Description Usage Arguments Details Value Author(s) References See Also Examples

Extract Common Trend(s) from a cointegration system according to
Gonzalo and Grange(1995). This method is also known as the
Permanent-Transitory Decomposition. Loading Matrix and Othogonal
Complement of *α* and *β* are also reported.

1 |

`data` |
Data used to construct the cointegration system |

`rank` |
Number of cointegration vectors specified |

`k` |
Lag order in VECM |

Currently functions `GG.ComT`

and `Kasa.ComT`

assume that no determinstic parts, such as
the constant and the trend, are in the Error-Correction Terms (ECT).
So that means we have to keep `ecdet = "none"`

in the `ca.jo`

function
(`ca.jo`

is the major function in package `urca`

to estimate
cointegration relations).
But it does allow the existence of constant term in the VECM (outside
ECT).

The method proposed by Gonzalo and Granger decomposes the time series
*X_{t}* as

*
X_{t}=α(β^{\prime}α)β^{\prime}X_{t}+β_{\perp}(α_{\perp}^{\prime}β_{\perp})^{-1}α_{\perp}X_{t}
*

where *α(β^{\prime}α)β^{\prime}X_{t}* is *I(0)*
and the transitory part, and *β_{\perp}(α_{\perp}^{\prime}β_{\perp})^{-1}α_{\perp}X_{t}*
is *I(1)* and the permanent part (see Equation 11 in Gonzalo
and Granger 1995). Be cafreful in Gonzalo and Granger's paper they use different notation for *α* and *β*.

Kasa's method decomposes the time series as

*
X_{t}=β(β^{\prime}β)^{-1}β^{\prime}X_{t}+β_{\perp}(β_{\perp}^{\prime}β_{\perp})^{-1}β_{\perp}X_{t}
*

where “the first part defines the stationary component and the second part then defines the common stochastic trend” (Kasa 1992) (see Equation 12 in Kasa 1992).

An object of class `ComT`

.

Fan Yang

Kasa, K., 1992. Common stochastic trends in international stock markets,
*Journal of Monetary Economics* **29**, 95-124.

Gonzalo, J., and C. Granger, 1995. Estimation of Common Long-Memory
Components in Cointegrated Systems, *Journal of Business & Economic Statistics* **13**, 27-35.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
data(benchmark)
x=seq(1,6689,by=23) ## monthly data
global=data.frame(benchmark[x,2:4])
GG.ComT (global,2,4)
## Plot the Common Trend
G=GG.ComT (global,2,4)
Date=benchmark[x,1]
plotComT(G,1,x.axis=Date,approx.ticks=12,
legend=c("S&P 500 Price index", "Common Trend"),
main="Extract Common Trend(s) from Benchmark Markets",
ylab="Price", xlab="Time" )
``` |

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