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Central banks have different objectives in the short and long run. Governments operate simultaneously at different timescales. Many economic processes are the result of the actions of several agents, who have different term objectives. Therefore, a macroeconomic time series is a combination of components operating on different frequencies. Several questions about economic time series are connected to the understanding of the behavior of key variables at different frequencies over time, but this type of information is difficult to uncover using pure time-domain or pure frequency-domain methods. To our knowledge, for the first time in an economic setup, we use cross-wavelet tools to show that the relation between monetary policy variables and macroeconomic variables has changed and evolved with time. These changes are not homogeneous across the different frequencies.

Central banks have different objectives in the short and long run and they operate simultaneously at different timescales — see Ref. [40]. Moreover, many economic processes are the result of the actions of several agents, who have different term objectives, with some agents focusing on daily movements and co-movements, while other agents are concerned about longer horizons. Therefore, a macroeconomic time series is a combination of components operating on different frequencies. On top of this, some interesting relations may exist between two macroeconomic time series at different frequencies. For example, it is possible that monetary policies have different impacts in the short or long run, therefore affecting the economy in different ways at different frequencies. Or, it is possible that monetary authorities react to inflation news in the short run, while, in the long run, the price level is essentially determined by the money supply. Finally, it is possible that the effects of a certain policy evolve with time, as institutions and policymakers change. While several questions about time series economic data are connected to the understanding of the behavior of key variables at different frequencies over time, this type of information is difficult to uncover using pure time-domain or pure frequency-domain methods. We use wavelets to analyze the impact of interest rate price changes on some macroeconomic variables: industrial production, inflation and the monetary aggregates M1 and M2.1 Specifically, we utilize the continuous wavelet power spectrum and three cross-wavelet tools: the cross-wavelet power spectrum, the wavelet coherency and the wavelet phase difference. With these instruments we are able to unravel some economic time–frequency relations that have remained hidden so far. With the wavelet power spectrum, we show that the “great moderation” is not a recent phenomenon and that the observed reduction in the volatility of production happened in the 1950s and not in the 1980s as has been erroneously assumed in the economic literature. The volatility was temporarily revived during the oil crisis of the 1970s and 1980s, mostly at business cycle frequencies. This is specifically one of the advantages of the wavelet analysis: the possibility of uncovering transient relations. The same conclusion is reached about inflation. The volatility of inflation decreased at the same time. Therefore, the great moderation is not just a real phenomenon but also a nominal phenomenon. With the cross-wavelet tools we show that the relation between monetary policy variables (money aggregates and interest rates) and macroeconomic variables (industrial production and inflation) has changed and evolved with time and is not homogeneous across the different frequencies. For example, in the 1970s and 1980s we observe that interest rates reacted procyclically with inflation at the business cycle frequencies, and that at lower frequencies this helped to control inflation. We also find evidence that in the 1950s interest rates were reacting to industrial production, in the 2–4 year period, but that in the 1970s and 1980s, especially in the 4–12 year period, we observe an anti-phase relation with interest rates leading, meaning that increases in the interest rates had contractionary effects, supporting the conclusions of some authors [1] and [30] who argued that monetary policy reinforced the recessionary effects of the oil shocks. We also find evidence of a structural break in the relation between the interest rates and the monetary aggregates. In the late 1970s, early 1980s, at the business cycle frequency, the broader definition of money, M2, stopped lagging the interest rates. At lower frequencies, corresponding to 15–20 year period oscillations, M1 started leading the interest rates. This suggest that the Fed has been following a monetary targeting type of policy, even if they are not explicit about it (except between the mid-1970s and mid-1980s). The paper proceeds as follows. In Section 2, we discuss the main advantages of wavelet analysis, its applications to economics and some of the typical difficulties in applying it to study economic relations. We present the continuous wavelet transform, and discuss its localization properties and the optimal characteristics of the Morlet wavelet. Section 3 describes the wavelet power spectrum, the cross-wavelet power spectrum, the wavelet coherency, and the phase difference. In Section 4, we apply these tools to study the effects and the effectiveness of monetary policy. Section 5 concludes.

In this paper, we claimed that wavelet analysis can be very useful for analyzing economic relations and that it is better suited for dealing with economic data than the Fourier transform. We illustrated how wavelet analysis can naturally be applied to the study of business cycles (given its periodic nature), or to any field of economics, or finance, especially when there is a distinction between short and long run relations. The main advantage of the wavelet approach is the ability to analyze transient dynamics, for single time series or for the association between two time series. We have also showed that some of the shortcomings that economists have found when applying wavelet techniques to study two or more time series disappear once the concept of the cross-wavelet is introduced. We used three tools that, to our knowledge, have not yet been used by economists: the cross-wavelet transform, the cross-wavelet coherency and the phase difference. While the wavelet power spectra quantifies the main periodic component of a given time series and its time evolution, the cross-wavelet transform and the cross-wavelet coherency are used to quantify the degree of linear relation between two non-stationary time series in the time–frequency domain. Phase analysis is a nonlinear technique that makes it possible to study synchronization and delays between two time series across different frequencies or timescales. This paper’s main contribution to the literature is to clearly demonstrate the utility of wavelets and cross-wavelets for the analysis of economic time series and to illustrate how relationships between macroeconomic variables change over time and across different frequencies. In fact, wavelets allowed us to detect transient effects which would be very difficult to detect using classical econometric techniques. For example, we were able to see that the reduction in the US output and inflation volatility decreased in the 1960s at all frequencies (and not in the 1980s as is usually claimed), but that it was temporarily revived in the 1970s (especially at the business cycle frequency) probably because of the oil price shocks. We observed the same behavior for inflation rates and concluded that the “great moderation” is also a nominal phenomenon. We were able to disentangle different short, medium and long run relations and to detail transient relations. For example, we showed that, in the 1970s and 1980s, at the business cycle frequency, inflation and interest rates were in phase, with the inflation rates leading, consistent with a central bank that follows a Taylor rule. In the long run, 12–20 year timescales, the phase difference showed that the variables are anti-phase, with interest rates leading, a result consistent with the notion that, in the long run, a restrictive monetary policy does help to control inflation. We saw that after 1980, coinciding with Paul Volcker being a chairman of the Federal Reserve, interest rates, in the 2–4 year time band, were in phase with and reacting to industrial production, having contractionary effects in the longer run. Our results also gave some support to authors like Barsky and Kilian[1] and Leduc and Sill [30] who argued that, during the 1970s and early 1980s, monetary policy reinforced the recessionary effects of the oil shocks. About the same time, there was a structural break in the relation between interest rates and the monetary aggregates, M1 and M2. This supports and illuminates the results of Ref. [34] which found evidence for a structural break, in the early 1980s, in the relation between money and real output.