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Sustainability in Portfolio Optimization
Author: Chamika Porage
(880926)
Autumn 2020
Statistics, Second Cycle and 15.0hp
Subject: Independent Project I, ST433A
Örebro University School of Business
Supervisor: Olha Bodnar
Examiner: Ann-Marie Flygare
i
Abstract
Sustainability has become a global trend due to the remarkable growth of the
demand for sustainable practices when controlling the risks associated with ESG
(Environmental, Social and Governance) aspects worldwide. Hence, the socially
responsible investors (or ESG investors) are curious to know whether investing in
portfolios containing sustainable assets would create better investment
opportunities compared to the portfolios consisted of both sustainable and non-
sustainable assets. In this study, mean-variance spanning tests that are based on
classical optimal portfolio theory, are performed to examine whether there is a
statistically significant difference between the minimum variance frontier of
sustainable stocks and the minimum variance frontier comprising both the
sustainable and non-sustainable stocks in OMXC25 index. For the analysis,
weekly returns of fifteen stocks of OMXC25 index are used over a three year
period from 2017 to 2019. Statistically significant results of the performed mean-
variance spanning tests suggest that inclusion of unsustainable stocks will
statistically significantly improve the performance of the portfolio compared with
a portfolio of sustainable stocks solely in the OMXC25 index of the Danish
capital market.
ii
Preface
This is a master‟s degree thesis worth 15 credits (ECTS) at Örebro University.
I am extremely grateful to my supervisor Olha Bodnar for her excellent guidance,
advices, and support during this process and for introducing this project to me.
iii
Abbreviations
CML Capital Market Line
ESG Environmental, Social and Governance
ETF Exchange Traded Fund
LM Lagrange Multiplier
LR Likelihood Ratio
MVP Minimum-Variance Portfolio
OMXC25 OMX Copenhagen 25
OMXS30 OMX Stockholm 30
SR Socially Responsible
SRI Socially Responsible Investment
TP Tangency Portfolio
UK United Kingdom
US United States
W Wald
iv
Contents
List of Tables ................................................................................................................................................ v
List of Figures ............................................................................................................................................... v
1 Introduction ................................................................................................................................................ 1
1.1 Background ................................................................................................................................... 1
1.2 Thesis objective and project outline .............................................................................................. 2
2 Literature Review ....................................................................................................................................... 3
3 Theoretical framework and methodology .................................................................................................. 4
3.1 Optimal portfolio theory ..................................................................................................................... 4
3.1.1 Minimum Variance portfolio ....................................................................................................... 5
3.1.2 Tangency portfolio ....................................................................................................................... 5
3.1.3 Two fund-Theorem ...................................................................................................................... 6
3.2 Sustainability....................................................................................................................................... 6
3.3 Mean-variance spanning tests ............................................................................................................. 7
3.3.1 The multivariate regression model ............................................................................................... 7
3.3.2 Model assumptions of the multivariate regression model ............................................................ 8
3.3.3 The maximum likelihood estimators of B and .......................................................................... 8
3.3.4 Hypothesis testing ........................................................................................................................ 9
3.3.5 Asymptotic mean-variance spanning tests ................................................................................... 9
3.3.6 Spanning test for small samples ................................................................................................ 10
3.3.7 The goodness-of-fit tests for multivariate normality. ................................................................ 10
3.4 Simulation study for robustness analysis .......................................................................................... 10
4 Data description ....................................................................................................................................... 11
5 Analysis and results ................................................................................................................................. 14
5.1 Screening........................................................................................................................................... 14
5.2 Results of mean-variance spanning tests .......................................................................................... 15
5.3 Checking for normality ..................................................................................................................... 18
5.4 Simulation study for robustness analysis .......................................................................................... 20
6 General discussion and conclusion .......................................................................................................... 22
6.1 Limitations of the study .................................................................................................................... 23
6.2 Improvements/ suggestions for future research ................................................................................ 23
v
7 References ................................................................................................................................................ 24
A Appendix ................................................................................................................................................. 26
List of Tables
Table 1 : Company, sector, ticker symbol, total ESG Score and total ESG risk rating of twenty five
stocks of OMCX25 index. .......................................................................................................................... 12
Table 2 : Number of test and benchmark assets under four screening criteria ........................................... 15
Table 3 : The test statistics and their corresponding p-values of Likelihood Ratio test (LR-test), Wald test
(W-test) and Lagrange Multiplier test (LM-test) on the weekly returns in OMCX25 index for three year
period from 2017 to 2019. .......................................................................................................................... 15
Table 4 : The p-values of asymptotic and exact mean-variance spanning tests on the weekly returns for
different values of T, N, K.. ...................................................................................................................... 17
Table 5 : Test statistics and p-values of the generalized Shapiro-Wilk‟s test and Mardia‟s test for
normality of weekly returns. ....................................................................................................................... 18
Table 6 : The probabilities of rejecting null hypothesis when H0 is true for the asymptotic mean-variance
tests, for normally distributed and t-distributed residuals using 1000 simulations for T=157 weekly
returns. ........................................................................................................................................................ 20
List of Figures
Figure 1 : The plot (a) indicates the distribution of total ESG score of fifteen stocks of OMXC25 index
and the plot (b) shows the number of assets falling under five categories of the total ESG risk ratings. ... 14
Figure 2 : Chi-squared QQ-plots of residuals from regression models for checking normality ................. 19
Figure 3 : Power curves for three spanning tests for rejecting null hypothesis when H0 is true for
significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of
multivariate normal distribution and the right power plot depicts the simulation from multivariate t-
distribution .................................................................................................................................................. 21
Figure 4 : Power curves for three spanning tests for rejecting null hypothesis when H1 is true for
significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of
multivariate normal distribution and the right power plot depicts the simulation from multivariate t-
distribution .................................................................................................................................................. 21
1
1 Introduction
1.1 Background
Harry Markowitz, the pioneer of the modern portfolio theory, theorized the construction of
optimal portfolio on risk-averse investors. Portfolio optimization is defined as the maximization
of the expected return for particular level of market risk or minimization of the risk for particular
level of market return. “Do not put all your eggs in one basket” is one of the famous phrases that
financial advisers give their investors as the first advice on portfolio diversification.
Diversification strategy minimizes the risk associated with portfolio construction.
Sustainability has become an increasingly significant aspect among global investors due to
increasing pressure and growing community concerns on issues such as climate conditions, labor
condition and human rights, energy use and conservation, waste management, cultural heritage,
natural resources etc. As a result, asset managers and asset owners are making allowances for use
of Environmental, Social and Governance (ESG) information or so called three dimensions of
Socially Responsible (SR) criteria when making their management and investment decisions in
portfolio construction.
The global trend towards “Responsible” and “Sustainable” finance has changed the attitude of
investors to balance the risk and return of their investment by “doing the right thing”. As stated
by KPMG in 2019, the estimated value of $30tn sustainable investing assets are managed
globally and it is 34% increase in two years. Further, it revealed $78bn of net inflow in
worldwide ESG strategies in 2018 and $400bn growth in ESG ETFs was forecasted over the next
decade. Moreover, it was indicated that Europe had 48.8% of sustainable investment relative to
total managed assets in 2018.
“Every dark cloud has the silver lining” is the best quote to describe the global financial crisis in
2008 and its impact on the rapid growth of ESG related sustainable investment around the
world during the last decade. Before 2008, many financial markets were not interested with ESG
principles. However, those financial institutions were accused after the crisis and they were
strongly requested to revise the capital allocation. As a result, investors were attracted to the
2
responsible and sustainable investment as reasonable number of long term investors desired to
mitigate risk associated with standard ESG performance.
As financial experts predicted, the current COVID-19 crisis would be a milestone of sustainable
investing due to the unstoppable interest of investors towards the ESG investing. Therefore it is
vital for investors to examine whether there are opportunities to gain positive returns on
sustainable investment.
1.2 Thesis objective and project outline
The primary objective of this study is to investigate whether the returns are statistically
significantly different between more sustainable stocks and all other stocks in OMXC25 index.
The findings of the study would provide contribution not only for the asset managers and asset
owners but also for the academic literature on financial markets.
The outline of the thesis is structured as follows. Analysis and summary of previous research on
this subject is discussed under the literature review in chapter 2.
Chapter 3 explains the theoretical framework and the methodology of the thesis. The total ESG
scores and total risk ratings from Sustainalytics are used to identify the sustainable stocks and the
Minimum Variance Portfolio (MVP) is used to reach the optimal portfolio.
Chapter 4 consists of description of data, the source of the data and variable definitions. For this
task, weekly stock returns in OMXC25 index are collected from Yahoo finance over three year
period from 2017 to 2019.
Lastly, the readers may find interesting analysis and results in Chapter 5 and the final chapter of
the thesis is a general discussion and conclusions. Thesis findings are summarized with the
limitations of the study and suggested improvements for future research.
3
2 Literature Review
Socially responsible investment (SRI) has become one of the very fashionable industries not only
among the investors but also the academics all over the world due to its growth sensitivity to
ESG issues. Accordingly, many researchers on this subject have set their main objective to
compare the financial performance between ethical and conventional funds.
Bauer, Koedijk, and Otten (2002) applied a multi-factor model of Carhart (1997) to identify
slightly noticeable significant differences in risk-adjusted returns between ethical and
conventional funds for the period 1990-2001 for a database consisting 103 German, UK and US
ethical mutual funds. Additionally, they found that these differences were insignificant after
controlling for common factors such as size, book to market and momentum. When explaining
the fund performance, they illustrated that the performance of ethical indices were worse than the
standard indices and “German and US funds under-perform while UK ethical funds out-perform
their conventional peers.”
Ortas, Moneva, and Salvador (2012) conducted their study in order to investigate whether social
and environmental processes influence the performance of SRI indexes compared to their official
benchmarks in terms of risk-adjusted returns and systematic risk levels. They found that
differences obtained by the SRI indexes and benchmarks are statistically insignificant. Moreover,
compared to the benchmarks, the results depicts that higher systematic risk were associated with
SRI indexes and that higher systematic risk is influenced by higher screening intensity.
As demonstrated in Pena and Cue Corte‟s (2017) study, there is no financial return sacrifice with
ethical investment policies. In their study, they considered 330 US and European SRI mutual
funds over the period 2003-2014 to identify the relationship between risk-adjusted performance
and screening strategies. Furthermore, the results depict a curvilinear relationship between
screening intensity and performance in US and Scandinavian mutual funds. According to the
study, it is evident that types of screening activities significantly influence the risk-adjusted
return. Finally, they concluded that positive screening impacts lower performance of the global
sample and for US funds since the funds are financially affected by meeting superior ESG
standards.
4
Anane (2019) has applied Markowitz mean-variance framework for a dataset consisting of 30
constituents of Dow Jones Industrial Average stocks over the period 2008-2018 and identified
that portfolio returns are higher for the selected stocks based on environmental and governance
ratings.
However, Wong‟s (2020) study is the most related and recent literature for this thesis. The
researcher has used mean-variance spanning tests for data consisting of weekly and monthly
returns of 21 constituents in OMX Stockholm 30 (OMXS30) index over the period 2008-2019.
The aim of the study was to investigate “..whether the differences between the efficient frontiers
of considered more sustainable assets and the efficient frontier of all 21 assets are statistically
significant.” After applying the mean-variance spanning tests with four screenings, statistically
non -significant differences between the efficient frontiers was obtained and the study concludes
that there is no need to include unsustainable assets into portfolio. A similar performance can be
reached when the portfolio consists of sustainable assets only.
3 Theoretical framework and methodology
This chapter comprises two sections. In the first section, theoretical background and some
concepts of the thesis are discussed and the second section contains the method that is used in the
analysis.
3.1 Optimal portfolio theory
This thesis is mainly based on optimal portfolio theory. An investor may always look into a
portfolio, a combination of financial investments containing both risky and risk-free assets such
as stocks, bonds, mutual funds etc that maximize expected return with minimum risk level.
Hence, the process of analyzing and assessing the assets in portfolio is known as “the portfolio
analysis” and a collection of risky or risk-free financial assets that produce the maximum
expected return for a given level of acceptable risk is referred to as “the optimal portfolio”. The
“efficient frontier” is the combination of optimal portfolios which are expected to offer the
highest returns for the given level of risk. Generally, the portfolio risk is measured by the
volatility of returns. Therefore, investors are interested to examine the optimal portfolio through
mean-variance analysis.
5
3.1.1 Minimum Variance portfolio
Minimum Variance Portfolio (MVP) is the portfolio of risky assets with the lowest variance for
the rate of expected return among all feasible portfolios.
Let us consider the portfolio of N risky assets.
Thus the vector of portfolio weights can be written as
w = (w1, w2, w3,…….wN)T , where wi indicates the weight of i
th risky asset.
Since the sum of the weights of N risky assets is 1,
wT1N=1, where 1N is a size N vector of ones.
The vector of expected returns of N risky asset is
µ = (µ1, µ2, µ3, …….µN)T
and the covariance matrix of the N risky assets
V=(
)
where Vij indicates the covariance between the returns of the ith
and jth
risky assets for
i=1,2,3,…,N and j=1,2,3…,N and V is symmetric and positive semi-definite matrix.
To find the MVP, the minimization problem is constructed as
subject to the constraint wT1N =1
The minimization problem is solved using the Lagrange multiplier method and the weights of the
MVP is obtained as
3.1.2 Tangency portfolio
The tangency portfolio (TP) is important with the presence of the risk free assets in the portfolio.
Generally, combinations of the tangency portfolio with risk free assets are selected by investors
that use mean-variance analysis. The capital market line (CML) graphically reflects the
portfolios that optimally combine risk and return and the slope of the CML is referred to as the
6
Sharp ratio. The portfolio of risky assets that maximizes the Sharp ratio is called the tangency
portfolio or in other words the most efficient portfolio, TP is created as a result of the intercept
(0, Rf) of CML and the efficient frontier. The expected return of the risk free assets is
symbolized as Rf and its standard deviation is zero.
Let us assume the feasible portfolio P of N risky assets and and are the expected return and
the standard deviation of portfolio P respectively. And also µ, w and V are the vector of expected
return, the vector of the weights and the covariance matrix respectively, as mentioned in the
3.1.1 section. Then the Sharp ratio is,
=
√
To find the weights of TP, the maximization problem is constructed as
√ subject to the constraint w
T1N =1
After solving the maximization problem, the weights of the TP is obtained as
3.1.3 Two fund-Theorem
Let the weights of the two different portfolios constructed by N risky assets that lie on the
minimum- variance frontier be two vectors of size N and denoted by w1 and w2. The expected
return of two portfolios are denoted µ1 and µ2, where µ1 ≠ µ2. Then w3, the weights of the third
portfolio that lies on the minimum frontier with expected return µ3 can be obtained by the
weights (w1 and w2) and the expected returns (µ1 and µ2) of two known portfolios by the
following formula
where ⁄
3.2 Sustainability
The concept of sustainability is mainly comprised of Environmental, Social and Governance
(ESG) factors. Socially responsible investors use the ESG concept to screen their investment.
7
Environmental (E): The environmental criteria emphasize the protection of environment.
Furthermore, it discusses the company‟s attitude and actions towards environmental risks
associated with pollution and waste, climatic changes, natural resources, energy
efficiency etc.
Social (S): The social criteria concentrate on people and relationships. Civil rights, labour
standards, Community affairs, Gender and diversity are among the social characteristics
of ESG factors
Governance (G): The Governance criterion consists of standards such as board
composition, audit committee, Executive compensation, bribery and corruption etc for
managing a company.
Socially responsible investors, or in other words ESG investors consider that company‟s ESG
information is important in portfolio construction.
3.3 Mean-variance spanning tests
Kan and Zhou‟s (2012) research paper on mean-variance spanning tests is the main reference for
this section and the notations and the technique mentioned by Kan and Zhou‟s in their paper are
closely followed in this thesis.
3.3.1 The multivariate regression model
Let us assume a dataset of returns of Y, of K+N assets over T time points. Then the regression
model
Y= XB + E
where, Y=[
] , X=[
] , B= [
] , E=[
]
The matrix Y is a matrix of the size TxN of the returns of N test assets over time points
t=1,2,….,T. The matrix X is a matrix of size Tx(K+1), the first column consists of ones and Rt
denotes the other columns that contain the returns of K benchmark assets, for t =1,2,….,T. Also,
B= [α,β]T is the parameter matrix of size (K+1) x N where α and β are constants over time, α is a
8
vector containing N intercepts and is a KxN matrix. Moreover, E is a TxN matrix of
disturbances, where is a vector of disturbances at time point t for t =1,2,….,T.
3.3.2 Model assumptions of the multivariate regression model
1. T ≥ N+K+1, this implies the existence of an inverse of the matrix XTX
2. The disturbances for t =1,2,….,T, conditioned on the returns of K benchmark assets
Rt for t =1,2,….,T are independent and identically distributed (i.i.d) and follow
N(0N, ) that is multivariate normal distribution with a mean vector comprising only
zeros, of size N and a covariance matrix . Thus, E is distributed as N(0TxN,IT ⊗ ),
where 0TxN is a zero matrix of size TxN and ⊗ indicates the Kronecker product.
3.3.3 The maximum likelihood estimators of B and
The unconstrained maximum likelihood estimators of B and matrices can be obtained by:
[ ]
= [ ]
( )
( )
The distribution of the maximum likelihood estimator of B conditioned on the K benchmark
assets is obtained as follows, assuming that the disturbances, are i.i.d and follow N(0N, )
|X ~ N [B, ⊗ ]
and the vectorization of T conditioned on X is also normally distributed as
vec( T)|X ~ N [vec (B
T), ⊗ ]
Additionally, under the normality assumption, T follows N-dimentional central Wishart
distribution with T-K-1 degrees of freedom and covariance matrix ,
T WN (T-K-1, )
9
3.3.4 Hypothesis testing
H0: There is no difference between the minimum-variance frontier of the K+N assets and the
minimum variance frontier of K assets,
H0:
where [ ] and and = 0 imply testing that TP and MVP of
the K+N assets have zero weights in N assets, respectively.
Let us write , where [
] is a matrix of size 2x(K+1) and [
] is a
matrix of size 2xN.
The estimator = A + C = [ ] produces the maximum likelihood estimator of ,
and the vectorization of T is normally distributed as
vec ( T)| X ~ N [vec ( ),( ⊗ ] with = TA A
T
Moreover, the matrix H is defined as
Let be the two eigen values of the matrix where ≥ 0.
3.3.5 Asymptotic mean-variance spanning tests
The following tests are taken from Kan and Zhou‟s (2012) study.
The asymptotic likelihood ratio test (LR- test) can be expressed as
LR = T [ ] = T ∑
Secondly, the asymptotic Wald test (W- test) can be expressed as
W = T
Thirdly, the asymptotic Lagrange multiplier test (LM-test) can be obtained from
LM = T ∑
10
3.3.6 Spanning test for small samples
If the sample is not sufficiently large, the results of asymptotic tests will be misleading. If so, a
monotone transformation of the likelihood ratio test is proposed.
Hence, when N 2 and T > K+N, the exact distribution of the test statistic under H0 is F-
distribution with the degrees of freedom 2N and 2(T-K-N).
(√ ) (
) ~F 2N, 2(T-K-N )
3.3.7 The goodness-of-fit tests for multivariate normality
The normality assumption of the multivariate regression model is examined using the
generalized Shapiro-Wilk‟s test, Mardia‟s test and graphical chi-squared QQ-plots.
The null- hypothesis of the generalized Shapiro-Wilk‟s test is that the tested data are normally
distributed and Mardia‟s test checks for both multivariate skewness and multivariate kurtosis. In
R, the two packages referred to as „mvShapiro test‟ and „MVN‟ perform the generalized Shapiro-
Wilk‟s test and Mardia‟s test, respectively. The chi-squared QQ-plot is a graphical approach to
check multivariate normality and the plot should reflect approximately a straight line, if the data
are multivariate normal.
3.4 Simulation study for robustness analysis
In this thesis, a simulation study is designed to examine the robustness of non-normality of the
asymptotic mean-variance spanning tests. The residuals are simulated from a multivariate normal
distribution and a multivariate t distribution in order to study the deviation of the performance of
spanning tests for i.i.d but non-normally distributed residuals.
A simulated sample of E and ϵt is defined as E* and ϵt*, respectively for t=1,2…..T and the steps
of the simulation procedure are as follows. It should be noted that the steps 2 to 6 are same for
both the simulations from multivariate normal and multivariate t-distribution.
1. Firstly, each row of ϵt* for t=1,2…..T, is simulated from N (0N, and from the
multivariate t- distribution with five degrees of freedom and covariance matrix to
produce the matrix E* for both multivariate normal and multivariate t-distribution,
respectively. can be any Nx N positive semi-definite matrix and in this study Σ is
11
defined as a matrix with same correlation ( and standard deviations drawn from
the Uniform (0.1,0.5) distribution.
2. Using the formula Y*= XB+E*, the new returns of the test assets (Y*) are obtained. In
this study, the matrix X is generated from the normal/t-distribution in each simulation run
and the matrix B of size Nx(K+1) is drawn from Uniform distribution and then it is
modified to the null hypothesis, that is the first row should be zero and the column sum
should be the vector of ones.
3. As mentioned in section 3.3.3, the maximum likelihood estimators of B and are
computed using X and Y*. The new eigen values are obtained as in section 3.3.4.
4. The asymptotic spanning tests are performed using the obtained eigen values (refer
section 3.3.5)
5. Steps 1 to 4 are repeated to have 1000 simulations.
6. For each spanning test, the number of times that the null hypothesis is rejected at 5%
significance level is counted. Then the each count is divided by 1000 to get the actual
probabilities of rejecting the null hypothesis when H0 is true.
4 Data description
In this thesis, primarily, a dataset of weekly returns of fifteen stocks in OMXC25 index is used
for three year period from January 2017 to December 2019. The OMXC25 index is a market-
weighted price index, which contains the largest and most traded stocks on Nasdaq Nordic
Exchange Copenhagen.
Firstly, a dataset containing total ESG scores and the total ESG risk ratings for twenty five
companies of OMXC25 index is obtained from Yahoo Finance.
12
Table 1 : Company, sector, ticker symbol, total ESG Score and total ESG risk rating of twenty
five stocks of OMCX25 index.
Company Sector Ticker Symbol Total ESG
score
Total
ESG risk
rating
1 AMBU Health care AMBU-B NA NA
2 Bavarian Nordic Health care BAVA NA NA
3 Carlsberg Group Food, beverage, tobacco CARL-B 18 Low
4 Chr. Hansen Health care CHR 19 Low
5 Coloplast Health care COLO-B 13 Low
6 Danske Bank Banks DANSKE 29 Medium
7 Demant Health care DEMANT NA NA
8 DSV Panalpina Industrial goods and services DSV 16 Low
9 FLSmidth & Co. Construction and material FLS NA NA
10 Genmab Health care GMAB 27 Medium
11 GN Store Nord Health care GN NA NA
12 ISS Industrial goods and services ISS 15 Low
13 Jyske Bank A/S Banks JYSK NA NA
14 Lundbeck Health care LUN 23 Medium
15 Maersk (class A) Industrial goods and services MAERSK-A NA NA
16 Maersk (class B) Industrial goods and services MAERSK-B 22 Medium
17 Novo Nordisk Health care NOVO-B 22 Medium
18 Novozymes Health care NZYM-B 21 Medium
19 Orsted Utilities ORSTED 21 Medium
20 Pandora A/S Consumer products and services PNDORA 12 Low
21 Royal Unibrew Food, beverage, tobacco RBREW NA NA
22 Rockwool International Construction and material ROCK-B NA NA
23 SimCorp Technology SIM NA NA
24 Tryg Insurance TRYG 21 Medium
25 Vestas Wind Systems A/S Energy VWS 16 Low
According to table 1, it is observed that twenty five stocks of OMXC25 belong to twenty four
distinct companies since the company Maersk has a Class A share and a Class B share. And also
the majority, 40% of the stocks of OMXC25 belongs to the health care sector.
The Yahoo finance provides company‟s total ESG score and total ESG ratings which is powered
by Sustainalytics, the prominent independent global provider of ESG research and ratings.
Company‟s total ESG score is an overall score and it measures the unmanaged risk level of the
13
company. The range of total ESG score is from 0 to 100, with 0 indicating no unmanaged ESG
risks and 100 indicating the highest unmanaged risk level driven by ESG factors. Therefore, it is
obvious that a company with lower ESG score attracts investors who are interested in sustainable
investing.
Moreover, the five categories of the total ESG risk ratings is created using the total ESG score as
a company with ESG score; 0-10 denotes negligible risk, 10-20 low risk, 20-30 medium risk,
30-40 high risk and the final category 40-100 represents severe risk. However, ten stocks are
excluded after some data cleaning due to missing values of company‟s ESG information in
Yahoo Finance and the final dataset of weekly return is consists of 15 stocks out of 25 stocks of
OMXC25 index.
Secondly, the weekly closing prices for fifteen stocks from 2017 to 2019 are obtained from
Yahoo finance to calculate weekly returns. Each stock is consisted of 157 observations for given
three year time period.
Furthermore, the asset based screening procedure and mean-variance spanning tests are used to
analyze if there exist a statistically significant difference between the efficient frontier of the
sustainable stocks and the efficient frontier of all fifteen stocks. The screening method contains
four categories as 10%, 25%, 50% and one other category refer to as “All”. The stocks are
screened based on the 90th
, 75th
and 50th
percentiles of total ESG scores to obtain 10%, 25% and
50% screening respectively. The test assets of the 10% screening are observed by screening all
the stocks with greater than the 90th
percentile of the total ESG score while the remaining stocks
are considered as benchmark assets. Therefore, it is obvious that those benchmark assets are
more sustainable compared to the test assets. Likewise all the stocks with higher than 75th
and
50th
percentiles of total ESG score are screened as the test assets and the rest of the stocks are set
as benchmark assets of 25% and 50% screening respectively. Moreover, the stocks are screened
on the total ESG risk rating to obtain the forth screening category which is referring to “All”.
The purpose of introducing “All” category is to perform screening on all fifteen stocks with the
stocks from medium to severe total ESG risk rating. In this category, all the stocks with a total
ESG risk rating from medium to severe are screened as test assets and the remaining assets
(assets with negligible and low total ESG risk rating) are set as benchmark assets.
14
Additionally, some notations are used throughout the thesis. The number of test assets and the
number of benchmark assets are indicated as N and K respectively, and T denotes the length of
the time series.
The statistical software R (version 3.6.1) is used for data analysis in this study (R codes used in
the thesis are available in Appendices).
5 Analysis and results
5.1 Screening
In this thesis, the screening is based on the total ESG score and the total ESG risk rating.
a.) Total ESG score
b.) Total ESG risk rating
Figure 1 : The plot (a) indicates the distribution of total ESG score of fifteen stocks of OMXC25
index and the plot (b) shows the number of assets falling under five categories of the total ESG
risk ratings.
0
1
2
3
4
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Num
ber
of
asse
ts
Total ESG score
0
2
4
6
8
10
Negligible Low Medium High Severe
Num
ber
of
asse
ts
Total ESG risk ratings
15
The plot (a) illustrates the range of the distribution of total ESG scores, which is from 12 to 29.
From plot (b), it can be seen that the total ESG risk ratings are from low to medium for all fifteen
stocks and there is no stock belongs to negligible, high or severe ESG risk ratings.
The 10% screening is based on the assets higher than the 90th
percentile of the distribution of
total ESG scores. Hence, there are two test assets for 10% screening as the 90th
percentile of the
distribution of total ESG score is 25.4. Likewise, 75th
, 50th
and 25th
percentile of the distribution
of total ESG score are 22, 21 and 16 respectively. Table 2 reveals the number of test and
benchmark assets falling under four screening criteria used throughout this thesis.
Table 2 : Number of test and benchmark assets under four screening criteria
Screening Number of assets
N (Test ) K ( Benchmark)
10% 2 13
25% 3 12
50% 7 8
All 8 7
The results of three asymptotic mean-variance spanning tests on the weekly returns in
OMXC25 index are summarized in table 3.
5.2 Results of mean-variance spanning tests
Table 3 : The test statistics and their corresponding p-values of Likelihood Ratio test (LR-test),
Wald test (W-test) and Lagrange Multiplier test (LM-test) on the weekly returns in OMCX25
index for three year period from 2017 to 2019.
Asymptotic tests
LR-test W-test LM- test
Screening N K T LR p-value W p-value LM p-value
10% 2 13 157 12.66 0.0131 13.11 0.0108 12.24 0.0157
25% 3 12 157 13.48 0.0360 13.99 0.0298 13.01 0.0429
50% 7 8 157 26.55 0.0220 28.13 0.0137 25.10 0.0335
All 8 7 157 49.76 2.50E-05 55.89 2.53E-06 44.58 0.0002
16
According to table 3, it is interesting to note that all three asymptotic tests on 10%, 25% and 50%
screening are statistically significant at 5% level or in other words statistically significant
difference between two efficient frontiers are discovered. Additionally, when all unsustainable
stocks (stocks with medium, high or severe total ESG risk rating) are screened, the null
hypothesis is rejected at 1% level of significance indicating that the seven sustainable stocks do
not span the fifteen assets at any significance level. Thus, it is evident that the difference between
the minimum-variance frontier of all the fifteen stocks and each of the minimum-variance
frontiers of the smaller number of stocks which are considered as more sustainable are highly
statistically significant in OMXC25 index. The rejection of the null hypothesis means that it is
not enough to consider only sustainable assets in the Danish capital market. The inclusion of
unsustainable stocks will statistically significantly improve the performance of the portfolio. This
is an interesting result, since the opposite was observed in the Swedish capital market where it is
enough to invest into the sustainable assets only (Wong 2020).
Furthermore, three asymptotic mean-variance spanning tests and an exact mean-variance
spanning LR test are performed on fifteen stocks of different dimensions and different shorter
series of observations. In table 4, three series of observations of 50, 100 and 157 are used and
exact mean- variance spanning LR test would be more appropriate in order to obtain accurate
results for fairly small samples as well. To construct the first row of table 4, initially, two stocks
with the highest ESG score are excluded as test assets. Secondly, the first two stocks from the
remaining thirteen stocks are obtained as benchmark assets (K=2). Then four spanning tests are
performed over three series of observations of 50,100 and 157.
17
Table 4 : The p-values of asymptotic and exact mean-variance spanning tests on weekly returns
for different values of T, N, K.
Asymptotic tests Exact test
N K T LR W LM LR
2
2
50 0.001 0.000 0.003 0.002
100 4.96E-05 1.02E-05 0.000 7.45E-05
157 9.19E-09 4.08E-10 1.21E-07 1.46E-08
5
50 0.011 0.006 0.018 0.023
100 0.019 0.014 0.024 0.026
157 8.47E-05 3.52E-05 0.000 0.000
10
50 0.011 0.006 0.020 0.041
100 0.021 0.016 0.026 0.036
157 0.002 0.002 0.003 0.004
13
50 0.072 0.054 0.092 0.192
100 0.037 0.032 0.043 0.069
157 0.013 0.011 0.016 0.022
5
2
50 0.001 1.46E-05 0.008 0.002
100 0.001 0.000 0.004 0.002
157 8.48E-08 1.46E-09 2.04E-06 1.74E-07
5
50 0.001 0.000 0.009 0.007
100 0.019 0.010 0.034 0.034
157 0.000 0.000 0.001 0.001
10
50 0.229 0.176 0.287 0.482
100 0.253 0.225 0.281 0.367
157 0.184 0.160 0.208 0.245
10
2
50 3.57E-09 2.92E-26 0.001 4.52E-07
100 1.25E-12 4.70E-23 2.55E-07 2.95E-11
157 2.39E-22 6.31E-40 3.16E-13 6.41E-21
5
50 3.70E-05 4.09E-11 0.013 0.002
100 8.63E-08 3.77E-12 3.78E-05 1.51E-06
157 9.75E-15 8.70E-23 8.26E-10 2.33E-13
As shown in table 4, when considering N=2, all four spanning tests for all dimensions over three
series of observations are statistically significant at 5% level except two situations. When K=13
and T=100 the exact LR test is significant at 10% level and when K=13 and T=50 all the
asymptotic mean-variance spanning tests are statistically significant at 10% level while the exact
18
mean-variance spanning test is insignificant. However, the results of asymptotic tests may be
misleading when the sample size is small.
Moreover, when N=5 and K= 10, it is interesting to note that all four spanning tests over three
sample sizes are statistically insignificant. Furthermore, it can be seen that all the tests are highly
statistically significant for all the dimensions over three sample sizes when the number of test
stocks is ten (N=10).
5.3 Checking for normality
When working with models, it is important to verify that the normality assumption of the
residuals in the multivariate regression model since the mean-variance spanning tests are based
on the multivariate regression model. Therefore, the generalized Shapiro-Wilk‟s test, Mardia‟s
test and graphical chi-squared QQ-plots are used to verify the normality assumption. Apart from
the normality, it should also be noted that the residuals are assumed to be independent and
identically distributed, homoscedastic and mean vector comprising only zeros in multivariate
regression model.
Table 5: Test statistics and p-values of the generalized Shapiro-Wilk‟s test and Mardia‟s test for
normality of weekly returns.
Generalized Shapiro-Wilk's
test Mardia's test
Screening Test
statistic p-value
Test statistic
for skewness p-value
Test statistic
for kurtosis p-value
10% 0.9424 < 2.2e-16 999.48 7.24E-43 20.03 0.000
25% 0.9481 < 2.2e-16 726.64 2.29E-26 18.12 0.000
50% 0.9412 < 2.2e-16 324.00 1.14E-20 20.97 0.000
All 0.9318 < 2.2e-16 276.97 1.90E-22 24.04 0.000
As indicated in table 5, both the generalized Shapiro-Wilk‟s test for normality and Mardia test
for both skeweness and kurtosis are highly significant. Hence, the null hypothesis of the
normality is rejected and it can be concluded the presence of non-normal residuals for the used
regression model.
19
(a) 10% screening (b) 25% screening
(c) 50% screening (d) All screening
Figure 2 : Chi-squared QQ-plots of residuals from regression models for checking normality
Furthermore, as per figure 2, it is obvious that the squared Mahalanobis distances of all the Chi-
squared QQ plots are considerably deviated from the straight line and it also confirms the
violation of normality assumption.
Therefore, the simulation study is designed in order to investigate the robustness of the non-
normality of the asymptotic mean-variance spanning tests. In this study, new weekly returns of
the test assets are simulated by simulating residuals from both the multivariate normal and the
multivariate t-distribution.
20
5.4 Simulation study for robustness analysis
Table 6 : The probabilities of rejecting null hypothesis when H0 is true for the asymptotic mean-
variance tests, for normally distributed and t-distributed residuals using 1000 simulations for
T=157 weekly returns.
Multivariate Normal
Distribution
Multivariate t-
Distribution
Screening N K LR W LM LR W LM
10% 2 13 0.074 0.082 0.067 0.075 0.081 0.067
25% 3 12 0.075 0.078 0.070 0.074 0.089 0.067
50% 7 8 0.061 0.091 0.047 0.073 0.093 0.061
All 8 7 0.09 0.129 0.061 0.093 0.119 0.064
As shown in table 6, it is obvious that the probabilities of rejecting null hypothesis under the
multivariate normal distribution are quite similar to corresponding probabilities from
multivariate t-distribution. Also, the probabilities from Wald test are higher compared to the
corresponding Likelihood Ratio test and Lagrange Multiplier test. However, for each asymptotic
spanning test, the probabilities of rejecting null hypothesis when H0 is true are slightly higher
than the expected 5% level of significance, but less than the 10% level of significance except for
the Wald test with all screening. Therefore, it can be concluded that getting the type 1 error is a
bit higher than the expected 5% level of significance level, if the residuals are i.i.d and non-
normal. However, these probabilities of rejecting null hypothesis when H0 is true will be lower if
we increase the number of simulations.
Furthermore, the power curves for three spanning tests are plotted for two scenarios of rejecting
null hypothesis when H0 is true and rejecting null hypothesis when H1 is true at significance
level, α = 0.05. The following plots depict the comparison between the simulations from
multivariate normal and multivariate t-distribution for newly simulated 157 weekly observations
for 10% screening when the number of test asset is 2 and the number of benchmark assets is 13.
21
(a) Multivariate normal distribution (b) Multivariate t-distribution
Figure 3 : Power curves for three spanning tests for rejecting null hypothesis when H0 is true for
significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation
of multivariate normal distribution and the right power plot depicts the simulation from
multivariate t-distribution.
According to figure 3, it is evident that the power curves of three spanning tests are coincided.
The power curve from multivariate normal distribution is a bit smoother than the multivariate t-
distribution.
(a) Multivariate normal distribution (b) Multivariate t-distribution
Figure 4 : Power curves for three spanning tests for rejecting null hypothesis when H1 is true for
significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation
of multivariate normal distribution and the right power plot depicts the simulation from
multivariate t-distribution
22
As indicated from figure 3, figure 4 also reveals the overlapping power curves for three
asymptotic spanning tests for rejecting null hypothesis when H1 is true. Although, the shapes of
curves are quite similar, the curves from the multivariate normal simulation are smoother than
the curves from multivariate t-distribution.
6 General discussion and conclusion
The purpose of this study was to examine whether there is a statistically significant difference, if
an investor would prefer invest in a portfolio containing more sustainable assets than the
portfolio consisting of both sustainable and unsustainable assets in OMXC25 Index. The total
ESG score and the total risk rating are collected from Yahoo finance as company‟s ESG
information. However, ten stocks are removed out of twenty five stocks of OMXC25 index due
to the inaccessibility of ESG information. Therefore, a dataset of weekly returns of fifteen stocks
in OMXC25 index for three year period from 2017 to 2019 is used for the analysis in this thesis.
In this study, Likelihood Ratio test, Wald test and Lagrange Multiple test were used as mean-
variance spanning tests under the four screening categories referred to as “10%”, “25%”, “50%”,
and “All” to investigate possible statistical significant difference between the minimum variance
frontier of more sustainable assets and the minimum variance frontier of all assets containing
both sustainable and unsustainable assets. It should be noted that these mean-variance spanning
tests are based on the multivariate regression model assuming that the residuals are multivariate
normal and i.i.d.
Interestingly, it is evident that all three asymptotic mean-variance spanning tests based on four
screening criteria results in statistically significance for fifteen stocks of OMXC25 index, or in
other words it reveals a statistically significant difference between the minimum variance frontier
containing more sustainable stocks and the minimum variance frontier consists of both
sustainable and non-sustainable stocks. Hence, it can be assumed that the investor may have
opportunity to improve his investment by adding more unsustainable stocks to the portfolio
rather than maintaining a portfolio of sustainable stocks solely.
Furthermore, the normality assumption of the residuals is tested using the generalized Shapiro-
Wilk‟s test, Mardia‟s test for skewness and kurtosis and graphical representation of Chi-squared
23
QQ plot. The results of the normality tests indicate that the normality assumption of the
residuals in multivariate regression model is violated. Therefore, the simulation study is designed
to examine the robustness of the non-normality assumption in residuals, by simulating new
weekly returns of the test assets. The residuals are simulated from multivariate normal and
multivariate t-distribution in order to simulate new weekly returns. The results depict that the
probabilities of rejecting null hypothesis when H0 is true for asymptotic mean-variance tests are
bit higher than the expected significance level of 0.05 for i.i.d and non-normal residuals. And
also the power curves illustrate that the power of the tests become little weaker when the
residuals are i.i.d and non-normal, compared to the normal and i.i.d residuals.
As mentioned in the literature review, this study is closely related to Wong‟s (2020) study. In her
study, she obtained non-significant difference between the efficient frontier of more sustainable
stocks and the efficient frontier containing both sustainable and non-sustainable stocks of twenty
one stocks in OMX Stockholm 30 index over the period 2008-2019 using monthly and weekly
returns. Therefore, she concluded that there is no opportunity for investor to make higher returns
by adding unsustainable stocks into the portfolio containing more sustainable assets. However, in
this study significant difference between the corresponding efficient frontiers were found. Hence,
investors may have better opportunities, if they would add unsustainable stocks into their
constructed portfolios of sustainable stocks in OMX Copenhagen 25 index. And also, comparing
the results of these two studies one may assume that the stocks in OMXS 30 are more sustainable
than the stocks in OMXC25.
6.1 Limitations of the study
When considering the ESG information, ten out of twenty five stocks in OMXC25 index are
removed due to the unavailability of ESG information which is a limitation of the study.
Therefore, the results may differ, if this limitation could be avoided.
6.2 Improvements/ suggestions for future research
In this study, weekly data of OMXC25 index is analyzed solely. It might be better to consider
other time frequencies such as daily and monthly when performing the spanning tests.
Additionally, this study has found a significant difference between the efficient frontier
containing sustainable assets and the efficient frontier consisted of both sustainable and non-
24
sustainable assets in OMXC25 index. Then, it is important to calculate the positive or negative
investment returns caused by the constructed sustainable portfolios in OMXC25 index due to
that significant difference between the efficient frontiers. Sometimes, one may be interested to
know whether this significant difference is caused due to the TP or MVP and it will also be an
interesting topic for a future study.
Moreover, socially responsible investors would be interested to compare the Nordic OMX
indices to make investment decisions on ESG investments. Therefore, this study can be extended
as a comparison of all the Nordic OMX indices to investigate the indices that may have better
investment opportunities towards sustainable investment.
Finally, it can be observed that the assumptions of performed spanning tests are violated in most
of the cases. In this study, the simulation study is designed only for the robustness analysis to the
non-normal assumption of mean-variance spanning tests. Thus, the simulation study can be
extended as the robustness analysis for other model violations such as heteroscedasticity and
autocorrelation.
7 References
1. Anane, A.K., 2019. Sustainability for Portfolio Optimization. [online] Available at:
<http://mdh.diva-portal.org/smash/get/diva2:1329098/FULLTEXT01.pdf> [Accessed 03
December 2020].
2. Bauer, R., Koedijk, K. and Otten, R., 2002. International evidence on ethical mutual fund
performance and investment style. Discussion paper series. [online] Available at:
<www.cepr.org/pubs/dps/DP3452.asp> [Accessed 03 December 2020].
3. Capinski, M. and Zastawniak, T., 2011 Mathematics for Finance: An Introduction to
Financial Engineering [e-book] Springer. Available at:
<http://poincare.matf.bg.ac.rs/~kmiljan/UFM.pdf> [Accessed 03 December 2020].
4. Friede, G., Busch, T. and Bassen A., 2015. ESG and financial performance: aggregated
evidence from more than 2000 empirical studies. Journal of Sustainable Finance &
Investment [online] Available at: <https://doi.org/10.1080/20430795.2015.1118917>
[Accessed 03 December 2020].
25
5. Herzel, S., Nicolosi, M. and Starica, C., 2012. The Cost of Sustainability in Optimal
Portfolio Decisions. The European Journal of Finance,18,3-4, pp. 333-349.
6. Kan, R., and Zhou, G., 2012. Tests of Mean-Variance Spanning, Annals of Economics
and Finance, 13(1), pp. 139-187.
7. KPMG, 2019.The numbers that are changing world. Revealing the growing appetite
for responsible investing, [pdf] KPMG, Available at:
<https://assets.kpmg/content/dam/kpmg/ie/pdf/2019/10/ie-numbers-that-are-changing-
the-world.pdf > [Accessed 03 December 2020].
8. Nasdaq, 2020.Index Methodology.OMX Copenhagen 25 Index, [pdf] Nasdaq, Available
at: <https://indexes.nasdaqomx.com/docs/Methodology_OMXC25EXP.pdf> [Accessed
03 December 2020].
9. Ortas, E., Moneva, J.M., and Salvador, M., 2012. Do social and environmental screens
influence ethical portfolio performance? Evidence from Europe. BRQ Business Research
Quarterly (2014),17, pp.11-21.
10. Pena, J. and Céu Cortez, M., 2017. Social screening and mutual fund performance:
international evidence, [online] Available at: <https://editorialexpress.com/cgi-
bin/conference/download.cgi?db_name=financeforum17&paper_id=96>[Accessed 03
December 2020].
11. Wong, V., 2020, Statistical Aspects of Sustainability in Optimal Portfolio Theory,
[online] Available
at:<http://kurser.math.su.se/pluginfile.php/20130/mod_folder/content/0/Master/2020/202
0_4_report.pdf>[Accessed 03 December 2020].
26
A Appendix
A.1 R code for screening criteria
#Screening
#calculating percentiles
x<-c(12,13,15,16,16,18,19,21,21,21,22,22,23,27,29)
quantile(x,probs =c(.25,.5,.75,.9) )
A.2 R code for importing data from excel and convert into data matrix
library(readxl)
weeklyret <- read_excel("D:/Orebro University/fall 2020/Master Thesis 1/weeklyret.xlsx")
M<-as.matrix(weeklyret)
A.3 R code for test statistics and their corresponding p-values of asymptotic mean-variance
spanning tests when T=157, N=2 and K=13 (table 3)
#N=2,K=13 and T=157
#creating Y
AD1 <- M[,-c(1,2,3,4,6,8,9,10,11,12,13,14,15,16)]
#creating X
BD1 <- M[,-c(5,7)]
#creating beta hat
CD1 <- solve(t(BD1)%*%BD1)%*%t(BD1)%*%AD1
# creating sigma
SD1 <- (1/157)*t(AD1-BD1%*%CD1)%*% (AD1-BD1%*%CD1)
#creating A
XD1 <- matrix(c(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1),nrow =2, ncol
= 14, byrow= TRUE)
#creating C
YD1 <- matrix(c(0,0,1,1), nrow =2,ncol = 2,byrow = TRUE )
#creating theta
OD1 <- (XD1%*%CD1) + YD1
#creating G
GD1 <- 157*(XD1%*% solve(t(BD1)%*%BD1)%*% t(XD1))
#creating H
HD1<-OD1%*%solve(SD1)%*%t(OD1)
# eigen value matrix
ED1<- HD1%*%solve(GD1)
eigen(ED1)
# Asympotic ratio tests
27
# test statistics are calulated after obtaining the eigen values
#LR Test
LR1= 157*(log(1.0562141944)+log(1.0001600834))
LR1
#critical value
qchisq(0.95,2*2)
#pvalue
pchisq(LR1,2*2,lower.tail = FALSE)
# Asymptotic Wald test
W1= 157*(0.0562141944+0.0001600834)
W1
pchisq(W1,2*2,lower.tail = FALSE)
# Asymptotic LM test
LM1= 157*((.0562141944/1.0562141944)+(.0001600834/1.0001600834))
LM1
pchisq(LM1,2*2,lower.tail = FALSE)
A.4 R code for simulation study
A.4.1 R code for calculating the actual probabilities of rejecting null hypothesis under the
multivariate normal distribution (table 6)
set.seed(2021)
#portfolios dimension and sample size: N=2 and K=13 and Ts=747
N<-2
K<-13
Ts<-157
#Number of repetitions in the simulation study: B=1000
r<-1000
#Model parameter \Sigma and B
#\Sigma can be theoretically any N\times N positive definite matrix. One can also take it from
the empirical study.
#Here, we define Sigma as a matrix with the same correlations \rho=0.6 and standard deviations
drawn from
#the uniform distribution on [0.1,0.5]
# matrix of standard deviations:
D<-diag(runif(N,0.1,0.5))
#correlation matrix
rho<-0.6
R<-(1-rho)*diag(rep(1,N))+rho*matrix(1,N,N)
#covariance matrix
28
Sigma<-D%*%R%*%D
#To generate the matrix of residuals E, we will also need the square root of Sigma. It is obtained
as the Cholesky root
sqSigma<-chol(Sigma)
## a - parameter used to model under H1, where the first row in B is obtained by replacing the
first row in B by
## the row whose all elements are a. The case a=0 corresponds to H0.
a<-seq(0,10,1)*0.05
a_l<-length(a)
#Similarly, we also compute the covariance matrix of X needed in the simulation
D_X<-diag(runif(K,0.1,0.5))
#correlation matrix
rho_X<-0.5
R_X<-(1-rho_X)*diag(rep(1,K))+rho_X*matrix(1,K,K)
#covariance matrix
Sigma_X<-D_X%*%R_X%*%D_X
sqSigma_X<-chol(Sigma_X)
# we will also have mean vector for X, which we generate from the uniform distribution on [-
0.5,0.5] here
mu_X<-runif(K,-0.5,0.5)
#### Two matrices used in the computation of the test statistics
#creating A
A <- cbind(c(1,0),rbind(rep(0,K),rep(-1,K)))
#creating C
C <- rbind(rep(0,N),rep(1,N))
#p-values of three tests will be saved in a three r\times a_l matrices called "results_LR",
"results_W", "results_LM".
results_LR<-matrix(0,r,a_l)
results_W<-matrix(0,r,a_l)
results_LM<-matrix(0,r,a_l)
for (j in 1:a_l) {
#Matrix B: N \times K+1 is drawn here from the uniform distribution (result B0: N \times K+1),
and then modified to
#correspond to the alternative hypothesis, that is it first row should have all elements "a" and
#the column sum should be the vector of ones.
#We can also take B0 from the empirical study and then modified it to get matrix B.
B0<-matrix(runif((K+1)*N),K+1,N)
29
B<-rbind(rep(a[j],N),B0[2:(K+1),]/(matrix(1,K,1)%*%colSums(B0[2:(K+1),])))
for (i in 1:r)
{
########Generation of data matrix Y consist of generating X and E
# generation of E: Ts \times N as independent sample from the multivariate normal distribution
N(0,Sigma)
E<-matrix(rnorm(Ts*N),Ts,N)%*%sqSigma
# generation of tX: Ts \times K as independent sample from the multivariate normal distribution
N(mu_X,Sigma_X)
tX<-matrix(1,Ts,1)%*%mu_X+matrix(rnorm(Ts*K),Ts,K)%*%sqSigma_X
# matrix X is obtained from tX by adding the vector of ones at the beginning
X<-cbind(rep(1,Ts),tX)
#Computation of Y
Y<-X%*%B+E
#######Computation of test statistics
#Estimation of B:
hB<-solve(t(X)%*%X)%*%t(X)%*%Y
#Estimation of Sigma
hSigma<-t(Y-X%*%hB)%*%(Y-X%*%hB)/Ts
#Computation Theta
Theta <- A%*%hB + C
#Computation G
G <- Ts*A%*%solve(t(X)%*%X)%*% t(A)
#Computation H
H<-Theta%*%solve(hSigma)%*%t(Theta)
# Computation of the eigenvalues of HG^{-1}
lambda<- eigen(H%*%solve(G))$values
#LR Test
LR<- Ts*sum(log(1+lambda))
results_LR[i,j]<-1-pchisq(LR,2*N)
#Wald test
W<- Ts*sum(lambda)
results_W[i,j]<-1-pchisq(W,2*N)
# LM test
LM<- Ts*sum(lambda/(1+lambda))
results_LM[i,j]<-1-pchisq(LM,2*N)
}
}
30
######Computation of actual probabilities of rejection the null hypothesis for significance level
alp=0.05
alp<-0.05
#Matrix res_dec consist of test decision: 1 -- rejection, 0 -- non-rejection
res_dec_LR<-results_LR<alp
res_dec_W<-results_W<alp
res_dec_LM<-results_LM<alp
##Actual probability of rejection:
prob_LR<-colSums(res_dec_LR)/r
prob_W<-colSums(res_dec_W)/r
prob_LM<-colSums(res_dec_LM)/r
rbind(a,prob_LR,prob_W,prob_LM)
#### figure
pdf("H1-alpha-nor.pdf", height=5.5, width=7)
plot(c(a,a,a),c(prob_LR,prob_W,prob_LM), type="n", main="", xlab=expression(a),
ylab="Empirical power ", ylim=c(0,1),cex=1.5, lwd = 1.6, axes=T,frame=T)
lines(a, prob_LR, lty=1, col="black", lwd = 1.6)
lines(a, prob_W, lty=2, col="red", lwd = 1.6)
lines(a, prob_LM, lty=3, col="green", lwd = 1.6)
legend("topleft", lty=c(1,2,3), col=c("black","red","green"), c("LR","W", "LM"), lwd = 1.6)
dev.off()
A.4.2 R code for calculating the actual probabilities of rejecting null hypothesis under the
multivariate t-distribution (table 6)
set.seed(2021)
#portfolios dimension and sample size: N=2 and K=13 and Ts=747
N<-2
K<-13
Ts<-157
#degrees of freedom fort-distribution
d<-5
#Number of repetitions in the simulation study: B=1000
r<-1000
#Model parameter \Sigma and B
#\Sigma can be theoretically any N\times N positive definite matrix. One can also take it from
the empirical study.
31
#Here, we define Sigma as a matrix with the same correlations \rho=0.6 and standard deviations
drawn from
#the uniform distribution on [0.1,0.5]
# matrix of standard deviations:
D<-diag(runif(N,0.1,0.5))
#correlation matrix
rho<-0.6
R<-(1-rho)*diag(rep(1,N))+rho*matrix(1,N,N)
#covariance matrix
Sigma<-D%*%R%*%D
#To generate the matrix of residuals E, we will also need the square root of Sigma. It is obtained
as the Cholesky root
sqSigma<-chol(Sigma)
## a - parameter used to model under H1, where the first row in B is obtaned by replacing the
first row in B by
## the row whose all elements are a. The case a=0 corresponds to H0.
a<-seq(0,10,1)*0.05
a_l<-length(a)
#Similarly, we also compute the covariance matrix of X needed in the simulation
D_X<-diag(runif(K,0.1,0.5))
#correlation matrix
rho_X<-0.5
R_X<-(1-rho_X)*diag(rep(1,K))+rho_X*matrix(1,K,K)
#covariance matrix
Sigma_X<-D_X%*%R_X%*%D_X
sqSigma_X<-chol(Sigma_X)
# we will also have mean vector for X, which we generate from the uniform distribution on [-
0.5,0.5] here
mu_X<-runif(K,-0.5,0.5)
#### Two matrices used in the computation of the test statistics
#creating A
A <- cbind(c(1,0),rbind(rep(0,K),rep(-1,K)))
#creating C
C <- rbind(rep(0,N),rep(1,N))
#p-values of three tests will be saved in a three r\times a_l matrices called "results_LR",
"results_W", "results_LM".
results_LR<-matrix(0,r,a_l)
results_W<-matrix(0,r,a_l)
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results_LM<-matrix(0,r,a_l)
for (j in 1:a_l) {
#Matrix B: N \times K+1 is drawn here from the uniform distribution (result B0: N \times K+1),
and then modified to
#correspond to the alternative hypothesis, that is it first row should have all elements "a" and
#the column sum should be the vector of ones.
#We can also take B0 from the empirical study and then modified it to get matrix B.
B0<-matrix(runif((K+1)*N),K+1,N)
B<-rbind(rep(a[j],N),B0[2:(K+1),]/(matrix(1,K,1)%*%colSums(B0[2:(K+1),])))
for (i in 1:r)
{
########Generation of data matrix Y consist of generating X and E. The factor sqrt((d-2)/d)
ensures that
########the covariance matrix is the same as in the normal distribution.
# generation of E: Ts \times N as independent sample from the multivariate t-distribution
(0,Sigma)
E<-sqrt((d-2)/d)*diag(sqrt(d/rchisq(Ts,d)))%*%matrix(rnorm(Ts*N),Ts,N)%*%sqSigma
# generation of tX: Ts \times K as independent sample from the multivariate t-distribution
(mu_X,Sigma_X)
tX<-matrix(1,Ts,1)%*%mu_X+sqrt((d-
2)/d)*diag(sqrt(d/rchisq(Ts,d)))%*%matrix(rnorm(Ts*K),Ts,K)%*%sqSigma_X
# matrix X is obtained from tX by adding the vector of ones at the beginning
X<-cbind(rep(1,Ts),tX)
#Computation of Y
Y<-X%*%B+E
#######Computation of test statistics
#Estimation of B:
hB<-solve(t(X)%*%X)%*%t(X)%*%Y
#Estimation of Sigma
hSigma<-t(Y-X%*%hB)%*%(Y-X%*%hB)/Ts
#Computation Theta
Theta <- A%*%hB + C
#Computation G
G <- Ts*A%*%solve(t(X)%*%X)%*% t(A)
#Computation H
H<-Theta%*%solve(hSigma)%*%t(Theta)
# Computation of the eigenvalues of HG^{-1}
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lambda<- eigen(H%*%solve(G))$values
#LR Test
LR<- Ts*sum(log(1+lambda))
results_LR[i,j]<-1-pchisq(LR,2*N)
#Wald test
W<- Ts*sum(lambda)
results_W[i,j]<-1-pchisq(W,2*N)
# LM test
LM<- Ts*sum(lambda/(1+lambda))
results_LM[i,j]<-1-pchisq(LM,2*N)
}
}
######Computation of actual probabilities of rejection the null hypothesis for significance level
alp=0.05
alp<-0.05
#Matrix res_dec consist of test decision: 1 -- rejection, 0 -- non-rejection
res_dec_LR<-results_LR<alp
res_dec_W<-results_W<alp
res_dec_LM<-results_LM<alp
##Actual probability of rejection:
prob_LR<-colSums(res_dec_LR)/r
prob_W<-colSums(res_dec_W)/r
prob_LM<-colSums(res_dec_LM)/r
rbind(a,prob_LR,prob_W,prob_LM)
#### figure
pdf("H1-alpha-t5.pdf", height=5.5, width=7)
plot(c(a,a,a),c(prob_LR,prob_W,prob_LM), type="n", main="", xlab=expression(a),
ylab="Empirical power ", ylim=c(0,1),cex=1.5, lwd = 1.6, axes=T,frame=T)
lines(a, prob_LR, lty=1, col="black", lwd = 1.6)
lines(a, prob_W, lty=2, col="red", lwd = 1.6)
lines(a, prob_LM, lty=3, col="green", lwd = 1.6)
legend("topleft", lty=c(1,2,3), col=c("black","red","green"), c("LR","W", "LM"), lwd = 1.6)
dev.off()