Silviase/jgov_v1.4_full_ann huggingface dataset:
First, compare the correlation coefficient of money with output before and after 1991. As expected, in Table 9 it can be seen that the correlation between money and output has weakened since 1991: the maximum correlation coefficient drops from 0.86 to 0.44 for nominal money stock, and from 0.86 to 0.61 for real money stock.

This relationship appears to have changed reflecting an economy awash in liquidity. Indeed, as seen in Table 10, the long-run relationship between money-growth and inflation rate is almost invisible. After 1991, the growth rate of money is negatively correlated with the inflation rate. This is consistent with the recent trend of disinflation in Japan, and is hence, is considered as one of the reasons why the economy had not recovered.

Second, the relationship between monetary base and money stock is examined. Much of the previous literature argues that the money multiplier has been decreasing since 1991 because of concerns about disinflation. According to the results in Table 11a showing the correlation between the level of monetary base and money stock confirms that the relationship obviously weakened since 1991. From the business cycle perspective, it seems that the money multiplier has been decreasing as well. These changes imply that both the propagation mechanisms from monetary base to money stock and from money stock to output have confronted serious problems since 1991.

Now we draw our attention to the impact of monetary policy on the business cycle. Suppose the Bank of Japan (BOJ) can control the growth rate of the monetary base as one of the means for monetary policy, what could we suggest about the effect of monetary policy on money stock or economy? Can BOJ control the growth rate of money stock as well? Table 11b shows that the growth rate of monetary base positively affects the growth rate of money stock. Roughly speaking, it means BOJ
could control the money stock somehow indirectly through the monetary base. However, after 1991, it seems much more difficult to control the money stock via the monetary base, since the correlation coefficient is small and the clear cyclicality present before 1991 disappears.

Still more interesting results are found when Table 9 is examined again. The growth rate of monetary base is counter-cyclical with a lag of one quarter, both before and after 1991. It means that BOJ has basically implemented the counter-cyclical type of monetary policy throughout the period. $ ^{13} $

Since the growth rate of monetary base is positively correlated with the growth rate of money stock, it is natural to regard money stock-growth as counter-cyclical. Indeed, Table 9 confirms that the growth rate of money stock is counter-cyclical after 1991. We tend to think that the counter-cyclicality of money stock-growth helps to stabilize the economy. However, before 1991, there is no counter-cyclicality of money stock-growth through the monetary base which we typically expect, probably due to the existence of the excess liquidity in economy. Unlike the growth rate of monetary base, the growth rate of money stock has changed its property substantially.

It is also meaningful to note that, after 1991, the growth rate of money stock can be interpreted as both counter-cyclical with a lag (the maximal correlation is -0.41 at k = -1 for nominal), and pro-cyclical with a lead (the maximal correlation is 0.40 at k = 6 for nominal). As we have mentioned before, the first part implies the stabilization mechanism of money stock-growth. The later part means that an increase in the money
stock-growth today leads to an increase in output more than one year later with maximum correlation. In sum, the business cycle has a negative impact on the growth rate of money stock, while the growth rate of money stock has a positive impact on the business cycle. One might argue that there exists a positive reverse causality, i.e. the positive effect of business cycle on the growth rate of money stock, however, our results show there exists a negative reverse causality.

At the last part of analyzing the statistical properties in money, we have alternatively implemented the Stock and Watson (1999) type of causality test, particularly concentrating on the relationship between money and output which seems to have changed substantially before and after 1991, to confirm our results from the cross-correlation analysis above.

As in Stock and Watson (1999), the test has been done by computing the marginal R2, assumed to come from regressing of future output onto current and past data on money, given current and past data on output. The results of the test for examining the causality between money and output, which are typically based on two sub-periods covering before and after 1991, are summarized in Table 12. It shows that the Stock and Watson (1999) type of test turns out to be positive in most of cases considered here: in fact, the apparent causality between money and output is confirmed which is to be consistent with the evidences in Stock and Watson (1999). $ ^{14} $ Furthermore, the degree of marginal R2 in money-output relation implies a presence of negative reverse causality, while money-inflation relation is almost invisible.

Effective exchange rate is counter-cyclical with a lag of one quarter. It implies that a
depreciation of yen causes an increase in exports, in turn grows economy. Considering the features of both exports and exchange rate together, one might say that in Japan output helps to predict exports, rather than exports leading output.

### I V. Conclusions

This paper characterizes business cycle fluctuations in Japan since 1980 to the present. Stock and Watson (1999) present robust stylized facts for the American economy. In an effort to provide a cross-national perspective, this paper presents analogous information for Japan. The results for the individual series are, (i) private consumption is least volatile; both private and public investment, exports, and imports are most volatile, (ii) employment, total hours worked, average hours worked (total), and labor productivity are less volatile, (iii) non-scheduled hours worked is much more volatile, implying it has been playing an important role as a buffer in labor input, (iv) non-scheduled wage is much more volatile, while scheduled wage is considerably stable, (v) both GDP deflator and consumer price index are less volatile, (vi) interest rates are less volatile regardless of its duration, and (vii) money is more volatile than output.

As regards cyclicality, (i) private consumption and private residential investment are pro-cyclical with a lead, while private non-residential investment is pro-cyclical with a lag, (ii) Public investment is counter-cyclical with a lag, (iii) both exports and imports are pro-cyclical, (iv) employment is pro-cyclical with a lag, while average hours worked (total) is pro-cyclical with a lead, (v) wage is essentially pro-cyclical with a lag, (vi) inflation rate measured by GDP deflator and consumer price index are pro-cyclical with a lag, (vii) basically money is pro-cyclical with a lead, while the growth rate of money
is counter-cyclical with a lag, although some properties of money have been changed since 1991 with important policy consequences, and (viii) effective exchange rate is counter-cyclical with a lag, that is, the depreciation of yen is positively correlated with output.

Taken as a whole, we believe that our findings may be useful for stimulating new studies on Japan’s business cycle.

## References

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Yoshizoe, Y., S. Ohira, E. Shioji, M. Katsuura, et al., “KEIKI SHIHYO NO ATARASHII DOKO,” [The new direction in the business indices], Keizai Bunseki 166 (Feb., 2003).
## Appendix 1

In this appendix, we briefly review the technique for isolating a business cycle component of macroeconomic time series which lies in a certain band of frequency, by introducing the approximate band-pass filter, one of the most broadly used in modern business cycle studies, developed in Baxter and King (1999) and Christiano and Fitsgerald (2003). $ ^{15} $

Band-pass filter is a linear filter which retains the cyclical component of each series within a specific band of frequency, and removes other components. It can be applied, therefore, to decompose a series into its low, high and remaining intermediate frequencies, which have been considered as trend, irregular, and business cycle component respectively, so as to distinguish between a business cycle component and other components.

Figure 1 in appendix 1 displays each component of real output along with the logarithm of its original series: Business cycle component follows the main stream of the Japanese business cycles. Trend component is much smoother than original series because of removing the irregular and business cycle component. On the other hand, irregular component is less volatile, but moves quickly comparing with other components.

Frequency domain analysis applying the band-pass filter is basically relying on the idea of Fourier transformation. Let $ y_t $ be a filtered series of $ x_t $ , an original and
pre-filtered series:

$$ y_{t}=B(L)\ x_{t}. $$

The band-pass filter $ B(L) $ , which is an ideal moving average with symmetric weights, has the following properties:

$$ B(L)=\sum_{k=-\infty}^{\infty}B_{k}L^{k}, $$

where L denotes lag operator and $ B_{k} = B_{-k} $ for all k.

The filter can be expressed as the frequency response function based on Fourier representation:

$$ \mathcal{B}(e^{-i\lambda})=\mathcal{B}_{0}+\mathcal{B}_{1}e^{-i\lambda}+\mathcal{B}_{2}e^{-i2\lambda}+,\qquad,+\mathcal{B}_{k}e^{-i k\lambda}, $$

where $ e^{-i\lambda} = \cos\lambda - i\sin\lambda. $

In order to isolate desired frequency in the series, here we assume that the frequency response function is one for our desirable band of frequency, while zero for else. That is,

$$ B(e^{-i\lambda})=\left\{\begin{aligned}&1\quad&if\lambda\in\left(\underline{\lambda},\overline{\lambda}\right)\cup\left(-\overline{\lambda},-\underline{\lambda}\right)\quad,\\ &0\quad&otherwise\end{aligned}\right.\quad\lambda\in\left[-\pi,\pi\right] $$

where $ \lambda $ denotes low cut frequency and $ \overline{\lambda} $ denotes high cut frequency. $ ^{16} $

Then, we can extract a business cycle component by band-pass filtering in the frequency domain. As discussed in Baxter and King (1999) and Christiano and Fitsgerald (2003), technically we are required to have an infinite number of observations to construct the band-pass filter, therefore the approximate band-pass filter has been applied in this study, with two imposed conditions, that is the filter's weights sum to zero and the filter is symmetric on leads and lags.
APPENTDIX 1 TABLE 1. Weights For Business Cycle Filters

Lag weights for the BP(6,32) filter with k=12
0	1	2	3	4	5	6	7	8	9	10	11	12
0.277	0.220	0.084	-0.052	-0.119	-0.102	-0.042	0.001	0.001	-0.028	-0.050	-0.043	-0.012

## APPENTDIX 1 FIGURE 1

Business Cycle, Trend And Irregular Component Of Real Output In Japan

Note: The filtered series are obtained using the approximate band-pass filter in the GAUSS program.
## Appendix 2

This paper documents the robust stylized facts for recent business cycle fluctuations in Japan. In this appendix, we additionally study the period 1955:q1-2002:q1, applying quarterly data on major aggregate series including real output and its components, inputs to production and nominal variables such as money stock.

Table 1 in appendix 2 summarizes the statistical properties of cyclical component. Our findings confirm the cyclical regularities in the properties of series after World War II are mostly consistent with those for the past two decades: private consumption is least volatile; both private and public investment, exports, and imports are most volatile. The cyclical properties of GDP components are also quite similar, with the possible exceptions of private consumption and public investment: private consumption has a clear lagging property, and public investment is pro-cyclical with a lead rather than counter-cyclical with a lag.

As regards production inputs, employment, total hours worked, average hours worked (total), and labor productivity are relatively stable. Moreover employment is pro-cyclical with a lag, while average hours worked (total) is pro-cyclical with a lead. The behavior of nominal variables is also consistent: money is more volatile and pro-cyclical with a lead, while inflation rate measured by GDP deflator and consumer price index is pro-cyclical with a lag and wages are weakly pro-cyclical with a lag. The negative reverse causality between money stock-growth and output is also observed.

In terms of discrepancies due to including the period enjoying rapid economic growth in the mid-1900s, the significant is perhaps that the relatively high volatilities of prices caused by the oil crises in the 1970s.
APPENTDIX 2 TABLE 1. Statistics Summary For Cyclical Component After World War II

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
Std. Dev.	Relative Std. Dev.	Auto-correlation	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
Gross Domestic Product	1.53%	1.00	0.93	-0.10	0.05	0.26	0.51	0.75	0.93	1.00	0.93	0.75	0.51	0.26	0.05	-0.10
GDP components
Private Consumption	1.21%	0.79	0.91	-0.18	-0.07	0.09	0.31	0.54	0.73	0.83	0.80	0.67	0.47	0.25	0.06	-0.08
Private Residential Investment	5.87%	3.83	0.89	-0.40	-0.33	-0.18	0.02	0.23	0.40	0.50	0.51	0.46	0.36	0.25	0.15	0.08
Private Non-Resi. Investment	6.71%	4.38	0.94	-0.03	0.17	0.40	0.61	0.77	0.85	0.83	0.71	0.53	0.34	0.16	0.02	-0.07
Government Consumption	1.12%	0.73	0.85	-0.14	-0.28	-0.36	-0.35	-0.26	-0.13	0.02	0.11	0.16	0.17	0.16	0.14	0.12
Public Investment	4.98%	3.25	0.90	0.10	-0.03	-0.11	-0.14	-0.09	0.01	0.12	0.21	0.25	0.25	0.21	0.16	0.10
Exports	4.18%	2.73	0.89	0.18	0.26	0.30	0.28	0.22	0.13	0.04	-0.04	-0.07	-0.07	-0.05	-0.01	0.02
Imports	6.17%	4.02	0.90	-0.13	0.01	0.16	0.32	0.47	0.58	0.65	0.62	0.52	0.38	0.22	0.10	0.02
Aggregate employment, Wages
Regular Employment Index	1.09%	0.71	0.96	0.30	0.41	0.50	0.56	0.56	0.51	0.42	0.30	0.17	0.06	-0.02	-0.08	-0.11
Total Hours Worked	1.35%	0.88	0.94	-0.11	0.03	0.20	0.37	0.52	0.61	0.64	0.59	0.49	0.37	0.25	0.15	0.08
Average Hours Worked Index(total)	0.80%	0.52	0.93	-0.60	-0.51	-0.35	-0.13	0.11	0.33	0.50	0.58	0.59	0.54	0.45	0.36	0.28
Labor Productivity	1.24%	0.81	0.93	0.01	0.04	0.11	0.22	0.36	0.47	0.52	0.49	0.38	0.22	0.04	-0.10	-0.20
Wage Index(Total cash earnings, nominal)	1.90%	1.24	0.96	0.30	0.32	0.30	0.25	0.17	0.08	-0.03	-0.13	-0.22	-0.28	-0.31	-0.31	-0.28
Deflators and Prices
GDP deflator(level)	1.75%	1.14	0.93	0.21	0.24	0.22	0.14	0.03	-0.10	-0.21	-0.30	-0.37	-0.41	-0.40	-0.37	-0.30
GDP deflator(growth rate)	2.22%	1.45	0.88	0.22	0.36	0.42	0.41	0.34	0.23	0.10	-0.03	-0.14	-0.24	-0.30	-0.32	-0.29
Consumer Price Index(level)	2.04%	1.33	0.95	0.33	0.30	0.21	0.07	-0.10	-0.26	-0.38	-0.46	-0.48	-0.46	-0.41	-0.34	-0.26
Consumer Price Index(growth rate)	2.44%	1.59	0.91	0.44	0.56	0.58	0.50	0.35	0.16	-0.02	-0.17	-0.26	-0.31	-0.31	-0.28	-0.24
Money
Money Stock(M2+CD, nominal)	1.69%	1.11	0.95	0.02	0.04	0.09	0.17	0.26	0.34	0.40	0.40	0.34	0.23	0.09	-0.06	-0.19
Money Stock(M2+CD, nominal, growth rate)	2.01%	1.31	0.89	-0.25	-0.30	-0.30	-0.23	-0.10	0.08	0.26	0.42	0.50	0.50	0.41	0.28	0.15
Money Stock(M2+CD, real)	2.53%	1.65	0.94	-0.14	-0.14	-0.09	0.02	0.15	0.29	0.41	0.48	0.49	0.44	0.34	0.21	0.08
Money Stock(M2+CD, real, growth rate)	3.22%	2.10	0.89	-0.27	-0.39	-0.44	-0.40	-0.29	-0.12	0.07	0.24	0.37	0.44	0.45	0.39	0.29
Note: The filtered series are obtained using the approximate band-pass filter in the GAUSS program.
TABLE 1. Business Cycle History In Japan

Peak	Trough	Business cycle/its duration in month
1951 II	1951 IV	-
1954 I	1954 IV	37
1957 II	1958 II	43
1961 IV	1962 IV	52
1964 IV	1965 IV	36
1970 III	1971 IV	74
1973 IV	1975 I	39
1977 I	1977 IV	31
1980 I	1983 I	64
1985 II	1986 IV	45
1991 I	1993 IV	83
1997 II	1999 I	63
2000 IV	2002 I	36

Note: Peaks and Troughs have been determined by ESRI, Cabinet Office.
TABLE 2. Statistics Summary For Cyclical Component: GDP components

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
Std. Dev.	Relative Std. Dev.	Auto-correlation	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
Gross Domestic Product	1.23%	1.00	0.93	-0.04	0.11	0.30	0.53	0.76	0.93	1.00	0.93	0.76	0.53	0.30	0.11	-0.04
GDP components
Private Consumption	0.68%	0.55	0.92	0.08	0.18	0.29	0.42	0.55	0.68	0.76	0.77	0.70	0.56	0.39	0.22	0.10
Consumption(Households)	0.71%	0.58	0.92	0.05	0.16	0.28	0.42	0.57	0.70	0.78	0.78	0.70	0.55	0.37	0.21	0.08
Consumption(Excluding Imputed Rent)	0.83%	0.68	0.92	0.00	0.12	0.25	0.39	0.55	0.69	0.78	0.79	0.71	0.57	0.39	0.22	0.09
Private Residential Investment	6.13%	5.00	0.91	-0.41	-0.41	-0.34	-0.19	0.02	0.25	0.44	0.56	0.60	0.57	0.49	0.41	0.34
Private Non-Resi. Investment	5.58%	4.55	0.94	0.06	0.25	0.46	0.67	0.83	0.91	0.88	0.75	0.56	0.35	0.16	0.01	-0.10
Change in Private Inventory(Relative to GDP)	0.24%	0.19	0.87	-0.36	-0.15	0.09	0.34	0.55	0.69	0.73	0.64	0.51	0.37	0.26	0.16	0.06
Government Consumption	0.89%	0.73	0.92	-0.33	-0.34	-0.30	-0.23	-0.14	-0.05	0.04	0.14	0.22	0.30	0.35	0.36	0.31
Public Investment	4.23%	3.44	0.88	0.05	-0.11	-0.27	-0.38	-0.42	-0.39	-0.31	-0.19	-0.07	0.04	0.12	0.16	0.16
Change in Public Inventory(Relative to GDP)	0.04%	0.03	0.83	-0.18	-0.18	-0.16	-0.11	-0.03	0.06	0.13	0.14	0.09	0.02	-0.07	-0.14	-0.18
Exports	3.84%	3.13	0.86	-0.08	0.00	0.09	0.20	0.32	0.44	0.53	0.51	0.39	0.21	0.01	-0.18	-0.34
Imports	5.28%	4.30	0.94	-0.34	-0.23	-0.08	0.10	0.31	0.52	0.71	0.83	0.86	0.82	0.72	0.56	0.37
Trade Balance(Relative to GDP)	0.38%	0.31	0.91	0.27	0.21	0.12	0.01	-0.10	-0.20	-0.27	-0.38	-0.50	-0.61	-0.67	-0.61
Gross domestic fixed capital formation	3.11%	2.53	0.94	-0.06	0.08	0.26	0.48	0.71	0.87	0.94	0.90	0.76	0.59	0.41	0.24	0.10
TABLE 3. Cross-correlation Of Consumption With Output, Before And After 1991

Consumption	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
Std. Dev.	k
-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
1. Before 1991: 1985: q2-1991: q1
Private Consumption	0.74%	0.12	0.15	0.22	0.36	0.54	0.75	0.92	0.94	0.84	0.64	0.41	0.20	0.08
2. After 1991: 1991: q1-2001: q3
Private Consumption	0.66%	-0.27	-0.11	0.07	0.24	0.41	0.56	0.68	0.68	0.60	0.45	0.26	0.06	-0.10

TABLE 4. Statistics Summary For Cyclical Component: Aggregate employment

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
TABLE 5. Statistics Summary For Cyclical Component: Wages

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
Std. Dev.	Relative Std. Dev.	Auto-correlation	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
Wages
Wage Index(Total cash earnings, nominal)	0.73%	0.59	0.94	0.22	0.40	0.57	0.71	0.79	0.78	0.69	0.54	0.36	0.16	-0.04	-0.22	-0.37
Wage Index(Contractual cash earnings, nom)	0.42%	0.34	0.92	0.00	0.12	0.27	0.44	0.57	0.64	0.62	0.51	0.36	0.18	0.00	-0.16	-0.30
Wage Index(Total cash earnings, nom, growth rate)	0.83%	0.67	0.90	-0.48	-0.30	-0.07	0.17	0.39	0.56	0.67	0.71	0.68	0.59	0.45	0.30	0.15
Wage Index(Total cash earnings, real)	0.79%	0.64	0.93	-0.32	-0.14	0.07	0.28	0.47	0.60	0.66	0.68	0.65	0.58	0.47	0.35	0.23
Wage Index(Contractual cash earnings, real)	0.72%	0.59	0.94	-0.58	-0.50	-0.35	-0.16	0.04	0.24	0.38	0.49	0.56	0.57	0.55	0.51	0.45
Wage Index(Total cash earnings, real, growth rate)	0.89%	0.73	0.89	-0.71	-0.66	-0.54	-0.38	-0.19	-0.01	0.16	0.29	0.38	0.41	0.39	0.35	0.32
Wage(Total cash earnings)	1.11%	0.91	0.74	0.28	0.44	0.55	0.60	0.58	0.50	0.39	0.27	0.13	-0.02	-0.16	-0.28	-0.36
Wage(Contractual cash earnings)	0.67%	0.54	0.91	0.17	0.31	0.40	0.44	0.43	0.36	0.26	0.16	0.04	-0.08	-0.20	-0.29	-0.35
Wage(Scheduled cash earnings)	0.71%	0.58	0.90	0.36	0.41	0.41	0.34	0.21	0.05	-0.12	-0.25	-0.34	-0.41	-0.46	-0.47	-0.45
Wage(Non-scheduled cash earnings)	4.05%	3.30	0.93	-0.33	-0.17	0.03	0.24	0.45	0.63	0.75	0.77	0.72	0.60	0.45	0.29	0.13
Wage(others, including bonus)	4.70%	3.83	0.59	0.24	0.41	0.54	0.61	0.59	0.51	0.37	0.24	0.11	-0.03	-0.14	-0.22	-0.27
Wage(Total cash earnings, growth rate)	1.31%	1.07	0.88	-0.24	-0.01	0.18	0.33	0.42	0.47	0.48	0.47	0.42	0.33	0.20	0.08	-0.02
Compensation of Employees(nominal)	1.21%	0.99	0.95	0.42	0.57	0.69	0.76	0.77	0.70	0.56	0.37	0.17	-0.03	-0.21	-0.35	-0.47
Compensation of Employees(nom, growth rate)	1.32%	1.08	0.91	-0.29	-0.07	0.17	0.40	0.58	0.70	0.75	0.72	0.63	0.51	0.38	0.23	0.08
Compensation of Employees(real)	0.93%	0.76	0.93	0.17	0.38	0.59	0.76	0.86	0.86	0.76	0.61	0.43	0.24	0.06	-0.09	-0.20
Compensation of Employees(real, growth rate)	1.10%	0.90	0.90	-0.56	-0.38	-0.14	0.12	0.35	0.53	0.62	0.62	0.56	0.47	0.36	0.25	0.16
TABLE 6. Statistics Summary For Cyclical Component: Deflators and Prices

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )
Std. Dev.	Relative Std. Dev.	Auto-correlation	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
Deflators and Prices
GDP deflator(level)	0.66%	0.54	0.95	0.62	0.61	0.53	0.39	0.22	0.04	-0.12	-0.26	-0.36	-0.43	-0.48	-0.52	-0.53
GDP deflator(growth rate)	0.70%	0.57	0.91	0.43	0.59	0.67	0.67	0.58	0.46	0.33	0.22	0.13	0.05	-0.03	-0.12	-0.20
Consumer Price Index(level)	0.78%	0.64	0.96	0.53	0.52	0.47	0.38	0.27	0.13	-0.02	-0.18	-0.32	-0.44	-0.51	-0.56	-0.58
Consumer Price Index(growth rate)	0.80%	0.66	0.91	0.30	0.43	0.53	0.60	0.61	0.58	0.51	0.40	0.27	0.15	0.03	-0.09	-0.21
Domestic Corporate Goods Price Index(level)	1.25%	1.01	0.94	0.15	0.22	0.29	0.35	0.39	0.41	0.39	0.32	0.22	0.08	-0.07	-0.23	-0.38
Domestic Corporate Goods Price Index(growth)	1.42%	1.16	0.87	-0.16	-0.10	-0.02	0.07	0.18	0.30	0.42	0.51	0.55	0.53	0.44	0.31	0.16

TABLE 7. Statistics Summary For Cyclical Component: Interest rates and Stock price

Macroeconomic time series	Cross-correlation with GDP ( Corr[xt, gdpt+k] )