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A computational algorithm to analyze unobserved sequential reactions of the central banks: inference on complex lead–lag relationship in evolution of policy stances

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Abstract

Central banks of different countries are some of the largest economic players at the global scale and they are not static in their monetary policy stances. They change their policies substantially over time in response to idiosyncratic or global factors affecting the economies. A very prominent and empirically documented feature arising out of central banks’ actions is that the relative importance assigned to inflation vis-a-vis output fluctuations evolves substantially over time. We analyze the leading and lagging behavior of central banks of various countries in terms of adopting low inflationary environment vis-a-vis high weight assigned to counteract output fluctuations, in a completely data-driven way. To this end, we propose a new methodology by combining complex Hilbert principle component analysis with state–space models in the form of Kalman filter. The CHPCA mechanism is non-parametric and provides a clean identification of leading and lagging behavior in terms of phase differences of time series in the complex plane. We show that the methodology is useful to characterize the extent of coordination (or lack thereof), of monetary policy stances taken by central banks in a cross-section of developed and developing countries. In particular, the analysis suggests that US Fed led other countries central banks in the pre-crisis period in terms of pursuing low-inflationary regimes.

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Fig. 1
Fig. 2
Fig. 3

Source of data: Datastream

Fig. 4

Source of data: OECD

Fig. 5

Source of data: Datastream

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Notes

  1. Unless in the time of extreme measures like the financial crisis where the policies were deliberately implemented to fight the recession.

  2. We restrict our first analysis till 2007. Post 2007, interest rate in the USA and most of the European countries touched zero lower bound or went marginally negative. Hence, it was not possible for the central bank to implement monetary policy response function mandated by the Taylor rule.

  3. Source: https://data.oecd.org/price/inflation-cpi.htm.

  4. Source: https://data.oecd.org/gdp/quarterly-gdp.htm.

  5. https://www.bis.org/statistics/cbpol.htm.

  6. https://data.oecd.org/interest/short-term-interest-rates.htm.

  7. We have conducted the time-varying state–space estimation using dlm package in R programming environment.

  8. See App. 4.1 for a brief description based on Cauchy integration.

  9. A matrix is Hermitian if it is a complex square matrix such that it is equal to its own conjugate transpose.

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Correspondence to Anindya S. Chakrabarti.

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

This research was partially supported by Institute Grant, IIM Ahmedabad. We thank Abhinav Sharma for preliminary work on the data. We are grateful to Wataru Souma and Irena Vodenska for some discussions on the methodology employed in this paper.

Appendix

Appendix

Hilbert transform and the complex plane

Here, we briefly motivate the discussion on Hilbert transform using Cauchy integration. The following discussion is partially based on Pipes and Harvill [26] and we review the description of Hilbert transformation. This appendix solely discusses the context and existence of the transform.

Consider an analytic function f(z) defined over region \(\Theta \). Due to the analytic nature of the function, it is single-valued, continuous and there exists definite derivative at every point in R. The Cauchy integral theorem asserts that

$$\begin{aligned} \oint _C f(z) \,\mathrm{d}z=0, \end{aligned}$$
(15)

where C is an arbitrary closed path (piecewise smooth) within \(\Theta \). This can be used to find the value of an analytic function at a point from the path which surrounds it. In particular, let us continue with the example. So f(z) is analytic in \(\Theta \) including a point \(z=a\). As in the earlier case, C is a closed path around it. The Cauchy integral result is that

$$\begin{aligned} f(a)=\frac{1}{2\pi i}\oint _C\frac{f(z)}{z-a} \,\mathrm{d}z. \end{aligned}$$
(16)

A generalization of this result is given as follows:

$$\begin{aligned} \oint _C\frac{f(z)}{z-a} \,\mathrm{d}z ={\left\{ \begin{array}{ll} 2i\pi f(a) ~~~~~~~\text{ for } a \text{ inside } \text{ closed } \text{ path } C\\ 0 ~~~~~~~~~~~~~~~~~\text{ for } a \text{ outside } \text{ closed } \text{ path } C. \end{array}\right. } \end{aligned}$$

If a is exactly on the path C, one can consider a new path \(C'\), with a simple pole at a and a path \(C_{\epsilon }\) with distance \(\epsilon \) from a. Then, one can have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\oint _{C'}\frac{f(z)}{z-a}\mathrm{d}z=PV \oint _{C}\frac{f(z)}{z-a}\mathrm{d}z+\lim _{\epsilon \rightarrow 0} \oint _{C_{\epsilon }}\frac{f(z)}{z-a}\mathrm{d}z, \end{aligned}$$
(17)

which can be shown to yield the result

$$\begin{aligned} PV \oint _{C}\frac{f(z)}{z-a}\mathrm{d}z=i\pi f(a). \end{aligned}$$
(18)

In order to link it to Hilbert transform, we consider a closed path comprising the x-axis and a semicircle in the upper half-plane. Under the conditions that f(z) is analytic and the contribution of the semi-circle tends to zero when the radius tends to infinity, we have

$$\begin{aligned} PV \int _{-\infty }^{\infty }\frac{f(z)}{z-a}\mathrm{d}z=i\pi f(a). \end{aligned}$$
(19)

Finally, if we write \(f(z)=r(z)+ic(z)\), then

$$\begin{aligned} r(z)=\mathscr {H}(c(z))~~~~\text{ and }~~~~ c(z)=-\mathscr {H}(r(z)). \end{aligned}$$
(20)

Details of the data

See Tables 1, 2, 3, 4, Figs. 6 and 7.

Table 1 Countries and their corresponding codes
Table 2 Datastream (1981–2007)
Table 3 OECD (1981–2007)
Table 4 Datastream (2010–2018)
Fig. 6
figure 6

Left panels: we have plotted three time-series generated from an identical autoregressive process of order 1. The middle series (colored red) is the benchmark series. Other two series are lagged (black dotted) and leading (blue) by \(\Delta t\) time points (top panel: \(\Delta t\)=10; bottom panel: \(\Delta t\)=5). Right panels: after application of complex Hilbert principle component method, we recover the phase difference from the cross-correlation matrix of the three time series. See text for details. Comparing the top panels with the bottom panels, we see that a higher lag and lead are manifested in terms of larger phase differences (see panels in the right)

Fig. 7
figure 7

This figure is created based on dataset identical to the one used in Fig. 4 except that the time horizon is 1981–1999. Left panel: components for inflationary weights (\(\beta _{\pi }\)) of different countries. The presence of the circle shows similar contribution of all countries. Right panel: components for output gap weights (\(\beta _{g}\)) of different countries

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Chakrabarti, A.S., Kumar, S. A computational algorithm to analyze unobserved sequential reactions of the central banks: inference on complex lead–lag relationship in evolution of policy stances. J Comput Soc Sc 3, 33–54 (2020). https://doi.org/10.1007/s42001-019-00052-w

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