Fisher's hypothesis in time-frequency space: a premier using South Africa as a case study

Abstract

Fisher hypothesis is universally accepted as an integral portion of monetary theory and practice, and yet the empirical evidence confirming a full Fisher effect remains scarce and the relationship has been challenged on several theoretical grounds referred to as 'puzzles'. Our paper suggests the use of continuous wavelet transforms as a unified analytical framework for confronting the different Fisher puzzles in a harmonious way. Taking South Africa as a case study, we focus on the inflation targeting period of 2002:01-2021:02 and use signal-image conversion tools such as wavelet power spectrum, wavelet coherence spectrum and phase-difference dynamics to extract signal features of nominal interest rates and inflation expectations and further explore their dynamic synchronization across a time-frequency plane/domain. Three unique findings emerge from our study. Firstly, across a time domain a full Fisher effect only holds in the pre-financial crisis period. Secondly, across the frequency spectrums, higher frequency oscillations gradually lose relevance to lower frequency oscillations providing evidence of volatility transfer in the Fisher effect. Lastly, the phase-dynamics indicate a consistent positive synchronization throughout the sample period which is line with the traditional Fisher effect. Overall, these findings highlight the success of the South African Reserve Bank in using inflation targeting to steer the expectations of economic agents under the tenures of the last three governors and provide important lessons for other Central banks.

Keywords: Continuous wavelet transforms; Fisher effect; Interest rates, Inflation expectations; Morlet wavelets.

Fig. 1

Repo rate and inflation expectations in South Africa (2002:q1–2020:q2). Source: Authors own computations using data sourced from SARB online database

Fig. 2

Repo rate vs disaggregated measures of inflation expectations

Fig. 3

Wavelet power spectrum for repo rate

Fig. 4

Wavelet power spectrum for all expectations

Fig. 5

Wavelet power spectrum for sector expectations

Fig. 6

Wavelet power spectrum for business representations expectations

Fig. 7

Wavelet power spectrum for trade unions representatives expectations

Fig. 8

Repo rate versus all expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 9

Repo rate versus financial sector expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 10

Repo rate versus business representations expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 11

Repo rate versus trade union representations expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 12

3–5 year bond yield and 5–10 year bond yield

Fig. 13

Wavelet power spectrum for medium-term bond yield (3–5 years)

Fig. 14

Wavelet power spectrum for long-term bond yield (5–10 years)

Fig. 15

3–5 year bond yield versus all expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 16

3–5 year bond yield versus financial sector expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 17

3–5 year bond yield versus business representatives expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 18

3–5 year bond yield versus trade union representatives expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 19

5–10 year bond yield versus all expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 20

5–10 year bond yield versus financial sector expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 21

5–10 year bond yield versus business representatives expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates

Fig. 22

5–10 year bond yield versus trade union representatives expectations. Note: The white contour line represents the significance level. The power of correlation ranges from blue colour (weak correlation) to red colour (strong correlation). The arrow notations, $\uparrow$, $\nearrow$, $\to$, $\searrow$, ($\downarrow$, $\swarrow$, $\leftarrow$, $\nwarrow$) indicate that a positive (negative) relationship between the series. The arrow notations,$\uparrow$, $\nearrow$, $\to$, $\swarrow$,$\leftarrow$, ($\searrow$,$\downarrow$,$\nwarrow$) indicate that nominal interest rates are leading (lagging) inflation rates