Smooth structural changes and common factors in nonstationary panel data: an analysis of healthcare expenditures^†

Abstract

This article suggests new panel unit root tests that allow for multiple structural breaks and control for cross-correlations in the panel. Breaks are modeled with a Fourier function, which allows for smooth or gradual change rather than abrupt breaks. Cross-correlations are corrected by using the PANIC procedure. The simulations show that our tests have good size and power properties and perform reasonably well when the nature of breaks or the factor structure is unknown. The new panel unit root tests support fresh evidence on the persistence of healthcare expenditures in OECD countries.

Keywords:

• Common factors
• Fourier function
• panel unit root
• structural change

JEL Classification:

• C12
• C23
• F31

Data availability statement

The data used in this study are available from the corresponding author upon reasonable request.

Notes

1 Alternatively, one may use a fixed value of the cumulative frequency with m_i = 1, 2, or 3. It is possible to impose the same frequencies for all cross-sections, or to consider heterogeneous frequencies. Also, if one uses a single representative frequency k_i as in (5)(5) $$d_{i,t}=c_i+{\gamma }_{i}t+a_i\hspace{0.17em}\cos \hspace{0.17em}(2\pi k_{i}t/T)+b_i\hspace{0.17em}\sin \hspace{0.17em}(2\pi k_{i}t/T).$$(5) , one can determine the optimal frequency by minimizing the sum of squared residuals in the following regression, and one iteration will suffice: $\Delta y_{i,t}=$ $$ ${\gamma }_i+a_i\Delta \cos (2\pi k_{i}t/T)+b_i\Delta \hspace{0.17em}\sin \hspace{0.17em}(2\pi k_{i}t/T)+w_{i,t}^{*}.$

2 We are grateful to an anonymous reviewer who raised this point and made helpful suggestions.

3 We have conducted a small-scale simulation experiment to investigate how often the suggested search procedure using the F-test and the BIC selects the correct Fourier frequency. The simulation results are provided in the online Appendix in Figure D1.a, for the case where a variable is non-stationary and in Figure D1.b and D1.c for the case where a variable is stationary. They indicate that the correct frequency is chosen nearly 100% of the time both under the null and alternative hypotheses, which highlights the importance of the trigonometric terms in the DGP. Hence our simulation results show that the usual standard inference on the significance of the Fourier terms yields consistent estimation results, even under the unit root null model.

4 We are grateful to an anonymous reviewer for pointing out that the parameters in the two-type Fourier expansion must be consistently estimated in the iteration procedure. Then, the constructed test statistic is the same as that in Enders and Lee (2012a). Bai (2009) shows that an iterative procedure can yield consistent estimates of the coefficients in the panel models. The same result is supported in the panel unit root tests with dummy variables in Bai and Carrion-i-Silvestre (2009). It seems evident that the key parameters of the Fourier function also can be consistently estimated by the iterative procedure. Indeed, the iterative procedure converges quickly with a very small margin of error (0.0001).

5 The suggested procedure is easy to implement, and the results for the response surface function also can be used to obtain the p-value of the LM test with the Fourier approximation of Enders and Lee (2012a).

6 Refer to Bai and Ng (2002) for the procedure of how to consistently estimate the number of common factors based on principal component analysis. Note that the ICp criterion does not depend on the choice of r_max (Bai and Ng, 2002, p. 201).

7 To perform a robustness check, we consider different values of r_max and obtain the size and power properties that are similar to those reported in Table 1. To save space, the simulation results for different numbers of maximum factors, $r_{\max }=(2,3,4,6,7)\prime ,$ are not reported here, but are available in Tables C5 a-f in the online Appendix.

8 In order to investigate how often this search procedure selects the correct Fourier frequency, we conduct a small scale simulation experiment, as follows. Specifically, the DGP is given by equation (22)(22) $$y_{i,t}=d_{i,t}+{\pi }_i{F}_t+e_{i,t},e_{i,t}={\rho }_i{e}_{i,t-1}+{\epsilon }_{i,t}$$(22) , with $d_{i,t}=c_i+{\gamma }_{i}t+{\sum }_{k=1}^{m_i}{a}_{ik}\hspace{0.17em}\cos \hspace{0.17em}\left(\frac{2\pi kt}{T}\right)+{\sum }_{k=1}^{m_i}{b}_{ik}\hspace{0.17em}\sin \hspace{0.17em}\left(\frac{2\pi kt}{T}\right).$ We test $m_i=0$ vs $m_i=1$ and $m_i=1$ vs $m_i=2$ for various values of $a_i\text{and}{\mathrm{b}}_{\mathrm{i}}.$ Then, we compare the percentage of F-tests that correctly estimate ${\widehat{m}}_i=m_i$ under both the null of a unit root and the alternative of stationarity. The simulation results are reported in Figures D1.a through D1.c of the online Appendix. See also footnote #2.

9 We thank an anonymous reviewer for raising this point.

10 In order to save space, we report the simulation results for these experiments in Section C of the online Appendix.

11 We are grateful to an anonymous reviewer for raising this issue.

12 For all of the tests except for the test of Bai and Carrion-i-Silvestre (2009), the lags are determined by using the general-to-specific sequential procedure with a maximum of 3 lags. We also use 3 lags in the autoregressive model of the regression residuals to estimate the long-run variance for the Bai and Carrion-i-Silvestre (2009) test. A maximum of 2 breaks is allowed and trimming is set to 0.15 for the sharp break procedures in Bai and Carrion-i-Silvestre (2009) and Lee and Tieslau (2019). For the tests with a common factor structure, a maximum of 5 factors are allowed, and the number of factors is estimated using the panel information criterion $(I{C}_p)$ developed by Bai and Ng (2002). In order to save space, we refer the interested reader to Bai and Ng (2002) for a detailed description of information criteria and small sample performance.

13 We use a parametric model based on the augmented version of the tests to control for autocorrelations. Using a non-parametric model can lead to different results. The types of breaks can be another factor. The plots of data appear to exhibit evidence of multiple smooth breaks that circulate over time. The number of cumulative frequencies does not affect the test results and using m = 1 gives similar approximations of breaks as m = 1, 2.

Additional information

Funding

Saban Nazlioglu gratefully acknowledges that this work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under grant number 215K086.