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A time-series investigation of the U.S. real health expenditure: evidence from nonlinear unit root tests.

Abstract This paper contributes to the literature on health expenditure studies by applying the non-linear unit root tests formulated recently by Kapetanios et al. (Journal of Econometrics 112(2):359-79, 2003) to empirically test whether the U.S. real health expenditure time-series are non-stationary or non-linear and globally stationary during a relatively long period from 1965 to 2009. For comparison purposes, it also reports the results of a battery of traditional linear unit root tests and the Lee and Strazicich's minimum LM unit root (Review of Economics and Statistics 85(4):1082-1089, 2003) test for structural breaks. The empirical findings of the paper show that in the United States during the period under investigation, the real health expenditure time-series are non-stationary in levels. Policy implications of the empirical findings are discussed.

Keywords Health expenditure * Nonclinear unit roots tests

Introduction

The growth of health expenditure in the United States has been a major concern for its politicians, policy makers, and ordinary citizens. The ratio of health care spending to gross domestic product (GDP) in the at United States stood at around 18 % in 2010. The stochastic nature of the U.S. health expenditure time series over the long-run and the factors that have led to a phenomenal increase in its level and rate need to be investigated. In fact, many health studies have examined the stochastic nature and determinants of the health expenditure over a long-period in the United States and other Organization of Economic Co-Operation and Development (OECD) countries (Potrafke 2010; Das and Martin 2010; Narayan 2006; Narayan and Narayan 2008; Carrion-i-Silvestre 2005; Jewell et al. 2003; Lee et al. 2008; Okunade and Murthy 2002; Gerdtham and Lothgren 2000; Murthy and Okunade 2000; Hansen and King 1996). The most important question that these above mentioned studies address in their cointegration analyses is whether the health expenditure series in these countries is stationary in levels. Most of these studies conclude that the health care expenditure series is non-stationary and hence its moments such as the mean, variance, and autocorrelation depend on time. The determination of the stochastic nature of the series has many implications for econometric modeling and forecasting. If the series is found to be non-stationary in levels and hence is characterized as an integrated I(1) process, then any shocks to the series would be permanent and the series will not be mean-reverting. Using such a non-stationary series, any econometric modeling would be unreliable and spurious and the consequent forecasting would yield extremely wide bands. On the other hand, if the series is stationary, I(0), then any shocks to the series would be temporary. Furthermore in that case, econometric modeling, inferential statistics, and forecasting using stationary series would be reliable. In fact, there has been recent controversy over the theory-first and the data-first approach to econometric methodology. The unit root testing has become a central issue in the debate. Katarina Juselius (2010, p. 3), who is one of the architects of the Cointegration Vector Regression (VAR) methodology, contends that "accounting for (near) unit roots (i.e., persistence) in the model is a powerful tool to make the statistical inference more robust." She emphasizes the need for testing the nature of data generating processes in economic modeling by stating that
  "When variables and relations are non-stationary rather than
  stationary many standard procedures and assumptions in economics
  become highly questionable, such as the use of the ceteris paribus
  assumption to simplify the economic model. This is because it is
  unlikely that conclusions from such a model would remain unchanged
  when the ceteris paribus variables are non-stationary rather than
  stationary."  (Juselius 2010)


Therefore, for structural analysis, forecasting, and policy guidance, there is an imminent need for an empirical analysis of the stochastic properties of time series of interest. Although much is written on issues relating to health expenditure in the United States, no studies on testing the stochastic properties of the U.S. health expenditure series by employing nonlinear unit root tests exist. The relevance of the application of the nonlinear unit root test is based on the grounds that intervention in health care, changing preferences, and transaction costs might generate a nonlinear path in the behavior of the health expenditure series (Jones and Hall 2007). Therefore, in order to fill in the void and extend the literature, the present paper empirically attempts to test the stochastic properties of the time series on the health expenditure in the United States over the period of 1965-2009 using the nonlinear unit root tests developed by Kapetanios et al. (2003)

Before we employ the nonlinear unit root tests of Kapetanios et al. (2003) to study the time-series properties of the U.S. health expenditure series, we apply a battery of unit root tests that are both linear and discern structural breaks. A brief survey of these tests is warranted. In the time-series literature, many tests are available to examine the presence of a unit root in the data generating process. The most widely used traditional linear unit root tests are the augmented Dickey-Fuller (ADF) unit root test, the augmented Dickey-Fuller Generalized Least Squares (ADF-GLS) unit root test, the Kwiatkowski et al.(KPSS 1992) unit root test and the Ng-Perron's [MZ.sub.[alpha]] and the MZ, unit root tests (Ng and Perron 2001). The ADF-GLS unit root test, proposed by Elliot et al. (1996), is a modified test of the ADF test, which is power invariant. The KPSS test considers the observed time series as the sum of both stationary and nonstationary components and tests the stationarity of the variance of the nonstationary component. The KPSS test, unlike the ADF, ADF-GLS, MZ and the MZ t unit root tests, maintains the null hypothesis of stationarity. Ng and Perron (2001) have proposed two modified tests, the MZ, and the MZ t unit root tests, to overcome the power problem associated with the traditional ADF and the Phillips-Perron (1988) unit root tests.

The above discussed traditional unit root tests have been criticized on several accounts, the main being that they do not consider the presence of structural breaks in the data generating processes. Often, they erroneously misinterpret a structural break as the presence of a unit root (Perron 1989). In order to overcome this shortcoming in the time-series literature, several unit root tests of the Dickey-Fuller type that incorporate structural breaks have been proposed. Some of the widely used tests to determine the presence of a unit root in the presence of structural breaks are the Zivot-Andrews unit root tests (1992) and the Lumsdaine-Papel (1997) unit root tests. However, these tests have been criticized by Nunes et al. (1997) and Lee and Strazicich (2003) on the grounds that in all these tests the null hypothesis is that the series of interest is a unit process with no structural breaks. In applications of these tests, if in reality a structural break exists under the null of a unit root, these tests have been known to suffer from size distortions and hence they often conclude that the null is rejected as well as estimating the break incorrectly. Thus when the null is rejected, these tests may erroneously conclude that the series is trend stationary, when in fact, the series is difference stationary with structural breaks. Therefore, Lee and Strazicich (2003) [L&S tests, hereafter] have proposed a more powerful unit root test, with one or two structural breaks under both the null and alternative hypothesis. The rejection of the null under the L & S unit root tests is a strong and genuine indication of stationarity of the data generating process.

Model Specification and Data

In the presence of nonlinearities in the data generating process, where the speed of adjustment towards the equilibrium is asymmetric, the traditional linear unit root tests, because of their low power, often fail to reject the null hypothesis of unit root behavior. To remedy this problem, Kapetanios et al. (2003) have recently developed a nonlinear unit root test, which has come to be called the KSS or the Nonlinear ADF (NLADF) unit root test. The NLADF test can be conducted for a data generating process with different deterministic terms including the demeaned data ([NLADF.sub.DM], [NLADF.sub.DT] and [NLADF.sub.RAw DATA]), de-trended data, and the raw data. In light of these recent developments in time-series econometrics, this paper attempts to test the stochastic nature of the U.S. health care expenditure series for the period 19652009 by specifying the following reparameterised exponential smooth transition autoregressive (ESTAR) model for the data generating process model, xt, as suggested by Kapetanios et al. (2003):

[[DELTA]x.sub.t] = [[phi]x.sub.t-1] + [[gamma]x.sub.t-1][(1-exp(-[delta][x.sub.t-d.sup.2])] + [[epsilon].sub.t]. (1)

We re-specify model (1), by imposing that (p=0 and d=1, as

[[DELTA]x.sub.t] = [[gamma]x.sub.t-1][(1-exp(-[delta][x.sub.t-1.sup.2])] + [[epsilon].sub.t]. (2)

In model (2) under the null hypothesis of the presence of a unit root, the speed of mean reversion parameter is zero ([theta]= 0) and it is positive ([theta] > 0) under the alternative hypothesis of a nonlinear but globally stationary data generating process [see, for further mathematical details, Kapetanios et al. (2003).] As it is mathematically clear that testing the null hypothesis is not feasible because [gamma] is not identified in (2) under the null, using the first-order Taylor approximation, Kapetanios et al. (2003) propose the following estimable auxiliary regression model, which can also be adjusted for the presence of serially correlated errors, with j augmentations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In model (3), under the maintained null hypothesis of the presence of a unit root, [sigma] = 0 against the alternative hypothesis of [sigma] < 0. To estimate the nonlinear model (3), we use the procedure of the Ordinary Least Squares (OLS). The test statistic [t.sub.NL], is obtained by dividing the estimated 5 in model (3) by its standard error and is reported as-[t.sub.NL] = [^.[sigma]]/ S.e.([^.[sigma]]). As the asymptotic standard normal distribution critical values are not defined for these tests, we use the 1 %, 5 %, and 10 % critical values, provided by Kapetonios et al. (2003), for testing the statistical significance of the estimated 5 for the raw, demeaned, and de-trended data series for various deterministic terms in the auxiliary regression model (3). We also present an estimate of the speed of mean reversion parameter [delta] and evaluate its magnitude, sign, and statistical significance.

The data used for testing the time series properties of the U.S. health care expenditure are gathered from the Economic Report of the President 2010 (2010) and the Statistical Abstract of the United States: 2010 (2009). The heath expenditure per capita are converted to 1982-84 U.S. Dollars using the medical care price index, which is a component of the U.S. consumer price index, with 1982-84=100. The time-series on real per capita health expenditure and the series in its natural log are denoted respectively, as HEXP and LHEXP.

Empirical Findings

Table 1 presents the summary statistics of the variable-series examined in the paper.
Table 1 Per capita health expenditure (HEXP) and natural
log of Per capita health expenditure (LHEXP)

Series      Mean   Maximum  Minimum   Standard         JB
                                     Deviation  Statistics

HEXP    1073.134  1407.853  530.878    257.600       4.470
(a)

LHEXP      6.944     7.249     6.29      0.276       7.990
(a)

(a)Expressed in 1982--84 dollars. 1965 2009


The graphs of the U.S. per capita health expenditure (HEXP) and natural log of U.S. per capita health expenditure (LHEXP) series are shown respectively, as Figs. 1 and 2 (see Appendix A). The graphs reveal an upward non-linear trend in both the series, with changes in the slope during the period 1994-2000.

In Table 2, we report the results of the traditional linear unit root tests for the level and first-differenced series. The ADF, ADF-GLS, [M.sub.x] and the [MZ.sub.t] unit root test results clearly show that both the LHEXP and HEXP series are non-stationary at the 1 % level. While the KPSS unit root tests indicate that the level series are non-stationary at the 5 % level, as we reject the null hypothesis of stationarity, for the differenced series of ALHEXP, we fail to reject the null at the 1 % level and for AHEXP series at the 10 % level. Thus, the overall tests results confirm that the level series are integrated of the order one, 1(1).
Table 2 Traditional linear unit root test results

Series           ADF  ADF-GLS   KPSS  [MZ.sub.[alpha]      MZ

HEXP          -1.464   -1.003  0.208           -2.071  -0.882

[DELTA]       -5.198   -5.211  0.099       -20-419 **  -3.195
                 (a)      (a)    (a)                       **

LHEXP         -2.916   -1.423  0.213           -6.999   1.771

[DELTA]LHEXP -4.818   -4.924  0.141       -19.975 **  -3.154
                 (a)      (a)     **                       **

(a.) Significant at the 10 % level in rejecting the null-hypothesis.
Optimal lags based on the Schwarz Bayesion criterion (S BC)
criterion. The 1 %, critical value for the ADF unit root tests is
-4.186. The 1 %, 5 %, and 10 % critical values for MZ . and MZt are
-23.80, -17.30, 14.20, and -3.42, -2.91, and 2.62, respectively.
Dcterministic terms include both the constant and time trend


As discussed above, ignoring the presence of structural breaks in the data generating process would result in size distortion and lower the power of the tests. Therefore, in order to control for this phenomenon, we test the presence of structural breaks in the HEXP and LHEXP series before we apply the Kapetanios et al. (2003) nonlinear unit root tests. This is done by applying the recent and most widely used Lee and Strazicich's endogenous two-break minimum LM unit root test (L&S unit root test, hereafter). The model estimated in this paper is Model CC, which incorporates two breaks both in the intercept and the slope of the trend function. [For details, see Lee and Strazicich (2003)]. The results of the L&S unit root test are reported in Table 3.
Table 3 Lee-Strazicich minimum LM unit root test of two structural
breaks

Series  [s.sub.t-1]  [B.sub.lt]  [DT.sub.lt]  [B.sub.2t]  [DT.sub.2t]

LHEXP        -0.358      -0.007       -0.037      -0.019       -0.023

           (-2.857)    (-0.42$)   (-9.011) *    (-2.978)     (-3.375)
                                                     ***

HEXP         -0.336       8.306        0.863     -21.160      -24.427

           (-2.768)    (-0.547)      (0.114)    (-1.445)     (3.079)*

Series  [TB.sub.1]  [TB.sub.2]

LHEXP         1979        1996

HEXP          1979        1996

Break locations, [lambda]l = TB1/T) and [lambda]2 = TB2/T. Critical
values for unit root tests on the co-efficient of St-1 for [lambda]l
and [lambda]2, at the 1 %, 5 %, and 10% levels respectively, are -6.32,
-5.71, and--5.33 [Model CC] (Lee and Strazicich 2003, Table 2). T-values
are reported in parentheses


The results from the L & S unit root tests presented in Table 3 vividly show that both the HEXP and LHEXP series are non-stationary, although they have undergone structural breaks during the period under investigation. The HEXP series has experienced two structural breaks in 1979 and 1996 with two statistically significant changes in both the intercept and the slope of its trend function. Similarly, on the other hand, the LHEXP series underwent a statistically significant change in its slope in 1979 at the 1 % level and a statistically significant change in its intercept in 1996 at the 10 % level. The structural breaks that HEXP series underwent in 1979 and 1996 can be broadly attributed respectively to a change in the structure of Medicare and the formation of Regional Health Alliance. The LHEXP series did experience structural breaks in 1979 and 1996. These structural breaks can be broadly attributed to the impact of the enactment of the National Health Planning Act and the Employee Retirement Income and Security Act of 1974, energy crisis in 1973, the long-term care reform of 1994, and the impact of technological changes in the 1970s and 1990s that led to health care expenditure. In 1996 in the United States, the Health Insurance Portability and Accountability Act (HIPAA) was enacted to continue health insurance protection for workers leaving employment. However, the overall conclusion from the results reported in Table 3 show that both of these series are a unit root or stochastic processes. Our results on the presence of structural breaks are broadly consistent with those of Jewell et al. (2003), Lee et al. (2008), and Sharma and Srivatsava (2011) on both structural breaks and the presence of a unit root in the health care expenditure series, despite the differences in the time-span of investigation.

As it has been demonstrated in the time-series literature, the traditional linear unit root tests lack power if the data generating process is nonlinear. Therefore, we test the presence of a unit root using the nonlinear unit root tests, NLADF tests, developed recently by Kapetanios et al. (2003). These tests have been shown to exhibit more power and less size distortions. The nonlinear unit root tests results for the raw data, demeaned and de-trended series are reported in Table 4. The results are similar for the tests with demeaned and raw data series. Using the NLADFRD, NLADFDm, and NLADFDT test statistics, we fail to reject the null of the presence of a unit root for HEXP and LHEXP series at the 1 % and 5 % levels of significance. As both the HEXP and LHEXP data series exhibit a trend, we focus on the results only for the de-trended series. In Table 5, we present estimates of the mean reversion parameters and their t-statistics for the de-trended series. It can be noted that the magnitude of the estimated mean reversion parameters is very small and their statistical significance is marginal, confirming that the series are non-mean reverting. Thus the traditional linear unit root tests, L&S unit root test, and the NLADF test results strongly confirm that the HEXP and the LHEXP series are non-stationary in levels.
Table 4 Kapetanios et al. nonlinear unit root tests

Series [NLADF.sub.DM]    [NLADF.sub.DT]    [NLADF.sub.RAW DATA]

HEXP           -2.013         -2.179(1)                  1.680

LHEXP          -2.056        -2.846 (2)                  1.735

The 1 % and 5 % critical values for the tNLRAW DATA, tNLT, and
tNLM test are respectively, -2.22, -2.82, -3.48, -2.93, -3.93,
and -3.40. Deterministic terms include both the constant and
time trend. The t NLT results are based on the de-trended data.
Optimal lags in parentheses

Table 5 Mean reversion coefficient for nonlinear unit root
tests (De-trended)

Series    [NLADF.sub.DT]    Estimated[theta]    T-Statistics

HEXP          -2.179 (1)               0.043         1.995 *

LHEXP         -2.846 (2)               0.036       -2.759 **

The asterisks * and ** denote significance at the 5 % and 1 %
levels. The 1 % and 5 % critical values for the tNLT test
are--3.93 and--3.40. Deterministic terms include both the
constant and time trend. The tNLT results are based on the
de-trended data. Optimal lags in parentheses


Kapetanios et al. assume that in Eq. (2), the location parameter, c, is zero. Recently, Kruse (2011) has extended the Kapetanios et al. (2003) test to allow for a nonzero location parameter. This test is known as the Kruse Tau test. The detailed mathematical and statistical derivations are discussed in Kruse (2011). Kruse (2011) demonstrates that this test performs extremely well in terms of power and size. In order to investigate whether the results presented in Table 4 are robust to instances where the location parameter is nonzero the results of the Kruse test are reported in Table 6. As it is clear from the results shown in Table 6, the observed Tau values for both the demeaned and de-trended series are less than the critical values at both the 1 % and 5 % levels, and therefore, we fail to reject the null hypothesis of non-stationarity.
Table 6 Kruse (2011) 'Tau' nonlinear unit root tests

Series    [Tau.sub.DM]    [Tau.sub.DT]

HEXP             5.952           4.713

LHEXP            6.121           4.706

The critical values for the demeaned and de-trended series at the
1 % and 5 % significance levels are 13.75,17.10, and 10.17, and
12.82, respectively (Kruse 2011)


Conclusions and Policy Implications

This paper, for the first time in the heath economics literature, examined the time series properties of the health care expenditure series, both in original units and logarithms, over the period 1965 to 2009 by applying the nonlinear unit root test developed recently by Kapetanios et al. (2003). For a comparative study, it also applied a battery of the traditional linear unit root tests and the widely used Lee and Strazicich's minimum LM unit root tests with two structural breaks. While the Lee and Strazicich (2003) LM unit root test helps us discern the presence of structural breaks without considering the nonlinearities in the data generating processes, the Kapetanios et al.(2003) test explicitly takes into account the possibilities of nonlinear adjustments in the unit root processes. The empirical results from all the tests reveal that, despite experiencing structural breaks, the health expenditure series in the United States during the period of investigation are non-stationary in levels and thus are integrated of the order one and are non-mean reverting. Furthermore, the series behaving as random walks would indicate that the health care market is efficient with a lack of presence of profitable opportunities. Over intervention in the form of policy changes and reforms would lead to difficulties in forecasting health spending. This finding has several policy implications, the main being that health-policy imparts permanent shocks to the series, and econometric modeling for structural analysis and forecasting necessitates cointegration. The dominant drivers of health care expenditure are aggregate supply-based or real factors such as changes in health care technology. This conclusion is consistent with the prediction of the real business cycle theory. The aggregate demand-based changes in consumers' preferences for more and better treatment using the available advanced medical technology may also have played a relatively less important role in raising the health care costs in the United States during the period under study. Although the results of the unit root tests reported in this paper confirm the non-stationarity nature of the stochastic properties of the U.S. health expenditure time series, as found in the previous literature using the traditional tests, the contributions of the paper lie in exploring the possibilities, using a longer span of data, that the data generating process might be stationary.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Acknowledgements This paper is a revised version of a paper presented at the 2011 Academy of Economics and Finance annual meeting on February 11, 2011 in Jacksonville, Florida.

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Int Adv Econ Res (2012) 18:429-438

Published online: 9 August 2012

[c] International Atlantic Economic Society 2012

V. N. R. Murthy ([??])

Department of Economics and Finance, Creighton University, Omaha, NE 68178 USA

e-mail: vmurthy@creighton.edu

DOI 10.1007/s11294-012-9374-z
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