CommonTrend (version 0.7-1)

GG.ComT: Permanent-Transitory Decomposition

Description

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 $\alpha$ and $\beta$ are also reported.

Usage

GG.ComT(data, rank, k)

Arguments

data

Data used to construct the cointegration system

rank

Number of cointegration vectors specified

k

Lag order in VECM

Value

• An object of class ComT.

Details

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

$$X_{t}=\alpha(\beta^{\prime}\alpha)\beta^{\prime}X_{t}+\beta_{\perp}(\alpha_{\perp}^{\prime}\beta_{\perp})^{-1}\alpha_{\perp}X_{t}$$ where $\alpha(\beta^{\prime}\alpha)\beta^{\prime}X_{t}$ is I(0) and the transitory part, and $\beta_{\perp}(\alpha_{\perp}^{\prime}\beta_{\perp})^{-1}\alpha_{\perp}X_{t}$ 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 $\alpha$ and $\beta$.

Kasa's method decomposes the time series as

$$X_{t}=\beta(\beta^{\prime}\beta)^{-1}\beta^{\prime}X_{t}+\beta_{\perp}(\beta_{\perp}^{\prime}\beta_{\perp})^{-1}\beta_{\perp}X_{t}$$ 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).

References

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.

See Also

ComT-class,plotComT

Examples

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"  )