Identifying Speculative Bubbles with an Infinite Hidden Markov Model

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Raftery’s (1995) criterion.5 Empirical Application: U.S Oil PriceThe second application investigates the existence of speculative bubble behaviors in the U.Soil price. Several papers have studied the evidence for bubbles in the oil price (among others,Phillips and Yu (2011), Sornette et al. (2009) and Shi and Arora (2011)). Although the sampleperiods and methodologies used in these papers are different, most studies have found evidenceof bubble existence.Our data is sampled from April 1983 to December 2010. The price of the nearest-monthWest Texas Intermediate futures contract, obtained from DataStream International, is usedas a proxy for the spot oil price.We deflate the oil price by U.S. Consumer Price Indexobtained from the U.S Bureau of Labour Statistics. The market fundamental is proxied bythe oil inventory, which is the ending stocks excluding Strategic Petroleum Reserve of crudeoil and petroleum products (thousand barrels) and comes from the U.S. Energy InformationAdministration.We apply both the MS2 model and the iHMM to the logarithmic real oil price and thelogarithmic oil inventory. 18 The priors are assumed to be the same as in Section 4. Theposterior probabilities of β st > 0 and the posterior mean of β st5, respectively.are presented in Figure 4 andFor the oil inventory, both models suggest that there is no evidence of explosive behavior.Namely, the posterior probabilities (posterior means) of the inventory from both models arebelow 0.5 (zero). In fact, the posterior mean of β stremains the same throughout the sampleperiod. 19 This supports that regime switching is not a feature of the oil inventory dynamic inthe sample period.For the oil price , the patterns of the posterior probability P (β st > 0|Y ) and the posterior18 For the oil inventory, we have estimated three versions of the data: the raw data, the data that is scaled by10 −5 and the log of the raw data. All results are the same for bubble detection and regime identification. So weonly report the results based on the log of the raw data.19 Notice that the posterior mean of β st from the iHMM is smaller than that from the MS2 model. This isbecause the iHMM uses hierarchical structures but the two-regime MS model does not.18
20 40 60 80 100 120Oil PriceP(β t>0 | Data,IHMM)P(β t>0 | Data,MS2)0.0 0.2 0.4 0.6 0.8 1.013.70 13.75 13.80 13.85 13.90Log oil inventoryP(β t>0 | Data,IHMM)P(β t>0 | Data,MS2)0.0 0.2 0.4 0.6 0.8 1.0198412 198709 199006 199302 199511 199808 200105 200401 200610 200907Figure 4: Posterior probabilities of β st > 0 for the logarithmic oil price and inventory20 40 60 80 100 120Oil PriceE(β t | Data,IHMM)E(β t | Data,MS2)−0.06 −0.04 −0.02 0.0013.70 13.75 13.80 13.85 13.90Log Oil inventoryE(β t | Data,IHMM)E(β t | Data,MS2)−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3198412 198709 199006 199302 199511 199808 200105 200401 200610 200907Figure 5: Posterior mean of β stfor the logarithmic oil price and inventory19
Page 2 and 3: 1 IntroductionBubbles, which are re
Page 4 and 5: involves nonstationary (especially
Page 6 and 7: 2 Infinite Hidden Markov ModelThe i
Page 8 and 9: This methodology is less subjective
Page 10 and 11: efficient than the individual sampl
Page 12: 3.2 Dating Algorithm of BubblesThe
Page 15 and 16: (money growth) is consistent with t
Page 20 and 21: mean E (β st |Y ) from the MS2 mod
Page 22 and 23: Ferguson. A bayesian analysis of so
Page 24 and 25: Y.W. Teh, M.I. Jordan, M.J. Beal, a
Page 26 and 27: ∑where n i is the number of occur
Page 28 and 29: A.2 Stick breaking processFor a ran
Page 30 and 31: where n ji is the number of {τ | s

Identifying Speculative Bubbles with an Infinite Hidden Markov Model

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