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European Quant Awards 2016

Detecting financial market bubbles with low-

frequency volatility models

The original research was carried out by the author during the preparation of

Master Thesis. The report is an adaptation of its main results.

2016

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Contents

1. Introduction........................................................................... 3

2. Research and empirical evidence.......................................... 3

3. Applications and further research......................................... 7

3.1. Brief summary of main results.......................................... 7

3.2. Applications for portfolio management............................ 7

4. Appendix................................................................................ 10

4.1. Appendix 1. MIDAS regressions........................................ 10

4.2. Appendix 2. GARCH-MIDAS model.................................... 11

4.3. Appendix 3. Actual estimation results............................... 12

5. References............................................................................. 15

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1. Introduction

Financial market bubbles are one of the main reasons behind stock market crashes

alongside with economic troubles and announcements concerning unexpected and/or adverse

political turns. Some of the most notorious and devastating market crashes were caused by

bubbles, for example the Wall Street crash of 1929, Japanese asset price bubble of 1991, the

Dot-com bubble, the housing market bubble and the corresponding crash of 2007, Chinese

stock bubble of 2007 etc. Therefore, any empirical tool that enables the stock market

participants to detect market bubbles is extremely useful and can contribute to the

methodology of market crash prediction. In this paper, we propose a new technique for market

bubble detection (and, consequently, market crash prediction) that might prove to be valuable

for stock market practitioners.

In the original research we use S&P 500 index returns from 01.01.1964 to 01.01.2014 to

detect market crashes in the aggregate US stock market, but, as it will be discussed in Section

3.2, the methodology has a high potential for extrapolation to industry stock markets and

individual stocks.

2. Research and empirical evidence

A distinctive feature of market bubbles is that they form without any solid background

other than investors’ overconfidence, or “euphoria”. Stock market volatility is generally

regarded as a measure of investors’ confidence, so it is quite natural to infer that it might

contain signs of market being overconfident prior to a bubble burst. Thus, “euphoria” can be

interpreted as a situation when stock market volatility behaves “unnaturally” – it is lower than

suggested by economic surrounding. With that in mind, we construct two volatility measures

for the same sample period of historical S&P 500 index returns:

The first measure is calculated from the GARCH-MIDAS model (see Appendices 1

and 2 for details). It has two components: the low-frequency one is modeled with

macroeconomic fundamentals and realized volatility, while the high-frequency one

follows a standard GARCH-type process. This measure represents the “true”

volatility given macroeconomic environment.

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The second measure is taken from a simple GARCH(1,1), which is most commonly

used in financial modeling, trading, risk management etc. It represents the “actual”

volatility as it is seen by market participants.

The details of GARCH-MIDAS estimation results are shown in Table 1 and Figures 5 and 6 in

Appendix 3. Figure 1 below exhibits the estimated low-frequency volatility component for

GARCH-MIDAS (moves quarterly).

Figure 1. Low-frequency volatility estimated from GARCH-MIDAS model

In its turn, Figure 2 below plots the total volatility together with squared returns.

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Figure 2. Squared returns and fitted volatility from the GARCH-MIDAS model

The details of GARCH(1,1) estimation results can be found in Table 2 of Appendix 3. The

series of S&P 500 logarithmic returns with superimposed standard deviations estimated from

GARCH(1,1) model is presented in Figure 3 below.

Figure 3. S&P500 log-returns with fitted GARCH(1,1)

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Then we construct an indicator variable that we call the Volatility Ratio (VR) – the ratio of

volatility measure of GARCH(1,1) to the volatility measure of GARCH-MIDAS:

𝑉𝑅𝑡 =(𝜎𝑡

𝐺𝐴𝑅𝐶𝐻(1,1))2

(𝜎𝑡𝐺𝐴𝑅𝐶𝐻−𝑀𝐼𝐷𝐴𝑆)2

(1)

The intuition behind this action is as follows. If VR is low, then current volatility is lower

than implied by macroeconomic factors, there is growing market overconfidence. On the

contrary, if VR is high, then current volatility is higher than implied by macroeconomic factors,

the market is nervous.

Figure 4 below shows the Volatility Ratio dynamics for the chosen sample period.

Figure 4. Volatility ratio dynamics

In these terms, we postulate the main hypothesis: the lower is VR today, the greater is the

probability of a market bubble burst tomorrow. We test this prediction using logit regression

with the lagged volatility ratio as explanatory variable. The dependent variable in the

regressions is a binary-choice variable that indicates the presence of a market crash on each

day. However, the definition of a market crash is ambiguous, so it is assumed in this paper that

the “crash” event corresponds to a daily stock index return of less than -5%. Of course, other

specifications are also possible. If our research hypothesis is correct, we expect the slope

coefficient to be significant and negative.

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The estimation results of logit regressions can be found in Table 3 of Appendix 3. There

is significant negative relationship between lagged VR and the probability of a market crash at

5% level. This is an encouraging finding since it may be useful for market crash prediction. The

change in tomorrow’s probability of a market crash can be determined as the predicted change

in the volatility ratio (based on a 1-day ahead GARCH(1,1) and GARCH-MIDAS forecasts)

multiplied by the marginal effect of the logistic regression estimated at today’s volatility ratio:

�̂�𝑡+1 = 𝑃𝑡 +𝑒−(�̂�0+�̂�1∗𝑉𝑅𝑡) ∗ �̂�1

(1 + 𝑒−(�̂�0+�̂�1∗𝑉𝑅𝑡))2∗ (𝑉�̂�𝑡+1 − 𝑉𝑅𝑡) (2)

Here the today’s probability of crash 𝑃𝑡 is just the today’s fitted value of the logit regression.

3. Applications and further research

3.1. Brief summary of main results

The main research question was whether market crashes can be predicted using the

estimated differences in the volatility measures from different models. The answer for now is

affirmative – we have discovered a significant negative relationship between the Volatility Ratio

and the probability of a market crash. Moreover, we have found a way to quantify and measure

investors’ overconfidence and show how it leads to market crashes.

This finding speaks against the predominant theoretical concept that investors are perfectly

rational and it might provide inspiration for development of new theoretical models of financial

markets.

3.2. Applications for portfolio management

The methodology used in our research is new to academic literature. Although it was

applied to a rather specific case – the aggregate US stock market, it might be feasible to

successfully extrapolate it to other situations, for example, financial bubbles forming in a

particular industry (leading to a crash of the industry index) or bubbles concerning individual

stocks. The main issue in developing any similar model is to identify some “fundamental”, or

“true”, volatility or correlation measure to be compared to the standard measure (e.g.

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GARCH(1,1) or DCC(1,1)). Let us consider the possible choices for individual stocks and

industrial aggregates.

In our paper we focus on S&P 500 market index, and our hypothesis is about capital market

aggregates, not individual stocks. In these terms, the diversification principle suggests that

general economic state variables provide the only source of influence on stock market return

properties. But that would not work for individual stocks, so we have to adjust our GARCH-

MIDAS model to include firm-specific fundamentals instead of macroeconomic ones. One can

extract relevant factors from the company’s financial statements. For example, financial

leverage and other ratios associated with financial risk are obvious candidates. If the

statements are published monthly, we can use the MIDAS framework (see Appendix 1) to

construct quarterly measure of low-frequency volatility including monthly lags of both RV and

the chosen ratios. In case if the statements are published quarterly, we can still use MIDAS

monthly lags for RV and leave the ratios sampled quarterly as simple regressors in the low-

frequency volatility component.

Also, it might be reasonable to augment the logit regression model with MIDAS structure.

That is, one may consider the probability of a market crash as a dependent variable sampled

weekly or monthly and the lagged Volatility Ratio as an independent variable sampled daily.

Such logit-MIDAS model will make forecasts of the probability of a market crash next

week/month given the values of Volatility Ratio on each day of the present week/month.

When it comes to sectors, the idea has to be generalized to a multivariate case. In these

terms, one can find the DCC-MIDAS model of Colacito, Engle & Ghysels (2011) extremely useful

for this purpose. It shares a similar approach with GARCH-MIDAS, but extends its methodology

to correlation. Namely, the correlation between the two assets is multiplicatively decomposed

into high- and low-frequency components, with the former following a usual DCC structure and

the latter having a MIDAS structure. We can use monthly MIDAS lags of common industry

factors and realized correlation to construct the quarterly low-frequency correlation

component for each pair of stocks. For example, for the oil industry we can use some of the

characteristics of oil prices as factors: monthly standard deviation, monthly return etc. When a

market bubble is forming in the sector, the prices of [most of] the stocks included in this sector

are subject to substantial growth. Thus, correlation between the stocks should be abnormally

high. Analogously, we can construct a new measure of overconfidence, the Correlation Ratio,

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for each pair of stocks. The Correlation Ratio will be defined as the ratio of correlation measure

from standard DCC(1,1) model to the correlation measure from DCC-MIDAS model. This time,

today’s increase in the correlation ratio will indicate an increase in the probability of an industry

bubble burst tomorrow, and vice versa. To test the hypothesis for the multivariate case, one

should run a logit regression for a panel of stock pairs. If the hypothesis turns out to be true,

the slope of the correlation ratio should be positive and significant.

In fact, a DCC-MIDAS model fit is a two-step procedure that includes the fit of univariate

GARCH-MIDAS models for each stock. Hence, it is capable of tracking both individual stocks and

the industry as a whole at the same time.

We believe that our results and further inferences will prove to be useful for market

practitioners in fields where market crashes are a great concern – portfolio management, risk

management and algorithmic trading. It is also worth mentioning that the suggested

methodology is rather flexible when being utilized for risk management. Every individual

investor and every investment fund has its own risk profile. By adjusting the definition of a

market crash (it was -5% or less in our research) and by varying the subjective benchmark

probability P* (if Pt+1 > P*, then short the stock/portfolio), any market practitioner can conceive

his own risk management policy.

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4. Appendix

4.1. Appendix 1. MIDAS regressions

In this section we briefly formulate the general MIDAS regression model as it was first

published in Ghysels, Santa-Clara & Valkanov (2002). Much of the technicalities are skipped

since they can be easily found in the mentioned paper and other works of the authors, for

example in Ghysels, Santa-Clara & Valkanov (2006) and Ghysels & Valkanov (2012).

Suppose that the dependent variable 𝑌𝑡 is sampled at some fixed frequency, while 𝑋(𝑚) is

sampled m times faster. Then a simple linear MIDAS regression will have the form

𝑌𝑡 = 𝛽0 + ∑ 𝛽𝑛𝑋𝑡−𝑛/𝑚(𝑚)

𝑁

𝑛=0

+ 𝜀𝑡(𝑚)

(3)

The implementation of MIDAS models makes the researcher face a tradeoff. On the one

hand, the model does not require aggregation of higher frequency data, so it better exploits the

available information. On the other hand, this advantage comes at the cost of increasing the

number of parameters. If the number of high-frequency lags N is large, some structure has to

be imposed on the weighting polynomial βn. The authors make use of the beta weighting

scheme: 𝛽𝑛 = 𝜃 ∗ 𝜑𝑛(𝑁, 𝜔1, 𝜔2), where

𝜑𝑛(𝑁, 𝜔1, 𝜔2) =𝑓(

𝑛𝑁

, 𝜔1, 𝜔2)

∑ 𝑓(𝑗𝑁

, 𝜔1, 𝜔2)𝑁𝑗=1

, (4)

𝑓(𝑧, 𝑎, 𝑏) =𝑧𝑎−1(1 − 𝑧)𝑏−1

𝐵(𝑎, 𝑏), (5)

θ is the scaling parameter and B(a,b) is the Beta function. Such restriction imposed on

coefficients diminishes the number of parameters significantly (there are only three parameters

for each independent variable) and is capable of producing a large variety of shapes. However,

other weighting structure specifications such as exponential or Almon lag polynomial are also

possible.

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4.2. Appendix 2. GARCH-MIDAS model

Manifold MIDAS models were applied to stock market volatility. However, we are

interested in those that focus on macroeconomic components of volatility. The GARCH-MIDAS

model was introduced by Engle, Ghysels & Sohn (2009) and represents one of the ways

decompose the stock market volatility into high- and low-frequency components, with the

former following a usual GARCH-type structure and the latter having MIDAS structure and being

driven by macroeconomic variables.

Suppose the unexpected return on day i of the quarter t is modeled as

𝑟𝑖𝑡 − 𝜇 = 𝜎𝑖𝑡 ∗ 𝜀𝑖𝑡 , (6)

where the volatility is multiplicatively decomposed into the daily component 𝑔𝑖𝑡 and the

quarterly component 𝜏𝑡, which is kept constant throughout the quarter:

𝜎𝑖𝑡 = √𝜏𝑡 ∗ 𝑔𝑖𝑡 (7)

𝑔𝑖𝑡 = (1 − 𝛼 − 𝛽) + 𝛼 ∗(𝑟𝑖−1,𝑡 − 𝜇)

2

𝜏𝑡+ 𝛽 ∗ 𝑔𝑖−1,𝑡 (8)

The constraint ω = 1 – α – β is imposed on the high-frequency component in order to

ensure that unconditional volatility is equal to its low-frequency component:

𝐸𝑡−1(𝑟𝑖𝑡 − 𝜇)2 = 𝜏𝑡𝐸𝑡−1(𝑔𝑖𝑡) = 𝜏𝑡 (9)

Therefore, the GARCH-MIDAS model relaxes the stationarity assumption of the volatility

process.

Now let us introduce macroeconomic variables into the GARCH-MIDAS model. Taking

into account the results of Engle & Rangel (2008) and Engle, Ghysels & Sohn (2009), we include

four macroeconomic variables (all sampled monthly): the inflation rate, inflation volatility,

industrial production growth rate and unemployment rate. Thus, our MIDAS structure for the

low-frequency volatility component is defined to be

ln 𝜏𝑡 = 𝑚 + 𝜑 ∑ 𝑎𝑗(3, 𝑢)𝑅𝑉𝑡−

𝑗3

3

𝑗=1

+ 𝜃1 ∑ 𝑎𝑗(3, 𝑣1)𝐼𝑃𝑡−

𝑗3

3

𝑗=1

+

+𝜃2 ∑ 𝑎𝑗(3, 𝑣2)𝜋𝑡−

𝑗3

3

𝑗=1

+ 𝜃3 ∑ 𝑎𝑗(3, 𝑣3)𝜎𝜋,𝑡−

𝑗3

3

𝑗=1

+ 𝜃4 ∑ 𝑎(3, 𝑣4)𝑢𝑡−

𝑗3

3

𝑗=1

(10)

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We follow an exponential lag weighting scheme, so that the lag weights depend on only one

parameter:

𝑎𝑗(𝑁, 𝑢) =𝑢𝑗

∑ 𝑢𝑖𝑁𝑖=1

(11)

Like any other GARCH-type model, GARCH-MIDAS can be estimated using maximum

likelihood approach. We treat the standardized residuals as normally distributed, and the

corresponding log-likelihood function for GARCH-MIDAS is defined as

𝑙𝑟 = ln 𝐿 = −1

2∑ (ln(𝑔𝑖𝑡 ∗ 𝜏𝑡) +

(𝑟𝑖𝑡 − 𝜇)2

𝑔𝑖𝑡𝜏𝑡)

𝑇

𝑖=1

(12)

4.3. Appendix 3. Actual estimation results

Table 1. GARCH-MIDAS estimation results

Parameter Estimate Standard error 95% CI lower

bound

95% CI upper

bound

α 0.074 2.973e-05 0.074 0.074

β 0.906 4.181e-05 0.906 0.906

m -10.502 0.001 -10.504 -10.500

θ1 -26.145 0.035 -26.214 -26.076

θ2 0.439 0.010 0.419 0.459

θ3 13.204 0.027 13.151 13.257

θ4 6.039 0.013 6.014 6.064

v1 1.000 0.003 0.994 1.006

v2 1.000 0.015 0.971 1.029

v3 128996.9 3.945 128989.168 129004.632

v4 60393.83 4.805 60384.412 60403.248

ϕ 63.308 0.050 63.21 63.406

u 1.927 0.003 1.921 1.933

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Figure 5. Plot of MIDAS lag weights for industrial production growth rate, CPI

volatility and monthly RV

Figure 6. Plot of MIDAS lag weights for unemployment rate, CPI rate and

monthly RV

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Table 2. GARCH(1,1) estimation results

Parameter Estimated Value Standard Error

µ 4.297e-04 6.086e-05

α 0.075 0.005

β 0.922 0.005

ω 5.795e-07 2.376e-07

Table 3. Logit regression estimation results

Variable Coefficient

estimate

Standard

Error

P-value 95% CI lower

bound

95% CI upper

bound

Intercept -1.001 2.178 0.646 -5.270 3.267

Lagged VR -5.448 2.239 0.015 -9.835 -1.060

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