The impact of the Global Financial
Crisis on the correlation between
Dutch stock market and the wealth
level of Dutch households
Yixiao Deng
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics Author: Yixiao Deng Student nr: 11371897
Email: dengyixiao722@gmail.com Date: July 15, 2018
Supervisor: Dr. Lu Yang Second reader: Dr. Umut Can
This document is written by Yixiao Deng who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Master’s Thesis — Yixiao Deng iii
Abstract
Although the Global Financial Crisis has passed for many years, the impact and after-shocks of this Great Recession are still of great interest to be investigated. This paper focuses on the impact of the Global Financial Crisis on the correlation between stock market and wealth level of people. The wealth level is represented by the consumption expenditure of people, which is measured by the Consumer price index (CPI). Since the volatility is an efficient measure for the risk, the correlation between volatilities of two time series data was studied systematically and comprehensively.
In this study, the Netherlands is treated as the subject, where the Dutch stock market index ratio is the AEX stock index ratio. Before the model specification, we have found that the historical AEX stock index ratio is negatively correlated with the Dutch CPI on average. Without loss of generality, we transformed both data by taking logarithms and differences for a more positive analysis. Both data became stationary after trans-formation.
GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model is suitable to model the time series data with time varying variances. The volatilities of stock market index and CPI price level are volatile in the past years, so the GARCH model is selected to analyze the data. First of all, the univariate GARCH(1,1) models were applied to both time series data respectively. We found that the GARCH models fit reasonably well for both time series data. While the correlation between the marginal residuals of both fitted models is negative, and since the volatility of financial market is usually dynamic, the Dynamic Conditional Correlation (DCC)-GARCH (1,1) model is employed to study the co-movements of the volatilities of both time series data. The conditional correlation of the residuals of the fitted DCC-GARCH (1,1) model shows that the Global Financial Crisis in 2008 indeed made the negative correlation even larger. The afterward effect, the European sovereign debt crisis, worsened the situation in the start of 2009. However, the Dutch government and DNB (Dutch Central Bank) have already taken actions to help the financial system back to the right track and avoid the breakdown of the financial system again.
Preface v
1 Introduction and Literature Review 1
2 Methodology 5
3 Descriptive Statistics 11
4 Empirical results 19
5 Conclusion and Discussion 25
Appendix 27
Bibliography 34
Preface
This thesis was conceived to investigate the impact of the Global Financial Crisis on the correlation between the stock market and wealth level of people. When I began work on the research topic, I have encountered several challenging problems. However, with the help of my supervisor and my friends, I have finished my Master’s thesis finally. Many people have contributed either directly or indirectly to this study. First and foremost, my deepest gratitude goes to my supervisor Dr. Lu Yang, for her constant encouragement and guidance. She has helped me through all the stages of the writing of this thesis. Without her consistent and illuminating instruction, this thesis could not reach its present form. Secondly, I would like to thank my friends and classmates for providing valuable comments on the earlier version of this thesis, and finishing all the courses of this Master’s degree in this year. I also owe my sincere gratitude to Jimmy who assisted in the English grammar check for me. Last my thanks would go to my beloved family for their loving considerations and great confidence in me all through these years.
Chapter 1
Introduction and Literature
Review
The stock market and wealth level of household are related
It is acknowledged by more and more people that the performance of the stock mar-ket is correlated with the wealth level of individuals. On the one hand, it is shown by Cho (2006) that the stock market has a statistically significant impact on the highest income bracket of households in Korea. Income bracket is defined as the level of all pos-sible income components, which is taken by governments to distinguish demographic data. The income bracket is also used to determine different level of taxation and ben-efits.Poterba (2000) found that the level of consumption of household is influenced by the change in the stock market values, known as the wealth effect. The wealth effect is that when the value of the investment portfolio of stocks increases dramatically, the investors often have more confidence in the security of their wealth, giving rise to the increase in the consumption. On the other hand, it was found byGarcia and Liu(1999) that one of the most important determinant of stock market development is the real income.Santos and Veronesi(2006) also made an investigation about the labor income and stock returns, and they found that the regression model of stock returns on lagged value of labor income to consumption ratio produced statistically significant coefficients, meaning that stock returns are highly impacted by the real income. Last but not least, Choudhry(2003) have shown that there exists causality from stock market volatility to the US level of consumption. When people belong to the high income bracket, they will tend to find an optimal investment strategy to accumulate more wealth and avoid risk. Stocks might be one of the investment tools they will consider, thus the stock market and the wealth level of individuals are closely correlated.The consumption level reflects the wealth level of household
The wealth level of household should be measured properly. Previous literature showed some interesting findings. Dvornak and Kohler (2007) conducted a research to show the changes in the wealth level of Australian household influenced the consumption expenditure. They found that the fluctuations in the housing wealth had a crucial im-pact on the level of consumption of Australian households. Thus, it is expected that the wealth level of households is closely correlated to the level of expenditure. Dynan and Maki (2001) also showed similar results in their research. They suggested that the direct wealth effects quickly appear and raise the consumption growth for a number of quarters. This implies that the consumption level of people would reflect the wealth level directly. Obviously, wealth level is the premise and basis of consumption. When theincome increases to a certain level, people will have more money and more willingness to expend. Therefore, in the following chapters of this paper, the consumption level representing the wealth level of households would be analyzed, which is reflected by the Consumer price index (CPI), a well-known indicator of the purchasing power. As CPI is published regularly, it is appropriate to find the change in the consumption level of people in terms of CPI.
The Global Financial Crisis impacted the stock market
world-wide
The economic environment has an extensive impact on the stock market. One well-known economic disaster is the 1997 Asian financial crisis, whichLim et al.(2008) have made an intensive study about. They are concerned about the effect of the inefficiency of stock market during the chaotic financial crisis in 1997 in 8 Asian stock markets. Another typical case is the Global Financial Crisis occurred in 2008, which had a long lasting and severe impact worldwide, and affected a majority of countries over the world, including European countries. In 2009, the DJIA (Dow Jones Industrial Average) fell more than 50% from its peak in 2008. While at the same time, the FTSE 100 (Financial Times and Stock Exchange), the Tokyo’s Nikkei 225 stock average and the HSI (Hang Seng Index) fell more than 50% over the same period. European financial systems are highly interacted with the US financial system, which is claimed to be the origin of this Great Recession. Obviously, this Global Financial Crisis caused even worse consequence than the 1997 Asian financial crisis. The worse performance of the financial market led to even worse situation of the society. Big companies have gone bankrupt, and thus the unemployment rate increased dramatically. The government has to rescue the financial markets, turning the financial system back into the right track. In addition to this, it can be assumed that the Global Financial Crisis in 2008 should have a larger and more serious impact on the stock markets over the world. The impact of this global financial crisis has been investigated by many researchers afterwards, and many pre-vious literature have shown significant and valuable results. A research conducted by Syllignakis and Kouretas(2011) proved that there is a significant increase in the condi-tional correlation between US market and stock returns of Central and Eastern Europe (CEE) especially the German market. In this paper, the Dynamic Conditional Correla-tion (DCC) multivariate GARCH model of Engle and Sheppard (2001) was employed, which lends the inspirations to this study. Frankel and Saravelos(2012) used the stock market performance to measure the crisis incidence. In this paper, they also defined the period of the Global Financial Crisis as running from late 2008 to early 2009. Both papers have suggested that the stock index ratio should be significantly affected by the financial crisis.
Further researches also showed significant results. That how strong is the relationship between the Global Financial Crisis and stock market is of great interest. Samarakoon (2011) found that in different regions, there exists bi-directional, interdependence and contagion in emerging markets. The shocks from US stock markets brought out more interdependence than other emerging markets, while contagion comes more from emerg-ing market shocks. On the other hand,Grammatikos and Vermeulen(2012) have shown that the crisis has transmitted from US non-financials to European non-financials. This suggested that the US markets and European markets are closely related. They also found that Economic and Monetary Union (EMU) financials react stronger to Greek CDS spreads after Lehman’s collapse. This could be the after-shocks of the financial crisis. Another significant finding comes from Dimitriou et al. (2013), they found that
Master’s Thesis — Yixiao Deng 3
the contagion appeared in almost all stock markets after Lehman’s collapse, especially from early 2009 onwards. This offers the evidence that there exists tail dependence in extreme cases, namely, larger dependence often occurred in the bullish markets rather than in bearish markets. It can be concluded that the stock market is heavily affected by the financial crisis, perhaps the afterward effect is even larger. As we know, The volatilities of stock index ratio can be assumed to be volatile, fluctuating before the crisis, during the crisis and after the crisis. The fluctuation becomes larger after the crisis.
The Global Financial Crisis impacted consumption level of
household
The performance of economic environment also strongly influences the consumption level of people. Looking back into the history, during the period of Great Depression, the society is unrest and the people suffered from poverty. The quality of living hit the historical lowest level. Whereas the Global Financial Crisis originated from the US and spread to the rest of the world quickly also has significant adverse effect on the quality of living of normal people. In terms of Gerstberger and Yaneva (2013), they studied the effect of financial crisis on the consumption expenditure of EU-27 countries, and they found that Baltic countries and Greece suffered most from this Great Recession, especially the household consumption of Greece decreased dramatically and the worst situation occurred in 2011. Even worse situation was found byBrinkman et al. (2010), which shows that the Global Financial Crisis resulted in the top of high food and fuel prices. Due to this effect, a large amount of households were forced to choose lower quality and quantity foods, which increased the risk of malnutrition, especially the pop-ulations with young children, pregnant and lactating women, and the chronically ill people. As we know, nutrition deficiency during the childhood will have life-long con-sequences, so the short-term price rises will have long-term effects. Therefore, previous findings indeed showed the severity of the impact of financial crisis on the consumption level of people.
The Global Financial Crisis affected the correlation
Based on the previous studies, we found several valuable outcomes. Firstly, we know that the stock market and the wealth level of household are closely related, and the wealth level of household can be represented by the consumption level of household. The consumption level of household can be measured by the CPI, one of the well-known economic indicator. Secondly, the Global Financial Crisis in 2008 has affected both the stock market and the consumption level of people dramatically and seriously. It is true that the financial system can help people to accumulate wealth and optimize risk management by providing useful investment portfolio strategies. However, this financial disaster has spread from the US to the European countries and developing countries faster than people expected. As a consequence, the financial system was damaged at first, and the breakdown has caused severe economic and social chaos. The economic en-vironment was influenced dramatically, and the unemployment rate rose to the peak in the past decades. It worsened the lives of many households, their wealth level fell down, even reached to the poverty line. The wealth of many individuals and families shrunk, their consumption expenditure decreased as well. In addition to the marginal effects, we investigate whether the dependence structure of the stock market and the consumption
expenditure is impacted by the Global Financial Crisis as well. More precisely, since the volatility of historical data would be very useful in the real world, like forecasting. The correlation of volatilities between the stock market index and consumption level is of interest to be analyzed, since if the correlation coefficient changes, the optimal investment portfolio of people should be changed consistently. Therefore, investigating the correlation of volatilities between the fluctuation in stock index ratio and the level of consumption of households during the Global Financial Crisis period is the main topic in this paper. Inspired by Lim et al. (2008), sub-periods of pre-crisis, crisis, and post-crisis are taken into consideration.
The Research topic
We study the Netherlands as the subject. The Dutch index ratio AEX is an effective and significant indicator to present the performance of the Dutch stock market. It is highly correlated with other stock markets worldwide, so the investigation of the impact of Global Financial Crisis should be significant. Meanwhile, as mentioned earlier, the consumption level reflects the wealth level properly, and the Netherlands is a developed European country whose income level of the individuals is relatively high, and the con-sumption level should be valid and significant. The Dutch monthly CPI has been chosen. The CPI is a famous and effective economic index for the level of expenditure. Both data can be found easily from the database, and these data are authoritative statistics, which should be reliable. Both data are monthly data and started from 1998 to 2018, since the impact of the Global Financial Crisis can be found precisely if the time series data are long enough, and the post effect might also be found, which is the well-known European debt crisis since the end of 2009.
With respect to these reasons, we will use the Dutch stock index ratio AEX and Dutch CPI price level, investigating the impact of the Global Financial Crisis on correlation of volatilities between stock market and wealth level of individuals. Obviously, the Global Financial Crisis has a long-lasting effect, affecting affecting both the current and the future situations. It is also true that the Global Financial Crisis would affect both the stock market and the consumption level of households respectively. Moreover, as we know there exists correlation between stock market and level of consumption of individ-uals, the effect of the Global Financial Crisis on the correlation is investigated in this paper.
The rest of the paper consists of four chapters. Chapter 2 explains about the Method-ology and Chapter 3 shows descriptive statistics. The empirical results are elaborated in Chapter 4. Chapter 5 would be the conclusion.
Chapter 2
Methodology
Literature review and financial background
Previous literature have shown that GARCH models are appropriate for analyzing fi-nancial data. Many researchers used GARCH models with macroeconomic determinants to model the volatilities of stock returns, e.g.,Flannery and Protopapadakis(2002). As mentioned before,Choudhry(2003) have applied the GARCH model to show that there exists long terms relationship between the consumption expenditure and the stock mar-ket volatility. WhileGrammatikos and Vermeulen (2012) also used individual GARCH models to estimate EMU financials and non-financials index.
Several classical GARCH models have been used to conduct the research on the dif-ferent stock markets. Alberg et al. (2008) have performed several GARCH models to check the mean return and conditional variance of Tel Aviv Stock Exchange (TASE) index, and they found that the asymmetric GARCH model with heavy-tailed densities improved the estimation, among these GARCH models, especially the EGARCH (ex-ponential generalized autoregressive conditional heteroskedastic) model using a skewed Student-t distribution worked most successfully. Another paper conducted by Chong et al. (1999) studied the performance of GARCH models and its modifications on the daily stock market indices of the Kuala Lumpur Stock Exchange (KLSE). They use maximum likelihood method to estimate the parameters of these models and variance processes. Several goodness-of-fit statistics were used to diagnose the performance of the within-sample estimation. They found that according to the goodness-of-fit statistics, exponential GARCH model was not the best model, but it performed best in describ-ing the often-observed skewness in stock market indices and in out-of-sample forecastdescrib-ing. In the modern finance, the volatilities of stock market and the correlation between financial system and economic environment are the crucial problems that a lot of re-searches focused on. It is true that volatilities, measured by the variances, can be used to depict the risks. The volatility of the time series data is very useful, which can be used to predict the future risk and calculate risk measures such as VaR (Value at Risk) and ES (Expected Shortfall). In the real world, the variances are always time-varying and uncertain. Hence we need to find a suitable technique to define the variance.
ARCH model and univariate GARCH model
Initially, the fundamental of GARCH models is the autoregressive conditional het-eroskedasticity (ARCH) model, imposed by Engle (1982). The idea is to model the time series data by assuming the variance of the current error term as a function of the actual sizes of the previous time periods’ error term. In the ARCH model, it is often
sumed that the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance follows an autoregressive (AR) model. If the error variance follows an autoregressive moving average model (ARMA), the model becomes a generalized autoregressive conditional heteroskedasticity (GARCH) model, which is proposed by Bollerslev (1986). GARCH model is a dynamic nonlinear model with basic concept that the variance of error at time t is assumed to dependent on the square of the past error at time t − 1, t − 2, etc.
It is apparent that in the real world, the conditional volatility on past periods of the stock index ratio or the consumption level of individual cannot be a constant over the time, and the processes should be conditionally heteroskedastic. Normally the processes is not stationary, we often transformed to log-returns when we use the data to analyze without lost of generality. The real world financial data usually has following stylized features: the time series data are mutually dependent, present value and past value are always highly correlated; the volatility is fluctuating over the time; and the distribution is not normally distributed, but it is often heavily-tailed. The time series data is also inclined to have excess peakedness at mean value. Generally, the change in the volatili-ties appears to be clustering: larger errors exist in a small period, while smaller errors exist in another period.
According to Posedel (2005), define Zt ∼ N (0, 1) and (Zt) be the i.i.d.( independent
and identically distributed) random variables, a process (Xt) is called GARCH (p, q)
process if
Xt= σtZt (2.1)
and conditional variance σt is such that
σt2= α0+ α1Xt−12 + ... + αqXt−q2 + β1σt−12 + ... + βpσt−p2 (2.2)
where t ∈ Z.
In addition, the GARCH (p, q) process should satisfy the conditions: α0 > 0, αi ≥ 0,
i = 1, ..., q, and βi ≥ 0, i = 1, ..., p, which ensure strongly positivity of the conditional
variance (2.2). Another restriction for stationarity is Pq
i=1αi+Ppi=1βi ≤ 1, standing
for the decaying coefficient, measuring the continuity of the impact from shocks on the future volatilities.
Multivariate GARCH model
Nevertheless, many researchers realized that the univariate GARCH model is not appli-cable enough in the real world. There often exists volatility spillovers among correlated factors, that is, the volatilities would have the tendency to move together, hence the univariate GARCH model is insufficient for this case. Another reason is that in the real application, the co-movements of different data could be captured by the variance covariance matrix, which we need to use as the explanatory variables, however, in the univariate GARCH model, we cannot achieve this goal. Thus, extending the univariate GARCH model to the multivariate GARCH model is necessary. As we intend to inves-tigate the relationship between the volatility of Dutch stock market index ratio and the fluctuation in the expenditure level of Dutch household, which can be computed based on Dutch CPI price level, the multivariate GARCH model is considered.
The GARCH model also provides optimal investment strategies for investors in the financial markets since the covariance matrix of the ratio can be estimated. The cor-relation between stock market and consumption level of household is very important,
Master’s Thesis — Yixiao Deng 7
as if the correlation coefficient changes, the investment strategies should be changed consistently. Hence we need to find the change in the correlation of volatilities to obtain the optimal asset strategies. As univariate GARCH (1,1) model is enough to capture the volatility pooling, the multivariate GARCH (1,1) model in 2-dimensional has been applied.
The unrestricted multivariate GARCH (1,1) model is defined as below. The vector of returns Xt is given by:
Xt= "X1t X2t # ="µX1t µX2t # +" t νt # (2.3) where ut= " t νt #
= Ht−1/2zt∈ Rd, t = 1, 2, ..., n, where d stands for dimension, is an i.i.d
vector with mean E[zt] = 0 and variance E[ztzt0] = It, and Htstands for the conditional
variance and covariance matrix.
Similar to the univariate GARCH model, the nonnegative conditions α0 > 0, αi ≥ 0,
i = 1, ..., q, and βi≥ 0, i = 1, ..., p are satisfied, the conditional variance and covariance
matrix for the multivariate GARCH model should be satisfied as well. To ensure the non-negative condition satisfied, let vech represents the vector-half operator, stacking the elements of the lower triangular part of the symmetric matrix Htto form a d(d+1)2 ×1
vector, where d = 2 is the dimension. Then the symmetric conditional variance Ht is
given by: vech(Ht) = h11,t h21,t h22,t = Vart−1(X1t) Covt−1(X1tX2t) Vart−1(X2t) = cX1t cX1tX2t cX2t + a1 a2 a3 a4 a5 a6 a7 a8 a9 2t−1 t−1νt−1 νt−12 + b1 b2 b3 b4 b5 b6 b7 b8 b9 Vart−2(X1,t−1) Covt−2(X1,t−1X2,t−2) Vart−2(X2,t−1) .
In order to estimate this model, one restriction is that the covariance matrix Ht =
E " t
νt
#
t νt should be positive definite. Followed from Engle and Kroner(1995),
as-sume that the matrix Ai =
a1 a2 a3 a4 a5 a6 a7 a8 a9 and Bi = b1 b2 b3 b4 b5 b6 b7 b8 b9
are positive definite, namely, a VEC model. This assumption ensures the elements in the equation of vech(Ht)
are all positive definite, then Ht itself is also positive definite by definition.
The estimation of the parameters of multivariate GARCH model can be done by using conditional maximum likelihood estimation (cMLE), the likelihood function is given by:
Lt(θ) = − n 2log(2π) − 1 2log(Ht(θ)) 1 2ut(θ) 0 Ht−1(θ)ut(θ) (2.4)
where θ is the vector of parameters and n is the number of equations.
If we set the condition on the initial values of z0 and H0, the likelihood function can be
written as L(θ) = T X t=1 Lt(θ) (2.5)
for the sample t = 1, ..., T .
Without loss of generality, we take the logarithm to get the log-likelihood function ` = ln(L(θ)) which is easier to deal with. In order to find the optimal solution, the most common numeric optimized method quasi-Newton optimizer Broyden-Fletcher-GoldfarbShanno (BFGS) can be applied, proposed by Orskaug(2009).
Dynamic Conditional Correlation (DCC)-GARCH model
The VECH model vech(Ht) has 21 parameters, which is difficult to apply efficiently inthe financial market. In order to investigate the correlation between stock market index and CPI price level more comprehensively and intensively, the multivariate GARCH model, called Dynamic Conditional Correlation (DCC)-GARCH model, is considered. The reason to choose DCC-GARCH model is that the estimation of optimal parameters based likelihood functions can be achieved efficiently, which will be shown later in this section. Another reason is that the variances of time series data are dynamic varying, DCC-GARCH model is one of the most-used multivariate GARCH model to solve the problem that the volatility varies over time. The model is introduced by Engle and Sheppard (2001) and we will follow the specification ofOrskaug(2009).
To be more precise, the conditional covariance matrix Ht will be modified, we will
modeled the conditional variances and correlations instead. The conditional covariance matrix Ht can be redefined as
Ht= DtRtDt (2.6)
Hence, the DCC-GARCH (1,1) model is the multivariate model with the specification of Ht. Xt= "X1t X2t # ="µX1t µX2t # +" t νt # ut= " t νt # = Ht−1/2zt Ht= DtRtDt
Here are the notations:
Xt: 2 × 1 vector of returns.
µt=
"µX1t
µX2t
#
: 2 × 1 vector of the mean of the returns Xt.
ut: 2 × 1 vector of mean-corrected returns, same as unrestricted multivariate GARCH model.
Ht: 2 × 2 conditional variances matrix.
zt: 2 × 1 vector of i.i.d. errors, same as unrestricted multivariate GARCH model.
Dt: 2 × 2 conditional standard deviations matrix.
Master’s Thesis — Yixiao Deng 9
Dtis the n×n dimension conditional standard deviations Dt=
√ h1t 0 . . . 0 0 √h2t . .. ... .. . . .. . .. 0 0 . . . 0 √hnt with hit = αi0+ Qi X q=1 αiqa2i,t−q+ Pi X p=1 βiphi,t−p. (2.7)
Similar to the univariate GARCH model, here every hitcan be defined as any univariate
GARCH process.
And the correlation matrix Rt=
1 ρ12,t ρ13,t . . . ρ1n,t ρ12,t 1 ρ23,t . . . ρ2n,t ρ13,t ρ23,t 1 . .. ... .. . ... . .. . .. ρn−1,n,t ρ1n,t ρ2n,t . . . ρn−1,n,t 1 .
Since the covariance matrix Ht is by definition positive definite, Rt should be
posi-tive definite. Moreover, all the elements in the correlation matrix must be equal or less than one. To meet the requirements, the correlation matrix Rtcan be decomposed into:
Rt= Q∗−1t QtQ∗−1t (2.8) Qt= (1 − a − b) ¯Q + at−1Tt−1+ bQt−1 (2.9) where εt∼ N (0, Rt), ¯ Q = Cov[εtεTt] = E[εtεTt] = 1 T T X t=1 εtεTt
is the unconditional covariance matrix of the errors εtand Q∗t =
√ q11t 0 . . . 0 0 √q22t . .. ... .. . . .. . .. 0 0 . . . 0 √qnnt .
Here the rescaling of Qt to Q∗t ensures that all the elements in the correlation matrix
are equal or smaller than 1. If Qt is positive definite, Rt is then also positive definite
from equation (2.8).
Similar to the univariate GARCH model, to ensure the positive unconditional vari-ances, the conditions a ≥ 0, b ≥ 0 and a + b < 1 must satisfy. Additional condition is that the initial value of Qt, Q0, should be positive definite.
The parameters optimization of DCC-GARCH (1,1) model is simplified by its defi-nition. The estimation can be achieved by two steps. We know that the εt∼ N (0, Rt),
and the log-likelihood function is given by: L = −1 2 T X t=1 (n log 2π + log Ht+ u0tH −1 t ut) = −1 2 T X t=1 (n log 2π + log DtRtDt+ u0tD −1 t R −1 t D −1 t ut) = −1 2 T X t=1
(n log 2π + 2 log Dt+ log Rt+ ε0tR−1t εt)
= −1 2 T X t=1 (n log 2π + 2 log Dt+ u0tD −1 t D −1 t ut− ε0tεt+ log Rt+ ε0tR −1 t εt).
Let θ stands for the parameters in D and φ represents the the parameters in R, then we have: L(θ, π) = Lv(θ) + Lc(θ, φ) where Lv(θ) = − 1 2 T X t=1 (n log 2π + 2 log Dt+ u0tD −2 t ut) and Lc(θ, φ) = 1 2 T X t=1 (ε0tR−1t εt− ε0tεt)
We can find that the likelihood function Lv(θ) can be regarded as the summation of the
log-likelihood function of single GARCH process. Rewriting it as:
Lv(θ) = − 1 2 T X t=1 n X i=1 (log 2π + log hit+ r2it hit )
Hence the maximization of Lv(θ) can be achieved by the maximization of every single
part, obtaining the estimated ˆθ = max Lv(θ). In the second step, given that ˆθ is known,
the optimal ˆφ is given by ˆφ = maxφLc(ˆθ, φ).
Under sufficient conditions, the estimation under first step should be consistent with the estimation under second step. This optimization method which reduces the number of parameters needs to be estimated efficiently, simplifying the calculation process. Hence the estimation for the higher-dimensions become possible.
In order to investigate the correlation of the volatilities between Dutch stock market and the level of consumption of individuals, the specific DCC-GARCH (1,1) model will be applied to the selected data.
Chapter 3
Descriptive Statistics
AEX Index Ratio
The effect of the Global Financial Crisis on the correlation of the volatilities between Dutch stock market and consumption level of Dutch people is the main research object in this paper. Hence the past period of 2008 and 2009 must be considered. In addition, the period before the Global Financial Crisis has been considered as well, as the his-torical data might show a general trend. Last but not least, the time horizon after the Global Financial Crisis is extremely important. As we know, the power of aftershocks was very strong. The long term effect of the Global Financial Crisis is globally strong. The European sovereign debt crisis, which started from the end of 2009, exerting signif-icant negative effect on the economic and labor market, is a multi-year effect originating from the Global Financial Crisis. Therefore, the time horizon from 1998 to 2018 of the selected data is used in this paper.
The AEX index is a stock market index ratio composed of Dutch companies that trade on Euronext Amsterdam (the former name is Amsterdam Stock Exchange) which orig-inated from Amsterdam Exchange index. This ratio started to be generated in 1983, and is composed of a maximum of 25 of the most frequently traded securities on the exchange. It is one of the main national indices of the stock exchange group Euronext in the European financial system. Other main national indices include Brussels’ BEL20, Paris’s CAC 40 and Lisbon’s PSI-20.
The monthly Dutch stock market index ratio AEX from 1998 to 2018 was used to analyze the stock market performance. The adjusted close prices were chosen to be an-alyzed. The summary statistics of AEX adjusted close price are shown in the Table 3.1. The mean of the historical data is about 429.47, whereas in the beginning of 2009, the close price hit the historical bottom point of 216.98, which can be assumed as the effect of the Global Financial Crisis. Notice that the variance of the adjusted close price is very volatile, as the variance is very important for risk management, the variance will be analyzed specifically.
Descriptive Statistics Value Year min 216.98 Feb, 2009 25% quantile 337.44 median 425.93 May, 2008 mean 429.47 75% quantile 509.04 max 689.52 Jul, 2000 std.dev. 106.50 variance 11341.63
Table 3.1: Descriptive statistics of AEX index ratio
Figure 3.1 shows a more intuitive impression. We can see that it is quite fluctuated over time. Consistently, the ratio reached to the bottom in the early 2009 in the figure, which is the later period of the Global Financial Crisis, as we expected.
Figure 3.1: The AEX Index Ratio adjusted close price
The Consumer Price Index (CPI)
The Consumer Price Index (CPI) is a widely used economic indicator which measures the change in prices of market basket of consumer goods and services. It is an important measure for inflation. Meanwhile, change in the CPI reflects effectively the price changes for cost of living. It can be considered as a deflater for other economic factors to find the purchase power of people. In this paper, since we want to investigate the change in
Master’s Thesis — Yixiao Deng 13
consumption level of individuals, the price level of CPI is used. For a single item, the CPI is calculated as:
CP I = Market Basket in Given Year
Market Basket in Base Year × 100 (3.1)
While for multiple items, we have different groups of consumer goods and services, n represents the total number of groups, the CPI is calculated as:
CP I = Pn i=1CP Ii× weighti Pn i=1weighti (3.2)
The monthly Dutch CPI price level is used to reflect the level of expenditure of Dutch household over the past years. The summary statistics are shown in the Table 3.2. Gen-erally, the CPI price level was gradually increasing, not affected by the Global Financial Crisis greatly. The minimum is found in the first month of the sample, whereas the maximum is found in the last month of the sample. The variation of the CPI price level is not very significant, meaning that the consumption level of Dutch household is rela-tively stable over the past years. The Figure 3.2 depicts the trend of the historical Dutch CPI over the year 1998 to 2018 intuitively. It can be seen that the CPI is indeed stably fluctuating upward over the past years. Thus the consumption expenditure of Dutch households should be not influenced by the financial diasters that much intensively.
Descriptive Statistics Value Year min 77.20 Jan, 1998 25% quantile 89.22 median 96.32 Feb, 2008 mean 96.48 75% quantile 106.83 max 111.88 Mar, 2018 std.dev. 10.20 variance 103.96
Figure 3.2: The Dutch CPI price level
The inspection of the correlation
As mentioned in the Chapter 2 Methodology, in order to apply the GARCH models on the time series data, we first need to check whether there exists a correlation between two time series data. We transformed both time series data to the log-returns and took differences respectively. The relation of the transformed data is shown in the Figure 3.3. It can be seen from this figure that the correlation between the transformed data is roughly between -0.1 and 0.1.
Master’s Thesis — Yixiao Deng 15
Figure 3.3: The historical relation between AEX adjusted close price and CPI price level
More precisely, the calculated correlation of the two original time series data is given as -0.3449449, and the correlation of the transformed data is -0.095, meaning that there exists a relatively strong and negative linear association between the two time series data.
Check stationarity
One strict condition for GARCH model is that the time series data must be stationary. We transformed the AEX index ratio for a better analysis by taking the logarithm and differences, and plot it against time horizon. Figure 3.4 below shows that the transformed data is reasonably stationary. It can be seen that the transformed AEX exists apparent volatility clustering. Furthermore, the ACF and PACF also suggest that the AEX index ratio is quite stationary, which is shown in the Figure 3.5.
Master’s Thesis — Yixiao Deng 17
Similarly, since the historical data of CPI had upward trend, we take the logarithm and differences on the CPI price level data as well. From the Figure 3.6 below we can find that the data is more volatile compared with AEX index ratio, but it is fluctuated around zero, and from ACF and PACF of the CPI, the data is stationary.
Figure 3.6: The transformed CPI
Another well-known method to check whether the data is identically independent dis-tributed is Ljung-Box test. The null hypothesis is stated as there exists no serial cor-relation, the data are independently distributed. We apply the Ljung-Box test on both time series data, the p-values is larger than 5%, which suggest that the null hypothesis of stationary white noise should not be rejected, indicating that both time series data are stationary.
Chapter 4
Empirical results
The fit of univariate GARCH (1,1) model on AEX index
ratio
First of all, univariate-GARCH(1,1) model has been applied to the difference of the logarithm of Dutch stock index ratio (AEX) to test the fluctuation of volatility of the AEX index ratio over time. This could give an overview of the performance of the Dutch stock market in the past years. The optimal estimated parameters are shown in the Table 4.1.
Parameter Estimate Std. Error t value P-values µ 0.001480 0.000146 10.1628 0.000000 ar1 0.975117 0.009726 100.2603 0.000000 ma1 -0.998644 0.000394 -2534.7279 0.000000 ω 0.000316 0.000166 1.9045 0.056840 α1 0.269420 0.100191 2.6891 0.007165 β1 0.637048 0.108376 5.8781 0.000000 shape 7.769080 3.583433 2.1681 0.030154
Table 4.1: Optimal Parameters of GARCH (1,1) on AEX index ratio The GARCH (1,1) model with optimal parameters can be written as:
Xt= σtZt (4.1)
σ2t = 0.000316 + 0.269420Xt−12 + 0.637048σt−12 (4.2) Almost all the parameters are statistically significant (except ω, the intercept, but the p-value is very close to 5%) from zero based on the p-values (smaller than 5%). The fact that α + β ≤ 1 indeed shows that the AEX index price is stable and the model is pre-dictable. Noticed that α1= 0.269 is less than β1 = 0.637. It is the result of the fact that
the external shocks are less than the internal market shocks. Moreover, the summation is close to 1, meaning that at a certain moment, the impact from a certain shock on the AEX index ratio has a long-lasting effect. In the GARCH model, it is assumed that the error variance follows an ARMA process, and the optimal estimated parameters suggest that the ARMA (1,1) process fits significant good, the ARMA-GARCH (1,1) model works good for Dutch stock market. The Weighted Ljung-Box Test also suggests that there exists no serial correlation. Thus, the Dutch stock market complies with the GARCH (1,1) model. The mean equation only consists of constant explanatory variates.
Looking at both the residuals and the squared of the residuals of the model (Figure 4.1 and Figure 4.2), the GARCH (1,1) model worked reasonably effective for the AEX index ratio. Based on the historical volatility, the future volatility of the AEX ratio can be forecasted in terms of the estimated coefficients. The estimated volatility of one month later is 0.01815567, so combined with the residuals from the model, we can compute the Value-at-Risk (VaR) of the AEX index ratio. The 95% VaR is about 0.0258366, meaning that there is the probability of 95% that the stock ratio will decrease by 0.0258366 at most.
Master’s Thesis — Yixiao Deng 21
Figure 4.2: Residual squared of fitted GARCH (1,1) model
Consistent with the calibration of the volatility of AEX, GARCH (1,1) model works for CPI as well. The equation with optimal parameters is given by
Xt= σtZt (4.3)
The fit of DCC-GARCH model on both AEX index ratio
and CPI
The marginal residuals and residuals squared of both univariate GARCH (1,1) models have supported this result. In addition, the correlation between the residuals of both model is approximately -0.05, showing a negative correlation between the residuals of two fitted models. As we know, the correlation could be dynamic conditional correlation, the DCC-GARCH (1,1) model is performed to study the co-movements of the correlation of AEX and CPI. The optimal parameters of DCC-GARCH (1,1) model is shown in the Table 2.
Paramter Estimate Std. Error t value P-values [AEX d].µ 0.005486 0.002327 2.35793 0.018377 [AEX d].ar1 0.001151 0.513924 0.00224 0.998213 [AEX d].ma1 -0.056388 0.500058 -0.11276 0.910218 [AEX d].ω 0.000288 0.000129 2.24342 0.024870 [AEX d].α1 0.318979 0.109542 2.91193 0.003592 [AEX d].β1 0.616304 0.085249 7.22942 0.000000 [CPI d].µ 0.001479 0.000372 3.97377 0.000071 [CPI d].ar1 0.063788 0.078160 0.81612 0.414431 [CPI d].ma1 0.244521 0.050636 4.82896 0.000001 [CPI d].ω 0.000000 0.000000 0.18067 0.856630 [CPI d].α1 0.000087 0.000255 0.34085 0.733216 [CPI d].β1 0.998813 0.000277 3604.29633 0.000000 [Joint]dcca1 0.019700 0.014495 1.35910 0.174115 [Joint]dccb1 0.969054 0.011317 85.62851 0.000000 Table 4.2: DCC-GARCH(1,1) We get µ =0.005486 0.001479 , Ω =0.000288 0 , α =0.318979 0.000087 and β =0.616304 0.998813 .
Based on the estimated coefficients, the [AEX d].α1 and [AEX d].β1 are used to check
the assumption of GARCH(1,1) joint significance. The p-values of the [AEX d].α1 and
[AEX d].β1 are smaller than 5%, meaning that the assumed GARCH (1,1) joint
signif-icance of α1 and β1 makes sense. Nonetheless, although the [CPI d].α1 is insignificant
and [CPI d].β1 is significant according to the p-values, the summation is still less than
1, meaning that the time series data is stationary and the GARCH (1,1) assumption still makes sense. The decaying coefficient is very close to 1, representing the result that the decaying speed of the volatility is very slow, the impact from the shocks will be extremely long-lasting. The [Joint]dcca1 and [Joint]dccb1 are used to evaluate DCC as-sumption. The p-value of [Joint]dcca1 is insignificant, but the coefficient of [Joint]dccb1 is significant. The DCC-GARCH (1,1) model fits the data not as well as we expected, but based on the performance and the basic knowledge of the DCC-GARCH model to capture the correlation effects, the fitted DCC-GARCH (1,1) model is accepted. In order to find the correlation of volatilities of AEX and CPI over the past periods, the correlation of the residuals of the fitted DCC-GARCH (1,1) model is presented in the Figure 4.3. The correlation matrix Rtis transformed to array format, and based on Engle and Sheppard (2001) and Orskaug(2009), we follow the method to assume that the mean ¯R is approximately equal to the unconditional covariance matrix, namely,
¯
R ≈ ¯Q. This is due to the fact that this method provides better bias properties for the correlation matrix. It can be seen that the correlation of volatilities over the past 20 years are heavily fluctuated between -0.25 (bottom) and 0.15 (peak). From January
Master’s Thesis — Yixiao Deng 23
1998, the correlation appears to be a dramatic downward fluctuated trend. It hit the lowest point in January 2005 and started to increase afterwards. The raise of the trend seems volatile. In the period of the Global Financial Crisis (from 2008 to 2009), the increase was affected, the correlation fluctuated downward in this smaller period. In the January 2010, the correlation hit the lowest point in the latest years. Before this point, there is another minimal point around 2009, which is perhaps due to the fact that the CPI is lag data, the influence of financial crisis would be reflected later.
Figure 4.3: The correlation of the volatilities between AEX index ratio and CPI
Another possibility is the European sovereign debt crisis. This debt crisis is extremely worse at the beginning of 2010 compared to the Global Financial Crisis. This might be the reason that the correlation of volatilities is even lower during the period of the Global Financial Crisis. From this figure, we do find the influence of the Global Financial Crisis on the correlation of volatilities between Dutch stock market ratio and level of expenditure of people. In people’s common sense, when the price of commodity increases, the stock price will increase and vice versa. Investing the money in the stock market might be a wise strategy to avoid inflation risk. If the CPI is stable over the recent years, the economic environment should be healthy, the unemployment rate should be low and the GDP should gradually increase. However, based on the research ofGraham (1996), it is not the case. Graham suggested that the negative correlation between real stock returns and inflation is spurious. The relation is influenced by the real-world events such as monetary policy or counter-cyclical policy. He believed that this relation should be considered as unstable.Onundu et al.(2016) found that the bad news always have stronger effect on the stock market volatility than the good news of the same magnitude. Furthermore, they found that the inflation rate and change in inflation
rate have significantly adverse impact on stock market volatility. In this study, the interpretation should be the Global Financial Crisis which has influenced the daily life of people, the amount of money they consumed is affected as well. On the other hand, the Global Financial Crisis have exerted the stock market undoubtedly. Meanwhile, the correlation between the Dutch stock market volatility and CPI was negative in the past periods, the Global Financial Crisis has aggravated this negative correlation. However, under the control of the Dutch government and Dutch Central Bank (DNB), the correlation of volatility changed from negative to positive steadily in the recent years.
Chapter 5
Conclusion and Discussion
The main contribution of this paper is to apply multivariate DCC-GARCH (1,1) model on the AEX index ratio and CPI price level to investigate the correlation of the volatil-ities between the Dutch stock market and the level of consumption of Dutch people. Specifically, we want to find whether the Global Financial Crisis started from 2008 in US had a dramatic impact on this correlation. The monthly data from 1998 to 2018 is employed. Before investigating the correlation, the data itself is checked. The AEX index price hit the lowest point in the early 2009, during the later period of the Global Financial Crisis, while the Dutch CPI appears to be a steady increase trend. The his-torical data are indeed on average negative correlated. Without loss of generality, we transformed both original data to the difference of logarithms. The transformed data are stationary, which are the prior constraint to apply GARCH models.
Initially, the univariate GARCH (1,1) model fit the AEX index price reasonably well, the optimal parameters are shown in the Table 4.1. We also find the fact that in the Dutch stock market, the external shocks are less than the internal market shocks. Then combined with transformed CPI, we established the DCC-GARCH (1,1) model to ex-amine the correlation of volatilities between AEX index price and CPI price level. The residual of the fitted DCC-GARCH (1,1) model is presented in Figure 4.3. This figure has shown that the correlation between volatilities are quite fluctuated over the past years. In the 2005, the trend hit the lowest historical point and started to increase af-terwards. The Global Financial Crisis has affected this increasing trend, the trend has suddenly dropped in the period of 2008 and 2009. Later in the January 2010, the trend dropped even deeper compared to previous decrease. This might be the result of the effect of time lag, as CPI is lag data.
In conclusion, based on the DCC-GARCH (1,1) model, the correlation between volatil-ities of Dutch stock markets and level of expenditures of Dutch people is inspected. The correlation on average is negative, and the Global Financial Crisis aggravated this negative correlation. After the Global Financial Crisis, the afterward effect was even stronger, showing the long-lasting effect of this financial crisis. However, under the con-trol of Dutch government, this negative correlation was not even worse and started to become positive in the recent years.
These findings help to build an overview of the relationship between the financial system and the daily life of normal people. It is true that if the disaster generated from the financial market extends to the daily life of people, and no efficient actions were taken, this would cause a huge adverse effect on the daily life of normal people. Conversely, the decrease in the consumption level of people would influence the financial markets as well. If the macroeconomic fundamentals is not stable and healthy, the financial system cannot be stable by consequence. Hence, it is suggested that the government
should take more control of the financial environments, supervising the behaviors of some too-big-to-fail companies, to avoid such incidents from happening again. Indeed, the market discipline is not enough to control such a huge risk. Therefore, after the Global Financial Crisis, DNB has used macro-prudential policies as a complementary tool. The macro-prudential policies are more efficient to ensure the financial stability and establish stable economic growth. The executive tools include counter-cyclical buffer requirements, macro stress tests, systemic risk buffer for large banks and etc.
Honestly, this paper has several limitations. In this paper, only Dutch stock market and Dutch CPI are considered. The result is not very systematic.Cai et al.(2009) has shown that the correlations between different international stock markets are time-varying and are closely related to cyclical fluctuations of inflation rate and stock volatility. Hence, including other European stock markets might give a more systematic and overall image of how the Global Financial Crisis occurred in the US affected the European financial system as a whole. Moreover, the DCC-GARCH (1,1) model was chosen in relation to the knowledge of previous literature to examine the correlation between Dutch stock markets and consumption level of Dutch model. Nevertheless, in the GARCH family, other classic GARCH models such as BEKK-GARCH and E-GARCH might also be used to investigate the correlation. Comparing the results of different models would give a more sufficient conclusion.
Appendix
> ### Master thesis > library(TSA) > library(tseries) > library(rmgarch) > library(parallel) > library(quantmod) > > ## data> AEX <- read.table("^AEX.csv", sep=",", header=T) > summary(AEX) Date Adj.Close 1/31/1998: 1 Min. :217.0 1/31/1999: 1 1st Qu.:337.4 1/31/2000: 1 Median :425.9 1/31/2001: 1 Mean :429.5 1/31/2002: 1 3rd Qu.:509.0 1/31/2003: 1 Max. :689.5 (Other) :237 > quantile(AEX$Adj.Close) 0% 25% 50% 75% 100% 216.98 337.44 425.93 509.04 689.52 > var(AEX$Adj.Close) [1] 11341.63 > sd(AEX$Adj.Close) [1] 106.4971
> plot(AEX$Adj.Close,xaxt=’n’,ylab = expression(AEX), type = "l") > axis(1, at=1:243, labels=AEX$Date)
> AEX_d<- diff(log(AEX$Adj.Close))
> plot(AEX_d, ylab = expression(AEX), xaxt=’n’,type = "l") > axis(1, at=1:243, labels=AEX$Date)
> Box.test(AEX_d,lag=20,type = "Ljung-Box") Box-Ljung test
data: AEX_d
X-squared = 17.731, df = 20, p-value = 0.6051 >
> CPI <- read.table("NLDCPIALLMINMEI.csv", sep=",", header=T) > plot(CPI$CPI,xaxt=’n’,ylab = expression(CPI), type = "l") > axis(1, at=1:243, labels=CPI$DATE)
> plot(log(CPI$CPI),xaxt=’n’,ylab=’Log(CPI)’,xlab="",type = "l") > CPI_d<- diff(log(CPI$CPI))
> plot(CPI_d, xaxt=’n’,ylab = expression(CPI), type = "l") 27
> axis(1, at=1:243, labels=CPI$DATE) > par(mfrow = c(1, 2)) > acf(CPI_d, main = "") > pacf(CPI_d, main = "") > plot(AEX_d,CPI_d) > cor(AEX_d,CPI_d) [1] -0.09504427 > cor(AEX$Adj.Close,CPI$CPI) [1] -0.3449449 > ## uni-garch > par(mfrow = c(1, 2)) > acf(AEX_d, main = "") > pacf(AEX_d, main = "")
> m1 = garch(x = AEX_d, order = c(1, 1), trace = FALSE) > m1
Call:
garch(x = AEX_d, order = c(1, 1), trace = FALSE) Coefficient(s):
a0 a1 b1
0.0002914 0.2949862 0.6321735
> acf(residuals(m1), na.action = na.omit) > pacf(residuals(m1), na.action = na.omit) > acf(residuals(m1)^2, na.action = na.omit) > pacf(residuals(m1)^2, na.action = na.omit) > > var <- m1$coef[1]+m1$coef[2]*AEX_d[242]^2;var^0.5 a0 0.01815567 > VAR <- 0+var^0.5*qnorm(0.95);VAR a0 0.02986342 > res <- quantile(residuals(m1),0.95,na.rm=TRUE,type = 1) > VAR1 <- 0+var^0.5*res;VAR1 a0 0.0258366 >
> m2 = garch(x = CPI$CPI, order = c(1, 1), trace = FALSE) > m2
Call:
garch(x = CPI$CPI, order = c(1, 1), trace = FALSE) Coefficient(s):
a0 a1 b1
9.356e+01 1.037e+00 8.576e-13
> acf(residuals(m2), na.action = na.omit) > pacf(residuals(m2), na.action = na.omit) > acf(residuals(m2)^2, na.action = na.omit)
Master’s Thesis — Yixiao Deng 29
> pacf(residuals(m2)^2, na.action = na.omit)
> df<-data.frame(residuals(m1)[2:242],residuals(m2)[2:242]) > plot(df[,1:2]) > cor(df,method="pearson") residuals.m1..2.242. residuals.m2..2.242. residuals.m1..2.242. 1.0000000 -0.0526359 residuals.m2..2.242. -0.0526359 1.0000000 > ## mgarch > > # 1. Fit DCC
> # First GARCH Specs.. GARCH(1,1)
> garch11.spec = ugarchspec(mean.model = list(armaOrder = c(1,1)), + variance.model = list(garchOrder = c(1,1),
+ model = "sGARCH"), distribution.model = "std")
> # dcc specification - GARCH(1,1) for conditional correlations
> dcc.garch11.spec = dccspec(uspec = multispec( replicate(2, garch11.spec) ), dccOrder = c(1,1), distribution = "mvnorm")
> > # SD
> garch.fit = ugarchfit(garch11.spec,
data = AEX_d, fit.control=list(scale=TRUE)) > print(garch.fit)
*---*
* GARCH Model Fit *
*---* Conditional Variance Dynamics
---GARCH Model : s---GARCH(1,1)
Mean Model : ARFIMA(1,0,1) Distribution : std
Optimal Parameters
---Estimate Std. Error t value Pr(>|t|) mu 0.001480 0.000146 10.1628 0.000000 ar1 0.975117 0.009726 100.2603 0.000000 ma1 -0.998644 0.000394 -2534.7279 0.000000 omega 0.000316 0.000166 1.9045 0.056840 alpha1 0.269420 0.100191 2.6891 0.007165 beta1 0.637048 0.108376 5.8781 0.000000 shape 7.769080 3.583433 2.1681 0.030154 Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|) mu 0.001480 0.000091 16.1880 0.000000 ar1 0.975117 0.007375 132.2270 0.000000 ma1 -0.998644 0.000328 -3048.2736 0.000000 omega 0.000316 0.000128 2.4799 0.013141 alpha1 0.269420 0.083576 3.2236 0.001266
beta1 0.637048 0.079509 8.0123 0.000000 shape 7.769080 2.753515 2.8215 0.004780 LogLikelihood : 383.7498 Information Criteria ---Akaike -3.1136 Bayes -3.0127 Shibata -3.1152 Hannan-Quinn -3.0730
Weighted Ljung-Box Test on Standardized Residuals ---statistic p-value Lag[1] 0.4280 0.5130 Lag[2*(p+q)+(p+q)-1][5] 0.8226 1.0000 Lag[4*(p+q)+(p+q)-1][9] 2.0941 0.9802 d.o.f=2 H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals ---statistic p-value Lag[1] 0.001727 0.9668 Lag[2*(p+q)+(p+q)-1][5] 1.476234 0.7457 Lag[4*(p+q)+(p+q)-1][9] 3.335718 0.7025 d.o.f=2
Weighted ARCH LM Tests
---Statistic Shape Scale P-Value ARCH Lag[3] 0.1383 0.500 2.000 0.7099 ARCH Lag[5] 1.3607 1.440 1.667 0.6297 ARCH Lag[7] 1.5095 2.315 1.543 0.8195 Nyblom stability test
---Joint Statistic: 0.7362 Individual Statistics: mu 0.04567 ar1 0.03142 ma1 0.03356 omega 0.31643 alpha1 0.19128 beta1 0.25267 shape 0.13344
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.69 1.9 2.35 Individual Statistic: 0.35 0.47 0.75
Master’s Thesis — Yixiao Deng 31
Sign Bias Test
---t-value prob sig Sign Bias 0.6392 0.5233 Negative Sign Bias 0.2601 0.7950 Positive Sign Bias 1.5126 0.1317 Joint Effect 2.5589 0.4647
Adjusted Pearson Goodness-of-Fit Test:
---group statistic p-value(g-1)
1 20 34.53 0.01591
2 30 40.64 0.07394
3 40 58.50 0.02315
4 50 61.72 0.10492
> garch.fit1 = ugarchfit(garch11.spec, data = CPI$CPI, fit.control=list(scale=TRUE)) > print(garch.fit1)
*---*
* GARCH Model Fit *
*---* Conditional Variance Dynamics
---GARCH Model : s---GARCH(1,1)
Mean Model : ARFIMA(1,0,1) Distribution : std
Optimal Parameters
---Estimate Std. Error t value Pr(>|t|) mu 77.122384 0.416182 185.309297 0.000000 ar1 1.000000 0.001757 569.130850 0.000000 ma1 0.321424 0.053598 5.996934 0.000000 omega 0.188230 0.054622 3.446022 0.000569 alpha1 0.085310 0.044096 1.934630 0.053036 beta1 0.000006 0.373838 0.000016 0.999987 shape 99.902872 141.060758 0.708226 0.478805 Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|) mu 77.122384 0.047930 1.6091e+03 0.000000 ar1 1.000000 0.001205 8.2962e+02 0.000000 ma1 0.321424 0.041443 7.7559e+00 0.000000 omega 0.188230 0.051055 3.6868e+00 0.000227 alpha1 0.085310 0.030927 2.7584e+00 0.005808 beta1 0.000006 0.272809 2.2000e-05 0.999982 shape 99.902872 16.669722 5.9931e+00 0.000000 LogLikelihood : -151.2681
Information Criteria ---Akaike 1.3026 Bayes 1.4032 Shibata 1.3010 Hannan-Quinn 1.3431
Weighted Ljung-Box Test on Standardized Residuals ---statistic p-value Lag[1] 0.2519 0.6157 Lag[2*(p+q)+(p+q)-1][5] 34.9574 0.0000 Lag[4*(p+q)+(p+q)-1][9] 63.2862 0.0000 d.o.f=2 H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals ---statistic p-value Lag[1] 0.09825 0.753945 Lag[2*(p+q)+(p+q)-1][5] 6.03806 0.088110 Lag[4*(p+q)+(p+q)-1][9] 13.12342 0.009994 d.o.f=2
Weighted ARCH LM Tests
---Statistic Shape Scale P-Value ARCH Lag[3] 0.5810 0.500 2.000 0.44594 ARCH Lag[5] 0.5969 1.440 1.667 0.85467 ARCH Lag[7] 8.4194 2.315 1.543 0.04229 Nyblom stability test
---Joint Statistic: 10.6415 Individual Statistics: mu 0.001414 ar1 1.478956 ma1 0.504082 omega 0.688058 alpha1 0.425232 beta1 0.918432 shape 0.815951
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.69 1.9 2.35 Individual Statistic: 0.35 0.47 0.75 Sign Bias Test
---t-value prob sig Sign Bias 1.0794 0.2815
Master’s Thesis — Yixiao Deng 33
Negative Sign Bias 1.0828 0.2800 Positive Sign Bias 0.6961 0.4870 Joint Effect 1.6725 0.6431
Adjusted Pearson Goodness-of-Fit Test:
---group statistic p-value(g-1)
1 20 39.22 0.004134
2 30 56.38 0.001703
3 40 62.35 0.010175
4 50 67.49 0.040947
> Dat<-data.frame(AEX_d,CPI_d)
> xspec = ugarchspec(mean.model = list(armaOrder = c(1, 1)), variance.model = list(garchOrder = c(1,1), model = ’sGARCH’),
distribution.model = ’norm’)
> uspec = multispec(replicate(2, xspec))
> spec1 = dccspec(uspec = uspec, dccOrder = c(1, 1), distribution = ’mvnorm’) >
> cl = makePSOCKcluster(2)
> multf = multifit(uspec, Dat, cluster = cl) >
> fit1 = dccfit(spec1, data = Dat, fit.control = list(eval.se = TRUE), fit = multf, cluster = cl)
> print(fit1) *---* * DCC GARCH Fit * *---* Distribution : mvnorm Model : DCC(1,1) No. Parameters : 15
[VAR GARCH DCC UncQ] : [0+12+2+1]
No. Series : 2 No. Obs. : 242 Log-Likelihood : 1344.312 Av.Log-Likelihood : 5.56 Optimal Parameters
---Estimate Std. Error t value Pr(>|t|) [AEX_d].mu 0.005486 0.002327 2.35793 0.018377 [AEX_d].ar1 0.001151 0.513924 0.00224 0.998213 [AEX_d].ma1 -0.056388 0.500058 -0.11276 0.910218 [AEX_d].omega 0.000288 0.000129 2.24342 0.024870 [AEX_d].alpha1 0.318979 0.109542 2.91193 0.003592 [AEX_d].beta1 0.616304 0.085249 7.22942 0.000000 [CPI_d].mu 0.001479 0.000372 3.97377 0.000071 [CPI_d].ar1 0.063788 0.078160 0.81612 0.414431 [CPI_d].ma1 0.244521 0.050636 4.82896 0.000001
[CPI_d].omega 0.000000 0.000000 0.18067 0.856630 [CPI_d].alpha1 0.000087 0.000255 0.34085 0.733216 [CPI_d].beta1 0.998813 0.000277 3604.29633 0.000000 [Joint]dcca1 0.019700 0.014495 1.35910 0.174115 [Joint]dccb1 0.969054 0.011317 85.62851 0.000000 Information Criteria ---Akaike -10.986 Bayes -10.770 Shibata -10.993 Hannan-Quinn -10.899 Elapsed time : 7.206691 > > stopCluster(cl) > > ts.plot(rcor(fit1)[1,2,]) > > rho.est.line <- list()
> rho.est.line = rcor(fit1, type="R") # plot(rho.est.line[1,2,])
> outputcorr <- matrix((rho.est.line[1,2,]), nrow = dim(Dat)[1], byrow = TRUE) > rho.est = data.frame((outputcorr))
> rho.est.zoo = zoo(rho.est, order.by=index(Dat[,1])) > dates <- seq(as.Date("01/01/1998", format = "%m/%d/%Y"), + by = "months", length = 242)
> plot(dates,rho.est.zoo,xaxt=’n’,ylab = expression(correlation), type = "l") > axis.Date(side = 1, dates, format = "%m/%d/%Y")
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