Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 Bio-based and Applied Economics BAE Copyright: © 2024 Esposti, R. Open access, article published by Firenze University Press under CC-BY-4.0 License. Firenze University Press | www.fupress.com/bae Citation: Esposti, R. (2024). Dating com- mon commodity price and inflation shocks with alternative approaches. Bio-based and Applied Economics 13(2): 171-201. doi: 10.36253/bae-14060 Received: December 9, 2022 Accepted: November 2, 2023 Published: July 25, 2024 Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The Author(s) declare(s) no conflict of interest. Editor: Simone Cerroni ORCID RE: 0000-0002-1656-0331 Dating common commodity price and inflation shocks with alternative approaches Roberto Esposti Department of Economics and Social Sciences, Università Politecnica delle Marche, Anco- na, Italy E-mail r.esposti@staff.univpm.it Abstract. This paper investigates the occurrence of common price shocks (co-exceed- ance) across different commodities. IMF monthly price series of 11 commodities are considered over the 1980-2021 period. The analysis considers two alternative stochas- tic processes. The first looks for common volatility clusters using individual GARCH models to detect whether and when respective clusters overlap. Through an appro- priate battery of tests, the second alternative looks for a common Bubble Generating Process (BGP) by searching for individual explosive roots and then dating them to identify the possible overlaps and first movers. Evidence emerging about these shock generating processes is linked to the analogous behaviour of the US Consumer Price Index (CPI) to assess to what extent inflation shocks can be associated to the observed commodity price spikes. Results show that the detection of temporary bubbles and vol- atility clusters only partially agrees on the episodes of exuberance, on the first-moving commodities and on the involvement of the CPI. This provides helpful suggestions on the development of a real-time surveillance tool supporting policy intervention in peri- ods of commodity price turbulence. Keywords: commodity prices, price volatility, explosive roots, GARCH models. JEL Codes: Q11, C32. 1. INTRODUCTION The large and rapid surge of most commodity prices that started in 2021 and lasted for the whole of 2022 points to two stylised facts that have been repeatedly investigated in previous episodes of price spikes: commodity prices move together; the rise of commodity prices transmits, somehow, to the Consumer Price Index (CPI). The consequent inflation rate rush largely impacts economies and societies and usually induces a quite vigorous pol- icy response (Ider et al., 2023). Nonetheless, the explanations of these price dynamics are still to be fully understood. The literature on the common movement (or co-movement) of commodity prices is vast (Byrne et al., 2020). One limit of this literature is that it implic- itly assumes that the communality of price dynamics has to be intended as the existence of a common Data Generation Process (DGP), usually represented https://doi.org/10.36253/bae-14060 http://www.fupress.com/bae https://doi.org/10.36253/bae-14060 https://orcid.org/0000-0002-1656-0331 mailto:r.esposti@staff.univpm.it 172 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 via some variant of Vector Auto-Regression (VAR) or Vector Error Correction (VEC) models or through more sophisticated representation of the underlying common drivers (for instance, common latent factors) (Esposti, 2021). But this may contrast with empirical evidence that suggests substantially different fundamentals across very diverse commodities, thus questioning the presence of common real determinants to justify commonality. In general terms, most representations of the common DGP and of the consequent price transmission process (like the conventional Granger causality, for instance) may be too simplistic to capture the real underlying interdependence across commodities, if any, thus providing misleading evi- dence on the actual causal linkages. However, a specific strand of the recent empirical lit- erature stresses that a common DGP is not strictly need- ed for a common temporary behaviour to be observed (Zhao et al., 2021, p. 781; Mutascu et al., 2022). In par- ticular, commonality may only occur within the periods of exuberance, also referred to as co-exceedance. When the price spike expires each series reverts to its own (pos- sibly different) normal-time DGP. This hypothesis can be also transferred to the second stylised fact, that of the CPI response: for a transmission of shocks to the CPI to occur we do not need a common DGP with the com- modity prices, but only some co-exceedance with them. The presents paper aims to contribute to this body of studies by proposing an original methodological approach which then leads to a novel policy tool. The main originality of the approach consists in juxtapos- ing and combining two alternative stochastic processes generating co-exceedance. The first resides in the occur- rence of common (but not interdependent, that is, multi- variate) volatility clusters whose behaviour is here mod- elled through appropriate Generalised Auto-Regressive Conditional Heteroskedasticity (GARCH) models. The second consists in the occurrence of common bubbles (a common Bubble Generating Process, BGP), that is, tem- porary explosive roots within the individual series but whose timing largely corresponds across commodities. Individual price series of very diverse commodities are thus separately investigated in order to assess whether and when volatility clusters (first) and temporary explo- sive roots (second) are found. Although these methodo- logical approaches have already been adopted in previ- ous empirical studies (Otero and Baum, 2021; Phillips and Shi, 2020; Zhao et al., 2021), this paper proposes a combination of these techniques to assess the co-exceed- ance of commodity prices without relying on some arbi- trary and unreliable common DGP. Monthly series of 11 commodity prices and the respective price indexes released by the International Monetary Fund (IMF) over the 1980-2021 period are considered. Co-exceedance is assessed by confront- ing the occurrence of these events across series. If some overlapping is observed, it supports the existence of some contagion (or transmission) across prices. The sequence of the events across prices can finally suggest the direction of this possible contagion. The same analy- sis is then repeated on the US CPI. The interest for this methodological approach even- tually lies in its application to design a suitable policy tool. Instead of concentrating on complex and possibly misleading causation processes, the proposed empirical strategy aims to identify when periods of rapid price ris- es occur and assesses whether they are common across commodities. Therefore, it allows to develop a real-time surveillance tool guiding a prompt policy response in the right direction, in particular by distinguishing inter- ventions that can be confined to the sectoral context from interventions that require an economy-wide spec- trum of actions. In order to be easily interpretable also by non-technical users, this tool is aimed to transfer results into a sort of periodically updatable dashboard visualizing the critical information under investigation: if a bubble is occurring for a given commodity, when it started, whether other commodities are involved by the same bubble, who moved first and, finally, if and to what extend this price surge is also reflected in the CPI. Con- tributing to the definition of such a policy tool repre- sents a further objective of the present study. The rest of the paper is structured as follows. Section 3 overviews the recent empirical literature in the field while Section 3 presents the adopted dataset and the main stylised facts. Section 4 details the adopted meth- odological approach, the results of which are illustrated in section 5. In Section 6 these results are discussed and juxtaposed with the evidence emerging from more con- ventional methodologies about the investigation of com- modity price dynamics. Section 7 draws some policy implications and concludes. 2. THE COMMON MOVEMENT OF COMMODITY PRICES: LITERATURE AND EVIDENCE The paper by Wang and Tomek (2007) may represent the first study that explicitly and extensively discussed the sequence of empirical issues to be tackled in inves- tigating the actual DGP of commodity prices. Though their main attention was on the stationarity proper- ties of agricultural commodities, their conclusions can be extended to other commodities and properties of the unknown DGP. The main argument is that, due to 173Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 their market fundamentals (on both the supply and the demand side), commodity prices are expected to be mean reverting, with the long-term mean value possibly mov- ing along a deterministic trend. So, prices are expected to follow a stationary DGP around a drift or a trend. The fact that in the empirical literature the pres- ence of a unit root is only occasionally rejected has to be attributed to the characteristics of the respective tests and/or to their misspecification. In particular, other characteristics of a stationary DGP can make it simi- lar to a unit-root process. One is that these prices often show long memory (that is, fractional integration) mak- ing it possible for a close-to-(but-lower-than)-one root to be confounded with a unit root. Another is the presence of a structural break that may shift the long-term value upward or downward and can itself generate a potential confusion as evidence of nonstationarity: the presence of a structural break within a stationary series may lead to accepting the presence of a unit root, thus wrongly con- cluding that the series is non-stationary (Baum, 2005; Glynn et al., 2007). A consistent body of recent studies concentrates on several different stochastic processes to explain the com- plex (i.e., non-linear) commodity price dynamics and the possible underlying co-movement. They are, in particu- lar, fractional integration and structural breaks. A recent example, though concerning stock market indices and not commodity prices is Caporale et al. (2020). Based on an approach originally proposed by Cuestas and Gil- Alana (2016), they argue that fractional integration is very much related to non-linearities.1 The possibility of structural breaks is also considered since many stud- ies argue that fractional integration might be artificial- ly generated by the presence of breaks in the data that have not been taken into account. In fact, the presence of structural breaks within commodity price series was already considered by Wang and Tomek (2007). However, it must be noticed that fractional integra- tion and/or structural breaks can hardly explain the behaviour of commodity prices and, in particular, the abovementioned co-exceedance, that is, their recurrent episodes of temporary exuberance as also emerging by simple visual inspection (see next section). They remain interesting and possibly relevant processes in the inves- tigation of individual DGP since they may significantly interfere with the investigation of temporary bubbles and/or GARCH effects. Therefore, although the approach here adopted considers other DGPs, the presence of structural breaks can not be excluded at least for some of 1 Another interesting strand of empirical literature on commodity price dynamics, and strongly linked to non-linearities and fractional integra- tion, consists in the so-called fractal approach (Cromwell et al., 2000). these commodities (Esposti, 2021) and will be considered here for comparative purposes (see Section 6). Concentrating on the stationarity properties these studies overlook another major characteristic of these price series that clearly emerges from a simple visu- al inspection: the presence of temporary exuberance. Therefore, their DGP is expected to also generate self- extinguishing periods of particularly high or low values. Most of the literature in the last 15 years has essentially focused on this issue also as a consequence of the 2007- 2008 price spike and of the following turbulent period. A lot of theoretical and empirical research has tried to investigate the origins of these price nonlinearities, jumps and spikes, as well as to put forward testing pro- cedures to assess their presence. We can summarize this research effort in three main directions and, then, in their possible combination. The first strand of research explains the observed price spikes and jumps as the consequence of a tempo- rary increase in their variability (or volatility). It is the formation of volatility clusters that eventually generates the observed highly irregular price dynamics. In most applications, this idea is implemented by specifying and estimating GARCH regression models possibly admit- ting asymmetric effects and non-stationary processes for the price level. See Li et al. (2017), Baur and Dimpfl (2018) and Esposti (2021), just to mention a few, for the application of different variants of GARCH modelling to commodity prices. Within the second body of studies the origin of the episodes of price turbulence is the formation of tempo- rary bubbles. Several tests have been originally proposed to detect temporary price bubbles within mean-revert- ing, thus stationary, processes (Gürkaynak, 2008). More recently, the presence of temporary bubbles has been admitted, and tested, within possibly non-stationary processes, that is, as temporary explosive roots emerg- ing within unit-root processes (Phillips et al., 2011, 2015; Phillips and Shi, 2020). Gharib et al. (2021) and Zhao et al. (2021) have recently used this battery of tests to assess the co-exceedance of some commodity prices and to date the respective bubbles. Co-existence of both processes is also possible. This is considered helpful for two complementary reasons. On the one hand, as already anticipated, it is always dif- ficult to clearly distinguish between the outcome of these two processes (Gürkaynak 2008, pp. 182-183; Chang, 2012). On the other hand, none of the two alterna- tive processes may totally capture all the features of the observed price dynamics. To reconcile these two alter- native processes, Chang (2012) adopts an Autoregressive Jump-Intensity(ARJI)-GARCH model. Originally pro- 174 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 posed by Chan and Maheu (2002), this model is in prin- ciple able to generate both temporary bubbles and vola- tility clusters within stationary processes. The third strand of empirical research in the field differs from the conventional time-series approaches as it is grounded on the spectral analysis and in time–fre- quency approaches. For the evolution of market prices, wavelet analysis has emerged as a useful and power- ful tool in assessing price co-movement cycles. With- out resorting on any theoretical causation (price trans- mission) process, it allows to explore how the series of prices are related at different frequencies admitting non- linearities like structural breaks.2 Mutascu et al. (2022) provide a valuable example of this kind of approach by investigating the co-movements of gasoline and die- sel prices in different countries at different frequencies. Though this approach is relatively new, interesting and promising, it is still based on the assumption of a per- manent interdependence between prices although flex- ible and not-linear. In the present study, as anticipated, we do not want to admit any persistent co-movement but only co-exceedance, therefore prices moving togeth- er only in specific periods of price spikes. Nonetheless, the combination of the co-exceedance analysis here pro- posed with wavelet analysis can open interesting devel- opments for future research in this area. Here, the aim is to investigate the commodity price dynamics following the first two relative recent strands of research by pointing to commodity price co-exceed- ance rather than co-movement. In particular, unlike Chang (2012) and Zhao et al. (2021) the objective is not to estimate the parameters of the actual DGP but to date the episodes of price turbulence by confronting, in this respect, two competing processes: GARCH within stationary processes (volatility clusters) and temporary explosive roots within non-stationary processes (bub- bles). Moreover, unlike Zhao et al. (2021) here we do not adopt Granger causality testing to assess the direction of the possible transmission of the price shocks across commodities.3 By dating these periods individually, we provide evidence on this transmission by solely juxta- posing the timing of the individual episodes. This is done not only on commodity prices and price indexes but also on the CPI series. While the empirical literature on the commodity price properties and behav- iour is vast and follows the abovementioned directions, 2 We wish to thank an anonymous reviewer for helpful suggestions on this aspect. 3 Granger causality tests imply a common linear DGP across series (VAR or VEC models) (Zhao et al., 2021, p. 783). But both commonal- ity and linearity may not hold in the present case. Nonetheless, for the sake of comparison and robustness check of results, in Section 6 we will present Granger causality tests. the investigation of the CPI dynamics (and its growth rate, the inflation rate) mostly follows other directions. It mainly concentrates on the common movement and possible interdependence with other macroeconomic variables and is only occasionally connected to com- modity prices (Garzón and Hierro, 2022; Ider et al., 2023). GARCH effects possibly occurring in the CPI or inflation rate series has been extensively analysed (Engle, 1982), but we are not aware of studies assessing the pres- ence of temporary bubbles within these series. In fact, visual inspection seems to suggest quite different prop- erties of CPI compared to commodity prices (see next section). Nonetheless, if a transmission from commodity prices to CPI is expected, especially in periods of price turbulence, this should imply some form of co-exceed- ance between these series. But there is a final original aspect of the present contribution with respect to the recent literature in the field. It concerns the policy implications of the proposed empirical approach. In previous studies either these implications are overlooked or they concentrate on the possible effect of policy interventions on the nature and scope of commodity price co-movement or co-exceed- ance like, for instance, the fuel tax system (Mutascu et al., 2022) or import tariffs (Esposti and Listorti, 2018). If the main objective of a policy in this context is to minimize the negative impact of a generalized rise of commodity prices, knowing the possible underlying causation and transmission process, that is the struc- tural linkages generating co-movement, might not be so critical. What seems important is rather a quick under- standing that a price “bubble” is forming and whether or not it is just sectoral (so it involves a limited number of commodities) or it is generalized across all markets, that is, it is a co-exceedence. Sectoral interventions to neutralize a momentary price surge are present in many contexts and are usually rapidly activated (in the case of agricultural commodities, for instance, the agricul- tural market-crisis interventions represent an interesting example (FAO et al., 2011)). When occurring on first- moving prices, these prompt sectoral responses may help to prevent a generalized “bubble”. Understanding if and when this latter is, in fact, occurring then becomes criti- cal to promptly activate system-wide actions, particular- ly intended to prevent or slow-down downstream impact on inflation rate surges (Ider et al., 2023). This real-time surveillance tool able to provide such an early warn- ing, as well as the generality and the first movers of the “bubble”, seems to be particularly helpful for a prompt policy response. 175Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 3. PRICE SERIES UNDER SCRUTINY The present analysis concerns the price of a selec- tion of 11 commodities belonging to three different cat- egories: 4 agriculture commodities (corn, wheat, soy- bean, beef); 3 energy commodities (crude oil, natural gas, coal); 4 metals (aluminium, copper, zinc; nickel).4 All price series are taken from the IMF commodity price dataset.5 All prices are monthly and cover the period January 1980 (1980M1)-December 2021(2021M12) (504 observations) with the only exception of natural gas whose series starts in 1985M1 (444 observations). Together with individual commodity prices, the IMF dataset also contains aggregate price indexes for groups of commodities. Here, three monthly price indexes are considered: food price index (FoodInd) covering the period 1991M1-2021M12; metals price index (MetInd) covering the period 1980M1-2021M12; fuel (energy) index (EneInd) covering the period 1992M1-2021M12. Annex 1 provides details about which product quality these prices refer to, where they have been collected and on which aggregates respective indexes have been defined. Table A1 also reports the respective descriptive statistics which include the con- ventional distributional indices suggesting that com- modity prices depart from the normal distribution mostly for a longer right tail depending on the excep- tionally high prices observed during temporary bubbles. The dynamics of commodity prices is investigated in combination with the evolution of the overall consum- er price index (CPI). Unfortunately, no worldwide (or global) CPI is available. Moreover, many available CPI are usually collected and released at a quarterly or year- ly basis. Here, the US monthly CPI series is used (see Annex 1 for more details).6 This series seems suitable in the present analysis not only for the concordant frequen- cy, but also because the US still represents the largest economy worldwide, so any impact of the global com- modity prices on inflation can be consistently assessed on this series. It must also be noticed that, as detailed in Annex 1, several price series concern US markets and, in any case, all prices are expressed in US $. Therefore, using the US CPI does not incur the risk of downscal- 4 Selected commodities are the most important worldwide (in terms of value) within the respective categories. In fact, nickel is the fifth in the list of metals after lead. But for this latter a sufficiently long series is not available. 5 These price series are proprietary and can not be made available with- in the paper’s material. However, they can be freely downloaded at htt- ps://data.imf.org/?sk=471DDDF8-D8A7-499A-81BA-5B332C01F8B9 or requested at https://www.imf.org/en/Research/commodity-prices. 6 This data can be freely downloaded from https://fred.stlouisfed.org/ series/CPIAUCSL. ing (if not neutralizing) the transmission of commodity price shocks to the CPI due to the exchange rate adjust- ment (Garzón and Hierro, 2022). Unlike many previous studies (Esposti, 2021), com- modity prices, as well the three price indexes, are not deflated. As here we want to investigate the possible impact of commodity price spikes on the CPI, it does not seem appropriate to purge inf lation from these series. The same strategy is followed for the possible presence of seasonality: no seasonal adjustment is per- formed on price series and indexes. The logic behind this choice is twofold. On the one hand, we prefer to analyse the price series that economic agents really con- front with. On the other hand, as stressed by Wang and Tomek (2007) and Corradi and Swanson (2006), any data transformation has to be taken with care as it could introduce artefacts within the series under investigation. However, we consider as appropriate a data trans- formation that is supported by the theory (Corradi and Swanson, 2006, p. 222). This is the case of the logarith- mic transformation of the price levels. This transforma- tion is largely used in empirical literature (Listorti and Esposti, 2012; Esposti and Listorti, 2013) and has two main motivations. First of all, price logarithms are more likely to show a normal distribution than price levels, and normality is usually required by the estimation and inference approaches. In other words, the log-normal statistical distribution of price levels has to be consid- ered as a main regular feature of these series (Listorti and Esposti, 2012; Esposti and Listorti, 2013). Secondly, the logarithmic transformation finds a robust theoretical justification in deriving the commod- ity price dynamics as Geometric Brownian Motions (GBM) (Diba and Grossman, 1988; Gürkaynak, 2008; Su et al., 2017). This tradition also includes the idea of “rational bubbles”, that is, periods of price exuberance entirely justified by agent’s expectations about commod- ity fundamentals (Diba and Grossman, 1988). Empiri- cally, this hypothesis implies that price logarithms might take the form of mean reverting processes (due to mar- ket fundamentals) plus a random walk, a mean-reverting non-constant volatility (GARCH) and, possibly, tempo- rary explosive roots.7 According to Ibrahim et al. (2021), a GBM can generate a stochastic process that assumes normally distributed price level growth rates (therefore, difference in the logarithms) while admitting both unit- root (with drift and/or deterministic trend) and GARCH effects (volatility clusters).8 However, these recent studies 7 Actually, Diba and Grossman (1988) exclude that, within this logic, a rational bubble can actually start: if it is observed it must always have existed. 8 See also Agustini et al. (2018) for a similar derivation. https://data.imf.org/?sk=471DDDF8-D8A7-499A-81BA-5B332C01F8B9 https://data.imf.org/?sk=471DDDF8-D8A7-499A-81BA-5B332C01F8B9 https://www.imf.org/en/Research/commodity-prices https://fred.stlouisfed.org/series/CPIAUCSL https://fred.stlouisfed.org/series/CPIAUCSL 176 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 do not admit temporary bubbles. Taking into account pros and cons of the logarithmic transformation (Cor- radi and Swanson, 2006; Wang and Tomek, 2007), the present paper considers both the price levels and the log- arithm of price levels and in parallel repeats the analysis for these two cases in order to assess which results are robust across the transformation. Annex 2 displays the time evolution of the three aggregate price indexes (Figure A1), the 11 individual commodity prices (Figures A2-A4) and the logarithms of these individual prices (Figures A5-A7) over the 1980M1-2021M12 period.9 Visual inspection points to some general characteristics of the price dynamics. With- in each group, commodity prices seem to show some common movements: periods of exuberance as well as collapses substantially correspond across different com- modities. This is only partially confirmed across groups: metals and agricultural commodities tend to share the same periods of rise and fall, while energy commodity prices seem more stable and less volatile at least until the very last years of the period under consideration. Howev- er, if aggregate price indexes instead of individual series are considered, it emerges that the three series largely overlap with a substantial correspondence of positive and negative spikes. What is common across commodities is also that price turbulence seems to sharply increase in the second half of the period under consideration and, in particular, from 2005 onwards. From this simple visual inspection, therefore, the hypothesis of common movement seems largely sup- ported. For all commodities, periods of temporary exu- berance are recurrently observed. During these periods, prices rapidly increase and then rapidly collapse to a level that does not differ much from the pre-exuberance level. Therefore, despite these “bubbles”, prices still seem to behave like mean-reverting processes. This does not exclude changes in the long-term mean level or a long- term trend in this respect (Esposti, 2021). But these changes or trends seem mild and are overshadowed by the large short-term instability. As could be expected, the logarithmic transformation does not change the gen- eral behaviour of the series. Qualitatively, price levels and their logarithms are similar even though the latter are obviously smoother and this seems particularly evi- dent for the energy commodity prices. At the same time, major differences emerge between commodity price series and the CPI series. Figure A8 (Annex 2) reports the CPI, its monthly growth rate (i.e., 9 The logarithmic transformation is not considered here for the price indexes and CPI. It would rather require a different aggregation of the elementary prices into the index and this would simply generate anoth- er kind of index possibly introducing a further artefact. the inflation rate) together with the oil price which argu- ably is one of its major drivers, but it is also one of the most stable commodity prices. The difference is evident. Oil price seems to follow a mean reverting process possi- bly with an increase of volatility in the second part of the period and an upward shift of the long-term mean value. CPI is much more stable, also in the second half of the period, and apparently moves along a deterministic trend. It follows that the inflation rate seems to behave like a mean-reverting process around an almost-zero long-term value with a limited, though appreciable, increase in the variability in the second half of the period. This purely visual inspection gives rise to the two key research questions underlying the present study. On the one hand, commodity prices seem to move together at least during periods of turbulence, but this would suggest a common stochastic process whose properties, however, are not self-evident. Most price series show some char- acteristics of mean-reverting processes, and this would indicate they are stationary processes around drifts or trends. But the large and quick shocks, though tempo- rary, do not seem consistent with this kind of processes. There should be some other underlying stochastic pro- cess, that may differ across prices but still admits their common movement at least in the periods of turbulence. On the other hand, the research challenge about the linkage between commodity prices and the CPI is quite the opposite. They apparently behave as very different stochastic processes, so commonality should be exclud- ed. Nonetheless, strong economic arguments, as well as an abundant empirical evidence (Garzón and Hierro, 2022), suggest that a common movement of many criti- cal commodity prices has to be transferred, somehow, to the CPI. 4. THE METHODOLOGICAL APPROACH The common theoretical framework of the investiga- tion of commodity price dynamics consists in price for- mation mechanisms (or equations), that is, reduced-form models expressing the respective underlying market equilibrium.10 Price formation equations represent the dynamic stochastic process as a mean-reverting or non- stationary process eventually generating the price level and volatility. These reduced form models have the fur- ther advantage of allowing a compact representation of cross-commodity price dynamics in the form of multiple 10 Fackler and Goodwin (2001) provide a common template based on linear excess demand functions embracing all dynamic regression mod- els from which an estimable reduced-form model can eventually be derived. 177Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 simultaneous equation models that may explain both co- movement and co-exceedance. The theoretical justification of these cross-commod- ity price transmission mechanisms, however, is not uni- vocal. The prevalent explanation is that also very differ- ent commodities (for instance oil and corn) may display interdependence in the respective fundamentals (i.e., demand and supply). For instance, on the supply side, one commodity (e.g., oil) may enter as an input (thus, a cost) in the production process or supply chain of anoth- er commodity (e.g., corn and, consequently, beef). On the demand side, consumption of one commodity may be directly (through substitution effect) or indirectly (through income effect) affected by the price of another commodity (Dawson et al., 2006; Listorti and Esposti, 2012; Esposti and Listorti, 2013). Sometimes, however, this interdependence through the fundamentals can be so indirect and remote that it seems more reason- able to provide another theoretical justification of price co-movement and co-exceedance: though prices are not interdependent, they still all respond to the same under- lying (often latent) common factors (Stigler, 2011; Byrne et al., 2020; Esposti, 2021). The research question underlying the present study, however, comes before these theoretical representations of price interdependence, that is nature and forms of price co-movement and co-exceedance. It rather looks for empirical support on the evidence of co-exceedance, its possible temporary nature and its dating. Therefore, we work on univariate models and not on multivariate models. On these premises, consider N commodities whose price is observed over T time periods (months in the present case). On the basis of rational agent’s expectation or efficient markets theory (Zhao et al., 2021), assume that for any i-th commodity there exists an unobserved fundamental price depending on the real market driv- ers (supply, demand, storage, expectations). The natural constraints applying to these drivers should make this market fundamental price nonexplosive. The actual (i.e. observed) price moves around this fundamental level but it usually deviates from it according to some underlying stochastic DGP expressed by the following univariate price formation equation: pit = αi + δit + bipit-1 + uit, ∀i∈N, ∀t∈T S 0, the price series has an explosive root implying a perma- nent and progressive departure from the fundamental price level unless it is temporary (a “bubble”). In prac- tice, such process would contradict the actual existence of a fundamental price level. Based on (2), distinct DGPs can be considered to represent the observed deviation of prices from the alleged fundamental level. Firstly, a Generalized Autore- gressive Conditional Heteroskedasticity effect on εit can be included to capture the presence and persistence of volatility clusters. This is obtained by reformulating (2) as follows (GARCH(p,q)) regression model): ∆pit = αi + δit + βipit-1 + θis∆pit-s + εit = γi + ρip -pεit + ωiq -q, ∀i∈N,∀t∈T S,P,QT (4) where r1 and r2 denote the starting and ending points, respectively, of the possible temporary bubble. r1 and r2 are expressed as fractions of T so that r2 = r1 + rW, where rW is the window size of the regression, also expressed as a fraction of T. The number of observations to estimate (4) is TW = [TrW], where [·] is the floor function which gives the integer part of the argument (Otero and Baum, 2021). For series showing temporary bubbles we should observe explosive roots for some sub-periods, that is, some [r1,r2] interval. This can be assessed through tests where the null hypothesis is H0: = 0, implying that the series shows a unit root, against the alternative hypothesis H0: > 0, implying that the series shows an explosive root in the [r1,r2] interval. A key contribution to a consistent formulation and implementation of this kind of tests was originally made by Phillips et al. (2011), then improved by Phillips et al. (2015) and Phillips and Shi (2020). The basic ver- sion of the test is the right-tailed ADF statistic based on 11 This is also called Integrated GARCH (IGARCH) process/model (Campbell et al., 1996; Chan, 2010). 12 Although their validity in generating reliable predictions is largely questioned, ARCH/GARCH models are usually quite successful in gen- erating in-sample projections (Taleb, 2009). the full range of observations, r1 = 0 and r2 = 1 (i.e., rW = 1), denoted . As it applies to the whole period of observations, this statistic may fail in detect short-time temporary bubbles. Therefore, a second statistic is based on the supremum t-statistic (SADF) that results from a forward recursive estimation of (4): SADF(r0) = (5) Also this statistic may fail in the case of multiple temporary bubbles within the series. A third statistic can be thus computed. It is the generalised supremum ADF (GSADF) test: GSADF(r0) = (6) Based on these statistics, it is firstly possible to ass- es if one or more temporary bubbles occur. Secondly, a backward testing procedure (backward SADF, or BSADF, statistics) allows dating these bubbles over the period T (Phillips et al., 2011; 2015). For any particular observation, i.e. the i-th commodity observed at time r2, it is possible to test whether it belongs to a phase of explosive behav- iour by performing a SADF test on a sample sequence where the endpoint is fixed at time r2, and expands back- wards to the starting point, r1, which varies between 0 and (r2 − r0). This backward SADF statistic is defined as: BSADFr2(r0) = (7) A further refinement of these tests has been recent- ly proposed by Phillips and Shi (2020) and takes into account both the presence of heteroskedasticity and the multiplicity issue in recursive testing. They thus recom- mend a wild bootstrap approach to compute the critical values of the abovementioned tests.13 The methodological approach followed here can thus be summarised as follows. Firstly, we look for the stochastic properties of the individual commodity price series and the CPI. In particular, the presence of a unit- root (with or without a drift or a trend) and of ARCH effects is investigated. Secondly, on the basis of the first- step evidence, GARCH effects are considered as the pos- sible explanation of the observed periods of price turbu- lence. GARCH regression models like (3) are estimated on individual series and in-sample volatility predictions are generated to assess and date the volatility clusters. 13 One limit of these tests is that they do not allow breaks in levels or time trends. As discussed, neither a trend nor a structural break can explain by itself the observed irregular price behaviour. However, they can not be excluded at least from some commodities (see Table 1) and might affect both the statistics and the critical values of these explosive root tests. 179Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 Thirdly, as an alternative to GARCH processes, we consider the formation of temporary bubbles as expressed by (4), therefore as a momentary departure from the fundamental process either stationary or non- stationary. , SADF and GSADF tests are performed on individual series and the temporary bubbles, if any, are consequently dated by performing the BSADF test. Finally, the beginning and the end of volatility clusters and of temporary bubbles are confronted both across the two alternative processes and among commodities (and CPI) in order to assess similarities and differences, as well as the presence of possible contagion effects. Thus the analysis of co-exceedance simply consists in seeing whether volatility clusters or bubbles are common (i.e., overlap) or not. In case of a positive answer, it is then legitimate to ask, and to assess, whether a contagion effect can be deduced, that is, which series (i.e. price) moves first possibly driving the movement of the others. Clearly, this investigation can not be confused with a formal causality assessment or testing. Usual time- series causality assessment in a multivariate context is performed via Granger causality testing. This latter, however, assumes a linear relationship across commodi- ties and does not seem consistent with the observed stochastic properties of these series and bubble forma- tion. In this respect, some recent developments in the field seem promising for future research (Shahzad et al., 2021; Esposti, 2022). It is worth stressing, however, that assessing causality is not so essential for the main policy implication of interest here. Investigating which commodities show a bubble formation earlier than oth- ers remains useful to build that real-time warning policy tool mentioned in previous sections. 5. RESULTS14 5.1. Stochastic properties of the series Table 1 reports the battery of unit root tests and of the ARCH tests on (2) for all series under investigation. In the case of price indexes, CPI included, it emerges that all series are stationary. The selected specification15 includes a drift in the case of the three commodity price 14 All testing and estimation procedures have been performed with soft- ware STATA 17. In particular: GARCH models have been estimated using the command Arch with arch(1) garch(1) specification; explosive roots have been tested using the Radf command; the structural break tests have been performed using commands Zandrews and Clem; pair- wise Granger causality tests have been performed by using, the Var and Vargranger commands. 15 The best specification has been selected following Enders (1995, p. 256-260). indexes and a trend in the case of CPI. At the same, all indexes here show an ARCH effect except FoodInd. Con- sequently, all indexes behave as mean-reverting processes (with the mean moving along a deterministic trend in the case of CPI) possibly with volatility clustering. Table 1. Unit root (ADF) and conditional heteroskedasticity (ARCH) tests on commodity price indexes and commodity prices (1980M1-2021M12)a. Series ADFb (w/o drift&trend) ADF (with drift) ADF (with trend) ARCHc Price indexes FoodIndd 0.475 -1.224† -2.439 0.126 MetInd -0.088 -1.404† -2.721 82.74* EneInde 0.150 -0.915† -3.138 106.9* CPI 6.988 0.539 -2.492† 72.91* Price levels Oil -0.502 -1.594† -2.906 92.63* Coal -0.327 -1.599† -3.415 160.9* Gasf 0.581 -0.409† -2.249 228.6* Aluminium -0.416 -3.063* -4.010†* 65.84* Copper 0.522 -0.621 -2.487† 112.8* Zinc -0.229 -2.023†* -3.804* 106.9* Nickel -1.041 -2.628†* -3.368 84.09* Wheat -0.375 -2.816†* -3.349 108.9* Corn -0.214 -1.872†* -2.891 29.86* Soy -0.322 -2.089†* -3.288 67.44* Beef 1.004 0.049 -1.739† 64.76* Logarithm of price levels Oil -1.199 -1.202† -2.631 78.04* Coal 0.429 -1.501† -2.919 79.51* Gasf -1.737 -1.445† -2.647 68.6* Aluminium 0.178 -3.169†* -4.322* 65.86* Copper 0.784 -0.858† -2.710 40.29* Zinc 0.693 -1.827†* -3.478* 19.18 Nickel 0.354 -2.008†* -3.102 19.62 Wheat 0.252 -2.410† -3.045 28.32* Corn 0.315 -1.755†* -2.855 7.38 Soy 0.212 -2.032†* -3.243 24.67* Beef 0.887 -0.167† -1.839 42.35* *Statistically significant at 5% confidence level. † Selected specification according to Enders (1995, p. 256-260). a The test specification in terms of lags included has been selected case by case on the basis of the AIC. b 5% Critical Value of the three ADF test specifications, respectively: -1.95; -1.65; -3.42. c Lagrange Multiplier (LM) test performed on the residuals of the ADF unit-root test equations; 5% Critical Value: 21.03. d 1991M1-2021M12. e 1992M1-2021M12. f 1985M1-2021M12. 180 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 Regarding the individual commodity price series, however, a differentiated picture emerges across the commodity groups. If the levels are considered, energy commodities are all stationary around a drift. Metals, on the contrary, show non-stationarity around a drift in the case of zinc and nickel, non-stationarity around a trend in the case of aluminium and stationarity in the case of copper. Finally, all agricultural commodities, except for beef, are non-stationary around a drift while beef is sta- tionary around a deterministic trend. Despite these dif- ference, all commodity prices show an ARCH effect. Interestingly enough, the logarithmic transforma- tion changes the evidence emerging from the tests only for four commodities and only in one case (wheat) does this change concern stationarity properties. aluminium remains non-stationarity but now around a drift. Also copper and beef downscale from a trend to a drift while maintaining stationarity. Wheat shows the most signifi- cant change passing from non-stationarity around a drift to a stationarity around a drift. Thus, unlike the respec- tive price level, the logarithm of the wheat price seems to behave like a mean-reverting process. The key point, here, is that while visual inspection of both price indexes and price series would indicate some common movement, tests indicate that such common- ality may occur for price indexes but not for individual prices where four different DGPs are observed, and this happens also within the same commodity group. This makes the hypothesis of common movement hardly ten- able, at least over the whole time period. At the same time, however, visual inspection also reveals the pres- ence of common periods of exuberance that are not necessarily compatible with the DGPs emerging from tests. The limited reliability of the DGPs emerging from the tests when compared to the actual price dynamics is confirmed by generating in-sample predictions from the estimated ADF regressions. Figure 1 compares these predictions with the real series for two cases that should express different DGPs: a stationary series around a drift (mean reverting) (oil) Figure 1. Oil (left scale) and Aluminium (right scale) prices: observed series and in-sample predicted series from respective ADF model estimation (1980M1-2021M12) (see Annex 1 for units of measure). 181Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 and a non-stationary series around a trend (aluminium). The two predicted series are quite similar, despite the dif- ferent DGPs, and, above all, in both cases these predic- tions largely diverge from the actual series especially in the last third of the observed period. Evidently, there is something more in the stochastic process generating these series and this has to do more with temporary effects than with constant properties of the series. As the ARCH test is concordant across all series (except for FoodInd), the presence of volatility clusters can be a serious candidate to explain these temporary processes. But also temporary explosive roots (bubbles) could be considered as they are compatible with both stationary and non-stationary series over the whole period (Diba and Grossman 1988, p. 529). 5.2. Volatility clusters Table 2 reports the estimates of parameters ρ and ω of the GARCH regression model (3) (with a GARCH(1,1) specification) for the different series in both price lev- els and logarithms. Two main facts emerge. First of all, with the only exception of FoodInd, in all series both estimated ρ and ω are statistically significant (Corn is the only case where ρ is not statistically different from 0). This confirms what was already obtained with ARCH tests presented in Table 2: volatility clusters occur in all series except for FoodInd. Secondly, many series vio- late the assumption of temporary clusters: for the price indexes EneInd and CPI, and price levels of natural gas, aluminium, zinc, wheat, corn, beef, we can not reject the hypothesis of ρ+ω =1. Therefore, in these cases volatil- ity follows a non-stationary process thus making clusters permanent rather than temporary as expected. Loga- rithms of prices partially confirm this evidence but some differences are worth noticing: non-stationary volatility is observed also for oil and nickel while it is now exclud- ed for aluminium, zinc and beef. Contradictory evidence emerges about the reliability of these GARCH processes as generators of the observed price dynamics. On the one hand, the existence of vola- tility clusters is consistent with the observed large vari- ability, or instability, of the commodity prices in specific periods of time. On the other hand, however, in sev- eral cases these processes support permanent volatility shocks thus becoming less compatible with the observed temporary episodes of turbulence. As discussed, once estimated, standard error in-sample predictions for these GARCH models can be generated. Figure 2 shows these predictions for the three price indexes and Figures 3a-3b for the individual price levels and logarithms, respective- ly. Figure A9 (panel a)) reports the same predictions for the CPI and its growth rate (i.e., inflation rate). As expected, volatility clusters do not emerge for FoodInd, while a significant increase of volatility can be appreciated in the second part of the period of (starting around 2005) for both MetInd and EneInd. For these indexes, this volatility dynamics seems consistent with the increased price turbulence observed in the same period as shown in Figure A1. In the case of individu- Table 2. GARCH(1,1) model estimation and persistency test on com- modity price indexes and commodity prices (1980M1-2021M12) (estimated standard errors in parenthesis)a. Series ρ Ω Test ρ+ω =1 (χ2(1)) Price indexes FoodIndb 0.036 (0.049) -0.171 (0.717) 3.96* MetInd 0.247 (0.049)* 0.675 (0.032)* 4.24* EneIndc 0.373 (0.081)* 0.641 (0.058)* 0.16 CPI 0.116 (0.022)* 0.879 (0.021)* 0.22 Price levels Oil 0.343 (0.041)* 0.726 (0.031)* 10.4* Coal 0.494 (0.062)* 0.664 (0.026)* 13.4* Gasd 0.304 (0.051)* 0.625 (0.044)* 3.52 Alumi- nium 0.276 (0.046)* 0.706 (0.038)* 0.39 Copper 0.241 (0.032)* 0.812 (0.019)* 8.74* Zinc 0.210 (0.031)* 0.815 (0.020)* 2.03 Nickel 0.427 (0.049)* 0.700 (0.027)* 19.6* Wheat 0.150 (0.022)* 0.865 (0.014)* 1.93 Corn 0.090 (0.013)* 0.901 (0.012)* 2.15 Soy 0.241 (0.030)* 0.718 (0.032)* 3.83* Beef 0.315 (0.043)* 0.699 (0.031)* 0.82 Logarithm of price levels Oil 0.441 (0.050)* 0.617 (0.038)* 3.26 Coal 0.214 (0.039)* 0.755 (0.036)* 3.95* Gasd 0.520 (0.063)* 0.491 (0.043)* 3.66 Alumi- nium 0.179 (0.041)* 0.748 (0.052)* 4.28* Copper 0.065 (0.023)* 0.878 (0.035)* 10.51* Zinc 0.064 (0.021)* 0.896 (0.028)* 6.35* Nickel 0.196 (0.029)* 0.799 (0.031)* 0.76 Wheat 0.062 (0.015)* 0.933 (0.013)* 0.58 Corn 0.016 (0.010) 0.942 (0.039)* 1.75 Soy 0.113 (0.031)* 0.676 (0.086)* 9.35* Beef 0.155 (0.049)* 0.613 (0.039)* 9.39* *Statistically significant at 5% confidence level. a Only estimates of parameters ρ and ω are reported. Other model parameter estimates are available on request. b 1991M1-2021M12. c 1992M1-2021M12. d 1985M1-2021M12. 182 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 al series, predictions show huge volatility variations for oil and for all mineral and agricultural commodities. Clusters seem to be relatively rare and quite temporary in the first part of the period, while they become more frequent and longer, thus possibly permanent, from 2005 onwards. This seems even more true for CPI and therefore, but less intensively, for the inflation rate. CPI volatility sharply rises in 2005 and remains higher than in the previous period with only a drastic drop during years 2013-2014. The question is whether the magnitude of this vola- tility clustering is consistent with the actual price turbu- lence or whether, in fact, we should look for alternative explanations. 5.3. Temporary bubbles Table 3 reports the sequence of tests for the pres- ence of temporary bubbles as expressed by equations (5) and (6). As discussed, moving from to GSADF the tests improve in terms of recursiveness and flex- ibility, therefore in precision, in detecting the temporary explosive roots.16 The presence of a temporary bubble is excluded in all cases (price indexes, individual price levels and logarithms of individual price levels) when the search of the bubble extends to the whole period ( ). Something emerges with SADF with a tempo- rary explosive root observed for MetInd and EneInd, and for the price level of all energy commodities, all minerals, and wheat. In the case of the logarithm of prices a bubble is detected only for oil. The generalised occurrence of temporary bubbles is eventually indicat- ed by the GSADF test. With the only exclusion of beef (both the price level and its logarithm), at least one tem- porary explosive root is found in all the series.17 16 It is worth noticing that the test in Table 3 (second column) corresponds to the ADF test with drift in Table 1 (third column) as the explosive bubble tests associated to equation (4) may include a drift but not a deterministic trend. However, strictu sensu, they are not the same test since the former is a right-tailed statistics so the critical values are different. The statistics itself slightly differs in some cases because the adopted specifications (i.e., lag structure) are not always the same. 17 Notice that the difference between the SADF and GSADF tests are larger here than what was presented in previous studies (see Gharib et al., 2021, p. 5, in particular) arguably because, despite the number Figure 2. GARCH(1,1) model standard error in-sample prediction for commodity price indexes (2000M1=1) (1992M1-2021M12). 183Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 a) b) c) Figure 3a. GARCH(1,1) model standard error in-sample prediction for energy commodities (a), metals (b) and agricultural commodities (c) price levels (2000M1=1) (1980M1-2021M12). 184 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 a) ) b) c) Figure 3b. GARCH(1,1) model standard error in-sample prediction for energy commodities (a), metals (b) and agricultural commodities (c) logarithm of prices (2000M1=1) (1980M1-2021M12). 185Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 In order to better appreciate how many bubbles occur and when, the BSADF tests (equation (7)) are computed. Results (with the critical values) are reported in Figure 4 for the three price indexes and in Figures 5a,b for the commodity price levels and logarithms, respectively. It appears that, for indexes, bubbles very sporadically emerge before and after the 2005-2008 of observations, the period covered here is quite long (more than 40 years). period. On the contrary, over these four years the tests exceed the critical values several times for all the index- es. MetInd is the index for which this exceedance is more often observed. In the case of individual price levels significant dif- ferences are found across the three groups. For energy prices, only in period 2005-2008 we observe one or more bubbles shared by the three prices. In the case of metals, beside that period, a common bubble is also observed in the mid-eighties. Agricultural commodities present a more composite situation: bubbles are more frequent and occur in the mid-eighties, mid-nineties, 2007-2008 and in the last decade. But they are often individual bub- bles and, again, only in 2007-2008 we observe a bubble shared by most (except for beef) agricultural commodi- ties. Qualitatively, results obtained with the price loga- rithms are similar even though, as could be expected, the bubbles are less frequent and, consequently, also the commonality of bubbles is more sporadic. 5.4. First movers and contagion As discussed in previous sections, the focus of the present study is not on commodity price interdepend- ence but on investigating the formation of temporary bubbles within individual price series in order to allow a real-time monitoring tool to inform on the formation of temporary bubbles, on the possible involvement of sever- al commodities and on the first moving prices. The com- bination of the two alternative approaches here proposed allows to present their results in a form that permits an intuitive visualization of all this information about if and how co-exceedance is occurring. Figures 6a,b aim to provide this easily interpret- able visualization by displaying the periods of exceed- ance (volatility clusters or bubbles) for price levels and logarithms, respectively. Bubbles are dated on the basis of the BSADF tests. In the case of volatility, fol- lowing Engle (1982, p 1003), exceedance is found any time the predicted volatility exceeds the double (in the case of price indexes) or the triple (in the case of indi- vidual commodity prices) of the average predicted volatility (i.e., standard error) over the two subperiods 1980M1-2000M12 and 2001M1-2021M12. Together with Table 4, these figures are also intended to provide an example of how the proposed approach can contribute to a real-time surveillance tool through an easily interpret- able and periodically updatable dashboard visualization. To summarize this evidence and better interpret it in terms of co-exceedance, Table 4 reports the beginning (the “exuberance date” to use the term of Gharib et al., 2021, p. 6) and the end (the “collapsing date”) months of Table 3. Temporary explosive root tests on commodity price index- es and commodity prices (1980M1-2021M12)a. Series SADF GSADF Price indexes FoodIndb -1.289 0.0487 3.644* MetInd -1.285 3.303* 7.170* EneIndc -1.167 4.907* 5.092* CPI 0.288 0.854 3.189* Price levels Oil -2.352 3.843* 4.417* Coal -1.166 8.177* 8.762* Gasd -0.456 4.164* 6.619* Aluminium -2.524 2.405* 5.132* Copper -0.883 2.972* 4.750* Zinc -2.937 3.871* 5.908* Nickel -2.656 3.426* 5.491* Wheat -2.517 3.360* 3.795* Corn -2.151 0.357 3.462* Soy -2.478 -0.553 2.981* Beef -0.134 0.937 1.211 Logarithm of price levels Oil -1.199 1.871* 2.275* Coal -1.500 0.414 2.511* Gasd -1.745 -0.135 2.891* Aluminium -3.168 -0.270 -2.816* Copper -0.858 -0.128 3.124* Zinc -1.827 0.057 3.583* Nickel -2.008 1.361 2.989* Wheat -2.410 1.120 3.098* Corn -1.756 0.513 2.306* Soy -2.032 -1.200 2.271* Beef -0.167 0.665 1.794 *Statistically significant at 5% confidence level with bootstrap criti- cal values computed with 200 repetitions. a All test specifications include 6 lags. b 1991M1-2021M12. c 1992M1-2021M12. d 1985M1-2021M12. 186 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 periods with at least three different commodities show- ing at least two consecutive months of exceedance, thus a co-exceedance in terms of volatility or common bubble for at least two consecutive months (Phillips et al., 2011). The table also reports: commodities showing this co- exceedance (second column); the commodity that can be identified as the first mover (third column), that is, the price whose exceedance started first before the period of co-exceedance; whether or not also CPI shows exceed- ance in the same period, and whether or not CPI can be considered the first mover (forth column). In the case of price levels, it emerges that bubbles concentrate in the 2005-2008 period though in differ- ent moments involving different commodities. Basically, we can identify two main episodes. The first goes from 2005M8 to 2006M9. It only involves energy commodi- ties and metals with oil and copper as first movers. CPI is itself involved in the bubble, as could be expected, but surprisingly it behaves as the first mover. The second episode is shorter and goes from 2008M2 to 2008M8. It involves all energy commodities wheat and corn but no minerals. Again, oil behaves as the first mover. This latter commodity actually seems to experience a single bubble from mid-2005 to the end of 2008. CPI is also involved but not as the first mover. Volatility clusters emerging from the GARCH regressions show a significant difference compared to the bubbles. A first episode is found from mid-1988 to mid- 1989, it concerns some minerals and agricultural com- modities but no energy commodities. Wheat seems to be the first mover. Other four episodes, in fact behaving as a single one, can be detected from 2006M8 to 2009M8. In the first part of this period, the cluster exclusively involves metals and wheat. Then, other prices enter the group included energy commodities and, finally, also oil. In the very last part of this episode, the volatility clusters involve most (9 out of 11) commodities. If we consider this whole period as a single episode, the first mover seems to be nickel which sounds a little surpris- ing. In the second part of the period, wheat and natural gas emerge as other possible candidates. Two other volatility clusters can be found in the last decade of the period under investigation. One con- cerns a very short period (two months in mid-2012) and only involves agricultural commodities. The other concerns the very last months of the period of obser- vation (2021M9-2021M12); it is short simply because it continues beyond the period of observation. This period of exceedance is not identified with the bubble testing arguably because the bubble has still to collapse. Future Figure 4. BSADF tests for indexes for commodity price indexes (1992M1-2021M12). 187Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 a) b) c) Figure 5a – BSADF tests for indexes for energy commodities (a), metals (b), agricultural commodities (c) price levels (1980M1-2021M12). 188 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 a) b) c) Figure 5b. BSADF tests for indexes for energy commodities (a), metals (b), agricultural commodities (c) logarithm of price levels (1980M1-2021M12). 189Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas Aluminium Copper Zinc Nickel Wheat Corn Soy Beef EN ER GY M ET AL S FO O D 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas Aluminium Copper Zinc Nickel Wheat Corn Soy Beef EN ER GY M ET AL S FO O D a) b) Figure 6a. Dating of explosive roots (a) and volatility clusters (b) for all commodity price levels (1980M1-2021M12) (see Table 4 for details). 190 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas Alum Copper Zinc Nickel Wheat Corn Soy Beef EN ER GY M ET AL S FO O D 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas Alum Copper Zinc Nickel Wheat Corn Soy Beef EN ER GY M ET AL S FO O D a) b) Figure 6b. Dating of explosive roots (a) and volatility clusters (b) for all commodity logarithms of price levels (1980M1-2021M12) (see Table 4 for details). 191Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 investigations will confirm the nature and scope of the current period of exuberance. The cluster identified here suggests that it concerns both energy prices and metals but it is likely driven by the former and, in particular, by natural gas. Two major facts seem to emerge from this analy- sis of co-exceedance. First of all, the correspondence between bubbles and volatility cluster detection is lim- ited. Periods correspond in the case of the major episode that occurred between 2005 and 2008. But for the rest of the sample, the detection of the episodes of co-exceed- ance does not correspond. Also the involved commodi- ties significantly differ and, consequently, the first mov- ers. Bubble detection seems to stress more the dynam- ics of energy commodities, and oil in particular, while volatility clusters point more to metals and agricultural Table 4. Dating of temporary explosive roots and volatility clusters for commodity price levels and logarithm of levels (1980M1-2021M12). Period Commodities First moverc (date) CPI: Y/N; first mover (Y/N & date)d Price levels Bubblesa 2005M8-2005M10, 2006M1, 2006M3, 2006M6 Oil, Copper, Zinc Oil, Copper (2005M6) Y; Y (2005M3) 2006M2, 2006M4-2006M5 Oil, Aluminium, Copper, Zinc 2006M7, 2006M9 Oil, Copper, Zinc Nickel 2006M10-2006M12 Oil, Copper, Zinc 2008M2-2008M7 Oil, Coal, Gas, Wheat, Corn Oil (2007M3) Y; N 2008M5, 2008M8 Oil, Coal, Gas, Corn Volatility Clustersb 1988M7-1988M10 Copper, Wheat, Beef Wheat (1988M4) N 1988M11-1989M7 Copper, Zinc, Nickel, Wheat 2006M8-2006M11 Copper, Zinc, Nickel, Wheat Nickel (2006M5) N 2007M4-2007M12 Copper, Zinc, Nickel, Wheat 2008M3-2008M8 Gas, Copper, Wheat, Corn Wheat (2006M8) N 2008M9-2009M8 Oil, Gas, Coal, Aluminium, Copper, Zinc, Corn, Soy, Beef Gas (2008M3) Y; N 2012M7-2012M8 Corn, Soy, Beef Soy (2012M8) N 2021M9-2021M12 Gas, Aluminium, Copper Gas (2021M7) Y; N/Y(2021M7) Logarithm of price levels Bubblesa Period Commodities First mover (date) CPI: Y/N; first mover (Y/N & date) 2005M12-2006M10 Aluminium, Copper, Zinc Zinc (2005M9) Y; Y (2005M3) 2008M6-2008M7 Oil, Coal, Wheat, Corn Wheat (2008M2) Y; N 2015M12-2016M6 Oil, Gas, Copper Copper (2015M8) N Volatility Clustersb Period Commodities First mover (date) CPI: Y/N; first mover (Y/N & date) 1988M4-1989M1 Aluminium, Nickel, Soy Aluminium (1987M12) Y; N 2008M11-2009M3 Oil, Coal, Aluminium, Copper, Nickel Coal (2008M3) Y; N a The dating of the bubble corresponds to periods when at least 3 commodities show explosive roots, that is BSADF test significant at 5% confidence level with bootstrap critical values computed with 200 repetitions. Only periods with at least two consecutive months of exceed- ance are reported. b The dating of the volatility clusters corresponds to periods when predicted volatility (i.e., standard error) is larger than three times the subperiod (1980M1-2000M12; 2001M1-2021M12) average volatility. Only periods with at least two consecutive months of exceedance are reported. c The first mover is the price of the group whose exceedance started first before the period of co-exceedance. d The first Y/N indicates whether or not also CPI shows exceedance in the same period; the second Y/N indicates whether or not CPI can be considered the forst mover (in parenthesis the date). 192 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 commodities. A final difference concerns the involve- ment of CPI that seems very limited in the case of clus- ters while it is more relevant for bubbles. The second notable fact is the difference between price levels and price logarithms. As the logarithmic transformation re-scales the data and, therefore, scales down their variability, respective results are expected to make the more robust evidence emerge: the number of individual episodes may slightly decline and the number of common episodes is expected to substantially reduce. It turns out that the number of episodes of co-exceed- ance detected on price logarithms is lower, as expected, for both bubbles and volatility clusters. But the nature of these episodes does not necessarily correspond with what is observed on price levels and this lack of robust- ness passing from levels to logarithms seems more evi- dent for bubbles than for volatility clusters. The involved commodities are not necessarily the same, as well as the first movers, and also the involvement of the CPI shows some difference. In general terms, when the logarithms are considered, it seems more difficult to find some gen- eral pattern in the results, especially in terms of a key role of some commodities like the energy ones. Looking for regularities, two special cases are worth noticing here. The first concerns the oil price. Proper- ties and behaviour of this commodity price emerging in the present work confirm the bubble detection and dat- ing reported in previous studies, particularly in Su et al. (2017) and Zhao et al. (2021).18 It could also be argued that oil price has to behave as a sort of upstream price since it enters as a production cost in most downstream production processes, included farming and mining activities. But this role of oil as first mover is not gener- ally observed and seems to emerge only in bubble testing with price levels. The second interesting role is that played by agri- cultural commodities. On the one hand, they can be considered as downstream prices compared to energy commodities and metals. But, for this reason, they can severely impact on CPI dynamics. From our results, it emerges that agricultural commodity prices seem to be a little more “stable” in the sense that episodes of exuber- ance (bubbles or volatility clusters) are less frequent and shorter. At the same time, while energy commodities and metals are apparently more interdependent, agricul- tural prices seem to follow more autonomous patterns and are less likely to act as first movers and, thus, to be suitable candidates to drive the other commodity prices and of the CPI. 18 Su et al. (2017, p. 6) conclude that “there are explosive multiple bub- bles in the WTI oil market in 1990, 2005, 2006, 2008 and 2015. Gener- ally, oil bubbles mostly occur during the period of price volatility”. What can we finally conclude about the evidence on the linkage between commodity prices and CPI? While results tend to confirm some stochastic proper- ties of the CPI that may explain periods of exuberance, the evidence that these periods are the consequence of analogous episodes in commodity prices is poor. The major episode of price exuberance between 2005-2008 confirms, as could seem obvious, a connection between commodity prices and CPI, maybe because this episode involves a large number of commodities, though in dif- ferent times. In fact, this connection seems quite weak beyond this period. And also within this 2005-2008 period it is not clear whether commodity price exuber- ance induced a CPI response or if it is actually the other way round. This lack of evidence should not be surpris- ing and evidently asks for further investigation. Other very recent empirical investigations (Lian and Freitag, 2022), for instance, suggest that oil price shocks do not always imply a shock on CPI and sometime this latter may move independently and also precede the former. 6. COMPARISON WITH OTHER STOCHASTIC PROCESSES AND APPROACHES For the sake of comparison and in order to vali- date the results here obtained, it is worth investigating the commonality of the commodity price dynamics also with more conventional approaches. Rather than focus- ing on co-exceedance, as in the present study, these approaches look for the commonality of the stochastic generation processes (i.e., co-movement and the conse- quent price interdependence) under the typical hypoth- esis of either stationary or non-stationary linear DGP, possibly with a drift and/or a trend (Esposti, 2021). As already discussed in Section 2, in order to capture the complexity and non-linearity of these series a further occurrence that can be considered consists in admit- ting that series undergo, in one or more points in time, a structural break in either the drift or the trend (or in both) (Baum, 2005). In principle, under multiple breaks, these stochastic processes could explain the presence of periods of extremely high (or low) prices (the “bubbles”) as a sequence of two structural breaks with the latter eventually compensating the former and thus making its effect only temporary. Table A2 (Annex 3) reports a battery of tests specifi- cally designed to assess whether these more conventional stochastic processes represent suitable alternatives to the two co-exceedance processes here considered. Four tests are reported. They all confront a unit-root process (the null hypotheses of the tests) with a stationary process 193Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 presenting one or two structural breaks in some terms of the process itself.19 All tests admit endogenous breaks, thus not only do they test their presence but they are also able to date these breaks. The first two tests consist in two specifications of the Zivot-Andrews (ZA) unit root test (Zivot and Andrews, 2002) admitting only one structural break in either the intercept or both intercept and trend. Test results largely accept unit-root processes without a break against the presence of a structural break within stationary series. The only exceptions are coal and soy in the price lev- els and only soy in the logarithms of prices. The break date is similar, late 2006-early 2007, and corresponds to one of the major periods of co-exceedance identified in previous sections. Not only is the break accepted for only two commodities but, more importantly, it can not explain why and how, once started, the period of turbu- lence then comes to an end since the structural break introduces a permanent change in the process. The other two tests, consisting in two variants of the Clemente-Montañés-Reyes unit root test (CMR) (Clem- ente et al., 1998), can be helpful in this respect. In this case, the statistical significance of the breaks themselves can be assessed as they enter in the test specification as time dummies with the respective coefficients. More importantly, this test admits two structural breaks with- in the stationary process thus allowing the combination of the two breaks to capture a temporary change in the process, like in the case of periods of price surge. The test can be performed under two different natures of the breaks: a sudden change in the series (the additive outli- ers, or AO, model) or a gradual shift in the mean of the series (the innovational outliers, or IO, model). Evident- ly, the former, much more than the latter, is expected to capture the short periods of price turbulence. Although CMR tests confirm how difficult it can be, over such a long period of time, to univocally identify clear stable stochastic processes for any given commodity and, even more, commonality across commodities in this respect, they still provide some interesting indications. One the one hand, the CMR test under the AO model seems to confirm the main evidence emerging from the ZA test results without any relevant difference between price levels and logarithms: for most commodity prices (oil being the only exception) a non-stationary process is accepted against a stationary process with structural breaks. When the IO model is considered, however, the CMR test indicates that for many commodities (all ener- gy prices, aluminium, zinc, corn and FoodInd itself) a mean-reverting process under two structural breaks 19 For more details on the ZA and CMR tests, also see Baum (2005). is accepted. Even more interestingly, this test indicates that, for both AO and IO cases, the structural breaks are always statistical significant (with only one exception). In some cases, the interval between the two breaks is too wide (more than three years) to really capture a period of price exuberance (see the CPI case, for instance). In other cases, however, the time window between the two breaks seems quite consistent with the periods of exceedance here identified, as shown in Figures 6a,b. This is the case, in particular, of coal and all metals. As already discussed, with respect to the purpose of the present study, the introduction of structural breaks may seem an unnatural way to capture co-exceedance: it still maintains the linear specification of the DGP pos- sibly with a permanent change while here the intention is to identify a DGP with temporary non-linearities. It follows that admitting structural breaks within the sto- chastic process representation may still confound short- term and long-term dynamics within the price series. Nonetheless, present results suggest that multiple breaks within an appropriate specification eventually constitute a sort of spline process capable to proxy temporary non- linearities. Even though not considered further here, this kind of approach, together with the introduction of multiple structural breaks within non-linear processes (Bai and Perron, 2003; Caporale et al., 2020), can repre- sent a promising alternative empirical strategy in future research in the field.20 There is a final aspect to be considered about the introduction of structural breaks as a valid alternative to capture co-exceedance. It concerns the identification of the first-moving commodities and the possible conse- quent contagion process. As shown, within the proposed approach, this identification is made only qualitatively by identifying and then visualizing when, commodity by commodity, the periods of exuberance start and end (see Figures 6a,b). Very often, however, within the empirical literature this identification is formally pursued using Granger causality testing (Esposti and Listorti, 2013). This approach must satisfy the prerequisite that series under investigation show the same stochastic properties (they are all either I(0) and I(1)), and then it requires the estimation of multiple-equation linear models (in the form of VAR or VEC models, respectively) represent- ing the common movement from which direction and nature of price interdependence (or transmission) can be assessed. Within this representation one or more struc- 20 The use of international or global commodity prices, as well as the widely heterogenous dating of these structural breaks across commodi- ties, makes it hard to speculate on the possible linkage between them and external shocks like, for example, policy regime changes. However, this investigation may represent a further direction of research for the future. 194 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 tural breaks can be included (as time dummies) to pos- sibly capture some changes in the linear relationships, thus admitting temporary non-linearities. Results here presented demonstrate how the pre- requisite of this empirical strategy to detect first mov- ers and contagion can be challenging. A common DGP is impossible to find when all price series are consid- ered. But even concentrating on individual commod- ity groups, Tables 1 and 5(??) suggest there is always at least one commodity showing a different underlying sto- chastic process compared to the others. Apparently, an interesting case is that of energy commodities under the IO specification of the CMR test: when two structural breaks are admitted they all behave as stationary pro- cesses with a drift and a deterministic trend. Therefore, an attempt to perform Ganger causality testing can be made here by estimating a VAR model with four endog- enous variables: the three energy prices (oil, coal and gas prices) and the CPI, since it quite robustly emerges as a I(0) process. The VAR specification also includes a drift, a deterministic trend and two time dummies represent- ing the two breaks at 2003M11 and 2013M5. Table A3 (Annex 3) reports the results of the respective Granger causality tests and the estimated coefficients of the two structural breaks.21 As often occurs with Granger causality testing, results are not easily interpretable. However, they con- firm some of the evidence obtained with our proposed approach. It seems hard to identify an indisputable driv- ing price and, in particular, this does not seem the case of the oil price. When price levels are considered, oil and coal show a reciprocal Granger causation, while natu- ral gas is only Granger caused by the coal price. Oil and coal price both Granger-cause the CPI response, while CPI itself does not Granger-cause any of the energy pric- es as could be expected. Coal rather than oil seems to be the driving price, if any, and this seems to be reinforced when the logarithm of prices are considered instead of the levels. The presence of structural breaks, though sug- gested by tests reported in Table A2, is not confirmed by VAR estimation coefficients associated to respective time dummies are mostly not statistically different from zero. Compared to the approach here proposed, which is based on the search of co-exceedance periods (thus admitting non-linearities in the DGP) rather than on linear price interdependence, these more conventional stochastic processes do not seems to provide any helpful additional information. On the contrary, they seem to fail in the search of common periods of exuberance over a large group of commodities, thus they do not seem 21 For the sake of space limitation, the VAR model estimates are not reported here but are available upon request. appropriate for designing a real-time surveillance dash- board informing a prompt policy response. Nonethe- less, even in these approaches recent contributions have opened new interesting perspectives that may deserve careful consideration in future research. For instance, the implementation of non-linear Granger causality test- ing seems particularly promising (Shahzad et al., 2021). 7. CONCLUSIVE REMARKS Periods of commodity price exuberance raise politi- cal concerns particularly for their possible impact on the inflation rate. Timely interventions by the appoint- ed institutions are often invoked but do not necessar- ily prove to be effective in preventing or neutralizing these episodes. After all, common price spikes (thus, co- exceedance) might not imply a common policy response since for some commodities exuberance tends to be motivated by real drivers while in other cases financial phenomena are prevalent. Understanding the mecha- nisms underlying generation, transmission and, then, collapse of co-exceedance remains relevant to design the proper, possibly differentiated, policy response. But in the shorter term an appropriate policy response may just need a timely detection of the price surge and of the degree of diffusion across commodities. The present paper aims to develop a single meth- odological approach, albeit based on alternative stochas- tic processes, that does not assume common movement and price interdependence but only co-exceedance, thus commonality occurring only within the periods of exu- berance. This approach is able to detect whether such a period occurs, when it starts and when it ends, the degree of diffusion across commodities, the possible presence of driving prices and, eventually, the transfer to the inflation rate. On this basis, the proposed methodol- ogy is intended to offer an easily interpretable visualiza- tion of the critical information it generates. Results presented indicate that the different approaches considered (bubbles and volatility cluster detection in both price levels and logarithms) are able to provide clear indications on when the exceedance occurs, on its overlapping across commodities and on possible first movers. However, this evidence is not con- cordant or, at least, robust across the different approach- es making the final outcome of the analysis, and the pol- icy implication itself, severely dependant on the analyst’s choices in this respect. Results do not even agree on the involvement of the CPI in these episodes of exuberance, therefore on the transmission of commodity price spike to inflation rate. 195Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 On the basis of this discrepancy, it seems wise to develop the abovementioned policy tool in a way that prudently admits both processes and elaborates informa- tion from a combination of them. At the same time, this discrepancy points to room for further methodological improvements. After all, both competing representations of the origin of exceedance, volatility clusters and tempo- rary bubbles, show pros and cons and this makes it dif- ficult to draw a general preference for one or the other. GARCH modelling seems to represent more permanent changes in volatility rather than short periods of exuber- ance. Furthermore, it hardly combines volatility clusters with a non-stationary process in the price levels. At the same time, bubble detection applies well to positive bub- bles, therefore periods of exuberance then followed by a collapse, but it does not necessarily succeed in case of negative bubbles, that is episodes that start with a price crash (Gharib et al., 2021, pp. 3-4).22 In fact, bubble detection can only by applied ex post, therefore when the bubbles have already collapsed. This substantially limits the actual applicability of the approach by analysts and policy makers. Moreover, currently available tests only apply to univariate bubble detection. Multivariate bub- ble testing has not yet been proposed and this prevents a direct investigation of contagion across commodities. Regarding all these aspects, results obtained in the present study also suggest the extension of the adopted tool to other stochastic processes, particularly those expressing non-linear dynamics of commodity prices in both level and variance. Multiple breaks, fractional inte- gration, fractal and wavelet analysis are some examples in this direction. Finally, it would be particularly helpful to replicate these results on higher frequency price data. Weekly or daily prices, if available, might definitely be useful to better refine this real-time surveillance policy tool making it more timely and accurate. However, these data might also bring about more statistical noise, thus making the identification of co-exceedance more dif- ficult and uncertain, and increasing the risk of false alarms. 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Journal of Business Economics and Statis- tics 20(1), 25–44. 197Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 ANNEX 1 - DESCRIPTION OF THE DATA USED IN THE ANALYSIS Individual commodity prices (source: IMF) Oil: Crude Oil (petroleum), Price index, 2005 = 100, simple average of three spot prices; Dated Brent, West Texas Intermediate, and the Dubai Fateh Gas: Natural Gas, Russian Natural Gas border price in Germany, US$ per Million Metric British Thermal Unit Coal: Australian thermal coal, 12,000- btu/pound, less than 1% sulfur, 14% ash, FOB Newcastle/Port Kembla, US$ per metric ton Aluminium: 99.5% minimum purity, LME spot price, CIF UK ports, US$ per metric ton Copper: grade A cathode, LME spot price, CIF Europe- an ports, US$ per metric ton Zinc: high grade 98% pure, US$ per metric ton Nickel: melting grade, LME spot price, CIF European ports, US$ per metric ton Wheat: No.1 Hard Red Winter, ordinary protein, Kansas City, US$ per metric ton Corn: U.S. No.2 Yellow, FOB Gulf of Mexico, U.S. price, US$ per metric ton Soy: U.S. soybeans, Chicago Soybean futures contract (first contract forward) No. 2 yellow and par, US$ per metric ton Beef: Australian and New Zealand 85% lean fores, CIF U.S. import price, US cents per pound Table A1. Descriptive statistics on commodity price indexes and commodity prices (1980M1-2021M12). Series Obs Minimum Maximum Mean Standard Deviation Skewness Kurtosis Price indexes FoodInda 372 76.04 194.1 121.9 60.20 -0.384 1.965 MetInd 504 44.16 256.2 101.3 53.69 -0.084 2.556 EneIndb 360 22.07 257.3 99.75 67.76 -0.656 2.244 CPI 504 40.79 146.2 92.17 28.09 -0.005 1.799 Price levels Oil 504 18.44 249.6 84.09 56.86 0.877 2.534 Coal 504 24.09 240.7 57.66 34.01 1.584 5.816 Gasc 444 1.444 32.91 5.398 4.199 1.979 10.42 Aluminium 504 918.8 3578 1712 468.1 0.778 3.325 Copper 504 1272 10308 3882 2520 0.742 2.116 Zinc 504 597.4 4381 1541 786.4 1.019 3.277 Nickel 504 3433 51783 11394 7270 1.867 8.130 Wheat 504 88.55 403.8 168.5 54.33 1.235 4.396 Corn 504 65.35 332.9 142.7 57.14 1.420 4.542 Soy 504 158.31 622.9 292.1 105.4 1.036 3.204 Beef 504 74.26 272.2 130.5 44.44 1.026 3.111 Price logarithms Oil 504 2.914 5.520 4.207 0.671 0.202 1.755 Coal 504 3.178 5.483 3.913 0.510 0.610 2.358 Gasc 444 0.367 3.493 1.461 4.250 1.255 6.286 Aluminium 504 6.823 8.182 7.409 0.265 0 .158 2.542 Copper 504 7.148 9.240 8.056 0.642 0.289 1.566 Zinc 504 6.392 8.385 7.222 0.477 0.370 1.948 Nickel 504 8.141 10.85 9.173 0.566 0.330 2.406 Wheat 504 4.483 6.000 5.081 0.296 0.489 2.882 Corn 504 4.179 5.807 4.894 0.353 0.663 2.868 Soy 504 5.064 6.434 5.619 0 .332 0.520 2.277 Beef 504 4.307 5.606 4.820 0.313 0.568 2.316 a 1991M1-2021M12. b 1992M1-2021M12. c 1985M1-2021M12. 198 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 Aggregate commodity price indexes (source: IMF) FoodInd: Food Price Index, 2016 = 100, includes Cereal, Vegetable Oils, Meat, Seafood, Sugar, and Other Food (Apple (non-citrus fruit), Bananas, Chana (legumes), Fish- meal, Groundnuts, Milk (dairy), Tomato (veg)) Price Indices MetInd: Metals Price Index, 2005 = 100, includes Cop- per, Aluminium, Iron Ore, Tin, Nickel, Zinc, Lead, and Uranium Price Indices EneInd: Fuel (Energy) Index, 2005 = 100, includes Crude oil (petroleum), Natural Gas, and Coal Price Indices Overall Consumer Price Index (source: US Federal Reserve) CPI: Federal Reserve Economic Data, Economic Research Division, Federal Reserve Bank of St. Louis. CPIAUCNS Consumer Price Index for All Urban Consumers: All Items in U.S. City Average, Index 1982-1984=100, Month- ly, Not Seasonally Adjusted. The Inflation rate is comput- ed as the monthly growth rate of this CPI. ANNEX 2 – COMMODITY PRICE DYNAMICS 0 50 100 150 200 250 300 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 FoodInd MetInd EnelInd Figure A1. Commodity price indexes (2005M1=100) (1980M1-2021M12). 0 100 200 300 400 500 600 700 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas Figure A2. Energy commodities prices (2005M1=100) (1980M1-2021M12). 0 50 100 150 200 250 300 350 400 19 80 M 1 19 81 M 1 19 82 M 1 19 83 M 1 19 84 M 1 19 85 M 1 19 86 M 1 19 87 M 1 19 88 M 1 19 89 M 1 19 90 M 1 19 91 M 1 19 92 M 1 19 93 M 1 19 94 M 1 19 95 M 1 19 96 M 1 19 97 M 1 19 98 M 1 19 99 M 1 20 00 M 1 20 01 M 1 20 02 M 1 20 03 M 1 20 04 M 1 20 05 M 1 20 06 M 1 20 07 M 1 20 08 M 1 20 09 M 1 20 10 M 1 20 11 M 1 20 12 M 1 20 13 M 1 20 14 M 1 20 15 M 1 20 16 M 1 20 17 M 1 20 18 M 1 20 19 M 1 20 20 M 1 20 21 M 1 Aluminium Copper Nickel Zinc Figure A3. Metals prices (2005M1=100) (1980M1-2021M12). 0 50 100 150 200 250 300 350 400 19 80 M 1 19 81 M 1 19 82 M 1 19 83 M 1 19 84 M 1 19 85 M 1 19 86 M 1 19 87 M 1 19 88 M 1 19 89 M 1 19 90 M 1 19 91 M 1 19 92 M 1 19 93 M 1 19 94 M 1 19 95 M 1 19 96 M 1 19 97 M 1 19 98 M 1 19 99 M 1 20 00 M 1 20 01 M 1 20 02 M 1 20 03 M 1 20 04 M 1 20 05 M 1 20 06 M 1 20 07 M 1 20 08 M 1 20 09 M 1 20 10 M 1 20 11 M 1 20 12 M 1 20 13 M 1 20 14 M 1 20 15 M 1 20 16 M 1 20 17 M 1 20 18 M 1 20 19 M 1 20 20 M 1 20 21 M 1 Beef Corn Soy Wheat Figure A4. Agricultural commodities prices (2005M1=100) (1980M1-2021M12). 199Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 20 40 60 80 100 120 140 160 180 200 220 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 Oil Coal Gas 80 85 90 95 100 105 110 115 120 19 80 M 1 19 81 M 1 19 82 M 1 19 83 M 1 19 84 M 1 19 85 M 1 19 86 M 1 19 87 M 1 19 88 M 1 19 89 M 1 19 90 M 1 19 91 M 1 19 92 M 1 19 93 M 1 19 94 M 1 19 95 M 1 19 96 M 1 19 97 M 1 19 98 M 1 19 99 M 1 20 00 M 1 20 01 M 1 20 02 M 1 20 03 M 1 20 04 M 1 20 05 M 1 20 06 M 1 20 07 M 1 20 08 M 1 20 09 M 1 20 10 M 1 20 11 M 1 20 12 M 1 20 13 M 1 20 14 M 1 20 15 M 1 20 16 M 1 20 17 M 1 20 18 M 1 20 19 M 1 20 20 M 1 20 21 M 1 Aluminium Copper Nickel Zinc 90 95 100 105 110 115 120 125 130 19 80 M 1 19 81 M 1 19 82 M 1 19 83 M 1 19 84 M 1 19 85 M 1 19 86 M 1 19 87 M 1 19 88 M 1 19 89 M 1 19 90 M 1 19 91 M 1 19 92 M 1 19 93 M 1 19 94 M 1 19 95 M 1 19 96 M 1 19 97 M 1 19 98 M 1 19 99 M 1 20 00 M 1 20 01 M 1 20 02 M 1 20 03 M 1 20 04 M 1 20 05 M 1 20 06 M 1 20 07 M 1 20 08 M 1 20 09 M 1 20 10 M 1 20 11 M 1 20 12 M 1 20 13 M 1 20 14 M 1 20 15 M 1 20 16 M 1 20 17 M 1 20 18 M 1 20 19 M 1 20 20 M 1 20 21 M 1 Beef Corn Soy Wheat Figure A5. Logarithms of the energy commodities prices (2005M1=100) (1980M1-2021M12). Figure A6. Logarithms of the metals prices (2005M1=100) (1980M1-2021M12). Figure A7. Logarithms of the agricultural commodities prices (2005M1=100) (1980M1-2021M12). 0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0 2 4 6 8 10 12 14 16 18 19 80 M 1 19 80 M 12 19 81 M 11 19 82 M 10 19 83 M 9 19 84 M 8 19 85 M 7 19 86 M 6 19 87 M 5 19 88 M 4 19 89 M 3 19 90 M 2 19 91 M 1 19 91 M 12 19 92 M 11 19 93 M 10 19 94 M 9 19 95 M 8 19 96 M 7 19 97 M 6 19 98 M 5 19 99 M 4 20 00 M 3 20 01 M 2 20 02 M 1 20 02 M 12 20 03 M 11 20 04 M 10 20 05 M 9 20 06 M 8 20 07 M 7 20 08 M 6 20 09 M 5 20 10 M 4 20 11 M 3 20 12 M 2 20 13 M 1 20 13 M 12 20 14 M 11 20 15 M 10 20 16 M 9 20 17 M 8 20 18 M 7 20 19 M 6 20 20 M 5 20 21 M 4 CPI Inflation rate (right scale) -5 -4 -3 -2 -1 0 1 2 3 4 5 19 80 M 1 19 81 M 1 19 82 M 1 19 83 M 1 19 84 M 1 19 85 M 1 19 86 M 1 19 87 M 1 19 88 M 1 19 89 M 1 19 90 M 1 19 91 M 1 19 92 M 1 19 93 M 1 19 94 M 1 19 95 M 1 19 96 M 1 19 97 M 1 19 98 M 1 19 99 M 1 20 00 M 1 20 01 M 1 20 02 M 1 20 03 M 1 20 04 M 1 20 05 M 1 20 06 M 1 20 07 M 1 20 08 M 1 20 09 M 1 20 10 M 1 20 11 M 1 20 12 M 1 20 13 M 1 20 14 M 1 20 15 M 1 20 16 M 1 20 17 M 1 20 18 M 1 20 19 M 1 20 20 M 1 20 21 M 1 CPI Inflation rate 90% CV 95% CV a) b) Figure A8. Oil price (2005=100), CPI (2005=100) and inflation rate (1980M1-2021M12). Figure A9. GARCH(1,1) model standard error in-sam- ple prediction (a) and BSADF test for CPI and Inflation rate (1980M1-2021M12). 200 Roberto Esposti Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 ANNEX 3 – UNIT-ROOT AND GRANGER-CAUSALITY TESTS WITH STRUCTURAL BREAKS Table A2. Testing with structural breaks within an ADF specification (unit-root testing) for commodity price indexes, price levels and loga- rithm of levels (1980M1-2021M12). Series ZA (break month)e CMR (b.reak months)f Intercept Intercept&Trend AO IO Price indexes FoodInda -4.780 -4.932 -3.844 (1990M2*; 2006M3*) -7.408* (1990M2*; 2006M3*) MetInd -3.826 -4.048 -5.176 (2005M6*; 2014M3*) -5.406 (2004M9*; 2013M7*) EneIndb -4.199 -4.305 -4.736 (1991M2*; 2004M9*) -4.831 (1990M11*; 2003M11*) CPI -3.461 -3.730 -3.079 (1992M5*; 2007M10*) -2.570 (1985M11*; 2002M11*) Price levels Oil -4.731 -4.755 -6.299* (2005M9*; 2015M2*) -6110* (2003M11*; 2013M5*) Coal -5.245* (2006M11) -5.253* (2006M11) -2.869 (2007M3*; 2008M1*) -5.791* (2006M9*; 2007M6*) Gasc -2.337 -2.454 -2.462 (2004M11*; 2014M5*) -1.718* (2003M8*; 2013M11*) Aluminium -4.532 -4.399 -3.447 (1987M4*; 2004M4*) -5.603* (2004M8*; 2007M6*) Copper -4.471 -4.464 -3.589 (2005M6*; 2014M3*) -4.131 (2004M4*; 2013M6) Zinc -4.236 -4.236 -4.209 (2005M6*; 2006M10*) -5.614* (2004M10*; 2007M4*) Nickel -4.027 -4.626 -4.155 (2005M9*; 2007M1*) -6.867 (2005M2*; 2006M4*) Wheat -3.663 -3.673 -3.747 (2007M4*; 2013M10*) -5.190 (2009M11*; 2013M1*) Corn -4.572 -4.577 -5.119 (2009M11*; 2013M1*) -5.688* (2009M5*; 2012M6*) Soy -5.091* (2006M10) -5.102* (2007M5) -4.749 (2007M3*; 2013M11*) -6.061 (2006M3*; 2013M3*) Beef -4.009 -4.574 -4.244 (2009M5*; 2018M8*) -4.532 (2008M9*; 2018M9*) Logarithm of price levels Oil -4.188 -4.363 -5.055 (1985M4*; 2003M9*) -4..888 (1998M1*; 2003M11*) Coal -4.273 -4.597 -3.476 (2007M3*; 2008M1*) -4..753 (2002M9*; 2005M9*) Gasc -2.598 -4.706 -4.327 (1993M10*; 2003M8*) -6..451* (1993M11*; 2003M9*) Aluminium -4.349 -4.639 -4.366 (1987M4*; 2003M4*) -5.307* (2004M5*; 2007M6*) Copper -4.584 -4.699 -4.392 (1987M1*; 2005M6*) -4.820* (1985M2*; 2002M8*) Zinc -4.701 -4.708 -4.580 (2005M6*; 2007M1*) -5.333 (1986M8*; 2004M6*) Nickel -3.958 -4.204 -4.401 (1987M1*; 2006M3*) -4.586 (1986M2*; 2002M3*) Wheat -3.972 -3.969 -3.456 (2006M10*; 2013M10*) -4.270 (2004M10*; 2013M4*) Corn -4.750 -4.772 -5.037 (2006M3*; 2013M10*) -4.969 (2005M7*; 2012M4*) Soy -5.343* (2006M10) -5.390* (2006M10) -4.472 (2007M3*; 2013M11*) -5.638* (2005M8*; 2013M3*) Beef -4.039 -4.886 -4.432 (1993M1*; 2009M5*) -4.169 (2002M4*; 2008M9*) *Statistically significant at 5% confidence level. a 1991M1-2021M12. b 1992M1-2021M12. c 1985M1-2021M12. e Zivot Andrews (ZA) unit-root test with one endogenous structural break in the intercept or in both the intercept and the deterministic trend; lags selected with AIC between 6 and 12 months; only statistically significant breaks are reported. f Clemente, Montanes and Reyes (CMR) unit-root test with two endogenous breaks (mean shifts) and deterministic trend; lags selected with AIC between 6 and 12 months; AO=Additive Outlier and IO=Innovational Outlier specifications; only statistically significant breaks are reported. 201Dating common commodity price and inflation shocks with alternative approaches Bio-based and Applied Economics 13(2): 171-201, 2024 | e-ISSN 2280-6172 | DOI: 10.36253/bae-14060 Table A3. Granger causality test (χ2) of VAR model estimates with Oil, Coal, Natural Gas and CPI as endogenous variables (1985M1-2021M12)a,b Price levels Logarithms of price levels Crude oil Coal 13.51* 8.529* Gas 3.558 9.221* CPI 5.138 8.354* Structural break dummies: 2003M11; 20013M5 4.955*; 1.273 -0.069*; -0.021 Coal Crude oil 16.09* 8.6054* Gas 7.41 0.01853 CPI 3.251 0.77946 Structural break dummies: 2003M11; 20013M5 2.131; -0.375 0.039*; 0.004 Gas Crude oil 0.739 1.253 Coal 56.59* 3.353 CPI 5.204 6.060 Structural break dummies: 2003M11; 20013M5 -0.257; -0.052 -0.034; -0.035 CPI Crude oil 67.25* 47.25* Coal 17.18* 15.91* Gas 5.573 0.876 Structural break dummies: 2003M11; 20013M5 0.067; 0.043 0.001; 0.000 *Statistically significant at 5% confidence level. a The period considered depends on natural gas data availability. b The VAR model specification includes a drift, a deterministic trend and lags decided on the basis of AIC. _Hlk149729446 _Hlk135142798 _Hlk135142780 _Hlk152021616 _Hlk136884351 _Hlk140394422 _Hlk149850987 _Hlk149823827 _Hlk116460470 _Hlk125295920 _Hlk125296237 _Hlk125295754 _Hlk94009644 Lessons learned and policy implications from 20 years of Swiss agricultural policy reforms: A review of policy evaluations Robert Huber1,*, Nadja El Benni2,3, Robert Finger1 The political economy determinants of agri-environmental funds in the European Rural Development Programmes Francesco Pagliacci1, Matteo Zavalloni2,* Economics sanction and barley price regime change in Iran Mohammad Mehdi Farsi Aliabadi*, Behzad Fakari Sardehaie Dating common commodity price and inflation shocks with alternative approaches Roberto Esposti An input-output hydro-economic model to assess the economic pressure on water resources Benedetto Rocchi1,*, Mauro Viccaro2,3, Gino Sturla1