RADBOUD UNIVERSITY Nijmegen School of Management
Master Thesis
The green bond market and its’ conventional counterpart, a comparison
ByHANNA KNIPPERS (S1014162).
Abstract
Climate change is one of the biggest challenges faced in the 21st century. The
topic has initiated a general trend in sustainability. A relatively recent area of this trend is the green bond market. Which was initiated a little over 10 years ago. This paper empirically investigates whether green bonds, from a return perspective, are comparable to conventional bond. And thereby, making the investment interesting for all bond investors and not only social investors. This is investigated by clustering and matching the bonds and testing the yield spreads. A second model uses Jensen’s alpha as a performance proxy to test whether the alphas of green bonds are significantly different to those of conventional bonds. No statistical significance is found that green bonds under-or-out perform conventional bonds in the daily data set tested from January 2018-March 2019.
Supervisor: Dr. Sascha Füllbrunn Department of Economics
Table of Contents
1 Introduction ... 4
2 Literature review ... 7
2.1 Sustainability, SRI and fixed income funds ... 7
2.2 The green bond market ... 7
3 Data and methodology ... 11
3.1 Data ... 11
3.2 Cluster analysis ... 11
3.2.1 Cluster attributes ... 11
3.2.2 Cluster method ... 12
3.3 Yield spread model ... 13
3.4 Jensen’s alpha ... 15
4 Results ... 17
4.1 Yield spread model ... 17
4.1.1 Yield spread model results discussion ... 18
4.2 Jensen’s alpha approach ... 20
4.2.1 Jensen’s alpha results discussion ... 22
5 Conclusion ... 25
5.1 Limitations and future research ... 26
6 Bibliography ... 27
7 Appendices ... 31
A. Descriptive statistics ... 31
B. Calinski Harabasz pseudo-F ... 31
C. Yield conventional versus green bonds ... 31
D. Clustering details ... 32
F. Radom effects regression ... 32
G. Correlation matrix ... 33
H. Variance inflation factor ... 33
I. Woolridge rest for autocorrelation in panel data ... 33
J. Random effects with AR (1) disturbances regression ... 34
K. Robust regression ... 35
L. Correlation matrix ... 36
M. Variance inflation factor ... 36
N. Jensen’s Alpha Regression ... 36
O. Ramsey test ... 36
P. Breach-Pagan / Cook-Weisberg test for heteroskedasticity ... 37
Abbreviations:
ESG Environmental Social Governance SRI Socially Responsible Investments
FE Fixed effects
RE Random effects
OLS Ordinary Least Squares
1 Introduction
Climate change is one of the most challenging issues faced by the world. The consequences are estimated to be of enormous impact. It leads to uncertainty of timing, distribution and the impact on climate in general. Due to these challenges, governments, scientists and private entities have to increasingly take these effects of climate change into consideration. (Farmer & Hepburn, 2015)
The challenges faced have led to a general and global trend in sustainability. In the field of finance trends in sustainably responsible investments (SRI) and initiations of sustainable investment markets are prime examples. (Horsch & Richter, 2017) These investments have been growing rapidly. Estimations indicate that in 2018 in the US alone one in every four dollars of professionally managed investments were under Environmental, Social and Governance (ESG) strategies. (Connaker & Madsbjerg, 2019)
These trends have surged into another type of bond investment options, the green bond market. This market is relatively young, the European Investment Bank issued the first form as a climate awareness bond in 2007. Simultaneously Swedish investors, pension funds, Skandinaviska Enskilda Banken and the World Bank issued the first green bonds. The following year the first set of green bonds was issued to a larger group of investors. Since then, several development banks and other financial institutions started bringing green bond offers to the market. In 2013 the first corporate bond was issued and since then the market has been growing rapidly. (Hachenberg & Schiereck, 2018) The outstanding value of green bonds in 2018 was $389 billion. (Climate Bonds Initiative, 2018) This growth is shown in the figures below.
Arguably these growth statistics in combination with the market recently surpassing its ten-year mark make it an interesting area of research. It can be questioned whether this market is already competing with conventional bonds. When considering market size, the green bond market is still a niche with $389 billion outstanding in 2018 compared to an estimated $100 trillion outstanding conventional bonds. Market size does not necessary entail lower returns and green bonds can potentially serve other areas. (Hachenberg & Schiereck, 2018)
Green investments and especially green technology face one major issue. Often these types of technologies require large investment sums in the development phase of the project. Due to this, average traditional investors might view green investment as riskier ventures relative to conventional investments. (Lam & Law, 2016) Currently, there are policy actions in place to overcome this gap. However, estimations are that these are not feasible in the long-run. This is where the option of a green bond can arguably incentivize the private industry to take a more active role in green investment. (International Finance Cooperation, 2016)
When comparing green bonds to conventional counterparts the main advantages identified in green bonds are that the pool of ESG investors is tapped into. (KPMG, 2014) Furthermore, green bonds are in some cases tax exempt in which case the bond investor does not have to pay taxes on the green bonds they hold, which often leads to the issuer getting a lower interest rate. (Climate bonds, 2019) The latter is interesting mainly for the issuer where the former can potentially be interesting for the investor. The combination of the rapid market growth and the potential advantages of green bonds make it an interesting research topic. The question that remains, is whether green bonds can already compete with their
conventional counterparts. Which led to the following research question; Are green bond investors paying a good will premium compared to conventional bond investors?
Two main models are used to test the above stated research question, daily data is gathered on green and conventional bonds. These are clustered by a cluster analysis and matched according to the matching procedure. These matched bonds are used to calculate the yield spread, which forms the dependent variable in the first model. In this model the constant should show whether there is a significant difference in yields between the two bond types. The constant is found not to be statistically significant and therefore, there is no significant difference found between the two.
The second model uses two-steps, first a model is used to calculate Jensen’s alpha for each bond. The second step constitutes of using this performance proxy alpha as a dependent variable to test whether green bonds outperform conventional bonds. The green bond dummy in the model shows whether the alphas of the green bonds are significantly different from those of the conventional bond. The results show no significant result for this dummy therefore, there does not seem to be a significant difference between the two alphas.
The remainder of this thesis is set-up as follows; first green bond literature thus far is outlined. The third section constitutes the data and methodology where the approach is discussed, and the data used is outlined. Afterwards, the results are presented and discussed. The last chapter provides a conclusion on the findings.
2 Literature review
2.1 Sustainability, SRI and fixed income funds
Neoclassical economics assumes that investments are exclusively chosen on two
characteristics; risk expectations and the return expectations. Sen (1999) argued that this classical model can be modified since there are several social, moral and ecological motives present as well. Thereby modifying classical market economic theory by adding weight to ethical decision making in investments. Classical financial theory suggests that the choice of efficient portfolio assets is based on investors utility function in terms of wealth, where each investor is only concerned with his or her own satisfaction and with that disregards the satisfaction of social nature. The classic hypothesis can be relaxed, and social returns can be added to the model. (Fernandez-Izquierdo & Matallin-Saez, 2008) This has led to a
discussion in economic literature on whether ethical investments lead to a financial sacrifice or a premium. When further exploring the topic and emphasizing fixed income funds this same discussion prevails. Thus far, contradicting results have been found. This discussion prevails largely around whether ethics in an investment decision affects the performance. The claim is that socially responsible investors limit their portfolio risk-return optimization by avoiding assets for ethical reasons. (Derwall & Koedijk, 2009) To an extend these models can be extrapolated to green bonds, especially in relation to conventional bonds. Green bonds incorporate, besides the risk return expectations, social, moral and ecological motives as well.
2.2 The green bond market
As outlined in the introduction the green bond market is relatively young. This is why the research on the topic is still limited, especially the number of published scientific studies. Most studies currently available are conducted by financial institutions. Both studies by financial institution and scientific studies are explored in this literature review.
Green bonds are essentially fixed income securities of which the proceeds are used to (re)finance new or existing green projects. (Horsch & Richter, 2017) These projects vary greatly from renewable energy projects, to sustainable buildings and clean transportation to waste management. In figure 3 the green bond commitments by sector are shown.
Figure 3: Green bond commitments per sector (Worldbank, 2018)
In order for the Green bond market to further flourish the returns should be similar and the risk levels should be no greater than those of conventional bonds. (Mathews & Kidney, 2012) Previous research shows that specific characteristics of the green bonds are dependent on the issuer. Where corporates and agencies tend to have the largest issue amounts, corporates and financial institutions offered higher coupon rates in comparison to agencies and
sub-sovereigns. The longest average maturity on green bonds are often issued by sub-sovereigns and the shortest average are those issued by agencies. The best rated green bonds tend to be those issued by national or international organizations that are backed by governments. On average the bonds that were rated worst are those issued by corporations. (Horsch & Richter, 2017) Green bonds are generally less liquid than their conventional counterparts. The main reasoning behind this is that most investors of green bonds are long-term institutional investors, with both social and environmental mandates. These investors are seeking
incentives for protection against inflation, default and market volatility risk. (Hyun, Park, & Tian, 2018)
Studies on green bonds are often conducted by investment banks. However, as mentioned there is some scientific literature available but due to the green bonds being relatively new the number of sources on these green bond aspects are very limited. Therefore, comparisons between results of studies tend to be challenging. The aspects researched in green bond
literature include topics such as liquidity, pricing, the role of the green bond market, a green premium and return aspects.
A pricing comparison has been conducted where it is empirically tested whether the pricing of green bonds is significantly different from that of conventional bonds. The empirical investigation result show no significant difference in the pricing of green bonds compared to conventional bonds. However, the analyisis is extended and there are differentiations on issuers which shows that green bonds issued by financial institutions seem to trade tighter and those issued by governments seem to trade marginally wider. (Hachenberg & Schiereck, 2018) The market has been analyzed where the labelling of green bonds is studied and the correlations with other assets is empirically tested. Furthermore, this study finds that the drivers of green bond prices are similar to those of conventional bonds. (Horsch & Richter, 2017) The liquidity risk on green bonds is addressed by Flaherty, Gevorkyan, Radpour, & Semmler (2017). This study empirically investicates whether the impact of liquidity risk is neglegable on the yield spread of green bonds. This study confirms that the liquidity risk is negligible on the yield spread of green bonds. (Febia, Schäfera, Stephan, & Sun, 2018) Furthermore, Zerbib (2019) researced the yield differential in a comparison study between green and conventional bonds. Where it is empirically tested if the yields of green bonds are comparable to those of coventional bonds. The results show that the yields of green bonds are on average -2bps lower than those of convetional bonds. (Bachelet, Becchetti, &
Manfredonia, 2019) address the green bond premium puzzle and find that green bonds have higher yields lower variance and tend to be more liquid. Studies have researched green bonds in a larger aspect as well. (Flaherty et al., 2017) address the financing gap that is currently present in combating climate change, an estimated $10 trillion is needed before 2030 to reach the set goals and this study researches what role green bonds can play in filling this financing gap.
When considering the approaches used of studies thus far, it is found that often a matching procedure is applied. The pricing comparison conducted by Hachenberg & Schiereck (2018) uses this approach. The same approach is used by Hyun et al. (2018) in their study on the price of greenness and by Zerbib (2019) in a study on a green bond premium. This matching approach has been used throughout different bodies of literature as well (Kreander, Gray, Power, & Sinclair, 2005; Renneboog, Horst, & Zhang, 2008). In this matching procedure the bonds are literally matched with its closest counterpart. These matched bonds can then be compared because of their similarities. This matching of bonds has to be applied because the attributes of bonds can significantly affect the returns. For example, yields on higher rated
bonds tend to be significantly lower than those on the lower rated bonds. Which can be explained by the risk-return tradeoff. (Bessembinder, Kahle, Maxwell, & Xu, 2009). Therefore, to make a comparison that excludes these influences the bonds are matched.
This study researches whether the returns of green bonds are different than those on
convectional bonds. Although there are a few studies that have researched a similar topic this thesis introduces a new approach in combination with a new dataset. Instead of directly matching the bonds, the bonds are first clustered. The cluster analysis is a prominent tool used is various fields such as mathematics and statistics. (Nanda, Mahanty, & Tiwari, 2010) The approach has already made its way into sustainability literature by Koellner, Weber, & Scholz (2005) who use a clustering approach to study sustainability ratings. This same approach is used in portfolio management studies (Nanda et al., 2010). The clustering of bonds introduces a new and efficient aspect to come to the matching of the bonds. This efficiency is because the matchings can be derived directly from the clusters which already consist of similar bonds. Therefore, providing a new and more effective way to come to the needed matches in a comparison study of green bonds.
Another new approach to the field of green bonds used in this study is Jensen’s alpha. This is a measure generally used to determine whether abnormal returns are present in a security or a portfolio. (Murthi, Choi, & Desai, 1997) The measure has already made its way into literature on sustainability. A prominent example of this is a study by Derwall & Koedijk (2009) who use the alpha to compare SRI funds to conventional funds. Thus far, this measure has not been used in the area of green bonds and is introduced in this study.
In order to formulate an appropriate hypothesis that will assist in answering the research question at hand, the results of previous literature are compared. These studies have thus far been showing conflicting results. Zerbib (2019) finds that green bonds generate slightly lower yields than the conventional counterparts, a difference of -2 basis points for his entire sample. Bachelet et al. (2019) find that green bonds generate higher yields, there is lower variance and they are more liquid than the conventional counterparts. Hyun et al. (2018) find no significant difference between the yields on the two bonds. Due to these conflicting results the following hypothesis is formulated; “Green bonds generate similar yields compared to their conventional counterparts”. The approach to how this hypothesis is tested is outlined in detail in the next chapter.
3 Data and methodology
3.1 Data
On April 15th there were 1342 active green bonds and about 900.000 active conventional
bonds listed in the Thompson Reuters data stream. Only plain vanilla, investment grade bonds issued for 150 million or more were included in the sample. (Hachenberg & Schiereck, 2018) The daily yields from January 2018 – March 2019 for this sample were generated again from the Thompson Reuters data stream. Any bond that was not active for the complete period stated above was excluded from the sample. For Jensen’s alpha a two-step approach is used, in the first step the alphas of each bonds are calculated. To do this various data on indexes are needed, these are gathered from the Factset database. The second step of the model uses the same data as described above.
3.2 Cluster analysis
In order to find similar bonds a clustering approach is used, as stated in the previous chapter. This analysis is conducted in order to find bonds that were relatively similar to each other. From these clusters the green bonds were matched to the closest conventional counterpart. From the matchings the yields spread between the two bond types are calculated. These then form the dependent variable in the first model.
3.2.1 Cluster attributes
In order to form the appropriate clusters a few important steps have to be considered. Cluster analysis is a technique in which relevant attributes are used to find natural groupings of data. These relevant attributes are highly important to form the appropriate clusters. Therefore, the first step of a cluster analysis is to select the appropriate attributes. The selected attributes are listed below; the second step is to scale and standardize the attributes. (Das, 2003) Since there are qualitative and quantitative measures in the set of attributes the qualitative measures are translated into quantitative measures. The approaches for these translations are listed beneath the attributes for which this standardization is needed.
Bond Rating; As addressed in the literature review, the rating of bonds can have a significant
impact on the way the bond performs (Zerbib, 2016; Hachenberg & Schiereck, 2018). The yield on highly rated bonds are significantly lower than those of lower rated bonds, due to the risk-return trade-off (Bessembinder et al., 2009). Therefore, this is the first clustering criteria. Additionally, the rating of a bond is the first measure that has to be standardized. The ratings
are recalculated using a numerical 17 grade scale (where AAA/Aaa=1, AA+/Aa1=2, …, CCC/Caa1 and below is 17). (Norden & Weber, 2004; Hachenberg & Schiereck, 2018) In this data set there are no ratings below BB+ therefore the 17-grade scale only goes up to 10.
Maturity; The maturity date of the bond affects the yield of that bond (Litterman &
Scheinkman, 1991; Houweling, Mentink, & Vorst, 2005). Especially if a bond is close to maturity these bonds have no representative trading. (Hachenberg & Schiereck, 2018) Therefore, these bonds are filtered out beforehand. For the clustering of the bonds the maturity rate is used as a second attribute. Bonds with similar maturities should be clustered together. In order to carry this out the remaining maturity on the bonds is calculated.
Currency; is the used as the third attribute for the cluster analysis. The currency in which the
bond is issued affects the bond. Often issuers issue the bond in currencies that are considered strong. (Cohen, 2005) Zerbib (2016) found that the issuance currency affected green bond performance. For this variable a dummy is used, which takes a value of 1 when the currency is either Euro or US Dollar and a value of 0 otherwise. (Hachenberg & Schiereck, 2018)
Issuer; Bonds can be issued by a number of entities, for example; governments, sovereigns,
municipals, financial institutions or corporates. The issuing party can potentially influence the yield and pricing of the bond. (Hachenberg & Schiereck, 2018) Furthermore, as discussed in the literature review the characteristics of bonds are dependent on the issuer of the bond. (Horsch & Richter, 2017)
Coupon rate; The coupon rate is also considered in the process of forming the clusters. The
yield to maturity of a bond is affected by the coupon rate (Elton, Gruber, Agrawal, & Mann, 2001). Using the coupon rate is in line with previous green bond literature Zerbib (2016) and Hachenberg & Schiereck (2018) have used the coupon rate as a control.
Issue size; The issue size of a bond is argued to play a role in the credit risk on that bond.
When the issue size is relatively low the risk appears to decrease simultaneously. (Edwards, Harris, & Piwowar, 2007) Therefore, this is the last attribute used to form the appropriate clusters. Because the issue size of bonds is often large and, in this sample, upwards of 150 million the natural logarithm is generated for each size. (Hachenberg & Schiereck, 2018)
3.2.2 Cluster method
There are several clustering methods, the first distinction is between hierarchical and non-hierarchical clustering. In non-hierarchical clustering; m nested classifications ranging from m
clusters of one member each to one cluster of m members. In non-hierarchical clustering the cluster data units into a single classification of k clusters. Which is where the clusters or k is determined before as part of the clustering method. (Das, 2003) In order to select the
appropriate clustering method several options are explored. Furthermore, previous studies that used cluster analysis are considered. From this analysis it was found that the non-hierarchical method of k-means clustering was most appropriate for this sample. The first reason is, in a non-hierarchical method there are no hierarchical relationships between the clusters. (Das, 2003) In this sample there is no hierarchy needed between the clusters. A second reason for selecting k-means clustering is that hierarchical methods are often time consuming especially for data with a large number of observations. (Schonlau, 2002) Therefore, the non-hierarchical method of k-means clustering is a better fit in this case. Furthermore, non-hierarchical clustering allows for reallocation of objects within the clusters if necessary. Which cannot be done within hierarchical clustering.
The k-means clustering approach partitions n entities into k sets, it does so by using the Euclidian distance measure. Each item is placed in on basis of the nearest mean in a cluster by the algorithm. (Das, 2003) The challenge in k-means clustering is that the number of clusters has to be determined before the cluster analysis can be conducted properly. In order to overcome this challenge and to determine the optimal number of clusters the Calinski-Harabasz pseudo F-statistic is used. (Caliński & Harabasz, 1974) This ratio reflects the within-group similarity and the between-group differences. From this statistic the optimal number of clusters for this particular data set can be determined. This statistic is tested in several different numbers of clusters and the results show that the optimal number of clusters is 52 in this sample. The results of the Calinski-Harabasz pseudo F-statistic can be found in appendix B.
3.3 Yield spread model
From the 52 clusters the green bonds are matched to their closest conventional counterpart in each of the clusters. The matching is done based on bonds being as similar as possible. (Hachenberg & Schiereck, 2018) However, identical bonds are unattainable. The currency, rating and issuer are identical. The issue size, coupon and maturity are as close as possible. This led to a sample of 61 matched pairs each with 305 observations. From these matchings the yield spreads between the bonds are calculated and these form the dependent variable for the first model. In order to ensure that no specific green bond data nor specific conventional
bond data is defined on the right-hand-side of the equation, the size coupon and maturity of the matched bonds are averaged out for the analysis. (Maul & Schiereck, 2017)
𝑌𝑖𝑒𝑙𝑑𝑠𝑝𝑟𝑒𝑎𝑑*+ = 𝛽/+ 𝛽1𝑆𝑖𝑧𝑒*++ 𝛽4𝑅𝑎𝑡𝑖𝑛𝑔*++ 𝛽9𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦*++ 𝛽=𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦*++ 𝛽?𝑖𝑠𝑠𝑢𝑒𝑟*+ + 𝛽@𝐶𝑜𝑢𝑝𝑜𝑛*++ 𝜀*+
The clustering attributes form the control variables in the model. Where the dependent
variable is the yield spread between the green and conventional bond on a specific date. If the yields spreads are increasing or widening the yield difference between the two bonds is increasing. Vice versa when the yields spreads are tightening the yield difference between the two bonds is decreasing. Given the construction of variables the constant in the models is the parameter of interest. When this variable takes a significant positive value, it indicates that the yield of green bonds is higher than those of its conventional counterparts. Similarly, if the parameter shows a negative significant value the green bond will generate lower yields than its conventional counterpart. (Bachelet et al., 2019)
The other variables in the model serve as control variables and are the same as the attributes used for the cluster analysis. Size is the natural logarithm of the issue size. In this model the average size between the matched green and conventional bond is used. (Hachenberg & Schiereck, 2018; Bachelet et al., 2019; Zerbib, 2019) Rating, is the rating of the bond which is identical for the green and the conventional bond. The ratings are based on the S&P, Fitch and Moody long-term issue credit ratings. (Hachenberg & Schiereck, 2018; Hyun et al., 2018; Zerbib, 2019) Maturity, represents the maturity left on the bond on the specific date. And although the maturity left on the green and conventional bonds are close in the
matchings, they are not identical. Therefore, the average maturity left on the matched bonds is used. (Maul & Schiereck, 2017) Currency, is a dummy that represents the currency in which the bonds are issued. This dummy takes a value of 1 if the currency is either US dollar or Euro and 0 otherwise. (Hachenberg & Schiereck, 2018; Hyun et al., 2018; Bachelet et al., 2019; Zerbib, 2019) Issuer, is a dummy that represents the party that issued the bond. Which can be an agency, governments/municipalities or corporate issuers. The dummy takes a value of 1 if the issuer is government related and 0 otherwise. (Hachenberg & Schiereck, 2018; Hyun et al., 2018; Zerbib, 2019) The last control variable is coupon, which represents the coupon rate on the bond. Again, to ensure no green or conventional bond specifics are specified at the right-hand side. The average coupon between the matched green and conventional bond is used. (Maul & Schiereck, 2017)
3.4 Jensen’s alpha
As outlined in the literature review Jensen’s alpha is an established measurement to test abnormal returns. This alpha has already made its way into sustainable investment literature. Typically, the alpha is comparable to the intercept term in equity fund performance models (a). The measurement is often seen as the contribution to active money management to fixed-income portfolio return. In this thesis the alpha is used to compare green bonds to
conventional bonds. A similar approach to (Derwall & Koedijk, 2009) is taken, who compare SRI mutual funds and conventional mutual funds. By slightly adapting the model a
comparison can be made, of which the results represent the abnormal returns of SRI funds compared to the abnormal returns of the conventional funds or in this case green bonds compared to conventional bonds. There is no consensus thus far as to which bond indexes combination is most fitting for explaining bond returns. The model is adapted from Elton, Gruber, & Blake, (1995) and Derwall & Koedijk (2009) and the indexes are adapted where needed. The following equation constitutes the first part of the model. By using this equation, the alpha for each bond is calculated, which is used as a performance proxy for each bond.
R*+ − RF+ = ai + b/*+HS&P+ − RF+L + b1*Default+ + b4*Option+ + e*+
The R*+ is the return on bond I on day t, which includes both green and conventional bonds. RF+ denotes the S&P treasury bill as a proxy for the risk-free rate. (Campbell, Huisman, & Koedijk, 2001)The first variable captures broad market sensitivity, by computing the returns on the S&P investment grade index and subtracting the risk-free rate. (Derwall & Koedijk, 2009) Default is the return spread between the S&P high yield index and the S&P US treasury index. Derwall & Koedijk (2009) used the high yield index by Merrill Lynch. However, this index has been unlisted therefore, the S&P high yield index is used in this model. (Pham, 2016) This variable is intendent to capture default risk. Option is implemented to capture features in specific bonds. This variable is generated by the calculated difference in return between the iBoxx mortgage index and the S&P treasury index. The parameters in the models are estimated daily (excluding days the indexes are closed). The time frame is the same as in the yield spread model 1 January 2018 – 1 March 2019. The alpha for each bond is calculated and these alphas form the dependent variable for the second step of the model.
𝛼*+ = 𝛼/++ 𝛾1*𝐺𝐵𝑑𝑢𝑚𝑚𝑦 + 𝛾4*𝑆𝑖𝑧𝑒 + 𝛾9*𝑅𝑎𝑡𝑖𝑛𝑔 + 𝛾=*𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 + 𝛾?*𝐶𝑜𝑢𝑝𝑜𝑛 + 𝛾@*𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 + 𝛾]*𝐼𝑠𝑠𝑢𝑒𝑟 + e*+
For the second step of the model controls are added to the model, these are the same controls as used in yield spread model. Furthermore, a green bond dummy is included which takes a value of 1 is the bond is green and a value of 0 if it’s a conventional bond. This dummy is added in order to test the effect of a bond being green on the alpha. Therefore, this dummy should show whether green bonds are out-performing or underperforming opposed to conventional bonds. The conventional alpha traditionally captures the added value of active management. (Derwall & Koedijk, 2009) The above adapted model is used to compare alphas and thus formally testing the hypothesis whether Green bonds generate similar returns
4 Results
The results chapter is split in two since two different approaches were applied. In the first part the results of the yield spread model are presented and thereafter discussed. In the second part the results of the Jensen’s alpha approach are outlined and discussed.
4.1 Yield spread model
Before the model is applied the returns of both green and conventional bonds in the sample are averaged and shown graphically in appendix C. On an average the yields do not seem to be that different from each other. This of course has to be formally tested.
When panel data is used a decision has to be made between a fixed effects (FE) and random effects (RE) model. In a FE model there is an assumption of individual differences over time. For the majority of variables in this equation this is not the case. In a RE model the variables that are constant over time can be estimated, which is needed in this model. (Studenmund, 2017) Therefore, the decision is made that the RE model is a better fit to the data. It does have to be tested whether the RE model is actually appropriate for this set or if an OLS regression would be a better fit. Therefore, the Breusch and Pagan Lagrangian multiplier test is carried out (appendix E). The null hypothesis states that the variances across entities is zero. The result show that the null hypothesis is rejected. Which entails that the modelling is continued with RE because the test shows that it is a better fit than an OLS regression. (Torres-Reyna, 2007) The results of the RE regression are presented in table 1. Before the results in the table can be discussed a few tests have to be carried out to ensure there are no issues with this RE regression.
The first test is to detect potential collinearity. This is tested by a correlation matrix (appendix G). This matrix shows some relatively high values therefore, a second test is employed. This test is called the VIF variance inflation factor (appendix H). The highest displayed value is 2.13 indicating that there is no concern for multicollinearity. The rule of thumb states that a beta larger than 5 should cause serious concern for multicollinearity. (Studenmund, 2017)
Another potential issue in panel data is autocorrelation/serial correlation. Therefore, the next test that is carried out is the Woolridge test for autocorrelation. (Drukker, 2003) The test shows significant results therefore, the null hypothesis cannot be rejected and autocorrelation
has been detected (appendix I). The reason behind these results could be that the panel data is unequally spaced. (Baltagi & Wu, 1999) Which in this case is because the market is closed on holidays and weekends, an issue that often occurs when using daily market data. An option to limit the impact of the unequally spaced data is a RE regression with AR 1
disturbances. (Baltagi & Wu, 1999) The results of this regression are presented in the table as well, in the regression 2 column.
Table 1: Random effects and Random effects with AR1 disturbances
Variables Regression 1 Regression 2
Size 2.004 1.8091 (0.80) (0.68) Bond Rating 0.8619* 0.8620* (1.95) (1.83) Maturity -0.8569*** -0.6435*** (-14.52) (-3.00) Currency 0.4184 0.2336 (0.10) (0.05) Issuer 2.7889 2.3221 (0.71) (0.56) Coupon 0.0153 -0.3829 (0.02) (-0.37) Constant -38.25597 -34.6950 (-0.78) (-0.66) No of observations 18605 18605 No of Groups 61 61 R-squared 0.0611 0.0681
Dependent variable: Yield spread
Notes: The table provided above are the results of the random effects model (RE) and the RE model with AR1 disturbances. The t-values are in the parentheses and *, ** and *** indicate significant coefficients at the 10% level, 5% level and 1% level respectively (*p<0.1, **p<0.05 and ***p<0.01).
4.1.1 Yield spread model results discussion
The results in the table above show insignificant results for the variable of interest, the constant. This indicates that there is no significant difference between the Yield of green bonds in comparison to conventional bonds. The control variables do generate some
significant results, the rating and maturity both seem to significantly influence on the yield spread. Where the rating of a bonds is found to have a significant positive effect and maturity is found to have a significant negative effect. The R-squared in the first model is relatively low, the model explains 6.11%. When the model with AR1 disturbances is used this reported value increases slightly to 6.81%. These low reported values could potentially be explained by the use of a matching procedure. Although the bonds are matched as closely as possible none are identical which could have had an influence on the relatively low explanatory power of the model.
Before the results are discussed in relation to other literature it is important to note that the models in these studies are similar but not exactly the same.
Hyun et al. (2018) for example use the ask yield spread. Altough they find some significant variables in their model the overall results do not confirm nor refute the presense of a green bond premium. These results can be considered to some extend similar to the results in the table above. The R-sqaured in their models differs a lot within the study from 13% in the first model to 31% in the last model. When considering the results of the control variables Hyun et al. (2018) also find a significant effect of the maturity, however, this effect is reported to be a positive one whereas the table above reports a negative one. As for the other sigificant control rating, Hyun et al. (2018) have split the ratings into 3: AA, A and BBB. Where the later two are reported to have a significant negative effect the first is reported to have a significant postive effect, similar to the result in this study.
Zerbib (2019) finds that there is a green bond premium that investors pay by deciding to invest in green bonds opposed to conventional bonds. The reported r-squared values here are comparable to those found by Hyun et al. (2018). The first few models report very low r-quared values at some point, 0.1% is explained by the model. However, they are able to increase these values with there other models to about 29%. In relation to the results reported above this study is dissimilar. Mainly because it reports that there is a green bond premium present. The main similairities between this study and the one conducted by Zerbib (2016) is that both use a matching procedure and the variable of interest is the yield on bonds.
Nonetheless, the reported results are different than the result in the table above. The
reasoning behind this finding can be that the data set used was different from the set used in this study. Furthermore, the model and the apporach are slightly different which can both be factors that attribute to the different findings.
The results of a study by Bachelet et al. (2019) show that the yields of green bonds are actually higher than those of their conventional counterpart. Which contradicts the results found in table 1 where, as stated, no statistically significant difference is found between the two sets of bonds. The set up of the model in both studies is similar, the parameter of interest is the constant and the other variables serve as controls. The differences in results are likely explained by the data set which spans from 2013-2017 whereas in this study 2018 and the first part of 2019 are used. The r-squared in the model by Bachelet et al. (2019) can also be considered relatively low at 20.08%.
Hachenberg & Schiereck (2018) consider the pricing of green bonds opposed to conventional bonds. The reason this is an interesting comparison is because they do incorperate the yield spread in their dependent variable. The daily I-spread is incorperated with Newton’s formula which forms the dependent variable. Overall the main results are insigificant however, once only the single A-rated bonds are considered the green bonds trade sigificantly tighter. Furthermore, government related bonds seems to trade marginally wider and financial bonds trade tighter again. In relation to the results above the overall result of no statistical
significant difference in the daily I-spread of the bonds is comparable to the no significant yield spread difference. When considering the results of the control variables in their study they find insignificant results for the variable maturity in most of their models. In the models that do present a significant result this effect is negative, similar to the results in the table above. The rating control in this study has been split up in 3 as well: AAA, AA and A. For the first no siginificant results are found the lattter two are significant where AA is negative and A has a positive effect.
4.2 Jensen’s alpha approach
As outlined in the data and methodology chapter in the second model Jensen’s alpha is used. The conventional alpha for funds tries to capture the effects of active money management. (Derwall & Koedijk, 2009) In this study the alphas of conventional bonds and green bonds are essentially compared. By including a dummy in the second step of the model, it can be seen whether the green bond has a significant effect. Thereby showing if green bonds outperform conventional bonds or vice versa. By calculating the alpha for each bond over time the alpha itself becomes time invariant and the alpha serves as a performance proxy of the bond at hand. These alphas form the dependent variable in the regression in the table 2.
After the alphas are calculated they are first visually inspected by using a graph to see if there are any notable differences already between the alphas. In order to carry this out the kernel density estimation is used shown in figure 4 below. The alphas of green and conventional bonds are compared.
Figure 4: Alpha's for green bonds and conventional bonds
From the graph above no notable differences in the alphas of both bonds can be seen yet. However, this is only a visual interpretation of the alphas and to draw formal conclusions these alphas need to be tested further.
In the second step of the approach the variables are time invariant, therefore the regression model used is an OLS regression. Before running this regression, a few tests have to be conducted to exclude certain modelling issues. The model is tested for correlation by using a correlation matrix (appendix L). The matrix indicates that there is no perfect
multicollinearity. However, similar to the yield spread model a second test is carried out. The VIF results (appendix M) shows that there is no serious concern for multicollinearity, the highest reported value in the table is 1.95. As noted, the rule of thumb is that a value higher than 5 should be cause for serious concern on multicollinearity. (Studenmund, 2017) The results of the OLS regression are presented in table 2. Before discussing the results of this regression two other tests have to be conducted in order to ensure that the there are no issues with the model. First, the Ramsey RESET test is carried out to test for omitted variable bias. The results (appendix O) show that there does not seem to be an issue with omitted variables, which can significantly affect the result of a model. (Studenmund, 2017)
The second test carried out is the Breusch-Pagan/Cook-Weisberg test for heteroskedasticity (appendix P). Which occurs when the error variance has a non-constant variance. The results of this test show that some heteroskedasticity is found in the model. Therefore, a new robust regression is applied where the model assumes heteroskedasticity. (Stock & Watson, 2008) The results can be found under regression 2 in table 2.
Table 2: OLS Regression bond Alphas
Variables Regression 1 Regression 2
GB-dummy 0.0124 0.0124 (0.83) (0.83) Size -0.0200 -0.0200 (-1.24) (-1.53) Rating -0.0089*** -0.0089*** (-2.72) -(2.30) Maturity -0.0006 -0.0006 (-0.34) (-0.33) Coupon 0.1053*** 0.1053*** (14.75) (15.91) Currency 0.5296* 0.5296* (1.85) (1.97) Issuer -0.0337 -0.0337 (-1.15) (-1.03) Constant 0.5544* 0.5544* (1.73) (2.13) No of observations 122 122 R-squared 0.7110 0.7277
Dependent variable: Alphahat
Notes: The table provided above are the results of OLS model with the bond alphas as the dependent variable. The first model is an OLS and the second is a robust OLS model. The t-values are in the parentheses and *, ** and *** indicate significant coefficients at the 10% level, 5% level and 1% level respectively (*p<0.1, **p<0.05 and ***p<0.01).
4.2.1 Jensen’s alpha results discussion
In the table above the results for the regression with Jensen’s alpha can be found. The result produces no significant value for the green bond dummy. From this result it can be concluded there is no significant difference in the bond return when it’s green as opposed to a
conventional bond. The adjusted R-squared in this model is quite high with 71.1% which indicates that the model is a relatively good fit. The robust regression shows a higher r-squared because, no adjusted r-r-squared is reported here and the r-r-squared is therefore reported. As reported in the results there is only a slight difference between the OLS regression (1) and the robust regression (2). This could be explained by the results of the Breusch-Pagan/Cook-Weisberg test. The heteroskedasticity in the first model is significant at the 5% level. Which could indicate that the heteroskedasticity is limited and by using the robust regression STATA assumes heteroskedasticity and accounts for it.
Before relating the results of the alpha approach to other literature it can be compared to the results of the yield spread model in this study. Both models empirically test the same data set but it is the approach to how both of these are tested that is different. For both models the variable of interest shows no statistically significant difference in the return of the bond types. The models to some extend serve as a check for each other. Since the first model showed no significant difference, expectations would be that the second model would not either.
When comparing these results to results in other literature this comparison is relatively challenging. Thus far, there are no studies in green bond literature that use the Jensen’s alpha approach, or a performance proxy closely related. Therefore, in order to relate the results, the first study used for a comparison is by Derwall & Koedijk, (2009). Altough the models in both studies test something different the general functionality of the model can be assessed with the model by Derwall & Koedijk, (2009). As stated the parameter of interest in this model does not generate a signficant result. However, the model does generate a high squared which shows that the model is a relatively good fit. In their models the reported r-squared is also quite high and in some models extremely high (98%). This can be compared to the r-squared reported in this model which at 71.1% is relatively high as well. The overall results cannot be related because this study empirically tests the return performance of green bonds versus conventional bonds. Whereas Derwall & Koedijk, (2009) empirically test the performance of sustainable mutual funds.
Literature that does empirically test the return of green bonds in comparison to their
conventional counterparts used different modelling types. The general consensus is however, similar to the literature discussed in the yield spread model part. Where (Hyun et al., 2018) generate insignificant differences between the two bonds similar to the results in table 2,
(Zerbib, 2019) finds that, different from these results, green bonds underperform in comparison to conventional bonds on a return aspect. Then there are the results found by (Bachelet et al., 2019) which indicate that green bonds are the fixed income securities that generate higher yields than their conventional counterparts. The details of these studies are elaborated upon in the discussion of the yield spread model in relation to other literature.
5 Conclusion
The relatively young green bond market has been growing significantly over the last years. Given de challenges faced by climate changes these fixed income securities can keep taking on an increasingly important role in the (re)financing of new or existing green projects. The market as it is now is still a niche market in comparison to the conventional bond market. However, it seems that over the course of the following years the market has further growth potential. In order for the green bonds to keep flourishing risks should be no greater and returns should be similar to those of conventional bonds.
In this study the returns of green bonds are formally compared to the return of conventional bonds. In order to find whether on a return basis green bonds are comparable investments to conventional bonds. Thereby, testing whether the green bond market is an interesting investment for traditional bond investors. Due to the market being relatively young the research on the topic thus far is limited. There are some comparison studies however, these papers focus on other aspects such as pricing, liquidity and only a handful address green bond premiums or returns. This study aims to fill the gap and by introducing an approach to the field of green bonds. First, a cluster analysis is introduced to cluster similar bonds based on size, rating, maturity, currency, issuer and coupon rate. From these clusters the bonds within each cluster are matched according to the matching procedure.
In order to further compare an approach from fund performance is used and employed on this data set. Jensen’s alpha is calculated from a model where the bond returns are tested against the market. These alphas are used as a performance proxy for all bonds in the set.
The second-step of this model tests whether the performance proxies (alphas) of green bonds are significantly different from those of conventional bonds by using a dummy.
The results of this study show that for neither model, the variable of interest is statistically significant. The first model shows an insignificant value for the constant thereby it cannot be confirmed that there is a significant difference. The second model shows an insignificant value for the green bond dummy, thereby showing that there is no statistically significant difference between the performance proxies (alphas) of green bonds compared to
conventional bonds. Therefore, both models show that there is no statistically significant out-or-under performing of green bonds opposed to conventional bonds.
Due to this lack of statistical significance in variables of interest the hypothesis cannot be supported nor rejected based on the data set. As for the main question “are green bond investors paying a good will premium compared to their conventional counterparts?” The results of this study show that there is no statistically significant difference between the two from a return aspect.
5.1 Limitations and future research
The models in this study used daily observations which could be a reason for the insignificant results that are found in both models. The green bond market, relative to the conventional bond market, is illiquid. Although, on this aspect there is no general consensus yet because in this research area the results are opposing as well. Using daily data could have affected the results of the study. A study with monthly data over a longer time-period could potentially generate different results. Thereby being an interesting topic to further research in this area. Furthermore, this study addresses the performance of green bonds on a return basis. Another aspect that is essential for the performance of a fixed income security, or any investment for that matter, is the risk aspect. Thus far, the amount of scientific studies on this subject in the field of green bonds are limited. In order for the green bond market to flourish further, both risk and returns are considered important aspects. Therefore, researching the risk aspects of green bonds would be an interesting insight for this young market as well.
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7 Appendices
A. Descriptive statistics
B. Calinski Harabasz pseudo-F Number of clusters Calinski/Harabasz pseudo-F 50 1852.75 51 1976.74 52 2032.66 53 1882.76
C. Yield conventional versus green bonds
Issuer 18605 .147541 .3546539 0 1 cur 18605 .8360656 .370226 0 1 coupon 18605 1.985861 1.233045 .05 5.975 Maturity 18605 7.072962 4.582272 .6205479 21.04795 Rating 18605 5.016393 2.944882 1 10 size 18605 20.07062 .5236296 18.90896 21.29981 Yieldspread 18605 1.025262 8.33998 -2.9179 112.5684 Variable Obs Mean Std. Dev. Min Max
1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 Jan-18Fe b-18 M ar-18 Apr-18M ay-18 Ju n-18 Jul-18Aug-18Sep-18 Oct-1 8 Nov-18 Dec-18 Ja n-19 Fe b-19 M ar-19
Yield green versus conventional
D. Clustering details
E. Breucsh and Pagan Lagrangian multiplier test for random effects
F. Radom effects regression range: 0 .
start: krandom k: 52
varlist: logsize rating maturity cur issuer coupon start(krandom)
other: cmd: cluster kmeans logsize rating maturity cur issuer coupon, k(52) measure(L2) vars: _clus_4 (group variable)
_clus_4 (type: partition, method: kmeans, dissimilarity: L2)
Prob > chibar2 = 0.0000 chibar2(01) = 2.2e+06 Test: Var(u) = 0 u 64.18209 8.011372 e 7.813286 2.795226 Yieldsp~d 69.55527 8.33998 Var sd = sqrt(Var) Estimated results:
Yieldspread[ising,t] = Xb + u[ising] + e[ising,t] Breusch and Pagan Lagrangian multiplier test for random effects
rho .89147516 (fraction of variance due to u_i)
sigma_e 2.7952256 sigma_u 8.0113723 _cons -38.25597 49.28293 -0.78 0.438 -134.8487 58.33679 coupon .015359 .9557445 0.02 0.987 -1.857866 1.888584 Issuer 2.788943 3.930375 0.71 0.478 -4.914449 10.49234 cur .4184856 4.032508 0.10 0.917 -7.485084 8.322056 Maturity -.8569515 .0590022 -14.52 0.000 -.9725937 -.7413093 Rating .8619821 .4412752 1.95 0.051 -.0029015 1.726866 size 2.004248 2.499591 0.80 0.423 -2.89486 6.903357 Yieldspread Coef. Std. Err. z P>|z| [95% Conf. Interval] corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000 Wald chi2(6) = 213.65 overall = 0.0587 max = 305 between = 0.0651 avg = 305.0 R-sq: within = 0.0114 Obs per group: min = 305 Group variable: ising Number of groups = 61 Random-effects GLS regression Number of obs = 18605
G. Correlation matrix
H. Variance inflation factor
I. Woolridge rest for autocorrelation in panel data
coupon -0.2502 0.4133 0.3706 0.1307 -0.2305 1.0000 Issuer -0.1352 -0.5203 -0.0583 -0.5649 1.0000 cur 0.5046 0.3032 0.1677 1.0000 Maturity 0.0664 0.2658 1.0000 Rating -0.1034 1.0000 size 1.0000 size Rating Maturity cur Issuer coupon
Mean VIF 1.68 Maturity 1.26 0.795999 coupon 1.47 0.679869 size 1.66 0.603443 Rating 1.66 0.602686 Issuer 1.90 0.526715 cur 2.13 0.470000 Variable VIF 1/VIF
.
Prob > F = 0.0000 F( 1, 60) = 110.323 H0: no first-order autocorrelation
J. Random effects with AR (1) disturbances regression
.
theta .58064261
rho_fov .99945484 (fraction of variance due to u_i) sigma_e .17907632
sigma_u 7.6675656
rho_ar .99895352 (estimated autocorrelation coefficient)
_cons -34.69507 52.23711 -0.66 0.507 -137.0779 67.68778 coupon -.3839965 1.043868 -0.37 0.713 -2.429941 1.661948 Issuer 2.322108 4.181029 0.56 0.579 -5.872558 10.51677 cur .233636 4.257632 0.05 0.956 -8.111169 8.578441 Maturity -.6435928 .2147899 -3.00 0.003 -1.064573 -.2226123 Rating .8620281 .4702586 1.83 0.067 -.0596617 1.783718 size 1.809156 2.650393 0.68 0.495 -3.385519 7.003831 Yieldspread Coef. Std. Err. z P>|z| [95% Conf. Interval] corr(u_i, Xb) = 0 (assumed) Prob > chi2 = 0.0999 Wald chi2(7) = 12.02 overall = 0.0661 max = 305 between = 0.0735 avg = 305.0 R-sq: within = 0.0114 Obs per group: min = 305 Group variable: ising Number of groups = 61 RE GLS regression with AR(1) disturbances Number of obs = 18605
K. Robust regression
rho .89147516 (fraction of variance due to u_i)
sigma_e 2.7952256 sigma_u 8.0113723 _cons -38.25597 49.08693 -0.78 0.436 -134.4646 57.95266 coupon .015359 .6514967 0.02 0.981 -1.261551 1.292269 Issuer 2.788943 3.784827 0.74 0.461 -4.629181 10.20707 cur .4184856 1.833847 0.23 0.819 -3.175789 4.01276 Maturity -.8569515 .9321105 -0.92 0.358 -2.683854 .9699514 Rating .8619821 .8794833 0.98 0.327 -.8617736 2.585738 size 2.004248 2.486822 0.81 0.420 -2.869834 6.878331 Yieldspread Coef. Std. Err. z P>|z| [95% Conf. Interval] Robust
(Std. Err. adjusted for 61 clusters in ising) corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.9785 Wald chi2(6) = 1.17 overall = 0.0587 max = 305 between = 0.0651 avg = 305.0 R-sq: within = 0.0114 Obs per group: min = 305 Group variable: ising Number of groups = 61 Random-effects GLS regression Number of obs = 18605
L. Correlation matrix
M. Variance inflation factor
N. Jensen’s Alpha Regression
O. Ramsey test Issuer 0.0068 -0.1238 -0.5273 -0.0579 -0.2256 -0.5649 1.0000 cur -0.0073 0.4622 0.3021 0.1664 0.1279 1.0000 coupon 0.0001 -0.2283 0.4041 0.3631 1.0000 Maturity -0.0096 0.0849 0.2673 1.0000 rating -0.0027 -0.0946 1.0000 size -0.0064 1.0000 Gbdummy 1.0000 Gbdummy size rating Maturity coupon cur Issuer
Mean VIF 1.53 Gbdummy 1.00 0.999824 Maturity 1.26 0.795589 coupon 1.42 0.702564 size 1.51 0.662289 rating 1.66 0.601180 Issuer 1.92 0.521511 cur 1.97 0.507011 Variable VIF 1/VIF
_cons .5544674 .3198361 1.73 0.086 -.0791255 1.18806 Issuer -.0337762 .0294349 -1.15 0.254 -.0920864 .0245341 cur .0529627 .0285972 1.85 0.067 -.0036881 .1096135 coupon .1053219 .0071398 14.75 0.000 .0911779 .1194658 Maturity -.0006289 .0018306 -0.34 0.732 -.0042554 .0029975 rating -.0089788 .0033032 -2.72 0.008 -.0155223 -.0024353 size -.0200816 .0162043 -1.24 0.218 -.0521823 .0120191 Gbdummy .012484 .0150804 0.83 0.409 -.0173902 .0423582 alphahat Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2.90267201 121 .023989025 Root MSE = .08327 Adj R-squared = 0.7110 Residual .790386032 114 .006933211 R-squared = 0.7277 Model 2.11228597 7 .301755139 Prob > F = 0.0000 F( 7, 114) = 43.52 Source SS df MS Number of obs = 122
.
Prob > F = 0.3371
F(3, 112) = 1.14 Ho: model has no omitted variables
P. Breach-Pagan / Cook-Weisberg test for heteroskedasticity
Q. Robust regression Prob > chi2 = 0.0172 chi2(1) = 5.68
Variables: fitted values of alphahat Ho: Constant variance
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
. _cons .5544674 .2600334 2.13 0.035 .0393432 1.069592 Issuer -.0337762 .0328905 -1.03 0.307 -.0989321 .0313798 cur .0529627 .0268216 1.97 0.051 -.0001707 .1060961 coupon .1053219 .0066217 15.91 0.000 .0922043 .1184394 Maturity -.0006289 .0019324 -0.33 0.745 -.004457 .0031991 rating -.0089788 .0039076 -2.30 0.023 -.0167198 -.0012378 size -.0200816 .0130859 -1.53 0.128 -.0460047 .0058415 Gbdummy .012484 .0150317 0.83 0.408 -.0172937 .0422618 alphahat Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust Root MSE = .08327 R-squared = 0.7277 Prob > F = 0.0000 F( 7, 114) = 70.89 Linear regression Number of obs = 122
