# Booms and Busts in the Oil Market: Identifying Speculative Bubbles Using a Continuous-Time Dynamic System

Authors: Kaizhi Yu; Yun Zhang
Journal: Complexity
 (2021)

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## Abstract

The sharp changes in oil prices since 2004 featured a nonlinear data-generating mechanism which displayed bubble-like behavior. A popular view is that such a salient pattern cannot be explained by shifts in economic fundamentals, but was driven by speculative bubbles as a consequence of the increased financialization of oil future markets. Testing this hypothesis, however, is challenging since the fundamental component of the oil price is unobservable. This paper attempts to isolate the contribution of speculative bubbles and fundamentals to the evolution of oil prices by providing a stylized model of commodity pricing. Motivated by our theoretical model, we adopt a continuous-time model with a random and time-varying persistence parameter to empirically investigate the presence of speculative bubbles in daily oil future prices over the period April 1983 to June 2020. We do not find any evidence in favor of speculative bubbles, although we indeed find that oil prices exhibit episodes of unstable behavior after 2004.


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## Body

## 1. Introduction

Figure1 displays two leading crude oil prices from 1986 to 2020, using daily data. As is evident from the figure, one striking characteristic of the oil prices has been the prolonged buildups and dramatic collapses over the past two decades. The most remarkable spike in the data series occurred from 2004 to mid-2008, followed by a rapid decline. The price of oil experienced another episode of slump between 2014 and 2016. In April 2020, it had plummeted to an all-time low, falling into the negative territory. Such salient empirical patterns have led many observers to attribute these steep price changes to the presence of speculative bubbles in the oil market [1–3]. One of their reasons is that the massive oil price fluctuations have coincided with the increased financialization of oil future markets since the early 2000s (for a comprehensive review of financialization of commodity markets, see Cheng and Xiong [4]). A similar argument is also made in Tang and Xiong [5]. Can the widespread claim be substantiated?Figure 1
Time series of daily spot prices for crude oil WTI and Brent.In this paper, we exploit the insights provided by a theoretical model of rational expectation commodity pricing in conjunction with empirical study to investigate the existence and duration of speculative bubbles in the volatile oil prices. To achieve this goal, we develop a reduced-form model of demand and supply for separating the contribution of speculative bubbles and traditional economic fundamentals to oil price movements. On this basis, our theoretical formulation decomposes the oil price into two components: the fundamental component and the unobserved bubble component. In particular, the model implies that bubbles, if they do exist, should manifest explosive dynamic features in prices. From an empirical perspective, this appealing property reduces bubble detection to examining evidence for stochastic explosive behavior in price series. To proceed, we use a continuous-time dynamic system to analyze daily oil prices over the period April 4, 1983, through to June 30, 2020. This novel approach provides a recursive test procedure that delivers a mechanism for real-time detecting various forms of unstable and explosive behavior and date-stamping the origination and termination of market exuberance. Overall, our methodology provides no evidence in support of speculative bubbles in oil prices over the sample period examined, despite the fact that the series displays periods of instability dynamics after 2004.This paper is related to different strands of the literature. Theoretically, rational expectation bubble models are well documented in long literature studying asset pricing [6–9]. Diba and Grossman [7] first pointed out, under a standard asset-pricing framework with rational, risk-neutral agents, that the detection of explosive (submartingale) behavior can be reflective of a bubble-like phenomenon. The similar argument has also been put forward by Pavlidis et al. [10], Martínez-García and Grossman [11], and Shi et al. [12], among many others, specifically in the context of the stylized asset-pricing model for housing. Our paper is not the first to look at whether bubbles are a key characteristic of the changes in oil prices. In fact, a growing body of empirical research has extensively applied the supremum augmented Dickey–Fuller (SADF) test of Phillips et al. [13] (PWY hereafter) and generalized supremum augmented Dickey–Fuller (GSADF) test proposed by Phillips et al. [14, 15] (PSY hereafter) to examine the possible presence of speculative bubbles [16–18]. The general idea of the SADF and GSADF methods involves a recursive evolving algorithm coupled with right-tailed unit root tests. In contrast to our conclusion, these studies provide some evidence of speculative bubbles. Finally, this paper also speaks to the extant literature that studies the role of speculation in the fluctuations of oil prices based on structural vector autoregressions (SVARs). These studies find strong evidence that historical oil price movements had suffered little, if any, speculation effects [19–22].Our paper makes several contributions. First, building upon prior analysis by Campbell and Shiller [23] and Pindyck [24], we contribute to the literature by refining the model of rational commodity pricing. From a theoretical perspective, the price evolution of storable commodities such as crude oil does not directly fit in the theory of classical rational bubbles whereby the fundamental value is approximated by a stream of expected future dividend payments. In order to determine the fundamental level of oil prices, our model makes the link between the latent convenience yield and a reduced-form model of dynamic demand and supply. Second, compared to active literature, which relies on a recursive method implemented in a mildly explosive discrete time model, we employ a novel econometric approach for detecting bubbles. Inspired by the work of Tao et al. [25], the present paper estimates a continuous-time model, a model that allows for a random persistence parameter and its endogeneity. The advantages of extending a discrete time model to continuous time are threefold. First, the persistence parameter, as is well known, is not determined by the choice of data frequency in the continuous time framework [26]. Consequently, the empirical results about the explosive behavior in continuous-time models are less sensitive to sampling frequency than that in a discrete time setting. Second, our continuous system formulations readily accommodate an initial condition and drift effects, giving a limit theory without nuisance parameters. By contrast, these additional unknown parameters appear in discrete time models, thereby complicating inference. Third, combined with high-frequency data, the continuous-time model enables us to gain substantial power in identifying bubbles that collapse periodically, especially when these bubbles are short-lived. To the best of our knowledge, this paper is the first to use the above methodologies to detect any potential extreme behavior in the oil market. Third, unlike the SVAR literature, our goal is not to precisely identify the sources of price shocks but rather to see whether speculation plays a central role in the price of oil. In this sense, this paper offers a complementary approach. Our findings also add further support to a fundamental-driven explanation for the recent high price volatility in the oil market.The remainder of the paper is organized as follows. After reviewing the relevant literature in the next section, we lay out a simple theoretical framework in Section3. Section 4 introduces the econometric methodology and describes the bubble detection strategy in more detail. Section 5 describes the data employed in the empirical analysis, while in Section 6, we present the main empirical results and evaluate the sensitivity of our findings. Section 7 discusses and compares the results with other studies. Section 8 contains some concluding remarks and draws out substantive policy implications.


## 2. Literature Review

Over the past decades, and particularly after the 2007-2008 global financial crisis, the rapid surge and subsequent collapse of asset prices, more loosely called as speculative bubbles, have reawakened interests in econometric tests of bubbles and have enlivened an emerging string of literature aimed at date-stamping strategies. Speculative bubbles have a long-standing history in asset markets. Well-known historical episodes comprise the 17th-century “Tulip Mania” and the Mississippi Bubble of 1718–1720. Such speculative behavior and crashes in the financial sector can trigger serious economic recessions such as the great stock market crash episode prior to the Great Depression or the recent subprime mortgage crisis followed by the global economic shock. There is a general consensus that speculative bubbles are defined as the divergence of an asset’s price from its intrinsic value [27].The classic rational bubble is the standard theoretical framework to model the presence and persistence of speculative bubbles, with seminal papers by Blanchard and Watson [6], Tirole [28, 29], Flood and Hodrick [30], and Froot and Obstfeld [31]. In this workhorse model of bubbles, it is economically rational for investors to buy overpriced securities today that are unjustified by fundamental factors as long as they believe that the selling price becomes high tomorrow. In spite of the central role of rational bubbles in the theoretical literature, empirical evidence in support of their existence remains elusive. The challenges in empirically testing for the potential bubbles lie in investigating their explosivity and dating their occurrence. Early empirical studies have designed and adopted various time series methods to capture exuberance in asset and commodity markets. This includes variance bound tests proposed by Shiller [32] and LeRoy and Porter [33], two-step tests initiated by West [34], and integration and cointegration tests developed by Diba and Grossman [7, 35], as well as intrinsic bubbles considered by Froot and Obstfeld [31]. Furthermore, Homm and Breitung [36] applied Chow and CUSUM-type tests. The list is by no means thorough and exhaustive. Gürkaynak [37] provided a detailed survey of different econometric tests on asset price bubbles. Most notable among these studies is the wide use of traditional unit root tests, e.g., augmented Dickey–Fuller. Although popularly applied, standard unit root tests have been criticized by Evans [38]. As Evans noted, when exploring to detect periodically collapsing bubbles, the unit root-based tests have extremely low power because the observed series may appear more like a stationary process than like an explosive process.The recent literature has proposed several other test methodologies to deal with this weakness. The two most prominent test procedures of which are the SADF put forward by PWY and its extension, the GSADF of PSY. In order to handle the effect of major downturns in the data trajectories on the testing power, the methods of PWY and PSY implement the forward recursive algorithm and the recursive evolving algorithm, respectively. Both of them estimate augmented Dickey–Fuller regression equations using subsamples of available data. An attractive feature of these two methods is that they allow dating the origination and termination of explosive periods in real time. The PWY and PSY techniques have gained increasing popularity in a wide variety of markets. Phillips and Yu [16] are the first to directly study the bubble characteristics in the crude oil price series from January 1999 to January 2009 based on the method of PWY. They found statistical evidence of the presence of a short price bubble between March and July 2008. Tsvetanov et al. [17], Figuerola-Ferretti et al. [18], and Caspi et al. [39] employed the GSADF test of PSY using different samples and reported multiple bubble episodes of oil price explosivity during the period under examination.Instead of taking the approaches of PWY and PSY, a number of studies have implemented some alternative methodologies towards speculative or rational bubbles in crude oil markets. Markov regime-switching regression models provide a method which assumes that the bubble component is in one of several regimes with positive probability. Shi and Arora [40] are the first to apply this approach to analyze the bubble issue in oil price dynamics. The authors used three different regime-switching models and reported favorable evidence for the claim of Phillips and Yu [16] that a short-lived bubble existed in 2008. Lammerding et al. [41] offered a Bayesian Markov switching approach to distinguish a stable phase in one regime and an explosive phase in another regime. Their two Markov-regimes test procedure uncovers significant evidence in support of the existence of bubbles in the oil prices. More recently, Pavlidis et al. [42] developed two novel approaches for bubble detection by exploiting differences between spot and forward (or futures) prices. Pavlidis et al. [43] applied them to crude oil markets and found no speculative bubbles in oil prices. The main advantage of their approach is that the need for approximating fundamentals of the oil price can be obviated by using the information incorporated in market expectations. However, the method is tied to an appropriate measure of market expectations. Altogether, it is clear that extant empirical research on speculative bubbles in the oil market has yielded mixed results.


## 3. Theoretical Model

Speculative bubbles in crude oil markets can be defined as systematic deviates from the fundamental price of oil. One of the most widely recognized models allowing for bubbles is the rational expectation asset-pricing model. Thus, our theoretical framework closely relates to that of Diba and Grossman [35], Gürkaynak [37], and Pavlidis and Vasilopoulos [44]. However, we extend the stylized asset-pricing model to explain why prices tend to show excess volatility in storable commodity markets. Consider that rational, infinitely-lived investors obtain utility from individual consumption in a standard endowment economy. The representative investor’s maximization problem is(1)maxEt∑i=0∞βiuCt+i,where Et denotes the rational expectation operator conditional on all the information available at time t, Ct+i is the consumption quantity at period t+i, and β stands for the discount rate of future consumption which is restricted to take values in (0, 1) such that the time preference of an agent is always positive. Consistent with the standard assumptions in economics, the instantaneous utility function, u·, is considered to be concave, continuously differentiable, and strictly increasing in Ct+i.At each time period, the investor subjects to the budget constraint(2)Ct+i≤wt+i+Pt+i+ψt+ixt−Pt+ixt+i,where xt is the storable asset, Pt represents the price of a commodity in the unit of the consumption good, and wt refers to an initial endowment that can be immediately consumed or physically held to gain the stream of current and expected discounted future benefits, called the convenience yield, ψt. In this case, the convenience yield is analogous to the payoff (dividend) on a stock. The concept of convenience yield is well known and has been widely used in analyzing commodity markets. Specifically, it measures a premium that accrues to inventory holders stemming from possible alternative inventory functions. Forward-looking investors usually establish inventories to smooth production requirements, avoid stockouts, and accommodate sales scheduling [45–48]. Changes in desired inventories reflect the shifts in the agents’ expectations about current and future market conditions. As such, our model links the latent convenience yield to a reduced-form dynamic model of demand and supply.The first-order condition for the representative investor’s utility optimization problem specified by (1) and (2) yields(3)Etβu′Ct+iPt+i+ψt+i=Etu′Ct+i−1Pt+i−1.Intuitively, equation (3) states that, for the optimal time path of x, an investor cannot increase expected utility by selling or buying an asset at time t and performing a reverse transaction at time t+1. For commodity pricing purposes, utility is often assumed to be linear that implies risk neutrality and constant marginal utility. In this setup, equation (3) can be simply written as(4)βEtPt+i+ψt+i=EtPt+i−1.Assuming further the presence of a risk-free bond available in clear financial markets with a time-invariant net interest rate,R, standard no arbitrage condition indicates(5)Pt=11+REtPt+1+ψt+1.Solving this first-order stochastic difference (equation (5)) by forward iteration derives the general solution(6)Pt=∑i=1∞11+RiEtψt+1+limi⟶∞Et11+RiPt+i.The last equation has decomposed the price into two parts. The first term of the right-hand side is regarded as a market fundamental component, which depends on the discounted sum of expected future convenience yields. The second term is commonly referred to as a periodically collapsing speculative bubble component. Imposing the transversality condition(7)limi⟶∞Et11+RiPt+i=0,rules out the existence of a bubble component. The price Pt reduces to the fundamental price which will be represented by Ptf, whereas if the transversality condition is not justified, there are infinitely many solutions to equation (5). They take the more general form(8)Pt=Ptf+Bt,where Bt is a rational bubble component that has to satisfy the submartingale property:(9)EtBt+1=1+RBt.If a bubble is present in the commodity market, equation (9) requires the bubble to grow, in expectation, geometrically at the rate of R in order for any rational investor willing to hold the asset. Investors may drive the price far higher than the fundamental value as long as they believe that they can compensate for the extra payment Bt through selling the asset to other market participants at an even higher price in the future. In other words, commodity price bubbles could be a self-fulfilling prophecy. Taken together, equations (8) and (9) imply that Bt will display explosive dynamics as well as the spot pricePt, irrespective of whether the fundamental component Ptf is or is not itself nonexplosive.As illustrated in the following, the contemporaneous future price at timet with maturity at t+T, FT,t, will also be explosive. Let F1,t denote the future price for delivery at t+1. To avoid arbitrage, ψt must satisfy(10)ψt=1+RPt−F1,t.Given this condition and equation (5), it follows that F1,t is an unbiased predictor of the expected future spot price:(11)F1,t=EtPt+1.Using equations (8) and (9), we obtain the following expression for the future price:(12)FT,t=EtPt+Tf+EtBt+T=EtPt+Tf+1+RTBt.We can easily see that the future price is a linear function of the explosive process. This statistical property has motivated our econometric tests for identifying explosive characteristics in prices.


## 4. Econometric Methodology

In this section, we give a brief description of bubble detection methods used in our empirical study so as to facilitate ease interpretation of the results. We first describe a continuous-time model with randomized persistence and then a statistical inference procedure in the continuous system. Finally, we discuss the model in the endogeneity setting.


### 4.1. Model Specification

Following Tao et al. [25], we begin our analysis with a variant version of the Ornstein–Uhlenbeck process, given by(13)dxt=xtμdt+σdWεt,x0=x0,where Wεt is a standard Brownian motion with constant variance 1, also termed a standard Wiener process, and μ is the crucial persistence parameter that is meant to govern the dynamic behavior of xt. When μ<0, the process xt is stationary; when μ=0, it is nonstationary; and when μ>0, it manifests explosive characteristics. In (13), μ is assumed to be time-invariant, which may not be corroborated by long-run data. Hence, the model studied in the paper takes the form of(14)dxt=xtμdt+σ˜Wμt+σdWεt,x0=x0,where Wμt and Wεt are two standard Brownian motions and the initial value x0 is viewed to be independent of Wμt and Wεt. For σ˜2≠0, we see that random shocks to the drift term have been incorporated into model (14). At this moment, we explore the situation, where Wμt and Wεt are independent noise processes, and subsequently allow for the endogenous case by relaxing the independence assumption.Solving model (14) yields the strong solution [49](15)dxt=expσ˜Wμt+μ−12σ˜2tx0+Vt,where(16)Vt=σ∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sdWεt∼MN0,σ2∫0te2σ˜Wμt−Wμs+2μ−1/2σ˜2t−sds,such that(17)EVt2=σ2E∫0te2σ˜Wμt−Wμs+2μ−1/2σ˜2t−sds=σ22e2μ+1/2σ˜2t−1μ+1/2σ˜2.Underμ+1/2σ˜2>0, EVt2 is expected to grow exponentially.The corresponding discrete-time representation of (14) is derived directly from the strong solution and is of the form(18)xtΔ=expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δxt−1,Δ+σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s,where Δ is the sampling interval over time span T and t=1,…,T/Δ. With respect to the random coefficient autoregression model [50], the autoregressive coefficient is defined by(19)ρtΔ=expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δ.Here, we introduce some notations to simplify the discrete-time system. Let(20)ϕ≔μ−12σ˜2,κ≔μ+12σ˜2,utΔ≔Wμ,tΔ−Wμ,t−1ΔΔ∼N0,1,ρtΔ≔expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δ=expϕΔ+σ˜ΔutΔ,ηtΔ≔∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s∼N0,e2κΔ−12κ.The discrete-time representation (18) can then be rewritten as(21)xtΔ=expσ˜ΔutΔ+ϕΔxt−1,Δ+σηtΔ=ρtΔxt−1,Δ+σηtΔ.Assumingσ˜2>0, models (14) and (21) have the following stationarity properties: (1) for μ+σ˜2/2<0, the process is asymptotically covariance stationary sinceμ<0 means that both EρtΔ=eμΔ<1 and EρtΔ2=exp2μΔ+σ˜2Δ=exp2κΔ<1; (2) for μ+σ˜2/2<0, the process remains asymptotically stationary but is not asymptotically covariance stationary any longer. Practically speaking, we have to impose the condition of κ=μ+σ˜2/2<0 to guarantee the covariance stationary. When κ=μ+σ˜2/2<0, it follows from (19) that the variance of Vt converges to −σ2/2κ<∞ as t⟶∞ When κ=0, the variance is equal to σ2t and diverges as t⟶∞; (3) for μ+σ˜2/2>0 and μ<0, the process is still asymptotically stationary but is not covariance stationary with evident persistence in the trajectory; (4) when μ=0, ϕ=μ−σ˜2/2<0 ensures the asymptotic stationarity, but leads to infinite second moments withκ=μ+σ˜2/2>0. The process exhibits apparent unstable behavior and larger variation; (5) when μ>0 and μ−σ˜2/2<0, the process continues to be covariance nonstationary. Contrast to the traditional explosive AR (1) model, the model retains asymptotically stationary. In this instance, bubble-like phenomenon should be clearly observed in actual data; (6) if μ−σ˜2/2>0, the model is no longer asymptotically stationary, and the explosive growth behavior is now obvious.As shown in Tao et al. [25], when σ˜2=0, model (21) collapses to a simple autoregression with a constant coefficient. It explains why a popular way to look for bubbles is to test ϕ=0 against ϕ>0 via standard or recursive right-tailed unit root tests [15–18, 27]. For the purpose of the present paper, we only examine three price behavior features that are divided into three categories:(1)
Unstable:κ=μ+σ˜2/2>0. Here, the model is asymptotically covariance nonstationary and applies to capture unstable behavior.(2)
Locally explosive:μ>0. The model in this case has the ability for generating both explosive and collapsing behavior.(3)
Explosive:ϕ=μ−σ˜2/2>0. Under the circumstance, the model is not asymptotically stationary and is capable of generating explosive behavior.Based on the above typology, explosive embodies locally explosive, while locally explosive embodies unstable.


### 4.2. Inference Scheme

This section is concerned with putting the continuous-time model specified above into the application of discretely sampled data. We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimate μ, σ˜2, and σ2. In the first stage, realized volatility is employed to provide a regression model for estimating the diffusion parameters, σ˜2 and σ2. In the second stage, we maximize the in-fill likelihood function to estimate the drift parameter μ.Suppose data are observed inN equally spaced blocks over the time interval 0,T, with sampling interval h. The overall time span of the data is T so that N=T/h. If each block has M observations, the total number of observations is M×N=T/Δ, where Δ is the sampling frequency. To develop asymptotics, we assume that, as Δ⟶0, M×N⟶∞. According to the algorithm of Phillips and Yu [51], the estimators based on realized variance (RV) and realized quarticity (RQ) are denoted by(22)θ^=argminθ∈ΘQΔθ,where θ^≔σ˜2,σ2′ and QΔθ=Δ∑n=1NlogRVn−logxn−1hnh+1/2sn22/sn2with sn2=max2ΔRQn2/RVn2,2/M.(23)RVn=∑i=1Mxn−1h+iΔ−xn−1h+i−1Δ2,n=1,2,…,N,RQn=13Δ∑i=1Mxn−1h+iΔ−xn−1h+i−1Δ4,n=1,2,…,N.In the second stage, the approximate logarithmic in-fill likelihood function is maximized with regard toμ (representing the resulting estimator as μ^):(24)μ^=argmaxμ1MNlogℓALFμ,where(25)ℓALFμ=∑t=1M×Nμxt−1Δσ˜2^xt−1Δ2+σ2^xtΔ−xt−1Δ−Δ2∑t=1M×Nμxt−1Δσ˜2^xt−1Δ2+σ2^.One important goal in the present empirical work is to check the presence of randomness in the persistence parameter of model (14), which is equivalent to test the null hypothesis σ˜2=0. To achieve this aim, we resort to the locally best invariant test (LBI test) statistic [52], namely,(26)Z˜N≔∑t=1M×Nε˜tΔ2−1/MN∑t=1M×Nε˜tΔ22x˜t−1Δ21/MN∑t=1M×Nε˜tΔ4−1/MN∑t=1M×Nε˜tΔ221/MN∑t=1M×Nx˜t−1Δ4−1/MN∑t=1M×Nx˜t−1Δ22,where x˜tΔ2=xtΔ/1+xtΔ2, ε˜tΔ=xtΔ−ρ˜xt−1Δ, and ρ˜=∑t=1M×Nx˜t−1Δxt−1Δ−1∑t=1M×Nx˜t−1Δxt−1Δ. Thus, as N⟶∞, MN−1/2Z˜N⟶LN0,1, under H0:σ˜2=0, and MN−1/2Z˜N⟶LN0,1, under H0:σ˜2=0.Relying on asymptotics with in-fill (as the sampling frequency tends to zero,Δ⟶0) and long time-span asymptotics, Tao et al. [25] developed new limit theory for the continuous-time system where the persistence parameter is random. The asymptotic properties of the diffusion parameter and drift parameter estimates can be used to construct test statistics for three types of unstable/explosive behavior, which are(27)tκ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2β^κ−βκ0⟶LN0,1,tμ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2ρ^−ρ0⟶LN0,1,tϕ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2β^ϕ−βϕ0⟶LN0,1,where(28)β^κ=expμ^+12σ˜2^Δ,β^ϕ=expμ^−12σ˜2^Δ,βκ0=expμ0+12μ02Δ,βϕ0=expμ0−12μ02Δ,ρ^=1+∑t=1M×Nxt−1ΔxtΔ−xt−1Δ/σ˜2^xt−1Δ2+σ2^∑t=1M×Nxt−1Δ2/σ˜2^xt−1Δ2+σ2^,ρ0=1.As can be seen, this test procedure does not directly verify whether or notκ=μ+σ˜2/2=0 or μ=0 or ϕ=μ−σ˜2/2=0, but transforms the null hypotheses to βk=1 or ρ=1 or βϕ=1. These three t-test statistics are accompanied by a date-stamping strategy that permits us to identify the exact origination and termination periods of price exuberance. Given this instrument, estimates of these dates are as follows:(29)r^unib=infs∈r^uni−1e,1s:tκs>Zα,r^unie=infs∈r^uni−1b,1s:tκs<Zα,r^leib=infs∈r^lei−1e,1s:tμs>Zα,r^unie=infs∈r^lei−1b,1s:tμs<Zα,r^eib=infs∈r^ei−1e,1s:tϕs>Zα,r^unie=infs∈r^ei−1b,1s:tϕs<Zα,where Zα is the right-tailed critical value of the standard normal distribution corresponding to a significance level of α; r^unib/r^leib/r^eib denote the beginnings on the dates of the ith extreme price behavior period; and similarly, r^unie/r^leie/r^eie denote the ends on the dates of the ith extreme price behavior period.


### 4.3. Extend the Model with Endogeneity

Now, we are concerned with the endogeneity case whereWμt is not independent of Wεt. To see this, let us suppose that data are generated from the following stochastic differential equation system:(30)dxt=xtdB˜t+σdBt,x0=x0,dB˜t=μdt+σ˜Wμt,dBt=σdWεt,where (Wμt,Wεt) is a two-dimensional Brownian motion with covariance parameter η. As Föllmer [53] noted, the strong solution obtained from this continuous system would be of the explicit form(31)dxt=expσ˜Wμt+μ−12σ˜2tx0+Zt,where(32)Zt=σ∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sdWεt−ω∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sds=Vt−Pt.Notice that when endogeneity is included in model (14), the driver process becomes Zt as a substitute for Vt. Zt consists of a component, Vt, and another one connected with the covariance parameter of B˜,B, ω, and variance of Vt. The strong solution gives the exact discrete-time form of model (31):(33)xtΔ=expσ˜ΔuΔ+μ−12σ˜2Δxt−1,Δ+ZtΔ=ρtΔxt−1,Δ+ZtΔ+σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s,where uti.i.d∼N0,1 and(34)ZtΔ=σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s−ω∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sds.Somewhat surprisingly, Tao et al. [25] indicated that endogeneity would not affect the sample path features of model (31). Our analysis therefore is the same as before:(1)
Unstable:κ=μ+σ˜2/2>0(2)
Locally explosive:μ>0(3)
Explosive:ϕ=μ−σ˜2/2>0We can estimateθ∗^≔σ˜2^,η^,σ2^′ in the same fashion as before. The likelihood ratio test (LR test) has been proposed by Tao et al. [25] to test the null hypothesis of endogeneity interest:(35)LR=Δ−1QΔr−QΔur∼χ21,underH0:η0=0.Tao et al. [25] also developed the asymptotic theory for μ^ and ρ^≔expμ^Δ. Based on their limit properties, the estimates of μ^ and ρ^ continue to follow asymptotic normal distributions. This good quality enables us to conveniently adopt the aforementioned testing procedures after making a small adjustment to the variance of the limit distribution.


## 4.1. Model Specification

Following Tao et al. [25], we begin our analysis with a variant version of the Ornstein–Uhlenbeck process, given by(13)dxt=xtμdt+σdWεt,x0=x0,where Wεt is a standard Brownian motion with constant variance 1, also termed a standard Wiener process, and μ is the crucial persistence parameter that is meant to govern the dynamic behavior of xt. When μ<0, the process xt is stationary; when μ=0, it is nonstationary; and when μ>0, it manifests explosive characteristics. In (13), μ is assumed to be time-invariant, which may not be corroborated by long-run data. Hence, the model studied in the paper takes the form of(14)dxt=xtμdt+σ˜Wμt+σdWεt,x0=x0,where Wμt and Wεt are two standard Brownian motions and the initial value x0 is viewed to be independent of Wμt and Wεt. For σ˜2≠0, we see that random shocks to the drift term have been incorporated into model (14). At this moment, we explore the situation, where Wμt and Wεt are independent noise processes, and subsequently allow for the endogenous case by relaxing the independence assumption.Solving model (14) yields the strong solution [49](15)dxt=expσ˜Wμt+μ−12σ˜2tx0+Vt,where(16)Vt=σ∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sdWεt∼MN0,σ2∫0te2σ˜Wμt−Wμs+2μ−1/2σ˜2t−sds,such that(17)EVt2=σ2E∫0te2σ˜Wμt−Wμs+2μ−1/2σ˜2t−sds=σ22e2μ+1/2σ˜2t−1μ+1/2σ˜2.Underμ+1/2σ˜2>0, EVt2 is expected to grow exponentially.The corresponding discrete-time representation of (14) is derived directly from the strong solution and is of the form(18)xtΔ=expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δxt−1,Δ+σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s,where Δ is the sampling interval over time span T and t=1,…,T/Δ. With respect to the random coefficient autoregression model [50], the autoregressive coefficient is defined by(19)ρtΔ=expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δ.Here, we introduce some notations to simplify the discrete-time system. Let(20)ϕ≔μ−12σ˜2,κ≔μ+12σ˜2,utΔ≔Wμ,tΔ−Wμ,t−1ΔΔ∼N0,1,ρtΔ≔expσ˜Wμ,tΔ−Wμ,t−1Δ+μ−12σ˜2Δ=expϕΔ+σ˜ΔutΔ,ηtΔ≔∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s∼N0,e2κΔ−12κ.The discrete-time representation (18) can then be rewritten as(21)xtΔ=expσ˜ΔutΔ+ϕΔxt−1,Δ+σηtΔ=ρtΔxt−1,Δ+σηtΔ.Assumingσ˜2>0, models (14) and (21) have the following stationarity properties: (1) for μ+σ˜2/2<0, the process is asymptotically covariance stationary sinceμ<0 means that both EρtΔ=eμΔ<1 and EρtΔ2=exp2μΔ+σ˜2Δ=exp2κΔ<1; (2) for μ+σ˜2/2<0, the process remains asymptotically stationary but is not asymptotically covariance stationary any longer. Practically speaking, we have to impose the condition of κ=μ+σ˜2/2<0 to guarantee the covariance stationary. When κ=μ+σ˜2/2<0, it follows from (19) that the variance of Vt converges to −σ2/2κ<∞ as t⟶∞ When κ=0, the variance is equal to σ2t and diverges as t⟶∞; (3) for μ+σ˜2/2>0 and μ<0, the process is still asymptotically stationary but is not covariance stationary with evident persistence in the trajectory; (4) when μ=0, ϕ=μ−σ˜2/2<0 ensures the asymptotic stationarity, but leads to infinite second moments withκ=μ+σ˜2/2>0. The process exhibits apparent unstable behavior and larger variation; (5) when μ>0 and μ−σ˜2/2<0, the process continues to be covariance nonstationary. Contrast to the traditional explosive AR (1) model, the model retains asymptotically stationary. In this instance, bubble-like phenomenon should be clearly observed in actual data; (6) if μ−σ˜2/2>0, the model is no longer asymptotically stationary, and the explosive growth behavior is now obvious.As shown in Tao et al. [25], when σ˜2=0, model (21) collapses to a simple autoregression with a constant coefficient. It explains why a popular way to look for bubbles is to test ϕ=0 against ϕ>0 via standard or recursive right-tailed unit root tests [15–18, 27]. For the purpose of the present paper, we only examine three price behavior features that are divided into three categories:(1)
Unstable:κ=μ+σ˜2/2>0. Here, the model is asymptotically covariance nonstationary and applies to capture unstable behavior.(2)
Locally explosive:μ>0. The model in this case has the ability for generating both explosive and collapsing behavior.(3)
Explosive:ϕ=μ−σ˜2/2>0. Under the circumstance, the model is not asymptotically stationary and is capable of generating explosive behavior.Based on the above typology, explosive embodies locally explosive, while locally explosive embodies unstable.


## 4.2. Inference Scheme

This section is concerned with putting the continuous-time model specified above into the application of discretely sampled data. We utilize a novel two-stage approach proposed by Phillips and Yu [51] to estimate μ, σ˜2, and σ2. In the first stage, realized volatility is employed to provide a regression model for estimating the diffusion parameters, σ˜2 and σ2. In the second stage, we maximize the in-fill likelihood function to estimate the drift parameter μ.Suppose data are observed inN equally spaced blocks over the time interval 0,T, with sampling interval h. The overall time span of the data is T so that N=T/h. If each block has M observations, the total number of observations is M×N=T/Δ, where Δ is the sampling frequency. To develop asymptotics, we assume that, as Δ⟶0, M×N⟶∞. According to the algorithm of Phillips and Yu [51], the estimators based on realized variance (RV) and realized quarticity (RQ) are denoted by(22)θ^=argminθ∈ΘQΔθ,where θ^≔σ˜2,σ2′ and QΔθ=Δ∑n=1NlogRVn−logxn−1hnh+1/2sn22/sn2with sn2=max2ΔRQn2/RVn2,2/M.(23)RVn=∑i=1Mxn−1h+iΔ−xn−1h+i−1Δ2,n=1,2,…,N,RQn=13Δ∑i=1Mxn−1h+iΔ−xn−1h+i−1Δ4,n=1,2,…,N.In the second stage, the approximate logarithmic in-fill likelihood function is maximized with regard toμ (representing the resulting estimator as μ^):(24)μ^=argmaxμ1MNlogℓALFμ,where(25)ℓALFμ=∑t=1M×Nμxt−1Δσ˜2^xt−1Δ2+σ2^xtΔ−xt−1Δ−Δ2∑t=1M×Nμxt−1Δσ˜2^xt−1Δ2+σ2^.One important goal in the present empirical work is to check the presence of randomness in the persistence parameter of model (14), which is equivalent to test the null hypothesis σ˜2=0. To achieve this aim, we resort to the locally best invariant test (LBI test) statistic [52], namely,(26)Z˜N≔∑t=1M×Nε˜tΔ2−1/MN∑t=1M×Nε˜tΔ22x˜t−1Δ21/MN∑t=1M×Nε˜tΔ4−1/MN∑t=1M×Nε˜tΔ221/MN∑t=1M×Nx˜t−1Δ4−1/MN∑t=1M×Nx˜t−1Δ22,where x˜tΔ2=xtΔ/1+xtΔ2, ε˜tΔ=xtΔ−ρ˜xt−1Δ, and ρ˜=∑t=1M×Nx˜t−1Δxt−1Δ−1∑t=1M×Nx˜t−1Δxt−1Δ. Thus, as N⟶∞, MN−1/2Z˜N⟶LN0,1, under H0:σ˜2=0, and MN−1/2Z˜N⟶LN0,1, under H0:σ˜2=0.Relying on asymptotics with in-fill (as the sampling frequency tends to zero,Δ⟶0) and long time-span asymptotics, Tao et al. [25] developed new limit theory for the continuous-time system where the persistence parameter is random. The asymptotic properties of the diffusion parameter and drift parameter estimates can be used to construct test statistics for three types of unstable/explosive behavior, which are(27)tκ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2β^κ−βκ0⟶LN0,1,tμ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2ρ^−ρ0⟶LN0,1,tϕ=1Δ∑t=1M×Nxt−1Δ2σ˜2^xt−1Δ2+σ2^1/2β^ϕ−βϕ0⟶LN0,1,where(28)β^κ=expμ^+12σ˜2^Δ,β^ϕ=expμ^−12σ˜2^Δ,βκ0=expμ0+12μ02Δ,βϕ0=expμ0−12μ02Δ,ρ^=1+∑t=1M×Nxt−1ΔxtΔ−xt−1Δ/σ˜2^xt−1Δ2+σ2^∑t=1M×Nxt−1Δ2/σ˜2^xt−1Δ2+σ2^,ρ0=1.As can be seen, this test procedure does not directly verify whether or notκ=μ+σ˜2/2=0 or μ=0 or ϕ=μ−σ˜2/2=0, but transforms the null hypotheses to βk=1 or ρ=1 or βϕ=1. These three t-test statistics are accompanied by a date-stamping strategy that permits us to identify the exact origination and termination periods of price exuberance. Given this instrument, estimates of these dates are as follows:(29)r^unib=infs∈r^uni−1e,1s:tκs>Zα,r^unie=infs∈r^uni−1b,1s:tκs<Zα,r^leib=infs∈r^lei−1e,1s:tμs>Zα,r^unie=infs∈r^lei−1b,1s:tμs<Zα,r^eib=infs∈r^ei−1e,1s:tϕs>Zα,r^unie=infs∈r^ei−1b,1s:tϕs<Zα,where Zα is the right-tailed critical value of the standard normal distribution corresponding to a significance level of α; r^unib/r^leib/r^eib denote the beginnings on the dates of the ith extreme price behavior period; and similarly, r^unie/r^leie/r^eie denote the ends on the dates of the ith extreme price behavior period.


## 4.3. Extend the Model with Endogeneity

Now, we are concerned with the endogeneity case whereWμt is not independent of Wεt. To see this, let us suppose that data are generated from the following stochastic differential equation system:(30)dxt=xtdB˜t+σdBt,x0=x0,dB˜t=μdt+σ˜Wμt,dBt=σdWεt,where (Wμt,Wεt) is a two-dimensional Brownian motion with covariance parameter η. As Föllmer [53] noted, the strong solution obtained from this continuous system would be of the explicit form(31)dxt=expσ˜Wμt+μ−12σ˜2tx0+Zt,where(32)Zt=σ∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sdWεt−ω∫0texpσ˜Wμt−Wμs+μ−12σ˜2t−sds=Vt−Pt.Notice that when endogeneity is included in model (14), the driver process becomes Zt as a substitute for Vt. Zt consists of a component, Vt, and another one connected with the covariance parameter of B˜,B, ω, and variance of Vt. The strong solution gives the exact discrete-time form of model (31):(33)xtΔ=expσ˜ΔuΔ+μ−12σ˜2Δxt−1,Δ+ZtΔ=ρtΔxt−1,Δ+ZtΔ+σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s,where uti.i.d∼N0,1 and(34)ZtΔ=σ∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sdWε,s−ω∫t−1ΔtΔexpσ˜Wμ,tΔ−Wμ,s+μ−12σ˜2tΔ−sds.Somewhat surprisingly, Tao et al. [25] indicated that endogeneity would not affect the sample path features of model (31). Our analysis therefore is the same as before:(1)
Unstable:κ=μ+σ˜2/2>0(2)
Locally explosive:μ>0(3)
Explosive:ϕ=μ−σ˜2/2>0We can estimateθ∗^≔σ˜2^,η^,σ2^′ in the same fashion as before. The likelihood ratio test (LR test) has been proposed by Tao et al. [25] to test the null hypothesis of endogeneity interest:(35)LR=Δ−1QΔr−QΔur∼χ21,underH0:η0=0.Tao et al. [25] also developed the asymptotic theory for μ^ and ρ^≔expμ^Δ. Based on their limit properties, the estimates of μ^ and ρ^ continue to follow asymptotic normal distributions. This good quality enables us to conveniently adopt the aforementioned testing procedures after making a small adjustment to the variance of the limit distribution.


## 5. Data

For our main empirical analysis, we apply daily WTI crude oil prices for the front-month (one-month) future contract on the New York Mercantile Exchange (NYMEX). The data are retrieved from the Energy Information Administration (EIA). Our sample spans the period from April 4, 1983, to June 30, 2020, and comprises a total of 9352 daily observations. The beginning and end dates of the sample period are dictated by the availability of the data.It is well known that WIT and Brent are grades of internationally traded crude oil that are widely used as benchmarks in oil pricing globally. The reason for this paper relying on the WTI price of crude oil is twofold. On the one hand, oil speculation is often blamed on people who trade futures in the U.S. market. WTI is the primary benchmark for oil extracted and consumed in the U.S. The WTI crude oil future market is also the world’s most liquid and largest crude oil future market. Nearly 1.2 million contracts trade a day, with more than 2 million in open interest. On the other hand, WTI future contracts have the longest available history dating back to 1983, giving us a large sample size to gain powers in detecting short-lived bubbles.Our sample period under investigation includes several major exogenous events that, to a large extent, are reckoned to have influenced oil markets. These contain the collapse of OPEC in 1986, the Persian Gulf War of 1990/91, the Asian financial crisis of 1997/98, the terrorist attacks on September 11, 2001, the Venezuelan crisis and Iraq war of 2002/03, the increased financialization of oil future markets since 2004, the OPEC oil supply announcements in 2008 and 2014, the 2019 novel coronavirus disease (COVID-19), and most strikingly, the 2007/08 global financial crisis. Figure2 plots the time series trajectories of the WTI nominal front-month future price and the corresponding real price. A look at the figure illustrates that the oil prices grew steadily from the starting of the sample until the early 2000s. The price series since then begun to surge, and the sharp increase in the price of oil continued until July 2008. As shown in Figure 2, it collapsed to about 30 dollars by the end of 2008. Between 2014 and 2016, the price of oil also experienced a crash. More recently, WTI crude price even went subzero for the first time in history. A common view is that the steep rise and decline in the price of oil in the 2000s could be attributed to speculation.Figure 2
Real and nominal prices of WTI front-month future contracts.


## 6. Empirical Results

### 6.1. Main Results

We start the empirical illustration with baseline model (14), expanding it to the endogeneity application later. In order to formally examine the existence and duration of unstable and explosive behavior in the crude oil market, we run our tests on the daily price data. Following the recommendation of PSY, the minimum window size set for the recursive test procedure is the first 5 years of the entire sample. Figure 3 plots the test empirical outcomes along with their corresponding 95% critical value sequences. Also plotted are the trajectories of test statistics for the random coefficient and the price data. Some interesting results are shown in this figure.Figure 3
Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity.
(a)(b)(c)(d)Looking at the recursive test results reported in the last panel of Figure3, we observe that, over the initial period of observation, the null hypothesis of a time-invariant autoregressive coefficient cannot be rejected at the 5% statistical significance level. The test statistic experiences a dramatic increase in the late 1980s and exceeds the 95% critical value in November 1990. Overall, the observed pattern of time variation rises and becomes much stronger as the time period expands. The evidence that the LBI test statistics for the majority of samples are greater than the 95% critical value suggests that the data-generating process is well characterized by the model with time-varying coefficients. One point worth noting is that the sequence falls in the second half of 2008 and slumps in early 2020. The styled fact seems to indicate that the crude oil market has undergone drastic changes. Interestingly, these located dates coincide with the 2007/08 global financial crisis and the COVID-19 pandemic.We now turn to our focus on different types of unstable and explosive behavior. The shaded areas in green in Figure3 are the identified periods where the t-test statistic crosses its 95% critical value. As is evident from the first panel, the procedure detects two unstable episodes. The first unstable episode is relatively short and only lasts two months (2005M08-M09). The second unstable episode appears in January 2006 and continues until March 2020. These unstable periods might be associated with the financialization of oil future markets from 2004. It can be seen in the second panel of Figure 3 that the detected episodes of locally explosive (submartingale) behavior are clustered in 2007-2008, 2009–2014, and 2018. As stated above, since local explosiveness implies instability and displays more extreme behavior, it should lead to a much shorter duration of exuberance than that of instability. The third panel of Figure 3 reflects that the t-test statistic is below the 95% critical value throughout the sample period. This points out that explosive behavior in crude oil price series does not exist. Table 1 summarizes time horizons for the three kinds of unstable and explosive behavior identified by our approach.Table 1
Time horizons of unstable and explosive periods identified by models assuming no endogeneity.
UnstableLocally explosiveExplosiveAug. 2005-Sep. 2005Dec. 2007–Sep. 2008—Jan. 2006–Mar. 2020Jun. 2009–Oct. 2014—Jun. 2018–Sep. 2018—To shed light on a more realistic scenario of data-generating process, we implement the extreme sample path behavior test by extending the model to the endogeneity case. Figure4 plots the empirical test results together with 95% critical values. A comparison of the LR test statistics with the critical value sequence provides novel evidence in support of the presence of endogeneity during the whole sample period. Thus, it is believed that endogeneity is essential to the oil price formation process. Furthermore, the last panel of Figure 4 highlights that the quadratic variance of the price process is related to the test statistic for endogeneity. This finding is in accordance with the fact that we have exploited the realized variance series to obtain the LR test statistics. In this sense, Figure 5 displays differences in the realized variance estimates, which are captured by the test statistic for endogeneity employing different methods. Similar to Figure 3, we again identify episodes of unstable dynamics and fail to detect explosive behavior when considering the model with endogeneity. In contrast, the statistic sequence for local explosiveness exhibits a different pattern. Previous evidence in favor of locally explosive behavior disappeared. Table 2 summarizes time horizons for the three forms of unstable and explosive behavior. These results suggest that endogeneity feedbacks play a significant role in investigating the existence of explosive price behavior. The conclusion that emerges is that there are no speculative bubbles in the crude oil market.Figure 4
Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity.
(a)(b)(c)(d)Figure 5
Realized variance and likelihood values for random coefficient regression.Table 2
Time horizons of unstable and explosive periods detected by models with endogeneity.
UnstableLocally explosiveExplosiveMay. 2008–Jul. 2008——Dec. 2010–Oct. 2014——Nov. 2016–Apr. 2017——Jul. 2017–Feb. 2020——In principal, our findings are in accordance with the explanations made by Kilian and Murphy [19], Baumeister and Hamilton [21], and Knittel and Pindyck [54] that the variations in crude oil prices are driven by economic fundamentals.


### 6.2. Comparison to Alternative Strategies

Rich literature has studied speculative bubbles in the crude oil market using the techniques developed by PWY and then extended by PSY. A brief description of the SADF methodology (used in PWY) and the GSADF methodology (used in PSY) can be found in Appendix. It is therefore helpful to compare results for alternative methods as the detection strategy in the present paper is quite different from them. Figure6 plots the backward ADF and backward SADF statistic sequences and the associated 95% critical value sequences for oil future prices. 95 percent critical values are taken from 2000 Monte Carlo simulations with the actual sample size. The initial window size is set to 5 years. A glance at Figure 6 reveals that explosive behavior occurred mainly in two occasions using PWY and PSY dating strategies. They are before and during 2007-2008 global recession. We also observe that PWY detects several short-lived episodes between 2011 and 2014, but PSY fails to do so. For daily oil future prices over April 4, 1983, to June 30, 2020, time horizons for explosive behavior identified by PWY and PSY are summarized in Table 3. In line with previous studies, we could conclude from both PWY and PSY tests that there is evidence of bubbles in the crude oil market data. However, as shown in Gürkaynak [37] and Pavlidis et al. [43], explosive dynamics in asset prices per se do not constitute evidence either in support of or against speculative bubbles since such explosive behavior may actually reflect explosive market fundamentals or both factors. As aforementioned, our random coefficient autoregressive model framework has considered endogeneity feedbacks that, to some extent, alleviate this problem. Moreover, the evidence in Figures 3 and 4 suggests the absence of explosive dynamics in the crude oil market. The fact permits us to safely ignore the distinction between bubbles or explosive fundamentals.Figure 6
Date-stamping explosive episodes in oil prices with the time-invariant coefficient model.
(a)(b)Table 3
Time horizons of explosive behavior identified by SADF and GSADF methodologies.
SADFGSADFAug. 2005–Aug. 2006Oct. 2011–Apr. 2012Sep. 1990-Oct. 1990Jun. 2007–Oct. 2008Aug. 2012-Sep. 2012Feb. 1991-Mar. 1991Jun. 2009-Jul. 2009Jul. 2013-Aug. 2013Oct. 2004-Nov. 2004Feb. 2011–Apr. 2011Feb. 2014-Mar. 2014Feb. 2005-Mar. 2005Jul. 2011–Aug. 2012Jun. 2014-Jul. 2014May. 2005–Jan. 2006Mar. 2006–Aug. 2006Jul. 2007–Aug. 2008


### 6.3. Robustness of the Results

In the following, we provide two robustness checks to assess whether our previous conclusions of the paper still hold.First, in our benchmark model, our use of front-month future price instead of other future prices at different maturities is justified by its being the most liquid future contract with the highest open interest. To explore the sensitivity of our results to the future contracts at different maturities, we repeat the analysis described above using the prices for longer-term contracts (i.e., 2, 3, and 4 months-ahead futures). It should be pointed out that the WTI future contract usually expires on the third business day before the 25th calendar day of the contract month proceeding the delivery month.Second, because WTI crude oil is by tradition traded in nominal US dollars per barrel, it might be of concern that inflation and other monetary variables may have real effects on oil price. It is therefore of interest to use real, rather than nominal price. In fact, recent work documents that the use of either price is inconsequential in practice [55]. Although we mainly focus on nominal prices, for robustness check, we shall report the results for real prices. To convert nominal daily series into daily real series, we use linearly interpolated monthly Consumer Price Index (CPI). Monthly CPI series for the United States is obtained from the websites of the Federal Reserve Bank of St. Louis.Further evidence, presented in Figures7–10, demonstrates that our results are remarkably robust to these modifications, and the conclusions remain qualitatively unchanged.Figure 7
Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 8
Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 9
Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 10
Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)


## 6.1. Main Results

We start the empirical illustration with baseline model (14), expanding it to the endogeneity application later. In order to formally examine the existence and duration of unstable and explosive behavior in the crude oil market, we run our tests on the daily price data. Following the recommendation of PSY, the minimum window size set for the recursive test procedure is the first 5 years of the entire sample. Figure 3 plots the test empirical outcomes along with their corresponding 95% critical value sequences. Also plotted are the trajectories of test statistics for the random coefficient and the price data. Some interesting results are shown in this figure.Figure 3
Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model assuming no endogeneity.
(a)(b)(c)(d)Looking at the recursive test results reported in the last panel of Figure3, we observe that, over the initial period of observation, the null hypothesis of a time-invariant autoregressive coefficient cannot be rejected at the 5% statistical significance level. The test statistic experiences a dramatic increase in the late 1980s and exceeds the 95% critical value in November 1990. Overall, the observed pattern of time variation rises and becomes much stronger as the time period expands. The evidence that the LBI test statistics for the majority of samples are greater than the 95% critical value suggests that the data-generating process is well characterized by the model with time-varying coefficients. One point worth noting is that the sequence falls in the second half of 2008 and slumps in early 2020. The styled fact seems to indicate that the crude oil market has undergone drastic changes. Interestingly, these located dates coincide with the 2007/08 global financial crisis and the COVID-19 pandemic.We now turn to our focus on different types of unstable and explosive behavior. The shaded areas in green in Figure3 are the identified periods where the t-test statistic crosses its 95% critical value. As is evident from the first panel, the procedure detects two unstable episodes. The first unstable episode is relatively short and only lasts two months (2005M08-M09). The second unstable episode appears in January 2006 and continues until March 2020. These unstable periods might be associated with the financialization of oil future markets from 2004. It can be seen in the second panel of Figure 3 that the detected episodes of locally explosive (submartingale) behavior are clustered in 2007-2008, 2009–2014, and 2018. As stated above, since local explosiveness implies instability and displays more extreme behavior, it should lead to a much shorter duration of exuberance than that of instability. The third panel of Figure 3 reflects that the t-test statistic is below the 95% critical value throughout the sample period. This points out that explosive behavior in crude oil price series does not exist. Table 1 summarizes time horizons for the three kinds of unstable and explosive behavior identified by our approach.Table 1
Time horizons of unstable and explosive periods identified by models assuming no endogeneity.
UnstableLocally explosiveExplosiveAug. 2005-Sep. 2005Dec. 2007–Sep. 2008—Jan. 2006–Mar. 2020Jun. 2009–Oct. 2014—Jun. 2018–Sep. 2018—To shed light on a more realistic scenario of data-generating process, we implement the extreme sample path behavior test by extending the model to the endogeneity case. Figure4 plots the empirical test results together with 95% critical values. A comparison of the LR test statistics with the critical value sequence provides novel evidence in support of the presence of endogeneity during the whole sample period. Thus, it is believed that endogeneity is essential to the oil price formation process. Furthermore, the last panel of Figure 4 highlights that the quadratic variance of the price process is related to the test statistic for endogeneity. This finding is in accordance with the fact that we have exploited the realized variance series to obtain the LR test statistics. In this sense, Figure 5 displays differences in the realized variance estimates, which are captured by the test statistic for endogeneity employing different methods. Similar to Figure 3, we again identify episodes of unstable dynamics and fail to detect explosive behavior when considering the model with endogeneity. In contrast, the statistic sequence for local explosiveness exhibits a different pattern. Previous evidence in favor of locally explosive behavior disappeared. Table 2 summarizes time horizons for the three forms of unstable and explosive behavior. These results suggest that endogeneity feedbacks play a significant role in investigating the existence of explosive price behavior. The conclusion that emerges is that there are no speculative bubbles in the crude oil market.Figure 4
Date-stamping unstable and explosive episodes in oil prices with the time-varying coefficient model with endogeneity.
(a)(b)(c)(d)Figure 5
Realized variance and likelihood values for random coefficient regression.Table 2
Time horizons of unstable and explosive periods detected by models with endogeneity.
UnstableLocally explosiveExplosiveMay. 2008–Jul. 2008——Dec. 2010–Oct. 2014——Nov. 2016–Apr. 2017——Jul. 2017–Feb. 2020——In principal, our findings are in accordance with the explanations made by Kilian and Murphy [19], Baumeister and Hamilton [21], and Knittel and Pindyck [54] that the variations in crude oil prices are driven by economic fundamentals.


## 6.2. Comparison to Alternative Strategies

Rich literature has studied speculative bubbles in the crude oil market using the techniques developed by PWY and then extended by PSY. A brief description of the SADF methodology (used in PWY) and the GSADF methodology (used in PSY) can be found in Appendix. It is therefore helpful to compare results for alternative methods as the detection strategy in the present paper is quite different from them. Figure6 plots the backward ADF and backward SADF statistic sequences and the associated 95% critical value sequences for oil future prices. 95 percent critical values are taken from 2000 Monte Carlo simulations with the actual sample size. The initial window size is set to 5 years. A glance at Figure 6 reveals that explosive behavior occurred mainly in two occasions using PWY and PSY dating strategies. They are before and during 2007-2008 global recession. We also observe that PWY detects several short-lived episodes between 2011 and 2014, but PSY fails to do so. For daily oil future prices over April 4, 1983, to June 30, 2020, time horizons for explosive behavior identified by PWY and PSY are summarized in Table 3. In line with previous studies, we could conclude from both PWY and PSY tests that there is evidence of bubbles in the crude oil market data. However, as shown in Gürkaynak [37] and Pavlidis et al. [43], explosive dynamics in asset prices per se do not constitute evidence either in support of or against speculative bubbles since such explosive behavior may actually reflect explosive market fundamentals or both factors. As aforementioned, our random coefficient autoregressive model framework has considered endogeneity feedbacks that, to some extent, alleviate this problem. Moreover, the evidence in Figures 3 and 4 suggests the absence of explosive dynamics in the crude oil market. The fact permits us to safely ignore the distinction between bubbles or explosive fundamentals.Figure 6
Date-stamping explosive episodes in oil prices with the time-invariant coefficient model.
(a)(b)Table 3
Time horizons of explosive behavior identified by SADF and GSADF methodologies.
SADFGSADFAug. 2005–Aug. 2006Oct. 2011–Apr. 2012Sep. 1990-Oct. 1990Jun. 2007–Oct. 2008Aug. 2012-Sep. 2012Feb. 1991-Mar. 1991Jun. 2009-Jul. 2009Jul. 2013-Aug. 2013Oct. 2004-Nov. 2004Feb. 2011–Apr. 2011Feb. 2014-Mar. 2014Feb. 2005-Mar. 2005Jul. 2011–Aug. 2012Jun. 2014-Jul. 2014May. 2005–Jan. 2006Mar. 2006–Aug. 2006Jul. 2007–Aug. 2008


## 6.3. Robustness of the Results

In the following, we provide two robustness checks to assess whether our previous conclusions of the paper still hold.First, in our benchmark model, our use of front-month future price instead of other future prices at different maturities is justified by its being the most liquid future contract with the highest open interest. To explore the sensitivity of our results to the future contracts at different maturities, we repeat the analysis described above using the prices for longer-term contracts (i.e., 2, 3, and 4 months-ahead futures). It should be pointed out that the WTI future contract usually expires on the third business day before the 25th calendar day of the contract month proceeding the delivery month.Second, because WTI crude oil is by tradition traded in nominal US dollars per barrel, it might be of concern that inflation and other monetary variables may have real effects on oil price. It is therefore of interest to use real, rather than nominal price. In fact, recent work documents that the use of either price is inconsequential in practice [55]. Although we mainly focus on nominal prices, for robustness check, we shall report the results for real prices. To convert nominal daily series into daily real series, we use linearly interpolated monthly Consumer Price Index (CPI). Monthly CPI series for the United States is obtained from the websites of the Federal Reserve Bank of St. Louis.Further evidence, presented in Figures7–10, demonstrates that our results are remarkably robust to these modifications, and the conclusions remain qualitatively unchanged.Figure 7
Date-stamping explosive episodes in real oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 8
Date-stamping explosive episodes in 2-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 9
Date-stamping explosive episodes in 3-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)Figure 10
Date-stamping explosive episodes in 4-month future oil prices with the time-varying coefficient model with endogeneity.
(a)(b)


## 7. Discussion

The oil price rally since the early 2000s has attracted great interest in the sources of oil price shocks. Some causal observers have stressed that financial speculation has become a significant driver of the dynamics of oil prices. A large body of literature emerged that has sought to investigate its existence. In this paper, we employ a novel approach for testing for speculative bubbles in oil prices. In synthesis, our empirical findings indicate that there is less evidence of exuberance in the oil future markets. Our results are very robust in several scenarios. We argue that the booms and busts in the oil price movements during the sample period are driven primarily by economic fundamentals rather than speculation.Overall, our results confirm the main findings obtained by structural VAR-based studies, including Kilian and Murphy [19], Kilian and Lee [20], Baumeister and Hamilton [21], and Zhou [22]. In the past decade, the use of structural VAR models to understand the evolution of the oil price has become common. The traditional SVAR literature documents that the fundamental drivers of oil price fluctuations have been dominated essentially by supply and demand shocks, see, e.g., Kilian [56], Lippi and Nobili [57], Kilian and Murphy [58], and Baumeister and Peersman [59]. Kilian and Murphy [19] focused on the role speculation by augmenting the SVAR framework of their earlier paper by global oil inventory data. They specified a four-variable model for the global oil market (the percentage change in global oil production, the real price of oil, a suitable proxy of cyclical variation in global real economic activity, and changes in global crude oil inventories). This setting and their sign-identified scheme allow them to separately explain variation in these data as four different types of shocks: (1) a shock to the amount of oil pumped out of the ground (“oil supply shock”), (2) an oil flow demand shock, (3) an oil-specific oil demand shock, and, especially, (4) a speculative (or inventory) demand shock. Their core finding is that the spike in the real price of crude oil prices during 2003–2008 was caused by unexpected increase in global oil consumption. Using two alternative proxies for global above-ground crude oil inventories, Kilian and Lee [20] reestimated the model of Kilian and Murphy [19] and ruled out the role of speculation in explaining the recent surge of the real oil price on the basis of their estimation results. Also, Zhou [22] assessed the sensitivity of the findings in Kilian and Murphy [19] to the introduction of various state-of-the-art oil market methods. She shows that the core conclusions reached by Kilian and Murphy [19] are highly robust to some refinements, corrections, and extension of the Kilian–Murphy model. Baumeister and Hamilton [21] recently contended that prior information plays a key role in any SVAR models, and traditional identification approaches can be generalized to a special case of a Bayesian perspective. They revisited the importance of supply and demand shocks in the oil market and demonstrated that speculative demand shock turns out to be only a small factor in accounting for most historical oil price shocks.For comparison purpose, we have also applied the tests of PWY and PSY, which are the most popular econometric techniques in the literature on speculative bubbles in the oil markets. If we were to follow the test procedures of PWY and PSY, our conclusions are also broadly consistent with the rich literature that supports the existence of speculative bubbles in oil prices. However, these cointegration-based bubble tests suffer from the joint hypothesis problem: “Every test of a bubble is a joint test of the presence of the bubble in the data and the validity of the model applied by the econometrician” (see Giglio et al. [60]). More specifically, the main drawback of such indirect, model-dependent tests of bubbles typically refers to the difficulty in distinguishing between the unknown true model for market fundamentals and the presence of bubbles (see, for example, Gürkaynak [37]; Pavlidis et al. [42]; Pavlidis et al. [43]). Given the fundamental value of oil price is unobservable, the literature has resorted to observe economic and financial variables which are used to estimate the intrinsic values. For example, both Phillips and Yu [16] and Caspi et al. [39] simply normalized crude oil prices by U.S. oil inventory data as the fundamental value. This measure is questionable since their method cannot sufficiently capture the information on market fundamentals and neglect the major drivers—like global economic activity—also affecting the fundamental value of crude oil. This issue of misspecified fundamentals (either in terms of model misspecification or omitted variables or both of these reasons) may make researchers fail to arrive at a conclusion from evidence of explosive prices alone. To make matters worse, even though we have captured the true market fundamentals, we may still fail to detect bubbles if the explosive behavior in the price of oil just stems from the explosive fundamentals.Compared to earlier studies, as briefly mentioned above, our methodology has the main advantage that, by considering the dependence between the random coefficient and system shocks, we have incorporated endogeneity into the continuous-time random coefficient setting, and thus, we do not need to prespecify the fundamental component of the oil price. The test results for endogeneity suggest that endogeneity is important in the data-generating process. In particular, our empirical analysis also finds that there are no explosive episodes in crude oil markets. This implies that it is not necessary to require the specification of a fundamental value of the oil price. Like Pavlidis et al. [43], their methods and ours have in common that the problem of the well-known joint hypothesis has, to be some extent, been effectively ameliorated. Although our approach is quite distinct, we draw the same substantive conclusion that speculative bubbles can be ruled out as a key determinant of the price of crude oil in the 2000s, reaffirming the conclusions of Kilian and Murphy [19], Baumeister and Hamilton [21], Zhou [22], and Fattouh et al. [61].


## 8. Conclusions

Repeated surges and crashes in crude oil prices during last decades have sparked a heated debate on their underlying causes among academics, commentors, and market regulators. One of the central questions concerns whether financialization of oil future markets creates a speculative bubble that subsequently collapses. Answering this question is difficult because one has to rely upon strong assumptions about the data-generating mechanism for the fundamental value of the oil price. In this paper, we present a stylized model of asset pricing for a commodity to address this issue. Our theoretical framework disentangles the effects of speculative bubbles from the traditional fundamental effects on oil price dynamics. We use this setting to illustrate that bubbles, if they are present, should demonstrate explosive behavior in prices. We then provide a novel approach for bubble detection that covers various explosive scenarios and find no evidence of speculative bubbles in oil markets even during episodes where the most popular SADF and GSADF methodologies identify a speculative bubble.Our conclusions have pertinent implications for policy makers. Given the historically important role of oil prices in the real economy, one of the serious concerns in policy circles over the past decade has been whether speculative trading activities in oil future markets are responsible for the large price swings. For example, in their 2008 testimonies to the U.S. Senate Commerce Committee, Michael Masters and George Soros posited that speculators had fueled price increases since 2003 and had caused a part of a “superbubble” in 2007 and 2008, resulting in significantly higher energy costs for consumers. This widespread bubble view dominated policy discussions, and efforts to limit speculative trading in the future market quickly followed. As a response to the excessive speculation pressures on prices, the Dodd–Frank Wall Street Reform and Consumer Protection Act was enacted by the U.S. Congress in 2010. As a direct policy implication, our results highlight that imposing stricter regulations on trading in oil derivative markets cannot be expected to lower the oil price, given the fact that the synchronized boom and bust cycles witnessed in the past 20 years are mainly due to economic fundamentals, which appear likely to resurge with the business cycle. The above analysis implies that there is no need to enforce additional regulatory limits against financial arbitrage in the commodity market.



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*Source: 8883416-2021-01-07.xml*