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**Method**

*In this chapter we present the research methodology and the data chosen. Further, we present the different methods and steps, which we use to answer our research questions.*

**Research Methodology**

There are different types of research designs, such as descriptive, explanatory and exploratory (Blumberg, Cooper, & Schindler, 2008). The most appropriate research designs for the purpose of this thesis are to use a descriptive and an explanatory study. This will be done by performing statistical test in order to define whether there exist any relationship between the markets or not.

The perspective of the research can be either deductive, inductive or a combination of both (Blumberg, Cooper, & Schindler, 2008). This study is deductive as it is based upon previous research and leads the authors towards the result and the conclusion.

Quantitative and qualitative studies are two different research approaches to use when collecting and analyzing data. The main difference between these two studies is the kind of information that is used. A quantitative study is based on quantitative information, such as numbers that are statistically tested, while a qualitative study is based on qualitative information, for example, interviews (Blumberg, Cooper, & Schindler, 2008). The best suitable approach for this thesis is a quantitative study since the authors intend to use already existing data to perform correlation and cointegration tests.

**Data**

The data that will be used is secondary data collected from Thomson Reuters Datastream and Statistics Sweden. We used OMX Stockholm as a market proxy for the stock market and we chose two indices, sold multi-dwelling and commercial buildings and sold manufacturers industries, for the commercial real estate market. Since the commercial real estate market includes renting office buildings, retail premises, warehouse storages, and industry buildings it is preferable to study both the indices. It could also be useful to study both indices to see if they distinguish from each other. When deciding which commercial real estate index to use in this thesis we chose between the indices of a direct investment and another index that consists of the largest listed commercial real estate companies. As the latter does not reflect the true commercial real estate values and is more correlated with the stock market we chose to use the indices that reflects the direct investments of commercial real estates.

The data that will be studied are quarterly and will be studied between the years 1994Q1 until 2013Q4. We believe that this time horizon investigated is sufficient as it covers the 1990s when the Swedish economy started to recover from the recession and the most recent crisis during 2007 and 2008.

**OMX Stockholm**

OMX Stockholm is a market index with all the listed companies on the Stockholm Stock Exchange and currently has 282 traded stocks (Nasdaq OMX Nordic, 2014). A common index to use as a market proxy is OMX Stockholm 30 but as it only covers the 30 most traded stocks it does not give a fair picture of the whole Swedish stock market. Therefore OMX Stockholm is an appropriate index to use in this thesis since this index includes all the stocks traded.

**Sold Multi-dwelling and Commercial Buildings**

Sold multi-dwelling and commercial buildings is one of the commercial real estate indices used in this thesis. The index is constructed by Statistics Sweden and it includes the average price for sold apartment buildings, offices and stores for each quarter. This index covers all purchases in the whole country and it is not weighted towards any specific geographical areas. Also, the prices are in nominal terms. This index does not include current earnings, such as rental prices, rather it reflects the real value of the appreciation of these commercial real estates. The index corresponds to a direct investment in commercial real estates and it is not exchange traded. Further in the thesis, this index will be referred to as commercial buildings.

**Sold Manufacturers Industries**

Sold manufacturers industries is the other commercial real estate index that we have used. This index is constructed in the same way as the sold multi-dwelling and commercial buildings index but instead it includes industry buildings. We will refer to this index as industry buildings.

**Discussion of Data**

The commercial buildings and the industry buildings indices are not weighted towards any specific geographical areas, which mean that there are large gaps in the prices between the cities and the countryside. Furthermore, the indices do not include rental prices, which are one of the main reasons to invest in commercial real estates. Therefore the results might be misleading and not give a fair view of the reality. However, commercial real estate indices in Sweden are limited. After careful consideration when choosing the indices we came to the conclusion that these two indices are the most appropriate for the purpose of this thesis, as the main purpose is to examine if there exists any relationship between the stock market and the commercial real estate market.

Since there are some flaws with these indices, we choose to compare the results with another commercial real estate index conducted by IPD (IPD, 2013). This index includes the total percentage return, both appreciation and the direct return, such as rent. However, the index only contains yearly returns and is therefore not suitable for performing cointegration, correlation and lead-lag relationship tests as these tests requires a certain number of observations. Even though, it can be useful to compare and discuss this index with the results from our own tests.

**Cointegration**

Testing for cointegration can be used to forecast future time series. In this thesis we will use cointegration tests to investigate if there exist any long-term relationship between OMX Stockholm and the commercial real estate indices. This relationship exists when the two time series move closely together and they appear to have the same stochastic trends. If a linear combination of the times series is stationary the markets are cointegrated. Time series that are non-stationary can move together in the long-run because they are affected by macroeconomic factors, and thus, are bounded by a relationship. It is also possible that the markets show no relationship in the short-run, but it appears in the long-run, thus indicating that the markets are cointegrated (Brooks, 2008).

**Stationarity**

To get a trustworthy result from the cointegration tests, it is important to state whether the data is stationary or non-stationary, since we then should treat the data differently. In the case where the data are stationary, this means that the mean, the variance and the autocovariances are constant for each given lag. Hence, it means that the future does not differ from the past, in a sense of probability distribution. On the other hand, if the data are non-stationary this implies that the data in period one are not related to the next period of the data. This in turn can lead to a spurious regression (Brooks, 2008). If the two markets are non-stationary and follows a stochastic trend over time they can show a close relation even though the markets are not related (Stock & Watson, 2012).

If the data is non-stationary it has to be transformed to become stationary. This is done with the following equations.

The new variable ∆ is now stationary and has no unit root. Depending on how many unit roots the non-stationary data has, this will determine how many times it needs to be differenced. By applying these equations *d* number of times, it leads to an I(0) process, which is a stationary series with no unit roots. A series with I(1) means that it is integrated of order one, thus contains one unit root and to become stationary it needs to be differenced once (Brooks, 2008).

To test for the degree of unit roots in OMX Stockholm and the commercial real estate indices the augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test will be used. By performing both tests it will provide us with a more reliable result.

The null hypothesis of the ADF test is ψ = 0, which means that the data contains a unit root, against the alternative hypothesis that the data has no unit root and is therefore stationary. To reject the null hypothesis the t-statistic has to be more negative than the critical values for Dickey Fuller tests at a significance level of 1 %. The test is performed with the following equation (Brooks, 2008):

∆ = ѱ _{ −1} + ∑ ∆ _{ −1} + =1

Where α is an unknown coefficient and *p* is the number of lags in the regression. The number of lags can be determined by either Akaike information criterion (AIC) or the Bayes information criterion (BIC). The difference between these two models is that with AIC the number of lags will be more than with the BIC. We will use the AIC model as it is recommended to have more lags when performing an ADF test. It is essential to use the right amount of lags since too few does not reduce the autocorrelation whereas too many will increase the uncertainty.

The KPSS test is also a test for stationarity. The difference between this test and the ADF test is that the null and the alternative hypothesis are reversed (Brooks, 2008). It has been found evidence that standard test of stationarity, like the ADF test, tend to fail to reject the null hypothesis, that the data contains a unit root, because there has to be strong evidence against the null hypothesis. The KPSS test is therefore a good complement to the ADF test since we then test both the null hypothesis that the data is stationary and the null hypothesis that the data contains a unit root (Kwiatkowski, Phillips, Schmidt, & Shin, 1992). To get a robust result of the two tests the outcome should be that one of the tests rejects the null hypothesis and the other test accepts the null hypothesis (Brooks, 2008).

**The Engle-Granger 2-step Method**

The Engle-Granger method is one method to use to examine if the stock market and the commercial real estate market are cointegrated. This method is used to measure if there is any stochastic trend in the time series (Brooks, 2008).

The first step in this method is to make sure that OMX Stockholm and the commercial real estate indices contain one unit root, which is done by performing the ADF and the KPSS tests. By running the OLS cointegration regression, the parameter values, β1 and β2, and the residuals will be estimated. We will run the OLS regression twice for each combination to have both indices as dependent and independent variables to get a more robust result. If the residuals in the OLS regression model are stationary one can proceed to the second step. To test the stationarity we will use the ADF and the KPSS tests. If the residuals contains one unit root there is no cointegration relationship between the variables.

In the second step the residuals from step one are put in the error correction model.

Where û_{ −1} is equal to _{ −1} − _{ −1} and is the error correction. The is known as the cointegrating vector, which measures the long-run relationship between the markets (Brooks, 2008).

**Correlation and Standard Deviation**

In this thesis we investigate if there is any correlation between the commercial real estate market and the stock market in order to see if there exists any short-run relationship between the two variables.

Correlation measures if there is a linear relationship between two variables, where the correlation coefficient is somewhere on a scale from -1 to +1. It describes the average relationship from one observation to the next observation, which in this study is quarterly. A correlation coefficient equal to zero would mean that there is no relationship between the variables. If the correlation coefficient is equal to one the linear relationship between the variables are perfect. This means that if one of the variables increases the other variable also increases. When the correlation coefficient is equal to minus one there is a perfect negative relationship, which means that the variables are moving in perfect opposite directions (Aczel & Sounderpandian, 2009). If this is the case an investor have the opportunity to create a diversified portfolio with no risk.

Where is the covariance, and are the average returns on the commercial real estate market and the stock market, and and is the standard deviations of the two markets. The standard deviation measures the spread of the data around its mean. A high standard deviation is an indication of high uncertainty of the investment.

The correlation coefficient along with standard deviation can be used to calculate the standard deviation of a multi-asset portfolio. By creating a portfolio with shares of the commercial real estate indices and shares of the OMX Stockholm an investor can choose a portfolio with low standard deviation and thus minimizing the risk in relation to the return. If the two assets have a low correlation the standard deviation in the portfolio will be lower because the investments do not move together (Elton, Gruber, Brown, & Goetzmann, 2011).

**Test for Lead-lag Relationship**

The Granger causality test is used to test the hypothesis that changes in the commercial real estate return is lagged by changes in the stock market returns. The F-statistic of a time series regression with multiple predictors tests the null hypothesis that the lags of one variable have no predictive content over the other variable in the model. The equation for the time series regression with multiple predictors is (Stock & Watson, 2012):

Where is the stock market at time t, is the commercial real estate market, and are unknown coefficients for the variables, and are the number of lags, and is the error term.

The Granger causality does not in fact really indicate causality, but the test is more used to see if a lead-lag relationship exists (Stock & Watson, 2012). If there exist Granger causality between commercial real estates and the stock market the strength of accepting a low or negative correlation weakens, hence the diversification benefits weakens. Investors with a short investment horizon could still take advantage of an existing lead-lag relationship. For example, if commercial real estate returns lag the stock market by a year an investor with a shorter time horizon could benefit from holding a multi-asset portfolio.

**Discussion of Methods**

When testing for unit roots we have chosen to use the ADF test and the KPSS test. Though, there also exists a unit root test called Phillips-Perron (PP). The ADF test and the PP test are very similar to each other as they often generate the same conclusions and they suffer from the same limitations. Therefore it is not necessary to conduct both these tests. The criticism of the ADF and the PP tests is that it is difficult to reject the null hypothesis, due to insufficient information in the sample. To avoid this problem we will perform both the ADF test and the KPSS test as the hypotheses of these tests are the opposites. If one of the null hypotheses is rejected and the other null hypothesis is accepted, it gives a more robust result (Brooks, 2008).

There are different methods to test for cointegration; Engle-Granger 2-step method, Johansen test and the Autoregressive distributed lag model (ADL). We have chosen to perform the Engle-Granger 2-step method as it does not require the data to be stationary and since previous research has used this method which makes it a proven method to use. Though there are some problems with this method. First, if there is not sufficient observations the unit root tests and the cointegration test is not reliable. Second, the test requires one variable to be dependent and the other to be independent which means that the variables need to be treated asymmetrically even though there is not a specific reason for this. We believe that these problems will not affect the output from the test as we have collected a relatively high number of observations and as we perform the test two times and use both assets as dependent and independent variable each time respectively (Brooks, 2008).

The Johansen test is used to test the hypothesis that the cointegrating vectors are of a minimum number. Though, there are some problems with this test as the vector autoregressive relationship (VAR) affects the serial correlation if it is too low, which in turn would bias the variance of the residuals and thus make the test unreliable. A too high VAR can lead to the test statistics to be biased upwards. Although the Johansen test is a proven method to use in cointegration test it is a very complex test and it can be difficult to interpret the results, which in turn can lead to misleading conclusions (Brooks, 2008).

The ADL model is an autoregression that includes an additional predictor with multiple lags. This model is good to use since it is possible to observe how much information the additional predictor provides. Though, this model can only be used if the data is stationary since using non-stationary data would lead to spurious regressions or biased results. Further, it is not the most preferable model to use when testing for cointegration as it only shows how much information the independent variable has on the dependent variable (Stock & Watson, 2012).

**Table of Contents**

**1 Introduction **

1.1 Background

1.2 Problem Discussion

1.3 Purpose

1.4 Research Questions

1.5 Academic Contribution

1.6 Delimitations

**2 Frame of Reference **

2.1 The History of the Swedish Commercial Real Estate Market and the Stock Market

2.2 Previous Research

2.3 Relationship between Real Estates and Stock Returns

2.4 Modern Portfolio Theory

2.5 Investment Alternatives

**3 Method **

3.1 Research Methodology

3.2 Data

3.3 Cointegration

3.4 Correlation and Standard Deviation

3.5 Test for Lead-lag Relationship

3.6 Discussion of Methods

**4 Empirical Results and Analysis**

4.1 Historical Data

4.2 Test for a Unit Root

4.3 Engle-Granger 2-step Cointegration Test

4.4 Lead-lag Relationship

4.5 Correlation and Standard Deviation

4.6 Integrated Analysis

**5 Concluding Remarks**

5.1 Conclusion

5.2 Discussion

**6 References**

GET THE COMPLETE PROJECT

Do the Stock Market and the Commercial Real Estate Market Cointegrate? A Study for Sweden