Testing for Hidden Information on the German Market Index
Author: Tichelea Sonia
Coordinator: assistant prof. Anamaria Aldea
Abstract
In this paper, I investigate whether the German capital market is characterized by efficiency,
by running a series of econometric tests and applying models in the ARMA and ARCH
family in order to research patterns in autocorrelation of returns and in the behaviour of
volatility. The logic behind this is to test whether any model can fit the data properly, in order
to provide an accurate forecast for future returns and for future volatility. The existence of
such a model would be inconsistent with the efficiency of the market, because it would
enable investors who trade strategies based on the model, to obtain superior gains from the
market. The results highlight some kinds of anomalies, such as the day of the week effect
(higher returns are obtained on average on Mondays and Tuesdays). However, the
ARMA(1,3) and the Exponential GARCH models, that have been found most representative
for the DAX returns, are not powerful enough to provide accurate forecasts. Moreover, the
Random Walk Hypothesis confirms the weak form of efficiency.
Key Words: efficient markets, modelling returns, volatility, DAX.
Introduction
One of the current popular issues in the financial world is whether the capital markets are
informationally efficient, and also, finding the factors that influence returns and volatility on
the markets. This line of research traces back to the 1970, when Fama first introduced the
Efficient Market Hypothesis in its three possible forms: weak form, semi-strong form and
strong form. Results of multiple empirical studies confirm of infirm market efficiency, in its
various forms, and more recently, due to progress made in behavioural finance, links aspects
of the efficiency to the behaviour of traders in the markets and to their capacity of evaluating
correctly the intrinsic value of the titles.
The subject of market efficiency is of great interest because evidence of inefficiency in
capital markets can attract investors, who are confronted in this case with the possibility of
obtaining excess returns by spotting the assets that are overvalued or undervalued. This
would lead to an increase in liquidity and therefore will help improve the efficiency.
Considering the economic situation of the past five years period (2008 to 2013), it is clear
that markets have had quite a difficult period, with very big losses and record lows in 2008
and 2009, due to events such as the Lehman Brothers collapse, that have thrown the financial
world into a deep recession. Contagion to the European economies has lead to the burst of
problems in the PIIGS economies. Major liquidity injections, both in the US economy and in
the EU, have been used as an incentive for economic growth, in order to get the capital
markets out of the very steep downward trend. In Germany's case, the market has recovered
very well, judging by the continuous upward trend that DAX has been following since 2010.
Moreover, the index has recently hit record highs.
The purpose of this paper is to investigate the behaviour of DAX, the main market index for
the German economy, during this very troubled period, by analyzing serial correlation,
volatility and different forms of market anomalies, such as the day of the week effect or the
month of the year effect. Any relevant findings to these research would come to argue with
the Efficient Market Hypothesis and would be a sign of hidden information in the market.
A set of econometric models (AR, MA, ARMA, ARCH, GARCH, EGARCH) are used in
order to analyze the return series of DAX, and the purpose is to find the models that have the
highest power in modelling the behaviour of returns and of the volatility in the market.
Literature Review
The Efficient Market Hypothesis was developed in 1970 by Fama, and, according to the
original model, a market can be in one of the following forms of efficiency: the weak form of
efficiency assumes that future returns cannot be predicted by analyzing past returns. There
should not be any significant serial correlation in the return series. The semi-strong efficiency
form keeps this assumptions and adds that prices should change very rapidly under new
information. The strong form states that prices fully reflect all information, public and
private, so that excess returns are impossible to obtain.
Other aspects of efficiency ('internal market efficiency') are trading costs, market liquidity
and competition between the traders. According to Sharpe (1999), a market should have low
trading costs and a high number of traders, in order to be highly liquid and therefore showing
the main characteristics of efficiency: prices adapting quickly to new information and
overpricings or underpricings are rapidly traded, so that prices adjust to show intrinsic value
of underlying assets.
The concept of informational asymmetry was developed in the 1970's by George Akerlof,
Michael Spence and Joseph Stiglitz, each of them contributing to it by studying a certain
aspect of the theory: adverse selection (Akerlof, 1970), signaling (Spence, 1973) and
screening (Stiglitz, 1975). This theory focused on the uneven quantities of information held
by market participants and on the process of distributing information across markets, in order
to explain inefficiencies that appear in the markets due to this uneven distribution of
information that alters the decision making.
Behavioural finance is another line of research that has contributed to the modern financial
theory. Developed by Kahneman and Tversky (1979) and improved during the past two
decades, behavioural finance comes to argue with the Efficient Market Hypothesis's
underlying assumption that investors are rational in their decision making process. Inferences
with psychology are made, in order to point out behavioural biases that are widely
encountered in markets.
The Theory of Adaptive Markets (Lo,2004) is a new approach to the classical Efficient
Market Hypothesis and combines classical efficiency theory with behavioural aspects, and,
for the first time, aspects of non-rationality encountered in the markets, coexist with the basic
assumptions of the EMH.
Seasonal market anomalies have been tested by Cross(1973) and French(1980), with results
of abnormally low returns on Mondays (the Monday effect), and higher returns in January
(the January effect). These anomalies have been tested ever since, but the amount of evidence
to support them has been decreasing in the recent period (Sewell 2011), sign of
improvements in the efficiency of markets around the world.
Empirical Review
The efficiency of capital markets is the subject for a multitude of empirical studies in the past
decades. There are a number of findings regarding financial time series, that are consistent
throughout the literature, and that have been named 'stylized facts'. A study comprising a
collection of such stylized facts is made by Sewell (2011), who concludes that in general, the
autocorrelation in returns is largely insignificant, and becomes significant only at high
frequencies. Most financial series are leptokurtic, non-stationary, have a unit root and present
non-linearity in variance. Some calendar effects exist, but their occurrence has decreased
significantly. Also, there is a higher probability of marker volatility exhibiting long term
memory, than the returns themselves.
Doyle and Chen (1976) study 76 market indexes in order to prove the multidimensional
aspect of market inefficiency. A large number of statistic methods are being used in order to
perform this analysis, and another purpose of the research is finding the relevance of using
each set of indicators for testing efficiency. A large number of market anomalies are found
and markets show signs of short and long term memory.
The models from the ARCH family (conditional volatility models) are used by Alagidede and
Panagiotidis (2009) to investigate the behaviour of African stock markets. The Random Walk
model has been rejected for all the major African economies, and also, the effect of volatility
clustering has been found. Because of the inefficiency in the markets, the authors have
concluded that it is indeed possible for investors to obtain higher returns by trading in the
African markets.
A number of Romanian studies on the issue of Romanian capital market efficiency have been
published after 1995. Stanculescu and Mitrica (2012) test the weak form of the market
efficiency and find evidence against it, concluding that it's possible to obtain higher returns in
the Romanian capital. Todea (2012) reaches the same conclusion regarding the lack of
efficiency in the market, by using fractal analysis. The results are useful for attracting
potential investors to our home country capital market.
The problem of efficiency in the 2008-2012 period for the PIIGS economies has been studied
by Kumar (2012) by using the market index for each of the countries. Weak form efficiency
has been tested both for daily and weekly data, and results regarding efficiency are better for
the weekly frequency (the weak efficiency form is accepted for all the five countries). All the
indexes show signs of nonlinear dependence
Case Study
The data base consists of daily closing prices for DAX, for a period of five years, from the
beginning of 2008 until April 2013. The period of time chosen is very relevant, as it
comprises both a period of market crash (2008-2009), and the uptrend that followed it (20102013). However, having there two very different trends in the dataset may alter the results.
The daily closing price series has one unit root, as it becomes stationary at the first difference.
The ADF test is used to test for stationarity:
Null Hypothesis: D_DAX has a unit root
Exogenous: None
Lag Length: 0 (Automatic - based on SIC, maxlag=23)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
Prob.*
-36.70041
-2.566629
-1.941052
-1.616546
0.0000
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(D_DAX)
Method: Least Squares
Date: 06/01/13 Time: 13:21
Sample (adjusted): 1/04/2008 4/30/2013
Included observations: 1388 after adjustments
Variable
Coefficient
Std. Error
t-Statistic
Prob.
D_DAX(-1)
-0.985338
0.026848
-36.70041
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.492670
0.492670
94.58601
12408815
-8283.706
1.998671
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
0.058293
132.7949
11.93762
11.94139
11.93903
ADF test for the first difference
This result is consistent with the Random Walk Hypothesis, and so we accept the weak form
of efficiency for the German capital market.
From the time series consisting of daily closing prices for DAX, the simple return time series
was calculated using the following formula:
𝑃𝑡
𝑅𝑡 = 𝑃
𝑡−1
- 1, where 𝑃𝑡 is the DAX closing price at time 't'.
The return series was then shown to be stationary, by using the ADF test. The next step was
to look for autocorrelation, with the help of the Ljung-Box test.
RDAX
.12
.08
.04
.00
-.04
-.08
I
II
III
2008
IV
I
II
III
2009
IV
I
II
III
2010
IV
I
II
III
2011
IV
I
II
III
2012
IV
I
II
2013
Different AR, MA and ARMA models were tested for the series of return. Results show that
AR(3), MA(3) and ARMA( 1,3) are the best fit. We chose the best model by looking for the
lowest values for Akaike Informational Criteria, which if found for ARMA(1,3).
Volatility was then modeled with the help or ARCH, GARCH and EGARCH models. The
best one was the EGARCH, as it account for the asymmetries present in the market.
While testing for market anomalies, neither Monday effect or January effect were found in
the DAX return series. However, results show that on average, higher returns are obtained on
Mondays and Tuesdays, and they tend to drop towards the end of the week. Regarding the
month of the year effect, the results were not conclusive and the January effect does not
characterise the studied return series.
Conclusions and proposal
The conclusions of the case study were that, first, the Random Walk Hypothesis cannot be
rejected for the DAX return series. Second, several models were fit to the series, but their
relevance is capturing the behaviour of the returns is quite small. From the ARMA family,
the best model found was ARMA (1,3) and from the ARCH family, the best volatility model
found was the Exponential GARCH with one asymmetrical order. Also, some anomalies
were found, such as higher returns on average, on Mondays and Tuesdays. However, the
results are not conclusive enough to reject the German market efficiency. Even though some
models were successfully fitted on the data, their forecasting power is rather small.
The proposal for future research consists of splitting the time series in two smaller samples,
according to the bear and the bull trend from the five year period. This may lead to
significantly better results in the time series modelling. Also, another way of improving this
research is to actually test the performance of trading strategies based on the models fitted to
the data. If indeed, trading according to those strategies is more profitable than a buy-andhold strategy, it would be the case to reject market efficiency for Germany.
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