الرئيسية
[World Scientific Series in Finance] Stock Market Crashes Volume 13 (Predictable and Unpredictable...
[World Scientific Series in Finance] Stock Market Crashes Volume 13 (Predictable and Unpredictable and What to do About Them)  The High Price–Earnings Stock Market Danger Approach of Campbell and Shiller versus the BSEYD Model
Ziemba, William T, Zhitlukhin, Mikhail, Lleo, Sebastienكم أعجبك هذا الكتاب؟
ما هي جودة الملف الذي تم تنزيله؟
قم بتنزيل الكتاب لتقييم الجودة
ما هي جودة الملفات التي تم تنزيلها؟
المجلد:
10.1142/10
عام:
2017
اللغة:
english
DOI:
10.1142/9789813223851_0004
ملف:
PDF, 12.43 MB
الشعارات الخاصة بك:
أبلغ عن مشكلة
This book has a different problem? Report it to us
إختار نعم إذا كان
إختار نعم إذا كان
إختار نعم إذا كان
إختار نعم إذا كان
نجحت في فتح الملف
يحتوي الملف على كتاب (يُسمح أيضًا بالرسوم الهزلية)
محتوى الكتاب مقبول
يتوافق الإسم و المؤلف و لغة الملف مع وصف الكتاب. تجاهل الخانات الأخرى لأنها ثانوية!
إختار لا إذا كان
إختار لا إذا كان
إختار لا إذا كان
إختار لا إذا كان
 الملف تالف
 الملف محمي بواسطة الوسائل التقنية لحماية حق المؤلف DRM.
 الملف ليس كتاب (علي سبيل المثال، xls, html, xml)
 الملف عبارة عن مقال
 الملف مقتطف من كتاب
 الملف عبارة عن مجلة
 الملف نموذج إختبار
 الملف سبام
هل تعتقد أن محتوى الكتاب غير مناسب ويجب حظره
لا يتطابق الإسم أو المؤلف أو لغة الملف مع وصف الكتاب. تجاهل الخانات الأخرى.
Are you sure you want to report this book? Please specify the reason below
Change your answer
Thanks for your participation!
Together we will make our library even better
Together we will make our library even better
سيتم إرسال الملف إلى عنوان بريدك الإلكتروني. قد يستغرق الأمر ما يصل إلى 15 دقائق قبل استلامه.
سيتم إرسال الملف إلى حساب كندل Kindle الخاص بك. قد يستغرق الأمر ما يصل إلى 15 دقائق قبل استلامه.
ملاحظة: أنت بحاجة للتحقق من كل كتاب ترسله إلى Kindle. تحقق من صندوق بريدك الإلكتروني بحثًا عن رسالة تأكيد بالبريد الإلكتروني من Amazon Kindle Support.
ملاحظة: أنت بحاجة للتحقق من كل كتاب ترسله إلى Kindle. تحقق من صندوق بريدك الإلكتروني بحثًا عن رسالة تأكيد بالبريد الإلكتروني من Amazon Kindle Support.
جاري التحويل إلى
التحويل إلى باء بالفشل
0 comments
يمكنك ترك تقييم حول الكتاب ومشاركة تجربتك. سيهتم القراء الآخرون بمعرفة رأيك في الكتب التي قرأتها. سواء كنت قد أحببت الكتاب أم لا ، فإنك إذا أخبرتهم بأفكارك الصادقة والمفصلة ، فسيجد الناس كتبا جديدة مناسبة لهم ولإهتماماتهم.
1
2


Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Chapter 4 The High Price–Earnings Stock Market Danger Approach of Campbell and Shiller versus the BSEYD Model We show how to use Campbell and Shiller’s work on pricetoearnings (P/E) ratio and the predictability of longterm returns to create a crash prediction measure: the high P/E measure. Next, we present a statistical procedure to test the accuracy of crash prediction models. We use this procedure to test the accuracy of the bond–stock earnings yield diﬀerential (BSEYD) and high P/E models on a 51year period on the US market, starting on January 1, 1962, and ending on December 31, 2014 (12,846 daily data points). At the end of the Chapter, we expand the analysis beyond the US market to look at the two main Chinese stock markets: Shanghai and Shenzhen. Material in this chapter is based on Lleo and Ziemba (2017) and Lleo and Ziemba (2016c). 4.1. 4.1.1. Exploring the predictability of longterm returns From P/E to PE10 Financial market practitioners routinely use the pricetoearnings (P/E) ratio to gauge the relative valuation of stocks and stock markets, and the explanatory power with respect to stock over and underperformance is well documented (Fama and French, 1992). Table 4.1 reports the evolution of the P/E–ratio over selected 20year periods with high annual returns. In each period, the P/E ends 1.6 to 4.7 times higher than it started. Furthermore, there is a 90% correlation between the annual returns and the ending P/E ratio. 55 page 55 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 56 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.1. Evolution of the P/E over selected 20year periods w; ith high annualized returns. Beginning year Ending year Annual rate of return Beginning P/E Ending P/E 1975 1977 1942 1983 1978 1981 1979 1982 1980 1994 1996 1961 2002 1997 2000 1998 2001 1999 9.6% 9.7% 9.9% 10.9% 11.9% 12.8% 12.9% 13.0% 14.0% 10.9 11.5 12.2 7.3 10.4 8.8 9.4 8.5 8.9 20.5 25.9 20.5 25.9 31.0 41.7 36.0 32.1 42.1 Source: Bertocch et al. (2010). In fact, Benjamin Graham and David Dodd were among the ﬁrst to use the P/E ratio to gauge the relative valuation of stocks, dedicating much of Chapter XXXIX of Security Analysis to the subject (Graham and Dodd, 1934). Graham and Dodd’s book was a source of inspiration for John Campbell and Robert Shiller when they set out to question the predictability of longterm equity returns. Campbell and Shiller (1988) proposed a vectorautoregressive model relating the log return on the S&P500 with the log dividend–price ratio, lagged dividend growth rate and average annual earnings over the previous 10 and 30 years. We can trace the use of average earnings to Graham and Dodd: they advocated taking the average earnings over 10 years to average out the eﬀect that unique or speciﬁc economic or business conditions might have on current earnings. Campbell and Shiller started by performing a regression of the log returns on the S&P500 over 1 year, 3 years and 10 years against each of these variables and the average annual earnings over the previous 10 years. Continuing their study, they found that the R2 of a regression of log returns on the S&P500 over a 10year period against the log of the P/E ratio computed using average earnings over the previous 10 and 30 years is signiﬁcant. Speciﬁcally, Campbell and Shiller deﬁne the oneperiod total return on the stock as beg Pt+1 + Dt beg , h1t := ln Ptbeg page 56 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 57 57 where, Ptbeg is the price of the stock at the beginning of period t and Dt is the dividend received during period t. Although Campbell and Shiller do not specify a present value or future value rule for the dividend, we will consider that all dividends received during period t are either carried or future valued to the end of period t, so that Dt = Dtend . The i period total return on the stock is hbeg it := i−1 h1,t+j . j=0 The regression in Campbell and Shiller (1988) is beg P t + , hbeg it = a + b ln beg BSEYDE t,−n (4.1) beg where Pt is the level of the S&P500 index at time t and BSEYDE t,−n is the average of past annual earnings over the last n years, namely beg BSEYDE t,−n = n−1 1 beg E . n i=0 t−i The R2 computed by Campbell and Shiller (1988) for n = 30 is 0.566, higher than the 0.401 computed for n = 10 and higher than the R2 of regressions against the log dividend–price ratio and lagged dividend growth rate. Hence, the regression based on 30year average earnings has a greater explanatory power than the regression based on 10 years of earnings, and than any other variable. To explain this diﬀerence, Shiller (1996) suggests that a 10year average may still be sensitive to changes in the business cycle, whereas a 30year average should be immune to shortterm shifts in the business cycle. Campbell and Shiller (1989) also conducted a Monte Carlo study to address the possibility of a small sample bias in their initial analysis. They found that the size of their sample did not provide a material explanation for their initial conclusions: the P/E ratio based on 30year average earnings and on 10year average earnings help explain the longterm performance of the S&P500. These ﬁndings led Shiller to suggest the use of a Cyclically Adjusted P/E ratio (PE10), a P/E ratio using an 10year average real (inﬂationadjusted) earnings, to forecast the evolution of the equity risk premium (see Shiller, 2015). July 31, 2017 10:48 58 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 4.1.2. Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Relating P/E and stock valuation Campbell and Shiller do not explain how their empirical results ﬁt with asset pricing theory. To ﬁll this gap and provide a ﬁrm theoretical footing for their observations, we use the Gordon (1959) growth model to derive an approximate linear relation between the oneperiod log return on the S&P500 and the earnings yield. On the assumptions of the Gordon Growth Model, and using standard notation, the price of a stock at time t equals Ptend = end Dt+1 + Pt+1 , 1+k where Ptend is the price of the stock at the beginning of period t, and k is the cost of equity. This relation implies that the holding period return is linear in the current earnings yield end Pt+1 + Dt = 1 + k − gdρt , end Pt where g is the constant growth rate of dividends, d is the dividend payout E end ratio and ρend = P tend . The logarithmic return is t t end Pt+1 + Dt+1 gd end end = ln(1 + k) + ln 1 − ρt . h1t = ln k Ptend We linearize this expression by performing a ﬁrstorder Taylor expansion around ρt = ρ̄, where ρ̄ is the average longterm earnings yield end hend (4.2) 1t ≈ a0 − a1 ρt , gd gd gd ρ̄ + 1+k−gd where a0 := ln(1 + k) + ln 1 − 1+k ρ̄ and a1 := 1+k−gdρ̄ . Provided that the price at the end of period t and at the beginning of beg period t + 1 are equal, that is Ptend = Pt+1 , then it follows that ρend = ρbeg t t+1 and beg end Pt+1 + Dt Pt + Dt beg = ln = hend h1t = ln (4.3) 1(t−1) end Pt−1 Ptbeg Therefore, the two models are equivalent, with a small notation change. Equation (4.2) diﬀers from the Campbell–Shiller regression model (4.1) in two important ways. First, the Campbell–Shiller model regresses log returns against log P/E while the Gordon model relates log returns to earning yield (no log). Second, the Campbell–Shiller model uses the P/E ratio whereas the Gordon model is based on the earnings yield. However, the page 58 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 59 59 impact of this second diﬀerence is minor. The properties of the logarithm imply that, E t,−n Pt + = a − b ln + . (4.4) ln (hit ) = a + b ln Pt E t,−n Hence, only the regression slope changes sign. The signiﬁcance of the model is not aﬀected otherwise. 4.1.3. Using the P/E ratio as a crash prediction model We saw in Chapter 3 that crash prediction models generate a signal to indicate a downturn in the equity market at a given horizon H. This signal SIGNAL(t) occurs whenever the value of a given measure M (t) crosses a time varying threshold K(t): SIGNAL(t) = M (t) − K(t) > 0. (4.5) The P/E ratio, calculated using either current earnings or using 10year average earnings, provides the measure M (t). The threshold K(t) is either standard conﬁdence interval around the measure based on 252day rolling horizon statistics, or an upper bound derived from Cantelli’s inequality. 4.2. A statistical test of accuracy for crash prediction models Testing the accuracy of crash prediction models is a ﬁvestep process, starting from deﬁning what we mean by equity market crash, to constructing a “hit” sequence, estimating the probability of a crash, performing a likelihood ratio test, and ﬁnally addressing small sample bias. We discuss each step in detail below. Step 1: Define An Equity Market Crash We deﬁne an equity market crash as a decline of at least 10% in the level of the S&P500 from peak to trough based on closing prices for the day, over a period of at most one year (252 trading days). A crash is formally identiﬁed on the day when the closing price crosses the 10% threshold. The crash identiﬁcation algorithm is as follows: (1) Identify all the local troughs in the data set. Today is a local trough if there is no lower closing price within ±d business days. July 31, 2017 10:48 60 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. (2) Identify the crashes. Today is a crash identiﬁcation day if all of the following conditions hold: (a) The closing level of S&P500 is down at least 10% from its highest level within the past year today, and the loss was less than 10% yesterday; (b) This highest level reached by the S&P500 prior to the present crash diﬀers from the highest level corresponding to a previous crash; (c) This highest level occurred after the local trough that followed the last crash. The objective of these rules is to guarantee that the crashes we identify are distinct. Two crashes are not distinct if they occur within the same larger market decline. The choice of the parameter d is not as important as might initially appear. In practice, selecting d in the range [1, 94], will still lead to the identiﬁcation of the same 22 crashes. The diﬀerence is in the number of local troughs and the amplitude of the various crashes. As d increases past a value of 94 days, our algorithm identiﬁes fewer crashes. For example, we identify 21 crashes for d = 95, 20 crashes for d = 100, and 15 crashes for d = 200. In this paper, we pick d = 90 in order to capture the full amplitude of some of the crashes, such as the 56% market meltdown of 2008. In practice, this choice will not aﬀect our statistical test, because the test is based on the number of crashes and crash identiﬁcation dates. These numbers are the same whenever we select d in the interval [1, 94]. Table 4.2 shows the 22 crashes that occurred between January 31, 1964 and December 31, 2014. We observe the October 1987 crash, the Russian currency default and LTCM in 1998, the dot.com collapse in 2000–2002 and the subprime crisis in 2007–2009. The average duration of a correction was 199 days, and the average decline was 20.3%. Table 4.2 gives the crash identiﬁcation date, deﬁned as the date at which the decline from the previous local peak reaches 10%. With this information, we deﬁne a crash indicator sequence C = {Ct , t = 1, . . . , T }, where Ct takes the value 1 if date t is a crash identiﬁcation date and 0 otherwise, and the associated vector c = (C1 , . . . , Ct , . . . CT ). Thus, the event “a crash is identiﬁed on day t” is represented as {Ct = 1}. This variable plays a leading role in the hypothesis test. We repeat this procedure for each day in our sample. The result is a binary sequence with “1” in 22 entries, one for each crash, and “0” everywhere else. page 60 07–10–1966 05–03–1968 29–07–1969 23–11–1971 03–10–1974 16–09–1975 06–03–1978 14–11–1978 07–11–1979 27–03–1980 25–09–1981 24–07–1984 04–12–1987 30–01–1990 11–10–1990 27–10–1997 31–08–1998 15–10–1999 04–04–2001 09–03–2009 02–07–2010 03–10–2011 Peaktotrough decline (%) Peaktotrough duration (in days) 73.2 87.72 89.48 90.16 62.28 82.09 86.9 92.49 99.87 98.22 112.77 147.82 223.92 322.98 295.46 876.99 957.28 1247.41 1103.25 676.53 1022.58 1099.23 22.2% 10.1% 17.4% 13.9% 48.2% 14.1% 19.4% 13.6% 10.2% 17.1% 19.7% 14.4% 33.5% 10.2% 19.9% 10.8% 19.3% 12.1% 27.8% 56.8% 16.0% 19.4% 240 162 242 209 630 63 531 63 33 43 301 288 101 113 87 20 45 91 376 517 70 157 b2882ch04 94.06 97.59 108.37 104.77 120.24 95.61 107.83 106.99 111.27 118.44 140.52 172.65 336.77 359.8 368.95 983.12 1186.75 1418.78 1527.46 1565.15 1217.28 1363.61 S&P Level at trough 9in x 6in 09–02–1966 25–09–1967 29–11–1968 28–04–1971 11–01–1973 15–07–1975 21–09–1976 12–09–1978 05–10–1979 13–02–1980 28–11–1980 10–10–1983 25–08–1987 09–10–1989 16–07–1990 07–10–1997 17–07–1998 16–07–1999 24–03–2000 09–10–2007 23–04–2010 29–04–2011 Trough date Stock Market Crashes 16–05–1966 05–03–1968 19–06–1969 04–08–1971 27–04–1973 08–08–1975 25–05–1977 26–10–1978 25–10–1979 10–03–1980 24–08–1981 13–02–1984 15–10–1987 30–01–1990 17–08–1990 27–10–1997 14–08–1998 29–09–1999 14–04–2000 26–11–2007 20–05–2010 04–08–2011 S&P Index at Peak 10:48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Peak date 61 page 61 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 Crash identiﬁcation date The High Price–Earnings Stock Market Danger Approach of Campbell Table 4.2. The S&P500 index experienced 22 corrections between January 31, 1964 and December 31, 2014. July 31, 2017 10:48 62 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Step 2: Construct the Hit Sequence X The construction process for the signal and hit sequence is crucial to ensure that the crash prediction models produce out of sample predictions free from lookahead bias. It also eliminates data snooping by setting the parameters ex ante during the signal construction, with no possibilities of changing them when we construct the hit sequence. More importantly, the construction of the hit sequence removes the eﬀect of autocorrelation, making it possible to test the accuracy of the measures using a standard likelihood ratio test. While predicting returns is extremely diﬃcult, earlier research suggests that it is possible to predict crashes based on a signal. To check this, we perform a simple and eﬀective statistical signiﬁcance test. All crash prediction models have two components: (1) a signal which takes the value 1 or 0 depending on whether the measure has crossed the conﬁdence level, and (2) a crash indicator which takes value 1 when an equity market crash occurs and 0 otherwise. From a probabilistic perspective, these components are Bernoulli random variables. We start by computing the signal St for each day period t over our historical sample t = 1, . . . , T . Next, we deﬁne a signal indicator sequence S = {St , t = 1, . . . , T }. This sequence records the ﬁrst day in a series of positive signals as the signal date and only counts distinct signal dates. Two signals are distinct if a new signal occurs more than 30 days after the previous signal. The objective is to have enough time between two series of signals to identify them as distinct. The signal indicator St takes the value 1 if date t is the starting date of a distinct signal and 0 otherwise. Thus, the event “a distinct signal starts on day t” is represented as {St = 1}. We express the signal indicator sequence as the vector s = (S1 , . . . , St , . . . , ST ). We denote by Ct,H the indicator function returning 1 if the crash identiﬁcation date of at least one equity market correction occurs between time t and time t + H. The relation between Ct,H and Ct is Ct,H := 1 − H (1 − Ct+i ) . i=1 We identify the vector CH with the sequence CH := {Ct,H , t = 1, . . . , T −H} and deﬁne the vector cH := (C1,H , . . . , Ct,H , . . . CT −H,H ). The accuracy of the crash prediction model is the conditional probability P (Ct,H = 1St = 1) of a crash being identiﬁed between time t and time t + H, given that we observed a signal at time t. The higher page 62 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 63 63 the probability, the more accurate the model. We use maximum likelihood to estimate this probability and to test whether it is signiﬁcantly higher than a random draw. We can obtain a simple analytical solution because the conditional random variable {Ct,H = 1St = 1} is a Bernoulli trial with probability p = P (Ct,H = 1St = 1). To estimate the probability p, we change the indexing to consider only events along the sequence {St St = 1, t = 1, . . . T } and denote by X := {Xi , i = 1, . . . , N } the “hit sequence” where xi = 1 if the ith signal is followed by a crash and 0 otherwise. Here, N denotes the total number of signals, that is, N= T St = 1 s, t=1 where 1 is a vector with all entries set to 1 and v denotes the transpose of vector v. The sequence X can be expressed in vector notation as x = (X1 , X2 , . . . , XN ). Step 3: Estimate p = P (Ct,H St ) Via Maximum Likelihood Methods The likelihood function L associated with the observations sequence X is L(pX) := N pXi (1 − p)1−Xi i=1 and the log likelihood function L is L(pX) := ln L(pX) = N Xi ln p + i=1 N− N Xi ln(1 − p) i=1 This function is maximized for p̂ := N i=1 N Xi (4.6) so the maximum likelihood estimate of the probability p = P (Ct,H St ) is the historical proportion of correct predictions out of all observations. Step 4: Perform a Likelihood Ratio Test We apply a likelihood ratio test to test the null hypothesis H0 : p = p0 against the alternative hypothesis HA : p = p0 . The null hypothesis reﬂects the idea that the probability of a random, uninformed, signal correctly July 31, 2017 10:48 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 64 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes predicting crashes is p0 . A signiﬁcant departure above this level indicates that the measure we are considering has some predictive value. The likelihood ratio test is L(p = p0 X) L(p = p0 X) = . (4.7) Λ= maxp∈(0,1) L(pX) L(p = p̂X) The statistic Y := −2 ln Λ is asymptotically χ2 distributed with ν = 1 degree of freedom. We reject the null hypothesis H0 : p = p0 and accept that the model has some predictive power if Y > c, where c is the critical value chosen for the test. We perform the test for the three critical values 3.84, 6.63 and 7.88 corresponding respectively to a 95%, 99% and 99.5% conﬁdence level. A key advantage of this test is that it can be applied to all crash prediction models, whether based on fundamental variables (P/E, BSEYD or Fed model) or on probabilistic models (Jarrow–Protter, Shiryaev–Zitlukhin– Ziemba) and could apply more generally to any model designed to predict a market event. Step 5: Perform a Monte Carlo Study for Small Sample Bias A limitation of the likelihood ratio test is that the χ2 distribution is only valid asymptotically. In our case, the number of correct predictions follows a binomial distribution with an estimated probability of success p̂ and N trials. Between January 31, 1964 and December 31, 2014, 22 crashes occurred. As a result, the total number of signals N should also be low and we will have a discrete empirical distribution of test statistics. The continuous χ2 distribution might not provide an adequate approximation for this discrete distribution: p̂ might appear signiﬁcantly diﬀerent from p0 under a χ2 distribution but not under the empirical distribution. This diﬃculty is an example of small sample bias. Monte Carlo methods are the method of choice to identify and correct small sample bias. The Monte Carlo algorithm is as follows. Generate a large number K of paths. For each path k = 1, . . . , K, simulate N Bernoulli random variables with probability p0 of obtaining a “success.” Denote by Xk := Xik , i = 1, . . . , N the realization sequence where xki = 1 if the ith Bernoulli variable produces a “success” and 0 otherwise. Next, compute the maximum likelihood estimate for the probability of success given the realization sequence Xk as p̂ := N i=1 N Xik , page 64 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 65 65 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. and the test statistics for the path as Yk = −2 ln Λk = −2 ln L(p = p0 Xk ) L(p = p0 Xk ) = −2 ln . maxp∈(0,1) L(pk Xk ) L(p = p̂k Xk ) Once all the paths have been simulated, we use all K test statistics Yk , k = 1, . . . , K to produce an empirical distributions for the test statistic Y . From the empirical distribution, we obtain critical values at a 95%, 99% and 99.5% conﬁdence level against which we assess the crash prediction test statistics Y . The empirical distribution also enables us to compute a pvalue for the crash prediction test statistics. Finally, we compare the results obtained under the empirical distribution to those derived using the asymptotic χ2 distribution. 4.2.1. Modelonmodel significance test In addition to testing each crash prediction model separately, the likelihood ratio test also allows us to perform a pairwise modelonmodel test to determine whether the accuracy pi of a given model i, i = 1, . . . 32 is signiﬁcantly higher than the estimated accuracy p̂j = argmaxpj ∈(0,1) L(pj X) of model j, j = 1, . . . 32, j = i. Treating p̂j as ﬁxed for the purpose of this test, we apply a likelihood ratio test to test the null hypothesis H0 : pi = p̂j against the alternative hypothesis HA : pi = p̂j . A signiﬁcant departure above this level indicates that model i contains more information than model j. Here, the likelihood ratio test is Λ= L(p = p̂j X) L(p = p̂j X) = . maxpi ∈(0,1) L(pi X) L(p = p̂i X) (4.8) The statistic Y := −2 ln Λ is still asymptotically χ2 distributed with ν = 1 degree of freedom. We reject the null hypothesis H0 : pi = pj and accept that model i has a higher predictive power than model j if pi > pj and Y > c, where c is the critical value chosen for the test. As we treat p̂j as ﬁxed, the test is not symmetric in i and j. The likelihood ratio only enables us to conﬁrm whether pi is signiﬁcantly diﬀerent from p̂j , but does not convey accurate statistical information about the signiﬁcance of pj relative to pi . 4.2.2. Robustness The question of robustness is critical for all empirical models. In our model, we distinguish two closely connected views of robustness: robustness with July 31, 2017 10:48 66 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. respect to the dataset (data robustness) and robustness with respect to the speciﬁc parameters of the model (model parameter robustness). Addressing Data Robustness: Subperiod Study Data robustness addresses the question of the sensitivity of model predictions when the underlying dataset changes. This is the more traditional view of robustness, and it is generally dealt with either by using an outofsample study or by analyzing the behavior of the model in several subperiods. Measuring Parameter Robustness: Robust Likelihood Statistics Though crucial, the question of robustness has mostly remained peripheral to the debate on the predictability of market downturns and crashes. To address this question, and quatify model speciﬁcation robustness, we can use a robust version of the likelihood ratio and of the associated test statistics. Measuring the robustness of a crash prediction model using a robust likelihood ratio has signiﬁcant advantages. First, the likelihood ratio provides a direct, objective measure of robustness that allows direct comparison between diﬀerent model speciﬁcations and even diﬀerent types of models. Second, we can use the robust test statistics to evaluate the signiﬁcance of the model using a χ2 distribution. The immediate beneﬁt is that we can equate loss of robustness over a set of model parameters with loss of prediction signiﬁcance. Third, the idea of robust likelihood ratio can be easily extended to account for small sample bias. We can use the Monte Carlo algorithm presented in an earlier section to compute the (standard) likelihood ratio under each speciﬁcation and then select the lowest one as a robust likelihood ratio. The robust likelihood ratio generalizes the (standard) likelihood ratio to account for the impact of the choice of parameters on the predictions of the model. Compared with the standard likelihood ratio, we need to enlarge the set of model parameters to consider both free parameters and model speciﬁcations. Free parameters are the parameters that we seek to estimate based on the model output, such as the success probability p. Model speciﬁcations are the set of parameters that specify the model: type of earnings, threshold calculation, type of bond yields. Denoting by Ω the enlarged set of parameters, Θ the set of free parameters and M the set of model speciﬁcations, we have Ω = (Θ, M). This enlarged formulation leads us to revisit the deﬁnition of likelihood function page 66 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 67 67 and likelihood ratio. The model speciﬁcation µ ∈ M directly aﬀects the signal indicator sequence, the total number of signals and the probability of successfully predicting a crash. We denote the signal indicator sequence by S µ := {Stµ , t = 1, . . . , T } and express the signal indicator sequence as a vector sµ := (S1µ , . . . , Stµ , . . . , STµ ). The total number of signals N µ is T Nµ = Stµ = 1 sµ , t=1 and the probability of successfully predicting a crash is p = P (Ct,H = 1Stµ = 1). As a result the “hit sequence” X µ := {Xiµ , i = 1, . . . , N } where xµi = 1 if the ith signal is followed by a crash and 0 otherwise also depends on the model speciﬁcation µ. The likelihood function L associated with the observations sequence X µ generated under model speciﬁcation µ is µ L(pX, µ) := N µ µ pXi (1 − p)1−Xi . i=1 In our case, there is only one free parameter, the probability p, and as a result the set of free parameters Θ is eﬀectively the interval (0, 1). Maximizing the log likelihood function gives us an estimate of the maximum likelihood of successfully predicting crashes based on model speciﬁcation µ p̂µ := Nµ i=1 Xi . Nµ (4.9) The standard likelihood ratio under model speciﬁcation µ and given the null hypothesis H0 : p = p0 is Λµ = L(p = p0 X, µ) L(p = p0 X, µ) = . maxp∈(0,1) L(pX, µ) L(p = p̂µ X, µ) (4.10) We deﬁne the robust likelihood ratio Λ̆ given the null hypothesis H0 : p = p0 := 12 as the supremum over all possible model speciﬁcations of the standard likelihood ratio L(p = p0 X, µ) µ∈M maxp∈(0,1) L(pX, µ) Λ̆ = sup Λµ = sup µ∈M L(p = p0 X, µ) . µ µ∈M L(p = p̂ X, µ) = sup (4.11) July 31, 2017 10:48 68 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Higher robust likelihood ratios are indicative of a lower statistical signiﬁcance. The associated robust test statistic is Y̆ := inf Y µ µ∈M = inf −2 ln µ∈M L(p = p0 X, µ) L(p = p̂µ X, µ) L(p = p0 X, µ) . µ µ∈M L(p = p̂ X, µ) = −2 ln sup (4.12) Because the test statistic for the (standard) likelihood ratio under speciﬁcation µ, Y µ , is asymptotically χ2 distributed, the test statistic for robust likelihood ratio Y̆ is also asymptotically χ2 distributed. 4.3. Testing the predictive ability of the BSEYD and P/E In this section, we test the predictions of 8 P/Ebased models and 32 BSEYDbased models on the S&P500, over a 51year period starting on January 31, 1964, and ending on December 31, 2014 (12,846 daily data points). 4.3.1. Prediction models The statistical procedure discussed above is modeldependent. We need to specify the model we are testing before performing the actual statistical test. Specifying the model requires deﬁning the measure M (t), the threshold K(t) and the horizon H. We ﬁx the horizon at two years, meaning H = 504 days. To compute the measure M (t), we use both current earnings and average earnings over the past 10 years. This is consistent with the observation that the cyclicality of earnings matters. Graham and Dodd (1934), for example, suggest using 10 years of earnings in stock and company valuation. Although Campbell and Shiller tested their model using average earnings over 30 years, we consider only 10 years because our objective is to predict mediumterm market downturns rather than longterm market returns. As a result, we do not want our measures to be immune from the movements of the business cycle. For the interest rates, we use the yield on either the 10year U.S. Treasury Note, or seasoned corporate bonds rated Aaa or Baa by Moody’s. page 68 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 69 69 To deﬁne the threshold K(t), we use both a standard conﬁdence rule and Cantelli’s inequality. Overall, this gives us four ways of computing the crash signals for the P/E and log P/E measure, based on the deﬁnition of earnings (current or 10year average) and the type of conﬁdence interval (standard or Cantelli). Taking also into consideration the type of interest rates used (yield on a 10year Treasury Note or on a seasoned Aaa corporate bond), we have 12 ways of computing the crash signals for the BSEYD and logBSEYD measure. Table 4.3 lists these 32 signal models. The P/E and log P/E measures serve as bases for our four models: (1) P/E1 and log P/E1 use current earnings and a standard conﬁdence level; (2) P/E2 and log P/E2 use current earnings and Cantelli’s inequality; (3) P/E3 and log P/E3 use average earnings over 10 years and a standard conﬁdence level; (4) P/E4 and log P/E4 use average earnings over 10 years and Cantelli’s inequality. The BSEYD and logBSEYD measures are used in eight models each: (1) BSEYD1 and logBSEYD1 use current earnings, a standard conﬁdence level and yields on 10year Treasury Notes; (2) BSEYD2 and logBSEYD2 use current earnings, Cantelli’s inequality; (3) BSEYD3 and logBSEYD3 use average earnings over 10 years, a standard conﬁdence level and yields on 10year Treasury Notes; (4) BSEYD4 and logBSEYD4 use average earnings over 10 years, a Cantelli’s inequality and yields on seasoned Aaa corporate bonds; (5) BSEYD5 and logBSEYD5 use current earnings, a standard conﬁdence level and yields on seasoned Aaa corporate bonds; (6) BSEYD6 and logBSEYD6 use current earnings, Cantelli’s inequality and yields on seasoned Aaa corporate bonds; (7) BSEYD7 and logBSEYD7 use average earnings over 10 years, a standard conﬁdence level and yields on seasoned Aaa corporate bonds; (8) BSEYD8 and logBSEYD8 are based on average earnings over 10 years, a Cantelli’s inequality and yields on seasoned Aaa corporate bonds. (9) BSEYD9 and logBSEYD9 use current earnings, a standard conﬁdence level and yields on seasoned Baa corporate bonds; 10:48 P/E1 P/E2 P/E3 P/E4 logP/E1 logP/E2 logP/E3 logP/E4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 BSEYD11 BSEYD12 log P/E BSEYD Cantelli’s inequality Yield on Baa corporate bond Dataset start date Dataset end date Jan. 1, 1962 Jan. 1, 1962 Jan. 31, 1964 Jan. 31, 1964 Dec. Dec. Dec. Dec. 31, 31, 31, 31, 2014 2014 2014 2014 Jan. 1, 1962 Jan. 1, 1962 Jan. 31, 1964 Jan. 31, 1964 Dec. Dec. Dec. Dec. 31, 31, 31, 31, 2014 2014 2014 2014 Jan. 1, 1962 Jan. 1, 1962 Jan. 31, 1964 Jan. 31, 1964 Jan. 4, 1983 Jan. 4, 1983 Jan. 4, 1983 Jan. 4, 1983 Jan. 2, 1986 Jan. 2, 1986 Jan. 2, 1986 Jan. 2, 1986 Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 b2882ch04 P/E Confidence interval Yield on Aaa corporate bond 9in x 6in Signal Yield on 10year Treasury Note Stock Market Crashes Current earnings Measure Average earnings over 10 years Stock Market Crashes page 70 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 70 Table 4.3. List of 16 Signal models tested. Jan. 1, 1962 Jan. 1, 1962 Jan. 31, 1964 Jan. 31, 1964 Jan. 4, 1983 Jan. 4, 1983 Jan. 4, 1983 Jan. 4, 1983 Jan. 2, 1986 Jan. 2, 1986 Jan. 2, 1986 Jan. 2, 1986 Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. Dec. 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 2014 9in x 6in Stock Market Crashes 10:48 b2882ch04 71 page 71 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 logBSEYD1 logBSEYD2 logBSEYD3 logBSEYD4 logBSEYD5 logBSEYD6 logBSEYD7 logBSEYD8 logBSEYD9 logBSEYD10 logBSEYD11 logBSEYD12 The High Price–Earnings Stock Market Danger Approach of Campbell logBSEYD July 31, 2017 10:48 9in x 6in b2882ch04 Stock Market Crashes 72 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Stock Market Crashes (10) BSEYD10 and logBSEYD10 use current earnings, Cantelli’s inequality and yields on seasoned Baa corporate bonds; (11) BSEYD11 and logBSEYD11 use average earnings over 10 years, a standard conﬁdence level and yields on seasoned Baa corporate bonds; (12) BSEYD12 and logBSEYD12 use average earnings over 10 years, a Cantelli’s inequality and yields on seasoned Baa corporate bonds. 4.3.2. Dataset We use daily S&P500, earnings and P/E data for the period January 31, 1964 to December 31, 2014. We obtained the data from Bloomberg and used Thomson DataStream to crossreference and check them. The daily yields on the 10year Treasury Note for the period January 1, 1962 to December 31, 2014, the daily yields on seasoned corporate bonds rated Aaa by Moody’s for the period January 3, 1983 to December 31, 2014 and the daily yields on seasoned corporate bonds rated Baa by Moody’s for the period January 2, 1986 to December 31, 2014 come from the Board of Governors of the Federal Reserve System. Because our models require the simultaneous use of stock market prices, P/E ratios and bond market yields, we are limited by the shortest data series. The longest period for which we can test the accuracy of crash prediction models is 50 years and 11 months, starting January 31, 1964, if we use 10year Treasury Note. It comprises 12,815 daily data points. The shortest period for which we can test the accuracy of crash prediction models is 29 years, starting January 2, 1986, if we use seasoned corporate bonds rated Baa. 4.3.3. Signal time series We use daily data from January 31, 1964 to December 31, 2014, 12,815 observations, to construct the time series for all the P/E and logP/E models as well as the BSEYD and logBSEYD models based on 10year Treasury yields. For the BSEYD and logBSEYD models computed with Aaa corporate yields, we use daily data from January 3, 1983 to December 31, 2014, 8,036 observations. Finally, for the BSEYD and logBSEYD models computed with Baa corporate yields, we use daily data from January 2, 1986 to December 31, 2014, 7,277 observations. The results are displayed in Figures 4.1–4.8. The line represents the evolution of the signal at time t, SIGN AL(t): a crash signal occurs whenever the signal series crosses the threshold when SIGN AL(t) > 0. page 72 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 73 73 Figure 4.1 presents the signal time series related to the P/E measure. Panel (a) shows the PE1 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the PE2 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the PE3 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the PE4 signal series based on average earnings and Cantelli’s inequality. PE1 and PE2 are nearly identical, and the same is true of PE3 and PE4. Figure 4.2 presents the signal time series related to the logP/E measure. Panel (a) shows the lnPE1 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) charts the lnPE2 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the lnPE3 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the lnPE4 signal series, which is based on average earnings and Cantelli’s inequality. Figure 4.3 presents the signal time series related to the BSEYD measure computed with Treasury yields. Panel (a) shows the BSEYD1 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the BSEYD2 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the BSEYD3 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the BSEYD4 signal series which is based on average earnings and Cantelli’s inequality. Figure 4.4 presents the signal time series related to the BSEYD measure based on corporate yields with a Aaa rating. Panel (a) shows the BSEYD5 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the BSEYD6 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the BSEYD7 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the BSEYD8 signal series, which is based on average earnings and Cantelli’s inequality. Figure 4.5 presents the signal time series related to the BSEYD measure based on the yield of Baarated corporate bonds. Panel (a) shows the BSEYD9 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the BSEYD10 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the BSEYD11 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the BSEYD12 signal series, which is based on average earnings and Cantelli’s inequality. July 31, 2017 10:48 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 74 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Figure 4.6 presents the signal time series related to the logBSEYD models calculated using Treasury yields. Panel (a) shows the logBSEYD1 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the logBSEYD2 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the logBSEYD3 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the logBSEYD4 signal series, which is based on average earnings and Cantelli’s inequality. Figure 4.7 presents the signal time series related to the logBSEYD models computed using the yields of Aaarated corporate bond. Panel (a) shows the logBSEYD5 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the logBSEYD6 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the logBSEYD7 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the logBSEYD8 signal series, which is based on average earnings and Cantelli’s inequality. Figure 4.8 presents the signal time series related to the logBSEYD models computed with Baa corporate yields. Panel (a) shows the logBSEYD9 signal series, which is based on current earnings and standard conﬁdence intervals. Panel (b) shows the logBSEYD10 signal series, which is based on current earnings and Cantelli’s inequality. Panel (c) shows the logBSEYD11 signal series, which is based on average earnings and standard conﬁdence intervals. Lastly, Panel (d) shows the logBSEYD12 signal series, which is based on average earnings and Cantelli’s inequality. The time series for all 32 signal models lead to similar conclusions: the signals are very noisy, exhibiting sharp, asymmetric changes. The positive signals, that is, when SIGNAL(t) > 0, indicate that a crash may occur. Signals based on the P/E and BSEYD measures are more visible than those based on the logP/E and logBSEYD measures. Similarly, the signal models based on current earnings generate more and clearer positive signals than those based on average earnings over 10 years. The length and behavior of the data series make it diﬃcult to calculate the relationship between signals and crashes. As a preliminary step to computing summary statistics of the joint behavior of signals and crashes, we construct the crash indicator sequence CH and an indicator sequence of distinct signals S. We express both sequences as vectors with upto 12,815 entries, one for each date, depending on the period covered by the model. The entry Ct is “1” if a crash is identiﬁed on date t, and 0 otherwise while page 74 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 75 75 4 Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 2 0 –2 –4 –6 –8 –10 –12 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model P/E1 4 Threshold Signal 2 0 –2 –4 –6 –8 –10 –12 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model P/E2 Fig. 4.1. Signal for Models P/E. This ﬁgure presents the signal time series related to the P/E measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 76 5 Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 0 –5 –10 –15 –20 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model P /E3 5 Threshold Signal 0 –5 –10 –15 –20 1966 1970 1974 1978 1982 1986 1990 1994 (d) Model P/E4 Fig. 4.1. (Continued ) 1998 2002 2006 2010 2014 page 76 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 77 77 0.2 Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 0.1 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model log P/E1 0.2 Threshold Signal 0.1 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model log P/E2 Fig. 4.2. Signal for Models logP/E. The ﬁgure presents the signal time series related to the logP/E measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 78 0.2 Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model log P/E3 0.2 Threshold Signal 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 1966 1970 1974 1978 1982 1986 1990 1994 (d) Model log P/E4 Fig. 4.2. (Continued ) 1998 2002 2006 2010 2014 page 78 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 79 79 3.0% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 2.0% 1.0% 0.0% –1.0% –2.0% –3.0% –4.0% –5.0% –6.0% –7.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model BSEYD 1. 3.0% Threshold Signal 2.0% 1.0% 0.0% –1.0% –2.0% –3.0% –4.0% –5.0% –6.0% –7.0% –8.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model BSEYD 2. Fig. 4.3. Signal for Models BSEYD computed using the yield on Treasury notes. The ﬁgure shows the signal time series related to the BSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 80 4.0% Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 2.0% 0.0% –2.0% –4.0% –6.0% –8.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model BSEYD 3. 4.0% Threshold Signal 2.0% 0.0% –2.0% –4.0% –6.0% –8.0% 1966 1970 1974 1978 1982 1986 1990 1994 (d) Model BSEYD 4. Fig. 4.3. (Continued ) 1998 2002 2006 2010 2014 page 80 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 81 81 2% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 1% 0% –1% –2% –3% –4% –5% –6% 1983 1987 1991 1995 1999 2003 2007 2011 (a) Model BSEYD 5. 2% Threshold Signal 1% 0% –1% –2% –3% –4% –5% –6% 1983 1987 1991 1995 1999 2003 2007 2011 (b) Model BSEYD 6. Fig. 4.4. Signal for Models BSEYD computed using Aaa corporate bond yields. The ﬁgure shows the signal time series related to the BSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 82 1% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 0% –1% –2% –3% –4% –5% –6% –7% 1983 1987 1991 1995 1999 2003 2007 2011 (c) Model BSEYD 7. 1% Threshold Signal 0% –1% –2% –3% –4% –5% –6% –7% 1983 1987 1991 1995 1999 2003 (d) Model BSEYD 8. Fig. 4.4. (Continued ) 2007 2011 page 82 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 83 83 2% Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 1% 0% –1% –2% –3% –4% –5% 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model BSEYD 9. 2% Threshold Signal 1% 0% –1% –2% –3% –4% –5% 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model BSEYD 10. Fig. 4.5. Signal for Models BSEYD computed using Baa corporate bond yields. The ﬁgure shows the signal time series related to the BSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 84 2% Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 1% 0% –1% –2% –3% –4% –5% 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model BSEYD 11. 2% Threshold Signal 1% 0% –1% –2% –3% –4% –5% 1986 1990 1994 1998 2002 (d) Model BSEYD 12. Fig. 4.5. (Continued ) 2006 2010 2014 page 84 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 85 85 40.0% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 20.0% 0.0% –20.0% –40.0% –60.0% –80.0% –100.0% –120.0% –140.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model logBSEYD 1. 40.0% Threshold Signal 20.0% 0.0% –20.0% –40.0% –60.0% –80.0% –100.0% –120.0% –140.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model logBSEYD 2. Fig. 4.6. Signal for Models logBSEYD computed using the yield on Treasury notes. The ﬁgure shows the signal time series related to the logBSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 86 40.0% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 20.0% 0.0% –20.0% –40.0% –60.0% –80.0% –100.0% –120.0% –140.0% –160.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model logBSEYD 3. 40.0% Threshold Signal 20.0% 0.0% –20.0% –40.0% –60.0% –80.0% –100.0% –120.0% –140.0% –160.0% 1966 1970 1974 1978 1982 1986 1990 1994 1998 (d) Model logBSEYD 4. Fig. 4.6. (Continued ) 2002 2006 2010 2014 page 86 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 87 87 20% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% 1983 1987 1991 1995 1999 2003 2007 2011 (a) Model logBSEYD 5. 20% Threshold Signal 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% –90% 1983 1987 1991 1995 1999 2003 2007 2011 (b) Model logBSEYD 6. Fig. 4.7. Signal for Models logBSEYD computed using Aaa corporate bond yields. The ﬁgure shows the signal time series related to the logBSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 88 20% Threshold Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Signal 0% –20% –40% –60% –80% –100% –120% 1983 1987 1991 1995 1999 2003 2007 2011 (c) Model logBSEYD 7. 20% Threshold Signal 0% –20% –40% –60% –80% –100% –120% 1983 1987 1991 1995 1999 2003 (d) Model logBSEYD 8. Fig. 4.7. (Continued ) 2007 2011 page 88 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 89 89 20% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% 1986 1990 1994 1998 2002 2006 2010 2014 (a) Model logBSEYD 9. 20% Threshold Signal 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% 1986 1990 1994 1998 2002 2006 2010 2014 (b) Model logBSEYD 10. Fig. 4.8. Signal for Models logBSEYD computed using Baa corporate bond yields. The ﬁgure shows the signal time series related to the logBSEYD measure. The jagged line is the signal at time t, SIGNAL(t). The horizontal line at 0 is the threshold for the signal: a crash signal occurs whenever the signal series crosses the threshold, meaning SIGNAL(t) > 0. July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 90 20% Threshold Signal Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% 1986 1990 1994 1998 2002 2006 2010 2014 (c) Model logBSEYD 11. 20% Threshold Signal 10% 0% –10% –20% –30% –40% –50% –60% –70% –80% 1986 1990 1994 1998 2002 (d) Model logBSEYD 12. Fig. 4.8. (Continued ) 2006 2010 2014 page 90 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 91 91 the entry St is “1” if date t corresponds to the start date of a new and distinct signal, and 0 otherwise. Now we compute the total number of distinct signals N as N= T St = 1 s. t=1 Table 4.4, Column 2, shows that over the full period (1964–2014), the total number of signals ranges from 36 for the P/E2 and logP/E2 model to 55 for the P/E3 model. P/E and logP/E models using average earnings (P/E3, P/E4, logP/E3 and logP/E4) generate between 20% and 50% more signals than models based on current earnings (P/E1, P/E2, logP/E1 and logP/E2). On the other hand, BSEYD and logBSEYD models based on current earnings generate slightly more signals than models using average earnings. Over the shorter period 1983–2014, models BSEYD5–8 generated between 18 and 24 signals while the models logBSEYD5–8 yielded between 20 and 27 signals. Finally, models BSEYD9–12 generated between 16 and 23 signals while the models logBSEYD9–12 produced between 21 and 22 signals over the shortest period 1986–2014. The total number of distinct signals tells only a small part of the story. In particular, it does not directly measure the accuracy of the crash prediction models. As a ﬁrst measure of accuracy, we compute the number of correct predictions n, which is deﬁned as a tally of the crashes identiﬁed no more than H = 504 days after a positive signal: n= T Ct,H = 1 cH t=1 We will later use it to compute a percentage of correct predictions. For models computed over the full period (1964–2014), Table 4.4, Column 3, shows that the number of correct predictions reaches a high of 37 for the P/E3 and a low of 24 for the P/E2 model. This statistic allows multiples signals for the same crash. The BSEYD and P/E models based on current earnings generated between 24 and 29 correct predictions, while the P/E models based on average earnings generated between 35 and 37 correct predictions, albeit out of a much larger number of signals. BSEYD and logBSEYD calculated with Treasury yields are consistent, with respective averages of 29 and 32.25 successful predictions. Over the shorter period (1983–2014), the BSEYD models based on Aaa corporate yields produced between 12 and 15 correct predictions while the logBSEYD models generated between 13 and 17 correct predictions. The BSEYD models based on July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 92 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.4. Proportion of correct and incorrect predictions for each signal model. Signal Model (1) PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 BSEYD11 BSEYD12 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 Total number of signals (2) Number of correct predictions (3) Proportion of correct predictions (%) (4) Number of incorrect predictions (5) Proportion of incorrect predictions (%) (6) 37 36 55 49 37 36 50 44 38 39 40 39 24 22 18 19 23 19 16 16 47 44 42 39 20 21 27 26 22 22 22 21 25 24 37 35 25 25 34 31 29 28 30 29 15 14 12 12 14 11 12 11 34 33 32 30 13 14 17 17 15 14 15 16 67.57% 66.67% 67.27% 71.43% 67.57% 69.44% 68.00% 70.45% 76.32% 71.79% 75.00% 74.36% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 75.00% 68.75% 72.34% 75.00% 76.19% 76.92% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% 12 12 18 14 12 11 16 13 9 11 10 10 9 8 6 7 9 8 4 5 13 11 10 9 7 7 10 9 7 8 7 5 32.43% 33.33% 32.73% 28.57% 32.43% 30.56% 32.00% 29.55% 23.68% 28.21% 25.00% 25.64% 37.50% 36.36% 33.33% 36.84% 39.13% 42.11% 25.00% 31.25% 27.66% 25.00% 23.81% 23.08% 35.00% 33.33% 37.04% 34.62% 31.82% 36.36% 31.82% 23.81% Baa corporate yields produced between 11 and 14 correct predictions while the logBSEYD models produced between 14 and 16 correct predictions over the period 1986–2014. Combining the total number of distinct signals and the number of correct predictions gives us a second measure of accuracy: the proportion n . This measure enables us to of correct predictions, deﬁned as the ratio N page 92 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 93 93 compare the accuracy of the various models directly. It also gives us an estimate of p = P (Ct,H = 1St = 1), the probability of correctly predicting a crash. Table 4.4, Column 4, shows that the BSEYD and logBSEYD models based on Treasury yields are consistently more than 70% accurate over the entire period 1964–2014. With the exception of P/E4 and logP/E4, the P/E and logP/E measures are between 65% and 70% accurate. The BSEYD and logBSEYD measures based on Aaa corporate bonds are between 62.50% and 66.67% accurate, while the BSEYD and logBSEYD measures based on Baa corporate bonds ﬂuctuate between 57.89% and 76.19% accuracy. The last two columns in Table 4.4 report the number of incorrect predictions N − n, that is, the number of signals that were not followed by a crash, and the proportion of incorrect crashes, that is, the percentage of false positive given by the signal and computed as NN−n . 4.3.4. Crash prediction test Although the proportion of correct signals computed in Table 4.4 is generally high, the central question is whether it is signiﬁcantly diﬀerent from the accuracy p0 of an uninformed signal. We construct this uniformed signal by picking a day at random between January 31, 1964 and December 31, 2014 and checking whether a crash is identiﬁed within 252 days. This is the same rule used to compute the accuracy of our other measures: P/E, logP/E, BSEYD and logBSEYD. The probability p0 is therefore the probability that a crash will be identiﬁed within 252 days of a randomly selected day. To compute p0 empirically, we tally the number of days that are at most 252 days before a crash identiﬁcation date and divide by the total number of days in the sample. For the entire period January 31, 1964 to December 31, 2014, we ﬁnd that p0 = 39.85%. We can conﬁrm this number heuristically. Given that 22 distinct crashes occurred, then at most 252 × 22 = 5,544 days in our sample fall within 252 days prior to a crash identiﬁcation date. Because there are 12,815 days in the dataset, the heuristic probability is 5,544 12,815 = 43.26%, just slightly higher than the empirical probability. The diﬀerence between the heuristic and empirical probability is due to the fact that in reality, equity market corrections are not spread evenly through the period. In fact, corrections might occur in quick succession, as was the case in the late 1990s when three crashes occurred within less than two years. Table 4.5 reports the uninformed probability p0 for all ﬁve periods considered in our study: the full period; Subperiod 1 from January 31, 1964 to December 31, 1981; Subperiod 2 from January 4, 1982 to December July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 94 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.5. Empirical probability p0 . Number of days in the period Number of days less 252 days prior to a crash identiﬁcation date Empirical probability p0 Full dataset: Jan. 31, 1964–Dec. 31, 2014 Subperiod 1: Jan. 31, 1964–Dec 31, 1981 Subperiod 2: Jan. 4, 1982–Dec 31st, 2014 Aaa dataset: Jan. 3, 1983–Dec 31, 2014 Baa dataset: Jan. 2, 1986–Dec 31, 2014 12,815 4,484 8,320 8,067 7,308 5,107 2,613 2,494 2,494 2,242 39.85% 58.27% 29.98% 30.92% 30.68% 31st, 2014, the period from January 3, 1983 to December 31, 2014 used for Aaa bonds; and the period January 2, 1986 to December 31, 2014 for Baa bonds. The empirical uninformed probability p0 ﬂuctuates from a high of 58.27% over the ﬁrst subperiod to a low of 29.98% over the second subperiod. The empirical probabilities over subperiod 2, the Aaa dataset and the Baa dataset are within a narrow 30–31% range. Because the empirical proportion of correct signals computed in Table 4.4 corresponds to the maximum likelihood estimate p̂ of signal accuracy, we can evaluate its signiﬁcance using a likelihood ratio test applied to the test statistics − 2 ln Λ = −2 ln L(p = p0 X) , L(p = p̂X) where p0 is the empirical probability reported in Table 4.5. Table 4.6 shows the maximum likelihood estimate of the signal p̂ in Column 4, the likelihood for p̂ in Column 5, followed by the likelihood ratio Λ in Column 6, the estimated test statistics −2 ln Λ in Column 7 and the pvalue. The estimated test statistic is asymptotically χ2 distributed with 1 degree of freedom. The degree of signiﬁcance and the pvalue indicated in Table 4.6 are based on this distribution. The critical values at the 95%, 99% and 99.5% conﬁdence level are respectively 3.84, 6.63 and 7.87. We ﬁnd that all the measures except BSEYD10 are signiﬁcant at a 99.5% level. BSEYD10, a measure computed using Baa corporate bonds, current earnings and Cantelli’s inequality is signiﬁcant at a 95% level and has a pvalue of 1.44%. page 94 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 95 95 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.6. Maximum likelihood estimate and likelihood ratio test: uninformed prior. Signal Model PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 BSEYD11 BSEYD12 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 Total Number of ML number correct Estimate of signals predictions p̂ 37 36 55 49 37 36 50 44 38 39 40 39 24 22 18 19 23 19 16 16 47 44 42 39 20 21 27 26 22 22 22 21 25 24 37 35 25 25 34 31 29 28 30 29 15 14 12 12 14 11 12 11 34 33 32 30 13 14 17 17 15 14 15 16 67.57% 66.67% 67.27% 71.43% 67.57% 69.44% 68.00% 70.45% 76.32% 71.79% 75.00% 74.36% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 75.00% 68.75% 72.34% 75.00% 76.19% 76.92% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% L(p̂) 7.50E11 1.12E10 7.91E16 1.86E13 7.50E11 2.38E10 2.44E14 2.52E12 9.25E10 8.40E11 1.70E10 2.28E10 1.27E07 5.46E07 1.06E05 3.71E06 2.06E07 2.42E06 1.24E04 4.83E05 9.18E13 1.80E11 9.73E11 7.08E10 2.38E06 1.57E06 1.87E08 5.21E08 1.06E06 5.46E07 1.06E06 9.87E06 Test Likelihood statistics ratio Λ −2 ln Λ 0.0031 0.0052 0.0002 0.0000 0.0031 0.0016 0.0003 0.0002 0.0000 0.0003 0.0000 0.0001 0.0063 0.0069 0.0078 0.0154 0.0117 0.0499 0.0013 0.0075 0.0000 0.0000 0.0000 0.0000 0.0074 0.0035 0.0029 0.0015 0.0015 0.0064 0.0015 0.0001 11.5744∗∗∗ 10.5318∗∗∗ 16.8358∗∗∗ 20.0039∗∗∗ 11.5744∗∗∗ 12.8682∗∗∗ 16.1408∗∗∗ 16.8448∗∗∗ 20.9073∗∗∗ 16.3034∗∗∗ 20.3806∗∗∗ 19.1244∗∗∗ 10.1190∗∗∗ 9.9454∗∗∗ 9.6971∗∗∗ 8.3431∗∗∗ 8.8913∗∗∗ 5.9940∗ 13.2951∗∗∗ 9.7846∗∗∗ 20.3449∗∗∗ 22.4187∗∗∗ 22.9420∗∗∗ 22.2148∗∗∗ 9.8012∗∗∗ 11.3133∗∗∗ 11.7151∗∗∗ 13.0279∗∗∗ 13.0563∗∗∗ 10.1063∗∗∗ 13.0563∗∗∗ 18.4227∗∗∗ pvalue 0.07% 0.12% 0.00% 0.00% 0.07% 0.03% 0.01% 0.00% 0.00% 0.01% 0.00% 0.00% 0.15% 0.16% 0.18% 0.39% 0.29% 1.44% 0.03% 0.18% 0.00% 0.00% 0.00% 0.00% 0.17% 0.08% 0.06% 0.03% 0.03% 0.15% 0.03% 0.00% Notes: ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. In addition to testing the measures against an uninformed signal with prediction probability p0 , we also test the measures to see whether they generate an accurate prediction more than half the time. To this end, we perform a likelihood ratio test against the arbitrary probability p = 12 using July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 96 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. the test statistics − 2 ln Λ = −2 ln L(p = 12 X) . L(p = p̂X) Table 4.7 has the same structure as Table 4.6: the maximum likelihood estimate of the signal p̂ is reported in Column 4, the likelihood for p̂ is in Table 4.7. Maximum likelihood estimate and likelihood ratio test: arbitrary 50% threshold. Signal Model PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 BSEYD11 BSEYD12 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 Total Number of ML number correct Estimate of signals predictions p̂ 37 36 55 49 37 36 50 44 38 39 40 39 24 22 18 19 23 19 16 16 47 44 42 39 20 21 27 26 22 22 22 21 25 24 37 35 25 25 34 31 29 28 30 29 15 14 12 12 14 11 12 11 34 33 32 30 13 14 17 17 15 14 15 16 Notes: ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. 67.57% 66.67% 67.27% 71.43% 67.57% 69.44% 68.00% 70.45% 76.32% 71.79% 75.00% 74.36% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 75.00% 68.75% 72.34% 75.00% 76.19% 76.92% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% L(p̂) 7.50E11 1.12E10 7.91E16 1.86E13 7.50E11 2.38E10 2.44E14 2.52E12 9.25E10 8.40E11 1.70E10 2.28E10 1.27E07 5.46E07 1.06E05 3.71E06 2.06E07 2.42E06 1.24E04 4.83E05 9.18E13 1.80E11 9.73E11 7.08E10 2.38E06 1.57E06 1.87E08 5.21E08 1.06E06 5.46E07 1.06E06 9.87E06 Test Likelihood statistics ratio Λ −2 ln Λ 0.0970 0.1302 0.0351 0.0096 0.0970 0.0611 0.0364 0.0225 0.0039 0.0217 0.0053 0.0080 0.4686 0.4366 0.3608 0.5139 0.5782 0.7883 0.1233 0.3157 0.0077 0.0032 0.0023 0.0026 0.4009 0.3044 0.3994 0.2862 0.2257 0.4366 0.2257 0.0483 4.6665∗ 4.0776∗ 6.7008∗∗ 9.2980∗∗∗ 4.6665∗ 5.5907∗ 6.6278∗∗ 7.5842∗∗ 11.0758∗∗∗ 7.6648∗∗ 10.4650∗∗∗ 9.6625∗∗∗ 1.5160 1.6573 2.0388 1.3314 1.0957 0.4757 4.1860∗ 2.3059 9.7232∗∗∗ 11.5115∗∗∗ 12.1189∗∗∗ 11.9296∗∗∗ 1.8280 2.3786 1.8357 2.5019 2.9769 1.6573 2.9769 6.0595∗ pvalue 3.08% 4.35% 0.96% 0.23% 3.08% 1.81% 1.00% 0.59% 0.09% 0.56% 0.12% 0.19% 21.82% 19.80% 15.33% 24.86% 29.52% 49.04% 4.08% 12.89% 0.18% 0.07% 0.05% 0.06% 17.64% 12.30% 17.55% 11.37% 8.45% 19.80% 8.45% 1.38% page 96 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 97 97 Column 5, followed by the likelihood ratio Λ in Column 6, the estimated test statistics −2 ln Λ in Column 7 and the pvalue. The estimated test statistic is asymptotically χ2 distributed with 1 degree of freedom and the critical values at the 95%, 99% and 99.5% conﬁdence level are respectively 3.84, 6.63 and 7.87. The accuracy of the P/E and logP/E measures is not particularly consistent, although all the measures are signiﬁcantly diﬀerent from p0 = 12 at least at a 95% conﬁdence level. On the other hand, seven out of eight BSEYD and logBSEYD measures computed with the yield of Treasury Notes over the full period are signiﬁcant at a 99.5% conﬁdence level. BSEYD3 misses the 99.5% mark by a narrow margin: it is in fact signiﬁcant at a 99.44% level. With the exception of BSEYD11 and logBSEYD11, the BSEYD models computed using Aaa and Baa bond yields are not signiﬁcant. A possible explanation for the low signiﬁcance is the shorter data history. The data robustness analysis in Section 4.4 partly resolves this issue and permits a more objective comparison across all models. 4.3.5. Monte Carlo study for small sample bias In addition to performing the likelihood ratio test using the χ2 distribution, we carry out a likelihood ratio test based on empirical distributions obtained through Monte Carlo simulations. Table 4.8 reports the total number of signals, calculated as the sum of all the entries of the indicator sequence S and the maximum likelihood estimate, p̂, which is the probability of correctly predicting a crash and is estimated by maximizing the likelihood function of the model. These statistics are identical to those reported in Table 4.9. Columns 4–6 in Table 4.8 present the critical values at a 95%, 99% and 99.5% conﬁdence level for the empirical distribution generated using K = 10,000 Monte Carlo simulations. These values are not uniform across all models as they depend directly on the total number of signals generated by each model. Column 7 and 8 report, respectively, the test statistics equal L(p= 12 X) 0 X) to −2 ln Λ = −2 ln L(p=p L(p=p̂X) and −2 ln Λ = −2 ln L(p=p̂X) . The level of signiﬁcance indicated is based on the empirical distribution. Column 8 and 10 report the pvalue, deﬁned as the probability of obtaining a test statistic higher than that actually observed assuming that the null hypothesis is true, for each of the models under its empirical distribution. The Monte Carlo study conﬁrms our observations regarding the likelihood ratio test using the χ2 distribution. All the measures but one (BSEYD10) are signiﬁcant at 99.5% level when assessed against the 10:48 Critical Value: 99% conﬁdence 67.57% 66.67% 67.27% 71.43% 67.57% 69.44% 68.00% 70.45% 76.32% 71.79% 75.00% 74.36% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 3.9737 3.6064 3.8139 3.8255 3.9737 3.6064 4.0703 3.8617 33.6670 3.5249 3.9147 4.3417 4.4920 3.5721 3.9728 4.4836 3.9346 4.3978 7.4726 6.8973 6.4684 6.6871 7.4726 6.6033 6.6821 6.6068 6.5886 6.6587 7.2228 6.6587 7.2008 6.0855 7.1985 7.8270 6.5576 7.7190 Critical Test Value: 99.5% statistics conﬁdence −2 ln Λ(p0 ) 7.4882 8.4455 8.0932 7.4286 7.4882 8.4455 7.2064 8.2601 8.0916 8.7070 8.3259 7.4412 7.6647 9.7455 7.1985 7.8270 8.8913 7.7190 11.5744∗∗∗ 10.5318∗∗∗ 16.8358∗∗∗ 20.0039∗∗∗ 11.5744∗∗∗ 12.8682∗∗∗ 16.1408∗∗∗ 16.8448∗∗∗ 20.9073∗∗∗ 16.3034∗∗∗ 20.3806∗∗∗ 19.1244∗∗∗ 10.1190∗∗∗ 9.9454∗∗∗ 9.6971∗∗∗ 8.3431∗∗∗ 8.8913∗∗ 5.9940∗ Test Empirical statistics Empirical pvalue −2 ln Λ(1/2) pvalue 0.13% 0.11% 0.00% 0.00% 0.07% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.24% 0.22% 0.27% 0.45% 0.53% 2.27% 4.6665∗ 4.0776† 6.7008∗ 9.2980∗∗∗ 4.6665† 5.5907∗ 6.6278∗ 7.5842∗ 11.0758∗∗∗ 7.6648∗∗∗ 10.4650∗∗∗ 9.6625∗∗∗ 1.5160 1.6573 2.0388 1.3314 1.0957 0.4757 4.88% 7.07% 1.31% 0.32% 5.21% 2.87% 1.43% 1.09% 0.16% 0.94% 0.00% 0.42% 31.56% 28.62% 24.03% 36.01% 40.72% 65.05% b2882ch04 37 36 55 49 37 36 50 44 38 39 40 39 24 22 18 19 23 19 Critical Value: 95% conﬁdence 9in x 6in PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 ML Estimate p̂ Stock Market Crashes Total number Signal model of signals Stock Market Crashes page 98 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 98 Table 4.8. Monte Carlo likelihood ratio test. 9.7846 6.8590 7.4161 8.2601 8.2873 7.4412 8.4614 9.1011 8.9292 8.3464 7.5500 6.9162 7.5500 8.9793 13.2951∗∗∗ 9.7846∗∗∗ 20.3449∗∗∗ 22.4187∗∗∗ 22.9420∗∗∗ 22.2148∗∗∗ 9.8012∗∗∗ 11.3133∗∗∗ 11.7151∗∗∗ 13.0279∗∗∗ 13.0563∗∗∗ 10.1063∗∗∗ 13.0563∗∗∗ 18.4227∗∗∗ 0.00% 0.44% 0.00% 0.00% 0.00% 0.00% 0.18% 0.14% 0.13% 0.04% 0.00% 0.07% 0.00% 0.00% 4.1860† 2.3059 9.7232∗∗∗ 11.5115∗∗∗ 12.1189∗∗∗ 11.9296∗∗∗ 1.8280 2.3786 1.8357 2.5019 2.9769 1.6573 2.9769 6.0595∗ 7.47% 21.26% 0.38% 0.12% 0.08% 0.00% 26.80% 18.67% 24.40% 16.65% 13.70% 28.62% 13.57% 2.46% 9in x 6in 6.8590 6.8590 7.3643 6.6068 6.5932 6.6587 7.1706 6.1486 6.9972 5.8557 5.9791 6.9162 5.9791 6.2718 Stock Market Crashes Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. 4.4686 4.4686 3.4037 3.8617 3.7950 3.4819 4.9574 4.1581 3.7521 4.0936 3.5580 4.7166 3.5580 4.2590 10:48 75.00% 68.75% 72.34% 75.00% 76.19% 76.92% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% b2882ch04 99 page 99 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 16 16 47 44 42 39 20 21 27 26 22 22 22 21 The High Price–Earnings Stock Market Danger Approach of Campbell BSEYD11 BSEYD12 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 July 31, 2017 10:48 Stock Market Crashes b2882ch04 Stock Market Crashes 100 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 9in x 6in uninformed probability p0 . The signiﬁcance of the measures varies more when we use the arbitrary probability p = 12 . The BSEYD and logBSEYD measures computed with Treasury yields on the full dataset perform best: they are all signiﬁcant at a 99.5% level. The signiﬁcance of the P/E and log P/E measures vary from 90% for P/E2 and logP/E1 to 99.5% for P/E4. With the exception of BSEYD11 and logBSEYD12, the BSEYD and logBSEYD measures based on a shorter timescale are not signiﬁcant. 4.3.6. Measureonmeasure significance test Next, we compare the accuracy of the two measures using a pairwise measureonmeasure test. Table 4.9 reports the test statistics L(p=p̂ X) −2 ln L(p=p̂ji X) , i = 1, . . . 32, j = 1, . . . , 32, j = i for all 992 pairs of models. Table 4.10 is a heat map of the measureonmeasure signiﬁcance test, based on the p − value. A darker background indicates a lower pvalue, while a white background is indicative of a pvalue above 20%. We ﬁnd that the measures based on corporate yields and shorter timescale, in particular BSEYD5, BSEYD6, BSEYD8, BSEYD9 and BSEYD10, are signiﬁcantly less accurate than the other measures. However, this is possibly a reﬂection of the short timescale used to estimate these measures. On the other hand, the accuracy of those measures estimated using the full dataset, is not signiﬁcantly diﬀerent, even with 22 crashes and nearly 51 years of data. 4.4. Robustness Data Robustness We test the robustness of the crash prediction models with respect to a change in the underlying data over two subperiods: (1) Subperiod 1 : January 31, 1964 to December 31, 1981; (2) Subperiod 2 : January 1, 1982 to December 31, 2014. Small sample bias is an important consideration here because of the limited amount of crashrelated data available for each subperiod. For Subperiod 1, we have access to all statistics for the P/E and logP/E models as well as the BSEYD and logBSEYD models calculated using Treasury yields. For Subperiod 2, we have statistics for all 32 models, with a caveat: data for the BSEYD and logBSEYD models based on corporate yields start on January 1, 1983. page 100 10:48 Stock Market Crashes 9in x 6in b2882ch04 101 page 101 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 (Continued ) The High Price–Earnings Stock Market Danger Approach of Campbell L(p=p̂ X) Table 4.9. Test statistics for the measureonmeasure test. The table reports the test statistics −2 ln L(p=p̂ji X) for all 992 pairs of models. 10:48 Stock Market Crashes 9in x 6in Stock Market Crashes b2882ch04 page 102 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 102 Table 4.9. (Continued ) 10:48 Stock Market Crashes 9in x 6in b2882ch04 (Continued ) 103 page 103 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 The High Price–Earnings Stock Market Danger Approach of Campbell Table 4.10. Heat map for the measureonmeasure test. The table reports a heat map of the measureon measure signiﬁcance test, based on the p − value. A darker background indicates a lower pvalue. A white background is indicative of a pvalue above 20%. 10:48 Stock Market Crashes 9in x 6in Stock Market Crashes b2882ch04 page 104 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 104 Table 4.10. (Continued ) July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. The High Price–Earnings Stock Market Danger Approach of Campbell page 105 105 Note that p = 12 is lower than p0 for subperiod 1, but higher than p0 for every other data set. This means that the signiﬁcance of the test statistics will increase for measures evaluated over the ﬁrst subperiod, but will decrease for the other periods. Tables 4.11 and 4.12 report likelihood statistics respectively, given the hypotheses p = p0 and p = 12 for the two subperiods using the asymptotic χ2 distribution. The total number of signals in Columns 2 and 6 is calculated as the sum of all the entries of the indicator sequence S. P/E and logP/E ratio models produced nearly twice as many signals in Subperiod 2 than in Subperiod 1, whereas BSEYD models generated roughly the same number of signals in both subperiods. The number of signals for logBSEYD models based on current earnings increased slightly, contrary to the logBSEYD models based on average earnings over 10 years, for which the number of signals remains constant. The Maximum Likelihood estimate p̂ reported in Columns 3 and 7 is the probability of correctly predicting a crash that maximizes the likelihood function of the model. It is equal to the ratio of the number of correct predictions to the total number of signals. Table 4.9 shows that all the models are statistically signiﬁcant at a 99.5% level in the ﬁrst subperiod. In the second subperiod, all the models are signiﬁcant at a 99% or 99.5% level. In fact, the highest pvalue across models is 1.44%, achieved by the BSEYD10 implementation. Tables 4.13 and 4.14 report empirical statistics, respectively, for subperiods 1 and 2 using an empirical distribution generated by K = 10,000 Monte Carlo simulations. The test statistics in Columns 7 and 9 respectively are L(p= 12 X) 0 X) equal to −2 ln Λ = −2 ln L(p=p L(p=p̂X) and −2 ln Λ = −2 ln L(p=p̂X) . The level of signiﬁcance indicated is based on the empirical distribution. Conducting the statistical eta of signiﬁcance using the empirical distribution further increases the diﬀerence between Subperiods 1 and 2. The results in Table 4.13 conﬁrm the initial observation made under the asymptotic χ2 distribution: all the measures are statistically signiﬁcant at a 99.5% level regardless of the null hypothesis. In the second subperiod, the results in Table 4.14 show that the accuracy of all the measures is signiﬁcantly diﬀerent from p0 at or around a 99% level. In fact, the highest pvalues are 1.33% for logBSEYD4 and 1.31% for BSEYD2. Furthermore, the accuracy of 14 models out of 32 is signiﬁcantly greater p = 12 : BSEYD7– 12 and logBSEYD5–12. With the exception of BSYED 10, these models are signiﬁcant at a 99.5% level. This result is noticeably diﬀerent from the observation made under the asymptotic χ2 distribution. Indeed, the critical ML Estimate p̂ Sample 2 Test statistics −2 ln Λ pvalue Total number of signals ML Estimate p̂ Test statistics −2 ln Λ 7.5993∗∗ pvalue 100.00% 100.00% 100.00% 100.00% — — — — — — — — 27 28 39 34 55.56% 57.14% 53.85% 58.82% 8.8619∗∗∗ 9.5940∗∗∗ 12.0986∗∗∗ 0.58% 0.29% 0.20% 0.05% lnPE1 lnPE2 lnPE3 lnPE4 9 8 15 12 100.00% 100.00% 100.00% 100.00% — — — — — — — — 28 28 35 32 57.14% 60.71% 54.29% 59.38% 8.8619∗∗∗ 11.2811∗∗∗ 8.9212∗∗∗ 11.8163∗∗∗ 0.29% 0.08% 0.28% 0.06% BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 BSEYD11 BSEYD12 17 16 19 18 100.00% 93.75% 94.74% 94.44% — 10.4672∗∗∗ 13.3534∗∗∗ 12.3845∗∗∗ — 0.12% 0.03% 0.04% 21 23 21 21 24 22 18 19 23 19 16 16 57.14% 56.52% 57.14% 57.14% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 75.00% 68.75% 6.6464∗∗ 6.9584∗∗ 6.6464∗∗ 6.6464∗∗ 10.1190∗∗∗ 9.9454∗∗∗ 9.6971∗∗∗ 8.3431∗∗∗ 8.8913∗∗∗ 5.9940∗ 13.2951∗∗∗ 9.7846∗∗∗ 0.99% 0.83% 0.99% 0.99% 0.15% 0.16% 0.18% 0.39% 0.29% 1.44% 0.03% 0.18% b2882ch04 10 8 16 15 9in x 6in PE1 PE2 PE3 PE4 Stock Market Crashes Signal Model Total number of signals 10:48 Sample 1 Stock Market Crashes page 106 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 106 Table 4.11. Robustness: maximum likelihood estimate and likelihood ratio test for each sample: Uninformed prior. 0.07% 0.04% 0.02% 0.03% 60.00% 61.54% 59.09% 60.00% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% 11.5431∗∗∗ 11.0329∗∗∗ 7.9710∗∗∗ 7.6954∗∗ 9.8012∗∗∗ 11.3133∗∗∗ 11.7151∗∗∗ 13.0279∗∗∗ 13.0563∗∗∗ 10.1063∗∗∗ 13.0563∗∗∗ 18.4227∗∗∗ 0.07% 0.09% 0.48% 0.55% 0.17% 0.08% 0.06% 0.03% 0.03% 0.15% 0.03% 0.00% 9in x 6in 30 26 22 20 20 21 27 26 22 22 22 21 Stock Market Crashes Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. 11.4222∗∗∗ 12.3845∗∗∗ 14.3281∗∗∗ 13.3534∗∗∗ 10:48 94.12% 94.44% 95.00% 94.74% b2882ch04 107 page 107 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 17 18 20 19 The High Price–Earnings Stock Market Danger Approach of Campbell lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 108 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.12. Data robustness of the crash prediction measures (Models 1–4): Arbitrary 50% threshold. Asymptotic (χ2 ) Crash prediction measure Robust likelihood ratio Model P/E logP/E BSEYD logBSEYD 0.1302 0.0031 0.0003 0.0000 P/E2 logP/E1 BSEYD2 logBSEYD1 Robust test statitstics 4.0776∗ 4.6665∗ 7.6648∗∗ 9.7232∗∗∗ Empirical (Monte Carlo) pvalue Robust test statitstics pvalue 4.35% 3.08% 0.56% 0.18% 4.0776† 4.6665∗ 6.2597∗∗ 20.3449∗∗∗ 7.07% 5.21% 0.94% 0.38% Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. values for the empirical distribution are lower than the critical values under χ2 distribution. The four measures are remarkably robust with respect to a change in the subperiod, when tested against a random uninformed signal with predictive probability p0 . All the measures are statistically signiﬁcant at a 99.5% level, while 31 of the 32 implementations are statistically signiﬁcant at least at a 99% level in the second subperiod. Model BSEYD10 has the lowest signiﬁcance in the second subperiod, with 98.56%. On the other hand, we see large discrepancies between the ﬁrst and the second subperiods when we compare the accuracy of our measures with an arbitrary p = 12 benchmark corresponding to 50% accuracy. While the signiﬁcance of most measures improves in the ﬁrst subperiod, only four measures are statistically signiﬁcant at the 90% level in the second subperiod. The explanation to this puzzle lies in Table 4.5. A random uninformed measure would have correctly predicted a crash 58.27% of the time in the ﬁrst subperiod, but only 29.98% of the time in the second subperiod. Crashes are not evenly spread over time. The ﬁrst subperiod lasts for about 18 years while the second subperiod lasts for 33 years, yet 11 crashes occurred in each subperiod. As a result, an arbitrary, static benchmark such as p = 12 is illadapted to test the accuracy of cash measures over multiple time periods. page 108 Critical value: 99% Test stat. −2 ln Λ(p0 ) Empirical pvalue Test stat. −2 ln Λ( 12 ) Empirical pvalue 2.1284 3.2800 2.7910 3.1926 2.2714 3.2800 3.1926 3.0285 2.4826 2.7910 3.5291 3.0434 2.4826 3.0434 2.6968 2.0053 3.2594 3.6512 3.8422 3.7856 4.1094 3.6512 3.7856 3.4830 3.6347 3.8422 3.6360 4.5427 3.6347 4.5427 4.2568 3.6360 6.1367 8.6403 6.5600 6.1516 8.7857 8.6403 6.1516 6.7444 7.3815 6.5600 5.9506 6.8549 7.3815 6.8549 6.6965 5.9506 — — — — — — — — — 10.4672∗∗∗ 13.3534∗∗∗ 12.3845∗∗∗ 11.4222∗∗∗ 12.3845∗∗∗ 14.3281∗∗∗ 13.3534∗∗∗ — — — — — — — — — 0.00% 0.09% 0.19% 0.00% 0.14% 0.06% 0.00% — — — — — — — — — 14.6994∗∗∗ 18.5043∗∗∗ 17.2292∗∗∗ 15.9606∗∗∗ 17.2292∗∗∗ 19.7853∗∗∗ 18.5043∗∗∗ — — — — — — — — — 0.00% 0.00% 0.00% 0.00% 0.02% 0.00% 0.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 93.75% 94.74% 94.44% 94.12% 94.44% 95.00% 94.74% b2882ch04 Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. 9in x 6in Critical value: 95% Stock Market Crashes 10 8 16 15 9 8 15 12 17 16 19 18 17 18 20 19 Critical value: 90% 10:48 PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 Total ML number Estimate of signals p̂ 109 page 109 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 Signal model The High Price–Earnings Stock Market Danger Approach of Campbell Table 4.13. Robustness: Monte Carlo likelihood ratio test for sample 1. 10:48 Critical value: 90% Critical value: 95% Critical value: 99% Test stat. −2 ln Λ(p0 ) 27 28 39 34 28 28 35 32 21 23 21 21 24 22 8 19 23 19 55.56% 57.14% 53.85% 58.82% 57.14% 60.71% 54.29% 59.38% 57.14% 56.52% 57.14% 57.14% 62.50% 63.64% 66.67% 63.16% 60.87% 57.89% 2.5086 2.3179 2.9641 2.7239 2.1627 2.1627 3.0544 3.1780 2.3786 3.2157 2.8317 2.8317 2.5581 3.4638 2.8341 2.3043 2.9480 2.3102 3.9085 3.6515 3.2182 4.3200 3.7756 3.7756 3.8305 3.1780 3.9849 3.6701 4.5671 4.5671 4.4920 3.5721 3.9728 3.8482 3.9346 4.3978 7.5993 7.3255 6.4074 6.3878 6.7346 5.9772 6.8540 6.3371 6.0595 6.9584 6.6464 6.6464 7.2008 6.0855 7.1985 7.8270 6.5576 7.7190 7.5993∗ 8.8619∗∗∗ 9.5940∗∗∗ 12.0986∗∗∗ 8.8619∗∗ 11.2811∗∗∗ 8.9212∗∗ 11.8163∗∗∗ 6.6464∗∗ 6.9584∗ 6.6464∗ 6.6464∗ 10.1190∗∗∗ 9.9454∗∗∗ 9.6971∗∗∗ 8.3431∗∗ 8.8913∗∗ 5.9940∗ Empirical pvalue Test stat. −2 ln Λ( 12 ) Empirical pvalue 1.21% 0.29% 0.09% 0.12% 0.54% 0.06% 0.52% 0.00% 0.74% 1.07% 1.54% 1.45% 0.32% 0.24% 0.14% 0.56% 0.54% 2.15% 0.3340 0.5734 0.2310 1.0644 0.5734 1.2957 0.2575 1.1317 0.4300 0.3924 0.4300 0.4300 1.5160 1.6573 9.6971∗∗∗ 8.3431∗∗∗ 8.8913∗∗∗ 5.9940∗ 70.51% 57.40% 75.71% 39.44% 56.87% 34.68% 73.62% 37.80% 67.27% 68.06% 66.97% 66.78% 31.56% 28.62% 0.00% 0.40% 0.27% 1.75% b2882ch04 ML Estimate p̂ 9in x 6in PE1 PE2 PE3 PE4 lnPE1 lnPE2 lnPE3 lnPE4 BSEYD1 BSEYD2 BSEYD3 BSEYD4 BSEYD5 BSEYD6 BSEYD7 BSEYD8 BSEYD9 BSEYD10 Total number of signals Stock Market Crashes Signal Model Stock Market Crashes page 110 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 110 Table 4.14. Robustness: Monte Carlo likelihood ratio test for sample 2. 6.8590 6.8590 6.9656 7.8212 5.7248 7.6954 7.1706 6.1486 6.9972 5.8557 5.9791 5.9791 5.9791 6.2718 13.2951∗∗∗ 9.7846∗∗∗ 11.5431∗∗∗ 11.0329∗∗∗ 7.9710∗∗ 7.6954∗ 9.8012∗∗∗ 11.3133∗∗∗ 11.7151∗∗∗ 13.0279∗∗∗ 13.0563∗∗∗ 10.1063∗∗∗ 13.0563∗∗∗ 18.4227∗∗∗ 0.06% 0.60% 0.07% 0.18% 0.82% 1.31% 0.00% 0.13% 0.21% 0.03% 0.00% 0.22% 0.00% 0.00% 13.2951∗∗∗ 9.7846∗∗∗ 1.2081 1.3972 0.7313 0.8054 9.8012∗∗∗ 11.3133∗∗∗ 11.7151∗∗∗ 13.0279∗∗∗ 13.0563∗∗∗ 10.1063∗∗∗ 13.0563∗∗∗ 18.4227∗∗∗ 0.11% 0.44% 35.61% 32.53% 51.88% 51.05% 0.20% 0.00% 0.01% 0.00% 0.00% 0.10% 0.07% 0.00% 9in x 6in 4.4686 4.4686 3.6809 4.5451 3.8459 4.6437 3.1490 4.1581 3.7521 4.0936 3.5580 3.5580 3.5580 4.2590 Stock Market Crashes Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. 2.9295 2.9295 2.8307 2.9920 3.2437 2.4356 2.7098 3.1330 2.1730 2.6395 3.4880 3.4880 3.4880 3.0558 10:48 75.00% 68.75% 60.00% 61.54% 59.09% 60.00% 65.00% 66.67% 62.96% 65.38% 68.18% 63.64% 68.18% 76.19% b2882ch04 111 page 111 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. July 31, 2017 16 16 30 26 22 20 20 21 27 26 22 22 22 21 The High Price–Earnings Stock Market Danger Approach of Campbell BSEYD11 BSEYD12 lnBSEYD1 lnBSEYD2 lnBSEYD3 lnBSEYD4 lnBSEYD5 lnBSEYD6 lnBSEYD7 lnBSEYD8 lnBSEYD9 lnBSEYD10 lnBSEYD11 lnBSEYD12 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes 112 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Model Parameter Robustness The last set of results presented in this section address the robustness of the four crash measures (P/E, logP/E, BSEYD and logBSEYD) with respect to a change in parameters. We temporarily leave aside the BSEYD and logBSEYD models based on corporate yields and the question of robustness with respect to a choice of interest rate. The main reason is that the choice of interest rates aﬀects not only the period for which data are available (1964– 2014 for Treasury Notes but only 1983–2014 for Aaa corporate bonds and 1986–2014 for Aaa corporate bonds), but also the number of crashes over which we can conduct the test. The best approach to this problem is to analyze the robustness of the models to a change in the underlying dataset, as we did previously. Column 2 in Table 4.15 reports the robust likelihood ratio Λ̆ for each of the measures under the null hypothesis H0 : p = p0 . We calculate the robust likelihood ratio using (4.11) as the highest likelihood ratio across all implementations of the measure. Higher robust likelihood ratios are indicative of lower statistical signiﬁcance. Column 3 identiﬁes the speciﬁc model, and therefore the set of parameters, corresponding to the likelihood ratio reported in Column 2. Columns 4 and 5 refer to signiﬁcance tests performed using the asymptotic χ2 distribution. Column 4 reports the robust test statistics, computed using formula (4.12), and level of signiﬁcance for the measure based on the χ2 distribution. Column 5 gives the associated pvalue. Columns 6 and 7 refer to signiﬁcance tests performed with the Table 4.15. Data robustness of the Crash prediction measures (Models 1–4): Uninformedprior. Asymptotic (χ2 ) Empirical (Monte Carlo) Crash prediction measure Robust likelihood ratio Model Robust test statitstics pvalue Robust test statitstics pvalue P/E logP/E BSEYD logBSEYD 0.0052 0.0031 0.0003 0.0000 P/E2 logP/E1 BSEYD2 logBSEYD1 10.5318∗∗∗ 11.5744∗∗∗ 16.3034∗∗∗ 20.3449∗∗∗ 0.12% 0.07% 0.00% 0.00% 10.5318∗∗∗ 11.5744∗∗∗ 16.3034∗∗∗ 20.3449∗∗∗ 0.16% 0.08% 0.99% 0.00% Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. page 112 July 31, 2017 10:48 Stock Market Crashes 9in x 6in b2882ch04 The High Price–Earnings Stock Market Danger Approach of Campbell page 113 113 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. Table 4.16. Data robustness of the crash prediction measures. Asymptotic (χ2 ) Crash prediction measure Robust likelihood ratio Model P/E logP/E BSEYD logBSEYD 0.1302 0.0031 0.0003 0.0000 P/E2 logP/E1 BSEYD2 logBSEYD1 Empirical (Monte Carlo) Robust test statitstics pvalue Robust test statitstics pvalue 4.0776∗ 4.6665∗ 7.6648∗∗ 9.7232∗∗∗ 4.35% 3.08% 0.56% 0.18% 4.0776† 4.6665∗ 6.2597∗∗ 20.3449∗∗∗ 7.07% 5.21% 0.94% 0.38% Notes: † signiﬁcant at the 10% level; ∗ signiﬁcant at the 5% level; ∗∗ signiﬁcant at the 1% level; ∗∗∗ signiﬁcant at the 0.5% level. empirical distribution generated by Monte Carlo simulations. Here we aim to correct for small sample bias. Column 6 reports the robust test statistics, computed using formula (4.12), and level of signiﬁcance for the measure based on the empirical distribution. Column 7 gives the associated pvalue. Table 4.16 reports the corresponding numbers under the null hypothesis H0 : p = 12 . The BSEYD and logBSEYD are robust crash prediction measures. Their robust test statistics are still signiﬁcant at a 99% and 99.5% level respectively, under the stricter null hypothesis H0 : p = 12 . By contrast, the signiﬁcance of the P/E and logP/E measures slides from 99.5% signiﬁcance under the null hypothesis H0 : p = p0 to between 90% and 95% under the null hypothesis H0 : p = 12 . However, the robust likelihood ratio and test statistics also have an important limitation, inherent in all summary measures: Because summary measures condense all available information into a single number, they are unable to give a nuanced view of a phenomenon. Having established an objective view of robustness with the robust likelihood ratio, we now discuss some of its subjective facets. Intuitively, a crash prediction model is robust if its accuracy, reﬂected in the total number of signals, the number of correct predictions and the proportion of correct predictions, does not change materially with a change in the model speciﬁcation. The total number of signals, number of correct predictions and proportion of correct predictions reported in Columns 1–3 of Table 4.4 suggest that all of the models are impacted by a change in the deﬁnition of earnings. For the P/E and logP/E ratios, the total number July 31, 2017 10:48 Stock Market Crashes Downloaded from www.worldscientific.com by AMERICAN UNIVERSITY OF BEIRUT UNIVERSITY LIBRARIES  JAFET LIBRARY on 06/24/18. For personal use only. 114 Stock Market Crashes 9in x 6in b2882ch04 Stock Market Crashes of signals jumps from 36 or 37 for models based on current earnings to between 44 and 55 for models using average earnings. The number of correct predictions also soars from about 25 to between 31 and 37. As a result, the proportion of correct predictions remains broadly constant across models. The BSEYD model is unaﬀected by the change in deﬁnition of earnings. On the other hand, changing from current earnings to average earnings reduces the total number of signals for the logBSEYD model while the number of correct predictions decreases only slightly. The end result is a small increase in the proportion of correct predictions by around 3%. The test statistics reported in Tables 4.6–4.8 conﬁrm this observation and explain the robustness of the BSEYD and logBSEYD measure. Next, we examine the eﬀect of a change in the deﬁnitions of the threshold. We observe that all the models seem robust to a change in the deﬁnition of the threshold: the total number of signals and the number of correct predictions reported in Table 4.4 typically do not diﬀer by more than 2. This also helps to explain the robustness of the measure. Hence, the use of Cantelli’s inequality to deﬁne the threshold does not add signiﬁcant value over a standard conﬁdence interval based on a normal distribution. An explanation could lie in the tail behavior of the model. Figure 4.9(a) shows that the empirical distribution of the BSEYD measure (based on current earnings) is not Gaussian: it is bimodal and skewed to the left. However, the tail behavior of the BSEYD measure in the last decile and especially in the last 5% is close to a Normal distribution: the R2 of two exponential regressions of the observations in the last decile against the quantiles and the quantiles squared reach 85% (see Figure 4.9(b)). 4.5. What can we conclude? Our ﬁndings show that all 32 implementations of the four measures predict equity market crashes better than a random uninformed signal with predictive probability p0 , under both asymptotic χ2 distribution a