3. MARKET EFFICIENCY
3.2. Random Walk Theory
According to Malkiel (1973), random walk hypothesis is a financial theory stating that stock markets evolve according to a random walk, implying that prices of the stock market cannot be predicted, prices have no memory and move randomly while adjusting to new information as it comes.
Formally, the random walk model can also be expressed as
(5) Pt Pt 1 at
Where Pt the asset price is observed at the beginning of timet and at is the error term with zero mean and is independent over time. The price change is simply equal to the error term as below.
(6) Pt Pt Pt 1 at
Price Pt can also be written as an accumulation of all purely random changes by successive backward substitutions (Mills 1999).
(7) Pt ti1ai
It has been described as “jibing” with the efficient market hypothesis. Economists have historically accepted the random walk hypothesis and have run several tests and continue to believe that stock prices are completely random because of the efficiency of the market. Tests of the theory are done by investigating whether any forecasting is possible. Persistence is found throughout nature, and it should be no great surprise that it appears to some degree in capital markets. New statistical models may help analyze such trends. The main point, however, is that the price activity of the market, assuming it is a complex adaptive system, would be similar to a classic random walk. The new models, however, would appear to do a better job of explaining persistence in returns to the extent that such persistence exists.
3.3. Volatility and the Risk-Return Relationship
Investors and portfolio managers have a certain degree of risk they can bear which has been termed as uncertainty in finance and is sometimes measured as standard deviation.
However, volatility is not defined the same way as risk but as standard deviation, ,of the continuously compounded rate of return in a given period or as variance ,2 estimated from a historical set of observations.
(8)
N
t
t r
N 1 r
2
2 ( )
1 1
where r is the mean return. The sample standard deviation, is the distribution free parameter representing the second moment characteristic of the sample.
Risk and volatility are key components in mean-variance portfolio theory and for the different asset pricing models like the famous CAPM. Increased volatility increases market risk and has several economic costs for example reduced trading activity and market arbitrage. Investors are also forced to postpone investment decisions in periods of uncertainty.
One element that relates between the level of asset prices and memory components of volatility is the risk-return trade off which has been studied using GARCH-type models.
Research work has shown that during crisis periods, stocks that co-vary with volatility are those that pay off and these stocks do require a smaller risk premium. Jointly, the risk-return trade off and serial correlation in volatility determine the level of stock prices, the stronger and higher the trade off and serial correlation are, the higher the elasticity of stock prices with respect to volatility. As a result, innovations from volatility die out very fast and affect absolute returns and stock prices in a short time.
(Porteba & Summers, 1986). Christensen & Nielsen (2007) examine the relation between risk-return trade off and serial dependence in volatility using the ARFIMA(p, d, q) model and their result are consistent with earlier results point to a strong significant risk-return trade off, long memory in volatility and a strong financial leverage effect.
According to the theory of efficient markets, the random walk model is too restrictive.
The expected rate of return from time t to t 1 of a portfolio with dividends reinvested is assumed to be the sum of the risk-free rate and a risk premium.
If the expected return is constant Et rt 1 =r then rt 1 is a fair game.
(9) Et rt 1 =
t t t t
t P
P D
E P 1 = rf,t xt
Where Pt, Dt, rf,t , xt are the stock price, dividend paid, nominal risk-free rate and the risk premium at period trespectively.
The link between equity premium and return volatility is expressed with the intertemporal CAPM model which implies a linear relationship between the equity premium and the market return variance.
(10) xt= Vt
Where Vt is the instantaneous variance of the market return and is the harmonic mean of individual investors’ Pratt-Arrow measures of relative risk aversion.
4. METHODOLOGY
4.1. Data Description
This study utilized daily observations from Thomson Financial database for the stock price indices of U.S, London, Japan and Finland. The stock indices used are FTSE ALL SHARE (United Kingdom), OMX HELSINKI (Finland), S&P 500 COMPOSITE (U.S.) and Tokyo Stock Exchange (Japan). For each of the four indices the data set starts from December 24, 2005 to February 17, 2009. Raw data from these indices is expressed in domestic currencies for example FTSE ALL SHARE closing prices in the Great Britain pound (GBP), S&P500 COMPOSITE in US dollars (USD), OMX HELSINKI in Euros (EUR) and Tokyo Stock Exchange in the Japanese Yen (JPY).
Table 1.Firms and European Stock Market Indices.
Stock Index No. Of Observations
S&P 500 COMPOSITE 1082
OMX HELSINKI (OMXH) 1082
FTSE ALL SHARE 1082
NIKKEI 225 STOCK AVERAGE 1082
Market indices and the number of observations from December 24, 2005 to February 17, 2009.
4.2. Market Description
4.2.1. FTSE All-Share Index
This market was originally called the FT actuaries All-Share index when it was founded in 1962. Two new sub-indices, FTSE 100 and FTSE 250 in the late ‘90’s were then added to this index. It represents the performance of all companies listed on the London
Stock Exchange’s major market. At present the market covers over 600 constituents with an approximate value of 98% of UK’s market capitalization.
The FTSE All-Share index accounts for 8.11% of the world’s equity market capitalization. In this market stocks are free-float weighted to ensure that only the investable opportunity set is included within the index.
Figure 2.Market series close prices.
4.2.2. S&P 500 Composite Index
The Standard & Poor’s composite index founded in 1957 is a value weighted index.
Stocks included in this index are those of large public companies that trade on either the New York Stock Exchange or NASDAQ. S&P 500 refers not only to the index but also to the 500 companies that have their common stock included in the index.
1600
It is widely known as the best single gauge of the U.S. equity markets and is also a proxy to the total market although it focuses on the large cap segment of the market.
4.2.3. Tokyo Stock Exchange
The Tokyo Stock Exchange index (TSE) is located in Japan and it is the second largest stock exchange market in the world by market value. By December 2007, this market comprised of 2,414 listed companies with a combined capitalization of 4.3 trillion US dollars. Stocks listed on this index are separated into sections depending on their sizes
4.2.4. OMX Helsinki
OMX Helsinki is a stock exchange located in Finland and is now a part of the NASDAQ OMX Group since February 2008. It operates in eight stock exchanges in the Nordic and Baltic countries.
4.3. Empirical Models
This study used E-views software package for the time series data, econometric multivariate GARCH and Component GARCH (CGARCH) models were used and compared to examine volatility shocks across the markets and to examine the memory components in these markets which in turn explains the degree of volatility persistence of these markets.
4.3.1. GARCH Model
The GARCH parameterization as introduced by Bollerslev (1986) gives parsimonious models that are easy to estimate and successful in predicting conditional variances. The simplest GARCH(1,1) specification used in the study is expressed as below.
(11) Yt Xt' t
(12) t2 t21 t21
Where the mean equation is written as a function of exogenous variables with an error term. t2is the conditional variance based on past information and is written as a function of three terms:
: Constant term
2 1
t : ARCH term measured as a lag of the squared residual from the mean equation
2 1
t : GARCH term (last period’s forecast variance)
This model is consistent with the volatility clustering in financial returns data, where large changes in returns are likely to be followed by further large changes. If the lagged variance is substituted on the right hand side of equation (12), then the conditional variance is expressed as a weighted average of all the lagged square residuals:
(13)
The GARCH(1,1) variance specification is similar to the sample variance but it down-weights more distant lagged squared errors. The error in the squared returns is given by
2 2
t t
t , substituting it in the variance equation and rearranging terms, the model in terms of errors as:
(14) t2 ( ) t21 t t 1
The autoregressive root which governs the persistence of volatility shocks is the sum of plus . If this root is close to unity, then shocks die out rather slowly. In addition,
the autoregressive root is important in determining the fourth and higher order moments of the unconditional distributions.
4.3.2. Component GARCH Model
Engle and Lee (1999) introduced a flexible alternative, the GARCH model variant, the Component-GARCH (CGARCH) which captures high persistence in volatilities; this model decomposes time-varying conditional volatility into a long-run component, and short run transitory component which reverts to trend following a shock.
The conditional variance in the GARCH(1,1) model below shows mean reversion to
(15) t2 ( t21 ) ( t21 )
which is a constant for all time. The component model allows mean reversion to a varying levelmt, modeled as:
(16) t2 mt ( t21 ) ( t21 )
(17) mt i i(mit 1 i) i( it21 it2 1)
2
t is the volatility, while qit replaces and is the time varying long run volatility and i (FTSE, S&P500, TSE, OMXH). Equation (16) describes the transitory component, t2 qt, which converges to zero with powers of ( ) where as equation (17) describes the long run component which converges to with powers of . typically between 0.99 and 1 so that qt approaches very slowly. Combining the tansitory and permanent equations gives the equation below;
(18) 2 (1 )(1 ) ( ) t21 ( ) ) t22 ( ) t21 ( ( ) ) t22
This shows that the component model is a non-linear restricted GARCH (2, 2) model.
4.4. Daily Return Series
The daily market return seriesRit is defined as the close to close prices on consecutive trading days. The market returns are calculated as the logarithmic difference in the in the daily stock indices.
(19) Rit 100(lnPit,close lnPit 1,close)
Where Rit is the return for the series and Pit is the closing price for the market series.
Figure 3.Market Returns.
4.5. Augmented Dickey-Fuller Test
To test for unit roots, the augmented Dickey-Fuller unit root test (1979) and Philips-Perron unit root test which allow for levels and trends are used; the optimal lag lengths are selected based on the Akaike information criterion (AIC).
(20)
Ho :Non stationary (Unit root)
-10
1 : 1
Ho : Stationary (Integrated of order zero)
at follows white noise distribution with mean of zero and constant variance i.e.
) , 0 (
~WN 2
at
4.6. VAR(p) Model
The VAR approach sidesteps the need for structural modelling by treating every endogenous variable in the system as a function of the lagged values of all of the endogenous variables in the system. Depending on the stationarity of the series, Vector autoregressive (VAR) modeling as written below with levels of differences will be applied to explain any possible linkages between the series.
(21) t i t
p
i t t
1
Where t=(all the market series) and p denotes the lag order of the system, and t' is a covariance stationary 4x1 vector of volatility time series, the 4x1 vector of intercepts, and t the 4x1 vector of white noise with zero mean and positive definite covariance matrix, and p denotes the lag order of the system.
The VAR(p) models has also proven to be a useful tool for analysis of term dynamics of several economic time series. The basic VAR model is just a multivariate generalization of the univariate autoregressive (AR) model.
(22) (L)yt et,
Where (L) I 1L 1L2 ... PLP is a matrix polynomial of order p. The vector yt is assumed to be centered for the sake of simplicity, et are random vectors of m time series k are m x m matrices (k=1,…….,p), and L is the lag operator. It is
assumed that each vector in et are (weak) white noise processes that, however, can be contemporaneously correlated. Formally,
E (et) = 0, for allt
(23) E (et et’) =
, ,
0 ,
t s
t s if if
Where the prime denotes transpose.
Determination of an appropriate lag order, p for the VAR system which depends on standard lag length criteria is an empirical issue. In this study, the order of the VAR is defined based on the standard length criteria and likelihood ratio tests. In addition, given that the residuals did not exhibit serial correlation additional tools for analyzing casual as well as feedback effects were used to examine cross dependencies amongst the market series.
4.7. Granger Causality and Impulse Response Analysis
Granger causality test, impulse response analysis and Variance decompositions were applied to interpret the estimated VAR (p) system. Granger causality tests identify potential lead-lag relationships between the estimated volatilities and the direction of the causalities.
Impulse response analysis using generalized standard deviation shocks in the volatility of the market series was performed to reveal the persistence of shocks in the system; it was also used to trace the impact of a shock in one market series to another market. The impulse response coefficients can be solved by inverting the VAR coefficient polynomial, giving
(24)
However, because the impulses are contemporaneously correlated, it is easier to interpret the results if each series own impulse effects are singled out. This can be accomplished by orthogonalizing the residuals. One of the most applied orthogonalization procedure is the so called Cholesky decomposition such that
(25) SS',
Where the S is a lower diagonal non singular m x m. Then the new residual shocks
k
Furthermore, because each component in the u-vectors has unit variances, the squared ijth component, ij2,k, of the k matrix indicates the fraction of the variance of yt,the innovation occurred the jth variable at lag k explains. For the sake of convenience these fractions are expressed in percentage terms.
4.7.1. Variance Decompositions
The uncorrelatedness of utSallows the error variance of theS step ahead forecast of yit to be decomposed into components accounted for by these shocks, or innovations (this is why this technique is usually called innovation accounting).
Because the innovations have unit variances besides the uncorrelatedness, the compone-nts of this error variance accounted for by innovations to yjis given by
(27)
Comparing this to the sum of innovation responses we get a relative measure how important variable jS innovations are in the explaining the variation in variable i at different step ahead forecasts, i.e.
(28) m
Thus, while impulse response functions trace the effects of a shock to one endogenous variable on to the other variables in the VAR, variance decompositions separate the variation in endogenous variable into the component shocks to the VAR.
Letting s increase to infinity one gets the portion of the total variance of yj that is due to the disturbance term jof yj .
4.8. Geweke’s Measures of Linear Dependence
Geweke’s measure of linear dependence is used to test for linear feedback from all markets for example if smaller (Japan and Finland) markets are dependant on larger (U.S. and United Kingdom) markets and vice versa.
Geweke suggested a measure of linear feedback between x and y based on the matrices
1 and 11 as shown below.
(29) Fx y ln( 1 / 11 ),
So that the statement “x does not granger cause y” is equivalent to F x y 0 Similarly the measure of linear feedback from y to x is defined by
(30) Fy x ln( 2 / 22 ),
The measure of linear dependence is defined byFx,y Fx y Fy x Fx.y.
With these estimates, particular dependencies within the markets can be tested as below.
01 Fx y 0
H : No Granger-causality between the markets (larger to smaller)
(31) (T p)Fx y ~ mkp2
02 Fy x 0
H : No Granger-causality between smaller markets to larger markets
(32) (T p)Fy x ~ mkp2
. 0
03 Fxy
H : No instantaneous feedback between the markets
(33) (T p)Fy x ~ mk2
. 0
04 Fyx
H : No linear dependence between the markets
(34) (T p)Fy x ~ mk2 (2p 1)
This is due to the asymptotic independence of the measuresFx y, Fy x andFx.y.
5. EMPIRICAL FINDINGS
Figure 2 indicates that the daily market prices are varying considerably overtime. All market series are non-stationary and are particularly volatile with an increasing pattern and large movements between 2005 to mid 2007. The considerable decline in volatility of the market prices from mid 2007 to date can be explained by the ongoing market crisis world wide. Besides this, Figure 1 indicates that market prices tend to move together very closely.
5.1. Descriptive Statistics
Table 2.Descriptive Statistics.
FTSE OMXH S&P500 TSE Mean -0.015536 -0.022953 -0.039148 -0.036650 Median 0.014326 0.018593 0.036229 0.000000 Maximum 8.810746 8.849971 10.95720 13.23458 Minimum -8.709914 -7.923905 -9.469514 -12.11103 Std. Dev. 1.336039 1.494813 1.469959 1.717081 Skewness -0.203172 -0.027485 -0.376405 -0.552627 Kurtosis 12.74399 8.524816 15.75390 13.78544 Jarque-Bera 4283.935 1374.969 7352.096 5294.525 Probability 0.000000 0.000000 0.000000 0.000000 Sum -16.79454 -24.81260 -42.31867 -39.61867 Sum Sq. Dev. 1927.801 2413.222 2333.642 3184.238
Observations 1081 1081 1081 1081
Table 2 reports the descriptive statistics for the daily market returns. The sample means for all the four markets are negative and significantly different from zero with standard deviation rather close to unity. Measures of skewness indicate that the series are all negatively skewed and highly leptokurtic (fat tails) implying that the distribution of
these series is non-symmetric which is also an indication of possible ARCH effect.
Koutmos (1996), argues that that non linear dependencies are due to the autoregressive conditional heteroskedasticity i.e. volatility clustering in market returns. The large magnitude of the Jarque-bera statistic enables us to reject normality distribution of all the series. Because of the ARCH effect still present in the squared residuals (see graphs in appendix), the series fail to be independently and identically distributed (i.i.d) over time.
Table 3. Contemporaneous Correlations.
FTSE OMXH S&P500 TSE
FTSE 1
OMXH 0.834648 1
S&P500 0.519722 0.453792 1
TSE 0.422207 0.419580 0.106712 1
Correlation structure is one of the important features for investors and portfolio managers since the strategies they employ require a measure of correlation (association). (Koutmos, 1996: 980). The considerable pair wise correlations as reported in Table 3 indicate market expectations of volatility are closely linked across markets and highest between FTSE and OMXH (0.834648).
To remove the ARCH effect before estimating the G(ARCH) models, a conservative strategy of adding AR and MA lags is used when running the autocorrelation and partial autocorrelation functions for each return series until the best specification is achieved when there is no remaining ARCH effects in the series.
After fitting the ARMA terms to all the market series, the resulting conditional mean models are reported in Table 4. Residual autocorrelations and related Q-statistics indicate no further autocorrelation left to the series as explained by the Durbin-watson test (approximately 2 for all series) indicating that these models give virtually a good fit.
However, the autocorrelations of the squared residuals suggest that there is still left non linear time dependency into the series which can only be removed with the G(ARCH) models.
Table 4.Choice of AR-MA lags.
FTSE OMXH S&P500 TSE
AR 7 2 2 2
MA - 2 1
-Squared residuals 1840.548 2374.933 2250.922 3154.203 Durbin-Watson stat 1.995591 1.982845 1.972141 2.009755 Schwarz criterion 3.428537 3.659174 3.591658 3.929996 Prob(F-statistic) 0.000000 0.001781 0.000000 0.006585
Together with the ARMA specifications, corresponding GARCH(1,1) model is estimated first for all the return series followed by the CGARCH(1,1) model and results compared.
Table 5 reports joint ARMA-GARCH(1,1) model estimates, resulting conditional mean models are ARMA(7,0) for FTSE, ARMA(2,2) for OMXH, ARMA(2,1) for S&P 500 and ARMA(2,0) for TSE. The drift term is significant for all series hence snubbing the random walk theory of market prices. Co-efficients describing conditional volatility are highly significant across all markets indicating that volatility is a function of past normalized residuals and the last period’s volatility. The magnitude of co-efficient which captures the impact an unexpected return has on volatility the next day ranges from 0.0742 for S&P 500 to 0.1334 for FTSE; this means that a return shock in FTSE causes almost twice as much volatility the next day as a return shock in S&P 500. The co-efficients capture tendency for shocks to persist, these are homogenous for all series and range from 0.8672 for FTSE to 0.9198 for S&P 500. Persistence of shocks measured by the sum of the estimated ARCH and GARCH parameters ( ) is 1 for FTSE and almost 1 for the remaining series, this indicates that the degree of volatility persistence in equity markets is high and greatest in FTSE hence shocks have a permanent effect on volatility and high persistence is also an indication that negative effects from increased market risk die out more slowly. The estimated Durbin-Watson
statistic is approximately two for all market series suggesting that there is no serial correlation amongst the residuals and the mean and variance are well specified.
statistic is approximately two for all market series suggesting that there is no serial correlation amongst the residuals and the mean and variance are well specified.