ECONOMICS SERIES SWP 2009/10. A Nonlinear Approach to Testing the Unit Root Null Hypothesis: An Application to International Health Expenditures

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1 Faculty of Business and Law School of Accounting, Economics and Finance ECONOMICS SERIES SWP 2009/10 A Nonlinear Approach to Testing the Unit Root Null Hypothesis: An Application to International Health Expenditures Paresh Kumar Narayan and Stephan Popp The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd.

2 A Nonlinear Approach to Testing the Unit Root Null Hypothesis: An Application to International Health Expenditures Paresh Kumar Narayan, School of Accounting, Economics and Finance, Deakin University. Stephan Popp, Department of Economics, University of Duisburg-Essen, Germany. Abstract In this paper, we examine the unit root null hypothesis for per capita total health expenditures, per capital private health expenditures, and per capital public health expenditures for 29 OECD countries. The novelty of our work is that we use a new nonlinear unit root test that allows for one structural break in the data series. We nd that for around 45 per cent of the countries we are able to reject the unit root hypothesis for each of the three health expenditure series. Moreover, using Monte Carlo simulations, we show that our proposed unit root model has better size and power properties than the widely used ADF and LM type tests. 1 Introduction Over the last decade there has been a surge in research on the time series properties of health expenditures (HE). One traditional branch of health economics research has focused on testing the unit root null hypothesis in health expenditure. Recent contributions to health economics research include examining common structural breaks in health expenditures (Narayan, 2006), convergence of health expenditures (Narayan, 2007; Hitiris, 1997; Barros, 199), and relative 1

3 importance of permanent and transitory shocks in explaining health expenditures (Narayan and Narayan, 200). In this paper, we do not contribute a new topic to health economics, rather we revisit the unit root null hypothesis for health expenditures. Our work here, however, is novel on three fronts. First, we examine the unit root null hypothesis for health expenditures by using a new nonlinear unit root test. It should be noted that in this literature, a nonlinear unit root test has not been used previously. This is an important contribution because, as we show later in the paper, based on Monte Carlo simulations, our new unit root test outperforms the conventional augmented Dickey and Fuller (ADF) and Lagrange Multiplier (LM) type tests used to study health expenditures. In other words, we show that the nonlinear test we use has superior size and power properties. The only work that comes closest to our work is Narayan (200), who examines evidence of asymmetric behaviour of per capita health expenditures. He uses the nonparametric Triples test and nds some evidence of asymmetric behaviour in per capita health expenditures. Second, we depart from the tradition of testing for a unit root in total health expenditures. Our empirical analysis is based on three health expenditure series: total per capita health expenditure, private per capita health expenditure, and public per capita health expenditure. The importance of considering this is sensible in light of the fact that the health care system in OECD countries di ers, based on some combination of private and public funded health care, but with di erent weights for each. 2

4 Our third contribution is that we study a wide range of OECD countries (29 countries) and drawing on the new OECD health dataset ensure that our time series sample is most up-to-date, allowing us to capture the more recent developments in health care systems in these countries. This is important, since recent developments (in terms of innovations and policies) can potentially be a source of structural breaks, which have implications for unit root testing. We organise the rest of the paper as follows. In the next section, we discuss the reasons why it is important to test for unit roots in health expenditures. In section 3, we present our nonlinear unit root test and explain the Monte Carlo design used to conduct size and power properties. In section 4, we discuss the data and results, including a comparison of the small sample performance of our test vis-a-vis the ADF type tests with and without a structural break and the minimum LM unit root test proposed by Lee and Strazicich (2004). In section 5, we provide some concluding remarks. 2 Importance of the unit root null hypothesis While testing the unit root null hypothesis in health expenditures dates back over two decades, dating from the work of Hansen and King (1996), and in this sense it is a traditional research topic, there is lack of understanding on the motivations for testing a unit root null hypothesis. As we will see below there are both economic (theoretical) as well as econometric modelling motivations behind this exercise, yet the absence of this understanding in the health economics 3

5 literature implies that testing the unit root null hypothesis is purely a statistical exercise. In this section, we attempt to dispel this view and make clear the motivation for unit root testing. There are three main reasons for the importance of testing the unit root null hypothesis. First, this hypothesis test pro ers knowledge on whether a shock has a permanent or a transitory e ect on health expenditures. When one rejects the unit root null, thus accepting the alternative hypothesis that there is no unit root, it implies that either the series is mean reverting or trend reverting. In other words, a series is deemed stationary when the unit root null is rejected. For a stationary health series, the implication is as follows. Over the time period for which the null was rejected, any health policies and any macroeconomic news, such as changes in interest rates and/or exchange rates, are interpreted to have a transitory e ect on the health expenditure series. This has implications for business cycles. The way this works is as follows. One branch of research has shown that as income increases health expenditures also tend to increase. In other words, there is clear empirical evidence that a positive association exists between per capita income and per capita health expenditures. Since the transitory component of income (or its growth rate or rst di erence) follows a cyclical pattern, depicting economic expansions and contractions, it follows that a similar pattern of expansions and contractions are likely for the health expenditure series. Second, the unit root null hypothesis has direct implications for the health expenditure convergence hypothesis. For instance, Narayan s (2007) conver- 4

6 gence hypothesis - whereby economies with low levels of health expenditures catch-up over time with economies with high levels of health expenditures - is by construction stochastic. This means that in a time series framework the di erence in health expenditures between any two countries should not contain a unit root. In other words, if the unit root null is accepted, then there is no convergence of health expenditures. Third, the unit root hypothesis has clear implications for econometric modelling. For example, one branch of the health economics literature has examined the long-run relationship (cointegration) between health expenditure and GDP. In testing for cointegration, a pre-requisite is that both variables contain a unit root; in other words, accepting the null hypothesis is a rst step toward conducting a test for a long-run relationship. Failure to accept the null hypothesis would only imply that health expenditures share a short-term relationship with GDP. In a related recent study, Narayan and Narayan (200) examine short-run comovement (common cycles) and long-run co-movement (common trends) among health expenditures. Their work suggests that one can only search for common cycles among stationary health expenditures and common trends among nonstationary health expenditures. So, a test for common cycles and common trends (whose motivations are explained in Narayan and Narayan, 200), requires as a pre-requisite, a test for the unit root null hypothesis. 5

7 3 Econometric Methodology 3.1 Nonlinear unit root test We apply the nonlinear unit root test proposed by Popp (200b). The test is based on the following DGP allowing for a smooth break in level and trend: y t = + t + (L) (DU 0 t + DT 0 t) + u t ; (1) u t = u t 1 + " t ; (2) " t = (L)e t ; e t iid(0; 2 e); (3) where DU 0 t = 1(t > T 0 B ) and DT t = 1(t > T 0 B )(t T B 0 ), 1(:) being the indicator function and T 0 B symbolizing the true break date. The parameters and indicate the magnitude of the level and slope break, respectively. It is assumed that all roots of the lag polynomial (L) lie outside the unit circle. The reduced form of the structural model serves then as test regression and has the following form: y t = y t t + D(T B ) t + DU t 1 + DT t 1 + e t (4) with = (1 ) + ; = (1 ); = ( + ), = ( ) and = : (5) This model allows for a break in level and slope for trending data and is de- 6

8 noted model 2 (M2). To restrict the test to a break in level for trending data (henceforth model 1, M1) we set = 0 in equation (1) and in the test regression (4). The null hypothesis = 1 of a unit root against the alternative hypothesis jj < 1 is tested using the pseudo t-statistic of ^. The form of the test regression is similar to the one used by Perron (1997) and Popp (200a) with the di erence that the regression equation (4) is nonlinear in the relevant coe cients, and. Taking the restrictions in (5) into account by estimating equation (4) using nonlinear least squares as described in Popp (200b) leads to e cient estimation of the coe cients and increases the power of the unit root test. The estimate of the unknown break date is that point in time for which the absolute t-value of ^ is maximised: ^T B = arg max T B jt^(t B )j: (6) The tests by Perron (1997) and Popp (200a) di er in the way they choose the break date. While Perron selects that point in time for which the absolute t-value of the coe cient of the slope dummy DT t is maximized, Popp focuses on the absolute t-value of the coe cient of the impulse dummy D(T B ) t as displayed in equation (6). A feature of both tests is that they permit a break under both the null and the alternative hypotheses. This is in contrast to the test by Zivot and Andrews (ZA, 1992), which only allows for a break under the alternative. Lee and Strazicich (2001) show that the ADF-type tests proposed by Perron 7

9 (1997) and ZA (1992) exhibit spurious rejections when a break occurs under the null hypothesis. Because they assign this shortcoming to the general design principle of the ADF-tests, they follow a di erent route by generalizing the LM unit root test of Schmidt-Phillips (1992) to structural breaks. But as shown by Popp (200a), this is not a common feature of all ADF-type unit root tests and also not of the nonlinear unit root test. 3.2 Monte Carlo design We generate the data to calculate the critical values and conduct the size and power analysis for models 1 and 2 according to equations (1) to (3) by using GAUSS version.0. All simulations are based on 10,000 replications of e t iidn(0; 1) each with T + 50 observations. Afterwards, we discard the rst 50 observations to avoid any e ect of the initial condition. To keep the simulations as concise as possible, we restrict our analysis to the case of a break fraction 0 = TB 0 =T of 0.5. We further assume (L) = 1 for the simulations, which means that the break takes e ect abruptly. The trimming factor is always 0.2 which means that we search for the break between the 20 and 0 percent quantile of the total sample, i.e. T B 2 [0:2T; 0:T ]. The critical values are computed for = 1 and T = 10, 20, 30, 40 and 50 under the assumption that no break has occured, i.e. = 0 for model 1 and = = 0 for model 2, based on equation (1). Critical values for further sample sizes can be found in Popp (200b). The size and power properties are calculated for = 1 and = 0:, respectively. In order to assess the e ect of

10 an increasing break magnitude to the test properties, we vary the level break parameter over the values 0, 3, 5 and 10 for model 1. For model 2, we consider all combinations of the level break sizes 2 f0; 5; 10g and the slope break sizes 2 f0; 2; 6; 10g. 4 Data and empirical results 4.1 Data We use three health expenditure data series obtained from the 2007 OECD health database: total per capita health expenditure, public per capita health expenditure, and private per capita health expenditure. For the empirical analysis, all data are converted into natural logarithmic form. Public health expenditure is expenditure incurred by public funds by the state, regional and local government bodies and social security schemes. By comparison private health care expenditure includes private sources of funds, such as out-of-pocket expenses (both over-the-counter and cost-sharing), private insurance programmes, charities and occupational health care (OECD, 2007). The three data series are in real values measured in local currencies. There are two reasons we prefer to work with local currencies rather than some common currency value, such as the US dollar. First, using local currency value ensures that we avoid any biasness in unit root results emanating from the exchange rate e ect. The unit root test hypothesis, motivated by purchasing power parity, is famous in international economics. Using a series based on say the US dollar 9

11 values, may mean that if one rejects (accepts) the null, this may be a result of the fact that the exchange rate variable was stationary (non-stationary). The second reason is that the health economics literature that uses US dollar denominated expenditure series or PPP based series nd mixed results (see results in Narayan, 2006). This con rms the earlier fear of distortionary e ects. It follows that it is more logical to work with domestic currencies. The data series are plotted in Figures 1 and 2. We notice two features of the data. First, a linear trend is observable in all the three series for all the OECD countries. Second, for most of the series, at least one structural break is also visible. In Table 1, we report the annual growth rates for the three health expenditure series. We observe the following. First, for 19 out of the 29 OECD countries the annual growth rate of total health expenditure, for 14 countries the growth rate of public health expenditures, and for 12 countries the growth rate of private health expenditures have been over 4 per cent per annum. 4.2 Results Critical values The critical values of our nonlinear unit root test for models 1 and 2 are reported in Table 2. As explained earlier, we generate critical values for T = 10, 20, 30, 40, 50. It is shown in Popp (200b) that the critical values of the nonlinear test under the assumption of an unknown and endogenously determined break date and those assuming a known break date both converge to the Dickey- Fuller critical values with increasing sample size. For empirical application, 10

12 we recommend the use of the critical values for exogenously given break dates because it leads to a test with empirical size close to nominal size when a break is present and to a test with high power. These critical values will be used for test decision and are displayed in Table Size and power properties In this section, we compare the size and power properties of our nonlinear structural break unit root tests with existing one break unit root tests used in the applied economics literature. In particular, we compare the performance of our nonlinear test with the Zivot and Andrews (1992), Lee and Strazicich (2004), Popp (200a), and Perron (1997) one structural break unit root test. For the sake of comparison, we also analyse the performance of the conventional Dickey and Fuller (1979) test which does not allow for any structural breaks in the data series. The aim of this comparison of the statistical performance of the tests is to show the strength of our test relative to those that are already available. The results for size and power are generated for both models 1 and 2. The results are based on T = 30, 50, 100 and = 0, 3, 5, 10. The results for model 1 are reported in Table 3. We notice that the empirical size of the ADF test is substantially undersized: with increasing break size and sample size, the empirical size converges to zero. By comparison, the Perron (1997) and the ZA (1992) tests are highly oversized with increasing break size. The LS (2004) and Popp (200a) tests, in contrast, have a nominal size close to the empirical 5 per cent level for medium sized breaks; even for large sized breaks 11

13 when T > 50, the size performance is relatively good. The size properties of our proposed nonlinear test, reported in the last columns of Table 3, suggest that the empirical size is close to the nominal 5 per cent level when a break is present ( > 0) even for small sample sizes, such as when T = 30. In terms of power, all tests show that with increasing sample size the power of the tests increases. Since our break date selection criteria is the same as that used in Popp (200a), the probability of detecting the true break is the same as for the Popp (200a) test. Hence, we compare our results directly with Popp (200a). It should be noted that the power of the nonlinear test is much better than that for Popp (200a). When compared with other tests, it is clear that the nonlinear test detects structural breaks more accurately than existing procedures. The results for the size properties of model 2 are reported in Table 4. We notice that the ADF and the Popp (200a) tests are mostly undersized, while the Perron (1997) and ZA (1992) tests are substantially oversized. The LS (2004) test is undersized in most of the cases, but gets considerably oversized with increasing slope break. The nonlinear test has stable size close to the nominal 5 per cent level. Equally important, the probability of detecting the true break date is close to 100 per cent with the Popp (200a) and the nonlinear test this performance is signi cantly superior to the rest of the one structural break test. In terms of the power of model 2, displayed in Table 5, all tests show high power but this power gain seems to have resulted from signi cant oversizing. The nonlinear test does not su er from this distortionary e ect. 12

14 4.2.3 Unit root test results We report the results from the total per capita health expenditure series in Table 6. The results are obtained from two models: M1 and M2, as explained earlier. The results are organised as follows: column 1 reports the list of countries, column 2 contains the sample size and the resulting number of time series observations is provided in column 3, column 4 reports the test statistic from M1 used to test the unit root null, columns 5 and 6 contain the break date and the break fraction, while column 7 reports the optimal lag lengths. The results from M2 are reported beginning column. The presentation of results for the per capita public health expenditure and per capita private health expenditure in Tables 7 and, respectively, are similarly organised. Beginning with the M1 results from the per capita total health expenditure series, we nd that the unit root null hypothesis is rejected at the 1 per cent level for Austria, Belgium, the Netherlands, and Sweden, at the 5 per cent level for Germany, Luxemburg, Portugal, Switzerland, and the UK, and at the 10 per cent level for Poland. In sum, the null hypothesis is rejected for 10 out of the 29 countries in our sample. Results obtained from the M2 model reveal that the unit root null hypothesis is rejected at the 1 per cent level for Australia, the Czech Republic, and the Netherlands, at the 5 per cent level for the UK, Spain, and Sweden, and at the 10 per cent level for Portugal and Switzerland. In total, the unit root null hypothesis from the M2 model is rejected for eight of the 29 countries. Considering the results from both models, for 13 countries the unit root null is 13

15 rejected. Turning to results obtained from the public health expenditure series shown in Table 7, we nd that the M1 model is able to reject the unit root null hypothesis at the 1 per cent level for Belgium, Hungary, Korea, Luxemburg, the Netherlands, Spain, and Sweden, at the 5 per cent level for the Slovak Republic, and at the 10 per cent level for Iceland, Italy, and Portugal. In sum, the unit root null hypothesis is rejected for 11 out of 29 countries. Results from the M2 model reveal that the unit root null hypothesis is rejected at the 1 per cent level for Hungary, Luxemburg, Spain, and Sweden, and at the 5 per cent level for Iceland, Belgium and Mexico. Results from the M2 model reveal that the unit root null hypothesis can be rejected for seven out of 29 countries. The results from the private health expenditure series are reported in Table. Results from M1 reveal that the unit root null hypothesis is rejected at the 1 per cent level for Canada, Ireland, Norway, and Poland, at the 5 per cent level for Australia, Denmark, and Germany, and at the 10 per cent level for Iceland, Japan, Portugal, and the UK. Taken together, the M1 model reveals that the unit root null hypothesis is rejected for 11 out of 29 countries. Results from M2 indicate that the unit root null hypothesis is rejected at the 1 per cent level for Canada and Ireland, at the 5 per cent level for Iceland, and at the 10 per cent level for Mexico and Germany. In all, for ve out of 29 countries the M2 model is able to reject the unit root null hypothesis. In summary, we observe the following. Models M1 and M2 together reveal 14

16 that the unit root null hypothesis can be rejected at conventional levels of signi cance for 13 out of 29 countries in the case of total health expenditure series, for 12 out of 29 countries in the case of public health expenditures and for 11 out of 29 countries in the case of private health expenditures. In general, then, there is evidence that for around 45 per cent of the countries the health expenditure series (total, private, and public) are stationary, while for just over half of the sample the series are non-stationary Discussion of results Table 9 provides a summary of the results relating to the rejection of the null hypothesis for the three health expenditure series. There are at least two reasons for the mixed results. First, the sample size of data di ers by country and is really dictated by data availability. Second, there is heterogeniety in terms of health systems in these OECD countries. In terms of funding for health-care, there is a considerable di erence in the amount of health-care expenditure across the sample. At the top end, in the United States, Switzerland, Germany and Belgium in 2005 health expenditure as a percentage of GDP was 15.3 per cent, 11.6 per cent, 10.3 per cent and 10.2 per cent respectively. By comparison, at the bottom end, in Ireland and Finland in 2005 health expenditure as a percentage of GDP was 7.5 per cent (OECD, 2007). Another source of heterogeneity between countries is in the relative importance of public and private health funding. The public sector is the main source of health funding in all OECD countries except Greece, Mexico and the United 15

17 States. However, private sector funding is more important in some OECD countries than others; see second last column of Table 1. Narayan and Narayan (200) examined permanent and transitory shocks in health-care expenditure in Canada, Japan, Switzerland, the United Kingdom, and the United States. They found that countries in which private funding dominated health-care expenditure, such as in Canada and the United States, were more likely to experience permanent or long-lasting shocks to health-care expenditure. We do nd this to be the case for the USA, Turkey and Greece where the role of the private sector is relatively high compared with the rest of the OECD countries. For these three countries, we are unable to reject the unit root null hypothesis for any of the three health expenditure series. Narayan and Narayan (200) found that for Japan and the United Kingdom, which have well-established public sectors, transitory shocks were more likely; in other words, health expenditure series for those countries dominated by public spending should be stationary. This is so because it seems that public spending of health care is less impacted by shocks in that shocks tend to have only shortterm e ects. This re ects, in large part, public expectations of a certain level of health-care delivery. On the other hand, private sector health providers react to shocks in a permanent way, implying that they make adjustments in response to shocks, thus shocks end up having long-term e ects. Our results to a large extent support the Narayan and Narayan (200) ndings. 16

18 5 Concluding remarks The goal of this paper was to examine the unit root null hypothesis for health expenditure series for OECD countries. The innovation of our work is threefold: rst, we identify the motivation for undertaking a test for the unit root null hypothesis; second, we depart from the tradition of testing the unit root null hypothesis for only the per capita total health expenditure and consider per capita private and public heath expenditures; and third, for the rst time in this literature, we propose a nonlinear approach to modelling the unit root null hypothesis. We study the size and power properties of our proposed nonlinear structural break unit root test with a range of existing one break unit root tests and con rm its statistically superior performance. Our results suggest that the unit root null hypothesis can be rejected at conventional levels of signi cance for 13 out of 29 countries in the case of total health expenditure series, for 12 out of 29 countries in the case of public health expenditures and for 11 out of 29 countries in the case of private health expenditures. In sum, then, there is evidence that for around 45 per cent of the countries the health expenditure series (total, private, and public) are stationary, while for the rest of the sample the series are non-stationary. The main implications of our ndings are that for at least 45 per cent of the OECD countries, shocks to health expenditures (either total, private or public) have only a transitory e ect. This means that shocks a ect health expenditures for only a short period of time. Given the positive link between per capita incomes and per capita health expenditures, in recessions when income levels 17

19 fall a negative shock to health expenditures then health expenditures will also fall. However, this fall in health expenditures is likely to be only for a short period of time, a behaviour consistent with the business cycle. Our results con rm this for at least 45 per cent of the countries. The second implication of our nding is embedded in econometric modelling, particularly cointegration analysis, where knowledge on the integrational properties of health expenditure series is a pre-requisite for the choice of econometric models and estimation techniques. 1

20 References Barros, P. (199): The black box of health care expenditure growth determinants, Health Economics, 7, Dickey, D., and W. Fuller (1979): Distribution of the Estimators for Autoregressive Time Series With a Unit Root, Journal of the American Statistical Association, 74(366), Hitiris, T. (1997): Health care expenditure and integration in the countries of the European Union, Applied Economics, 29, 1 6. Lee, J., and M. Strazicich (2001): Break Point Estimation and Spurious Rejections with Endogenous Unit Root Tests, Oxford Bulletin of Economics and Statistics, 63(5), (2004): Minimum LM Unit Root Test With One Structural Break, Working Paper 04-17, Department of Economics, Appalachian State University. Narayan, P. (2006): Examining Structural Breaks and Growth Rates in International Health Expenditures, Journal of Health Economics, 25, (2007): Do health expenditures "catch-up"? Evidence from OECD countries, Health Economics, 16, (200): Are health expenditures and GDP characterised by asymmetric behaviour? Evidence from 11 OECD countries, Applied Economics, DOI: /

21 Narayan, P., and S. Narayan (200): The role of permanent and transitory shocks in explaining international health expenditures, Health Economics, DOI: /hec OECD (2007): OECD Health Data. CREDES, OECD, Paris. Perron, P. (1997): Further Evidence on Breaking Trend Functions in Macroeconomic Variables, Journal of Econometrics, 0, Popp, S. (200a): New Innovational Outlier Unit Root Test With a Break at an Unknown Time, Journal of Statistical Computation and Simulation, forthcoming. (200b): A Nonlinear Unit Root Test in the Presence of an Unknown Break, Ruhr Economic Papers 45, Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Essen. Schmidt, P., and P. Phillips (1992): LM Tests for a Unit Root in the Presence of Deterministic Trends, Oxford Bulletin of Economics and Statistics, 54(3), Zivot, E., and D. Andrews (1992): Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis, Journal of Business and Economic Statistics, 10(3),

22 Figure 1: Logarithms of total health expenditure (solid line), public health expenditure (dashed line) and private health expenditure (dotted line) in 15 OECD countries 7 Australia 7 6 Austria 7 6 Belgium Canada Finland Czech Republic France Denmark Germany Greece Hungary Iceland Ireland Italy Japan

23 Figure 2: Logarithms of total health expenditure (solid line), public health expenditure (dashed line) and private health expenditure (dotted line) in 15 OECD countries Korea Luxembourg 7 Mexico Netherlands New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey United Kingdom United States

24 Table 1: Annual growth rates (in percent) for total, public and private health expenditure in 30 OECD countries Country Sample TotHE Sample PubHE Sample PriHE AvRatio Range Australia Austria Belgium Canada Czech Republic Denmark Finland France Germany Greece Hungary Iceland Ireland Italy Japan Korea Luxembourg Mexico Netherlands New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey UK USA Note: AvRatio is the average ratio of public HE to total HE. Range is the di erence between maximum and minimum ratio. 23

25 Table 2: Critical values of nonlinear unit root test for small samples Model 1 Model 2 T 1% 5% 10% 1% 5% 10%

26 Table 3: Size and power comparison of popular one-break unit root tests; Model 1 T ADF Perron Prob ZA Prob LS Prob Popp Prob NL Prob Prob: Probability of detecting the true break date P ( ^TB = T 0 B ) 25

27 Table 4: Size comparison of popular one-break unit root tests; Model 2 T ADF Perron Prob ZA Prob LS Prob Popp Prob NL Prob Prob: Probability of detecting the true break date P ( ^TB = T 0 B ) 26

28 Table 5: Power comparison of popular one-break unit root tests; Model 2 T ADF Perron Prob ZA Prob LS Prob Popp Prob NL Prob Prob: Probability of detecting the true break date P ( ^TB = T 0 B ) 27

29 Table 6: Results of nonlinear one break unit root test for total health expenditure in 30 OECD countries Country Sample T t M ^ 1 ^T B ^ k t M 2 ^ ^T B ^ k Australia Austria Belgium Canada Czech Republic Denmark Finland France Germany Germany Greece Hungary Iceland Ireland Italy Japan Korea Luxembourg Mexico Netherlands New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey UK USA Note: */**/*** denotes signi cance at the 10%/5%/1% level. 2

30 Table 7: Results of nonlinear one break unit root test for public health expenditure in 30 OECD countries Country Sample T t M ^ 1 ^T B ^ k t M 2 ^ ^T B ^ k Australia Austria Belgium Canada Czech Republic Denmark Finland France Germany Germany Greece Hungary Iceland Ireland Italy Japan Korea Luxembourg Mexico Netherlands New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey UK USA Note: */**/*** denotes signi cance at the 10%/5%/1% level. 29

31 Table : Results of nonlinear one break unit root test for private health expenditure in 30 OECD countries Country Sample T t M ^ 1 ^T B ^ k t M 2 ^ ^T B ^ k Australia Austria Belgium Canada Czech Republic Denmark Finland France Germany Germany Greece Hungary Iceland Ireland Italy Japan Korea Luxembourg Mexico Netherlands New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey UK USA Note: */**/*** denotes signi cance at the 10%/5%/1% level. 30

32 Table 9: Rejection of the null hypothesis Country Total HE Public HE Private HE Australia X - X Austria X - - Belgium X X - Germany X - X Luxemburg X X - Netherlands X X - Poland X - X Portugal X X X Sweden X X - Switzerland X - - UK X - X Czech Rep. X - - Spain X X - Mexico - X X Slovak Rep. - X - Korea - X - Italy - X - Iceland - X X Hungary - X - Canada - - X Denmark - - X Ireland - - X Japan - - X Norway - - X 31