https://github.com/cadam00/corbouli
Corbae-Ouliaris Frequency Domain Filtering
https://github.com/cadam00/corbouli
frequency-domain-filtering r r-package time-series-analysis
Last synced: 5 months ago
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Corbae-Ouliaris Frequency Domain Filtering
- Host: GitHub
- URL: https://github.com/cadam00/corbouli
- Owner: cadam00
- License: gpl-3.0
- Created: 2024-09-01T10:54:54.000Z (9 months ago)
- Default Branch: main
- Last Pushed: 2024-12-07T16:31:58.000Z (6 months ago)
- Last Synced: 2024-12-07T17:25:17.155Z (6 months ago)
- Topics: frequency-domain-filtering, r, r-package, time-series-analysis
- Language: R
- Homepage: https://cadam00.github.io/corbouli/
- Size: 2.27 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: NEWS.md
- License: LICENSE
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# **Corbae and Ouliaris ([2006](#ref-corbae2006)) Frequency Domain Filter in R**
## **Install**
The official [(CRAN)](https://cran.r-project.org/) version of the package can be
installed using
``` r
install.packages("corbouli")
```
Alternatively, the development version of the package can be installed via
``` r
if (!require(remotes)) install.packages("remotes")
remotes::install_github("cadam00/corbouli")
```
## **Citation**
To cite the official [(CRAN)](https://cran.r-project.org/) version of the
package, please use
> Adam, C. (2024). corbouli: Corbae-Ouliaris Frequency Domain Filtering.
> R package version 0.1.3. Available at
> .
Alternatively, to cite the latest development version, please use:
> Adam, C. (2024). corbouli: Corbae-Ouliaris Frequency Domain Filtering
(v0.1.3). Zenodo. Available at
## **Corbae-Ouliaris Frequency Domain Filtering**
Corbae and Ouliaris ([2006](#ref-corbae2006)) Frequency
Domain Filter is used for extracting cycles from either both on stationary and
non-stationary time series. This is one approximation of the ideal band pass
filter of the series. The result is close to the one of the Baxter-King
([1999](#ref-baxter1999)) filter, but end-points are directly estimated and so
facing the end-point issue is not faced.
The main idea of this filtering algorithm is illustrated
in Fig. [1](#ref-Figure1) and [2](#ref-Figure2). The main idea of the DFTSE
subroutine is shown in Fig. [1](#ref-Figure1), where DFT (Discrete
Fourier Transform) of the times series, then frequencies lower and higher
by periods of oscillation thresholds are assigned to zero and finally IDFT
(Inverse Discrete Fourier Transform) are performed. Additional implementation
details of this subroutine can be found at source code of the function
`corbouli::dftse`.
Fig. 1: DFTSE subroutine.
The final algorithm is described in Fig. [2](#ref-Figure2),
where filtered series is the residuals of the regression of $DFSTE(x)$ on
$DFSTE$ of the normalized trend.
Fig. 2: Corbae-Ouliaris main algorithm.
The minimum and the maximum periods of oscillation should be determined when
using this method, so as to irrelevant to filtering frequencies are removed.
As an example from Shaw ([1947](#ref-shaw1947)), a business cycle usually has
a lower period of 1.5 years and a higher period of 8 years. This information can
be used while for filtering as expressed from the following
Table [1](#ref-table1).
Sample Frequency
Lower
Higher
Month
18
96
Quarter
6
32
Year
2
8
Table 1: Lower and higher periods of oscillation.
The same table in fragments of $\pi$ can be transformed into the Table
[2](#ref-table2) using the $lower\ frequency = 2 / higher\ period$ and
$higher\ frequency = 2 / lower\ period$. For instance, for quarterly sampled
time series, we have $lower\ frequency = 2 / 32 = 0.0625$ and
$higher\ frequency = 2 / 6 = 0.3333$.
Sample Frequency
Lower
Higher
Month
0.0208
0.1111
Quarter
0.0625
0.3333
Year
0.25
1
Table 2: Low and high frequency in fragments of $\pi$.
The longer the series, the more the long run frequency is expressed by a
lower frequency as fragment of $\pi$ equal to 0. Moreover, the output gap can be
expressed using higher frequency as fragment of $\pi$ equal to 1
([Ouliaris, 2009](#ref-ouliaris2009)).
## **Example**
For this example, the quarterly US GDP in billions of chained 2017 dollars
(Seasonally adjusted) will be used.
``` r
# Import package to workspace
library(corbouli)
# Import "USgdp" dataset
data(USgdp)
plot(USgdp, main = "Quarterly US GDP in billions of chained 2017 dollars
(Seasonally adjusted)", ylab = "", lwd = 2)
```
Fig. 3: USgdp dataset.
```r
# Extract cycles
co <- corbae_ouliaris(USgdp, low_freq = 0.0625, high_freq = 0.3333)
# Plot cycles of filtered series
plot(co,
main = "Corbae-Ouliaris FD Filter cycles for USgdp",
ylab = "",
lwd = 2)
```
Fig. 4: Corbae-Ouliaris FD Filter cycles.
```r
# Plot real data with the ones after removing cycles
# Removing cycles from original data
USgdp_rmco <- USgdp - co
# Plot Original vs Decycled data
plot(USgdp, main = "Quarterly US GDP in billions of chained 2017 dollars
(Seasonally adjusted)", col = "black", lwd = 2, ylab = "")
lines(USgdp_rmco, col = adjustcolor("red", alpha.f = 0.7), lwd = 2)
legend(x = "topleft", lwd = 2, text.font = 2,
col= adjustcolor(c("black","red"), alpha.f = 0.7),
legend=c("Original data", "Decycled data"))
```
Fig. 5: Original vs Decycled USgdp data.
As noted by Ouliaris ([2009](#ref-ouliaris2009)), for setting `high_freq = 1`
the output gap series than business cycle one will have higher volatility (Fig.
[6](#ref-Figure6)).
```r
# Extract output gap
og <- corbae_ouliaris(USgdp, low_freq = 0.0625, high_freq = 1)
# Plot Business cycle vs Output gap
plot(co, main = "Business cycle vs Output gap for USgdp",
col = adjustcolor("blue", alpha.f = 0.7), lwd = 2, ylab = "")
lines(og, col = adjustcolor("orange", alpha.f = 0.7), lwd = 2)
legend(x = "bottomleft", lwd = 2, text.font = 2,
col= adjustcolor(c("blue","orange"), alpha.f = 0.7),
legend=c("Business cycle", "Output gap"))
```
Fig. 6: Business cycle vs Output gap.
## **References**
Baxter, M., & King, R. (1999),
Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time
Series. Review of Economics and Statistics 81(4), pp.
575-593.
Corbae, D., Ouliaris, S., & Phillips, P. (2002),
Band Spectral Regression with
Trending-Data. Econometrica 70(3), pp. 1067-1109.
Corbae, D. & Ouliaris, S. (2006),
Extracting Cycles from Nonstationary Data,in Corbae D., Durlauf S.N.,
& Hansen B.E. (eds.). Econometric Theory and Practice: Frontiers of
Analysis and Applied Research. Cambridge: Cambridge University Press, pp.
167–177. https://doi.org/10.1017/CBO9781139164863.008.
Ouliaris, S. (2009), Ideal Band Pass
Filter For Stationary/Non-Stationary Series.
Pérez Pérez, J. (2011), COULIARI: Stata
module to implement Corbae-Ouliaris frequency domain filter to time series
data. Statistical Software Components, S457218, Boston College
Department of Economics.
Shaw, E.S. (1947), Burns and Mitchell on
Business Cycles. Journal of Political Economy, 55(4):
pp. 281-298. https://doi.org/10.1086/256533.