Abstract
This paper introduces an economically important new idea for detrending macroeconomic time series and examines the Spanish business cycle pattern with respect to potential asymmetries. To address difficulties in the trend and cycle decomposition, a nonparametric trend estimation approach is introduced and exemplary applied to the Spanish GDP data for the period 1850 to 2015. The application of an iterative plug-in (IPI) algorithm for endogenous bandwidth selection solves the problem of choosing an adequate smoothing parameter for nonparametric regression. The algorithm identifies continuously Moving Trends (MT) with a time length of 34 years. After we estimate the trend nonparametrically, we fit several time series models to the residuals for further analysis. Although asymmetry during expansion and recession phases is indicated, it is not unambiguous.
Acknowledgments
We would like to thank the anonymous referee for helpful comments and suggestions that clearly improve the paper.
Appendix A
Estimated AR(p) models for the standardized Spanish GDP growth rates.
| Series Coeff. | Spain GDP | ||||
|---|---|---|---|---|---|
| Model | AR(1) | AR(2) | AR(3) | AR(4) | AR(5) |
| 0.0002 | 0.0001 | 0.0001 | 0.0000 | 0.0000 | |
| (0.0035) | (0.0030) | (0.0026) | (0.0021) | (0.0019) | |
| 0.0054 | 0.0059 | –0.0155 | –0.0427 | –0.0652 | |
| (0.0778) | (0.0767) | (0.0772) | (0.0760) | (0.0774) | |
| −0.1653** | −0.1674** | −0.1998*** | −0.2138*** | ||
| (0.0766) | (0.0759) | (0.0754) | (0.0757) | ||
| −0.1275* | −0.1311* | −0.1518** | |||
| (0.0768) | (0.0750) | (0.0761) | |||
| −0.2111*** | −0.2158*** | ||||
| (0.0754) | (0.0751) | ||||
| −0.1032 | |||||
| (0.0768) | |||||
| 0.0020 | 0.0020 | 0.0019 | 0.0018 | 0.0018 | |
| Log-likelihood | 278.08 | 280.38 | 281.75 | 285.56 | 286.46 |
| AIC | −550.17 | −552.76 | −553.49 | −559.13 | −558.92 |
Notes: Model parameters, standard errors in parentheses and p-values, *p(z)<0.1, **p(z)<0.05, ***p(z)<0.01.
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