In Section 2.2 we make several predictions regarding the characteristics of a bubble and the likelihood of a subsequent crash, based on the theoretical model of Abreu and Brunnermeier (2003). First, we hypothesize that a stronger growth rate of the bubble increases the likelihood of a crash. Second, if a bubble is difficult to date, the length of the bubble should be unrelated to the probability of a crash. In this section we analyze these hypotheses empirically.
We use two measures for the strength of the bubble. The first measure, labeled
STRENGTH1, is simply the t-ratio of αi1t in Eq. (2.1) for the candidate window that is selected as a bubble. It indicates the value with which the bubble exceeds the threshold. However, since our bubble search procedure is based on maximizing the length of the bubble period and not the strength, we also slightly modify our search procedure to find the strongest bubble during each estimation period. Instead of choosing the longest bubble during each estimation period, we select the bubble with the maximum t-ratio given that it fulfills the minimum and maximum length requirements. This procedure leads to our second measure of the strength of a bubble calledSTRENGTH2. For each of the two measures of bubble strength we also compute the corresponding length of the bubble, calledLENGTH1andLENGTH2, respectively.
The average value forSTRENGTH1is 2.57. As expected, the average t-statistic of
STRENGTH2, 3.10, is larger thanSTRENGTH1. LENGTH1 equals on average 39 months, while the average of LENGTH2 is shorter with 29 months. The respective standard deviations are 0.34, 0.69, 17 and 15. We use these at a later stage to determine the economic significance of the variables with respect to crash likelihood.
We analyze how the different bubble characteristics affect the probability of a crash in a standard logit model:
P r[ηit+1≤ −1.65σit|xit] = F (a + b0xit) F (y) = ey
1 + ey, (2.4)
where ηit+1 is the abnormal return of sector i at time t + 1, −1.65σit reflects the category zero threshold and xitis a vector of explanatory variables. As explanatory variables we include a dummy for the presence of a bubble during the previous six months (BUBBLE0), a dummy if there was a bubble during the last two to six months (LBUBBLE), the number of months between the crash and the last bubble (LAGS) as well as the bubble characteristics that we discussed earlier.
The results of this analysis are presented in Table 2.5. In panel (a) we focus on the bubble characteristics STRENGTH1 and LENGTH1. For crash category 2 to 5 our results confirm that the strength of the bubble affects the likelihood of a crash. We calculate that an increase in STRENGTH1 by one standard deviation (from 2.57 to 2.91) causes a relative increase in the likelihood of a crash by 28%.11 For more severe crashes the effect of strength becomes larger. For category 5 crashes, we find that for every increase of one standard deviation in the t-statistic the probability of a crash increases by 63%. If the volatility remains at a similar level of say 5% per month, and the bubble has an average length of 39 months, this means that the bubble yields an extra return of 0.34 ·√
39 · 5% = 10% over its life span. In none of the regressions do we find a significant relation between the length of the bubble and the probability of a crash.
In Table 2.5, panel b, we investigate the relation between crash likelihood and
STRENGTH2 and LENGTH2. We find again that the strength of a bubble is signifi-cantly related to the likelihood of a crash, whereas the length is not. For the other explanatory variables such as BUBBLE0 and LAGS, the results are also similar to our prior findings. In panel c, we report the estimates for the model with STRENGTH1
andSTRENGTH2 included. The results confirm the findings above, and indicate that
STRENGTH2has more explanatory power thanSTRENGTH1.
The logit model enables us to investigate whether the influence of a bubble be-comes less and less if it stopped growing longer ago, or that it only matters whether a bubble is still inflating or not. If the influence becomes gradually less, the coefficient for the LAGSvariables should be significant. If there is mainly a difference between still inflating and stopped growing, the coefficient onLBUBBLEshould be significant.
Since the coefficient on LBUBBLEis not significant in any of the settings, while the coefficient on LAGSis in most, we interpret this as evidence supporting a gradually decreasing influence of a bubble.
Our results in this section support the hypothesis that the strength of the bubble increases the likelihood of a crash. We also find evidence that a bubble is difficult to date. Finally we find that the effect of a bubble diminishes gradually after it stopped growing. These conclusions are robust to a different construction of the variables.
11We base this calculation on the first order approximation of the logistic model in Eq. (2.4). For STRENGTH1and category one crashes we find an increase of 0.56%, which is a relative increase of 28% compared to the situation when a bubble is absent (which has a probability 2.0% according to Table 2.4).
Table2.5:Logitmodelsfortheprobabilitythatafirstcrashoccurs Cat.0Cat.1Cat.2Cat.3Cat.4 modela constant−3.13∗∗(0.03)−3.88∗∗(0.04)−4.35∗∗(0.05)−4.81∗∗(0.06)−5.56∗∗(0.09) BUBBLE’0.62∗∗(0.07)0.84∗∗(0.09)0.99∗∗(0.11)1.09∗∗(0.13)1.01∗∗(0.20) LBUBBLE−0.33(0.21)0.06(0.26)0.03(0.33)−0.05(0.40)0.73(0.56) LAGS0.02(0.06)−0.18∗(0.09)−0.27∗(0.12)−0.26(0.14)−0.63∗(0.26) STRENGTH10.13(0.17)0.42∗(0.17)0.52∗∗(0.18)0.47∗(0.21)0.77∗∗(0.26) LENGTH10.001(0.004)−0.002(0.005)−0.002(0.006)0.000(0.007)−0.017(0.010) logL−4474.8−2689.7−1891.6−1347.6−723.73 modelb constant−3.13∗∗(0.03)−3.88∗∗(0.04)−4.35∗∗(0.05)−4.81∗∗(0.06)−5.56∗∗(0.09) BUBBLE’0.59∗∗(0.07)0.82∗∗(0.09)0.96∗∗(0.11)1.03∗∗(0.14)1.00∗∗(0.20) LBUBBLE−0.31(0.20)0.05(0.26)0(0.33)−0.05(0.40)0.71(0.56) LAGS0.04(0.06)−0.16(0.09)−0.24∗(0.12)−0.23(0.14)−0.60∗(0.26) STRENGTH20.22∗∗(0.08)0.29∗∗(0.09)0.35∗∗(0.11)0.41∗∗(0.12)0.39∗(0.17) LENGTH1−0.007(0.004)−0.005(0.005)−0.007(0.007)−0.005(0.008)−0.010(0.011) logL−4470.4−2687.6−1889.6−1344.4−724.84 modelc constant−3.13∗∗(0.03)−3.88∗∗(0.04)−4.35∗∗(0.05)−4.81∗∗(0.06)−5.56∗∗(0.09) BUBBLE’0.58∗∗(0.07)0.85∗∗(0.09)0.97∗∗(0.11)1.04∗∗(0.13)1.08∗∗(0.18) LAGS−0.06(0.03)−0.15∗∗(0.05)−0.24∗∗(0.07)−0.24∗∗(0.08)−0.29∗(0.12) STRENGTH1−0.14(0.19)0.10(0.21)0.17(0.23)−0.01(0.26)0.24(0.36) STRENGTH20.29∗∗(0.10)0.27∗(0.12)0.30∗(0.14)0.44∗∗(0.15)0.30(0.23) logL−4474.7−2690.9−1890.3−1345.3−726.92 Weestimatetheprobabilitythatafirstcrashoccursusingstandardlogitmodelswithdifferentbubblecharacteristicsasexplanatoryvariables.We includeaconstantandadummyforabubblethatmayendatmostsixmonthsbeforethelastobservation(BUBBLE0).LBUBBLEequalsoneifthe bubbledoesnotincludethelastobservation.LAGScountsthenumberofmonthssincethebubblehasended.Further,weconsiderthet-statisticof thebubble(STRENGTH1),itslength(LENGTH1),themaximumt-statisticoverasubperiodwithinthebubble(STRENGTH2)anditslength(LENGTH2). Allcharacteristicshavebeendemeaned.Asadependentvariableweconsiderthenumberoffirstcrashesinthefivecategories.Wereportthe estimateswithstandarderrorsinparentheses,andthevalueoftheloglikelihoodfunction.Asingle(double)asteriskafteranestimateindicates significanceatthe5%(1%)confidencelevel.
Table 2.6: Distribution of market crashes and crashes for the average industry
market crashes industry crashes
all crashes first crashes aftershocks all crashes first crashes aftershocks
cat. 0 41 22 19 45 24 21
cat. 1 24 59% 8 36% 16 84% 25 55% 13 52% 12 59%
cat. 2 20 49% 7 32% 13 68% 17 38% 8 34% 9 44%
cat. 3 18 44% 7 32% 11 58% 12 27% 5 22% 7 32%
cat. 4 10 24% 4 18% 6 32% 7 15% 3 11% 4 20%
Market crashes are defined in similar way as for sectors. Abnormal returns are constructed by subtracting the long run average market return from the observed returns. The average industry crashes are constructed by dividing the pooled crashes (see Tables 2.2 and 2.10) by the total number of observations (37,800) and multiplying it by the number of market observations (822). We consider all crashes, first crashes and aftershocks.