A contribution to the analysis of historical economic fluctuations (1870–2010): filtering, spurious cycles, and unobserved component modeling

Abstract

Time series filtering methods such as the Hodrick–Prescott (HP) filter, with a consensual choice of the smoothing parameter, eliminate the possibility of identifying long swing cycles (e.g., Kondratieff type) or, alternatively, may distort periodicities that are in fact present in the data, giving rise, for example, to spurious Kuznets-type cycles. In this paper, we propose filtering Maddison’s time series for the period 1870–2010 for a selection of developed countries using a less restrictive filtering technique that does not impose but instead estimates the cutoff frequency. In particular, we use unobserved component models that optimally estimate the smoothing parameter. Using this methodology, we identify cycles of periods, primarily in the range of 4–7 years (Juglar-type cycles), and a number of patterns of cyclical convergence. We analyze the historical processes underlying this last empirical finding: Peacetime periods, monetary arrangements, trade and investment flows, and industrial boosts are confluent forces driving the economic dynamism. After 1950, we observe a common business cycle factor that groups all economies, which is consistent with the consolidation of the so-called second globalization.

Appendices

Appendix 1: Spectral gains of the filters for the unobserved components

The spectral gain of a filter measures the increase in amplitude of any specific frequency component of a time series. It is obtained by the Wiener–Kolmogorov (WK) formula (Whittle 1983). To this end, we depart from (1a), in which the signal $\mu_{t} = \frac{{\eta_{t - 2} }}{{(1 - L)^{2} }}$. The WK filter (of a doubly infinite realization of a time series) that provides the minimum mean-squared error of the signal is given by the ratio of the autocovariance generating functions of the signal $\mu_{t}$ and the series $y_{t}$. For the trend component, the filter is

$$\hat{\mu }_{t} = \frac{{\frac{{\sigma_{\eta }^{2} }}{{(1 - L)^{2} (1 - L^{ - 1} )^{2} }}}}{{\frac{{\sigma_{\eta }^{2} }}{{(1 - L)^{2} (1 - L^{ - 1} )^{2} }} + \sigma_{\varepsilon }^{2} }}y_{t} = \frac{q}{{q + (1 - L)^{2} (1 - L^{ - 1} )^{2} }}y_{t} = \frac{q}{{q + \left| {1 - L} \right|^{4} }}y_{t}$$

(6)

where $L^{ - 1}$ is the forward operator ($L^{ - k} y_{t} = y_{t + k}$) and the convention $(1 - L)(1 - L^{ - 1} ) = \left| {1 - L} \right|^{2}$ is adopted. The spectral gain of the filter of $\mu_{t}$ is obtained by doing $L = e^{ - i\omega }$ in (6), where $i = \sqrt { - 1}$ is the imaginary number and $\omega$ the frequency, obtaining

$$G_{\mu } (\omega ) = \frac{q}{{q + 4(1 - \cos \omega )^{2} }}.$$

(7)

For the growth component $g_{t}$, (1a) is expressed as $y_{t + 1} = \frac{{g_{t} }}{1 - L} + \varepsilon_{t + 1}$ with the signal $g_{t} = \frac{{\eta_{t - 1} }}{1 - L}$. The resulting WK filter is

$$\hat{g}_{t} = \frac{{\frac{{\sigma_{\eta }^{2} }}{{(1 - L)(1 - L^{ - 1} )}}}}{{\frac{{\sigma_{\eta }^{2} }}{{(1 - L)^{2} (1 - L^{ - 1} )^{2} }} + \sigma_{\varepsilon }^{2} }}y_{t + 1} = \frac{{(1 - L)(1 - L^{ - 1} )q}}{{q + (1 - L)^{2} (1 - L^{ - 1} )^{2} }}y_{t + 1} = \frac{{\left| {1 - L} \right|^{2} q}}{{q + \left| {1 - L} \right|^{4} }}y_{t + 1}$$

(8)

for which the spectral gain is

$$G_{g} (\omega ) = \frac{2(1 - \cos \omega )q}{{q + 4(1 - \cos \omega )^{2} }}$$

(9)

This gain has a maximum at the frequency

$$\omega_{\hbox{max} } = \arccos \left( {1 - \left( {\frac{q}{4}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right)$$

(10)

For example, for $q = \lambda^{ - 1} = \left\{ {0.001,\, \, 0.01, \, \,0.1,\, \, 1,\, \, 10} \right\}$, the corresponding periods $p = \frac{2\pi }{{\omega_{\hbox{max} } }}$ are $p = \left\{ {35.3, \, 19.8, \, 11.0, \, 6.0,\;2.9} \right\}$ units of time.

For the acceleration component, (1a) is expressed as $y_{t + 2} = \frac{{a_{t} }}{{(1 - L)^{2} }} + \varepsilon_{t + 2}$ with the signal $a_{t} = \eta_{t}$; then, the WK filter is

$$\hat{a}_{t} = \frac{{\sigma_{\eta }^{2} }}{{\frac{{\sigma_{\eta }^{2} }}{{(1 - L)^{2} (1 - L^{ - 1} )^{2} }} + \sigma_{\varepsilon }^{2} }}y_{t + 2} = \frac{{(1 - L)^{2} (1 - L^{ - 1} )^{2} q}}{{q + (1 - L)^{2} (1 - L^{ - 1} )^{2} }}y_{t + 2} = \frac{{\left| {1 - L} \right|^{4} q}}{{q + \left| {1 - L} \right|^{4} }}y_{t + 2}$$

(11)

In addition, in the frequency domain,

$$G_{a} (\omega ) = \frac{{4(1 - \cos \omega )^{2} q}}{{q + 4(1 - \cos \omega )^{2} }}$$

(12)

The HP filter is the optimal filter when the trend follows an IRW (King and Rebelo 1993). In the context of HP filtering, the cycle (growth cycle) is defined as the deviation with respect to the trend $\varepsilon_{t} = y_{t} - \mu_{t}$ and the corresponding WK filter is

$$\hat{C}_{t}^{HP} = y_{t} - \hat{\mu }_{t} = \left( {1 - \frac{q}{{q + \left| {1 - L} \right|^{4} }}} \right)y_{t} = \frac{{\left| {1 - L} \right|^{4} }}{{q + \left| {1 - L} \right|^{4} }}y_{t}$$

(13)

A comparison of (7), (9), (12), and (13) allows the verification of

$$G_{HP} (\omega ) = \frac{4}{q}(1 - \cos \omega )^{2} G_{\mu } (\omega ) = \frac{2}{q}(1 - \cos \omega )G_{g} (\omega ) = \frac{1}{q}G_{a} (\omega )$$

(14)

with $G_{\text{HP}} (\omega )$ the spectral gain of the HP filter given $q$.

Appendix 2: State space representations of the univariate and the multivariate models

State space representation consists of two equations. The measurement equation relates the observed variable with the unobserved components and the observation noise. For IRW model (1), which basically coincides with Eq. (1a),

$$\left[ {y_{t} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mu_{t} } \\ {g_{t} } \\ {a_{t} } \\ \end{array} } \right] + \left[ {\varepsilon_{t} } \right]$$

(15)

The state transition equation represents the dynamics of the unobserved components. According to Eqs. (1b) to (1d), this is

$$\left[ {\begin{array}{*{20}c} {\mu_{t} } \\ {g_{t} } \\ {a_{t} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mu_{t - 1} } \\ {g_{t - 1} } \\ {a_{t - 1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {\eta_{t} } \\ \end{array} } \right]$$

(16)

Gaussianity and orthogonality assumptions of the error terms imply that $\varepsilon_{t} \sim N\left( {0,\sigma_{\varepsilon }^{2} } \right)$ and $\xi_{t} \sim N\left( {0_{3 \times 1} ,Q} \right)$, where $\xi_{t} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {\eta_{t} } \\ \end{array} } \right]$ and $Q = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & {\sigma_{\eta }^{2} } \\ \end{array} } \right]$. By doing $H = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} } \right]$, $\beta_{t} = \left[ {\begin{array}{*{20}c} {\mu_{t} } \\ {g_{t} } \\ {a_{t} } \\ \end{array} } \right]$, and $F = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]$, the state space representation of the system (15)–(16) in compact form is

$$\left. {\begin{array}{*{20}c} {y_{t} = H\beta_{t} + \varepsilon_{t} } \\ {\beta_{t} = F\beta_{t - 1} + \xi_{t} } \\ \end{array} } \right\}$$

(17)

Estimation of the vector of variances $\left\{ {\sigma_{\varepsilon }^{2} ,\sigma_{\eta }^{2} } \right\}$ is obtained by maximizing the likelihood function of the one-step ahead prediction errors (Harvey 1989; Durbin and Koopman 2001).

For the multivariate IRW model (3) and for illustrative purposes, let us assume two time series, $i = 1, \, 2$, with specific acceleration components $a_{1,t}$ and $a_{2,t}$, and that the common acceleration component, $a_{t}$, follows an autoregressive model such as $a_{t} = \phi a_{t - 1} + \eta_{t}$. From Eq. (3a), we have that the measurement equation relating observed variables with unobserved components is

$$\left[ {\begin{array}{*{20}c} {y_{1,t} } \\ {y_{2,t} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mu_{1,t} } \\ {g_{1,t} } \\ {\mu_{2,t} } \\ {g_{2,t} } \\ {a_{t} } \\ {a_{1,t} } \\ {a_{2,t} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\varepsilon_{1,t} } \\ {\varepsilon_{2,t} } \\ \end{array} } \right]$$

(18)

According to Eqs. (3b) to (3e), the state transition equation is

$$\left[ {\begin{array}{*{20}c} {\mu_{1,t} } \\ {g_{1,t} } \\ {\mu_{2,t} } \\ {g_{2,t} } \\ {a_{t} } \\ {a_{1,t} } \\ {a_{2,t} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & {\gamma_{1} } & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & {\gamma_{2} } & 0 & 1 \\ 0 & 0 & 0 & 0 & \phi & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mu_{1,t - 1} } \\ {g_{1,t - 1} } \\ {\mu_{2,t - 1} } \\ {g_{2,t - 1} } \\ {a_{t - 1} } \\ {a_{1,t - 1} } \\ {a_{2,t - 1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ {\eta_{t} } \\ {\eta_{1,t} } \\ {\eta_{2,t} } \\ \end{array} } \right]$$

(19)

Gaussianity and orthogonality assumptions of the error terms imply that $\varepsilon_{t} \sim N\left( {0_{2x1} ,R} \right)$ and $\xi_{t} \sim N\left( {0_{7x1} ,Q} \right)$, where $\varepsilon_{t} = \left[ {\begin{array}{*{20}c} {\varepsilon_{1,t} } \\ {\varepsilon_{2,t} } \\ \end{array} } \right]$, $\xi_{t} = \left[ {\begin{array}{*{20}c} {0_{4x1} } \\ {\eta_{t} } \\ {\eta_{1,t} } \\ {\eta_{2,t} } \\ \end{array} } \right]$, $R = \left[ {\begin{array}{*{20}c} {\sigma_{{\varepsilon_{1} }}^{2} } & 0 \\ 0 & {\sigma_{{\varepsilon_{2} }}^{2} } \\ \end{array} } \right]$, $Q = \left[ {\begin{array}{*{20}c} {0_{4 \times 4} } & {0_{4 \times 1} } & {0_{4 \times 1} } & {0_{4 \times 1} } \\ {0_{1 \times 4} } & 1 & 0 & 0 \\ {0_{1 \times 4} } & 0 & {\sigma_{{\eta_{1} }}^{2} } & 0 \\ {0_{1 \times 4} } & 0 & 0 & {\sigma_{{\eta_{2} }}^{2} } \\ \end{array} } \right]$, and $0_{m \times n}$ is a $m \times n$ matrix of ceros. By doing $y_{t} = \left[ {\begin{array}{*{20}c} {y_{1,t} } \\ {y_{2,t} } \\ \end{array} } \right]$, $H = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$, $\beta_{t} = \left[ {\begin{array}{*{20}c} {\mu_{1,t} } \\ {g_{1,t} } \\ {\mu_{2,t} } \\ {g_{2,t} } \\ {a_{t} } \\ {a_{1,t} } \\ {a_{2,t} } \\ \end{array} } \right]$, and $F = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & {\gamma_{1} } & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & {\gamma_{2} } & 0 & 1 \\ 0 & 0 & 0 & 0 & \phi & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$, the state space representation of the system (18)–(19) in compact form follows the general form (17). Estimation of the vector of parameters $\left\{ {\sigma_{{\varepsilon_{i} }}^{2} ,\sigma_{{\eta_{i} }}^{2} ,\gamma_{i} ,\phi } \right\}$ is obtained as previously outlined for the univariate model.