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Modelling Time-Varying Parameters in Panel Data State-Space Frameworks: An Application to the Feldstein–Horioka Puzzle

Abstract

In this paper, we develop a very flexible and comprehensive state-space framework for modeling time series data. Our research extends the simple canonical model usually employed in the literature, into a panel-data time-varying parameters framework, combining fixed (both common and country-specific) and varying components. Under some specific circumstances, this setting can be understood as a mean-reverting panel time-series model, where the mean fixed parameter can, at the same time, include a deterministic trend. Regarding the transition equation, our structure allows for the estimation of different autoregressive alternatives, and include control instruments, whose coefficients can be set-up either common or idiosyncratic. This is particularly useful to detect asymmetries among individuals (countries) to common shocks. We develop a GAUSS code that allows for the introduction of restrictions regarding the variances of both the transition and measurement equations. Finally, we use this empirical framework to test for the Feldstein–Horioka puzzle in a 17-country panel. The results show its usefulness for solving complexities in macroeconomic empirical research.

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Notes

  1. The code can be downloaded at http://econweb.ucsd.edu/~jhamilto/KALMAN.ZIP.

  2. Excellent textbook treatments of state-space models are provided in Harvey (1993, 1989), Hamilton (1994a, b), West and Harrison (1997), or Kim and Nelson (1999), among others. They all use different conventions, but the notation used here is based on James Hamilton’s, with slight variations.

  3. A very simple example proposed by Faragher (2012) is what happens to the trajectory of a rocket when fuel injection is activated during flight.

  4. Parameter instability in dynamic econometric models has often been integrated in the form of structural change models (see Perron 1989). Markov-switching models, as in Hamilton (1989), constitute a less ambitious approach, despite the advantage of easily allowing the parameters to change gradually over time. The simplest TVP framework can be estimated by rolling regressions, while the Kalman filter is the most popular framework due to its simplicity. Our proposal, compared to the other alternatives, has the advantage of implementing a smooth time-varying transition overtime instead of discreet changes.

  5. See Rosenberg (1973), Hsiao (1974), Hsiao (1975), Ck and Zellner (1993), Swamy and Mehta (1977), Zellner et al. (1991) among others.

  6. Also known as “dispersed coefficient models” (Schaefer et al. 1975) or “mean reverting models”.

  7. Ohlson and Rosenberg (1982) formulation of the MRV model that allows for both autocorrelated (predictable) and random (unpredictable) variation within the same model, combining mean reversion to a random mean for parameters, where \(\left( \beta _{t}-\bar{\beta }\right) -\xi _{t}={\varPhi }\left[ \left( \beta _{t-1}-\bar{\beta }\right) -\xi _{t}\right] +\nu _{t}\). In this model, the constant “true” mean of the parameter, \(\bar{\beta }\), is affected by a random variable, \(\nu _{t}\), with zero mean and a variance \(\lambda \) (if \(\lambda =0\), then this model becomes the MRV presented above). \(y_{t}=\left( \bar{\beta }+\xi _{t}\right) x_{t}+\left[ \beta _{t}-\left( \bar{\beta }+\xi _{t}\right) \right] x_{t}+\omega _{t}\). This model allows for a heteroskedastic variance in the measurement equation, induced by the tendency of the parameter’s mean to fluctuate randomly about its “true” value, with \(u_{t}=x_{t}\xi _{t}+\omega _{t}\).

  8. Note that our Gauss code allows for multiple common-factors as well as the inclusion of potential restrictions on them.

  9. Increasing signal-to-noise ratio would weigh the observation heavier in the correction equations of the Kalman filter.

  10. See Wyplosz (1999) , Pagoulatos (1999) and Camarero et al. (2002) for a detailed description of this process.

  11. Exhaustive reviews can be found at Coakley et al. (1998) or Apergis and Tsoumas (2009).

  12. See also Tw (2002), Telatar et al. (2007), Mastroyiannis (2007), Özmen and Parmaksiz (2003) Kejriwal (2008) or Ketenci (2012).

  13. The predictions of the partial equilibrium inter-temporal theory of the current account refers to idiosyncratic components (country-specific or regional shocks) of saving and investment rates that, as they do not affect all countries similarly, are unlikely to generate imbalances in the world capital market.

  14. Examples of this are Glick and Rogoff (1995) and Ventura (2003).

  15. Harberger notes that the difference between gross domestic saving and investment to GDP has greater variability and larger absolute value for small countries than for large ones.

  16. An alternative variable for global risk aversion is the CBOE Volatility Index, known by its ticker symbol VIX; this variable was not available for the whole period. Note that CBOE VIX measures the stock market’s expectation of volatility implied by S&P 500 index options, calculated and published by the Chicago Board Options Exchange. Moreover, although the literature finds a relevant role for both proxies, while BAA spread measures risk appetite, a variation in implied volatility on a market may stem from a change in the quantity of risk on this market and not necessarily from a change in the investor’s risk aversion.

  17. The data has been obtained from https://data.worldbank.org/. Gross fixed capital formation (formerly gross domestic fixed investment) includes land improvements (fences, ditches, drains, and so on); plant, machinery, and equipment purchases; and the construction of roads, railways, and the like, including schools, offices, hospitals, private residential dwellings, and commercial and industrial buildings. According to the 1993 SNA, net acquisitions of valuables are also considered capital formation. Gross domestic savings are calculated as GDP less final consumption expenditure (total consumption). As in the literature, both variables have been expressed as a percentage of GDP.

  18. Adapting Bai and Perron (2003) methodology to a panel data framework.

  19. Following Bai and Ng (2004) and Moon and Perron (2004).

  20. An important part of the literature on time-varying parameter models would make this choice.

  21. See Bai and Carrion-i Silvestre (2009) for details.

  22. See Zivot and Andrews (2002), Perron (1997), Vogelsang and Perron (1998), Perron and Vogelsang (1992a, b), among others.

  23. The conventional wisdom is for capital to flow in the opposite direction: insufficient domestic saving is augmented by foreign saving to match investment demand, i.e., capital flows in, and this should be reflected in a current account deficit.

References

  • Apergis, N., & Tsoumas, C. (2009). A survey of the Feldstein–Horioka puzzle: What has been done and where we stand. Research in Economics, 63(2), 64–76.

    Google Scholar 

  • Bai, J., & Carrion-i Silvestre, J. L. (2009). Structural changes, common stochastic trends, and unit roots in panel data. Review of Economic Studies, 76(2), 471–501.

    Google Scholar 

  • Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70(1), 191–221.

    Google Scholar 

  • Bai, J., & Ng, S. (2004). A PANIC attack on unit roots and cointegration. Econometrica, 72(4), 1127–1177.

    Google Scholar 

  • Bai, J., & Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47–78.

    Google Scholar 

  • Bai, J., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18(1), 1–22. https://doi.org/10.1002/jae.659.

    Article  Google Scholar 

  • Bautista, C., & Maveyraud-Tricoire, S. (2007). Saving-investment relationship, financial crisis and structural changes in east Asian countries. Économie internationale, 3, 81–99.

    Google Scholar 

  • Bernoth, K., & Erdogan, B. (2012). Sovereign bond yield spreads: A time-varying coefficient approach. Journal of International Money and Finance, 31(3), 639–656.

    Google Scholar 

  • Bos, T., & Newbold, P. (1984). An empirical investigation of the possibility of stochastic systematic risk in the market model. Journal of Business, 57(1), 35–41.

    Google Scholar 

  • Broto, C., & Perez-Quiros, G. (2015). Disentangling contagion among sovereign cds spreads during the european debt crisis. Journal of Empirical Finance, 32, 165–179.

    Google Scholar 

  • Camarero, M., Carrion-i Silvestre, J. L., & Tamarit, C. (2009). Testing for real interest rate parity using panel stationarity tests with dependence: A note. The Manchester School, 77(1), 112–126.

    Google Scholar 

  • Camarero, M., Carrion-i Silvestre, J. L., & Tamarit, C. (2010). Does real interest rate parity hold for oecd countries? New evidence using panel stationarity tests with cross-section dependence and structural breaks. Scottish Journal of Political Economy, 57(5), 568–590.

    Google Scholar 

  • Camarero, M., Ordóñez, J., & Tamarit, C. (2002). Tests for interest rate convergence and structural breaks in the ems: further analysis. Applied Financial Economics, 12(6), 447–456.

    Google Scholar 

  • Carrion-i Silvestre, J. L., Kim, D., & Perron, P. (2009). GLS-based unit root tests with multiple structural breaks under both the null and the alternative hypotheses. Econometric Theory, 25(Special Issue 06), 1754–1792. https://doi.org/10.1017/S0266466609990326.

    Article  Google Scholar 

  • Min, C. K., & Zellner, A. (1993). Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates. Journal of Econometrics, 56(1–2), 89–118. https://doi.org/10.1016/0304-4076(93)90102-B.

    Article  Google Scholar 

  • Coakley, J., Kulasi, F., Smith, R., et al. (1998). The Feldstein–Horioka puzzle and capital mobility: A review. International Journal of Finance & Economics, 3(2), 169.

    Google Scholar 

  • Faragher, R. (2012). Understanding the basis of the Kalman filter via a simple and intuitive derivation [lecture notes]. IEEE Signal processing magazine, 29(5), 128–132.

    Google Scholar 

  • Feldstein, M., & Bacchetta, P. (1991). National saving and international investment. In: National saving and economic performance (pp. 201–226). University of Chicago press.

  • Feldstein, M. (1983). Domestic saving and international capital movements in the long run and the short run. European Economic Review, 21(1–2), 129–151.

    Google Scholar 

  • Feldstein, M., & Horioka, C. (1980). Domestic saving and international capital flows. The Economic Journal, 90(358), 314–329. https://doi.org/10.2307/2231790.

    Article  Google Scholar 

  • Giannone, D., & Lenza, M. (2010). The Feldstein–Horioka fact. In: NBER International Seminar on Macroeconomics 2009 (pp. 103–117). University of Chicago Press.

  • Glick, R., & Rogoff, K. (1995). Global versus country-specific productivity shocks and the current account. Journal of Monetary Economics, 1(35), 159–192.

    Google Scholar 

  • Gomes, F. A. R., Ferreira, A. H. B., & Filho, Jd J. (2008). The Feldstein–Horioka puzzle in South American countries: A time-varying approach. Applied Economics Letters, 15(11), 859–863.

    Google Scholar 

  • Gourieroux, C., & Monfort, A. (1997). Time series and dynamic models. Cambridge: Cambridge University Press.

    Google Scholar 

  • Granger, C. W. J. (2008). Non-linear models: Where do we go next—Time varying parameter models? Studies in Nonlinear Dynamics & Econometrics, 12(3), 1–9.

    Google Scholar 

  • Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica: Journal of the Econometric Society, 57(2), 357–384.

    Google Scholar 

  • Hamilton, J. D. (1994a). State-space models. In R. F. Engle & D. L. McFadden (Eds.), Handbook of econometrics (Vol. 4, pp. 3039–3080). New York: Elsevier.

    Google Scholar 

  • Hamilton, J. D. (1994b). Time series analysis (Vol. 2). Princeton: Princeton University Press.

    Google Scholar 

  • Harberger, A. C. (1980). Vignettes on the world capital market. The American Economic Review, 70(2), 331–337.

    Google Scholar 

  • Harvey, A. (1989). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.

    Google Scholar 

  • Harvey, A. C. (1993). Time series models (2nd ed.). London: Harvester Wheatsheaf.

    Google Scholar 

  • Hildreth, C., & Houck, J. P. (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association, 63(322), 584–595. https://doi.org/10.2307/2284029.

    Article  Google Scholar 

  • Ho, T. W. (2003). The saving-retention coefficient and country-size: the Feldstein–Horioka puzzle reconsidered. Journal of Macroeconomics, 25(3), 387–396.

    Google Scholar 

  • Hsiao, C. (1974). Statistical inference for a model with both random cross-sectional and time effects. International Economic Review, 15(1), 12. https://doi.org/10.2307/2526085.

    Article  Google Scholar 

  • Hsiao, C. (1975). Some estimation methods for a random coefficient model. Econometrica, 43(2), 305. https://doi.org/10.2307/1913588.

    Article  Google Scholar 

  • Ito, M., Noda, A., & Wada, T. (2014). International stock market efficiency: a non-bayesian time-varying model approach. Applied Economics, 46(23), 2744–2754.

    Google Scholar 

  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Fluids Engineering, 82(1), 35–45. https://doi.org/10.1115/1.3662552.

    Article  Google Scholar 

  • Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of basic engineering, 83(3), 95–108.

    Google Scholar 

  • Kejriwal, M. (2008). Cointegration with structural breaks: An application to the Feldstein–Horioka puzzle. Studies in Nonlinear Dynamics & Econometrics, 12(1), 1–39.

    Google Scholar 

  • Ketenci, N. (2012). The Feldstein–Horioka puzzle and structural breaks: Evidence from EU members. Economic Modelling, 29(2), 262–270.

    Google Scholar 

  • Khan, S. (2017). The savings and investment relationship: The Feldstein–Horioka puzzle revisited. Journal of Policy Modeling, 39(2), 324–332.

    Google Scholar 

  • Kim, C. J., & Nelson, C. R. (1999). State-space models with regime switching: Classical and Gibbs-sampling approaches with applications. Cambridge: MIT Press Books.

    Google Scholar 

  • Kim, D., & Perron, P. (2009). Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses. Journal of Econometrics, 148(1), 1–13. https://doi.org/10.1016/j.jeconom.2008.08.019.

    Article  Google Scholar 

  • Lenza, M. (2018). Special feature C: An empirical assessment of the Feldstein–Horioka’s saving-retention coefficient as a measure of financial integration in the Euro area. In: Financial integration in Europe, May 2018, ECB, pp 119–126.

  • Litterman, R. B., & Scheinkman, J. (1991). Common factors affecting bond returns. The Journal of Fixed Income, 1(1), 54–61.

    Google Scholar 

  • Ma, W., & Li, H. (2016). Time-varying saving-investment relationship and the Feldstein–Horioka puzzle. Economic Modelling, 53, 166–178.

    Google Scholar 

  • Mastroyiannis, A. (2007). Current account dynamics and the Feldstein and Horioka puzzle: The case of greece. The European Journal of Comparative Economics, 4(1), 91.

    Google Scholar 

  • Moon, H. R., & Perron, B. (2004). Testing for a unit root in panels with dynamic factors. Journal of Econometrics, 122(1), 81–126. https://doi.org/10.1016/j.jeconom.2003.10.020.

    Article  Google Scholar 

  • Murphy, R. G. (1984). Capital mobility and the relationship between saving and investment rates in oecd countries. Journal of International Money and Finance, 3(3), 327–342.

    Google Scholar 

  • Ohlson, J., & Rosenberg, B. (1982). Systematic risk of the CRSP Equal-Weighted Common Stock Index: A history estimated by stochastic-parameter regression. The Journal of Business, 55(1), 121–145.

    Google Scholar 

  • Özmen, E., & Parmaksiz, K. (2003). Exchange rate regimes and the Feldstein–Horioka puzzle: The French evidence. Applied Economics, 35(2), 217–222.

    Google Scholar 

  • Pagoulatos, G. (1999). Financial repression and liberalization in Europe’s southern periphery: From “growth state” to “stabilization state”. Technical report, European Community Studies Association (ECSA) Sixth Biennial International Conference, Pittsburg.

  • Paniagua, J., Sapena, J., & Tamarit, C. (2017). Sovereign debt spreads in EMU: The time-varying role of fundamentals and market distrust. Journal of Financial Stability, 33(Supplement C), 187–206. https://doi.org/10.1016/j.jfs.2016.06.004.

    Article  Google Scholar 

  • Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica, 57(6), 1361–1401.

    Google Scholar 

  • Perron, P. (1997). Further evidence on breaking trend functions in macroeconomic variables. Journal of Econometrics, 80(2), 355–385. https://doi.org/10.1016/S0304-4076(97)00049-3.

    Article  Google Scholar 

  • Perron, P., & Vogelsang, T. J. (1992a). Nonstationarity and level shifts with an application to purchasing power parity. Journal of Business & Economic Statistics, 10(3), 301–320. https://doi.org/10.1080/07350015.1992.10509907.

    Article  Google Scholar 

  • Perron, P., & Vogelsang, T. J. (1992b). Testing for a unit root in a time series with a changing mean: corrections and extensions. Journal of Business & Economic Statistics, 10(4), 467–70.

    Google Scholar 

  • Rao, C. R. (1965). The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. Biometrika, 52(3/4), 447. https://doi.org/10.2307/2333697.

    Article  Google Scholar 

  • Rosenberg, B. (1973). Random coefficients models: The analysis of a cross section of: Time series by. NBER Annals of Economic and Social Measurement, 2(4), 397–428.

    Google Scholar 

  • Rubin, H. (1950). Note on random coefficients. Statistical Inference in Dynamic Economic Models, 10, 419–421.

    Google Scholar 

  • Sachs, J. D., Cooper, R. N., & Fischer, S. (1981). The current account and macroeconomic adjustment in the 1970s. Brookings Papers on Economic Activity, 1981(1), 201–282.

    Google Scholar 

  • Sargan, J. D., & Bhargava, A. (1983). Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica, 51(1), 153–174. https://doi.org/10.2307/1912252.

    Article  Google Scholar 

  • Schaefer, S., Brealey, R., Hodges, S., & Thomas, H. (1975). Alternative models of systematic risk (pp. 150–161). International capital markets, North Holland, Amsterdam.

  • Swamy, P. A. V. B. (1970). Efficient inference in a random coefficient regression model. Econometrica, 38(2), 311–323. https://doi.org/10.2307/1913012.

    Article  Google Scholar 

  • Swamy, P. A. V. B., & Mehta, J. S. (1977). Estimation of linear models with time and cross-sectionally varying coefficients. Journal of the American Statistical Association, 72(360a), 890–898. https://doi.org/10.1080/01621459.1977.10479978.

    Article  Google Scholar 

  • Swamy, P., & Tavlas, G. S. (2003). Random coefficient models. In B. H. Baltagi (Ed.), A companion to theoretical econometrics (pp. 410–428). Hoboken: Blackwell Publishing Ltd.

    Google Scholar 

  • Telatar, E., Telatar, F., & Bolatoglu, N. (2007). A regime switching approach to the feldstein-horioka puzzle: Evidence from some european countries. Journal of Policy Modeling, 29(3), 523–533.

    Google Scholar 

  • Tw, Ho. (2002). A panel cointegration approach to the investment-saving correlation. Empirical Economics, 27(1), 91–100.

    Google Scholar 

  • Ventura, J. (2003). Towards a theory of current accounts. The World Economy, 26(4), 483–512.

    Google Scholar 

  • Vogelsang, T. J., & Perron, P. (1998). Additional tests for a unit root allowing for a break in the trend function at an unknown time. International Economic Review, 39(4), 1073–1100. https://doi.org/10.2307/2527353.

    Article  Google Scholar 

  • Wells, C. (1996). The Kalman filter in finance. New York: Springer Science & Business Media.

    Google Scholar 

  • Westerlund, J. (2006). Testing for panel cointegration with multiple structural breaks. Oxford Bulletin of Economics and Statistics, 68(1), 101–132.

    Google Scholar 

  • West, M., & Harrison, J. (1997). Bayesian forecasting and dynamic models (2nd ed.). New York, NY: Springer.

    Google Scholar 

  • Wyplosz, C. (1999). International financial instability (pp. 152–89). Global Public Goods: International Cooperation in the 21st Century.

  • Zellner, A., Hong, C., & Ck, Min. (1991). Forecasting turning points in international output growth rates using Bayesian exponentially weighted autoregression, time-varying parameter, and pooling techniques. Journal of Econometrics, 49(1–2), 275–304. https://doi.org/10.1016/0304-4076(91)90016-7.

    Article  Google Scholar 

  • Zivot, E., & Andrews, D. W. K. (2002). Further Evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business & Economic Statistics, 20(1), 25–44. https://doi.org/10.1198/073500102753410372.

    Article  Google Scholar 

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Acknowledgements

The authors are indebted to James D. Hamilton and J. LL. Carrion-i-Silvestre for providing them with the Gauss codes to implement some of the tests used in the paper. They also thank comments and suggestions from participants in the 5th ISCEF Symposium 2018 (Paris).

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Correspondence to Juan Sapena.

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The authors gratefully acknowledge the financial support from AEI/Ministerio de Economía, Industria y Competitividad (MINEIC) and FEDER Project ECO2017-83255-C3-3-P and the Generalitat Valenciana (PROMETEO/2018/102 and GV/2017/052). Authors are also indebted to the Chair “Betelgeux” for a Sustainable Economic Development, for its specific funding of this research. This paper has been developed within the research thematic network ECO2016-81901-REDT financed by MINEIC. The usual disclaimer applies.

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Camarero, M., Sapena, J. & Tamarit, C. Modelling Time-Varying Parameters in Panel Data State-Space Frameworks: An Application to the Feldstein–Horioka Puzzle. Comput Econ 56, 87–114 (2020). https://doi.org/10.1007/s10614-019-09879-x

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Keywords

  • Feldstein–Horioka puzzle
  • Panel unit root tests
  • Multiple structural breaks
  • Common factors
  • Kalman Filter
  • Time varying parameters

JEL Classification

  • C23
  • F32
  • F36