Exploring US Business Cycles with Bivariate Loops Using Penalized Spline Regression
Tóm tắt
The phrase business cycle is usually used for short term fluctuations in macroeconomic time series. In this paper we focus on the estimation of business cycles in a bivariate manner by fitting two series simultaneously. The underlying model is thereby nonparametric in that no functional form is prespecified but smoothness of the functions are assumed. The functions are then estimated using penalized spline estimation. The bivariate approach will allow to compare business cycles, check and compare phase lengths and visualize this in forms of loops in a bivariate way. Moreover, the focus is on separation of long and short phase fluctuation, where only the latter is the classical business cycle while the first is better known as Friedman or Goodwin cycle, respectively. Again, we use nonparametric models and fit the functional shape with penalized splines. For the separation of long and short phase components we employ an Akaike criterion.
Tài liệu tham khảo
Akaike H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6): 716–723. doi:10.1109/TAC.1974.1100705 MR0423716.
Akritas M., Politis D. (2003) Recent advances and trends in nonparametric statistics. Elsevier, Amsterdam
Atkinson A. (1969) The timescale of economic models: How long is the long run?. Review of Economic Studies 36: 137–152
Baxter M., King R. (1999) Measuring business cycles: Approximate band-pass filters for economic time series. The Review of Economics and Statistics 81(4): 575–593
Breslow N. E., Clayton D. G. (1993) Approximate inference in generalized linear mixed model. Journal of the American Statistical Association 88: 9–25
Brockwell P. J., Davis R. A. (1987) Time series: Theory and methods. Springer, Berlin
Burns A. F., Mitchell W. C. (1946) Measuring business cycles. National Bureau of Economic Research, New York
Chiarella C., Flaschel P., Franke R. (2005) Foundations for a disequilibrium theory of the business cycle. Qualitative analysis and quantitative assessment. Cambridge University Press, Cambridge
Christiano L. J., Fitzgerald T. J. (2003) The band pass filter. International Economic Review 44(2): 435–465
Currie I., Durban M. (2002) Flexible smoothing with P-splines: A unified approach. Statistical Modelling 2: 333–349
de Boor C. (1978) A practical guide to splines. Springer, Berlin
Durban M., Currie I. (2003) A note on P-spline additive models with correlated errors. Computational Statistics 18: 251–262
Eilers P., Marx B. (1996) Flexible smoothing with b-splines and penalties. Statistical Science 11(2): 89–121
Einbeck J., Tutz G., Evers L. (2005) Local principle curves. Statistics and Computing 15(4): 301–313
Eubank R. L. (1989) Spline smoothing and nonparametric regression. Dekker, New York
Fan J., Gijbels I. (1996) Local polynomial modelling and its applications. Chapman & Hall, London
Fan J., Yao Q. (2003) Nonlinear time series: Nonparametric and parametric methods. Springer, New York
Fisher N. (1995) Statistical analysis of circular data (3rd ed.). Cambridge University Press, Cambridge
Flaschel, P., Kauermann, G., & Teuber, T. (2008). Long cycles in employment, inflation and real unit wage costs. Qualitative analysis and quantitative assessment. American Journal of Applied Sciences, 69–77.
Friedman M. (1968) The role of monetary policy. American Economic Review 58: 1–17
Gallegati, M., Ramsey, J. B., & Semmler, W. (2006). The decomposition of the inflation-unemployment relationship by time scale using wavelets. In C. Chiarella, R. Franke, P. Flaschel, & W. Semmler (Eds.), Quantitative and empirical analysis on nonlinear dynamic macromodels, Vol. 277 of Quantitative and empirical analysis of nonlinear dynamic models (Chap. 4, pp. 93–112). Amsterdam: Elsevier.
Goodwin R. (1967) A growth cycle. In: Feinstein C. (eds) Socialism, capitalism and economic growth. Cambridge University Press, Cambridge, pp 54–58
Härdle W., Marron J. S. (1991) Bootstrap simultaneous error bars for nonparametric regression. The Annals of Statistics 19(2): 778–796
Harvey A., Jaeger A. (1993) Detrending, stylized facts and the business cycle. Journal of Applied Econometrics 8: 231–247
Hastie T., Stützle W. (1989) Principle curves. Journal of the American Statistical Association 84: 502–516
Hastie T., Tibshirani R. (1990) Generalized additive models. Chapman and Hall, London
Härdle W., Lütkepohl H., Chen R. (1997) A review of nonparametric time series analysis. International Statistical Review 65: 49–72
Hodrick R., Prescott E. (1997) Postwar U.S. Business Cycles: An empirical investigation. Journal of Money, Credit, and Banking 29: 1–16
Kauermann G. (2005) Penalised spline fitting in multivariable survival models with varying coefficients. Computational Statistics and Data Analysis 49: 169–186
Kauermann G., Krivobokova T., Fahrmeir L. (2009) Some asymptotic results on generalized penalized spline smoothing. Journal of the Royal Statistical Society, Series B 71(2): 487–503
Kauermann, G., Krivobokova, T., & Semmler, W. (2011). Filtering time series with penalized splines. Studies in Nonlinear Dynamics and Econometrics, 15(2), Article 2.
Koopmans T. C. (1947) Measurement without theory. Review of Economic Statistics 29: 161–172
Krivobokova T., Kauermann G. (2007) A note on penalized spline smoothing with correlated errors. Journal of the American Statistical Association 102: 1328–1337
Kydland, F. E., & Prescott, E. C. (1990). Business cycles: Real facts and a monetary myth. Federal Reserve Bank of Minneapolis Quarterly Review, 14(Spring), 3–18.
Li Y., Ruppert D. (2008) On the asymptotics of penalized splines. Biometrika 95: 415–436
Lindstrom M., Bates D. (1990) Nonlinear mixed-effects models for repeated measures data. Biometrics 46: 673–687
Long J., Plosser C. (1983) Real business cycles. Journal of Political Economy 91(1): 39–69
Lotka A. (1925) Elements of physical biology. Williams and Wilkins Co, Baltimore
Lucas, R. E. J. (1977). Understanding business cycles. In: K. Brunner, & A. H. Meltzer (Ed.) Stabilization of the domestic and international economy, Vol. 5 of Carnegie–Rochester Conference series on Public policy (pp. 7–29). Amsterdam: North-Holland.
Mao W., Zhao L. (2003) Free-knot polynomial splines with confidence intervals. Journal of the Royal Statistical Society, Series B 65: 901–919
Opsomer J., Wang Y., Yang Y. (2001) Nonparametric regression with correlated errors. Statistical Science 16: 134–153
O’Sullivan F. (1986) A statistical perspective on ill-posed inverse problems (c/r: P519–527). Statistical Science 1: 502–518
Paige R., Trindade A. (2010) The Hodrick–Prescott filter: A special case of penalized spline smoothing. Electronic Journal of Statistics 4: 856–874
Pedregal D., Young P. (2001) Some comments on the use and abuse of the Hodrick–Prescott filter. Review on Economic Cycles, International Association of Economic Cycles 3(1): 93–104
Proietti T. (2005) Forecasting and signal extraction with misspecified models. Journal of Forecasting 24: 539–556
Ruppert D. (2002) Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics 11: 735–757
Ruppert D. (2004) Statistics and finance—An introduction. Springer, New York
Ruppert D., Wand M., Carroll R. (2003) Semiparametric regression. Cambridge University Press, Cambridge
Ruppert D., Wand M., Carroll R. (2009) Semiparametric regression during 2003–2007. Journal of the American Statistical Association 3: 1193–1256
Schlicht E. (2005) Estimating the smoothing parameter in the so-called Hodrick–Prescott filter. Journal of the Japanese Statistical Society 35(1): 99–119
Solow R. (1990) Goodwin’s growth cycle: Reminiscence and rumination. In: Velupillai K. (eds) Nonlinear and multisectoral macrodynamics. Essays in Honour of Richard Goodwin. Macmillan, London, pp 31–41
Stock J. H., Watson M. W. (1999) Business cycle fluctuations in us macroeconomic Vol. 1A. In: Taylor J. B., Woodford M. (eds) Handbook of macroeconomics. Elsevier, North-Holland, pp 3–64
Tschernig R. (2004) Nonparametric time series modelling. In: Lütkepohl H., Krätzig M. (eds) Applied time series econometrics. Cambridge University Press, New York
Volterra V. (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118: 558–560
Wahba G. (1990) Spline models for observational data. SIAM, Philadelphia
Wand M. (2003) Smoothing and mixed models. Computational Statistics 18: 223–249
Welham S., Cullis B., Kenward M., Thompson R. (2006) The analysis of longitudinal data using mixed model L-splines. Biometrics 62: 392–401
Wood S. N. (2006) Generalized additive models: An introduction with R. Chapman and Hall/CRC, Boca Raton