Faculteit Economie en Bedrijfskunde, Amsterdam School of Economics Bachelorscriptie en Afstudeeerseminar Econometrie
The influence of structural dependence
between health insurers on the market
distribution
Zooey Bossert (10748725)
22 December - 2017
Abstract
In this paper the market distribution of the health insurers in the Netherlands are investigated while focusing on the spatial autoregressive model. The SAR model is made to take the spatial dependence between observation into the re-gression. However this model is often used with structural dependence. For this research the SAR model is used to investigate the structural dependence with other explanatory variables.
Statement of Originality
This document is written by Zooey Bossert who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are 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 supervision of completion of the work, not for the contents.
Contents
1 Introduction 2
2 Theoretical Review 3
2.1 The Spatial Autoregressive Model . . . 3 2.2 Influences on the Market Share . . . 5
3 Data and Model Specification 7
3.1 The Data . . . 7 3.2 The Model . . . 9
4 Results 10
4.1 Maximum Likelihood . . . 11 4.2 Explanatory Variables . . . 14 4.3 Likelihood Ratio Test . . . 16
5 Conclusion 16
1
Introduction
Every year health insurance company DSW Zorgverzekeraar is the first to announce the premium price for the coming year and sets the tone for the other insurers (Boogaars Brink, 2016). Even though DSW Zorgverzekeraar is not one of the four biggest health insurers in The Netherlands, it leads the trend when it comes to the premium. Ac-cording to the NOS (’DSW verlaagt onverwacht zorgpremie en eigen risico’, 2017) this year DSW Zorgverzekeraar startled everyone with a reduction of the premium of the so called "Basisverzekering" which is the minimal health insurance cover that every Dutch person, by law, has to take out with one of the private insurance companies. The reduction in the premium came as a surprise since DSW indicated earlier that there probably would be an increase caused by limited reserves.
When the other insurers do not reduce their premium, the result could be a shift in the market. By law every Dutchman has to be insured and for this reason a shift in the market, for example an increase in customers of DSW Zorgverzekeraar, leads to a decrease in customers of one, or multiple, of the other insurers. Not the same insurers are affected when the shift is caused by another health insurer. The way these health insurers influence each other has strong resemblence of structural dependence.
In this paper the market distribution of the health insurers will be examined. The market distribution has, apart from the premium, other explanatory variables. The main focus is on the structural dependence between the companies. What is the in-fluence of structural dependence on the market distribution of health insurers? The spatial autoregressive model will be used to take the dependence in to account.
In the second chapter the spatial autoregressive model will be introduced. Sub-sequently is the theoretical discussion of the other variables that will be used in the regression. In the third chapter the data is explained. In this chapter there is an
ex-tensive version of the model and the method to estimate the structural dependence. In Chapter 4, the results are presented and examined. Lastly, in the fifth chapter the conclusion is discussed.
2
Theoretical Review
In this chapter the theory of this research will be explained and discussed. First the spatial autoregressive model is introduced. Then the market distribution of the health insurance and its influences will be discussed.
2.1
The Spatial Autoregressive Model
Spatial autoregressive (SAR) models are made to take into account spatial dependence of observations (LeSage, 2008). This means that observations from one location influ-ence observations from another location that is geographically close. These observa-tions could represent income, unemployment levels, tax rates and so on. In standard regression models used to analyse cross-section or panel data it is assumed that these observations or regions are independent. This leads to biased and inconsistent estima-tors. To include this spatial dependence in the regression, an n x n weight matrix is made:
y = λW y + Xβ + (1)
y = (In− λW )−1Xβ + (In− λW )−1 (2)
∼ N (0, σ2I n)
Spatial dependence in a model, spatial interaction, can be expressed in two ways. The error term of different observations correlate (2), this is called spatial error. The
other model is called spatial lag, when dependent variable y of observation i is influ-enced by the independent variable of a company j (1). This matrix W demonstrates the connection between the regions and is specified based on the distance between the regions. λ is the scalar parameter that demonstrates the extent of the spatial depen-dence. In the SAR model, the λ will be estimated to investigate the spatial dependepen-dence. There are many methods to estimate the λ as is shown by Anselin (1996). However this paper will use maximum likelihood.
SAR models are becoming increasingly common in applied econometric work. For example it is used when researching peer effects in Lin (2010), or when researching the effects on house prices in Lan Zhang (2011). When the model is applied to struc-tural dependence there are two potential problems when applied. The first problem is specifying the dependence structure between companies in a matrix. Specifying the de-pendence between companies is more complicated than specifying spatial dede-pendence. Spatial dependence is based on the geographic distance between points which is rather simple to construct in a matrix. However this is not the case with structural dependence. The second problem arises from the first problem; asymmetric relationships between the companies are far more likely than the symmetrical relationships such as spatial dependence. This means that the eigenvalues of the weight matrix is considerable more complex.
Furthermore there are some issues that are generally not discussed but, as Anselin (2002) points out, are extremely important. The first issue that Anselin discusses is the importance of making a distinction in the data. The distinction is between objects, the things you can physically touch, and fields, price- and riskfactors. The next issue involves the weight matrix. There is often a lack of identification of the parameters of the full covariance matrix and the correct specification of the spatial weight matrix. Lastly, the SAR model is based on asymptotic properties that only hold under a restrictive set
of assumptions. These assumptions reduce the degree of heterogeneity and dependence on the spatial stochastic process.
2.2
Influences on the Market Share
The SAR model will be applied on the structural dependence of health insurers in the Netherlands. The Dutch health care aims at providing everybody with quality care and to create solidarity through a mandatory and accessible health insurance (Rijksoverheid, 2016). This means that every Dutchman has to be insured by law, therefore the number of potential customers is finite. Naturally there are deaths and births every year. However this has such a small influence on the population in one year that it will be disregarded in this research1.
As explained in the introduction, structural dependence determines the shift in the market. The other explanatory variables in the regression are influences that cause shifts in the market distribution. The variables used in this research are the price pre-mium of the Basisverzekering2, the difference in price between the lowest and highest offered premium, the investments, the number of different brands within the concern, NPS, the number of branches and the sustainability of the investments.
The premium of the Basisverzekering is used for the reason of basic economics. When the price of a product decreases, the demand will increase3. Since the market
1In the numbers of the Central Bureau for Statistics, you can find that the birth surplus is about 0.1 percent of the entire population, https://www.cbs.nl/nl-nl/cijferstheme=bevolking
2The basisverzekering is the basic care package. The content of it is legally established to com-pensate the basic needs in care. The government determines the content every year and publishes it on their site, https://www.rijksoverheid.nl/onderwerpen/zorgverzekering/vraag-en-antwoord/wat-zit-er-in-het-basispakket-van-de-zorgverzekering
share is based on the percentage of consumers, the price is an important variable. It is expected that there will be a negative correlation between the price and market share. Most health insurers have more than one care package. The more different packages, the more consumers can appeal to the health insurer. With every different care package, there is a different targeted audience. To include this in the regression there is a variable premium difference between the lowest and highest priced package. Targeting a larger audience causes more consumers and therefore a bigger market share.
The insurers in the market, apart from one, manage multiple brands. Each brand targets a different audience by distinguishing itself. When an insurer owns more brands, it will have a bigger audience hence a bigger market share (Morgan Rego, 2009).
Besides investments in stocks, firms invest in their own company. These investments are made with the purpose of improvement within the company, which leads to growth of the company (Massell, 1962). This shows an active presence in the market and will have a positive effect on the market share.
As mentioned above, companies invest in stocks to increase their equity. The com-pany itself decides how the invest, it can be in all different kind of stocks. However the Verbond van Verzekeraars made an agreement (’Code Duurzaam Beleggen’, 2011) with all the firms that are member of this interest association. This code set rules for the investments of the firms, with sustainability as main purpose. The variable for sustain-ability in the regression is in this research a dummy variable, that is 1 if a health insurer is part of this association and is inclined to adhere to the code. Consumer’ s choices are influenced by the sustainability; in The Netherlands there is a growing interest in sustainability (’Duurzaam bij consument in de lift’, 2017). Therefore a health insurer that invests sustainable attracts more consumers. This increases the market share of the insurer.
by Reicheld (2003). The score can vary between -100 and 100. A positive score is seen as a good result. To calculate the NPS according to Reicheld, the customer is asked if the person would recommend, in this case, the insurer. The customer can answer to give a score from 0 to 10. When the customer gives a score of 9 or 10, the customer is a promoter. When the customer gives a score of 7 or 8, the customer is neutral. If the score is a 6 or lower, the customer is a criticaster. The NPS is calculated by subtracting the percentage criticasters from the percentage promoters. Insurers will score higher when customers are more satisfied with their service. A high NPS means that customers will recommend the insurer, which leads to more customers and a higher market share. The last variable that is included in the regression is the number of branches. When a company has multiple branches spread over the country, it will be more accessible for the customer. It is expected that more branches will have a positive effect on the market share.
In the following chapter, the variables are further explained and the model specified.
3
Data and Model Specification
In this chapter the research design will be introduced. The models will be expressed in formulas.
3.1
The Data
The market of health insurers exist of nine concerns: Achmea, VGZ, CZ, Menzis, DSW, ONVZ, Zorg Zekerheid, ASR and Eno. The data of the health insurers is obtained by the annual report4. The dependent variable is the market share, this is expressed in percentage. The explanatory variables premium of the Basisverzekering (PREMIUM),
price difference (DIFF) and investment (INVEST) will be variables expressed in euros. The number of branches (BRANCHES), brands (BRANDS) and the NPS (NPS) will be integers. Lastly, the variable sustainability (SUST) is a dummy variable. The dummy will be 1 when an insurer complies with the Code Duurzaam Beleggen.
In this paper, we specify two different structural dependence matrices. The first structural dependence matrix W is as follows:
W1 = 0 0.34 0.3 0.19 0.05 0.04 0.04 0.03 0.01 0.4 0 0.27 0.18 0.05 0.04 0.03 0.03 0.01 0.38 0.3 0 0.17 0.04 0.03 0.03 0.03 0.01 0.35 0.28 0.24 0 0.04 0.03 0.03 0.02 0.01 0.04 0.04 0.03 0.01 0 0.34 0.3 0.19 0.05 0.04 0.03 0.03 0.01 0.4 0 0.27 0.18 0.05 0.03 0.03 0.03 0.01 0.38 0.3 0 0.17 0.04 0.03 0.03 0.02 0.01 0.35 0.28 0.24 0 0.04 0.03 0.03 0.02 0.01 0.32 0.25 0.22 0.14 0
The W1 matrix is based on the probability that a customer leaving insurer i is
going to insurer j. The four largest health insurers have more than 88 percent of the insured. These companies are considerably larger compared to the other health insurers. Therefore exert the larger companies more influence on each other than on the smaller companies. This also applies to the smaller health insurers. With that in mind, the W matrix is constructed with the market distribution to specify the chance that a customer switches to one of the other insurers.
The second W matrix is as follows: W2 = 0 0.333 0.333 0.33 0 0 0 0 0 0.333 0 0.333 0.333 0 0 0 0 0 0.25 0.25 0 0.25 0.125 0.125 0 0 0 0.25 0.25 0.25 0 0.125 0.125 0 0 0 0 0 0.125 0.125 0 0.25 0.25 0.125 0.125 0 0 0.125 0.125 0.25 0 0.25 0.125 0.125 0 0 0 0 0.333 0.333 0 0.167 0.167 0 0 0 0 0.2 0.2 0.2 0 0.4 0 0 0 0 0.2 0.2 0.2 0.4 0
The second W matrix is based on the market distribution of the last 5 years, 2011-2016, by observing the shifts in the market. When the market distribution of one of the health insurers increases, which of the other insurers also increases or decreases. Matrix W2 reflects these connections.
3.2
The Model
As mentioned in Chapter 2.1 the SAR model is used:
y = Sλ−1Xβ + Sλ−1 (3)
With Sλ = I - λ W. Assuming is normally distributed, the likelihood is: l(β, σ2, λ) = −n 2log(2π) − n 2log(σ 2) + log(det|S λ|) − 1 2σ2(Sλy − Xβ) 0 (Sλy − Xβ) (4)
There are three unknown parameters, β, σ2 and λ with the dimensions respectively
first, to obtain a likelihood that is a function of λ only. This leads to an expression for β:
β = (X0X)−1X0Sλy (5)
The expression is substituted in the likelihood after which it is differentiated with respect to σ2. With the first order condition gives us the expression for σ2:
σ2 = 1
n(MxSλy)
0
(MxSλy) (6)
With Mx = I − X(X0X)−1X0. When both expressions for β and σ2 are put in the
likelihood, this results in:
l(λ) = −n 2log(2π) − n 2 − n 2log( 1 n(MxSλy) 0 (MxSλy)) + log(det|Sλ|) (7)
The likelihood is numerically optimized to obtain its maximum and maximum like-lihood estimator (MLE). When Sλ is not invertible the loglikelihood function will go to
−∞. For Sλ to be not invertible, the determinant of Sλ has to be equal to zero. This
is when λ = 1v, with v the eigenvalues of weight matrix W. Between these λ’s, there are local maxima and one global maximum. Furthermore, a confidence interval for the global maximum is determined and a Likelihood Ratio test is performed.
In the following chapter the results of the research is shown and discussed.
4
Results
In this chapter the results are discussed. First the maximum likelihood estimator, followed by the explanatory variables and lastly the likelihood ratio tests.
4.1
Maximum Likelihood
Equation 2 is the eventual likelihood function that is used with the X variable as follows: X = [1 PREMIUM DIFF BRANDS NPS SUST BRANCHES INVEST]. With these X variables, the likelihood functions with W1 and W2 resulted in the following graphs:
Figure 1: The likelihood functions, on the left with W1 and on the right W2
Figure 1 shows multiple maximum likelihood estimators for lambda between striped vertical lines. These lines are at the eigenvalues of W1, where the likelihood will go to
∞. In chapter 3, it was discussed why these points go to −∞.
The first thing that stands out, is that the global maximum of the likelihood function is one vertical line, this is at λ1 = 0.7182 and λ2 = 0.8432 5. It is expected that every
maximum has a parabola kind of shape, since the likelihood is based on a normal distribution. Both of the likelihood functions a similar shape at the global maximum. This means that the odd shape is not caused by the kind of matrix but rather by a variable. When looking at the data, the variable INVEST immediatly stands out. This is caused by the big differences in the amount of money the companies invest.
5The estimators in the research are marked with a 1 or 2 to indicate with which matrix it belongs. For example, λ1is the MLE of the likelihood function with W1.
Figure 2: Investments of the health insurers
In Figure 2 the health insurers Achmea, VGZ, CZ, Menzis, DSW,ONVZ, Zorg Zek-erheid, ASR and Eno are respectively 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is obvious that there are big differences between the amount each company invests in themselves. The highest amount is invested by Achmea. Achmea invests more than 10 billion and the lowest amount is A.S.R. with a small 200,000. In both the likelihood functions is IN-VEST insignificant, with W1 the p-value is 0.05941 and with W2 the p-value is 0.07799.
Therefore the variable INVEST is omitted. That gives the following likelihood func-tions:
Figure 3: The likelihood function for W1 and W2, respectively
In Figure 3 the likelihood functions without the variable INVEST are displayed. This likelihood function has lower values than the function with the omitted variable. In Figures 5 and 6 in the Appendix, the differences between the function with and without INVEST are very clear.
The functions for W1 and W2 are very similar. However the eigenvalues of the two
matrices are different, this causes the functions to go to ∞ at different values of λ. Both graphs show that there is one obvious global maximum that are not near the other maxima. The λ’s at the global maximum is for W1 at 0.4453 and for W2 at
0.4768. The likelihood function with W1 has a standard deviation 0.0158 and for W2 is
0.0167. Both the standard deviations are very small, which indicates that the data is closely distributed around the mean value.
Figure 4: The confidence interval of the likelihood function for W1 and W2, respectively.
For both of the likelihood functions a 95% confidence interval was constructed, Figure 8. The under bound for λ1 is 0.28814 and the upper bound is 0.59406. The
under bound of λ2 is 0.28979 and the upper bound is 0.64927.
4.2
Explanatory Variables
In this section the results of the parameters are discussed. The results of the likelihood function with W1 are as follows:
Variable Beta Standard Error T-Statistic P-Value
C -0.1687 0.0111 -15.1430 0.0046 PREMIUM 0.0011 0.0001 9.6383 0.0054 DIFF 0.0002 0.0001 1.8405 0.2128 BRANDS 0.0229 0.0022 10.4398 0.0088 NPS -0.0008 0.0005 -1.5961 0.2816 SUST -0.0092 0.0149 -0.6159 0.6104 BRANCHES 0.0037 0.0012 3.0588 0.0921
Table 1 shows that PREMIUM, NPS and SUST have an unexpected output. PRE-MIUM was expected to have a negative effect on the market share. This result suggests that a higher premium increases a health insurer’s market share. The coefficients of NPS and SUST were expected to be positive, the result shows this is not the case. However both the variables are insignificant. All the variables, apart from PREMIUM and BRANDS, are insignificant.
Variable Coefficient Standard Error t-Statistic P-Value
C -0.1477 0.0118 -12.5404 0.0076 PREMIUM 0.0011 0.0001 9.1100 0.0118 DIFF 0.0003 0.0001 1.8850 0.2003 BRANDS 0.0191 0.0023 8.2362 0.0157 NPS -0.0014 0.0006 -2.5291 0.1378 SUST -0.0093 0.0158 -0.5887 0.6237 BRANCHES 0.0042 0.0013 3.3414 0.0850
Table 2: Results of the parameters of the likelihood with W2
Table 2 shows the output of the parameters for likelihood function with W2. It
immediately stands out that these results are very similar to those of the likelihood function with W1. The variable PREMIUM has an unexpected output. However this
is one of the only two significant outputs. NPS and SUST also have not the expected output, still the outputs are insignificant. This also applies to DIFF and BRANCHES. The estimated β of PREMIUM in both the likelihood function are the same. However in the function with W1 the significance level is higher. This is also the case with the
4.3
Likelihood Ratio Test
To test the estimators in the SAR model, a likelihood ratio test is performed. The first test is with the null hypothesis that β = 0. Which gives a restricted maximum likelihood that is equal to a pure SAR model. When the restricted maximum likelihood is too far from the unrestricted maximum likelihood, the null hypothesis gets rejected. The p-value of the test for both the likelihood functions was zero, which is strong evidence that the unrestricted model fits the data better. The null hypothesis gets rejected which means that a pure SAR model is not preferred with the data.
The next likelihood ratio test is performed with null hypothesis λ = 0. The un-restricted model is equal to a linear regression. The p-value of this test for both the likelihood functions is also equal to zero. The null hypothesis is rejected which means a linear regression is not favoured.
5
Conclusion
This paper has investigated the market distribution of health insurers in The Nether-lands. The focus is on the structural dependence between the health insurers and to take this into account the spatial autoregressive model is used. By law every Dutch-man has to be insured and for this reason a shift in the market always affects the other health insurers. Apart from the structural dependence, other explanatory variables are included in the regression.
Figure 1 shows that the shape of the global maximum, for both the likelihood functions, is very strange. When omitting the variable INVEST, both functions change into a more expected shape. When collecting the data, investments was not easily found. This was caused by the different titles these investments have in the annual reports. While collecting the data the variety in the amount invested should have been
noticed.
It is remarkable that both the likelihood functions have one global maximum. The other maxima are not near the maximum. The maximum on the right of the global maximum, Figure 9 in the Appendix, is also very odd. As mentioned before, the maxima are expected to have similar shape like a parabola, since is normal distributed in the SAR model. Since it is not the global maximum this is not investigated. However the cause of this would be interesting to examine in further research.
Of the explanatory variables, only the premium of the Basisverzekering and the number of brands were significant. From an economic point of view, a lower premium will attract more customers. Tables 1 and 2 show that this does not apply here. This can perhaps be explained by the larger companies. Considering that they have a large share in the market, they do not feel the pressure to lower the premium.
The results show that there is indeed a structural dependence between health insur-ers. This probably does not only apply in the insurer market. For further research, it would be interesting to apply the model to different markets. I believe this would not only explain the market distribution better, but is also a way to investigate the SAR model better.
6
Appendix
Figure 5: The likelihood function W1, blue is without INVEST and red is with INVEST
Figure 7: Likelihood functions, blue is W1 and red is W2
References
Achmea. Annual Report 2016. Advised from
http://www.achmeabank.nl/Downloads/Jaarverslag%20Achmea%20Bank%202016.pdf Anselin, L. (2002). Under the hood issues in the specification and interpretation of spatial regression models. Agricultural economics, 27 (3), 247-267.
Anselin, L., Bera, A. K., Florax, R., Yoon, M. J. (1996). Simple diagnostic tests for spatial dependence. Regional science and urban economics, 26 (1), 77-104.
ASR Nederland. Annual Report 2016. Advised from
http://asrnederland.nl/media/2567/asr-annual-report-2016.pdf
Boogaars, L., Brink,R. 2016, 27 August. Hoe DSW al jaren de trend bepaalt [NOS]. Advised from https://nos.nl/artikel/2134713-hoe-dsw-al-jaren-de-trend-bepaalt.html
Coöperatie Eno U.A.. Jaarverslag 2016. Advised from
https://www.eno.nl/data/ck/jaarverslag-co
Coöperatie VGZ.. Maatschappelijk Jaarverslag 2016. Advised from https://www.cooperatievgz.nl/cooperatie-vgz/jaarverslag
DSW verlaagt onverwacht zorgpremie en eigen risico. 2017, 26 September. [NOS]. Advised from https://nos.nl/artikel/2194825-dsw-verlaagt-onverwacht-zorgpremie-en-eigen-risico.html
DSW Zorgeverzekeraar. Jaarverslag 2016. Advised from
https://web.dsw.nl/consumenten/jaarverslagen
Lan, F., Zhang, Y. (2011). Spatial Autoregressive Model of Commodity Housing Price and Empirical Research. Systems Engineering Procedia, 1, 206-212.
LeSage, J. P. (2008). An introduction to spatial econometrics. Revue d’économie industrielle, 123 (3), 19-44.
autoregressive models with group unobservables. Journal of Labor Economics, 28 (4), 825-860.
Massell, B. F. (1962). Investment, innovation, and growth. Econometric: Journal of the Econometric Society, 30 (12) , 239-252.
Menzis. Jaarverslagen en Kengetallen 2016. Advised from https://www.menzis.nl/over-menzis/jaarverslagen-en-kengetallen
Morgan, N. A., Rego, L. L. (2009). Brand portfolio strategy and firm performance. Journal of Marketing, 73 (1), 59-74.
MVO Nederland, Duurzaam bij consument in de lift, 27th of March in 2017, https://mvonederland.nl/nieuws/duurzaamheid-bij-consument-de-lift
Onderlinge Waarborgmaatschappij CZ groep Zorgverzekeraar U.A. Jaarverslag 2016. Advised from https://www.cz.nl/∼/media/over-cz/cz-groep-jaarverslag-2016.pdf?la=nl-nlrevid=7d8a1ca5-5359-4603-a442-eb3efdf0052f
ONVZ Zorgeverzekeraar. Jaarverslag 2016. Advised from https://onvz.maglr.com/nlNL/3514/58110/onvzjaarverslag2016.html
Reichheld, F. F. (2003). The one number you need to grow. Harvard business review, 81 (12), 46-55.
Rijksoverheid, Het Nederlandse Zorgsysteem, 9th of February 2016, https://www.rijksoverheid.nl/documenten/brochures/2016/02/09/het-nederlandse-zorgstelsel
Verbond van verzekeraars, Code Duurzaam Beleggen, 21st of December 2011,https://www.verzekeraars.nl/media/2223/code_duurzaam_beleggen.pdf Zorg en Zekerheid Groep. Jaarbericht 2016. Advised from
