relative importance of different factors for the oil equation y₁. The US 10 year government bond yield (lag 13) factor has a relative importance weighting of 0.687. As the frequency with which a factor is selected increases, the disparity between the conditional and unconditional estimates will decrease (as evidenced by the conditional and unconditional estimates for the US 10 year government bond yield (lag 0) factor). The percentage difference between the bootstrap unconditional and conditional standard errors for this factor is 164%.

These bootstrap weights can be used in a couple of ways. For example, the weights used here could be used to make weighted average predictions out of sample as part of a confirmatory study. Alternatively, the bootstrap model weights can be used to determine a set of candidate models a priori, ranked according to their bootstrap relative importance. Out of sample weighted average predictions using Bayesian model averaging techniques or Akaike weights could then be performed. Multi-model inferences are shown to have better predictive performance than those made from a single model. See Eicher et al. (2011) for a discussion on improved predictive performance and Lavoué and Droz (2009) for practical implementations of multi-model inferencing using bootstrap and Bayesian model averaging. Anderson and Burnham (2003, 2004) explain a multi-model inferencing approach that utilises Akaike weights after first identifying a small a priori set of candidate models. In an exploratory context, the bootstrap weights can be used to help identify a set of a priori candidate models when researchers are seeking to identify relationships between variables that are not already well recognised by the market.

Initial results suggest that the combination of variables in Table 8 with a relative importance weighting of 0.590 is the most important.

Table 8.

Oil price variation attributed to top speculative exogenous factors.

Factor	Coefficient	Std error	t-value	Partial R²
Constant	−1615.48	21.61	−74.80	0.583
US10GB	55.87	1.58	35.40	0.238
USInfl	41.51	2.71	15.30	0.055
USInfl_11	−24.80	2.72	−9.12	0.020

We measure the partial R² of the speculative exogenous factors US10GB, USInfl, and USInfl_11 with a relative importance weight of approximately 85% (i.e. the three predictors chosen as a group together 85% of the time). We find that the partial R² is 23.84% for US10GB while USInfl and USInfl_11 have partial R² of 5.54% and 2.03%. In total, these three speculative factors explain 31.41% of the variation in the price of oil for the given data set.

While the analysis above assists us in developing a better understanding of which variables and combinations of variables are ‘important’ in terms of frequency of selection, it does not give any indication of the relative impact that each variable has on the oil market given a unit change in those variables. A more complete understanding of the factors driving the oil market will come with an analysis of the relative size of coefficients and standard errors. However, we remind readers that this analysis is exploratory in nature; we caution against making inferences about the size of the coefficients based on the results presented. All results should be interpreted as possible hypotheses to be tested in a more confirmatory analysis. As such we do not interpret the size of the estimated coefficients in this study.

5. Conclusion

Identifying speculation in the world oil markets is an important topic due to the policy implications, however, one must be careful due to the presence of model uncertainty. Our paper is the first study of this kind to address the presence of model uncertainty and is therefore robust to model mis-specification. This allows us to provide further evidence to the hypothesis that speculation is playing a role in the determination of prices.

The oil market has evolved from one dominated by pure commercial interests into an asset class providing inflation hedging opportunities, diversification, and alpha resulting from speculative bets. This motivates our interest in identifying possible new factors of oil prices beyond the supply and demand imbalances driving relative inventories.

Coefficient estimates and their standard errors have traditionally been determined by conditioning on a model form assumed to be correct. However, the model form is a source of variation in itself and adds to the uncertainty in our coefficient estimates. Traditional model selection techniques underestimate standard errors, overestimate coefficient estimates, select noise variables, and exclude true factors from the model because they only account for within-model uncertainty and ignore between-model uncertainty. We use a model-based resampling technique known as the bootstrap to more accurately identify important new speculative drivers of the global oil market. Using the bootstrap we are able to quantify the variability associated with model uncertainty as well as rank models and variables according to their relative importance. We propose a set of five possible new factors for their possible speculative relationships with the oil market: industrial production, US fed funds target rate, inflation, 10 year government bond yield, and US business (manufacturing) confidence indicator. Our results explicitly identify a set of new important speculative factors, rank the factors by relative importance, and identify the time value of shocks to these factors.

Our results suggest that the 10 year government bond yield at lag 0 and the inflation rate at lag 11 (around the two-trading-week mark) are the most important factors given that they were chosen 100% of the time. Next, inflation at lag 0 was selected 84% of the time, followed by business confidence at lag 14, business confidence at lag 8, and then the 10 year government bond yield at lag 4. Further, we see that the combination of US 10 year government bond yield (lag 0), US inflation rate (lag 11), and US inflation rate (lag 0) yield a relative importance weighting of 0.843. Almost 85% of the time, these three variables turned up together as a result of the model selection process. In Table 8, we measure the partial R² of the speculative exogenous factors US10GB, USInfl, and USInfl_11. We find that these factors explain 31.41% of the variation in oil prices for the given data set. This further supports the claim that speculative factors are now intimately tied with the dynamics of oil prices.

We believe our results lend support to the notion that asset allocation decisions, inflation concerns, portfolio diversification considerations, and speculation are influencing the oil market. In particular, the compelling evidence for the importance of the inflation rate at lag 11 lends support to the idea that oil is often considered an investable asset class that acts as an inflation hedge. Similarly, the importance of the 10 year government bond yield supports the hypothesis that alternative asset classes and their relationship to the oil market are now part of the asset allocation decision. The importance of the business confidence factor implies that speculation about oil prices, whilst less important, is still a major consideration in the determination of oil prices. These results lend support to the hypothesis that financial market participants are indeed playing a role in the determination of oil prices.

Footnotes

Appendix

The conditional bootstrap coefficient estimate for $b_{τ}^{i, j}$ is determined by

(5)

b_{τ}^{*, i, j} = \frac{\sum_{l = 1}^{L} I_{l} {\hat{b}}_{τ}^{i, j, l}}{\sum_{l = 1}^{L} I_{l}}

Here, we find the average coefficient estimate over the set of models that report an estimate for $b_{τ}^{i, j}$ . For example, if a factor was chosen only 10 times out of the set of L replications, then the denominator in (5) will sum to 10. That is, there will be 10 different coefficient estimates of $b_{τ}^{i, j}$ , represented as $I_{l}$ for ${\hat{b}}_{τ}^{i, j, l}$ , in (5).

The conditional standard error estimate for $b_{τ}^{i, j}$ is calculated as

(6)

s e_{L}^{*} (b_{τ}^{i, j}) = \sqrt{\frac{\sum_{l = 1}^{L} {(I_{l} {\hat{b}}_{τ}^{i, j, l} - \frac{\sum_{l = 1}^{L} I_{l} {\hat{b}}_{τ}^{i, j, l}}{\sum_{l = 1}^{L} I_{l}})}^{2}}{\sum_{l = 1}^{L} I_{l} - 1}}

where I_l is 1 if the parameter $b_{τ}^{i, j}$ is estimated for model l and 0 otherwise. This is the standard deviation of the coefficient estimates over the set of models that selected the factor with coefficient estimate ${\hat{b}}_{τ}^{i, j}$ and estimates variability conditional on the chosen model being selected. We will be comparing these estimates to the bootstrap unconditional standard error calculations to estimate the variability associated with model uncertainty. We can compare this to the standard error calculated in the traditional way given by

(7)

s e (b_{τ}^{i, j}) = \frac{s_{v}}{\sqrt{(n - 1) Var (X)}}

where S_V is the standard error of the estimate: the standard deviation of the residuals. This is the standard error calculated directly from the fitted model.

We use (8) to determine the unconditional bootstrap coefficient estimate for $b_{τ}^{i, j}$ . Here, we find the average coefficient over all l bootstrap replicates. Where a model fails to select a factor with coefficient estimate ${\hat{b}}_{τ}^{i, j}$ , that replication contributes 0 to the summation in the numerator:

(8)

b_{τ}^{i, j} = \frac{\sum_{l = 1}^{L} I_{l} {\hat{b}}_{τ}^{i, j, l}}{L}

Equation (9) is the unconditional standard error estimate for $b_{τ}^{i, j}$ . This is the standard deviation of the coefficient estimates over all L simulations:

(9)

s e_{L} (b_{τ}^{i, j}) = \sqrt{\frac{{\sum_{l = 1}^{L} (I_{l} {\hat{b}}_{τ}^{i, j, l} - \frac{\sum_{l = 1}^{L} I_{l} {\hat{b}}_{τ}^{i, j, l}}{L})}^{2}}{L - 1}}

Final transcript accepted 21 March 2014 by Hodaka Morita (AE Economics)

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Aastveity

Bjornlandz

Thorsrud

(2012) What drives oil prices? Emerging versus developed economies. Working paper, Norges Bank, Norway. Available at: http://www.norges-bank.no/pages/91303/Norges_Bank_Working_Paper_2012_11.pdf (last accessed May 2012).

Alfaro

Huelsenbeck

(2006) Comparative performance of Bayesian and AIC-based measures of phylogenetic model uncertainty. Systematic Biology 55(1): 89–96.

Anderson

Burnham

(2003) Model Selection and Multi-Model Inference. New York, NY: Springer.

Anderson

Burnham

(2004) Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods Research 33(2): 261–304.

Anderson

Burnham

Gould

. (2001) Concerns about finding effects that are actually spurious. Wildlife Society Bulletin 29(1): 311–316.

Austin

(2008) Using the bootstrap to improve estimation and confidence intervals for regression coefficients selected using backwards variable elimination. Statistics in Medicine 27: 3286–3300.

Bernanke

Gertler

Watson

(2004) Reply: Oil shocks and aggregate macroeconomic behavior: The role of monetary policy. Money, Credit and Banking 36(2): 287–291.

Buckland

Burnham

Augustin

(1997) Model selection: An integral part of inference. Money, Credit and Banking 53(2): 603–618.

Chatfield

(1995) Model uncertainty, data mining and statistical inference. The Royal Statistical Society 158(3): 419–466.

10.

Chen

Pinsky

(2003) Invest in direct energy. The Journal of Investing 12(2): 64–71.

11.

Clyde

(2000) Model uncertainty and health effect studies for particulate matter. Environmetrics 11(6): 745–763.

12.

Clyde

George

(2004) Model uncertainty. Statistical Science 19(1): 81–94.

13.

Cox

Reid

(2000) The Theory of the Design of Experiments (Monographs on Statistics & Applied Probability, vol. 86). Boca Raton, FL: Chapman & Hall/CRC.

14.

Cunado

Perez de Gracia

(2005) Oil prices, economic activity and inflation: Evidence for some Asian countries. The Quarterly Review of Economics and Finance 45(1): 65–83.

15.

Doornik

(2007) Autometrics. Working paper, University of Oxford, UK. Available at: http://www.economics.ox.ac.uk/hendryconference/Papers/DoornikDFHVol.pdf (last accessed May 2012).

16.

Efron

Tibshirani

(1994) An Introduction to the Bootstrap (Monographs on Statistics & Applied Probability, vol. 57). Boca Raton, FL: Chapman & Hall/CRC.

17.

Eicher

Papageorgiou

Raftery

(2011) Default priors and predictive performance in Bayesian model averaging, with application to growth determinants. Applied Econometrics 26(1): 30–55.

18.

Frankel

(1986) Expectations and commodity price dynamics: The overshooting model. American Journal of Agricultural Economics 68(2): 344–348.

19.

Frankel

(2008) The effect of monetary policy on real commodity prices. Working paper, National Bureau of Economic Research, MA. Available at: http://www.nber.org/papers/w12713 (last accessed February 2013).

20.

Frankel

Rose

(2010) Determinants of agricultural and mineral commodity prices. In: Fry

Jones

Kent

(eds) Inflation in an Era of Relative Price Shocks. Sydney: Reserve Bank of Australia. Available at: http://EconPapers.repec.org/RePEc:rba:rbaacv:acv2009-02 (last accessed February 2013).

21.

Froot

(1995) Hedging portfolios with real assets. Journal of Portfolio Management 21: 60–77.

22.

Gisser

Goodwin

(1986) Crude oil and the macroeconomy: Tests of some popular notions: Note. Journal of Money, Credit and Banking 18(1): 95–103.

23.

Gooijer

JGD

Hyndman

(2006) 25 years of time series forecasting. International Journal of Forecasting 22(3): 443–473.

24.

Hamilton

(1983) Oil and the macroeconomy since World War II. The Journal of Political Economy 91(2): 228–248.

25.

Wang

Lai

(2010) Global economic activity and crude oil prices: A cointegration analysis. Energy Economics 32(4): 868–876.

26.

Hendry

Krolzig

(2004) Sub-sample model selection procedures in general-to-specific modelling. In: Becker

Hurn

(eds) Contemporary Issues in Economics and Econometrics: Theory and Application. UK: Edward Elgar, pp. 53–74.

27.

Hinkley

Davison

(2006) Bootstrap Methods and their Application (Cambridge Series in Statistical and Probabilistic Mathematics, No. 1). 8th edn. Cambridge, NY: Cambridge University Press.

28.

Huang

Masulis

Stoll

(1996) Energy shocks and financial markets. Journal of Futures Markets 16(1): 1–27.

29.

International Monetary Fund (2006) World economic outlook: Globalisation and inflation. Available at: http://www.imf.org/external/pubs/ft/weo/2006/01/pdf/weo0406.pdf (last accessed May 2012).

30.

Jennings

(2012) Energy stocks as a separate portfolio allocation. The Journal of Wealth Management 14(4): 70–86.

31.

Juvenal

Petrella

(2011) Speculation in the oil market. Working paper, Federal Reserve Bank of St. Louis, MO. Available at: http://research.stlouisfed.org/wp/2011/2011-027.pdf. (last accessed May 2012).

32.

Kilian

Murphy

(2011) The role of inventories and speculative trading in the global market for crude oil. Available at: http://www-personal.umich.edu/lkilian/MS2010126R1.pdf.

33.

Krolzig

(2000) General-to-specific reductions of vector autoregressive processes. Department of Economics and Nuffield College, Oxford University, UK. Available at: http://www.nuffield.ox.ac.uk/economics/papers/2003/W15/hmk03a.pdf (last accessed May 2012).

34.

Krolzig

(2003) General-to-specific model selection procedures for structural vector autoregressions. Available at: http://www2.wiwi.hu-berlin.de/wpol/schumpeter/seminar/pdf/krolzig2.pdf (last accessed May 2012).

35.

Lavoué

Droz

(2009) Multimodel inference and multimodel averaging in empirical modeling of occupational exposure levels. Annals of Occupational Hygiene 53(2): 173–180.

36.

Manera

Longo

Markandya

. (2007) Evaluating the empirical performance of alternative econometric models for oil price forecasting. Working paper, Fondazione Eni Enrico Mattei, Italy (last accessed May 2012).

37.

Merino

Ortiz

(2005) Explaining the so-called ‘‘price premium’’ in oil markets. OPEC Review 29(2): 133–152.

38.

Miller

(1990) Subset Selection in Regression. London: Chapman and Hall.

39.

Pang

. (2011) Forecasting the crude oil spot price by wavelet neural networks using OECD petroleum inventory levels. New Mathematics and Natural Computation 7(2): 281–297.

40.

Roache

(2008) Commodities and the market price of risk. Working paper, International Monetary Fund, DC (last accessed May 2012).

41.

Sachs

Cooper

Fischer

(1981) The current account and macroeconomic adjustment in the 1970s. Brookings Papers on Economic Activity 1981(1): 201–282.

42.

Smith

Koper

Francis

. (2009) Confronting collinearity: Comparing methods for disentangling effects of habitat loss and fragmentation. Landscape Ecology 24: 1271–1285.

43.

Srinivas

Srinivasan

(2001) Post-blackening approach for modeling periodic streamflows. Hydrology 241(3–4): 221–269.

44.

Uri

Boyd

(1997) An evaluation of the economic effects of higher energy prices in Mexico. Energy Policy 25(2): 205–215.

45.

Warnock

(2009) International capital flows and US interest rates. Journal of International Money and Finance 28(6): 903–919.

46.

Zyren

Shore

(2002) Forecasting crude oil spot price using OECD petroleum inventory levels. International Advances in Economic Research 8(4): 324–333.

47.

Zyren

Shore

(2005) A monthly crude oil spot price forecasting model using relative inventories. International Journal of Forecasting 21(3): 491–501.

48.

Zyren

Shore

(2006) Forecasting short-run crude oil price using high- and low-inventory variables. Energy Policy 34(17): 2736–2743.

49.

Zucchini

(2000) An introduction to model selection. Mathematical Psychology 44: 41–61.