How to improve the quality of pension systems throughout the world

(1)

How to improve the quality of

pension systems throughout

the world

Kim de Bakker

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Fi-nance

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Kim de Bakker Student nr: 11378719

Email: kim debakker@hotmail.com Date: July 14, 2017

Supervisor: dr. T.J. Boonen Second reader: dr. S. van Bilsen

(2)

Statement of originality

This document is written by Student Kim de Bakker who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervi-sion of completion of the work, not for the contents.

(3)

1 Introduction

Pension systems are very different around the world. Some countries have a mainly public pension system where the government regulates the pensions. Other countries have a mainly private pension system where the private sec-tor regulates the pension. There are also countries who have a part private and a part public pension system.

Each pension system works differently under various circumstances. This means that a certain system can perform well in one country but not in an-other. Even though pension system can not be the same everywhere, there are some components which have proven to be effective. For example a part public pension system provides security for low income worker and a part private pension system provides a more individually fitted pension.

Lately there has been an increasing amount of pensioners compared to work-ers. Many pension systems use a pay-as-you-go system to finance part of the pensions. A pay-as-you-go pension system lets the current workers pay for the current pensioners. It has been extra difficult to pay for pensions with this kind of system because of the increase in pensioners compared to workers.

This problem is researched in this thesis. We want to know if an ageing population effects the quality of the pension system. We have received data of the survival probability to live till you are 65 years old from the World Bank. We use an increase in this data as an indication of ageing population. We also want to find a possible solution for the decrease in pension system quality. We analyse tax revenue as a possible solution because taxes are used as a payment method for certain pensions. We have received the data of the percentage of the GDP which consists of tax revenue from the World Bank. We hypothesize the following:

• The survival probability to live till you are 65 years old has a negative effect on the quality of a pension system.

• It is possible to negate any decrease in quality of the pension system with an increase in tax revenue.

The Melbourne Mercer Global Index is used as a measure of the quality of a pension system. This index is an overall score given to the quality of a pension system which consist of an adequacy, a sustainability and an integrity component.

The data is analysed with a panel data analysis where possible fixed effects and random effects are taken into account. Some control variables are added in order to estimate the effect of the main variables (survival probability to

(5)

live till 65 years for men, tax revenue) better.

Finally after our analysis we found a negative effect of an increase in survival probability to live till 65 years for men on the quality of a pension system. We also found an positive effect of an increase in tax revenue on the quality of a pension system. This positive effect is too small to negate the negative effect of the increase in survival rate. This is why a partial switch from a pay-as-you-go system to a funded system is advised for countries with a large pay-as-you-go pension system.

(6)

2 Literature review

There does not exist a similar research to compare the research of this thesis to. However there are elements of this thesis which could be compared to other studies and the conclusions are also based on existing knowledge. This thesis is mainly based on the Melbourne Mercer Global Index reports1_.

These reports are publicized every year since 2009. The reports discuss the quality of pension systems around the world. This is discussed further in Section 3.2 of this thesis. The reports are funded by The Victorian Govern-ment. The research and the reports are executed by the Australian Centre for Financial Studies in partnership with Mercer. These are both businesses with experts in for example the fields of pension fund management and fi-nancial research which makes the reports more reliable.

The study in this thesis is a panel data study. Hsiao (2007) discusses the use of a panel data study and it’s advantages. Some of the advantages which are mention are more degrees of freedom compared to cross-sectional data and it is better at capturing complex behaviour compared to cross-section or time series data. This explains why it is wise to use a panel data study.

Stauvermann and Kumar (2016) discuss the effect of the ageing population and other factors on the sustainability of a pay-as-you-go pension system. They concluded that these factors do not effect the sustainability of a purely pay-as-you-go system. This result is interesting but might be different for systems which are only partly a pay-as-you-go system. This is the case for most of the pension systems involved in our research.

Cardoso and van Praag (2003) actually discuss the sustainability of pensions and assume that pensions are a part pay-as-you-go system and a part funded system. They conclude that the ageing population does have an effect on the welfare of the population. They also show that this problem is mostly present in the Western world.

Artige et al. (2014) discuss the effects of ageing on both a defined benefit (DB) pay-as-you-go system and a defined contributions (DC) pay-as-you-go pension system. They find that a DC pay-as-you-go system leads to the best outcome when dealing with the ageing problem compared to the DB pay-as-you-go-system.

Gollier (2008) discusses intergenerational risk sharing which is usually named as one of the positive factors of a pay-as-you-go system. He discusses other options beside the pay-as-you-go system which also contain intergenerational risk sharing.

The switch from a pay-as-you-go pension system to a funded system is

(7)

cussed by World Bank (2005). The article discusses how the population is ageing and suggests a funded system as a better system for the population.

(8)

3 Data

3.1 Types of pension systems

Each country has a different pension system. This is not necessarily a bad thing because there is not one pension system which works for every county. There are a lot of factors which influence how well a pension system fits with a country. Some of these factors, for example the type of population and the culture, can not be changed which is why pension systems can not be the same in every country.

This research uses the Melbourne Mercer Global Index (explained in Section 3.2) as a measure of the quality of a pension system. The countries who are analysed for the Melbourne Global Pension Index are different from each other. This does not mean that there is no scheme which the pension systems should follow in order to get a well functioning pension system.

The World Bank contemplated in 1994 and defined a three pillar system. This system took multiple factors like old age poverty and long term security into account. The World Bank (2008) also defined a five pillar system. This system separates the first pillar of the three pillar system in a zero pillar and a first pillar. It also adds a fourth pillar which takes the financial support outside of the pension system into account. The five pillars are given by:

• Zero pillar

A very basic pension level with the purpose of preventing poverty. This is usually financed by the government.

• First pillar

This is a mandatory pension level mainly meant for people with a low income and who do not plan ahead. The contribution are based on the level of income and it is usually pay-as-you-go where the young pay for the old.

• Second pillar

An mandatory occupational or personal pension plan.

• Third pillar

This is a voluntary occupational or personal pension plan which exists to fill in any missing funds from the other pillars.

• Fourth pillar

This exists of financial support which takes place outside of the pension system. This is for example family support, social housing and home ownership.

(9)

A pension system should consist of multiple pillars instead of one pillar. A multiple pillars pension system will be much more stable when problems arise.

Suppose a pension system is only a public pay-as-you-go system. This sys-tem will face many problems especially with the ageing population.

Suppose a pension system is fully privately funded pension system. This sys-tem will have problems because there are no regulations to prevent poverty. A multiple pillar pension system solves these problems by diversification. The distribution across the pillars depends on the type of country where the pension system is applied.

3.2 Sustainability, Adequacy and Integrity

The Melbourne Mercer Global Pension Index is a score given to pension sys-tems. This score is an indicator for the quality of pension systems for different countries. Their research has started in 2009 with 11 different countries and has grown till 2016 to 27 countries. The countries and the years that are participating in the research are displayed in Table 1.

As seen in the table, not every country has participated every year in the research. This causes an unbalanced data set. The data can still be used but it would have been ideal if it was a balanced data set.

The overall score of the Melbourne Mercer Global Pension Index is calcu-lated using three sub-indices. The score consists of 35% sustainability, 40% adequacy and 25% integrity. Each of these indices gives a different indication of a well functioning pension system.

The first index we discuss is the sustainability index. This measures the long-term sustainability of a pension system. One of the most resent prob-lem which effects this index is the ageing population and the way countries plan to account for this problem. Also government debt plays a role in the sustainability of a pension system.

The second index we discuss the adequacy index. This index is dependent on how adequate a pension system is. This adequacy is dependent for example on the minimal income level of the pensioners, how easily pension money can be withdrawn before the pensionable age and how a change of employment affects the pension.

The last index we discuss is the integrity index. This index is dependent on the integrity of the pension system in the private sector. This is dependent for example on the amount of information which is shared with the policy-holders and the prudential regulations which are set up.

Figure 1 show the average and the spread (minimum and maximum) of the overall index per year. The figure for the sustainability, adequacy and

(10)

in-Table 1: Participating countries per year Year 2009 2010 2011 2012 2013 2014 2015 2016 Country Argentine Australia Austria Brazil Canada Chili China Denmark Finland France Germany India Indonesia Ireland Italy Japan Malaysia Mexico Netherlands Poland Singapore South Africa South Korea Sweden Switzerland UK US

(11)

tegrity index is displayed in Figure 2, 3 and 4 of Appendix A.

The figure of the overall index shows that the average index is slightly going down throughout the years but the spread of the index is also going up. The decline in the average overall index could be caused by the resent ageing problems. There are some countries which have a better score than previous years. These countries probably have a more stable pension system which could handle the resent problems. Some of the increase and decrease could also be explained by the inclusion of more countries.

The sustainability figure in Appendix A shows the same kind of effect but here it is even clearer. The maximum score is going up while the average and the minimum score is going down. The decrease in the minimum could be because of the inclusion of countries like Italy with a very low sustainability index.

The adequacy figure is similar to the previous two figures but the integrity figure is different. The integrity shows an increase in all three aspects, aver-age, maximum and minimum. This in combination with the meaning of the integrity index means that the private pensions systems have become more trustworthy and reliable.

The sub-indices are also correlated with each other. These correlations are displayed in Table 2.

Table 2: Correlation between the sub-indices Sustainability Adequacy Integrity Sustainability 1 0.308 0.530

Adequacy 0.308 1 0.524

Integrity 0.530 0.524 1

This table shows positive correlation between the integrity index and the other indices of more than 0.5. This means that a pension system which is adequate or sustainable is also likely to be integer. The adequacy index and the sustainability index have a much lower correlation of 0.3 between each other. This means that an adequate pension system is not necessary sustainable or vice versa.

3.3 Independent variables

In this research, the Melbourne Mercer Global Pension Index which is ex-plained in Section 3.2 is used as a measure of the quality of a pension system. These indices are used to see if certain variables have a significant effect on

(12)

Figure 1: The overall index throughout the years

the quality of a pension system. Section 5 discusses which independent vari-ables have a significant effect on the quality of a pension system and why they are chosen.

The main independent variables are not the only independent variables which are added to the model. There are also control variables which are added in order to determine the effect of the main independent variable better. The data of these independent variable and other possibly useful variables were taken from the World Bank Open Data2. The data from this database was not always complete. Some countries do not keep track of every possible variable or they do it once every few years. Most of the missing data has been filled in with the use of linear interpolation and extrapolation. Some of the data is still missing which causes the data to be unbalance. This will be a problem if the missing data is deliberate in order to hide bad observations. It does not seem like that is the case in this dataset.

Table 3 displays variables which we have received from the World Band and which we have considered as possible indicators for the quality of a pension system. The table shows the average, minimum and maximum value to give an indication of the spread and the value of each variable.

(13)

Table 3: Indication of certain factors

Variable Average* Minimum* Maximum*

GDP (per capita) 37,585 1,447 88,615

Total population 183.2 M. 4.6 M. 1,378 M. Central government debt (%GDP) 72.68 22.22 214.63 Compulsory education duration (years) 10.53 6 15.97 Current account balance (%GDP) 2.06 -6.19 23.44 Death rate (# per 1000) 7.89 4.3 12.46 Completed lower secondary education (%) 84.16 37.57 100 Completed upper secondary education (%) 69.25 22.32 100 Completed short-cycle tertiary education (%) 29.27 7.14 79.21 Fertility rate (births per woman) 1.73 1.15 2.56 Gross savings (%GDP) 26.41 11.79 51.68

Income highest 10% 27.95 20.47 42.76

Income lowest 10% 2.71 0.97 3.90

Land area (km2) 2.6 M. 700 9.4 M.

Life expectancy at birth 79.27 57.18 83.88 Population density (# per km2₎ _540.25 _2.82 _7,955

Survival probability to reach 65 years old (male) 82.67 40.31 90.03 Survival probability to reach 65 years old (female) 89.17 48.44 94.68

Tax revenue (%GDP) 17.09 7.94 36.89

Urban population (%total) 77.83 31.28 100 *The average, minimum and maximum of the data which we received from the World Bank for the participating countries.

(14)

4 Model

4.1 Panel Data

The data we use in this thesis is panel data which includes multiple periods. A panel data model is given by:

yit = β0+ β1xit1+ β2xit2+ · · · + βkxitk+ ai+ uit, (1)

where i denotes the individual, t denotes the time period, k denotes the in-dependent variable.

In this model yit depicts the dependent variable the overall index, the

sus-tainability index, the adequacy index or the integrity index which we want to approximate. The xitk depicts independent variables which are used to

approximate the dependent variable. The βk is the coefficient of independent

variable xitk which displays the effect of a change in the independent variable

on the dependent variable. The ai and uit represent the error terms. There

is a time constant error term ai and a time varying error term uit.

The time constant error term ai is also called the fixed effect or the

unob-served effect. It measures the unobunob-served effects that are time constant and effect the dependent variable yit. The time varying error term uit is also

called the idiosyncratic error term. This error term measures the unobserved effect that are time varying and effect the dependent variable yit. There are

different methods to dealing with these error terms which are discussed in Section 4.1.1, 4.1.2 and 4.1.3.

The first independent variable (xit1) which we add is the main independent

variable. This is the variable whose effect we want to approximate. The next independent variables (xit2,. . . ,xitk) are control variables which we add to

improve the approximation of the effect from the main independent variable. The control variables are variables which have a high correlation with the dependent variable and the main independent variable.

4.1.1 Pooled model

One of the most basic methods of dealing with the error terms of (1) is the pooled ordinary least squares (pooled OLS) method. This method pools the error terms together into one error term:

vit = ai+ uit. (2)

and assumes the errors vit are i.i.d. with mean zero and variance σ2v for all i

and t. This is substituted into (1) which results in:

(15)

The coefficients of (3) can be estimated with OLS. These estimation will be unbiased if the error term vit is uncorrelated with the independent variables.

This means that the time varying error term uit and the time constant error

term ai have to be uncorrelated with the independent variables.

The pooled OLS model is usually not the best model but it is a good model to compare to. A comparison with the pooled OLS model can give information about the error terms.

4.1.2 Fixed effects model

The fixed effects method estimates the coefficients of (1) and assumes the errors uit are i.i.d. with mean zero and variance σu2 for all i and t. The

fixed effects method deals with the time constant error term by eliminating it with a fixed effects transformation. This fixed effects transformation also eliminate the time constant independent variables and the constant term β0.

The first step in the fixed effects transformation is to take the average over time of (1). This results in:

¯

yi = β0+ β1x¯i1+ β2x¯i2+ · · · + βkx¯ik+ ai+ ¯ui. (4)

The time average of a general term pit is defined as:

¯ pi = 1 T T X t=1 pit, (5) where pit ∈ {yit, xit, uit}.

The next step is to subtract (4) from (1). This results in the fixed effects transformed model:

¨

yit = β1xït1+ β2xït2+ · · · + βkxïtk+ üit. (6)

Here, ¨pit = pit− ¯pi for a general term pit∈ {yit, xit, uit}.

This new fixed effects transformed model does not include the time constant error term ai and can be estimated by pooled OLS.

These estimation of the coefficients for (6) are the same estimations of the coefficients for (1) with the assumption that the errors uit are i.i.d. with

mean zero and variance σ2

u for all i and t.

The estimated coefficients will be unbiased if the time varying error term is not correlated with the independent variables for all time periods. The time constant error term is no longer used to estimate the dependent variable which means that it may be correlated with the independent variables.

(16)

4.1.3 Random effects model

In Section 4.1.2 the time constant error term was eliminated because it would make the coefficients unbiased if the term was correlated with the indepen-dent variables. This is the right thing to do if there is a non-zero correlation between ai and one or more independent variables. If there is no correlation

between ai and any independent variables than it would not be wise to

ex-clude the time constant error term. The exclusion would result into wrong estimations of the coefficients of the independent variables.

The alternative to a fixed effects model is a random effects model. The ran-dom effects model assumes the same as the fixed effects model except that the time constant error term ai is not correlated with any independent

vari-able for all time periods and all individuals. It assumes that the errors ai are

i.i.d. with mean zero and variance σ2

a for all i.

The time varying and time constant error terms can be pooled together just like (2). A model with pooled together error terms is (3). The model is transformed because the error terms vit are serially correlated. It is serially

correlated because of the time constant error term ai. The correlation within

the error term vit is given by:

Corr(vit, vis) = σ_a2 σ2 a+ σ2u , (7) where t 6= s.

This correlation will always cause positive serial correlation in the error term of the model which is why the model is transformed. The serial correlation can be solved using a General Linear Squares (GLS) transformation. This transformation uses the following parameter:

λ = 1 − s σ2 u σ2 u + T σa2 . (8)

This parameter, (3) and (4) are used for the following transformed model:

yit− λ¯yit =β0(1 − λ) + β1(xit1− λ¯xi1) + β2(xit2− λ¯xi2) + . . .

+ βk(xitk− λ¯xik) + (vit− λ¯vi),

(9)

where λ = [0, 1] which follows from (8).

If λ = 0, (9) represents the pooled OLS model which is described in (3). If λ = 1, (9) represents the fixed effects which is described in (6).

(17)

used in a practical research. For a practical research the following estimation is used: ˆ λ = 1 − s 1 1 + Tσˆa2 ˆ σ2 u , (10)

with estimated standard deviations.

4.1.4 Comparing models

Pooled OLS and Fixed effects We compare the pooled OLS model and the fixed effects model with the help of the F-test. The F-test tests the following hypothesis:

H0 : Pooled OLS model preferred

H1 : Fixed effects model preferred

The F test has the following test statistic (Wooldridge, 2009, p.145-146):

F = RSSOLS−RSSF E dfOLS−dfF E RSSF E dfF E . (11)

where RSS is the residual sum of squares and df is the degrees of freedom. The test statistic (11) has a F distribution under H0 with (dfOLS − dfF E,

dfF E) degrees of freedom:

F ∼ FdfOLS−dfF E,dfF E (12)

It is compared to the F distribution critical value with significance α. Here, H0 is rejected when the test statistic is higher than the critical value.

Fixed effects and Random effects We compare the fixed effects model to the random effects model with the help of the Hausman test (Hausman, 1978, summarized in Greene, 2008, p.208-209). It tests the following hypoth-esis:

H0 : Random effects model preferred

H1 : Fixed effects model preferred

The Hausman test tests this hypothesis using the following test statistic:

H = ( ˆβ_{F E}− ˆβ_RE)0[VF E − VRE]−1( ˆβF E − ˆβRE). (13)

The ˆβ is the estimated coefficient vector. The vector of the random effects model should not include the time constant variables because the fixed effects

(18)

model does not include these variables. The V is the covariance matrix. The test statistic which results from (13) has a χ2 _{distribution under H}

0 with

the degrees of freedom equal to rank(VF E− VRE) and is compared to the χ2

critical value with significance α. If the Hausman test statistic (H) is lower than the chi-square critical value, the random effects model is the preferred model. Otherwise, the fixed effects model is the preferred model.

4.2 Comparing nested models

The Wald test is a test which can be applied for multiple purposes. In this research it is used to compare nested models.

Definition 4.1 (Nested models). Models are nested if all the independent terms of the smaller model are included in the larger model. Suppose models M1, . . . , Mn have P1, . . . , Pn number of independent parameter respectively

with P1 < · · · < Pn and they have the same dependent variable. Models

M1, . . . , Mnare nested if Mnincludes all independent terms of M1, . . . , Mn−1

and Mn−1 includes all independent terms of M1, . . . , Mn−2 etc.

The Wald test compares two or more models and it concludes which one is a significantly better fit for the data. It decides if the reduction in variance is large enough to make up for the loss in the degrees of freedom. This explains if a certain independent variable should be included in the model or if it should be excluded.

We are going to use the test to compare a model to the same model without the last control variable. We can conclude if the last control variable should be included in or excluded from the model. We are going to repeat this process for all control variables.

4.3 Heteroskedasticity

The application of the previous sections of Chapter 4 results in a model but this model might have a problem with heteroskedasticity. The first step is to test if the model suffers from heteroskedasticity and the second step is to use a robust covariance matrix to account for the heteroskedasticity.

We use the Breusch-Pagan test to look for heteroskedasticity. It tests for the presents of heteroskedasticity by looking at the variance of the error. If this variance is dependent on the independent variable, there is heteroskedasticity. The hypothesis of the Breusch-Pagan test is given by:

H0 : The variance of the error term is constant: Homoskedasticity

(19)

The Breusch-Pagan test statistic is:

BP = n ∗ R2 (14)

where n is the sample size and R2 _{(coefficient of determination) is the R}2 _of

the regression of the squared residuals of the model. This regression is given by:

e2_it = α0+ α1zit,1+ · · · + αmzit,m+ vit, (15)

where e2

it is the squared residuals of the original model and z are all the

explanatory variables.

The test statistic (14) follows a χ2_k distribution under H0 with k number of

degrees of freedom and significance α. If BP is higher than the critical value of the chi-square distribution, there is an indication for heteroskedasticity. Otherwise, there is no indication for heteroskedasticity and the error term is homoskedastic.

Suppose the Breusch-Pagan test indicates homoskedasticity then the model does not have to change. If the test indicates heteroskedasticity then this should be solved. A possible solution is the inclusion of heteroskedastistic-consistent (HC) standard errors (Zeileis 2004).

Suppose β is the coefficients vector and it is estimated by ˆβ. The covariance matrix of ˆβ is given by:

VAR( ˆβ) = (X0X)−1X0ΩX(X0X)−1, (16)

where X is the vector of independent variables. The Ω can be estimated using ˆΩ = diag(ω1. . . ωn). Suppose that the error terms are homoskedastic.

The variance of the error term is constant which means that ωi = ˆσ2 and

ˆ

Ω = ˆσ2_I

n. This results in the following estimation of the variance:

[

VAR( ˆβ) = ˆσ2(X0X)−1. (17)

This estimation will be biased if there is heteroskedasticity because the vari-ance of the error term is not constant. There are five types of HC standard errors which can be used instead of ωi = ˆσ2 to account for heteroskedasticity.

The most important type is:

ωi =

ˆ u2_it (1 − hi)δi

, (18)

where hi are the diagonal elements of the hat matrix H = X(X0X)−1X0, ˆuit

are the OLS residuals and δi = min{4, hi/¯h} with ¯h the mean of hi.

This was proven (Cribari-Neto 2004) to be the best HC standard error be-cause it takes influential observations into account.

(20)

5 Results

Section 5.1 and 5.2 show our results of applying the methods, that we ex-plained in Chapter 4, to the data set which we exex-plained in Chapter 3. These results will help us test our hypothesis. In these results, we use a significant of α = 0.10 because the dataset is not very large.

5.1 Survival probability to live till 65 years for men

5.1.1 Hypothesis

The first possible indicator for the quality of a pension system which we analyse is the survival probability of men who are born to live till they are 65 years old (abbreviated by S65M ). The age of 65 was chosen by the World Bank. It is also the most common pensionable age.

There is an explanation for why S65M has an effect on the quality of a pension system. A pension system needs to work no matter how many people reach the pensionable age. If people have a high chance of reaching the pensionable age, it means more money needs to be available to cover these pensions. For example some systems work with a pay-as-you-go system where the current workers pay part of the pensions of the current pensioners. If there are more pensioners, there is more money which needs to be paid. This causes a problem for many pension systems which is why S65M could have a negative effect on the adequacy of a pension system.

The integrity of a pension system is correlated with the adequacy of a pension system. A system which is not adequate might not want to inform their policyholders about everything because it is usually negative information. This will make the pension system have less integrity. This is a reason why S65M could also negatively effect the integrity level of a pension system.

5.1.2 Model

Survival probability to live till 65 years for men is the main independent vari-able which we add to the model. Other varivari-ables are also added as control variables. These variables have a high correlation with the main indepen-dent variable and with the depenindepen-dent variable. The results of our analysis are displayed in Appendix B.

The first dependent variable which we want to estimate is the overall index (OI) which was described in Section 3.2. The control variables are the sur-vival probability to live till 65 years for women (S65W ), the life expectancy (LE), the logarithm of the GDP per capita (LGDP ), the percentage of the income which belongs to the richest 10% of the population (I10H) and the

(21)

percentage of the population which is urban population (U P ). The squared version of the main independent variable (S65M2_{) is also added because the}

variable shows signs of a non-linear relation. The model for the overall index (OI) is given by:

OIit =β0+ β1· S65Mit+ β2· S65Mit2 + β3· S65Wit+ β4 · LEit+

β5· LGDPit+ β6· I10Hit+ β7· U Pit+ ai+ uit.

(19)

We estimate the model using the pooled OLS method of Section 4.1.1, the fixed effects method of Section 4.1.2 and the random effects method of Section 4.1.3. The estimated coefficients by these methods are displayed in Table 4. We can find the best fitting method using the F-test and the Hausman test which were described in Section 4.1.4 and are displayed in Table 8. Using the F-test, we find significant reason to reject the pooled OLS model in favour of the fixed effects model. We do not find a significant reason to use the fixed effects model over the random effects model. This is why we use the random effects.

Subsequently we use the Wald test described in Section 4.2 to determine which control variables make a significant difference to the model and should stay in the model. The results are displayed in Table 10. The first, third and sixth variable from the right of (19) reduce the variance of the model enough to be added to the model. The resulting model for the overall index (OI) is given by:

OIit = β0+ β1· S65Mit+ β2· S65Mit2+ β3· LGDPit+ β4· U Pit+ ai+ uit. (20)

The estimated coefficients are given in Table 13 and show a significant rela-tion between the survival probability of men who are born to live till they are 65 years old and the overall index.

Lastly, we check for heteroskedasticity using the Breusch-Pagan test de-scribed in Section 4.3. The results we found are displayed in Table 17. We found a significant reason to belief that there is heteroskedasticity in the model.

We correct for this heteroskedasticity using the method described in Section 4.3. This does not effect the estimations of the coefficients but it can ef-fect the variance and thus the significance of the coefficients. The results we found are displayed in Table 18. These are the estimations for the final model.

The same steps we did for the overall index, we also apply the sustainability, adequacy and integrity index.

The first sub-index is the sustainability index (SI). The control variables that we add are life expectancy (LE), survival probability to live till 65

(22)

years for women (S65W ), the logarithm of the GDP per capita (LGDP ), the percentage of the population which is urban population (U P ), the per-centage of the income which belongs to the richest 10% of the population (I10H) and the percentage of the population who have finished the lower secondary education (LSE).

We estimate the model using the pooled OLS, fixed effects and random ef-fects methods. The estimations are displayed in Table 5. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 8. We continue to use the random effects model. We apply the Wald test which is displayed Table 10 and find that U P and LSE should stay in the model and the other control variables should be taken out of the model. Lastly we test the model for heteroskedasticity and display the results in Table 17. We find heteroskedasticity and take this into account. The final model for the sustainability index (SI) is given by:

SIit = β0+ β1· S65Mit+ β2· U Pit+ β3· LSEit+ ai + uit. (21)

The estimated coefficients are given in Table 19 and show no significant relation between the survival probability of men who are born to live till they are 65 years old and the sustainability index.

The second sub-index is the adequacy index (AI). The control variables that we add are life expectancy (LE), survival probability to live till 65 years for women (S65W ), the logarithm of the GDP per capita (LGDP ), the percentage of the income which belongs to the richest 10% of the population (I10H), the fertility rate (F R) and the percentage of the population which is urban population (U P ). The squared version of the main independent variable (S65M2_{) is also added because the variable shows signs of a}

non-linear relation.

We estimate the model using the pooled OLS, fixed effects and random effects methods. The estimation are displayed in Table 6. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 8. We conclude that the fixed effects method fits the model the best. We apply the Wald test which is displayed in Table 11 and find that S65M2 _{and I10H}

should stay in the model and the other control variables should be taken out of the model. Lastly we test the model for heteroskedasticity and display the results in Table 17. We find heteroskedasticity and take this into account. The final model for the adequacy index (AI) is given by:

AIit= β0+ β1· S65Mit+ β2· S65Mit2 + β3· I10Hit+ ai+ uit. (22)

The estimated coefficients are given in Table 20 and show a significant rela-tion between the survival probability of men who are born to live till they

(23)

are 65 years old and the adequacy index.

The last sub-index is the integrity index (II). The control variables that we add are the same as for the adequacy index.

We estimate the model using the pooled OLS, fixed effects and random effects method. The estimation are displayed in Table 7. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 8. We conclude that the fixed effects method fits the model the best. We apply the Wald test which is displayed in Table 12 and find that S65M2_{, S65W ,}

I10H and F R should stay in the model and the other control variables should be taken out of the model. Lastly we test the model for heteroskedasticity and display the results in Table 17. We find heteroskedasticity and take this into account. The final model for the integrity index (II) is given by:

IIit=β0+ β1 · S65Mit+ β2· S65Mit2 + β3· S65Wit+ β4· I10Hit+

β5· F Rit+ ai+ uit.

(23)

The estimated coefficients are given in Table 21 and show a significant rela-tion between the survival probability of men who are born to live till they are 65 years old and the integrity index.

5.1.3 Summary of the results

The survival probability to live till 65 years for men has a negative effect on the adequacy and the integrity of a pension system. This is in line with our expectations of Section 5.1.1. The effect becomes less negative when the sur-vival rate increases but the effect will not become positive because that would only happen with a survival rate of more than 100% which is not possible. The negative effect of the independent variable is bigger in absolute terms for integrity compared to adequacy but it goes faster towards zero when the survival rate increase.

The survival rate has the same kind of effect on the overall index as it has on the integrity or the adequacy index. The effect is negative and it becomes less negative with larger survival rate.

5.2 Tax revenue

5.2.1 Hypothesis

The second possible indicator for the quality of a pension system which we analyse is the percentage of the GDP which exists of tax revenue (abbrevi-ated by T R).

(24)

There is an explanation for why we expect an effect of T R on the quality of a pension system. In Section 3.2 we explained that the adequacy of a pension system was partly determined by the minimal income level of the pensioners. The minimal income level is mainly determined by the zero pillar (or first pillar) pension which is explained in Section 3.1. This pillar is a tax financed pension. If the tax income increase, it will be easier to finance the first pillar. This means that when the tax revenue is high, government will have a better opportunity to provide a solid basic income level for pensioners and the adequacy of a pension will be high.

5.2.2 Model

The variable T R is chosen as the main independent variables. Other variables which have a high correlation with this main independent variable and with the dependent variable are added as control variables. The result of our analysis are displayed in Appendix C.

The first effect which we analyse is the effect of TE on the overall index (OI) which was discussed in Section 3.2. The control variables which we add are the logarithm of the total population (LT P ), the percentage of the income which belongs to the richest 10% of the population (I10H), the logarithm of the GDP and the percentage of the income which belongs to the poorest 10% of the population (I10L).

The model for the overall index (OI) is given by:

OIit =β0+ β1· T Eit+ β2· LT Pit+ β3 · I10Hit+ β4· LGDPit+

β5· I10Lit+ ai+ uit.

(24)

We estimate this model with the pooled OLS, fixed effects and the random effects method which were discussed in Section 4.1.1, 4.1.2 and 4.1.3 re-spectively. The results we find are displayed in Table 22. We compare these methods using the F-test and the Hausman test we discussed in Section 4.1.4. The results we find are displayed in Table 26. Using the F-test, we find a significant reason to reject the H0 hypothesis which means the fixed effects

method is preferred over the pooled OLS method. Using the Hausman test, we also find a significant reason to reject the H0 hypothesis which means the

fixed effects method is preferred over the random effects method. This why we use the random effects method for this model.

Subsequently we use the Wald test which was discussed in Section 4.2 in order to decide which control variables decrease the variance of the model enough compared to the loss in degrees of freedom. The results we find are displayed in Table 27. The variables LT P and I10H should stay in the model and

(25)

the other control variables should be taken out of the model. The resulting model for the overall index (OI) is given by:

OIit = β0+ β1· T Eit+ β2· LT Pit+ β3· I10Hit+ ai+ uit. (25)

The estimated coefficients are displayed in Table 31.

Lastly we test the model for heteroskedasticity with the Breusch-Pagan test which was discussed in Section 4.3. The results that we find are displayed in Table 35. We find a significant indication to reject the H0 hypothesis of

homoskedasticity in favour of the H1 hypothesis of heteroskedasticity. We

control this heteroskedasticity with heteroskedastistic-constant standard er-rors which were discussed in Section 4.3. The final model is given by (25) and the estimates of the coefficient are displayed in Table 36. These estimates show a significant relation between the percentage of the GDP which exists of tax revenue and the overall index.

The overall index consists of the sustainability, adequacy and integrity index. Next, we are applying the same test we did on the overall index but now on the other indices.

The first sub-index is the sustainability index (SI). The control variables which we add to the model are the logarithm of the total population (LT P ), the percentage of the income which belongs to the richest 10% of the pop-ulation (I10H), the logarithm of the GDP (LGDP ), the death rate (DR) and the percentage of the income which belongs to the poorest 10% of the population (I10L).

We estimate the model with the pooled OLS, fixed effects and random ef-fects method. The results we find are displayed in Table 23. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 26. We conclude that the random effects method fits the model the best. We apply the Wald test to determine which control variables should re-main in the model. The results are displayed in Table 28. We find that LT P and I10H should stay in the model and the other control variables should be taken out of the model. The approximation of the remaining model is given in Table 32. Lastly we test the model for heteroskedasticity and display the results in Table 35. We find heteroskedasticity and take this into account. The final model for the sustainability index (SI) is given by:

SIit = β0+ β1· T Eit+ β2· LT Pit+ β3· I10Hit+ ai+ uit. (26)

The estimated coefficients are displayed in Table 37. These estimations show no significant relation between the percentage of the GDP which exists of tax revenue and the sustainability index.

(26)

which we add to the model are the logarithm of the total population (LT P ), the percentage of the income which belongs to the richest 10% of the popula-tion (I10H), the gross savings as a percentage of GDP (GS), the logarithm of the GDP (LGDP ), the death rate (DR) and the percentage of the income which belongs to the poorest 10% of the population (I10L).

We estimate the model with the pooled OLS, fixed effects and random ef-fects method. The results we find are displayed in Table 24. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 26. We conclude that the random effects method fits the model the best. We apply the Wald test to determine which control variables should remain in the model. The results are displayed in Table 29. We find that I10H and LGDP should stay in the model and the other control variables should be taken out of the model. The approximation of the remaining model is given in Table 33. Lastly we test the model for heteroskedasticity and dis-play the results in Table 35. We find heteroskedasticity and take this into account. The final model for the adequacy index (AI) is given by:

AIit= β0+ β1· T Eit+ β2· I10Hit+ β3· LGDPit+ ai+ uit. (27)

The estimated coefficients are displayed in Table 38. These estimations show a significant relation between the percentage of the GDP which exists of tax revenue and the adequacy index.

The third and last sub-index is the integrity index (II). The control variables which we add to the model are the logarithm of the total population (LT P ), the total land area in km2 _{(LA), the percentage of the income which belongs}

to the richest 10% of the population (I10H), the gross savings as a percentage of GDP (GS), the logarithm of the GDP (LGDP ) and the percentage of the income which belongs to the poorest 10% of the population (I10L).

We estimate the model with the pooled OLS, fixed effects and random effects method. The results we find are displayed in Table 25. We compare the methods using the F-test and the Hausman test. The results are displayed in Table 26. We conclude that the fixed effects method fits the model the best. We apply the Wald test to determine which control variables should remain in the model. The results are displayed in Table 30. We find that I10H and GS should stay in the model and the other control variables should be taken out of the model. The approximation of the remaining model is given in Table 34. Lastly we test the model for heteroskedasticity and display the results in Table 35. We find heteroskedasticity and take this into account. The final model for the integrity index (II) is given by:

(27)

The estimated coefficients are displayed in Table 39. These estimations show no significant relation between the percentage of the GDP which exists of tax revenue and the integrity index.

5.2.3 Summary of the results

The models for the overall index and adequacy index in the previous section show a positive influence between the tax revenue (T R) and the quality of a pension system. This is in line with our expectations of Section 5.2.1. The models show if T R increases with one percentage point, the adequacy index will increase with 0.39 percentage points and the overall index will increase with 0.57 percentage points.

(28)

6 Conclusion

In this thesis, we analysed the Melbourne Mercer Global Index with the use of a panel data study. We hypothesized the following statements:

• The survival probability to live till you are 65 years old has a negative effect on the quality of a pension system.

• It is possible to negate any decrease in quality of the pension system with an increase in tax revenue.

We found a positive link between the quality of a pension system and the survival probability to 65 years for men. We also found a negative link be-tween the quality of a pension system and the tax revenue.

These influences exist mainly because of the zero and first pillar. Many coun-tries use a pay-as-you-go system to finance the public part of the pensions of the current pensioners. These types of systems can not handle the trend of the increasing survival rate. These countries could decide to increase the the tax to compensate for this increase in pensioners. However the positive effect of an increase in tax revenue is much smaller than the negative effect of an increase in the survival probability to live till 65 years for men.

Another possible solution for this problem could be a switch from a pay-as-you-go system to a funded system. In a funded system, people are contribut-ing to the future pension instead of the pension of the current pensioners. This kind of system will not have a problem with the increasing survival rate. There is one problem with this switch and that is the transition costs. The current pensioners have contributed to the pay-as-you-go system when they were working and they are expecting a certain pension level. The transition costs could be limited by making the transition as fast and smooth as possi-ble. The cost could be paid by the government by either borrowing, cutting spending or increasing tax on income or consumption. It is also possible to promise the government some of the profits from investing the funded pen-sions.

The pay-as-you-go system does provide some security against poverty which is why the system should not be fully replaced by a funded system. There should still be a system in place to provide some security but this system should not be so large that it can not be paid.

(29)

7 Discussion

This part will discuss some aspects of the study in the thesis which could be improved upon and show an opportunity for future research.

The pension study of the Melbourne Centre of Financial Studies is still a relatively new study as it has only existed for a few years. This might have an influence on the results of this thesis because results from a large dataset are more trustworthy. A larger dataset can be looked at with a smaller sig-nificance than the significant of α = 0.10 which is used in the thesis.

Some countries do not register everything every year from the variables dis-played in Table 3 which causes missing data. We tried to fill in the missing data with the use of linear interpolation and extrapolation but this was not possible for the entire dataset. It would be better if the information was made public to everyone in order to avoid any chance of non random missing data. Non random missing data could make results biased because the data which is left does not represent the situation correctly. There is no indication of non random missing data in the dataset used but this is not for certain.

(30)

8 Appendix

8.1 Appendix A

(31)

Figure 3: The adequacy index throughout the years

(32)

8.2 Appendix B: Survival probability to live till 65

years for men

Table 4: Dependent variable: overall index (OI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept 513.13 (0.000 ***) 445.47 (0.005 **) S65M -12.69 (0.001 ***) -11.68 (0.065 .) -9.43 (0.031 *) S65M2 _{0.08 (0.000 ***)} _{0.08 (0.025 *)} _{0.06 (0.016 *)} LE -1.15 (0.373) -1.17 (0.480) -1.30 (0.284) S65W 0.79 (0.363) -3.20 (0.271) 0.13 (0.909) LGDP 1.99 (0.306) -1.10 (0.729) 2.81 (0.233) I10H -0.35 (0.058 .) -0.91 (0.010 *) -0.55 (0.039 *) U P 0.55 (0.000 ***) 0.47 (0.488) 0.47 (0.002 **) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 5: Dependent variable: sustainability index (SI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept 181.55 (0.008 **) 157.86 (0.084 .) S65M 2.79 (0.002 **) 1.34 (0.600) 2.45 (0.097 .) LE -1.99 (0.512) -1.22 (0.641) -1.86 (0.388) S65W -2.67 (0.172) -2.11 (0.595) -2.97 (0.144) LGDP -6.97 (0.165) 3.00 (0.528) 3.02 (0.443) U P 0.97 (0.000 ***) 1.94 (0.032 *) 1.18 (0.002 **) I10H -0.32 (0.459) 0.15 (0.783) -0.16 (0.739) LSE 0.51 (0.029 *) -0.25 (0.297) -0.16 (0.427) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

(33)

Table 6: Dependent variable: adequacy index (AI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept 453.67 (0.001 ***) 259.96 (0.217) S65M -13.87 (0.001 ***) -14.96 (0.181) -8.98 (0.130) S65M2 _{0.08 (0.001 ***)} _{0.10 (0.126)} _{0.05 (0.128)} LE 0.24 (0.874) -0.83 (0.777) -0.65 (0.699) S65W 1.45 (0.244) -0.93 (0.857) 2.08 (0.197) LGDP 4.63 (0.028 *) -1.12 (0.852) 3.85 (0.223) I10H -0.31 (0.111) -0.56 (0.422) -0.26 (0.439) F R -4.87 (0.269) 23.42 (0.144) -0.25 (0.971) U P 0.24 (0.018 *) 0.31 (0.810) 0.04 (0.837) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 7: Dependent variable: integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Pooled OLS Fixed effects Random effects Intercept 1206.41 (0.000 ***) 896.17 (0.001 **) S65M -29.34 (0.000 ***) -21.93 (0.043 *) -25.80 (0.001 ***) S65M2 _{0.19 (0.000 ***)} _{0.15 (0.016 *)} _{0.16 (0.001 ***)} LE 0.59 (0.771) -1.93 (0.492) -1.49 (0.472) S65W -0.28 (0.863) -0.74 (0.881) 3.32 (0.111) LGDP -1.62 (0.554) 3.89 (0.502) 1.50 (0.706) I10H -0.18 (0.483) -0.88 (0.192) -0.74 (0.092 .) F R -19.39 (0.001 **) 67.66 (0.000 ***) 14.77 (0.110) U P 0.67 (0.000 ***) 0.54 (0.663) 0.20 (0.420) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 8: Comparing models Test statistics with p-values in brackets Index F-test Hausman test Overall F = 39.93 (0.000) χ2 = 10.58 (0.158) Sustainability F = 93.21 (0.000) χ2 _{= 3.74 (0.809)}

Adequacy F = 12.08 (0.000) χ2 _{= 13.93 (0.084)}

(34)

Table 9: Wald test

Dependent variable: overall index (OI)

Residual degrees of freedom Variable which is tested χ2 _P-value

94 95 U P 9.73 0.002 ** 96 I10H 1.92 0.166 97 LGDP 3.87 0.049 * 98 S65W 2.18 0.140 99 LE 0.002 0.964 100 S65M2 5.66 0.017 * Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 10: Wald test

Dependent variable: sustainability index (SI)

(could not test LSE) 95 96 I10H 0.001 0.971 97 U P 10.85 0.001 *** 98 LGDP 1.85 0.173 99 S65W 0.11 0.743 100 LE 0.28 0.597 Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 11: Wald test

Dependent variable: adequacy index (AI)

Residual degrees of freedom Variable which is tested χ2 P-value 72 73 U P 0.06 0.809 74 F R 2.20 0.138 75 I10H 2.94 0.087 . 76 LGDP 0.25 0.619 77 S65W 0.74 0.390 78 LE 0.01 0.916 79 S65M2 4.80 0.028 * Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

(35)

Table 12: Wald test

Dependent variable: integrity index (II)

72 73 U P 0.19 0.662 74 F R 20.60 0.000 *** 75 I10H 12.22 0.000 *** 76 LGDP 0.03 0.873 77 S65W 2.75 0.097 . 78 LE 0.07 0.791 79 S65M2 _9.43 _{0.002 **} Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 13: Model after Wald test for dependent variable overall index (OI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) Intercept 119.31 (0.009 **) S65M -3.10 (0.007 **) S65M2 _{0.02 (0.004 **)} LGDP 2.73 (0.126) U P 0.08 (0.545) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 14: Model after Wald test for dependent variable sustainability index (SI)

Estimations of coefficients with p-values in brackets. Variable Estimate (p-value)

Intercept -8.46 (0.689) S65M 0.06 (0.852) U P 0.85 (0.001 ***) LSE -0.13 (0.383)

(36)

Table 15: Model after Wald test for dependent variable adequacy index (AI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) S65M -17.03 (0.034 *) S65M2 0.10 (0.037 *) I10H -0.98 (0.098 .)

Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 16: Model after Wald test for dependent variable integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) S65M -25.82 (0.004 **) S65M2 0.172 (0.001 ***) S65W 0.13 (0.976) I10H -0.91 (0.147) F R 63.71 (0.000 ***) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 17: Test for Heteroskedasticity

Index Breusch-Pagan test P-value Overall BP = 75.58 0.000 Sustainability BP = 127.69 0.000 Adequacy BP = 68.35 0.000 Integrity BP = 336.63 0.000

Table 18: Model controlled for heteroskedasticity Dependent variable: overall index (OI)

Estimations of coefficients with p-values in brackets. Variable Estimate Intercept 119.31 (0.022 *) S65M -3.10 (0.033 *) S65M2 0.02 (0.035 *) LGDP 2.73 (0.080 .) U P 0.08 (0.564) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

(37)

Table 19: Model controlled for heteroskedasticity Dependent variable: sustainability index (SI) Estimations of coefficients with p-values in brackets.

Variable Estimate Intercept -8.46 (0.817) S65M 0.06 (0.893) U P 0.85 (0.061 .) LSE -0.13 (0.573) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 20: Model controlled for heteroskedasticity Dependent variable: adequacy index (AI) Estimations of coefficients with p-values in brackets.

Variable Estimate

S65M -17.03 (0.034 *) S65M2 0.10 (0.040 *) I10H -0.98 (0.340)

Table 21: Model controlled for heteroskedasticity Dependent variable: integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Estimate S65M -25.82 (0.018 *) S65M2 0.17 (0.008 **) S65W 0.13 (0.978) I10H -0.91 (0.239) F R 63.71 (0.034 *) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

(38)

8.3 Appendix C: Tax revenue

Table 22: Dependent variable: overall index (OI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept 29.19 (0.452) 97.37 (0.034 *) T R 0.16 (0.207) 0.54 (0.096 .) 0.11 (0.614) LT P -3.22 (0.000 ***) -31.36 (0.086) -3.10 (0.034 *) I10H 0.59 (0.046 *) -1.03 (0.008 **) -0.38 (0.227) LGDP 5.92 (0.000 ***) 0.58 (0.858) 2.41 (0.259) I10L 3.46 (0.092 .) 0.39 (0.828) 0.80 (0.622) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 23: Dependent variable: sustainability index (SI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept 14.79 (0.881) 86.58 (0.341) T R 0.14 (0.639) 0.07 (0.888) 0.08 (0.845) LT P -2.78 (0.168) 23.48 (0.398) -3.38 (0.278) I10H 0.19 (0.835) 0.001 (0.999) -0.08 (0.884) LGDP 9.32 (0.019 *) 2.52 (0.646) 3.02 (0.454) DR -4.76 (0.020 *) -0.05 (0.982) -1.38 (0.479) I10L 7.47 (0.133) -0.76 (0.786) 1.09 (0.666) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

(39)

Table 24: Dependent variable: adequacy index (AI) Estimations of coefficients with p-values in brackets. Variable Pooled OLS Fixed effects Random effects Intercept -37.41 (0.422) 20.07 (0.744) T R 0.34 (0.017 *) 1.18 (0.030 *) 0.36 (0.212) LT P -0.29 (0.757) -23.57 (0.435) -1.07 (0.532) I10H 0.52 (0.206) -1.39 (0.056 .) -0.26 (0.602) GS 0.08 (0.540) 0.36 (0.255) 0.11 (0.562) LGDP 6.92 (0.000 ***) -4.20 (0.497) 5.82 (0.032 *) DR 0.48 (0.601) -0.26 (0.925) 0.40 (0.776) I10L 3.70 (0.092 .) -1.14 (0.709) -0.52 (0.831) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 25: Dependent variable: integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Pooled OLS Fixed effects Random effects Intercept 265.58 (0.000 ***) 243.33 (0.002 **) T R -0.04 (0.836) 0.43 (0.463) -0.04 (0.892) LT P -7.65 (0.000 ***) -134.43 (0.001 ***) -7.01 (0.002 **) LA 5.07 × 10−07 (0.315) -0.003 (0.014 *) 9.76 × 10−07 (0.231) I10H -0.35 (0.370) -2.95 (0.000 ***) -0.63 (0.222) GS -0.25 (0.121) 0.86 (0.010 *) 0.05 (0.805) LGDP -3.28 (0.131) -2.75 (0.645) -3.10 (0.344) I10L -2.79 (0.347) 3.40 (0.348) 0.07 (0.980) Significance levels: *** = 0.001; ** = 0.01; * = 0.05; . = 0.1

Table 26: Comparing models Test statistics with p-values in brackets Index F-test Hausman test Overall F = 39.79 (0.000) χ2 _{= 15.03 (0.010)}

Sustainability F = 103.11 (0.000) χ2 = 3.16 (0.789) Adequacy F = 14.68 (0.000) χ2 _{= 8.79 (0.268)}

(40)

Table 27: Wald test

Dependent variable: overall index (OI)

72

73 I10L 0.05 0.827

74 LGDP 0.02 0.881

(could not test I10H) 124

125 LT P 6.03 0.014 *

Table 28: Wald test

Dependent variable: sustainability index (SI)

90

91 I10L 0.19 0.665

92 DR 0.48 0.489

93 LGDP 0.87 0.351

149 LT P 6.25 0.012 *

Table 29: Wald test

Dependent variable: adequacy index (AI)

89

90 I10L 0.05 0.830

91 DR 0.08 0.777

92 LGDP 6.46 0.011 *

93 GS 0.04 0.851

149 LT P 1.99 0.158

(41)

Table 30: Wald test (integrity index) Dependent variable: integrity index (II)

70

71 I10L 0.89 0.345

72 LGDP 0.36 0.548

73 GS 9.05 0.003 **

124 LA 0.04 0.842

125 LT P 0.60 0.441

Table 31: Model after Wald test for dependent variable overall index (OI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) T R 0.57 (0.061 .) LT P -30.17 (0.084 .) I10H -1.08 (0.001 ***)

Table 32: Model after Wald test for dependent variable sustainability index (SI)

Intercept 128.18 (0.013 *) T R 0.14 (0.717) LT P -4.45 (0.109) I10H -0.13 (0.745)

(42)

Table 33: Model after Wald test for dependent variable adequacy index (AI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) Intercept -5.75 (0.807) T R 0.39 (0.098 .) I10H -0.24 (0.404) LGDP 6.84 (0.001 ***)

Table 34: Model after Wald test for dependent variable integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) T R -0.41 (0.453) I10H -2.18 (0.000 ***) GS 0.99 (0.003 **)

Table 35: Test for Heteroskedasticity

Index Breusch-Pagan test P-value Overall BP = 43.29 0.004 Sustainability BP = 96.67 0.000 Adequacy BP = 55.44 0.000 Integrity BP = 251.21 0.000

Table 36: Model controlled for heteroskedasticity Dependent variable: overall index (OI)

T R 0.57 (0.053 .) LT P -30.17 (0.393) I10H -1.08 (0.191)

(43)

Table 37: Model controlled for heteroskedasticity Dependent variable: sustainability index (SI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) Intercept 128.18 (0.068 .) T R 0.14 (0.886) LT P -4.45 (0.231) I10H -0.13 (0.852)

Table 38: Model controlled for heteroskedasticity Dependent variable: adequacy index (AI) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) Intercept -5.75 (0.889) T R 0.39 (0.017 *) I10H -0.24 (0.640) LGDP 6.84 (0.030 *)

Table 39: Model controlled for heteroskedasticity Dependent variable: integrity index (II) Estimations of coefficients with p-values in brackets.

Variable Estimate (p-value) T R -0.41 (0.549) I10H -2.18 (0.365) GS 0.99 (0.259)

(44)

9 References

Artige, L., Cavenaile L. & Pestieau, P. (2014). “The Macroeconomics of PAYG Pension Schemes in an Aging Society”, CORE Discussion Paper, 2014/33

Cardoso, P. D. M. L. & van Praag, B. M. S. (2003). “How sustainable are old-age pensions in a shrinking population with endogenous labour supply?”, CESifo Working Paper, No. 861

Cribari-Neto, F. (2004). “Asymptotic Inference Under Heteroskedasticity of Unknown Form.”, Computational Statistics & Data Analysis, 45, 215-233. Gollier, C. (2008). ”Intergenerational risk-sharing and risk-taking of a pen-sion fund”, Journal of Public Economics, Vol. 92, Issues 5-6, 1463-1485. Greene, W.H. (2008). ”Econometric Analysis”, 6th edition, Pearson Prentice

Hall, New Jersey.

Hausman, J.A. (1978). “Specification Tests in Econometrics.”, Economet-rica, Vol 46, 1251-1271.

Hsiao, C. (2007). “Panel Data Analysis — Advantages and Challenges.”, TEST, Vol 16, Issue 1.

Stauvermann, P.J. & Kumar, R.R. (2016). “Sustainability of A Pay-as-you-Go Pension System in A Small Open Economy with Ageing, Human Capital and Endogenous Fertility”, Metroeconomica, Vol 67, Issue 1, 2-20. Wooldridge, J.M. (2009). ”Introductory Econometrics: A Modern

Ap-proach”, 4th edition, South-Western, a part of Cengage Learning.

World Bank (2005). “Transition : Paying for a Shift from Pay-as-You-Go Financing to Funded Pensions.”, World Bank Pension Re-form Primer Series. World Bank, Washington, DC. c World Bank. https://openknowledge.worldbank.org/handle/10986/11242 License: CC BY 3.0 IGO.

World Bank (2008). “The World Bank Pension Conceptual Framework.”, World Bank Pension Reform Primer Series. Washington, DC. c World Bank. https://openknowledge.worldbank.org/handle/10986/11139 License: CC BY 3.0 IGO.”

Zeileis, A. (2004). “Econometric Computing with HC and HAC Covariance Matrix Estimators”, Journal of Statistical Software, Vol 11 (2004), Issue 10.

How to improve the quality of pension systems throughout the world