In this appendix, we check whether the results described above are affected by the choice of an alternative filter. There are two main bandpass filters: the Baxter-King (BK) filter (Baxter and King, 1999) and the Christiano-Fitzgerald (CF) filter (Christiano and Fitzgerald, 2003).
Wavelet filters are both time and frequency localized. Contrastingly, the BK filter and the CF filter have properties formulated in the frequency domain only. Both the BK filter and the CF filter approximate the ideal filter through a two sided moving average of the time series. They differ with respect to the symmetry of the weights. The BK formulates symmetric weights.
This implies that the filter does not produce phase shift in the filtered series. However, this requires to drop observations at the beginning and at the end of the sample. The CF filter uses asymmetric weights, which imply that no observations are lost. Contrastingly to wavelets, both filters also make assumptions regarding the data generating process (iid for BK and random walk for CF). We choose the CF filter as the filtered series have the same number of observations as the wavelet filter, which makes the comparison more straightforward. In addition, the assumption that variables follow a random walk imply that the CF filter better captures the low frequency rather than the BK filter. The power spectrum of a random walk is defined as a constant over frequency squared (Everts, 2006).
The parameters of the CF filter are chosen to correspond to the frequencies of the wavelet filter. FigureC.1displays the labor share series corresponding to the ‘16-32 years’ frequency as well as to the ‘32 years and beyond’ frequency under both the wavelet filter and the CF fil-ter. A striking features is the similarity between the two series. The main difference is that the CF filtered series display larger amplitudes than the wavelet series for all frequencies. This may be due to the fact that wavelets are localized in the time dimension while CF filters are global. This implies that one extreme value, such as the extreme values for the labor share during the two World Wars would be transmitted to all the filtered series in the case of the CF filter but not in the case of the wavelet filters. There are two smaller differences between wavelets and CF filters. There are small phase shifts between the two filtered series, espe-cially for France and for the United Kingdom. In addition, the boundaries of the series differ slightly depending on the filter used. The same issues arise when looking at GDP growth (not represented here due to space limitation).
The main result highlighted in Section4, i.e. the sign of the coefficient associated with the labor share changes from negative at high frequency to positive at low frequency, remains under the alternative filter. In that respect, TableC.1is almost identical to Table1. There are two small differences. The coefficient associated with the ‘beyond 32 years’ frequency turns insignificant in the United States. On the contrary, the sign of the coefficients associated with the frequency ‘8 to 16 years’ and frequency ‘16 to 32 years" are now significant (and positive) in France. In addition, the sign associated with the labor share in the long run is also robust to the inclusion of control variables under the CF bandpass filter as shown in TableC.2. Note that the sign associated with the labor share is positive and significant in France when GDP per capita level and/or investment rate are included as control variables. The coefficient
as-FIGUREC.1: LABOR SHARE- CHRISTIANOFITZGERALD BANDPASS FILTER
1900 1920 1940 1960 1980 2000
−0.03−0.02−0.010.000.010.020.03
1900 1920 1940 1960 1980 2000
−0.03−0.02−0.010.000.010.020.03 Wavelet D4Christiano Fitzgerald bandpass
(a) United States:D416-32 years
1900 1920 1940 1960 1980 2000
0.650.700.750.800.85
1900 1920 1940 1960 1980 2000
0.650.700.750.800.85 Wavelet S4Christiano Fitzgerald bandpass
(b) United States:S4>32 years
1850 1900 1950 2000
−0.04−0.020.000.020.04
1850 1900 1950 2000
−0.04−0.020.000.020.04
Wavelet D4
Christiano Fitzgerald bandpass
(c) United Kingdom:D416-32 years
1850 1900 1950 2000
0.650.700.750.800.85
1850 1900 1950 2000
0.650.700.750.800.85 Wavelet S4Christiano Fitzgerald bandpass
(d) United Kingdom:S4>32 years
1900 1920 1940 1960 1980 2000
−0.04−0.020.000.020.040.06
1900 1920 1940 1960 1980 2000
−0.04−0.020.000.020.040.06 Wavelet D4Christiano Fitzgerald bandpass
(e) France:D416-32 years
1900 1920 1940 1960 1980 2000
0.650.700.750.800.85
1900 1920 1940 1960 1980 2000
0.650.700.750.800.85 Wavelet S4Christiano Fitzgerald bandpass
(f ) France:S4>32 years
This figure displays the wavelet and Christiano Fitzgerald bandpass filter for the labor share for two frequenciesD416-32 years andS4Beyond 32 years.
sociated with the labor share is0.293∗∗∗when controlling for GDP per capita level and0.130∗∗∗
adding investment rate as well. These coefficients become0.306∗∗∗and0.148∗∗∗with lagged labor share. The sign turns insignificant when the filtered series for price inflation is added to the other control variables. The reason for this sudden change and whether it is related to the quality of the filtered series needs to be further researched.
TABLEC.1: REGRESSION ACROSS FREQUENCIES- CFBANDPASS FILTER
Notes:This table presents the regressions results across each time scale and for each country with and without dummies for the two world wars (DW W IandDW W II). The equation estimated is equation2.
The estimation method is HAC-OLS. The weights follows Newey-West. The constant is suppressed to
TABLEC.2: REGRESSION USING THE SMOOTH COMPONENTS4 - CFBANDPASS FILTER
Dependent variable∆ys4,t
cst lst lst−1 Controls R2 US: 1899-2010
−0.249∗∗∗ 0.348∗∗∗ yes 0.55
−0.289∗∗∗ 0.402∗∗∗ yes 0.58 UK: 1898-2010
−0.02 0.038 yes 0.19
−0.03∗∗ 0.049∗ yes 0.20 UK: 1857-2010
0.014 −0.043 yes 0.10
0.005 0.008 yes 0.10
France: 1898-2010
−0.118∗∗∗ 0.13∗∗∗ yes 0.67
−0.131∗∗∗ 0.148∗∗∗ yes 0.69 Panel: 1899-2010
No 0.109∗∗∗ yes 0.56
No 0.121∗∗∗ yes 0.57
Notes:This table presents the regressions using the smoothed componentS4for each coun-try. The equation estimated is (2) with∆ythe growth rate of real GDP per capita,lsthe labor share. The control variables are real GDP per capita from previous period, lagged invest-ment share in GDP, lagged CPI inflation andDW W IandDW W II the dummies for World War I and II. In France, lagged CPI inflation is not included as a control variable. See the text for a discussion.tandt−1indicate the lag structure. The estimation method is HAC-OLS. The weights follow Newey-West.∗∗∗p <0.01,∗∗p <0.05,∗p <0.1