Multiple speculative Bubbles and Trading in the Gold Market

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Multiple speculative Bubbles and Trading in the Gold Market

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Finance

at the Faculty of Economics and Business at the University of Groningen

Submitted by Referee:

Verena Mercedes Schlosser Prof. Dr. Y.R. Kruse

S3068781

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2

Multiple speculative Bubbles and Trading in the Gold Market

VERENA M. SCHLOSSER*

January 2017

ABSTRACT

The detection and identification of asset price bubbles in financial markets has not just since the most recent global financial crisis been an ever present topic in current research. Phillips et al. (2015) have developed a state-of-the art econometric procedure applying recursive regressions based on flexible windows that provides valuable real time results. Following this approach, various gold price series are analyzed in this study. In the context of a rational asset pricing model, several price bubbles are found in the history of the commodity gold. The robust results are highly relevant for regulators, central banks, and other market partici-pants.

Keywords: Bubble Phenomena, Asset Pricing, Commodities, Gold.

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3 Table of contents List of abbreviations 4 List of tables 6 List of figures 8 1. Introduction 9

2. Bubbles: A general overview 10

2.1. Definition 10

2.2. Theoretical foundations and financial implications 12

3. The case of gold 17

3.1. Specifics 17

3.2. Gold as a safe haven asset 19

4. Bubble tests, applying PSY procedure 19

4.1. Methodology and data 19

4.2. Results 27

5. Interpretation: Fundamentals versus speculative behavior 41

5.1. Convenience yield 41

5.2. Gold lease rates 44

6. Conclusion 49

Appendix 52

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4 List of abbreviations

ADF Augmented Dickey-Fuller AIC Akaike Information Criterion

AVG Average

BSADF Backwards Supremum Augmented Dickey-Fuller

CF Cash Flow

CY Convenience Yield

ETF Exchange Traded Fund

FADF Forward Augmented Dickey-Fuller

FX Foreign Exchange

GSADF Generalized Supremum Augmented Dickey-Fuller ICE Intercontinental Exchange

LEASE1M Gold lease rate for 1 month maturity LEASE2M Gold lease rate for 2 months maturity LEASE3M Gold lease rate for 3 months maturity LEASE6M Gold lease rate for 6 months maturity LEASE12M Gold lease rate for 12 months maturity LIBOR London Interbank Offered Rate

M Month

PPI Producer Price Index

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5 SADF Supremum Augmented Dickey-Fuller

SDR Special Drawing Right

Sup Supremum

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6 List of tables

Table 1 Nominal gold spot price series: The SADF test and the GSADF test 29 Table 2 Real (U.S. PPI deflated) gold spot price series: The SADF test and the GSADF test 30 Table 3 Real (SDR basket deflated) gold spot price series: The SADF test and the GSADF

test 31

Table 4 Nominal gold futures price series: The SADF test and the GSADF test 33 Table 5 Real (U.S. PPI deflated) gold futures price series: The SADF test and the GSADF

test 34

Table 6 Real (SDR basket deflated) gold futures price series: The SADF test and the GSADF

test 35

Table 7 Nominal gold spot price series: Date-stamping procedure (BSADF) under relaxed

min. duration conditions 36

Table 8 Real (U.S. PPI deflated) gold spot price series: Date-stamping procedure (BSADF)

under relaxed min. duration conditions 37

Table 9 Real (SDR basket deflated) gold spot price series: Date-stamping procedure

(BSADF) under relaxed min. duration conditions 38

Table 10 Nominal gold futures price series: Date-stamping procedure (BSADF) under

relaxed min. duration conditions 38

Table 11 Real (U.S. PPI deflated) gold futures price series: Date-stamping procedure

Table 12 Real (SDR basket deflated) gold futures price series: Date-stamping procedure

Table 13 Robustness check: SADF and GSADF test statistics of the different gold price

series on the basis of automatic lag length selection via AIC (max. lag number = 12) 40 Table 14 Gold convenience yield series: The SADF test and the GSADF test 42

Table 15 Gold price bubbles 43

Table 16 Robustness check: SADF and GSADF test statistics of the nominal gold convenience yield series on the basis of automatic lag length selection via AIC (max. lag

number = 12) 44

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8 List of figures

Fig. 1 The SADF test versus GSADF test: Sample sequences and window widths 22

Fig. 2 The BSADF test 23

Fig. 3 BSADF sequence of the nominal gold spot price series 25

Fig. 4 Nominal gold spot price BSADF sequence 28

Fig. 5 Real (U.S. PPI deflated) gold spot price BSADF sequence 30 Fig. 6 Real (SDR basket deflated) gold spot price BSADF sequence 31

Fig. 7 Nominal gold futures price BSADF sequence 32

Fig. 8 Real (U.S. PPI deflated) gold futures price BSADF sequence 34 Fig. 9 Real (SDR basket deflated) gold futures price BSADF sequence 35

Fig. 10 Gold convenience yield BSADF sequence 42

Fig. 11 Gold lease rates for 1 month to 12 months maturities 45

Fig. 12 Gold 12 months lease rates BSADF sequence 47

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9 1. Introduction

The last decades in financial markets can be characterized by a sequence of various crises triggered by financial asset price bubbles among other things. In particular, when an asset bubble finally bursts, the negative effects resulting from that cannot only be very damaging for the society as a whole but also on a personal investor’s level. One of the most prominent examples is the so-called dot.com bubble and the associated sharp up- and downward move-ments of internet stock prices in the 1990s. As a result, the overall economy and many inves-tors’ wealth suffered massively. The second and probably most prevalent example in recent history is the 2007/2008 real estate bubble in the USA which caused a devastating worldwide financial crisis. The effects of the subsequent global economic crisis and recession are still perceptible today. In this regard, the European sovereign debt crunch is just one of the nega-tive impacts that are not entirely resolved yet (e.g. Figuerola-Ferretti and McCrorie, 2016; Porras, 2016).

All in all, showing the interconnectedness of events, when the price bubble finally collapses, volatile capital markets, contagion effects from one market to the next and the rebalancing of portfolios leading to increased trading volumes are substantial outcomes. The associated total costs of the increased uncertainty, risk, loss of trust, and a general fear in the markets induced by the events following the bubble bursting cannot be quantified. This can be explained by the wide-ranging repercussions that are hard to conceive today. Higher funding costs will affect and harm the economy’s growth for years (Porras, 2016).

Given the far-reaching consequences and danger, prevention and early detection of such de-velopments is of main interest not only to regulators and governments but also to society in general. However, it is important to note that market participants pursue different objectives and that those interests might contradict each other. In other words, there are also actors who profit from the described market conditions by being able to anticipate the events early enough (e.g. Long et al., 2016; Porras, 2016).

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10 other policymakers and investors, a major future aim is to be able to discover potentially de-structive price dynamics in advance or rather early enough to counteract with appropriate means (e.g. Porras, 2016).

In this thesis, one of the newest econometric techniques for bubble testing developed by Phil-lips et al. (2015) is presented and applied. As previously mentioned, the bursting of the hous-ing bubble and the subsequent financial crisis had contagion effects on other markets and as-set classes. In the direct aftermath, especially commodity markets were affected. One of the hit markets in which the impact was eminently clear was the gold market. In general, the commodity gold has always been regarded as a solid investment that does not lose its value over time. It is sometimes also attributed a so-called safe-haven status (e.g. Beckmann et al., 2015). Therefore, the commodity gold is the asset of interest on which the empirical analysis of this thesis is based on.

The remaining part of this thesis is structured into four main parts. Section 2 outlines a gen-eral background of bubble phenomena. Then, the commodity gold and its specific properties are presented in section 3. Section 4 introduces the bubble-testing method by Phillips et al. (2015) and documents the results of the empirical anlysis. Thereafter, interpretations of the obtained findings are provided in section 5. Section 6 concludes the thesis.

2. Bubbles: A general overview

2.1.Definition

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11 In general, firm value depends on its wealth creation potential. For example, in the case of a listed company, the profits earned by the firm are distributed to the shareholders relative to their individual shares in the corporation they own. So if one holds a number of stocks, one owns a certain part of a company. The return that the equity holder then receives is deter-mined by the expected growth. Accordingly, the fundamental value of a firm is constituted by a number of factors that influence and help maintain a certain growth level over time in order to assure the proportional reward for the stock owners. On the whole, the fundamental value can be determined by the rate at which dividends and earnings grow the ratio of dividends and net earnings, capital costs, and cash flow risk (Porras, 2016).

Thus, a development where the asset trades at a substantial premium in comparison to its fun-damental value qualifies as a bubble. This is for instance happening if the rate at which the stock price moves up within a certain sequence of time is considerably higher than the earn-ings growth rate. In that case, the fundamentals cannot justify the explosive price movement. The foundation provides Tirole’s basic valuation model (Tirole, 1985). Here, the current fun-damental asset value is represented by future expected returns in form of dividend payments discounted to today. Tirole suggests that the current asset price (present value), consists of two components, i.e. a fundamental part and a bubble element. In the face of normal market conditions without explosiveness, the market price equals the price element which refers to the fundamentals, whereas if a bubble is present a second component is added (Porras, 2016; Tirole, 1985). Unfortunately, finding the correct fundamental value is one of the hardest tasks in asset valuation. In this regard, having realistic expectations about future market develop-ments and circumstances probably poses the most difficult challenge. Furthermore, uncertain-ty concerning the estimates used for the pricing process and potential model or variable value mistakes make it even more sensitive (Tirole, 1985; Tirole, 1982).

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12 to this type of explosivity and systematic deviation of market prices from the according un-derlying fundamental value (Porras, 2016).

After giving a brief introduction into the world of bubbles, the next subsection provides theo-retical explanations.

2.2.Theoretical foundations and financial implications

Financial bubbles have been extensively studied for years. Prevailing literature identifies two main model categories, rational and behavioral models (e.g. Scherbina and Schlusche, 2014). Beginning with a rational model under symmetric information, Tirole (1982) sets the subject of bubbles into a general equilibrium context and reasons that the existence of bubbles is not possible if it is common knowledge that resources are provisionally allocated in a pareto-efficient way at the starting point. Thus, if a bubble is beneficial for the market participant that offers the according asset, it has to be at the disadvantage for the one acquiring the asset. As a result, no agent would be interested in purchasing the bubble asset (Brunnermeier, 2008; Tirole, 1982).

More formally, scholars find that in a frictionless regime of rational expectations and symmet-ric information, asset psymmet-rice bubbles can only occur if the rate at which the bubble is growing over time ( ) equals the discount rate (r), i.e. the required rate of return of the asset. This is based on the assumption that the asset price consists of a fundamental component and a sec-ond factor referring to the bubble element. In summary, the current price of an asset ( ) with infinite maturity includes the present value of all future expected cash flows (CF) and the dis-counted bubble factor ( ) (Scherbina and Schlusche, 2014, p. 591):

= ₍ ₎ + lim → ₍ ₎ (1)

Furthermore, the bubble growth can be derived as

= (1 + ) " _. ₍₂₎

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13 the existence of the bubble part, without collapsing incorporated in the asset’s price, is only possible if the two rates equal each other (Scherbina and Schlusche, 2014).

The presented argument is just one of a number of theoretical explanations that limit the set of potential rational bubble phenomena in some particular settings. For instance, if there is an upper bubble size limit imposed, a positive bubble cannot occur. Commodity bubbles are an example for this since there might be other resources that could substitute the inflated asset. When faced with a seemingly infinite price explosion for a certain commodity rendering the asset more and more costly, buyers will prefer close substitutes at some point instead. Another example where a bubble is not possible is, when the required return rate of an asset which price is not an equilibrium one is higher than the rate at which the overall economy expands. The reason behind this is that the bubble would otherwise rise above the total economy’s wealth. Thus, bubbles can only arise in an environment where the required rate of return is at most as high as the rate of economic growth. Furthermore, the existence of rational bubbles can also depend on trading opportunities being available because of the bubble occurrence resulting in a different equilibrium. A well-known example is fiat money which is traded at a non-negative price although its intrinsic value is equal to zero. Within a model of overlapping generations, shifting wealth from one generation to the next is only possible if faced with a positive price. In this setting, a negative bubble related to an asset with limited liability cannot emerge because this would be only possible if the market price of the asset was expected to fall below zero in the future. Hence, when the bubble dissipates at some point, it does not build up again and therefore stays equal to zero. As a conclusion, the presence of a rational bubble must be already given on the first trading day of an asset, it cannot arise in the middle of an asset-pricing process (Brunnermeier, 2008).

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14 this into account, there are co-integration dynamics present between the stock price and the related expected dividend stream discounted to today. To examine this hypothesis, Diba and Grossman (1988) perform a number of different tests, namely procedures testing for unit roots, co-integration and autocorrelation. Their results document that the null hypothesis of no bubble being present has to be accepted (Diba and Grossman, 1988).

So far, a rational agent is only willing to keep an overpriced asset in his portfolio as long as the explosive development is expected to grow to infinity. By contrast, the other frameworks that shall be discussed in the following paragraphs allow for an asset buyer to hold on to an overvalued asset if he is positive that he can transfer it to a different market player that has access to less information or whose perception is not unbiased (Brunnermeier, 2008).

Moving on to an asymmetric information setting where investors do not share common knowledge yet, they initially have the same distribution. In this type of modelling, prices mainly have two functions. First, they work as an indicator for how scarce a certain resource is. Second, they partially signal the level of information that other dealing parties hold (e.g. Brunnermeier, 2001). While under the condition of no information asymmetries all agents need to be aware of the bubble exposure, this is not the case with asymmetric information bubbles. A practical example could be that all market players recognize that a particular asset is overvalued in comparison to the underlying cash flows in form of dividends, but the investors might not realize that their competitors anticipate that as well. This is the reason why in such a situation bubbles that end at some point can emanate under certain circumstances (Brunnermeier, 2008; Allen et al., 1993).

A first condition is that the information asymmetry has to be somewhat permanent, meaning that investors cannot deduce complete information from settlement prices and net trades. Thus, prices are not completely informative and therefore do not disclose everything. Secondly, investing parties must be subject to a short-selling constraint. Lastly, the fact that resources are allocated in an interim pareto efficient way at the beginning is not public understanding. The explanation for this is that all investors would otherwise be aware of making no profit from such a deal when the market player who acquires the overvalued asset knows that the process is at the advantage of the rational trader who offers the bubble asset (Brunnermeier, 2008; Tirole, 1982).

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15 perspective, it can be profitable to invest in too expensive bubble assets since particular trading activities enable him to make his investors presume that he possesses superior insights. By not pursuing any trades, an uninformed manager of funds would, thus, signal the opposite. As a result, dubious fund managers artificially produce bubbles paid for by their customers that invest in their funds without having adequate information (Brunnermeier, 2008; Allen and Gorton, 1993). Moreover, asset managers whose liability is limited due to certain legal constructs could be likely to engage in trading activities targeted at bubble assets because they might be incentivized by being able to transfer potential downside risks if necessary, while at the same time always profiting when the trade turns out favorably (Brunnermeier, 2008).

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17 point out that the group of rational players is more likely to sit out the bubble instead of betting against it. Riding bubbles makes only sense if a trader is truly convinced that the price explosion will further move on (Abreu and Brunnermeier, 2003). Empirical results suggest that bubble-riding is sometimes preferred. Hedge funds, for instance, strongly invested in equities of technology firms in the years from 1998 till 2000, although they were clearly overvalued (Brunnermeier and Nagel, 2004). This is in contrast to the hypothesis of efficient markets (Fama, 1970) since they could not sufficiently correct the price despite their well-known proficiency and closeness to rational arbitrage (Brunnermeier, 2008).

After having examined rational bubbles within different information scenarios and behavioral phenomena in connection with limited arbitrage opportunities, the last category centers on bubbles in the presence of heterogenous beliefs and short-selling constraints. In this regard, psychological biases can be a source for different initial beliefs of investors. Overconfidence, for example, can lead to various prior distributions concerning the accuracy of investors’ individual signals. In such a scenario, it is possible that there is no general concent reached among investors even after exchanging all available information. As opposed to a situation where information is asymmetrically distributed, investment parties do not seek to derive knowledge held by other traders captured in prices. In summary, overpricing issues can arise if heterogenous beliefs are paired with short-selling restrictions, since optimistic investors drive the asset price to higher levels and pessimistic agents fail to offset the development due to short-sale constraints (Brunnermeier, 2008; Miller, 1977). In addition to this, high trading volumes and volatility in prices are characteristics of heterogeneous belief bubbles (Scheinkman and Xiong, 2003).

In the following section, the particular role of the commodity gold in this context is presented.

3. The case of gold

3.1. Specifics

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18 protect their countries’ currencies. However, its traditional role has changed to a more modern view of gold as a source of different hedging strategies (Baur and McDermott, 2010).

Primarily, investors hold gold to hedge inflation or currency risk. As gold is typically de-nominated in dollars, its nominal market price rather goes up if the dollar devaluates. There-by, its real value is protected. By this means, the exchange-rate risk in dollar positions can be hedged by gold (Baur and McDermott, 2010; e.g. Capie et al., 2005).

Another factor why gold has been experiencing such an increase in popularity is the straight-forwardness of the gold market. This is especially true when uncertainty is high and the reluc-tance to perform trades leads to elusive asset valuations. In contrast to other assets, the pricing process of gold is less complex and more comprehensible. In times of financial turmoil, gold could be considered as a serious investment alternative featuring greater stability. In addition to that, in its capacity of a physical metal, gold is processed in industrial production, jewelry manufacturing, and in dental care. This number of real world applications makes it a lot easier to acknowledge the intrinsic value of gold (Baur and McDermott, 2010). Yet, the determina-tion of this fundamental value is delicate and not without controversy. Since commodities do not pay out dividends, some strands of literature take the sheer price explosion as adequate enough to hint at bubble events due to speculative trading activities (Phillips and Yu, 2011). Other researchers solve this problem by applying Pindyck’s convenience yield model (Pindyck, 1993) as a replacement for the dividend yield (e.g. Figuerola-Ferretti and McCrorie, 2016).

Moreover, there is no default risk associated with gold investments. Similarly, debt and future earnings potential do not play a role. In this regard, the value of gold is completely independ-ent. The difference to other commodities is how gold reacts when other asset markets decline. According to latest research and experience, the gold price apparently responds to negative market developments in the opposite direction. Another characteristic of the gold market is that while demand changes in different economic conditions, the supply side stays approxi-mately constant on a fixed level. The entirety of the described characteristic factors support the notion that gold can be interpreted as a ‘store of value’ and replace other assets as a stable investment in the event of negative market shocks (Baur and McDermott, 2010).

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19 3.2. Gold as a safe haven asset

A typical behavior of investors after financial crises and subsequent stock market crashes is the flight from risky asset classes in allegedly safer assets such as U.S. Treasuries or German Bunds, for instance (e.g. Upper, 2000). In this connection, gold is also frequently labelled as a so-called ‘safe haven asset’ (e.g. Baur and McDermott, 2010). In order to evaluate whether this is true, it is crucial to define the term ‘safe haven’ in a distinct way. Baur and Lucey (2010) differentiate it from a hedge or a diversifier asset. They define a safe haven asset as one that that does not correlate with both other single assets or portfolios of assets in stressed markets. The correlation can even be negative in those times. In contrast to this, a hedge fea-tures the same properties but only on average. However, a hedge is not able to diminish losses incurred in such extreme market conditions. Similarly, a diversifying asset is subject to the same properties as a hedge with the difference that the correlation is positive to some extent meaning that there is not a perfect co-movement at hand (Baur and Lucey, 2010). All in all, there is manifold evidence in the literature that gold qualifies as such a safe haven (e.g. Beckmann et al., 2015). Baur and McDermott (2010), for example, document that gold can take both the form of a safe haven and a hedge for specific markets such as the U.S. and im-portant stock markets in Europe. The findings suggest that investors can use gold as a protec-tion device to preserve their wealth in times of negative or adverse condiprotec-tions in the markets. Since gold functions best if things get bad by limiting the loss potential, it has the capability to work as a stabilizer for the entire financial system (Baur and McDermott, 2010). In addition to that, institutional investors have been identified to be one of the main drivers of increasing trade volumes in commodities markets representing the relative attractiveness of such invest-ments (Baur and Glover, 2015; Tang and Xiong, 2012).

After giving a general introduction to bubble phenomena and the specific case of gold, the next section depicts the empirical analysis.

4. Bubble tests, applying PSY procedure

4.1. Methodology and data

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20 proves to be complex when applied to extensive historical data. In particular, dealing with characteristics such as break mechanisms and non-linear structures with multiple bubbles per sample period is especially challenging from an econometric point of view (Phillips et al., 2015). In order to overcome these difficulties, Phillips et al. (2015) (PSY) introduced a state-of-the-art methodology that makes use of window sizes as fractions of the total number of observations being adjusted in a flexible way and applied recursively. Since this technique is a lot more practicable for long time series of historical data (Phillips et al., 2015), the empirical part of this thesis is based on this procedure. In the following subsection, the employed methodology and the underlying data base are described in detail.

The proposed method by PSY (2015) consists of three main steps. First, the null hypothesis (H0) of no mildly explosive periods is tested based on the underlying sample. Second, if H0 is

rejected, it remains to determine the exact starting and end dates of the mildly explosive phas-es. Third, after having determined the starting and end dates, the last part of the procedure concerns the classification of the results. For that purpose, the analysis of particular proxy variables that represent the fundamental value of the asset, here: the commodity gold, is con-ducted. Similarly as in a rational asset pricing model, it is of main interest to check if the doc-umented periods of mild explosiveness are reflected in the fundamental as well, or if the as-set’s price truly departs from its fundamental value. Only if the latter holds, the detected epi-sodes are to be classified as bubbles (Phillips et al., 2015; e.g. Figuerola-Ferretti and McCrorie, 2016).

Following PSY (2015) approach, an Augmented Dickey-Fuller (ADF) test (Dickey and Fuller, 1979) is performed. Specifically, the rolling window regression is modelled by Phillips et al. (2015, p. 1047) as follows:

∆$% = &'_(),(++ -._(),(+$%")+ / 0.2_{1 )} _(),(+1 ∆$%"1+ 34% , (3)

where 5 represents the variable of interest, i.e. the gold price at time t, 64 _{, 7} is a constant,

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21

(+>?(@, )A

regression: B₌ = CB₌D and EFG 7 denotes the corresponding ADF statistic (Phillips et al., 2015). The hypothesis test performed in the first step then is H0 (unit root): 89 , 7 = 1 vs. H1

(explosive root): 89 _{, 7} > 1 (e.g., see Phillips et al., 2015; Phillips et al., 2011).

Phillips et al. (2015) find this kind of rolling regression model to be useful and suitable for the identification of bubble-like episodes, especially for the detection of multiple bubble-like phenomena. Since the so-called PSY (2015) strategy already comprises the afore developed procedure by Phillips et al. (2011) (PWY), the methodology description herein mainly follows PSY (2015) (Phillips et al., 2015).

First of all, PWY (2011) and the associated Supremum (sup) Augmented Dickey-Fuller (SADF) test, which is a specific right-sided unit root test, is based on the rationale of iteratively computing ADF estimates. The central component of the SADF test is therefore the underlying forward recursive regression on which the sup statistic is based on. In short, it expands the used sample sequence in a forward looking manner, see Fig. 1 (a). The individual window sizes are fractions of the total sample size and the concrete process relies on expan-sions of different window sizes. The window size rw ranges from r0 , the “smallest sample

window width fraction” (Phillips et al., 2015, p. 1048) to 1, the largest one of the recursive procedure. As shown in Fig. 1 (a), the sample sequence starts at r1 which is fixed at 0,

where-as the point where each sample ends, represented by r2, is equal to rw and shifts from r0 to 1.

In this regard, EFG_H7 is the notation for a sample with a starting point 0 and an ending point at r2 (Phillips et al., 2015; Phillips et al., 2011). The PWY (Phillips et al., 2015, p. 1048;

Phillips et al., 2011) test result stemming from the according sequence of the ADF statistic is called the sup value and is expressed by

IEFG ( H) = JKL EFGHM. (4)

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22

7N?H, 1A

N?0, 7P HA

Fig. 1

The SADF test versus GSADF test: Sample sequences and window widths

The figure shows how sample sequences and window spans are chosen within the SADF and the GSADF re-gimes (Phillips et al., 2015, p. 1049).

Next, the GSADF test (PSY), an extension of the SADF method (PWY) is outlined. First, note that the PSY method is a rolling window procedure. The basic idea behind the test is to repeatedly apply the recursive ADF regressions to different data subsamples. In comparison to the previously discussed SADF test, the extent of the run fractions of data is much greater though. Another major difference to the SADF test is that when using the GSADF test, it is possible to vary the starting point r1 from 0 to r2 - r0. Fig. 1 (b) displays these alternatives

(Phillips et al., 2015). Phillips et al. (2015) establish the GSADF statistic as the maximum ADF statistic over all permissible ranges of starting and endpoints of the regression, resulting in the following definition (Phillips et al., 2015, p. 1049):

QIEFG H JKL REFG_SMT. (5)

The size of the smallest window r0 plays a superordinate role for both the asymptotic GSADF

distribution and the prior SADF statistic. From a practical point of view, the choice of r0

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cho-23 sen. In that case, it is ensured that also early periods of explosiveness can be tracked if exist-ent (Phillips et al., 2015). In order to get satisfactory conditions for the SADF and GSADF test implementations, Phillips et al. (2015) suggest the following functional rule for setting r0

(Phillips, et al., 2015, p. 1050):

H = 0.01 _√.V. (6)

This standard is adopted in all of the empirical tests carried out in this thesis. The finite sam-ple critical values for each price series are generated by Monte Carlo simulations with 1,000 replications. This computation has proven to be quite time intensive. Thus, a large number of existing studies in this field rely on standard critical values provided by PSY instead of per-forming the according simulations themselves (e.g. Long et al., 2016; Phillips et al., 2015). Secondly, the date-stamping process is considered. In this step, the main interests lie in as-sessing whether single real-time data points (observations) are part of a potential price bubble and in identifying the exact start and end dates of a bubble. Following the strategy of PSY (2015), this is done by the so-called Backward Sup ADF (BSADF) test. Here, the sample se-quence upon which the test statistic is based on expands backwards. The sample selection process works similarly as described in the previous paragraph but the other way round, in backward instead of in forward direction, see Fig. 2 (Phillips et al., 2015).

Fig. 2

The BSADF test

Fig. 2 displays sample sequences of the BSADF test (Phillips et al., 2015, p. 1052).

Phillips et al. (2015, p. 1051) give the following definition for the corresponding test statistic:

IEFG_M H JKL REFGSMT. (7)

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24 Recapping the aforementioned notation of the GSADF statistic (Phillips et al., 2015, p. 1053), (5) can be rewritten as

QIEFG ( _H) = JKL R IEFGM ( H)T. (8)

This means that the BSADF statistic is created by a sampling process with different window sizes. Subsequently, the repetition of this procedure leads to the GSADF statistic. Thus, the outcome of the hypothesis test relies on the GSADF statistic, and the actual dating process is based on the BSADF statistic. The BSADF test identifies the exact points where mildly ex-plosive behavior starts occurring and also when it ends, based on the assumption of a particu-lar minimum duration condition (Phillips et al., 2015). In this analysis, minimum duration conditions of 6 months, 3 months, and lastly 4 weeks have been imposed on the different time series. The origination and termination points, _Y,Z and _Y,[, identified subject to the explosive-ness lasting for at least 6 months, are indicated by the black arrows, see Fig. 3 as an example. Whenever the sequence of the BSADF test statistic moves above the 95% critical value se-quence, this indicates mildly explosive behavior. Once the first explosive period is found, the procedure is repeated again and again (Phillips et al., 2015).

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25 Fig. 3

BSADF sequence of the nominal gold spot price series

The green line represents the nominal gold spot price (monthly basis). The red (95% critical value sequence) and the blue line (BSADF sequence) depict the detection and subsequent dating of mildly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, there are four exuberant periods discovered. The dated periods of mildly explosive behavior are 1) 09/1972 - 07/1975, 2) 01/1979 – 01/1981, 3) 10/2005 – 10/2008 and 4) 12/2008 – 06/2013. The black arrows point to the starting and ending points of periods of explosive behavior existing in the underlying nominal gold spot price series, detected assuming a minimum duration of 6 months.

Having described the first two elements of the PSY (2015) procedure, the hypothesis testing and subsequent dating mechanism, the third and final part of the procedure deals with the ra-tional asset pricing model and how the obtained results can be interpreted in that context. Analyzing bubbles in financial markets often departs from the basic formula of asset pricing (Phillips et al., 2015, p. 1046; Tirole, 1985):

= ∑ ]Y H _^_YΕ (F Y + a Y) + , (9)

where is the current asset price, F represents the dividend, _[ denotes the risk-free interest rate, U is the fundamentals component, and B stands for the bubble element. The difference between the present value of an asset and the bubble component is commonly labelled as the market fundamental. B is characterized by the following equation (submartingale property of

B ), (Phillips et al., 2015, p. 1046; Tirole 1985):

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26 If _[ is positive, i.e. greater than zero, explosiveness is given. In this case, classification of a detected period of mild explosiveness as a bubble depends on the development in the funda-mental. To that end, two possible representatives of the fundamental value of gold, the gold convenience yield and the gold lease rates, are given in section 5. In short, the difference in the degree of non-stationarity between the current asset price and the fundamental is an indi-cator for bubble declaration (Figuerola-Ferretti and McCrorie, 2016; Phillips, et al., 2015). The data used for the present empirical analysis consists of 12 different gold price series. The first three time series are based on the weekly gold spot price per troy ounce in the London Bullion market valued in USD. The included time period lasts from September 23, 1968 to September 26, 2016. In order to get a broader economic picture and following Figuerola-Ferretti and McCrorie’s (2016) approach, two real gold price series are added to the basic nominal gold price series described above. The nominal gold spot price series is deflated 1) by the U.S. Producer Price Index (PPI) for commodities and 2) by the Special Drawings Rights (SDR) basket, yielding two real gold spot price series. Due to different data availability of the deflators, the datasets are reduced accordingly. The SDR basket, i.e. the value of USD currency units per SDR, is only documented from January 3, 1994 on, and is downloaded from the Internationional Monetary Fund website. The monthly U.S. PPI quotes were retrieved from FRED, the economic research division of the Federal Reserve Bank of St. Louis. In addition to the spot prices, monthly one-months futures prices ranging from November 26, 1979 to September 19, 2016 are evaluated. Again, the same deflation procedure as with the nominal spot price series is being performed. For the calculation (see 5.1.) of the nominal gold convenience yield, the one month U.S. Dollar (USD) Intercontinen-tal Exchange (ICE) London Interbank Offered Rate (LIBOR) weekly frequency is used. The convenience yield series covers the time period from January 6, 1986 to September 19, 2016. If not otherwise specified, all quotes were extracted from Datastream, with the exception of the daily gold lease rates (07/17/2012 - 09/26/2016) which were obtained from Reuters. Due to memory limits and the computational burden in the simulation of the finite sample critical values, all weekly data have been converted to monthly data (by taking averages of the weekly data).

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27 4.2. Results

The intention of the PSY (2015) procedure and the according GSADF statistical test is to as-certain if the sample of interest contains minimum one period of mild explosiveness (Phillips et al., 2015). Figures 4 to 9 display the BSADF sequences of the 6 different gold price series that are analyzed in this section. The results of the SADF and GSADF tests for each gold price series are reported in Tables 1 to 6. The test critical values are computed taking into account the particular size of the sample and the value of r0 which is based on a general rule

(see 4.1.).

For all six different gold price series (nominal & real gold spot prices (deflated by U.S. PPI/ SDR basket), and nominal & real gold futures prices (deflated by U.S. PPI/ SDR basket)) the null hypothesis of no mildly explosive period is being rejected by both the SADF (see A.1 and A.2 in the Appendix as an example) and the GSADF statistic (see Tables 1 to 6). All test statistics are greater than their respective test critical values at the 1% significance level. After having obtained such strong evidence for the presence of mildly explosive episodes, the second stage of the PSY method entails the actual date-stamping procedure via the BSADF sequence. For this purpose, as a starting point a condition that the explosive period has to persist for at least 6 months (minimum duration condition) is imposed. If this minimum duration condition was adjusted it could also be the case that not a single period of explosiveness is being documented. Outcomes vary with different minimum duration conditions. With a smaller minimum duration condition it is likely that a larger number of explosive periods are found.

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28 due to rebalancing portfolios away from more risky investments, e.g. equities, towards seemingly more solid and less volatile assets (e.g. Phillips and Yu, 2011). This way, nervous markets and participants set ground for such price explosions. Nevertheless, to classify the detected periods of mild explosiveness as actual price bubbles, it is necessary to be able to differentiate whether the development arises from the price itself or whether the behavior is inherent to the fundamental value as well. If the fundamental value, e.g. the gold convenience yield or the gold lease rates, which will be discussed in the next section, exhibits the same price jumps as the asset price itself, the seemingly explosive price development cannot be categorized as a bubble. In contrast, if only the price experiences explosiveness, but the fundamental or intrinsic value does not reflect a similar movement, it is likely that a price bubble is occuring (Figuerola-Ferretti and McCrorie, 2016; Phillips et al., 2015).

Fig. 4

Nominal gold spot price BSADF sequence

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29 Table 1

Nominal gold spot price series: The SADF test and the GSADF test

Test Critical Values

Prob. (right-tailed test) Test Stat. 90% 95% 99%

SADF 0.0000 8.139206 1.271682 1.493250 2.114202

GSADF 0.0000 8.139206 2.025789 2.240443 2.927219

BSADF Min. Duration Condition Period No.

6 months 1 1972: M09 - 1975: M07

2 1979: M01 - 1981: M01

3 2005: M10 - 2008: M10

4 2008: M12 - 2013: M06

The finite sample critical values of both the SADF and GSADF test for all three levels are estimated via Monte Carlo simulation with 1000 replications. Lag length is fixed and set to zero. The sample size amounts to 577 and the initial window size is 49 observations. The BSADF sequence has detected four periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 6 months.

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30 Fig. 5

Real (U.S. PPI deflated) gold spot price BSADF sequence

The green line represents the real (deflated by the U.S. PPI) gold spot price (monthly basis). The red (95% criti-cal value sequence) and the blue line (BSADF sequence) depict the detection and subsequent dating of mildly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, there are four exuberant periods discovered. The dated periods of mildly explosive behavior are 1) 09/1972 - 10/1973, 2) 12/1973 - 04/1975, 3) 02/1979 - 01/1981 and 4) 11/2005 - 11/2013.

Table 2

Real (U.S. PPI deflated) gold spot price series: The SADF test and the GSADF test

SADF 0.0000 8.050653 1.270934 1.488544 2.114202

GSADF 0.0000 8.050653 2.025789 2.240443 2.927219

6 months 1 1972: M09 - 1973: M10

2 1973: M12 - 1975: M04

3 1979: M02 - 1981: M01

4 2005: M11 - 2013: M11

The finite sample critical values of both the SADF and GSADF test for all three levels are estimated via Monte Carlo simulation with 1000 replications. Lag length is fixed and set to zero. The sample size amounts to 577 and the initial window size is 49 observations. The BSADF sequence has detected four periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 6 months.

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31 documented from January 1994 on. Consequently, the sample size reduces accordingly. As indicated by the black ellipse in the chart below (see Fig. 6), the date-stamping process identi-fies only one period of mild explosiveness (see also Table 3). At first, this episode appears to be longer compared to the other graph (see Fig. 5). However, this is only due to size distor-tions. Essentially, the dated period is not different from the one of the U.S. PPI deflated gold spot price series.

Fig. 6

Real (SDR basket deflated) gold spot price BSADF sequence

The green line represents the real (deflated by the SDR basket) gold spot price (monthly basis). The red (95% critical value sequence) and the blue line (BSADF sequence) depict the detection and subsequent dating of mild-ly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, the BSADF sequence date-stamps one period of mildly explosive behavior: 10/2005 - 06/2013, highlighted by the black ellipse in the graph.

Table 3

Real (SDR basket deflated) gold spot price series: The SADF test and the GSADF test

SADF 0.0000 5.864008 1.197023 1.406404 1.787506

GSADF 0.0000 5.864008 1.927138 2.119510 2.757886

6 months 1 2005: M10 - 2013: M06

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32 Moving on to the one month futures prices, the BSADF sequence locates two areas of explo-siveness for the nominal gold futures prices. The periods of interest are circled in black in Fig. 7. While the first explosive period from 07/1997 to 03/1998 is different from the ones detect-ed in the spot price series, the second episode from 10/2005 – 10/2013 has been identifidetect-ed in all of the other spot price series as well. Table 4 displays the respective test statistics and ac-cording simulated critical values for the nominal gold futures prices. Overall, the results for the spot price series are comparable to the ones based on the futures price series, which does not come as a surprise since the futures can be seen as a predictor of future spot prices. The additionally detected episode from 07/1997 - 03/1998 was simply a spike not fulfilling the initial minimum duration condition of 6 months in the spot price series.

Fig. 7

Nominal gold futures price BSADF sequence

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33 Table 4

Nominal gold futures price series: The SADF test and the GSADF test

SADF 0.0000 5.011970 1.151136 1.411934 1.889312

GSADF 0.0000 6.910806 2.008177 2.245443 2.874008

6 months 1 1997: M07 - 1998: M03

2 2005: M10 - 2013: M10

The finite sample critical values of both the SADF and GSADF test for all three levels are estimated via Monte Carlo simulation with 1000 replications. Lag length is fixed and set to zero. The sample size amounts to 443 and the initial window size is 42 observations. The BSADF sequence has detected two periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 6 months.

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34 Fig. 8

Real (U.S. PPI deflated) gold futures price BSADF sequence

The green line represents the real (deflated by the U.S. PPI) gold futures price (monthly basis). The red (95% critical value sequence) and the blue line (BSADF sequence) depict the detection and subsequent dating of mild-ly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, there are two exuberant periods discovered. The dated periods of mildly explosive behavior are 1) 07/1997 - 03/1998 and 2) 11/2005 - 11/2013.

Table 5

Real (U.S. PPI deflated) gold futures price series: The SADF test and the GSADF test

SADF 0.0000 5.423712 1.186329 1.485330 2.100944

GSADF 0.0000 7.433933 2.008177 2.245443 2.874008

6 months 1 1997: M07 - 1998: M03

2 2005: M11 - 2013: M11

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35 Fig. 9

Real (SDR basket deflated) gold futures price BSADF sequence

The green line represents the real (deflated by the SDR basket) gold futures price (monthly basis). The red (95% critical value sequence) and the blue line (BSADF sequence) depict the detection and subsequent dating of mild-ly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, the BSADF sequence date-stamps one period of mildly explosive behavior: 10/2005 - 06/2013, highlighted by the black ellipse in the graph.

Table 6

Real (SDR basket deflated) gold futures price series: The SADF test and the GSADF test

SADF 0.0000 5.600777 1.186854 1.400154 1.777366

GSADF 0.0000 5.600777 1.927138 2.119510 2.757886

6 months 1 2005: M10 - 2013: M06

The finite sample critical values of both the SADF and GSADF test for all three levels are estimated via Monte Carlo simulation with 1000 replications. Lag length is fixed and set to zero. The sample size amounts to 273 and the initial window size is 32 observations. The BSADF sequence has identified one period of mildly explosive behavior at the 5% significance level under a minimum duration condition of 6 months.

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36 condition is reduced to 3 months and to 1 month, respectively. The according results are presented in Tables 7 to 12.

In concrete terms, the BSADF sequence identifies three additional periods of mild explosiveness under the imposition of a 3-months minimum duration condition for the basic nominal gold spot price series. Assuming a minimum duration of 1 month, four more sequences are observed (see Table 7). Additional periods that have been added after the particular reduction of the minimum duration condition are highlighted in each table.

Table 7

Nominal gold spot price series: Date-stamping procedure (BSADF) under relaxed min. duration conditions

3 months 1 1972: M09 - 1975: M07 2 1979: M01 - 1981: M01 3 1997: M11 - 1998: M03 4 1999: M06 - 1999: M09 5 2003: M12 - 2004: M04 6 2005: M10 - 2008: M10 7 2008: M12 - 2013: M06

1 month 1 1972: M09 - 1975: M07 2 1979: M01 - 1981: M01 3 1997: M11 - 1998: M03 4 1998: M06 - 1998: M08 5 1999: M06 - 1999: M09 6 2003: M12 - 2004: M04 7 2004: M11 - 2004: M12 8 2005: M10 - 2008: M10 9 2008: M12 - 2013: M06 10 2013: M08 - 2013: M10 11 2014: M03 - 2014: M04

The BSADF sequence has dated seven periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 3 months and eleven episodes after imposing a 1-month minimum duration condition. The periods that have been added after the relaxation to 3 months and 1 month respectively are highlighted.

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37 Table 8

Real (U.S. PPI deflated) gold spot price series: Date-stamping procedure (BSADF) under relaxed min. duration conditions

3 months 1 1972: M09 - 1973: M10 2 1973: M12 - 1975: M04 3 1979: M02 - 1981: M01 4 1997: M11 - 1998: M03 5 1999: M04 - 1999: M09 6 2003: M12 - 2004: M04 7 2005: M11 - 2013: M11

1 month 1 1972: M09 - 1973: M10 2 1973: M12 - 1975: M04 3 1979: M02 - 1981: M01 4 1997: M11 - 1998: M03 5 1998: M06 - 1998: M08 6 1999: M04 - 1999: M09 7 2003: M12 - 2004: M04 8 2004: M11 - 2004: M12 9 2005: M11 - 2013: M11 10 2014: M03 - 2014: M05 11 2014: M07 - 2014: M08

The BSADF sequence has dated seven periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 3 months and eleven episodes after imposing a 1-month minimum duration condition. The periods that have been added after the relaxation to 3 months and 1 month respectively are highlighted.

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38 Table 9

Real (SDR basket deflated) gold spot price series: Date-stamping procedure (BSADF) under relaxed min. dura-tion condidura-tions

3 months 1 1999: M06 - 1999: M09

2 2005: M10 - 2013: M06

1 month 1 1997: M11 - 1998: M01

2 1999: M06 - 1999: M09

3 2003: M01 - 2003: M02

4 2005: M10 - 2013: M06

The BSADF sequence has dated two periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 3 months and four episodes after imposing a 1-month minimum duration condition. The periods that have been added after the relaxation to 3 months and 1 month respectively are highlighted.

Regarding the futures price series, there are five periods in total documented for a 3-months condition and six for a minimum duration condition of four weeks (see Tables 10 and 11).

Table 10

Nominal gold futures price series: Date-stamping procedure (BSADF) under relaxed min. duration conditions

3 months 1 1997: M07 - 1998: M03

2 1998: M06 - 1998: M09

3 1999: M05 - 1999: M09

4 2003: M12 - 2004: M04

5 2005: M10 - 2013: M10

1 month 1 1997: M07 - 1998: M03 2 1998: M06 - 1998: M09 3 1999: M05 - 1999: M09 4 2003: M12 - 2004: M04 5 2004: M11 - 2004: M12 6 2005: M10 - 2013: M10

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39 Table 11

Real (U.S. PPI deflated) gold futures price series: Date-stamping procedure (BSADF) under relaxed min. dura-tion condidura-tions

3 months 1 1997: M07 - 1998: M03

2 1998: M05 - 1998: M09

3 1999: M05 - 1999: M09

4 2003: M11 - 2004: M04

5 2005: M11 - 2013: M11

1 month 1 1997: M07 - 1998: M03 2 1998: M05 - 1998: M09 3 1999: M05 - 1999: M09 4 2003: M11 - 2004: M04 5 2004: M11 - 2004: M12 6 2005: M11 - 2013: M11

The BSADF sequence has dated five periods of mildly explosive behavior at the 5% significance level under a minimum duration condition of 3 months and six episodes after imposing a 1-month minimum duration condition. The periods that have been added after the relaxation to 3 months and 1 month respectively are highlighted.

For the gold futures prices deflated by the currency basket, the date-stamping procedure un-covers two mildly explosive periods for a 3-months minimum duration condition and five episodes respectively under the assumption of at least four weeks of the development (see Table 12).

Table 12

Real (SDR basket deflated) gold futures price series: Date-stamping procedure (BSADF) under relaxed min. duration conditions

3 months 1 1999: M06 - 1999: M09

2 2005: M10 - 2013: M06

1 month 1 1997: M11 - 1998: M01

2 1998: M12 - 1999: M01

3 1999: M06 - 1999: M09

4 2003: M01 - 2003: M02

5 2005: M10 - 2013: M06

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40 Up to this point of the conducted empirical evaluation, the lag length was fixed and set to ze-ro. Furthermore, as part of the process the finite sample critical values were estimated via Monte Carlo simulations, replicated 1,000 times.

In order to check robustness, both the SADF and GSADF tests of the various gold price series are repeated under different conditions and settings. The lag length for both statistical tests is varied and selected automatically by the Akaike Information Criterion (AIC) with the maxi-mum lag number set to 12. The maximaxi-mum lag number is chosen in accordance with the fre-quency of the underlying data which is monthly (Brooks, 2008). For comparison purposes, the standard critical values listed in Phillips et al. (2015) are used.

On the whole, the (new) test results correspond with the initial outcomes. All test statistics for both the SADF and GSADF tests for the six different gold price series are greater than the used standard critical values (see Table 1 in Phillips et al., 2015) at the 1% significance level (see Table 13). All in all, the rejection of the null hypothesis of no mildly explosive period is collectively confirmed.

Table 13

Robustness check: SADF and GSADF test statistics of the different gold price series on the basis of automatic lag length selection via AIC (max. lag number = 12)

Test Stat.

SADF GSADF

Nominal gold spot price 5.675754 7.094006

Real (U.S. PPI deflated) gold spot price 5.455848 7.468793 Real (SDR basket deflated) gold spot price 5.940779 5.940779

Nominal gold futures price 5.881623 6.910806

Real (U.S. PPI deflated) gold futures price 6.094372 7.433933 Real (SDR basket deflated) gold futures price 5.844809 5.844809

The SADF and GSADF statistics are obtained using AIC for automatic lag length selection with a max. lag number of 12. Finite sample critical values are 1.19 (10%), 1.49 (5%) and 2.05 (1%) for the SADF test and 1.92 (10%), 2.20 (5%) and 2.80 (1%) for the GSADF test (see Table 1 in Phillips et al., 2015).

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41 5. Interpretation: Fundamentals versus speculative behavior

5.1.Convenience yield

The last part of the method developed by PSY (2015) addresses the relation to the rational asset pricing model and tries to answer the question if the identified periods of mild explo-siveness can be classified as bubbles (Figuerola-Ferretti and McCrorie, 2016; Phillips et al., 2015). As Figuerola-Ferretti and McCrorie (2016) state, this task heavily depends on having available the right data to create fundamental proxy variables from. When dealing with commodities, this is of particular difficulty, as finding high frequency data for such natural proxies is challenging. For example, data availability is often limited to monthly or quarterly frequency. On the other hand, data featuring relatively high frequency is considered desirable and beneficial for the bubble-dating process (Figuerola-Ferretti and McCrorie, 2016).

Hence, when considering stock markets, the measure that is mainly used is the dividend yield. Since there is no such thing as dividends in storable commodities markets, the convenience yield serves as an equivalent instead (Figuerola-Ferretti & McCrorie, 2016). In general, the term ‘convenience yield’ refers to the implied value of payoff streams originating from inventory holdings. This description includes, for example, any gains associated with having the commoditiy in stock, but also the according costs for doing so (e.g. Figuerola-Ferretti and McCrorie, 2016; Pindyck, 1993). From an equity owner’s perspective, the convenience yield (Fama and French, 1988) can be seen as the premium one is willing to pay for being able to directly access a specified asset stored at a known place (Figuerola-Ferretti & McCrorie, 2016). Following Figuerola-Ferretti and McCrorie’s (2016, p. 731) practical approach, a standard approximization excluding costs for warehousing is applied. The convenience yield cy is therefore then defined as

f5gLL hi,j = ( k)k " , (11)

where St represents the nominal gold spot price and Ft the nominal one month gold futures price, both at time t. Since the convenience yield is calculated on an interest-rate adjusted ba-sis here, rt stands for the one month U.S. Dollar (USD) Intercontinental Exchange (ICE) Lon-don Interbank Offered Rate (LIBOR), weekly frequency.

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42 of the simulation of the critical values in particular, the weekly convenience yield data were converted into monthly data (averages of weekly data).

The results of the SADF and GSADF tests for the gold convenience yield series based on nominal prices are reported in Table 14. Both test statistics are smaller than their respective critical values. This outcome implies that the null hypothesis of no presence of a mildly ex-plosive period cannot be rejected. In other words, it appears that there was no exex-plosive movement in the convenience yield time series. The BSADF sequence displayed in Fig. 10 confirms this conclusion.

Table 14

Gold convenience yield series: The SADF test and the GSADF test

SADF 0.8910 -0.700341 1.156598 1.436970 2.016833

GSADF 0.4940 1.200978 1.973993 2.205061 2.930934

The finite sample critical values of both the SADF and GSADF test for all three levels are estimated via Monte Carlo simulation with 1000 replications. The sample size amounts to 369 and the initial window size is 38 observations.

Fig. 10

Gold convenience yield BSADF sequence

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43 As a consequence, if both types of prices, i.e. spot and futures, share a common discovered period of explosiveness, but this period is not reflected in the convenience yield, the price is suggested to be the sole source of the observed exaggeration. In this case, the convenience yield process is not driving the price explosion. Based on this analysis, it can be concluded that the explosive prices themselves caused a development in which prices depart from the fundamental value of the asset (Ferretti and McCrorie, 2016). Note that Figuerola-Ferretti and McCrorie (2016) also argue in favor of such an explanation. Hence, the above detected periods of explosiveness that both the spot and futures price series share can be interpreted as bubbles. Table 15 shows the different bubble times under different minimum duration conditions for the nominal gold spot and futures prices.

Table 15

Gold price bubbles

Min. Duration Condition Bubble No. Start End

6 months 1 2005: M10 - 2008: M10

2 2008: M12 - 2013: M06

3 months 1 1997: M11 - 1998: M03

2 1999: M06 - 1999: M09 3 2003: M12 - 2004: M04 4 2005: M10 - 2008: M10

5 2008: M12 - 2013: M06

1 month 1 1997: M11 - 1998: M03 2 1998: M06 - 1998: M08 3 1999: M06 - 1999: M09 4 2003: M12 - 2004: M04 5 2004: M11 - 2004: M12 6 2005: M10 - 2008: M10 7 2008: M12 - 2013: M06 8 2013: M08 - 2013: M10

Table 15 displays the periods of mildly explosive behavior that are common both to the nominal gold spot and futures prices that qualify as a bubble in relation to different minimum duration conditions.

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se-44 verely, since those standard values are always based on a set number of observations, which is not necessarily identical to the number of observations of the convenience yield series used here.

Table 16

Robustness check: SADF and GSADF test statistics of the nominal gold convenience yield series on the basis of automatic lag length selection via AIC (max. lag number = 12)

Test Stat.

SADF GSADF

Nominal gold convenience yield 0.190010 2.399401

The SADF and GSADF statistics are obtained using AIC for automatic lag length selection with a max. lag number of 12. Finite sample critical values are 1.19 (10%), 1.49 (5%) and 2.05 (1%) for the SADF test and 1.92 (10%), 2.20 (5%) and 2.80 (1%) for the GSADF test (see Table 1 in Phillips et al., 2015).

5.2.Gold lease rates

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45 Fig. 11

Gold lease rates for 1 month to 12 months maturities

Tables 17 to 21 (Appendix A.3) display the results of both the SADF and the GSADF simulations for the different lease rate series. Regarding the PWY (SADF) procedure, all test statistics are significantly higher than their particular critical test value (right-tail) at the 1% level. This outcome indicates that at least one explosive period exists in each of the analized time series (Figuerola-Ferretti and McCrorie, 2016; Phillips et al., 2015). The GSADF test statistics show a slightly different picture. Only the two longer maturities, 6 months and 12 months lease rates, exhibit significant test statistics at the 1% level (see Tables 20 and 21 in Appendix A.3). In contrast, the 1 month to 3 months lease rates produce GSADF test values that only exceed their respective test critical values at the 5% significance level (see Tables 17 to 19 in Appendix A.3). Under the assumption of a randomnly chosen (e.g. Phillips et al., 2015) minimum duration condition for mildly explosive movements of 6 months, only the BSADF sequence (see Fig. 12) for the 12 months lease rate series succeeds in date stamping a period of mild explosiveness. The detected period of exuberance lasts from 02/2013 to 09/2013, illustrated by the black ellipse in Fig. 12.

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46 be especially crucial to see whether the development has already been present in the years around the global financial crisis (2007/2008). Only then the existing exuberant behavior of both the spot and the futures prices could be classified as a bubble. For this reason, the preferred fundamental used in this analysis is the convenience yield.

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47 Fig. 12

Gold 12 months lease rates BSADF sequence

The green line represents the gold 12 months lease rates. The red and the blue line depict the detection and sub-sequent dating of mildly explosive behavior under PSY method (Phillips et al., 2015). Based on a minimum duration of 6 months, there is one exuberant period discovered from 02/2013 - 09/2013, highlighted by the black ellipse in the chart. The black arrow on the long end points to another potential period of exaggeration (already 5 months fulfilled) if the particular market movement has continued in the same fashion in October 2016.

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48 Fig. 13

Gold 2 months lease rates BSADF sequence

The green line illustrates the gold 2 months lease rates. The red and the blue line depict the detection and subse-quent dating of mildly explosive behavior under PSY method (Phillips et al., 2015). None of the sections where the BSADF sequence (blue line) exceeds the 95% critical value sequence (red line) meets the requirement of a minimum duration of 6 months. At the long end there might be a potential period of exaggeration (already 5 months fulfilled) if the particular market movement has continued unchanged till end of October 2016.

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49 All in all, if gold lease rates can indeed be considered as an adequate representative of the fundamental or intrinsic value of the asset gold (e.g. Lucey and O'Connor, 2013; Barone-Adesi et al., 2014), any mildly explosive movement of the gold price should also be reflected in the fundamental, i.e. the gold lease rates, if there are no no price bubbles (Figuerola-Ferretti and McCrorie, 2016). Figuerola-Ferretti and McCrorie (2016) conclude that this was not the case around 2008. Since the analysis of the lease rates does not further the same development as for the gold price, this phase can be seen as a bubble which is possibly caused by speculative actions beyond fundamental contemplations (Figuerola-Ferretti and McCrorie, 2016).

6. Conclusion