The Stock Price of the ThyssenKrupp AG - A Time Series Analysis Using the Box Jenkins Approach

Content

List of Figures

List of Tables

Introduction

1. The Box-Jenkins-Approach

2. Modeling A Stationary Time Series For The Stock Price Of The ThyssenKrupp AG Using The Box-Jenkins-Approach
2.1. Test for Stationarity
2.2. Postulate a General Class of Models
2.3. Identification of an Adequate Model
2.4. Estimation and Diagnostic Checking
2.5. Validation and Forecast

3. Conclusion

References

Appendix A

List of Figures

Figure 1: The Box-Jenkins Forecasting Approach¹

Figure 2: Line Plot of the Weekly Opening Stock Price of ThyssenKrupp

Figure 3: Line Plot of the First Difference of the Stock Price

Figure 4: Plot of the ACF for the First Difference of the Stock Price

Figure 5: Plot of the PACF for the First Difference of the Stock Price

Figure 6: Bartlett’s Periodogram-Based White Noise Test

Figure 7: Actual and Forecasted Weekly Opening Stock Price of ThyssenKrupp

List of Tables

Table 1: Augmented Dickey-Fuller Test on the Stock Price

Table 2: Augmented Dickey-Fuller Test on the First Difference of the Stock Price

Table 3: ACF and PACF for the First Difference of the Stock Price

Table 4: Trial-and-Error- Process to find the Best Model

Table 5: ARIMA Regression

Table 6: Correlogram of the Residuals

Table 7: Portmanteau White Noise Test

Table 8: Augmented Dickey-Fuller Test on Estimated Residuals

Table 9: Forecast of the Weekly Stock Price for the next 10 Weeks

Table 10: Validity Test for the Stock Price from 2003-01-06 - 2006-07-24

Introduction

In this paper the Box- Jenkins forecasting technique should be applied to the stock price of the ThyssenKrupp AG. ThyssenKrupp arose from the merger of the “Thyssen AG” and the “Friedrich Krupp AG Hoesch-Krupp” in 1999. The main focus of the trust lies on steel, industrial goods and services with its five sections Stainless, Steel, Technologies, Elevator and Services. With 191,350 employees in over 70 countries and a turnover of 51.7 billion Euro p.a., ThyssenKrupp is one of the largest industry and technology groups in the world. At the same time it is Germany’s biggest steel and armaments manufacturer. I chose the stock price of the ThyssenKrupp trust for several reasons. First, it is a blue chip listed on the stock exchange since 1999 allowing me easy access to a sufficient and reliable amount of data. Second, I have no reason to believe that this trust underlies any influence of seasonality since it has so many different segments that contribute to its economic performance. Third, since the steel demand and thus prices are steadily increasing in the last years it is not surprising that the stock price of the ThyssenKrupp AG does this too (see figure 2 further down) giving me a reason to question financial market theories. In particular, financial investors and researches are very often interested to predict future values of stock prices. On the one hand they do so, to gain profits from investing in stocks from buying at a low price and selling at a higher price and on the other hand to verify if financial markets work efficiently. For the latter reason, financial research for a long time believed stock prices to follow a random walk and thus that prices of the stock market cannot be predicted². This implies that financial markets are at least weak form efficient and excess returns cannot be earned by using investment strategies based on historical share prices, i.e. time- series analyses can not be used to predict future values of stock prices. In this paper I will try to find out if the stock price of the ThyssenKrupp AG follows a random walk or not using the Box-Jenkins forecasting approach. For my analysis I use the weekly opening stock price starting at 01/01/2003 and ending at 04/21/08 and thus I have 278 observations. My paper consists of three parts. In the first, I will shortly explain the Box- Jenkins forecasting approach, in the second part I will apply the approach to the weekly opening stock price of the ThyssenKrupp AG and finally I will present some concluding remarks. To analyze the data I use the software STATA (version 9.1.).

1. The Box-Jenkins-Approach

The Box- Jenkins approach (see Box and Jenkins, 1976) to forecasting time series³ was first developed as a theoretical framework and became more and more sophisticated by the practical experience in forecasting data. That is why, the approach became very popular among researchers who recognized that accurate forecasts can be obtained very easily through- time series methods. The approaches’ strength lies partly in its strong theoretical foundations, partly in its success in empirical comparisons as it allows forecasts as accurate as many very complex econometric methods and partly in its generality since it can almost handle all time- series data (see Makridakis and Wheelwright, 2004).

The forecasting approach generally consists of three different stages that can be seen in figure 1. Initially, the researcher has to check his time- series for stationarity⁴, since the approach can only be applied to stationary time series. Moreover, the researcher postulates a general class of models that could fit the data. In the second stage, the researcher picks one of the postulated models he thinks is likely to fit his data best, the so-called tentative model. Afterwards, in stage 2, the researcher estimates the parameters of his tentative model with his given historical data and determines whether the model fits his data adequately in a diagnostic check on the estimated residuals of the tentative model. The successive values of residuals should be randomly related to each other to have a model that is adequately identified. If the model is adequate, the validity of the model is checked,i.e. the actual values and the forecasted values of the estimated model are compared with each other to evaluate the ability of the model to predict the best future values possible. Then, we can forecast future values of the time series. However, if the model is not adequate we need to start over and identify another tentative model that might fit the data better.

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Figure 1: The Box-Jenkins Forecasting Approach Source: On the basis of Makridakis and Wheelwright, 2004.

2. Modeling A Stationary Time Series For The Stock Price Of The ThyssenKrupp AG Using The Box-Jenkins-Approach

In the following paragraphs I will go into the details of the Box-Jenkins approach to forecasting time series of which I have only presented the surface up to this part of the paper. Thereby, the Box-Jenkins approach is basing on autoregressive integrated moving average processes (ARIMA models), a general class of models used to describe and to explain the characteristics of time series.

2.1. Test for Stationarity

The first thing to do before we really can go on with the Box-Jenkins approach is to check our time series for stationarity. If we interpret a time series as realizations of one random variable out of an infinite universe of random variables we can call this random variable a stochastic process. A time series is stationary if the underlying stochastic process that generated that series is invariant with respect to time, i.e. mean, variance and covariance are constant over all data points of the time series. A stationary series allows us to model its underlying process via an equation with fixed coefficients that can be estimated from the given historical data. Another important issue is that the time series has no seasonality included. Otherwise, we have to deseasonalize the series beforehand and apply the Box-Jenkins approach to the data without seasonality. Having a closer look at the plot of the opening stock price of ThyssenKrupp from 01/01/2003 until 04/21/2008 in figure 2 suggests that there is no seasonality but a trend included in the time series. A trend yields the long term movement of the time series, i.e. the upward or downward tendency of it.

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Figure 2: Line Plot of the Weekly Opening Stock Price of ThyssenKrupp

Moreover, a trend signalizes that the series is not stationary. Another common method to check for stationarity is to use the Dickey-Fuller or the Augmented Dickey-Fuller test to test for a unit root. Basically, the Dickey-Fuller test approximates the time series by an AR

(1) - process while the null hypothesis is indicating a unit root between the variable and its

first lagged value. That means, we test

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t, i.e. for the existence of a random walk. Since the

approximation with an AR(1)- process is very inaccurate the Augmented Dickey-Fuller test tries to test for a unit root using an AR(p)- process. Thus, we test the same hypotheses as

above but for

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the Augmented Dickey-Fuller test is more precise I just present the results of this test here.

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Table 1: Augmented Dickey-Fuller Test on the Stock Price

In table 1 we can see that the weekly opening stock price of ThyssenKrupp is non- stationary because for all levels of significance the test statistic yields a greater value than the critical value and furthermore the p-value is also 0.9292. Hence, we can not reject the null hypothesis that the weekly opening stock price follows a random walk, i.e. is not non- stationary. However, a common method is taking successive differences [first difference, e.g. is Y t − Y t −1= Δ Y t ] of the data to remove the trend and thus make the series stationary.

We have to take so many differences until the series is stationary. Let the number of lags we need to make the time series stationary be x. Then an initially non-stationary time series is called integrated of order x [I(x)]. For example, a stationary time series is integrated of order zero [I(0)] or when the first differences of a non-stationary time series are stationary we call it integrated of order one [I(1)]. The number of lags we should take can be checked using the information criteria of Akaike or Schwartz-Bayes. These criteria yield that we should take the first or second difference. Performing an Augmented Dickey-Fuller test on the first difference like in table 2 yields that we can reject the null hypothesis for a unit root since the p-value is zero and thus using one lag makes the time series stationary. Hence, the weekly opening stock price of ThyssenKrupp is integrated of order one [I(1)].

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Table 2: Augmented Dickey-Fuller Test on the First Difference of the Stock Price

That the first difference is stationary is also suggested by a visual check of the line plot of the first difference of the ThyssenKrupp weekly opening stock price in figure 3.

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¹ Please note, that all figures where the “Time” is on the x-axis are coded by STATA. Hence, the date 1960w2 yields the 01/01/2003 and 1965w1 yields 12/17/2007.

² Sometimes this is also referred to the saying that “the best prediction of the stock price of today is the stocks’ price of yesterday”.

³ A time series is a sequence of observations of a certain attribute for a statistical element or the extent of a statistical bulk ordered over time.

⁴ For an explanation of stationarity see page 7 under point 3.1. Test for Stationarity.