identification of bubble in gold and study of its propagation to equity markets

Identification of Bubble in Gold and Study of its Propagation to Equity

Markets1

Sandip Chakraborty

Vishal Jagtap

Abstract

Prior research has stipulated that financial markets all over the world are getting more

and more interconnected and the volatility in one market spills over the volatility in other

markets. The US stock market (S&P500) is claimed to be one of the most efficient equity

markets in the world. The major Asian equity markets – Hong Kong, Singapore and India

are well influenced by S&P500 and successively by each other’s. Traditionally gold has

been treated as hedge for the inflation; however since 2000 the volatility in the gold

prices has risen tremendously2. Identifying a bubble situation in the equity markets has

been a topic of interest for many researchers. In past decade we have seen the adverse

impact of successive shifts in bubbles in equity market. The volatility patterns in gold and

associated shifts in the conditional volatility to the dependent sensitivities from gold to

equity indices have triggered our interest to find out whether we can identify the bubble

in gold market and study its propagation to the equity markets. We have considered five

variables – Gold price(GOLD), S&P index (SNP), Hang Seng Index(HKI), Straits Times

Index (STI) and Bombay Stock exchange index (SENSEX) for our study. We have used

Markov Switching Augmented Dickey-Fuller (MSADF) test to identify the bubble in

gold prices. To study the propagation of bubble 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡 time series was constructed

after constructing a Diagonal Vector model (DVEC) of Multivariate GARCH class,

1 Second Draft Version, dated 4/6/2013 2 Gold volatility from 1995 till 2013 is shown in Figure 1.

representing the S&P500 return sensitivity with respect to the gold market. We have then

constructed three more 𝛽 series- 𝛽𝐻𝐾𝐼𝑤𝑟𝑡𝑆𝑁𝑃𝑡, 𝛽𝑆𝑇𝐼𝑤𝑟𝑡𝑆𝑇𝐼𝑡 and 𝛽𝑆𝐸𝑁𝑆𝐸𝑋𝑤𝑟𝑡𝑆𝑇𝐼𝑡which

represented the sensitivity part over time of one market’s return with respect to the other

market. We have the found six important sub-periods in 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡using Bai & Perron

(2003) methodology and analyzed each sub-period to understand the long run causality

relationship between 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡and the other variables which represented relationship

between equity market volatility and gold market volatility. We have found that there

exist negative long run causality between the equity market return sensitivity and gold

market volatility pre financial crises; however post financial crises this relationship was

broken. The study concludes that pre-crisis situation gold accelerates the process of

bubble migration; however, post-crisis cycles are random and noisy.

Keywords: Bubble Migration, Gold, Causality, Conditional Beta, Regime Switching

I. Introduction

Numerous studies have found from empirical research that the markets all over the world

have become more interconnected over past decade. The shock or information in one

market affects the other markets. For the portfolio managers and investment practitioners

it is becoming increasing important to understand how these different markets are

interconnected. As a portfolio manager one should be aware of which market is the

largest news or shock producer and how the volatility in that market spills over the other

markets? What is the mechanism of transmission of this volatility? Is the volatility

transmitted to other markets directly or via some other markets? Is the volatility

transmission contemporaneous or non-contemporaneous? Are the relation between

markets and different financial assets which existed in past are still valid today? These

questions has been attempted to be answered using multivariate time series techniques for

modeling the volatility among different financial markets and asset classes.

The general definition of financial bubble is the phenomenon when the prices of

underlying assets rise rapidly over their intrinsic value which is mostly due to

speculation. There is no economic substance supporting such high prices. Eventually such

prices will be corrected and this is refereed as “bursting of bubble”. Studies of such

financial bubbles formation and burst are ever interesting topic for many economist,

researchers and policy makers. Extensive research has been done on how to identify the

bubble formation, exploring the reasons why the bubble are formed and then what

policies should one implement to avoid such bubbles. With ever interconnected markets

when such a phenomenon occurs in one market then its repercussions are felt in other

markets as well. In their recent article Philip & Yu (2011), have investigated three

financial time series – housing price index, crude oil price, and bond prices. Their

empirical research concludes that the bubble emerged in housing market in 2002 then

traversed to selective commodity market and bond market and finally busted in 2008.

Gold is a very special asset class. Gold has proven from time to time that it is as a real

currency in times of wars and crisis. The gold has four important applications - industry,

alternative currency during uncertain times, speculative investment and central banks of

many emerging countries. Many countries are piling up the gold in their foreign reserve

to hedge against the unforeseen circumstances3. As per the article by Ghosh, et all (2004),

the demand for gold can be classified into two major categories – “use demand” & “asset

demand”. Gold is extensively used in industries such as jewelry, electronics etc. This is

the use demand of gold. The other major driver for gold as an “asset class” is by the fund

managers. Ghosh, et al (2004), concludes that Gold is an inflation hedge in the long term

but the prices are influenced by the short run influences. We ran a univariate GARCH

model on daily gold prices (1995-2013) and found conditional variance in gold as shown

in Figure 1. We can see observe the volatility was quite high from the period of 2000 till

2012; moreover, the volatility in gold was tremendously high post 2007 subprime crises.

Figure 1 Volatility in GOLD return since 1995 to 2013.

Source : World Gold Council : http://www.gold.org/investment/statistics/

We have indentified three important markets in ASIA as key markets – Hong Kong,

Singapore and India. The major indices in these three markets are Hang Seng Index

(HKI), Straits Times Index (STI), and SENSEX, respectively. Sariannidis et al (2009)

concluded that US equity markets (such as S&P500) are the largest news, volatility

3 Total reserves in terms of gold and US dollars can been seen from World Bank website -

http://data.worldbank.org/indicator/FI.RES.TOTL.CD

CondVGOLD

02468

10121416

1/5

/1995

1/5

/1996

1/5

/1997

1/5

/1998

1/5

/1999

1/5

/2000

1/5

/2001

1/5

/2002

1/5

/2003

1/5

/2004

1/5

/2005

1/5

/2006

1/5

/2007

1/5

/2008

1/5

/2009

1/5

/2010

1/5

/2011

1/5

/2012

1/5

/2013

CondVGOLD

http://www.gold.org/investment/statistics/

http://data.worldbank.org/indicator/FI.RES.TOTL.CD

generator and found that volatility spillover effects are significant from US markets to

Hong Kong, Singapore and India.

In this study we have first studied the volatility spillover among the major equity markets

– SNP, HSI, STI and SENSEX along with volatility in gold spot prices (GOLD) to

understand how the volatility in one market spills over the volatility in other market. The

𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡 time series constructed representing the SNP return sensitivity part over

time with respect to the gold market. Similarly, three more conditional three more 𝛽

series- 𝛽𝐻𝐾𝐼𝑤𝑟𝑡𝑆𝑁𝑃𝑡, 𝛽𝑆𝑇𝐼𝑤𝑟𝑡𝑆𝑇𝐼𝑡 and 𝛽𝑆𝐸𝑁𝑆𝐸𝑋𝑤𝑟𝑡𝑆𝑇𝐼𝑡 were constructed, which represented

the sensitivity part over time of one market’s return with respect to the other market.

MSADF test was applied on the GOLD prices to find out different regimes in gold prices

with unit root. The structural break were found out in 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡 using Bai & Perron

(2003) methodology and each sub-period was studied for understanding the long run

causality relationship between ∆𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡 and the other conditional 𝛽 series which

represents the propagation of bubble from gold market to the equity market.

This work contributes to the existing literature of market interconnectedness, bubble

identification and volatility migration study in following ways. Traditionally, gold is

always referred to as inflation hedge. The study first tries to identify the bubble in gold

prices and then tries to find out long run causality between equity market sensitivity over

time and gold volatility pre, post and during financial crises. The relationship between

∆𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡 and the other conditional 𝛽 series would provide us useful information

about pre, post and during financial crises.

The rest of the work is arranged in following way. Section II discusses the literature

review on the market interconnectedness and methods on how to identify the bubble in

financial markets. Section III discusses the empirical approach and framework that will

be used for this study. Section IV discussed the Data analysis and empirical results.

Section V summarizes the conclusion of the study.

II. Literature Review

Analysis of volatility and modeling

The most of the financial instruments which are traded in financial markets exhibit

volatility which changes over time. The volatility represents the risk the asset carries and

modeling this risk is an important aspect of risk management. Volatility is represented by

the variance term (σ2) which is the square root of standard deviation (σ).

Traditional econometric models considered constant one period forecast variances until

the revolutionary paper published by Engle R (1982) which brought in a new

revolutionary model for modeling the time varying volatility– ARCH4 . The ARCH

model provided a way to study the changes in variance over the time by modeling the

variance as an autoregressive process which is related to the square of the previous error

terms in the series (Egnel, 1982). ARCH provided a systematic way to model the

variances, thus the risk factors in the time series data.

4 ARCH – Autoregressive Conditional Heteroskedasticity

Bollerslev (1986) came up with generalized ARCH model (GARCH5). Many researches

were carried out to improve the ARCH models which gave rise to variety of flavors of

ARCH models such as Exponential GARCH or EGARCH (Nelson, 1991), NGARCH,

QGARCH etc. (Villaverde & Ramírez, (2010)). Bauwens, et al (2006) has summarized

the most important development in multivariate GARCH models, their specifications and

inference methods.

Globalization and market integration

Many studies have been conducted to study how the equity markets are correlated with

each other. Baele, L. (2005) studied the effect of globalization and market integration

with special focus on Western European markets. Xiao, L., & Dhesi, G. (2010) studied

volatility spillover effects across various stock indices in USA using MGARCH6 BEKK7

(Engle & Kroner, 1995) and DCC8 model (Engle 2002). Sariannidis, et al (2009) studied

volatility linkages among three Asian stock exchange markets, namely India, Singapore

and Hong Kong, during the period July 1997 to October 2005 using multivariate diagonal

BEKK GARCH model and found that all examined markets are highly integrated,

reacting to common information which was mostly derived from the USA market.

There are few studies done on the volatility spillover between oil prices, USD index, and

equity markets. Basher et al. (2011) studied the relationship between oil prices, USD

index and emerging market stock prices using SVAR (Structural Vector Autoregression)

5 GARCH – Generalized Autoregressive Conditional Heteroskedasticity 6 MGARCH – Multivariate GARCH 7 BEKK - Baba, Engle, Kraft and Kroner Model 8 DCC - Dynamic Conditional Correlation

model and found that emerging market stock prices get depressed if there are positive

market shocks in oil market and vice versa. Another finding of their study was increasing

stock index in emerging markets has positive spillover effect on oil markets. Sari et al.

(2010) studied volatility between four precious metals (gold, silver, platinum, palladium),

oil prices and US Euro exchange rate and found strong feedbacks in short run but weak

long run equilibrium relationship. Garefalakis et al (2011) tried to determine the effect of

equity, energy, gold, currency exchange rates on Han Seng Index using GJR-GARCH

model and found that the volatility of gold returns influence negatively to mean returns of

Han Seng index. Alom et al.(2010) studied the mean and volatility spillover effects of

food prices across different APAC markets using component GARCH method and did

not found any significant spillover effect among the APAC countries.

Prior research has stipulated that (e.g Sariannidis et al. 2009) the BEKK model would be

most suitable for running multivariate GARCH model to study volatility spillover among

various asset time return series. The mean returns equations for the BEKK model can be

written as shown in Equation 1

𝑅𝑡 = Θ + Φ𝑅𝑡−1 + 𝜀𝑡, 𝑅𝑡 = [𝑟1𝑡

𝑟2𝑡] is the vector of asset returns,

Equation 1 Mean equation for BEKK model

Θ is the constant matrix; 𝜀𝑡 represents the random error vector in the equation; and the

diagonal components of the matrix Φ represents the lagged returns for the own variable

𝑅𝑡. The off diagonal components of the matrix 𝐵 = (𝑏𝑖𝑖 𝑏𝑖𝑗

𝑏𝑗𝑖 𝑏𝑗𝑗) represent the mean

spillover effects across the variables under consideration. The Variance equation of the

BEKK model is shown in Equation 2 as follows:

(ℎ𝑖𝑖,𝑡 ℎ𝑖𝑗,𝑡

ℎ𝑗𝑖,𝑡 ℎ𝑗𝑗,𝑡) = (

𝑐𝑖𝑖 𝑐𝑖𝑗

𝑐𝑗𝑖 𝑐𝑗𝑗) (

𝑐𝑖𝑖 𝑐𝑖𝑗

𝑐𝑗𝑖 𝑐𝑗𝑗)′

+ (𝑎𝑖𝑖 𝑎𝑖𝑗

𝑎𝑗𝑖 𝑎𝑗𝑗)′

(𝜀𝑖𝑖,𝑡−1 𝜀𝑖𝑗,𝑡−1

𝜀𝑗𝑖,𝑡−1 𝜀𝑗𝑗,𝑡−1) (

𝜀𝑖𝑖,𝑡−1 𝜀𝑖𝑗,𝑡−1

𝜀𝑗𝑖,𝑡−1 𝜀𝑗𝑗,𝑡−1)′

(𝑎𝑖𝑖 𝑎𝑖𝑗

𝑎𝑗𝑖 𝑎𝑗𝑗)

+ (𝑏𝑖𝑖 𝑏𝑖𝑗

𝑏𝑗𝑖 𝑏𝑗𝑗)

′

(ℎ𝑖𝑖,𝑡−1 ℎ𝑖𝑗,𝑡−1

ℎ𝑗𝑖,𝑡−1 ℎ𝑗𝑗,𝑡−1)(

ℎ𝑖𝑖,𝑡−1 ℎ𝑖𝑗,𝑡−1

ℎ𝑗𝑖,𝑡−1 ℎ𝑗𝑗,𝑡−1)

′

(𝑏𝑖𝑖 𝑏𝑖𝑗

𝑏𝑗𝑖 𝑏𝑗𝑗)

Equation 2 Volatility equation for BEKK model

In the Equation 2, matrix C is a constant matrix, matrix A represents the shocks or news

(ARCH component) and matrix B represents the past volatility affects across the markets

(GARCH term). The BEKK model was a generalized model and more specific models of

BEKK are available such as Diagonal BEKK and Scalar BEKK.

According to Leeb & Pötscher (2009) setting Equation 2 to B = A * D where D is a

diagonal matrix becomes

t t 1 1 t 1 1 t 2H [ | ]

t tC C A A DE A A D

Equation 3 Volatility equation for Diagonal BEKK model

The Equation 3 can model the conditional variance and covariance of the different

combinations of asset returns or time series under consideration. We have chosen

Diagonal BEKK for this study which as it resulted into better model fit.

Studies on Gold

Gold has been always treated as the inflation hedge and safe heaven in times of crises.

Ghosh, et al (2004) concludes that the gold does act as inflation hedge in long term.

Despite the importance of gold not many studies were conducted to analyze the volatility

in gold prices and its impact on other markets. Tully & Lucey (2007) applied APGARH9

model on the cash and gold future prices and found that this model was a better fit.

During the study they have found that the asymmetric component in gold was statistically

insignificant. Batten & Lucey (2009) studied the volatility in gold futures using Garman

Class Estimator and found that the gold prices are not just function of supply and demand

but are also dependent on how other inter connected markets behave. Ewing & Malik

(2012) employed univariate and bivariate GARCH models to model volatility between

gold and oil returns and found that with structural breaks in variances accounted, the

spillover effect was significant; however, ignoring the structural break the spillover effect

was not significant. Thus we can conclude that the gold prices are not just the function of

supply and demand and in more globalized world the gold volatility can affect other

markets and vice-versa.

Studies on financial bubble

There are many methods proposed by researchers on how to detect the financial bubble in

an asset. Hamilton & Whiteman (1985) proposed the methodology to detect the financial

bubble in asset by looking at the order of integration of pair of variables under

consideration. If the price of asset is rising exponentially compared to its intrinsic value

which is determined using dividends etc then one can conclude the existence of the

bubble. Hall et al. (1999) have developed the method for detecting periodically collapsing

bubbles by using Dickey-Fuller test which makes use of the Markov regime-switching

models. We will use a similar concept in identifying the bubble like situation in the gold

prices. Philip & Yu (2011) have investigated three financial time series – housing price

9 APGARCH - asymmetric power GARCH model

index, crude oil price, and bond prices. They have used forward recursive regression

along with sequential right-sided unit root tests. Their empirical research concludes that

the bubble emerged in housing market in 2002 then traversed to selective commodity

market and bond market and finally busted in 2008.

Our approach for financial bubble detection would be to first find the regimes in gold

prices using the MSADF test. If there exist unit root in the identified Regimes then it

signifies that the gold prices during that period were not having mean reverting trend

which is very similar to a bubble like situation. The date stamping on the regimes would

help us map these bubble like situation with the equity market return sensitivity over time

using the constructed conditional β series. Our base line conditional β series will be

SNPwrtGOLDt . We will use method proposed by Bai & Perron (2003) to find structural

breaks in SNPwrtGOLDt . We will then work on each sub-period separately. We will run

Augmented Dickey Fuller Test for testing the unit root in the time series, Johansen

Cointegration test for testing the cointegration between the variables, VECM model to

study the long run and short run causality between the conditional β series and finally

performing Wald Test for testing the significance and impact of the coefficients

associated with the variable. Following section briefly explains the above mentioned

terminologies.

Regime switching model and Structural breaks

Regime switching model was first proposed by Hamilton (1989). A simple regime

switching model can be modeled as

𝑅𝑒𝑔𝑖𝑚𝑒 0 ∶

∆𝑦𝑡(𝑆𝑡 = 0) = 𝜇0(𝑆𝑡 = 0) + 𝜌(𝑆𝑡=0)0𝑦𝑡−1 + [∑𝛾𝑖∆𝑦𝑡−𝑖

𝑘0

𝑖=2

(𝑆𝑡 = 0)] + 𝜖0𝑡(𝑆𝑡 = 0)

𝑅𝑒𝑔𝑖𝑚𝑒 1:

∆𝑦𝑡(𝑆𝑡 = 1) = 𝜇1(𝑆𝑡 = 1) + 𝜌(𝑆𝑡=1)1𝑦𝑡−1 + [∑𝛾𝑗∆𝑦𝑡−𝑗

𝑘1

𝑗=2

(𝑆𝑡 = 1)] + 𝜖1𝑡(𝑆𝑡 = 1)

𝑊ℎ𝑒𝑟𝑒 𝐸 = [𝜖0𝑡

𝜖1𝑡] ~𝑀𝑢𝑙𝑡𝑖𝑣𝑎𝑟𝑖𝑎𝑡𝑒 𝐺𝑎𝑢𝑠𝑠𝑖𝑎𝑛 𝐼𝐼𝐷(0, [

𝜎0𝑡

𝜎1𝑡]

Equation 4 Regime Switching Model

It was method to model the non linear dynamics in the financial time series. Markov

regime switching model uses an unobserved random variable St which follows the

Markov Chain. The Markov Chain defines the transition probabilities within the defined

N states. The probabilities are defined by the equation 5.

1p|q , p,q 0,... 1[ | .]t tp P s p s q N

Equation 5 Markov Chain probabilities

The probability of moving from q state to p state is only dependent on its past state. .

Once we have obtained the Markov regimes then we will check in which regime we see

unit root behavior as per Equation 6,

𝑅𝑒𝑔𝑖𝑚𝑒 0 ∶ ∆𝑦𝑡(𝑆𝑡 = 0)𝜇0(𝑆𝑡 = 0) + 𝜌(𝑆𝑡=0)0𝑦𝑡−1 + ∑ 𝛾𝑖∆𝑦𝑡−𝑖 𝑘0𝑖=2 (𝑆𝑡 = 0) +

𝜖0𝑡(𝑆𝑡 = 0)

𝑅𝑒𝑔𝑖𝑚𝑒 1: ∆𝑦𝑡(𝑆𝑡 = 1) = 𝜇1(𝑆𝑡 = 1) + 𝜌(𝑆𝑡=1)1𝑦𝑡−1 + ∑ 𝛾𝑗∆𝑦𝑡−𝑗 𝑘1𝑗=2 (𝑆𝑡 = 1) +

𝜖1𝑡(𝑆𝑡 = 1)

𝐼𝑓 [𝜌(𝑆𝑡=0)0], [𝜌(𝑆𝑡=1)1] → 0, ∈ 𝑎 𝑢𝑛𝑖𝑡𝑟𝑜𝑜𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑖𝑚𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 𝑝𝑟𝑜𝑐𝑒𝑠𝑠.

Equation 6 Unit root check

The issue with the regime switching model is that they assume fixed N states within

which the shifts will occur. Structural breaks are unexpected shifts seen in the time series.

If structural breaks are not accounted for then it will lead to inaccurate results. When

compared with the regime switching models structural break models provide better

flexibility in the sense that the number of states are assumed to be infinite10. Bai & Perron

(2003) has discussed the method of computation and analysis for finding the multiple

structural breaks in a time series. The previous methods of finding structural break were

having a major issue of limiting distribution of estimators. This issue is overcome by the

model introduced by Bai & Perron (2003) (abbreviated BP model). The primary

advantage of the BP model is that it uses effective algorithm to obtain global minimizers

of the sum of squared residuals. Secondly, the new algorithm is based on the principals of

dynamic programming and required least number of operations to find out the structural

breaks. Thirdly, the method works well for the data with pure and partial structural

breaks. Fourthly, the algorithm has inbuilt methods to construct the confidence intervals

for providing the structural breaks with date stamping. As this method overcomes the

most of the issues of previous methods of finding structural break we will also consider

this model for our study.

10 http://personal.strath.ac.uk/gary.koop/Song.pdf

http://personal.strath.ac.uk/gary.koop/Song.pdf

We used MSADF test to find the regimes in GOLD prices and BP model for finding

structural breaks in 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡.

Augmented Dickey Fuller Test for unit root – ADF Test

Once we have obtained the structural breaks in the data series then we will run ADF test

on the data series to see if the data series have any unit root. The ADF test is used to find

out whether the time series is stationary or exhibiting random walk behavior. The ADF

test incorporates ARMA (p,q) model with unknown order which conventional unit root

test do not. As per the article, Cheung & Lai (1995) typical ADF regression equation for

ta time series 𝑋𝑡 can be written as shown below.

Δ𝑋𝑡 = 𝜇 + 𝛾𝑡 + 𝛼𝑋𝑡−1 + ∑ 𝛽𝑗Δ𝑋𝑡−𝑗 + 𝜁𝑡𝑘𝑗=2

Equation 7 ADF Regression Equations

ADF Test Hypothesis

H0 = There is no unit root

Ha = There is unit root.

The unit root exists in the time series if we can reject the null hypothesis. Theoretically, if

there exist a unit root in the time series it means that the time series does not have mean

reverting trend. It signifies the explosive behavior in asset price.

Johansen-Juselius (JJ) Cointegration Test & VECM Model

If the variables are integrated with same order, generally of order I(1), then we can run JJ

cointegration test on these variables to see if the variables are cointegrated. The result

would be the cointegration vector with error correction term. As pointed out by Darrat &

Zhong (2002) The JJ cointegration test is preferred over the Engle-Granger test as the JJ

cointegration test allows having more than one cointegration equation. As pointed out by

Kleiman et al. (2002) the JJ test is used to determine the number of cointegrating vectors

based on following VAR or VECM model can be modeled by Equation 8 –

Δ𝑌𝑡,𝑗 = ∑Γ𝑖,1Δ𝑌𝑡−𝑖,𝑗 + Π𝑌𝑡−𝑖,𝑗

𝑁

1=1

+ Ε𝑗𝑡

Equation 8 Cointegration Equation

where Δ𝑌𝑡,𝑗 represents the conditional 𝛽 time series matrix for our study. Δ is the

difference operator. The Term Π𝑌𝑡−𝑖,𝑗 is the Error correction term which represents the

long run causality relationship between the variables. The matrix can be split into two

components 𝛼𝛽 where the 𝛼 represents the vector error correction coefficients while the

𝛽 represents the cointegrating equation with all the variables.

We have considered 4 conditional β series as -𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡, 𝛽𝐻𝐾𝐼𝑤𝑟𝑡𝑆𝑁𝑃𝑡

, 𝛽𝑆𝑇𝐼𝑤𝑟𝑡𝐻𝐾𝐼𝑡and

𝛽𝑆𝐸𝑁𝑆𝐸𝑋𝑤𝑟𝑡𝑆𝑇𝐼𝑡 in six sub-periods. We will run the JJ test on these variables over six sub-

periods. Thus vector Δ𝑌𝑡,𝑗 can be represented by following expression:

Δ𝑌𝑡,𝑗 =

[ 𝛽𝑆𝑁𝑃𝑤𝑟𝑡𝐺𝑂𝐿𝐷𝑡

𝛽𝐻𝐾𝐼𝑤𝑟𝑡𝑆𝑁𝑃𝑡

𝛽𝑆𝑇𝐼𝑤𝑟𝑡𝐻𝐾𝐼𝑡

𝛽𝑆𝐸𝑁𝑆𝐸𝑋𝑤𝑟𝑡𝑆𝑇𝐼𝑡]

Equation 9 Cointegration test matrix

where i represents the sub-periods from 1 to 6 in our case. The JJ test provides two test

statistics – Maximum eigenvalue test and trace test. The null and alternative hypothesis

of both the test statistics are given below

Max Eigenvalue Test Hypothesis

H0 = Number of cointegrating vectors are r

Ha = Number of cointegrating vectors are r+1

Trace Test Statistics Hypothesis

H0 = Number of cointegrating vectors are r

Ha = Number of cointegrating vectors are > r

If we found that the time series are cointegrated then we can run VECM model to find out

long run causality and short run causality between tSNPwrtGOLD and other β series -

tHKIwrtSNP , tSTIwrtHKI ,

tSENSEXwrtSTI

The VECM model for the tSNPwrtGOLD is shown in Equation 10

(t i) (t i)

(t i) (t i) 1 1

1 1

1 1

1 1

1 1

1*

t

t t

SNPwrtGOLD i SNPwrtGOLD i HKIwrtSNP

i STIwrtH

p p

i i

KI i SENSEXwrtSTI

p p

i i

Z ECT

Equation 10 VECM Model for tSNPwrtGOLD

Similarly, the VECM model for the other conditional β series is shown from Equation 11

to 13.

(t i) (t i)

(t i) (t i) 1 1

1 1

1 1

1 1

1 1

1*

t

t t

HKIwrtSNP i HKIwrtSNP i SNPwrtGOLD

i

p p

i i

p p

STIwrtHKI i SENSEXwrtSTi i

I Z ECT

Equation 11 VECM Model for tHKIwrtSNP

(t i) (t i)

(t i) (t i) 1 1

1 1

1 1

1 1

1 1

1*

t

t t

p p

i i

p p

i

STIwrtHKI i STIwrtHKI i SNPwrtGOLD

i HKIwrtSNP i SENSEXwrtSTIi

Z ECT

Equation 12 VECM Model for tSTIwrtHKI

(t i) (t i)

(t i) (t i) 1 1

1 1

1 1

1 1

1 1

1*

t

t t

SENSEXwrtSTI i SENSEXwrtSTI i SNPwrtGOLD

i HKIwrtSNP

p p

i i

i STIwrtHKI

p p

i i

Z ECT

Equation 13 VECM Model for tSTIwrtHKI

The 1t

ECT

term represents the long run causality relationship among the variables while

the symbols α, , γ and δ are the representative of coefficient of short run causality

among the variables. The term 1t is residual term in the equation. The Wald test will be

conducted on the coefficients obtained to see if together (lagged variable coefficient) they

can have long run or short run causality with the dependent variable. These coefficients

will help us to understand how the variables are related with each other.

III. Empirical Approach & Framework

Data

The variables under consideration are gold prices (GOLD), S&P500 index (SNP), Hang

Seng Index (HKI), Straits Times index (STI) and SENSEX. The daily closing prices for

each of these variables from 5th Jan 2004 till 31st Jan 2013 will be used for the study.

Framework & Approach

Our approach for financial bubble detection would be to first find the regimes in gold

prices using the MSADF test. If any of the regimes can satisfy Equation.6 then we can

conclude that we are seeing unit root in gold price. The date stamping on the regimes

would help us map these bubble like situation with the equity market return sensitivity

using the constructed conditional β series.

The relationship between the gold and the equity markets is studied using framework

shown in Figure 2.

Figure 2 Framework of study

We have first ran the multivariate GARCH Diagonal BEKK (1,1) model on the 5

variables and find out the conditional covariance among these variables and their

individual conditional variance (representative of direct volatility of an asset). In our

case, we got 10 covariance terms and five conditional covariance terms – one for each

variable. This has enabled us to calculate the return sensitivity of one variable with

respect to the volatility of other variable. The β is calculated using the Equation given

below.

Step 1. Run Multivariate GARCH on daily returns data GOLD, SNP,

HKI, STI & SENSEX from 1st Jan 2004 to 31st Jan 2013

Step 2.Find Covariance among all 5 variables & conditional Variance

for each variable.

Step 3. Calculate = COV( gold,

SNP)/condVariance( GOLD)

Similary, calculate , &

Step 4. Find out structural break periods in and

arrange the , & as

per the break periods.

Step 5. For each sub period test for unit root, cointegration and obtain

VECM or VAR model. Test the coefficients.

Step 6. Analyze the results and conclude the effects with date

stamping.

(

(t t

tt

COV

Variance

Equation 14: β formula

In step 3 we have calculated four conditional β series - tSNPwrtGOLD ,

tHKIwrtSNP ,

tSTIwrtHKI and tSENSEXwrtSTI . The series

tSNPwrtGOLD represents the returns

sensitivity part over time of SNP market return with respect to the volatility in the GOLD

market. tHKIwrtSNP represents the return sensitivity part over time of HKI market with

respect to the volatility in SNP market. tSTIwrtHKI represents the return sensitivity over

time of STI market returns with respect to the volatility in HKI market and

tSENSEXwrtSTI represents the return sensitivity part over time of SENSEX market

returns with respect to the volatility in STI market.

tSNPwrtGOLD time series is the baseline time series and is used to study how the

volatility in gold market affects the sensitivity of returns of the world’s largest news

producer and most efficient equity market - SNP market. We are especially interested to

find out if there are any changes in long run causality between the “other conditional β

series”tHKIwrtSNP ,

tSTIwrtHKI ,tSENSEXwrtSTI and

tSNPwrtGOLD during pre or post

financial crises. For simplicity of interpretations and calculations we have considered the

volatility flow from GOLD to SNP, then SNP to HKI, then HKI to STI and then from STI

to SENSEX markets. Past studies suggest that SNP, HKI, STI and SENSEX are

interconnected.

In step 4, we have used the methodologies introduced by Bai & Perron (2003)

(abbreviated as BP Model) to estimate the structural breaks in time series. We have

applied this methodology to find structural break in the tSNPwrtGOLD . If there are n

structural breaks found in the series then we can map the series into n+1 sub-periods. The

other conditional β series will then be mapped into these sub-periods accordingly with

date stamping. Thus, we will get 4 conditional β time series with different sub-periods.

In step 5 and 6, we will work on each individual sub-period to find out how these

conditional β series behave together. We will run following tests one after other to

eventually find out their long run causality relationship. We will first run ADF Test on

each conditional β series to find out if each variable exhibit random walk phenomenon,

followed by Johansen Cointegration Test to find if they are cointegrated. If the

conditional β series are cointegrated then we will run VECM model else VAR model to

find out long run and short run causality relationship between the other conditional β

series andtSNPwrtGOLD . The long run causality is represented by the Error Correction

Term in VECM model and it will provide important relationship between the gold

volatility and equity market return sensitivity before, after and during financial crises.

The analysis of these sub-periods will be date matched with the unit root regimes found

in gold prices using MSADF test. This enables us to analyze the propagation of bubble

from GOLD to equity market.

IV. Data Analysis & Empirical Results

To identify bubble like situation in GOLD prices we ran MSADF test and found out

following regimes in gold prices. The two regimes are labeled as Regime0 and Regime1.

Regime 0 Regime 1

9-Dec-05 3-Jan-06 21-Jan-04 8-Dec-05

18-Apr-06 23-Aug-06 4-Jan-06 13-Apr-06

29-Aug-06 15-Sep-06 24-Aug-06 28-Aug-06

29-Sep-06 5-Oct-06 18-Sep-06 28-Sep-06

4-Jan-07 5-Jan-07 6-Oct-06 3-Jan-07

22-Feb-07 5-Mar-07 8-Jan-07 21-Feb-07

2-Nov-07 25-Jun-09 6-Mar-07 1-Nov-07

24-Jul-09 31-Jan-13 26-Jun-09 23-Jul-09

Table 1 Regimes in gold prices using Markov-Switching Augmented Dickey Fuller

test

We found that in Regime1 the gold prices were definitely having random walk

phenomenon and thus bubble like situation existed in Regim1 compared to Regime0.

Thus, we can say that we have identified Regime1 as bubble regime in gold prices. The

regimes are then sub divided into major regime periods as follows –

Following two periods where the major Regime1 period.

1. 21-Jan-04 to 13-Apr-06 (Regime1,1)

2. 18-Sep-06 to 1-Nov-07 (Regime1,2)

Following two periods were major Regime0 period.

1. 2-Nov-07 to 25-Jun-09 (Regime0,1)

2. 24-Jul-09 to 31-Jan-13 (Regime0,2)

The empirical results for framework mentioned in Figure 2 are provided below. In Step 1,

we have run multivariate Diagonal BEKK(1,1) GARCH model on the GOLD, SNP,

HKI, STI and SENSEX . Table 1 & Table 2 summarizes ARCH and GARCH results.

ARCHGOLD ARCHSNP ARCHHKI ARCHSTI ARCHSENSEX

GOLD 0.190265 NA -0.055297 NA NA

SNP -0.032285 0.295362 -0.064055 NA NA

HKI NA 0.101171 0.163048 NA 0.044785

STI NA 0.074807 NA 0.204013 0.045883

SENSEX NA 0.099299 -0.113720 NA 0.294441

Table 2 ARCH Matrix

From Table 1 we can see that the news produced in GOLD market impacts negatively on

SNP market with coefficient of -0.032285. The news produced in SNP market does not

affect GOLD market but affects positively on SNP, HKI and SENSEX market. The news

produced in HKI markets affects negatively for both GOLD and SNP market. The news

in STI does not affect any of the market. The news produced in SENSEX does not affect

GOLD or SNP but positively affects the Asian markets.

GARCHGOLD GARCHSNP GARCHHKI GARCHSTI GARCHSENSEX

GOLD 0.972626 NA NA NA NA

SNP 0.016249 0.935787 0.040245 NA -0.014268

HKI NA 0.042454 1.003476 NA -0.017070

STI 0.010411 0.035813 0.053220 0.928660 -0.014372

SENSEX 0.021527 0.041850 0.096434 -0.080913 0.9428

Table 3 GARCH Matrix

From Table 2 we can see that the volatility in GOLD market affect SNP volatility

positively; however the volatility in SNP does not affect the GOLD but affects all other

equity market positively. The volatility in HKI has high impact on SENSEX volatility

than on STI and SNP volatility. The volatility in STI does not impact the GOLD, SNP

and HKI but does affect SENSEX negatively. The volatility in SENSEX affects

negatively on SNP, HKI and STI.

In Step 2 and 3 we have calculated the four conditional β series - HKIwrtSNP t , STIwrtHKI t

,

SENSEXwrtSTI t and SNPwrtGOLDt

. In Step 4 we have found following structural breaks in

SNPwrtGOLDt series using BP Model.

Sub Period From To

S1 6-Jan-04 27-May-05

S2 31-May-05 5-Oct-06

S3 6-Oct-06 12-Feb-08

S4 13-Feb-08 2-Jul-09

S5 6-Jul-09 23-Nov-10

S6 24-Nov-10 31-Jan-13

Table 4 Structural breaks in SNPwrtGOLDt using BP Model

Taking this as a reference, six sub-periods from S1 to S6 were formed and all β were then

split into six sub-periods. These four structural breaks obtained using MSADF test were

mapped onto the structural breaks obtained from the BP Model as shown in Table 5.

Gold prices

Regimes

obtained using

MSADF Test

Regime1,1 Regime1,2 Regime0,1 Regime0,2

SNPwrtGOLDt

Subperiods

using BP Model

S1 & S2 S3 S3&S4 S5&S6

Table 5 Comparison of break periods

Clearly, we can see that the regimes Regime1,1, Regime1,2 represents the pre-financial

crises period are mapped to S1, S2 and S3 sub period while the regimes Regime0,1,

Regime0,2 represents the post financial crises period and can be mapped to S3, S4, S5 &

S6 period.

The results of step5 are shown for each sub-period from S1to S6 below.

Sub-Period S1 from 6-Jan-04 to 27-May-05

Table 6 summarizes the test results conducted on S1 sub-periods.

S1 period ADF Test Does series has Unit

Root ? Johansen Cointegtration

Test

SNPwrtGOLDt

Yes

Yes with 1 cointegrated equation found

HKIwrtSNP t

Yes

STIwrtHKI t

Yes

SENSEXwrtSTI t

Yes

Table 6 Sub-period S1 - ADF, Johansen Cointegration Test results

As ADF test suggests the individual series are having unit root we can run Johansen

cointegration test. We found that the series are cointegration within S1 sub-period. We

have run VECM model and obtained the following model for 1t s

SNPwrtGOLD

1 11

1 1 1

( ) ( )

( ) ( )

1 1

1 21

0.00145 0.055 0.00204 *

0.0128 * 0.01725 * 0.249633*

s ss

s s s

t tt

t t t

SNPwrtGOLD SNPwrtGOLD HKIwrtSNP

STIwrtHKI SENSEXwrtSTI STIwrtHKID

Equation 15: Sub-period S1 tSNPwrtGOLD equation

The Error Correction Term (ECT) in the model is which represents long run causality

relationship if following.

1 1 1

1 1

( ) ( )

( )

1 1

1 1

0.00145 0.055 * 0.00204 *

0.0128 * 0.01725 *

t t

t

s

st

s s

s

t SNPwrtGOLD HKIwrtSNP

STIwrtHKI SENSEXwrtSTI

ECT

Equation 16: Error Correction Term Sub-period S1 tSNPwrtGOLD equation

In sub-period S1, we see 1t s

SNPwrtGOLD has negative long run causality with its own

lag 1 values, as well as with other conditional β series. Also, 1t s

SNPwrtGOLD has short

run negative causality with STI return sensitivity with respect to HKI market volatility

with coefficient of -24.9%. Thus we conclude that 1t s

SNPwrtGOLD has overall

negative long run causality all equity markets in consideration.

Sub-Period S2 from 31-May-05 to 5-Oct-06

Table 7 summarizes the test results conducted on S2 sub-periods.


Root ? Johanses Cointegtration

Test

SNPwrtGOLDt

Yes


HKIwrtSNP t

Yes

STIwrtHKI t

Yes

SENSEXwrtSTI t

Yes




have run VECM model and obtained the following model for 2t s

SNPwrtGOLD

2 2 2

2 2

( 1) ( )

( 1 (

1

)1)

0.01012 0.1225 0.8713

0.00287 0.0077

s s s

s

t

s

t t

t t






2 2 2

2 2

1( ) ( )

( ) ( )

1

1 1

0.01012 0.1225 0.8713

0.00287 0.0077

s s s

s

t t

t t s



ECT


In period S2, we see that 2t s

SNPwrtGOLD was having negative long run causality with

its own lag 1 values, as well as with other conditional β series. It is observed that the

coefficients of all conditional β series has became more negative compared to S1 period.

This signifies the negative long run causality has increased over S2 period compared to

S1 period. The period S1 & S2 overlaps with the gold regime -Regime1,1 which

represents unit root. We can thus conclude that the gold could have accelerated the

process of bubble migration to the equity market.

Sub-Period S3 from 6-Oct-06 to 12-Feb-08

Table 8 summarizes the test results conducted on S3 sub-periods



Test

SNPwrtGOLDt

Yes


HKIwrtSNP t

Yes

STIwrtHKI t

Yes

SENSEXwrtSTI t

Yes




have run VECM model and obtained the following model for tSNPwrtGOLD .

3 3 3

3 3

( ) 1 1

1 1

( )

( ) ( )

0.04169 0.1175 0.01947 *

0.06257 0.02607

s s s

s s

t t t

t t






3 3 3

3 3

1 1

1

( ) ( )

( ) ( )1

0.04169 0.1175 0.01947 *

0.06257 0.02607

t t

t

s s s

s st



ECT


In period S3, we see that 3t s

SNPwrtGOLD was having negative long run causality with

its own lag 1 values, as well as with other conditional β series. It is observed that the

coefficients of all conditional β series has became less negative compared to S2 period.

This was the period of sub primes crises which has affected US stock markets heavily.

The dates of major crashes in the US stock markets11 during this sub-period were– 27 Feb

2007, 11 Oct 2007. From the analysis of S1, S2 and S3 period we can conclude that the

long run causality was negative from equity markets to the SNP return sensitivity with

respect to gold markets. We observed the negative causality increased tremendously in S2

period and subsided in S3 period. The Regim1,2 transitioned into Regime0,1 during S3

period. From the major crash dates of US equity market we can observe that S2 period

represents the formation of bubble in financial markets.

11 Data is taken from following articles -

http://online.wsj.com/article/SB121460787893112069.html?mod=googlenews_wsj

http://www.nytimes.com/2007/03/04/business/yourmoney/04count.html?_r=2&st=cse&sq=%22Black+Mo

nday%22&scp=5&oref=slogin&

http://online.wsj.com/article/SB121460787893112069.html?mod=googlenews_wsj

http://www.nytimes.com/2007/03/04/business/yourmoney/04count.html?_r=2&st=cse&sq=%22Black+Monday%22&scp=5&oref=slogin&

http://www.nytimes.com/2007/03/04/business/yourmoney/04count.html?_r=2&st=cse&sq=%22Black+Monday%22&scp=5&oref=slogin&

Sub-Period S4 from from 13-Feb-08 to 2-Jul-09

Table 9 summarizes the test results conducted on S4 sub-periods



Test

SNPwrtGOLDt

Yes

No Cointegration found ! HKIwrtSNP t

Yes

STIwrtHKI t

Yes

SENSEXwrtSTI t

Yes



cointegration test. Interestingly, we found that in sub-period S4 the series are no longer

cointegrated. As series are not cointegrated we can not run VECM hence we used VAR

to model the relationship between the other 3 β series and ΔβSNPwrtGOLD and found

following equation for ΔβSNPwrtGOLD

4 4 4( ) ( )1 2 0.884902 * 0.057916

s s st t tSNPwrtGOLD SNPwrtGOLD SNPwrtGOLD


In Period S4, the beta series were not cointegrated yet each had unit root. The

4

stSNPwrtGOLD was only dependent on its past return sensitivity with respect to

volatility in gold market. This means that the long run causality relationship between

these beta series was broken during this period. The equity market return sensitivity with

respect to the volatility in other market is not having any long run or short run causality

with the return sensitivity of SNP with respect to the gold market volatility. This could be

termed as flight to safety phenomenon where the investors have realized that the gold

might be a safe heaven to invest in the period of financial crises.

Sub-Period S5 & S6 from 6-Jul-09 to 31-Jan2013

Table 10 summarizes the test results conducted on S5 & S6 sub-periods

S5 & S6 period ADF Test Does series has Unit


Test

SNPwrtGOLDt

Yes

1 Coint eq found HKIwrtSNP t

Yes

STIwrtHKI t

Yes

SENSEXwrtSTI t

Yes

Table 10 Sub-period S5 & S6 - ADF, Johansen Cointegration Test results


cointegration test. Interestingly, we found that in sub-period S5 & S6 there was no

significant long run causality relationship or for tSNPwrtGOLD and the other β

series.

Few short run causality were found as below –

For S5 sub-period

5 5

)1( 0.157387 *s st tSNPwrtGOLD SENSEXwrtSTID


For S6 sub-period

6 6 6

2( )1) ( 0.476487 * 0.345132 *s s st t tSNPwrtGOLD STIwrtHKI STIwrtHKID D


In the period S5, and S6 phenomenon of flight to safety continued and we did not observe

any long run causality between tSNPwrtGOLD with other conditional β series. The

results suggest that from 13 Feb 2008 until Jan 2013, gold volatility has little impact on

the SNP return sensitivity and gold is treated as a different asset class.

V. Conclusion

MSADF test on gold prices revealed two regimes – Regime0 and Regime1. Unit root was

found in Regime1 gold prices. The Regime1 was divided into two main periods –

Regime1,1 (21-Jan-04 to 13-Apr-06) and Regime1,2 (18-Sep-06 to 1-Nov-07). Both these

periods indicate existence of bubble situation in the gold. Overall, Regime1 overlaps with

sub-period S1, S2 and S3 of tSNPwrtGOLD series. We have also observed that in S1,

S2 and S3 sub-periods all conditional β series were cointegrated and each series had unit

root. It was found that there was a negative long run causality relationship between

tSNPwrtGOLD and other conditional β series during these sub-periods; moreover, the

coefficients obtained from VECM equation for S1, S2 and S3 suggests that the negative

causality was at its peak in S2 sub-period compared to S1 sub-period and then it subsided

to moderate negative value in S3 sub-period. This suggests the bubble in gold market

would have migrated to the equity market during S2 period (31-May-05 to 5-Oct-06) as it

represented the peak negative causality relationship among conditional β series.

Empirical results suggest that during sub-period S4 the conditional β series were not

cointegrated. The relationship between conditional β series that was broken in S4 sub-

period did not recover in S5 & S6 sub-period thus we can term S4 sub-period as “burst

period” and conclude that post financial crises in equity markets (represented by S4, S5

and S6 sub-periods) volatility in gold prices did not affect the return sensitivity of equity

market significantly. One reason could be “flight to safety” phenomenon - post financial

crises of 2007-2008 investors have realized that gold is safe heaven and treated it as a

separate asset class from equity markets. We thus conclude that pre-crisis situation gold

accelerates the process of bubble migration; however post crisis the cycles are random

and noisy.

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identification of bubble in gold and study of its propagation to equity markets

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